5. Two-Level Atom Interacting with a Classical Field¶
PDF pages 151–268
5.1 Atom-Field Interaction¶
Chapter 5 Two-Level Atom Interacting with a Classical Field The interaction of a two-level atom is one of the canonical problems in quantum optics. We will now be accounting for quantum coherences between the two levels, which we ignored in our rate-equation treatment in Chapter 3. The approaches we will use here are widely applicable, since a two-level atom is essentially equivalent to a qubit or a spin-1/2 particle. Less obvious is that equivalent phenomena also occur in seemingly different problems such as tunneling in the double-well potential, Bragg diffraction, and neutrino oscillations.1 5.1 Atom–Field Interaction We begin our treatment with a general description of the atom-field interaction. We will assume the field is monochromatic with angular frequency \omega to model the field due to a laser:
(5.1) Here, ˆ\epsilon is the unit polarization vector of the field. Note that we are ignoring the spatial dependence of the field, only writing down the field at the location of the atom. This is appropriate in the dipole approximation or long-wavelength approximation, where we assume that the wavelength of the field is much longer than the size of the atom, so that we can neglect any variations of the field over the extent of the atom. This is generally appropriate for optical transitions, since atomic dimensions have Å scales, while optical wavelengths are hundreds of nm. Formally, as the name suggests, the dipole approximation corresponds to the lowest-order contribution in a multipole expansion of the atom–field interaction. As before, it is convenient to decompose the field into its positive- and negative-rotating components E(+) and E(−):
ei\omegat
(5.2) That is, E(\pm) ∼e−i(\pm\omega)t. We will treat the atom as a two-level atom. This is clearly an approximation to a true atom, which has an infinite set of bound states. The justification is that we will consider near-resonant interactions, so that the transitions to other levels are negligible. We will label the ground and excited levels as |g\rangle and |e\rangle , respectively, and we will denote the resonant frequency by \omega0 (that is, the energy splitting of the pair of states is ¯h\omega0). 1Alexander Friedland, ‘‘Evolution of the neutrino state inside the Sun,’’ Physical Review D 64, 013008 (2001) (doi: 10.1103/PhysRevD.64.013008).
5.1.1 Parity and the Dipole Operator¶
Chapter 5. Two-Level Atom Interacting with a Classical Field D w0 w |og‚ |oe‚ We will define ∆:= \omega −\omega0 to be the detuning of the laser field from the atomic resonance. We can write the total Hamiltonian for the atom and field as a sum of the free atomic Hamiltonian HA and the atom–field interaction Hamiltonian HAF:
(5.3) The atomic free-evolution Hamiltonian is given by
(5.4) (free evolution) if we take the ground-state energy to be zero. The atom-field interaction Hamiltonian in the dipole approx- imation is
(5.5) (dipole interaction) where d is the atomic dipole operator, given in terms of the atomic electron position re as d = −ere, (5.6) if we assume a single electron for the atom (i.e., the field interacts predominantly with one electron). We denote the fundamental charge by e, so that the electron charge is q = −e. We will not justify this form of the interaction Hamiltonian for now, except to note that it seems reasonable as a dipole-field interaction, and also it is consistent with the classical case (see Problem 5.1). 5.1.1 Parity and the Dipole Operator We can then use a simple parity argument to gain more information about the form of the dipole operator. The parity operator \Pi flips the sign of the position operator, and is thus defined by the operator transfor-
gives \Pire = −re\Pi, and thus the anticommutator of the parity operator with re vanishes:
(5.7) The matrix elements of the anticommutator clearly vanish,
(5.8) but we can also write the matrix elements in the energy basis as
(5.9) where \pia and \pib are eigenvalues of \Pi. We can define these because \Pi commutes with the atomic Hamiltonian, which has the form p2 e/2me −\alpha/|re|, and thus \Pi and H have simultaneous eigenstates. But \Pi2 = 1, so the possible eigenvalues of \Pi are \pm1, corresponding to even (+) and odd (−) parity. For both (5.8) and (5.9) to hold, either \pia + \pib = 0 or \langle a|re|b\rangle = 0. This argument obviously applies just as well to the dipole operator instead of re. We can then see that the diagonal matrix elements of d vanish, since \pig and \pie are both nonzero:
(5.10)
5.1.2 Rotating-Wave Approximation¶
5.1 Atom–Field Interaction The off-diagonal matrix elements \langle g|d|e\rangle = \langle e|d|g\rangle ∗are nonvanishing, however, provided that the states have opposite parity, \pie = −\pig. The point is that the dipole operator couples the ground and excited states, but does not produce any first-order shift of either state. By applying the identity |e\rangle \langle e| + |g\rangle \langle g| on both sides of d, we see that dipole operator admits the decomposition
(5.11) We can choose the phase of the dipole matrix element \langle g|d|e\rangle such that it is real, in which case we can write the dipole operator as
(5.12) (dipole operator in terms of \sigma) where \sigma := |g\rangle \langle e| is the atomic lowering operator. For clarity when we get to more complicated atoms, we will always write the dipole matrix element \langle g|d|e\rangle with the excited state to the right. We can thus write the total atom–field Hamiltonian as
, (5.13) where \sigma\dagger\sigma = |e\rangle \langle e| is the excited-state projection operator. 5.1.2 Rotating-Wave Approximation Just as we decomposed the field into positive- and negative-rotating parts, we can do the same for the dipole operator in the form (5.12):
(5.14)
time dependence e−i\omega0t (since this is the evolution of |e\rangle under the free atomic Hamiltonian), and thus corresponds to a positive frequency. Including the same decomposition of the field, the atom–field Hamiltonian becomes
(5.15) Recalling the time dependences d(\pm) ∼e∓i\omega0t; E(\pm) ∼e∓i\omegat, (5.16) we see that the first two terms oscillate rapidly as e\pmi(\omega+\omega0)t, while the last two (cross) terms oscillate slowly
This approximation focuses on slow dynamics, replacing terms rotating at optical frequencies are replaced by their zero average value, which amounts to a coarse-graining on fs time scales. This is reasonable since optical detectors don’t respond on fs time scales anyway. Something that is sometimes not appreciated is that the two-level approximation and the RWA are at the same level of accuracy. That is, it makes no sense to throw one out and keep the other. Both amount to discarding interactions that are far off resonance (the RWA amounts to a very far detuned interaction of a positive-frequency resonance with a negative-frequency field). If the detuning is large enough that the counter-rotating term is not negligible, then neither are the couplings to the other levels. 2The name ‘‘rotating-wave approximation’’ comes from nuclear magnetic resonance. A spin-1/2 particle in a magnetic field precesses (rotates) naturally. When you hit it with a linearly polarized microwave field, the field is a superposition of two waves with opposite circular (‘‘rotating’’) polarization. Only one of the polarization rotates in the same direction as the spin precession, and so the RWA amounts to ignoring the counter-rotating field component. Said another way, it amounts to replacing the linearly polarized wave with a rotating wave. Note that in NMR, the RWA is exact for circular polarization, a property that does not carry over to the optical case.
5.1.5 Rotating Frame¶
Chapter 5. Two-Level Atom Interacting with a Classical Field 5.1.3 Rabi Frequency Thus, the atom–field interaction Hamiltonian in the RWA becomes
(5.17) (RWA dipole interaction) Using Eq. (5.2) for the explicit time-dependence of the field along with Eq. (5.14) for the dipole operator, we can write
E(−)
= ¯hΩ
, (5.18) where we have assumed E(+) to be real, and we have defined the Rabi frequency3 as
¯h
¯h . (5.19) (Rabi frequency) Note that we generally choose the phase of the dipole matrix element so that Ω> 0. The Rabi frequency characterizes the strength of the atom–field coupling. In the case of a linearly polarized field, the Rabi frequency simplifies to
¯h (5.20)
5.1.4 Schrödinger Equation Let’s write the atomic state as¶
(5.21) where cg and ce carry all the time dependence of the state. With the atomic Hamiltonian HA (5.4) and
2 ei\omegatce|g\rangle −iΩ
(5.22) Projecting with \langle g| and \langle e| gives the pair of coupled differential equations,
2 ceei\omegat
2 cge−i\omegat, (5.23) which we can now in principle solve for the atomic evolution. 5.1.5 Rotating Frame We now have a set of coupled equations that involve oscillatory terms at optical frequencies. However, at resonance the precessions are phase-locked and should disappear in the proper coordinates. It it therefore convenient to transform into a corotating frame to eliminate the fast rotation. Thus, we make a transforma- tion into the rotating frame of the laser field by defining the slowly varying excited-state amplitude ˜ce := ceei\omegat. (5.24) (rotating-frame transformation) 3after Isador Isaac Rabi, who pioneered the field of nuclear magnetic resonance. He was awarded the 1944 Nobel prize for this work.
5.1 Atom–Field Interaction We can then rewrite the equations of motion as¶
2 ˜ce \partial t˜ce = i∆˜ce −iΩ 2 cg. (5.25) In fact, these are the same equations of motion generated by the effective, rotating-frame atomic Hamilto- nian
(5.26) (rotating-frame free evolution) where recall that ∆:= \omega −\omega0 is the detuning of the laser from the atomic resonance, and the effective, rotating-frame interaction Hamiltonian
= ¯hΩ
(5.27) (rotating-frame dipole interaction) where we have defined the stationary field amplitudes
(5.28) In making the rotating-wave approximation, we have discarded the two terms that would have an explicit time dependence of e\pmi2\omegat in Eq. (5.27), and in transforming to the rotating frame, we have removed all of the explicit time dependence from this problem. Essentially, we are hiding the time dependence of the field in the slowly varying amplitude ˜ce, where it cancels most of the natural state evolution. Notice also that |e\rangle is still an eigenstate of ˜HA, with eigenvalue ¯h\omega0 −¯h\omega = −¯h∆, so that the rotating-frame transformation has the effect of shifting the excited state down in energy by an amount ¯h\omega. This representation of the problem in the laser frame is interesting when we look at it this way: it shows that this ac interaction is equivalent to the problem of two states separated in energy by ¯h∆interacting with a dc electric field (in the RWA). Because we have eliminated any explicit time dependence, this problem will be easy to solve. 5.1.5.1 Unitary Transformations Suppose that we have a unitary transformation U, which induces the transformation | ˜\psi\rangle = U|\psi\rangle . Then how does the Hamiltonian transform? Both the original and transformed states must satisfy the Schrödinger equation,
(5.29) where ˜H is the transformed Hamiltonian. We can write the first equation here as
(5.30) Then we can expand the time derivative and operate on the left by U:
(5.31)
(5.32) Comparing this to the Schrödinger equation in the transformed variables, we can identify the transformation law
(5.33) (time-dependent transformation)
The following equations are shown in the figure, with \(a_i\) being one of several eigenstates:
5.2.1 Resonant Interaction¶
5.2 Rabi Flopping which is obviously missing here. We can transform out of the interaction picture by using the transformation
(5.40) The resulting Hamiltonian is
(5.41) where the tildes indicate operators after transformation (i.e., in the Schrödinger picture), and we have hidden constant phase factors in the atom–field Hamiltonian. This Hamiltonian is then in the Schrödinger picture with respect to the field, where the field time dependence is generated by the presence of HF. Note that we didn’t shift the energy of the excited state down, but in a sense we shifted the energy of the ground state up, which amounts to the same thing. This happened because the Hamiltonian here couples states of the form
(5.42) where the integer refers to photon-number states. The splitting between these states is −¯h∆, as we saw in the semiclassical rotating frame. In the classical limit, the average photon number N of the laser field is very large, and in a coherent (Gaussian) state of the field, the fractional uncertainty in N becomes vanishingly small. Hence, the field operator a can be replaced by \sqrt N. With the argument above, the free atom and field Hamiltonians conspire to give an atomic Hamiltonian with the correct splitting in the rotating frame. Upon making the identification
\sqrt N, we also recover the correct form for the rotating interaction Hamiltonian (5.27). Hence we have shown that the transformation to the rotating frame also arises as a transformation from the interaction picture to the Schrödinger picture with respect to the quantized field. We have also established explicitly how the fully quantized atom–field treatment reduces to the present semiclassical model in the classical-field limit. With the expressions (5.37) and (5.38) for the atom–field interaction in hand, we can make one final remark about the RWA. Since E(+) annihilates a photon from the laser field, the terms left in the interaction Hamiltonian from the RWA correspond to raising the atomic state while lowering the field state (d(−) z E(+) −\rightarrow \sigma\daggera) and lowering the atomic state while raising the field state (d(+) z E(−) −\rightarrow \sigmaa\dagger). Near resonance, these interactions are energy-conserving. The rotating terms that we neglected are of the form (d(+) z
and (d(−) z E(−) −\rightarrow \sigma\daggera\dagger), corresponding to lowering the atom but annihilating a photon, and raising the atom but creating a photon, respectively. These processes violate energy conservation by an energy of about two photons’ worth, and should thus be much less important than the energy-conserving ones. Invoking the RWA, then, amounts to keeping only the energy-conserving (resonant) terms in the interaction Hamiltonian. 5.2 Rabi Flopping Now we can solve the coupled amplitude equations (5.25) in the rotating frame to look at the driven atomic dynamics. We will do this first in the case of exact resonance, and then in the more general, nearly resonant case. 5.2.1 Resonant Interaction
2 ˜ce
2 cg. (5.43)
Chapter 5. Two-Level Atom Interacting with a Classical Field We can easily decouple these by differentiating them and substituting in the original equations. For example, \partial 2 t cg = −iΩ
Ω 2 cg. (5.44) This has the form of an undamped harmonic oscillator of frequency Ω/2. The equations of motion are invariant under the exchange g \leftarrow \rightarrow e, and so ce satisfies the same uncoupled equation as cg. Thus, we may write the uncoupled equations as \partial 2 t cg = − Ω 2 cg \partial 2 t ˜ce = − Ω 2 ˜ce. (5.45) The general solution for cg is
1 2Ωt + B cos 1 2Ωt . (5.46)
2Ω A cos 1 2Ωt −B sin 1 2Ωt = −i 2Ω˜ce(t). (5.47)
general solution for ˜ce comes from Eq. (5.47), which gives
1 2Ωt −i˜ce(0) sin 1 2Ωt
1 2Ωt −icg(0) sin 1 2Ωt
(5.48) as the general solution for the two amplitudes. 5.2.1.1 Example: Initially Unexcited Atom
1 2Ωt
1 2Ωt . (5.49) The ground- and excited-state populations are thus
1 2Ωt = 1 2(1 + cos Ωt)
1 2Ωt = 1 2(1 −cos Ωt). (5.50) Thus, we see explicitly the significance of the Rabi frequency: the population oscillates between the ground and excited levels at the angular frequency Ω. This oscillation phenomenon is referred to as Rabi flopping.
5.2.2 Nearly Resonant Interaction¶
5.2 Rabi Flopping Wt 2p 3p 4p 5p 6p p P Pooo(t) e Pooo(t) g Roughly, the ‘‘upswings’’ in population from |g\rangle to |e\rangle correspond to light absorption, while the downswings correspond to stimulated emission back into the original field (thus far we are ignoring spontaneous emission). The period of the oscillation is T = 2\pi/Ω, and thus some particular times are important. If the field is turned
state with unit probability. A pulse of this form is called a \pi-pulse. On the other hand, if the field is turned
of the ground and excited states. A pulse of this form is called a \pi/2-pulse. Of course, these ‘‘pulses’’ refer to square-profile pulses, since we have assumed a constant field ampli- tude E0, but more generally we can consider the above analysis to be valid even when the field amplitude is time-dependent, so long as it varies slowly on the optical time scales. In this case, the Rabi frequency itself is time-dependent, and we can generally define a \pi-pulse to be any pulse with an ‘‘area’’ of \pi, Z
(5.51) and a \pi/2 pulse is any pulse with area \pi/2. 5.2.2 Nearly Resonant Interaction In the case of nonzero detuning ∆, we need to solve Eqs. (5.25). We can again decouple these by differentiating and eliminating appropriate variables, with the result \partial 2
cg = 0 \partial 2
˜ce = 0. (5.52) Rewriting the decoupled equations as
\partial t −i∆ 2 + i ˜Ω ! \partial t −i∆ 2 −i ˜Ω ! cg = 0
\partial t −i∆ 2 + i ˜Ω ! \partial t −i∆ 2 −i ˜Ω ! ˜ce = 0, (5.53) where ˜Ωis the generalized Rabi frequency, ˜Ω:= p Ω2 + ∆2, (5.54) it is easy to see that any function that causes either factor to vanish will solve the equation, and thus the solutions are linear combinations of functions of the form exp(i∆t/2\pmi˜Ωt/2). We will thus write the solutions as
Ag cos 1 ˜Ωt + Bg sin 1 ˜Ωt
Ae cos 1 ˜Ωt + Be sin 1 ˜Ωt . (5.55)
Chapter 5. Two-Level Atom Interacting with a Classical Field Setting t = 0 in these equations gives
(5.56) while differentiating the solutions and comparing to the equations of motion gives
2 cg(0) + ˜Ω 2 Bg = −iΩ 2 ˜ce(0)
˜Ω [∆cg(0) + Ω˜ce(0)]
2 ˜ce(0) + ˜Ω
˜Ω [∆˜ce(0) −Ωcg(0)] . (5.57) Thus, the general solution in terms of initial conditions reads
cg(0) cos 1 ˜Ωt −i ˜Ω [∆cg(0) + Ωce(0)] sin 1 ˜Ωt
˜ce(0) cos 1 ˜Ωt + i ˜Ω [∆˜ce(0) −Ωcg(0)] sin 1 ˜Ωt . (two-level atom solution, arbitrary ∆) (5.58) 5.2.2.1 Example: Initially Unexcited Atom
cos 1 ˜Ωt −i∆ ˜Ω sin 1 ˜Ωt
˜Ω sin 1 ˜Ωt . (5.59) The excited-state population is thus
˜Ω2 sin2 1 ˜Ωt = Ω2 ˜Ω2 1 2 −1 2 cos ˜Ωt . (5.60) We can thus notice two things. First, the Rabi oscillations now occur at the generalized Rabi frequency
Wt D = 5W D = 2W D = W
D = 0 2p 3p 4p 5p 6p p Poo(t) e The other thing to notice is that the amplitude of the oscillations is reduced. The modulation depth (maximum excitation probability) is Ω2 ˜Ω2 = Ω2 Ω2 + ∆2 , (5.61) which reduces to Ω2/∆2 for weak excitation and to 1 for strong excitation.
5.3 Dressed States¶
5.3 Dressed States D/W Wt = p Wt = 2p
-2 -4 -6 -8 Poo(t) e 5.3 Dressed States One thing that we can conclude from the Rabi-flopping behavior is that the old eigenstates |g\rangle and |e\rangle are no longer eigenstates of the coupled system. However, we can still find the new eigenstates. We start by writing the equations of motion (5.23) for the coefficients in the matrix form \partial t ˜ce cg = −i " −∆ Ω/2 Ω/2 # ˜ce cg = −i ¯h ˜H ˜ce cg , (5.62) where we identify the rotating-frame Hamiltonian ˜H = ˜HA + ˜HAF = ¯h " −∆ Ω/2 Ω/2 # (5.63) in the uncoupled energy basis. We leave the diagonalization of this Hamiltonian as an exercise, but it can be shown that the eigenvalues are E\pm = −¯h∆ 2 \pm ¯h˜Ω 2 , (5.64) (dressed-state energies) and the corresponding eigenvectors are given by an effective rotation of the uncoupled states,
(5.65) (dressed states) where by convention the state |+\rangle has the higher energy, and the Stückelberg angle \theta is defined via
∆
. (5.66) (Stückelberg angle) These are the dressed states of the atom, and we see from Eq. (5.64) that the coupling to the field causes an avoided crossing in the energy level structure of the atom. That is, in the uncoupled case, the energies of the ground and excited states—the bare states—are 0 and −¯h∆, respectively. The energy curves cross when ∆= 0. D E E ooo= -hDooo e E oo= 0ooo g
5.3.1 Rabi Oscillations in the Dressed-State Picture¶
Chapter 5. Two-Level Atom Interacting with a Classical Field The coupling between the two levels lifts the degeneracy, however, converting the level crossing into a hyperbolic avoided crossing. Avoided crossings happen commonly in quantum mechanics, and here we have a simple model for this phenomenon. D W W~ E Eò E– In the limit of large detuning, when the coupling is small, we can approximately identify the coupled eigen- states with the uncoupled eigenstates. Near resonance, however, the states are mixed, and the energies are shifted. The energy shifts are the ac Stark shifts, or lamp shifts, which we will treat in more detail later. 5.3.1 Rabi Oscillations in the Dressed-State Picture We saw that due to the interaction of the field, the atom undergoes Rabi oscillations, where the magnitudes of the probability amplitudes oscillate in time. How does this work in the dressed-state picture, where the only possible evolution of the amplitudes is a change in phase? As an illustration, we’ll work this out explicitly for the resonant case ∆= 0. In this case, the dressed states are even and odd superpositions of the bare states,
(5.67) (we are dropping the normalization factor 1/ \sqrt 2 here), while the bare states are
(5.68) in terms of the dressed states. Then if the atom starts in the ground state |g\rangle = |+\rangle + |−\rangle , the dressed-state phases will evolve according to the appropriate energies:
(5.69) Dropping an irrelevant overall phase,
(5.70) Thus, it is the relative phase of the two dressed states that changes in time, and causes the Rabi flopping. The frequency splitting between the dressed states is just Ω, so the evolution must be periodic with frequency Ω. At time Ωt = \pi,
(5.71) and the atom is in the excited state. At time Ωt = 2\pi,
(5.72) and the atom is back in the ground state. This is precisely what we found in our analysis above. Of course, the same general picture holds off resonance. In this case, the dressed-state splitting becomes ˜Ω, so the oscillations occur at the generalized Rabi frequency. Also, from Eq. (5.65) the bare states are not equal superpositions of the dressed states, so the off-resonant Rabi flopping will not result in a complete transfer of population, say from |g\rangle to |e\rangle . This intuitive picture gives a simple way of looking at the analytic, off-resonant solutions we found above.
5.3.2 Adiabatic Passage and Landau-Zener Crossings¶
5.3 Dressed States 5.3.2 Adiabatic Passage and Landau–Zener Crossings Suppose you have an atom in the ground state, and you want to tranfer it exactly into the excited state. Sure, you can just use a resonant \pi-pulse. But lasers aren’t perfect: they have jitter in both frequency and amplitude, the amplitude jitter often being more important directly on resonance. Instead of working really hard to stabilize the laser, there is an easy trick that you can use, provided you can chirp (sweep) the laser frequency. The idea is to chirp the laser frequency across the atomic resonance. As long as the chirp is slow enough, it turns out that the atom will flip to the excited state with nearly unit probability. This effect is called adiabatic passage. We can see how this works as follows. Suppose we start the chirp well below resonance, ∆≪−Ω
thus |−\rangle \approx |g\rangle . As the chirp slowly proceeds, the adiabatic theorem guarantees that the atom will remain in the |−\rangle dressed state. When the chirp ends well above resonance, ∆≫Ω, we can see that the phase angle
D |gÒ |gÒ |-Ò |+Ò |eÒ |eÒ (start) (end) Thus, by mixing the two states, the atom adiabatically passes from |g\rangle to |e\rangle : the avoided crossing exchanges the identities of the uncoupled eigenstates. Of course, we could have chirped the other way, in which case the atom would have followed the |+\rangle dressed state from |g\rangle to |e\rangle . Clearly this happens if the chirp is ‘‘slow enough.’’ For a very fast chirp, the atom–field coupling has effectively no time to become manifest, and the atom starting in |g\rangle will jump (‘‘tunnel’’) across the avoided crossing as if it weren’t there. For intermediate chirps, we might expect the atom to end up in a superposition of |g\rangle and |e\rangle , since some of the probablity will tunnel through the gap. but how slow is slow enough? This problem can be solved completely, and the solution was given independently by Landau, Zener, and Stückelberg; this is commonly called the Landau–Zener crossing problem.4 To solve the problem, recall the atom–field interaction Hamiltonian from Eq. (5.18) in the nonrotating frame: HAF = ¯hΩ
. (5.73) This came from a monochromatic field E(+) with phase e−i\omegat. To generalize this to a chirped field, we note that the frequency is the rate at which phase accumulates, so that in general we can write the phase as e−i\phi(t), with \omega ≡d\phi(t)/dt. Thus, the generalized interaction Hamiltonian has the same form as above, but with the replacement e−i\omegat −\rightarrow exp −i Z t \omega(t′) dt′ , (5.74) where \omega(t) is the instantaneous frequency. We carry through the derivation of the Schrödinger equation as before, defining the slowly varying excited-state amplitudes by ˜ce = ce exp i Z t \omega(t′) dt′ (5.75) 4L. D. Landau, ‘‘Zur Theorie der Energieübertragung. II.,’’ Physikalische Zeitschrift der Sowjetunion 2, 46 (1932); Clarence Zener, ‘‘Non-Adiabatic Crossing of Energy Levels,’’ Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 137, 696 (1932); E. C. G. Stueckelberg, Helvetica Physica Acta 5, 369 (1932). See also Jan R. Rubbmark, Michael M. Kash, Michael G. Littman, and Daniel Kleppner, ‘‘Dynamical effects at avoided level crossings: A study of the Landau–Zener effect using Rydberg atoms,’’ Physical Review A 23, 3107 (1981) (doi: 10.1103/PhysRevA.23.3107).
Chapter 5. Two-Level Atom Interacting with a Classical Field so that the rotating-frame atomic and atom–field Hamiltonians become
˜HAF = ¯hΩ
, (5.76)
2 ˜ce
2 cg. (5.77) Decoupling by differentiation and back substitution as usual gives \partial 2
˜ce = 0 \partial 2
cg = 0. (5.78) Now we transform into yet another rotating frame defined by the new variables c′ e := ˜ce exp −i Z t ∆(t′) dt′ c′ g := cg exp −i Z t ∆(t′) dt′ , (5.79) in terms of which the equations (5.78) become \partial 2 t −i
4 + Ω2 c′ e = 0 \partial 2 t + i
4 + Ω2 c′ g = 0. (5.80)
\partial 2 t + Ω2 4 −i\alpha
4 t2 c′ e = 0 \partial 2 t + Ω2
4 t2 c′ g = 0, (5.81) and then defining the new variables
(5.82) the uncoupled equations of motion become \partial 2
2 −z2 c′ e = 0 \partial 2
2 −z2 c′ g = 0. (5.83) These are both in the form of Weber’s equation, \partial 2
2 −z2 y = 0, (5.84)
5.3 Dressed States whose solutions are the parabolic-cylinder functions D\nu(z), D\nu(−z), D−(\nu+1)(iz), and D−(\nu+1)(−iz).5 We will now be concerned with the solution for the excited-state amplitude c′ e. The leading-order asymptotic expansion of Dn(z) (for large |z|) is
(5.85) for | arg(z)| < 3\pi/4. Letting \nu −\rightarrow −(\nu + 1) and z −\rightarrow i|z|e−i\pi/4,
(5.86) We can conclude from this that D−(\nu+1)(−iz) is a function that vanishes for large |z|, if z follows the direction
we can write our solution for the excited-state amplitude c′
(5.87) where A is an undetermined constant, since this represents a particular solution to Weber’s equation with the correct initial condition c′
(5.88) Note that we are idealizing the chirp, assuming that it extends to all frequencies, and thus we treating this as a scattering-type problem, where the boundary conditions are applied at t = \pm\infty. Now to determine the coefficient. We use the bare-state equations of motion (5.77), which in terms of new variables becomes c′ g = i \sqrt\nu
c′ e. (5.89) We can then use the asymptotic expansion (5.86) to find |c′ g| in the limit t −\rightarrow −\infty: c′ g = |A| p |\nu|
(5.90) Our other boundary condition is that the ground state is initially fully populated, c′ g(t −\rightarrow −\infty) = 1, (5.91) and thus we have |A| = p
(5.92) which fixes the form of the solution, up to an overall phase. To look at the t −\rightarrow \inftylimit, we use an alternate form of the asymptotic expression,
\sqrt 2\pi
(5.93)
Letting \nu −\rightarrow −(\nu + 1) and z −\rightarrow −i|z|e−i\pi/4, and keeping only the leading-order term, we find
∼ \sqrt 2\pi
(5.94) in which case the excited-state population becomes
=
(5.95) 5The theory of Weber’s equation and the parabolic-cylinder functions that we will use here is covered by E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, 4th reprinted ed. (Cambridge, 1945), section 16.5.
5.4 Bloch Sphere¶
Chapter 5. Two-Level Atom Interacting with a Classical Field in the limit t −\rightarrow \infty. Thus, the fraction that didn’t adiabatically follow the dressed state is
−\piΩ2 2|\alpha| = exp −\piΩ2 2|\partial t∆| . (Landau–Zener tunnel probability) (5.96)
lost—all the population adiabatically follows the dressed state and makes the transition. In the opposite
everything stays in the original state. The simulated evolution for an atom undergoing a chirp from ∆/Ω= −20 to 20 for two different chirp
D/W -20 -10 Pe The Landau–Zener predictions for the final excited-state population are 0.9996 and 0.544, respectively. What we have discussed here is just adiabatic passage. We also need to consider the effects of the fact that the excited state will decay due to spontaneous emission. Thus, we also need adiabatic rapid passage,6 since the chirp must be fast compared to decay rate of excited state for the analysis to be valid. The field must also be strong enough so that the dressed states are well resolved, in spite of homogeneous broadening of the levels. Thus, if the Rabi frequency is much larger than the decay rate, and the field is chirped quickly enough, the atom can still make a complete transition. 5.4 Bloch Sphere Consider the Pauli operators
" #
" −i i # = i
" −1 #
(5.97) 6Adiabatic rapid passage was first observed in NH3 by sweeping the resonant frequency via the dc Stark effect (shifting the resonant frequency by applying an external dc electric field). See Michael M. T. Loy, ‘‘Observation of Population Inversion by Optical Adiabatic Rapid Passage,’’ Physical Review Letters 32, 814 (1974) (doi: 10.1103/PhysRevLett.32.814).
5.4 Bloch Sphere which satisfy the commutation and anticommutation relations¶
(5.98) These operators work on (rotating-frame) states with the ordering ˜ce cg . (5.99) The idea behind the Bloch sphere7 is to use the expectation values \langle \sigma\alpha\rangle as dynamical coordinates for the atomic evolution. Let’s first connect these variables to the density matrix for the atom. It is sufficient to use the relations (in the rotating frame)
\sigma\dagger
(5.100) Here the twiddles indicate coherences in the rotating frame. To make this more explicit, ˜\rho is the slowly varying state, which in a pure state has the form | ˜\psi\rangle \langle ˜\psi|. The corresponding density matrix for a pure state then has the form ˜\rho\alpha\beta = ˜c\alpha˜c∗ \beta (with ˜cg ≡cg). Then the rotating-frame populations are independent of the choice of frame,
e = cec∗
(5.101) while the coherences in the rotating frame differ from the usual coherences by a rotating phase factor,
(5.102) and thus the rotating-frame coherences are called the slowly varying coherences. Now using the evolution equations for the coefficients ˜ce and cg [Eqs. (5.23)], we can compute the equations of motion for the excited-state population,
= i∆˜ce˜c∗ e −iΩ 2 cg˜c∗ e + c.c. = iΩ
(5.103) the ground-state population,
= −iΩ
(5.104) 7after Felix Bloch, who derived the equation of motion for a spin in a magnetic field, which has the spherical representation. See F. Bloch, ‘‘Nuclear Induction,’’ Physical Review 70, 460 (1946) (doi: 10.1103/PhysRev.70.460).
Chapter 5. Two-Level Atom Interacting with a Classical Field and the coherences,
e + ˜c∗ e\partial tcg = −i∆cg˜c∗ e + iΩ 2 cgc∗ g −iΩ 2 ˜ce˜c∗ e
ge
(5.105) Of course, these four equations for the density matrix elements are equivalent to the Schrödinger–von Neu- mann equation
¯h h
i (5.106) in the rotating frame. Again, without the transformation to the rotating frame, the equations of motion would have explicit time dependences, representing the relative precession of the atomic and field phases. Given the above relations for the density matrix elements, how many degrees of freedom are there? There are four matix elements, and if each is complex, then there are eight possible independent, real numbers. The populations must be real, so this removes two free variables. The populations much further sum up to unity, removing another free variable. Finally, the constraint \rhoge = \rho∗ eg removes two more free variables, leaving only three independent, real numbers to represent the quantum state. This motivates the idea of using a three-vector (in R3) to represent the atomic state. To proceed with this idea, we start with the relations
\sigma\dagger
\sigma\dagger
(5.107) for the Bloch variables, and then we use the equations of motion for the density matrix elements (5.103)- (5.105) to write
(5.108) (Bloch-vector equations of motion) Note that these equations may be rewritten in terms of the Bloch vector\langle \sigma\rangle :=\langle \sigmax\rangle ˆx +\langle \sigmay\rangle ˆy +\langle \sigmaz\rangle ˆz as
(5.109) (Bloch-vector equation of motion) which we can also write this as a torque equation in terms of a single ‘‘precession vector’’ ℘as8
(5.110) (Bloch-vector equation of motion) where ℘:= Ωˆx −∆ˆz, (5.111) (precession vector) in analogy with \tau = \partial tL = Ω\times L, with L the angular momentum, \tau the torque, and Ωthe angular frequency vector for the precession (or the magnetic field vector in the case of Larmor precession of 8The symbol ℘is pronounced ‘‘squiggle.’’
5.4 Bloch Sphere a magnetic moment).9 This picture of the two-level atom in terms of a precessing spin is known as the Feynman–Vernon–Hellwarth representation.10 One property that follows immediately from this representation is that the length of the Bloch vector is a constant of the motion. That’s because the change in \langle \sigma\rangle is always normal to it. Furthermore, if we assume a pure quantum state, we can see that
(5.112) since ˜\rhoeg˜\rhoge = \rhoee\rhogg for a pure state. Thus, the Bloch vector for a pure state lies on a sphere of unit radius, which we call the Bloch sphere. Thus, of the three independent real numbers in the density matrix, one of these is fixed by the purity of the state. The remaining two numbers, as we will see, correspond to the degree of atomic excitation and a phase angle. We can see that the FVH representation is a compact, handy way to visualize the evolution. The trajectories lie on the unit sphere, and their evolution is generated simply by a constant-speed rotation of the whole sphere. The rotation axis and angular speed are determined by the precession vector ℘, whose magnitude is simply ˜Ω. From Eqs. (5.107), we can interpret the meanings of the components of the Bloch vector. The vertical (z) component of the Bloch vector represents the degree of atomic excitation (population inversion): the Bloch vector points straight down for an atom in |g\rangle and straight up for an atom in |e\rangle . When the driving
9See Richard Feynman, Robert B. Leighton, and Matthew L. Sands, The Feynman Lectures in Physics (Addison–Wesley, 1963), Chapter 20. 10Richard P. Feynman, Frank L. Vernon, and Robert W. Hellwarth, ‘‘Geometrical Representation of the Schrödinger Equation for Solving Maser Problems,’’ Journal of Applied Physics 28, 49 (1957) (doi: 10.1063/1.1722572).
Chapter 5. Two-Level Atom Interacting with a Classical Field If the atom is initially in the ground state, the trajectory follows the great circle given by the intersection of the Bloch sphere with the x = 0 plane. Thus, it passes through the excited state. Now we turn to the transverse (x and y) components of the Bloch vector. Recalling that the dipole operator has the form
(5.113) we can easily see that
(5.114) Thus, \langle \sigmax\rangle represents the atomic dipole. The other transverse component \langle \sigmay\rangle , represents the alternate ‘‘quadrature’’ of the dipole moment. That is, \langle \sigmax\rangle represents the real part of
d(+) ,
hD d(+)Ei , (5.115) while \langle \sigmay\rangle represents the imaginary part of
d(+) ,
hD d(+)Ei . (5.116) Note that there can only be a dipole moment when the atom is in a superposition of |g\rangle and |e\rangle , since the diagonal matrix elements of d vanish. The other picture of the transverse Bloch-vector components is as follows. When the atom is in a superposition of |g\rangle and |e\rangle , the azimuthal angle (represented by the x and y components of the Bloch vector) represents the relative phase of the ground and excited states. In the absence of an external field, Ω= 0. Then the precession vector is ℘= −∆ˆz.
5.4.1 Atomic Timekeeping and Ramsey Fringes¶
5.4 Bloch Sphere Thus, the Bloch sphere simply spins about the z-axis, causing azimuthal rotation of the trajectories. This evolution represents the relative phase evolution of the ground and excited states. But remember that we are in the rotating frame, where the rotation rate is the field frequency \omega. To go back to the original variables, you just have to add \omega to the precession frequency to get \omega0ˆz in the stationary coordinates. We can therefore see that the free evolution is just the precession of the excited state phase relative to that of the ground state. With a nearly resonant driving field with nonzero detuning, the rotation axis is tilted, being a combi- nation of the previous two rotations. If the atom starts in the ground state, the trajectory never quite makes it exactly to the excited state. This a nice way to visualize how the off-resonant excitation ends up being incomplete. Furthermore, the rate at which the Rabi oscillations occur is given by the magnitude |℘| = p
(5.117) as we saw from the direct solution to the Schrödinger equation. 5.4.1 Atomic Timekeeping and Ramsey Fringes One nice application of the Bloch sphere is to understanding Ramsey fringes,11 which form the basis for atomic time and frequency standards as well as the benchmark for demonstrating quantum coherence. Suppose that we want to let a field interact with an atom in order to compare their frequencies. The main limitation in doing this is the interaction time, since long interaction times are required to resolve small frequency splittings (as dictated by the ‘‘time-frequency uncertainty relation’’). In principle, one can let the atom and field interact for a long time, but this poses a number of difficult problems. For example, for configurations such as the atomic beam, it is difficult to maintain a uniform interaction over the entire length of the beam, since the interaction region must be large (say, meters) for a sensitive measurment. Furthermore, constraints on the apparatus itself (such as the vacuum system) may not permit the driving field to enter in certain regions. And even if it is possible to have a uniform field, the nearly resonant field will cause an energy (Stark) shift of the transition (as we saw from the dressed-state solutions), so that a precise comparison isn’t possible anyway. So what do we do? Well, there is a clever trick called Ramsey’s method of separated, oscillatory fields. Suppose we have an beam of atoms moving at velocity v. We allow two identical fields (laser or microwave fields, depending on the transition, but both fields are derived from the same source) of width ℓto cross the beam a distance L apart. Mathematically, we will idealize the fields as spatially uniform, but it is straightforward to generalize this to arbitrary beam profiles. The beams are then followed by a state-sensitive detector (Stern–Gerlach apparatus) to measure the excitation probability. 11Ramsey, a former student of Rabi, shared the 1989 Nobel prize for the method of separated, oscillatory fields. See Norman F. Ramsey, ‘‘A New Molecular Beam Resonance Method,’’ Physical Review 76, 996 (1949) (doi: 10.1103/PhysRev.76.996); Norman F. Ramsey, ‘‘A Molecular Beam Resonance Method with Separated Oscillating Fields,’’ Physical Review 78, 695 (1950) (doi: 10.1103/PhysRev.78.695); Norman F. Ramsey, Molecular Beams (Oxford, 1956).
PDF page 172¶
To see how this works, we will assume that the field is very close to the atomic resonance, so that \(|\Delta| \ll \mathcal{W}\), where \(\mathcal{W}'\) is the Rabi frequency for each field. In doing so, we can ignore the fact that the Rabi oscillations do not quite occur about the x-axis. Now letting \tau = \ell / v be the interaction time of each field with the passing atoms, the first field causes a Rabi oscillation with an accumulated phase of \(\widetilde{\mathcal{W}}\,\tau \approx \mathcal{W}\tau\). We assume the atoms start in the ground state, and we will choose the field amplitude \mathcal{W}^ such that the field drives a `\(\pi/2\)-pulse (i.e., \(\mathcal{W}\tau = \pi/2)\)). Then the interaction with the first field puts the atom in an equal superposition of the ground and excited states.
Then in between the fields, the atom undergoes free evolution—precession about the -z-axis at rate \Delta$—for a timeT = L/v`. The accumulated phase is thus \(-\Delta T\).
The final field causes another \(\pi/2\)-pulse, but its effect depends on the state of the atom after the precession stage. If the atom ends up with its initial phase after the precession$, which happens if \DeltaT is an integer multiple of 2$\pi, then the effect of the second `$$`puls continues the evolution from before and promotes the atom to the excited state.
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5.4 Bloch Sphere On the other hand, if the atom ends up with the opposite phase after precession, as happens when ∆T is an odd-integer multiple of \pi, then the second \pi/2-pulse has the opposite effect: the atom returns to the ground state. For other final phases after the precession stage, the final excited-state population interpolates sinusoidally between these extreme values. We thus see that the output signal (the excited-state population) is sinusoidal in T with period 2\pi/∆. Similarly the output signal is sinusoidal in ∆with period 2\pi/T. DT 2p 3p p -p -2p -3p Poooe These oscillations are what are referred to as Ramsey fringes. Essentially, we have built something like an optical Mach–Zehnder interferometer, but where the two arms of the interferometer correspond to the internal states of the atom, and the beamsplitters correspond to the \pi/2-pulses. Alternately, we can think of this experiment as a sort of Young double slit, but where the slits are separated in time (and thus the fringes appear as a function of frequency). Since the output signal varies between 0 and 1, we can write Pe = cos2 ∆T = 1 2 (1 + cos ∆T) , (5.118) and we can see that the width (FWHM) of the central fringe is \pi/T in angular frequency. Thus, the accuracy of the comparison of the atom and field frequencies increases as T increases, as we expect.
5.4.2 Spin Echoes and Photon Echoes¶
Chapter 5. Two-Level Atom Interacting with a Classical Field In the more general case, where we are not restricted to very small detunings, the excitation probability can be written (Problem 5.6) Pe = 4Ω2 ˜Ω2 sin2 ˜Ω\tau ! " cos ∆T cos ˜Ω\tau ! −∆ ˜Ω sin ∆T sin ˜Ω\tau !#2 , (Ramsey excitation proability) (5.119) where \tau is the interaction time of each of the two laser pulses of Rabi frequency Ω, and T is the time interval (time spent ‘‘in the dark’’) between pulses. For larger detunings, then, the tipped Rabi oscillations lead to a reduction in the fringe contrast. A more serious reduction in the fringe contrast for larger detunings occurs in atomic beams due to the longitudinal velocity spread, which causes slightly different interaction times and thus slightly different fringe spacings. This method is precisely the one used for atomic clocks. The current frequency standard corresponds to the energy difference between two spin states of an isolated 133Cs atom, which is defined to have a transition frequency of 9.192 631 770 GHz. In typical cesium frequency standards, a ‘‘flywheel’’ oscillator with very good short-term stability, such as a crystal oscillator or a hydrogen maser, has its frequency periodically compared to the cesium transition frequency by a similar atomic beam measurement. Of course, there are many tricks in getting this method to work well and to compensate for systematic effects that we won’t get into here. One of the best atomic-beam clocks was NIST-7,12 a cesium-beam atomic clock with a drift region of L = 1.53 m, an interaction region of ℓ= 2.3 cm, and a mean beam velocity of 230 m/s. It operated from 1993-1999, and had an uncertainty of 5 \times 10−15. The current U.S. standard clock operated by the National Institute of Standards and Technology (NIST) is NIST-F1, a ‘‘fountain’’ clock, where a sample of ultracold atoms is tossed upwards and returns to the interaction region under free fall to reach long interaction times, and as of 2005, the uncertainty is about 5 \times 10−16. The measure of clock stability is the ratio \omega0/\delta\omega, where \delta\omega is the frequency uncertainty. Traditional approaches to cesium clocks has focused on making \delta\omega as small as possible. However, the ratio can also be made large by choosing a much larger transition frequency, such as in the optical regime. Candidates for future standards include optical transitions in single trapped ions, where the transition shift due the trapping fields averages to zero, or atoms trapped in ‘‘magic wavelength’’ optical lattices, which we will discuss soon. 5.4.2 Spin Echoes and Photon Echoes Ramsey-type experiments work a lot less well when dephasing between members of an ensemble occurs. This type of issue crops up with inhomogeneous broadening, where each atom effectively has a slightly different resonance frequency, as happens with Doppler broadening in atomic vapors or local field effects on atoms embedded in crystalline media (and has something of a similar effect of a velocity spread in the Ramsey experiment, which causes effectively different drift times for different atoms). To see the problem, let’s walk through the Ramsey experiment in the presence of inhomogeneous broadening. The first step is the \pi/2-pulse to put the atoms in a superposition of the ground and excited states. 12J. H. Shirley, W. D. Lee, and R. E. Drullinger, ‘‘Accuracy evaluation of the primary frequency standard NIST-7,’’ Metrologia
5.4 Bloch Sphere During the free-drift time, each atom precesses at a slightly different frequency, leading to a spread in phase angles that increases with time. As shown here, the blue vectors precess more quickly than the green vectors. Now when the second \pi/2-pulse (the ‘‘interrogation pulse’’) comes, the Bloch sphere rotates appropriately. But in a situation that would put all the atoms in the ground state, only a small fraction of the atoms actually makes it to the right state. In the limit of large dephasing, the atoms are spread uniformly around the equator, and thus after the interrogation pulse, the average excited-state population is just 1/2, independent of the drift time. The Ramsey fringes damp away in a drift time of order 1/\delta\omega0, where \delta\omega0 measures the inhomogeneously broadened width of the atomic transition. This damping of the ensemble-averaged dipole moment due to dephasing is sometimes called free-induction decay.
Chapter 5. Two-Level Atom Interacting with a Classical Field The fact that the Ramsey fringes damp away makes it look like irreversible decoherence of the atomic polarization. But just how reversible is it? Let’s try something else—after the atoms have dephased, hit the atoms with a \pi-pulse. This effectively reflects the orientation of the dipoles. Now as the evolution continues, the dipoles begin to come back together to the same phase. In the diagram, the faster (blue) dipoles are now behind the slower (green) ones, and thus the slower ones can now ‘‘catch up.’’ The other way to look at this is that the reflection due to the \pi-pulse is effectively equivalent to flipping the precession axis, and thus reversing the direction of time. The dipoles thus come back to their common original location. Actually, that’s not quite right: they come back to the mirror image of the original orientation, if we account fully for the effect of the reflection of the \pi-pulse. When the dipoles rephase, the Ramsey fringes become visible again. If the \pi-pulse is applied a time T after the original preparation pulse, the spin echo occurs at time 2T. This phenomenon in the case of nuclear spins is called the spin echo.13 t drift p-pulse echo P In the optical case, there is a more direct, dramatic manifestation. Suppose that an atomic sample (a ruby crystal in the original experiments) is irradiated by two coherent light pulses separated by some time 13E. L. Hahn, ‘‘Spin Echoes,’’ Physical Review 80, 580 (1950) (doi: 10.1103/PhysRev.80.580).
5.4.3 Adiabatic Following¶
5.5 Optical Bloch Equations T. The atoms then spontaneously emit another pulse of light a time T after the second pulse. We can see how this works based on the above analysis. The first laser pulse comes in and polarizes the atoms. After the dipoles dephase, the atoms can only emit incoherently, so the radiation is nondirectional and even suppressed, as we have seen is the case for classical dipoles radiating out of phase. The other way to say this is that the polarization wave decays away. However, upon applying the \pi-pulse, the dipoles rephase, and the polarization wave recurs. The recurrent polarization wave emits a pulse of light, an ‘‘echo’’ of the original excitation. This is called the photon echo.14 5.4.3 Adiabatic Following We will close this discussion of the Bloch sphere by revisiting the problem of rapid adiabatic passage. Suppose a bunch of atoms begin in the ground state, so that the Bloch vector points along −z. Now apply
that the precession vector ℘\approx |∆|ˆz is aligned with the Bloch vector. The precession of the Bloch vector is very simple, since it just stays in place. Now start sweeping the detuning through resonance, so that the precession vector moves through the x-axis and up towards the +z-axis. As long as we change the direction of ℘slowly on the time scale of the Rabi frequency Ω, the Bloch vector will follow the precession vector. When the detuning is swept until it is far below resonance, the precession vector has flipped by this time, and the Bloch vector has flipped as well. This is the ‘‘classical’’ view of the adiabatic passage problem that we treated quantum mechanically in Section 5.3.2. 5.5 Optical Bloch Equations Now let’s return to the evolution of the density operator. Recall that we have the Schrödinger–von Neumann equation in both the ‘‘laboratory’’ frame,
(5.120) and in the rotating frame,
¯h h
i . (5.121) In the latter case, we have already worked out the equations of motion for the density-matrix elements in Eqs. (5.103)-(5.105):
(5.122) To model spontanous emission, we need to add extra terms. We will do so now by simply putting them in, but we will justify them later. With ∆= Ω= 0, the extra terms have the form
(5.123) 14Photon echoes with pulse separations of around 100 ns (with 10 ns pulses) were observed in ruby (T ∗ 2 ∼0.1 ns) by I. D. Abella, N. A. Kurnit, and S. R. Hartmann, ‘‘Photon Echoes,’’ Physical Review 141, 391 (1966) (doi: 10.1103/PhysRev.141.391).
Chapter 5. Two-Level Atom Interacting with a Classical Field Let’s look at these and understand them. The excited-state population now decays at a rate of \Gamma, and to compensate for this, a similar term puts the decayed population into the ground state. These terms have exactly the same form as the rate-equation terms for spontaneous emission, if we identify \rhoee and \rhogg as the relative number densities Ne/N and Ng/N in the excited and ground states, respectively. We thus identify \Gamma = A21 as the excited-state decay rate. Since \Gamma is the rate of relaxation of the z-component of the Bloch vector to the ground state, it is also called the longitudinal decay rate. The coherences also damp at the rate \gamma⊥, which are introduced phenomenologically now, but which we will justify later via the quantum theory of damping. For now we note that in general \gamma⊥\ge \Gamma/2, and in fact we can write
(5.124) where \gammac models additional coherence decay beyond the minimum rate of \Gamma/2 needed for consistency with spontaneous emission. Thus \gammac models dephasing effects such as atom–atom collisions that do not affect the populations. Since \gamma⊥is the rate at which the coherences damp it is also the rate at which the transverse components (transverse to z) of the Bloch vector damp, and hence \gamma⊥is called the transverse decay rate. The original and still common notation15 for these decay rates is in terms of the longitudinal relaxation time T1 = 1/\Gamma and transverse relaxation time T2 = 1/\gamma⊥. Note that the notation T ∗ 2 is used when there is inhomogeneous broadening, and would include inhomogeneous dephasing as well as other sources of damping (e.g., collisions), so that T ∗ 2 \le T2. We can thus combine the damping terms with the Hamiltonian-evolution terms in (5.122) to obtain the optical Bloch equations:
(5.125) (optical Bloch equations) That is, these are the extension of Bloch’s original equations for nuclear magnetic resonance to the optical regime. Note that we may also write the damped optical Bloch equations in terms of the Bloch vector as
, (optical Bloch equations, Bloch-vector form) (5.126)
rates for the three Bloch-vector components as we discussed above (note that there is no implied summation in the \gamma\alpha\langle \sigma\alpha\rangle term. Writing the components out separately gives
, (optical Bloch equations, Bloch-vector form) (5.127) where we can see explicitly that the damping terms push the transverse components towards zero, while they push the longitudinal component towards the ground-state value \langle \sigmaz\rangle = −1. 15F. Bloch, op. cit.
5.5.1 Steady State¶
5.5 Optical Bloch Equations 5.5.1 Steady State¶
iΩ \gamma⊥−i∆ \rhoee −1
\gamma2 ⊥+ ∆2 \rhoee −1 . (5.128) The complex conjugate of this equation is
iΩ
\rhoee −1
\gamma2 ⊥+ ∆2 \rhoee −1 , (5.129) which we can subtract from the previous equation to obtain
2iΩ\gamma⊥ \gamma2 ⊥+ ∆2 \rhoee −1 . (5.130) Now we can set \partial t\rhoee = 0 to obtain
\gamma2 ⊥+ ∆2 \rhoee −1 . (5.131) Solving for \rhoee, we find the steady-state excitation
Ω2 2\gamma⊥\Gamma 1 + ∆2 \gamma2 ⊥ + Ω2 \gamma⊥\Gamma . (5.132) (steady-state excitation) We can put this result into Eq. (5.128) to obtain the steady-state coherence
2\gamma⊥ 1 + i∆ \gamma⊥ 1 + ∆2 \gamma2 ⊥ + Ω2 \gamma⊥\Gamma . (5.133) (steady-state coherence) The other elements of the density matrix are of course given by \rhoee + \rhogg = 1 and ˜\rhoge = ˜\rho∗ eg. We can simplify the notation here somewhat by defining the saturation parameter s :=
⊥ . (5.134) (saturation parameter) The saturation parameter is proportional to the intensity, and it has a Lorentzian frequency profile with full width at half maximum of 2\gamma⊥. We can then write the steady-state solutions as
1 + s
\Gamma 4\gamma⊥ s (1 + s)2 . (steady-state solutions to optical Bloch equations) (5.135) In this form it is easier to see that we get generally the same result that we got for the rate equations in Eq. (3.6): for small intensities, the excitation increases linearly with s (as s/2), and in the limit of large intensity (s −\rightarrow \infty), the largest possible excitation is half the population (\rhoee −\rightarrow 1/2). Furthermore, although the excitation \rhoee increases monotonically with s, we can see that the expectation value of the
5.5.2 Damped Rabi Oscillations¶
Chapter 5. Two-Level Atom Interacting with a Classical Field dipole moment, which is proportional to the real part of ˜\rhoge, increases as \sqrts for small excitations but decreases back to zero for very large excitations. You might get the false impression from this that a highly excited atom does not radiate! This is not quite true, and we will return to this point shortly. Most often we will be concerned with the ‘‘pure’’ case of homogeneous (natural) broadening, with \gamma⊥= \Gamma/2. That is, there is no additional damping of the coherences due to collisions. In this case, the saturation parameter becomes s := 2Ω2/\Gamma2
(saturation parameter, homogeneous broadening) (5.136) the steady-state population becomes
Ω2/\Gamma2 1 + 2∆ \Gamma 2 + 2Ω2 \Gamma2 , (steady-state excitation, homogeneous broadening) (5.137) and the steady-state coherence is
\Gamma 1 + i2∆ \Gamma 1 + 2∆ \Gamma 2 + 2Ω2 \Gamma2 . (steady-state coherence, homogeneous broadening) (5.138) These solutions will be important in our discussion of resonance fluorescence. 5.5.2 Damped Rabi Oscillations 5.5.2.1 Laplace Transform Now let’s consider solutions to the optical Bloch equations. Recall (from Section 4.1) that we can write the Liouville–von Neumann equation for the density operator as
(5.139) where L is the Liouvillian superoperator, and effectively has larger tensor rank than the density operator. We can write this in component form as
(5.140) where \alpha is a composite index, so that the density matrix is a column vector (i.e., for a two-level atom, \alpha takes on the values ee, eg, ge, and gg). The Liouvillian then acts as a matrix in this notation. In the general case of a linear, time-independent equation of this form, we can obtain a solution via the Laplace transform. To review this method, start with the identity
Z t dt′ ˙y(t′) + y0, (5.141)
5.5 Optical Bloch Equations¶
Z \infty dt e−sty(t) = Z \infty dt e−st y0 + Z t dt′ ˙y(t′) = y0 s + Z \infty dt′ ˙y(t′) Z \infty t′ dt e−st = y0 s + 1 s Z \infty dt′ e−st′ ˙y(t′) = y0 s + 1 sL ˙y. (5.142) Here, we used the identity (5.141), and the fact that the two-dimensional integral is over the t′ < t region, so that we can interchange the order of integration via Z \infty dt Z t dt′ = Z \infty dt′ Z \infty t′ dt. (5.143) Thus, we can solve our result (5.142) for L [ ˙y] to find the transform of the time derivative L [ ˙y] = sL [y] −y0. (5.144) (Laplace transform of time derivative) We can now use this result to take the Laplace transform of the Liouville–von Neumann equation to find
(5.145) assuming L is time-independent. Thus, the Laplace transform conveniently changes a system of coupled differential equations into an algebraic problem. Now we can solve for L [˜\rho], with the result
s −L ˜\rho(0). (5.146) Note that the addition of a scalar and an operator here should be interpreted in the sense
(5.147) The operator (s −L)−1 is called the resolvent of the Liouvillian, and gives the decoupled form for the Laplace transform of the solutions in terms of the initial condition. The solutions are then given by inverse Laplace transforms:
s −L ˜\rho(0) . (5.148) (general solution to master equation) Again, we have assumed that L is time-independent: this is where it helps to use the density operator ˜\rho in the rotating frame. 5.5.2.2 Torrey’s Solutions To apply this method to the two-level atom, it is useful to do so in a slightly different form.16 Starting with the optical Bloch equations in the form (5.127), we can write these in matrix form as
−\gamma⊥ ∆ −∆ −\gamma⊥ −Ω Ω −\Gamma
\Gamma
(5.149) 16Here we follow H. C. Torrey, ‘‘Transient Nutations in Nuclear Magnetic Resonance,’’ Physical Review 76, 1059 (1949) (doi: 10.1103/PhysRev.76.1059). See also L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, 1975), section 3.5, p. 62.
Chapter 5. Two-Level Atom Interacting with a Classical Field We thus have a smaller matrix to deal with than the Laplacian, but at the expense of an extra constant in the equation. Now taking the Laplace transform of this equation, and using the fact that L [1] = 1/s, as we used in Eq. (5.142), we find
s. (5.150) Rearranging this, we get a slightly modified form of the resolvent solution:
(5.151) The analog of the resolvent operator here is s −Q = f(s)
(s + \Gamma)∆ −∆Ω −(s + \Gamma)∆
−∆Ω
, (5.152) where
. (5.153) Looking at this solution, we can see that each component of L [\langle \sigma\rangle ] can be written in the form g(s) sf(s), (5.154) where f(s) is a cubic polynomial in s, and g(s) is also at most cubic in s [and thus of smaller degree than f(s)]. We can thus write f(s) in terms of three roots −a1,2,3,
(5.155) and comparison to the form (5.153), where all the polynomial coefficients are positive, tells us that the product a1a2a3 > 0. For this to hold, as well as to maintain the positivity of the other coefficients, one of these numbers must be positive, and the other two must either also be real and positive, or they must form a complex-conjugate pair. Thus, we may write
, (5.156) where a, c > 0, and b is real for conjugate roots, but could otherwise be imaginary. This form for f(s) implies the partial-fraction decomposition g(s)
A
s , (5.157) where A, B, C, and D are constants that depend both on the initial conditions and on which component of \langle \sigma\rangle we are considering. Computing the inverse Laplace transform of this expression, we can see that the solution for any component of \langle \sigma\rangle can be written in the form
b e−at sin bt + D. (general form of solution to optical Bloch equations) (5.158) We can thus see that the general solution is reasonably simple, and even without finding the explicit forms of the coefficients, we can interpret the different terms. The first term represents exponential decay of the populations and coherences, as we would expect from the damping terms. The second and third terms represent exponentially damped Rabi oscillations with different phases. The final term is just the steady-state solution that we derived in Section 5.5.1.
5.5 Optical Bloch Equations 5.5.2.3 Exact Resonance Unfortunately, since the general solution depends on the roots of f(s), which do not have a simple form, we don’t gain much intuition by trying to write them down. Torrey17 identified three regimes where the
(5.159)
c = \Gamma 2 , (5.160) giving the decay rate of the damped but nonoscillating term. The other two roots are s = −3\Gamma 4 \pm iΩ\Gamma, (5.161) where Ω\Gamma := s Ω2 − \Gamma 2 (5.162) is the Rabi flopping frequency in the presence of damping. Note that in the limit Ω−\rightarrow 0, the roots become s = −\Gamma and s = −\Gamma/2, which is what we expect for the longitudinal and transverse decay rates in the absence of driving. Thus, we have fixed the other two roots, a = 3\Gamma 4 , b = Ω\Gamma, (5.163) in the notation above, giving the decay rate and oscillation frequency, respectively, of the oscillating terms. x-component: Now we need to determine the coefficients of the different terms for the three compo- nents. First for the \langle \sigmax\rangle case. Starting with the steady-state coefficient D, we note from Eq. (5.138) that on resonance the steady-state value of ˜\rhoeg is purely imaginary. But \langle \sigmax\rangle is the real part of ˜\rhoeg, so D = 0. From (5.157), we see that we can recover A from A = lim s\rightarrow −c g(s) sf(s)(s + c) = 0, (5.164)
derivative is
(5.165) Similarly differentiating Eq. (5.158),
(5.166) and comparing these two expressions gives C = 0. Thus,
(5.167) as we expect from the resonant case. 17H. C. Torrey, op. cit.
Chapter 5. Two-Level Atom Interacting with a Classical Field
find D = 2Ω/\Gamma 1 + 2Ω2 \Gamma2 . (5.168) The A coefficient is again given by A = lim s\rightarrow −c g(s) sf(s)(s + c) = lim s\rightarrow −\Gamma/2
s = 0. (5.169)
initial time derivatives of \langle \sigmay(0)\rangle from Eqs. (5.149) and (5.158) gives
3Ω/2 1 + 2Ω2 \Gamma2
Ω2 −\Gamma2/4
. (5.170) The complete solution is thus
Ω\Gamma
1 −e−(3\Gamma/4)t cos Ω\Gammat −Ω2 −\Gamma2/4 \GammaΩ\Gamma sin Ω\Gammat .
(5.171) We can clearly see the damped oscillations in the dipole moment at the frequency Ω\Gamma.
steady-state solution, we find
1 + 2Ω2 \Gamma2 −1 = − 1 + 2Ω2 \Gamma2 = Ω2
(5.172) Note that D −\rightarrow 0 as Ω−\rightarrow \infty, as we expect it to. The A coefficient is once again given by A = lim s\rightarrow −c g(s) sf(s)(s + c) = lim s\rightarrow −\Gamma/2
s = 0. (5.173)
Ω2
(5.174) Finally, again matching the initial time derivatives of \langle \sigmaz(0)\rangle from Eqs. (5.149) and (5.158),
(5.175) The complete solution is thus
Ω2
1 −e−(3\Gamma/4)t cos Ω\Gammat + 3\Gamma 4Ω\Gamma sin Ω\Gammat .
(5.176) We can again clearly see the damped Rabi oscillations at the damped Rabi frequency Ω\Gamma. Wt G = 5W G = W
G = 0 2p 3p 4p 5p 6p p Poo(t) e
5.5.4 Orders of Magnitude¶
5.5 Optical Bloch Equations¶
5.5.3 Operator Form The optical Bloch equations (5.125) can be written in the equivalent and compact operator form of a master equation for the density operator as
¯h h
i
(optical Bloch equations, operator form) (5.177) where we have defined the Lindblad superoperator
. (5.178) (Lindblad superoperator) The last two terms in the master equation correspond to decay of the excited state and extra dephasing due to collisions, respectively (the latter turns out to have the form of a measurement of \sigmaz, and thus causes increased uncertainty in the transverse spin components). Again, this is a ‘‘superoperator’’ on \rho because it operates on it from both sides. We will see that this form is universal, in the sense that all Markovian master equations (those where \partial t\rho(t) depends on \rho at time t and not any other time) always have damping terms in this Lindblad form—that is, any form where the master equation is written purely in terms of Hamiltonian commutators and Lindblad superoperators. Note also that in many cases, these damping terms must be written as separate terms from the Hamiltonian part. For example, take the naturally damped atom,
¯h h
i
(5.179) It is common in the literature to define an effective, non-Hermitian Hamiltonian by ˜Heff := ˜HA + ˜HAF −i¯h\Gamma
(5.180) In this way, we can absorb some of the Lindblad terms into the Hamiltonian:
¯h h
eff i
(5.181) However, one term remains. What is the meaning of this? Consider that
(5.182) Thus, this operator only enters the \partial t\rhogg equation, and in fact it is the term that returns the decayed population to the ground state. Thus, the non-Hermitian Hamiltonian can correctly handle both the decay of the excited state and the coherence decay. However, on its own, it does not preserve the trace of the density operator, as we can see from the form exp(−iHefft/¯h) of the time-evolution operator. Renormalization at each time step, then, introduces an extra term that cannot be written in terms of a Hamiltonian. 5.5.4 Orders of Magnitude Now that we have introduced a couple of time scales, it is useful to estimate their magnitude. First, the Rabi frequency. Suppose we have a Gaussian beam with beam-waist parameter (1/e2 intensity radius) w0.18 In terms of the total beam power P, the intensity at the beam center is I = 2P \piw 2 . (5.183) 18Daniel A. Steck, Classical and Modern Optics (2006), chapter 6, available online at http://steck.us/teaching.
5.6.1 Classical Limit¶
Chapter 5. Two-Level Atom Interacting with a Classical Field In terms of the field amplitude, the intensity is I = E 2 2\eta0 , (5.184) where \eta0 is the impedence of the vacuum, so that E0 = p 2\eta0I. (5.185) We may thus write the Rabi frequency as
¯h =
\pi¯h2w 2 1/2 . (5.186) The dipole matrix element \langle g|dz|e\rangle is of the order of ea0, where e is the fundamental charge and a0 \approx 0.529 Å is the Bohr radius. On the D2 transition (780 nm) in 87Rb, depending on the details of the interaction, the matrix element can be as high as about 3.0 ea0 \approx 2.5 \times 10−29 C \cdot m. A beam power of 10 mW is achievable by a very modest diode laser, and a beam waist of w0 = 1 mm is fairly typical. Putting in these numbers,
couple of W for the largest cw lasers at 780 nm) and smaller beam waists (on the order of 10 µm is easy to achieve), giving Rabi frequencies in the tens of GHz range. Even larger values can be achieved with pulsed lasers, until the fields are strong enough that the whole two-level treatment breaks down. We have also introduced the spontaneous decay rate \Gamma. Back when we studied the rate equations, we related this quantity to the Einstein B coefficient. Shortly, we will connect this to the dipole matrix element. But typically, for atomic transitions in the optical, decay rates \Gamma/2\pi are on the order of several MHz (6.1 MHz for 87Rb, corresponding to a lifetime of 26 ns). The decay rates turn out to scale as \omega 3 (Chapter 11), so these can become larger as the transition becomes more energetic. For ‘‘dipole forbidden’’ transitions, the decay rates can be substantially smaller. 5.6 Consistency with Other Models 5.6.1 Classical Limit We will now show the connection between the optical Bloch equations and the classical Lorentz atom. Formally, these two problems are equivalent in the case of a weak drive. The general idea is to construct the damped version of the quantum harmonic oscillator, and then show that it reduces to both the Lorentz model and the optical Bloch equations in the weak-excitation limit. 5.6.1.1 Review: Harmonic Oscillator in Quantum Mechanics We will take the following results for the quantum harmonic oscillator to be given. The Hamiltonian is H = p2 2m + 1 2m\omega 2
, (5.187) where the creation (a\dagger) and annihilation (a) operators are defined via the relation a = \sqrt x x0 + ix0p ¯h (5.188) and its Hermitian adjoint. The length scale x0 is given by x0 = r ¯h m\omega0 . (5.189)
5.6 Consistency with Other Models The phase-space operators can be written in terms of the ladder operators as x = x0 \sqrt
p = ¯h i \sqrt 2 x0 a −a\dagger . (5.190) We have the usual commutation relations [x, p] = i¯h
(5.191) which are equivalent to each other via the above definitions. We will denote the eigenstates of the Hamiltonian (‘‘Fock states’’) by |n\rangle for nonnegative n, with corresponding eigenvalues
n + 1 . (5.192) These states all have \langle x\rangle = \langle p\rangle = 0 and moments
n x2 n
= ¯h m\omega0 n + 1 ,
n p2 n
n + 1 . (5.193) In this basis, the ladder operators have the effect
\sqrt
(5.194) The eigenstate of the annihilation operator is the coherent state
\infty X n=0 \alphan \sqrt n!
(5.195) with eigenvalue \alpha:
(5.196) Note that the occupation probabilities for the number states form a Poisson distribution of mean \alpha. The effect of the creation operator on the coherent state is more complicated:
(5.197) The ground state |0\rangle is a special case of a coherent state. The general coherent state |\alpha\rangle has the same Gaussian probability-density profile as the ground state, but the centroid oscillates with frequency \omega0 and amplitude \sqrt 2 x0|\alpha| in position and \sqrt 2 ¯h|\alpha|/x0 in momentum. 5.6.1.2 Evolution of the Means: Damped Quantum Harmonic Oscillator We can add damping to the harmonic oscillator by adding an extra component in Lindblad form to the master equation:
(damped quantum harmonic oscillator) (5.198) Again, the Lindblad superoperator is given by
. (5.199) We already know from the discussion of the Moyal bracket that the Hamiltonian part of this evolution is classical. So now let’s check this again and show the correspondence of this oscillator with damping with the
Chapter 5. Two-Level Atom Interacting with a Classical Field classical damped oscillator. The master equation implies the equation of motion for the expectation value of an arbitrary operator A:
= −i
a\daggerAa −1
= − i
¯h
. (5.200) Recall that we are in the Schrödinger picture, so the time dependence is contained only in \rho, and not in A.
can also use [x, a] = −x0/ \sqrt 2 = −[x, a\dagger] to evaluate the dissipation term:
= −x0 \sqrt
(5.201) The resulting position equation is
m −\gamma
(5.202)
the Hamiltonian term and [p, a] = [p, a\dagger] = ¯h/i \sqrt 2 x0 to evaluate the dissipation term. The resulting equation is
(5.203) It may look funny to have damping terms on both Eqs. (5.202) and (5.203), but differentiating Eq. (5.202) and eliminating \partial t\langle p\rangle gives \partial 2
\omega 2
(centroid evolution, quantum damped harmonic oscillator) (5.204) for the wave-packet centroid. This has the same form as for a classical damped oscillator:
0 x = 0. (5.205) Note that we identify the frequency \omega0 in Eq. (5.204) as the renormalized oscillation frequency \omega\gamma of the damped oscillator, given by \omega 2
0 −\gamma2/4, and not the true resonance frequency \omega0 that appears in the classical formula (5.205). Adding a dipole interaction Hamiltonian for coupling to an external applied field,
(5.206)
−i¯heE(+)
eE(+)
. (5.207) Rewriting the equations as a second-order equation, we find \partial 2
\omega 2
m
(centroid evolution, with external drive) (5.208) which is the equation for the driven Lorentz atom in the dipole approximation, if we again associate the wave-packet centroid with the classical electron position and interpret the frequency \omega0 properly. Of course, for typical atomic dipole transitions in the optical, \gamma/\omega ∼10−8, so that the difference between the resonance and damped oscillation frequencies is negligible.
5.6 Consistency with Other Models Note also that the damped harmonic oscillator in the weak-driving limit occupies mostly the ground state, with small population in |1\rangle . The populations in higher-energy states are negligible, so we can identify the harmonic oscillator with the weakly driven, two-level atom by taking |1\rangle −\rightarrow |e\rangle , |0\rangle −\rightarrow |g\rangle , and
(5.209) which generates the usual optical Bloch equations for the two-level atom if we identify \gamma = \Gamma. The same interaction Hamiltonian above gives the dipole coupling of the two-level atom to a monochromatic field.
of a quantum-mechanical atom interacting with a classical monochromatic field. Note that there is a subtlety involved in introducing the atom–field interaction HAF. Really, in Eq. (5.208), we should have ended up with the same result, but with a factor of the oscillator strength f0 multiplying the right-hand side. This is the same replacement e/m −\rightarrow ef0/m that we discussed in the classical treatment, which turns out to be necessary to get the quantitatively correct answer. How does this work out here? We have to be more careful about the replacement \sigma −\rightarrow a in going from the two-level atom to the harmonic oscillator. Quantum mechanically, the dipole operator is
, (5.210) while the classical dipole moment is ex = ex0 \sqrt
= s e2¯h 2m\omega0
. (5.211) To put in the oscillator strength, we let e/m −\rightarrow ef0/m: ex = s e2¯hf0 2m\omega0
, (5.212) to make this expression quantitatively correct. Thus, to make the identification \sigma −\rightarrow a, we must also identify the coefficients
2m\omega0 , (5.213) which is the correct relation between the oscillator strength and dipole matrix element. (See also Prob- lem 5.10, where the same result comes out of comparing the classical and quantum expressions for the polarizability.) 5.6.1.3 Evolution of the Variances The solution to a driven, damped harmonic oscillator turns out to be a coherent state, so let’s see this explicitly. Let’s evaluate the equation of motion for \langle x2\rangle . Using [x2, p2] = 2i¯h[x, p]+ for the Hamiltonian
\sqrt 2 x0x = −[x2, a\dagger], we find \partial t
x2 = 1
hx0 2¯h a\daggerx −xa i = 1
x2 − ¯h 2m\omega0 . (5.214) Using the variance definition Vx :=
x2
(5.215)
Chapter 5. Two-Level Atom Interacting with a Classical Field the equation of motion becomes
x2
mCxp −\gamma Vx − ¯h 2m\omega0 , (5.216) where the symmetrized covariance is Cxp := 1
(5.217) Similarly, for the momentum variance Vp :=
p2
(5.218) we can use the commutators [p2, a] = [p2, a\dagger] = −2i¯hp/ \sqrt 2x0 to obtain
0 Cxp −\gamma Vp −m\omega0¯h . (5.219) For the covariance, the Hamiltonian part requires the commutators [x, p]+, x2 = −4i¯hx2 [x, p]+, p2 = 4i¯hp2
= i¯ha. (5.220)
[a\dagger2, a] = −2a\dagger and [a2, a\dagger] = 2a. The equation of motion for the anticommutator is
m
p2 −2m\omega 2
x2
(5.221) so that the covariance equation becomes
mVp −m\omega 2 0 Vx −\gammaCxp. (5.222) Collecting all the equations for the means and variances together,
mCxp −\gamma Vx − ¯h 2m\omega0
0 Cxp −\gamma Vp −m\omega0¯h
mVp −m\omega 2 0 Vx −\gammaCxp. (damped quantum harmonic oscillator, evolution of means and variances) (5.223) This is sufficient to completely characterize a Gaussian state for the damped harmonic oscillator. Of course to consider forcing, we still need to add a dipole interaction Hamiltonian. But as a check, the steady state here should be the ground state of the harmonic oscillator. However, when we compute the steady state of the above equations, we find
(5.224) (steady-state means)
5.6.2 Rate-Equation Limit¶
5.6 Consistency with Other Models while Vx = ¯h 2m\omega0 ,
, Cxp = 0. (5.225) (steady-state variances) This is a state of minimum uncertainty (where the generalized uncertainty relation is VxVp −C 2
and thus must be Gaussian.19 The variances are indeed the same as for |0\rangle from Eq. (5.193), so the oscillator damps to the ground state. In the driven case, note that the means and variances of Eqs. (5.223) are uncoupled. For a sinusoidal drive of the form (5.206), the means evolve according to the classical equation of motion (5.208), so that in steady state the wave packet centroid oscillates with an amplitude given by the steady state of Eq. (5.208). Hence, the steady state is just a coherent state |\alpha\rangle with amplitude
e|E(+) |/m
(5.226) again showing the classical nature of the solution. 5.6.2 Rate-Equation Limit We can also show that under certain conditions, the optical Bloch equations (5.125) reduce to the rate equations of Chapter 3. Recall the form of the optical Bloch equations:
(5.227) In the case of strong collisional damping \gamma⊥≫Ω, \Gamma, we can note that the coherences will be damped very quickly, whereas the populations will continue to evolve on much longer time scales. We can exploit this separation of time scales and make the adiabatic approximation, where we focus only on the slow pop- ulation dynamics by assuming the coherences are always approximately in equilibrium. Thus, approximate steady state of the coherence equations give
iΩ
(5.228) Adding these two equations together gives
\gamma⊥
(5.229) while subtracting them gives
(5.230) Combining these two relations to obtain the adiabatic difference of the coherences, we find \gamma⊥ 1 + ∆2 \gamma2 ⊥
(5.231) We can now put this into the first population equation to obtain the adiabatic evolution equation
Ω2
(5.232) 19Eugen Merzbacher, Quantum Mechanics, 3rd ed. (Wiley, 1998), pp. 219-20.
Chapter 5. Two-Level Atom Interacting with a Classical Field This result is now formally equivalent to the rate equation for nearly monochromatic light [Eq. (3.17),
¯h\omega (N2 −N1) , (rate-equation limit of optical Bloch equations) (5.233)
To compare the Einstein equation to the adiabatic result, we can clearly identify
\Gamma −\rightarrow A21. (5.234) Comparing the coefficients of the stimulated emission and absorption terms is less straightforward but very useful. Recall from Eq. (3.18) that the resonant cross section is
\lambda2 4 s(\omega), (5.235) while for a Lorentzian line shape s(\omega), we have from Eq. (3.19)
∆\omega
(5.236) Comparing the denominator of s(\omega) to the similar denominator of Eq. (5.232), we conclude that we must identify the transverse decay rate with the line width:
2 . (5.237) We may thus write the line-shape function as
\gamma⊥ \pi (\gamma2 ⊥+ ∆2). (5.238) Now identifying the coefficients of the spontaneous emission and absorption terms, \gamma⊥Ω2 2 (\gamma2
¯h\omega =
(5.239) Thus, we have
¯h\omega3 . (5.240) Using Ω= −\langle g|dz|e\rangle E0/¯h for z-polarized light, and I = E 2
0 , we find \Gamma = \omega3
(5.241) This analysis is only valid near resonance, so \omega \approx \omega0, and we take the atom to be spherically symmetric, so that |\langle g|dx|e\rangle |2 = |\langle g|dy|e\rangle |2 = |\langle g|dz|e\rangle |2, with the result \Gamma = \omega 3
(relation between decay rate and dipole matrix elements) (5.242) This is, in fact, the correct relation between the atomic decay rate and the atomic dipole matrix element that comes out of a full quantum electrodynamics calculation, as we will later see in Section 11.4.
5.6 Consistency with Other Models 5.6.2.1 Saturation Intensity We can make a connection to the earlier notion of the saturation intensity. From our rate equation analysis, specifically Eq. (3.27), the population inversion for an exactly resonant drive is given by N2 −N1 N = − 1 + 2\sigma0I ¯h\omega0A21 = − 1 + I Isat , (5.243) where we again have ignored any degeneracy. We have also defined the saturation intensity as
2\sigma0 , (5.244) (saturation intensity) as we did before in Eq. (3.31). We can do the same thing for the optical Bloch equations. Recalling that the steady-state population in the excited state from Eq. (5.132) is
Ω2 2\gamma⊥\Gamma 1 + ∆2 \gamma2 ⊥ + Ω2 \gamma⊥\Gamma , (5.245) we can write
Ω2/\Gamma2
(5.246)
inversion is given by
(5.247) Since Ω2 scales as the intensity, we can similarly define the saturation intensity for the two-level atom to match (5.243): I Isat ≡2Ω2 \Gamma2 . (saturation intensity related to Rabi frequency) (5.248) Using Ω= −\langle g|ˆ\epsilon\cdotd|e\rangle E0/¯h for arbitrary light polarization and I = E 2
0 , this relation gives Isat = cϵ0\Gamma2¯h2
(5.249) (saturation intensity) Thus, for example, we can write
1 I/Isat
, (5.250) (steady-state population) for the excited-state population in the case of homogenout broadening. Similarly, we may write s = I/Isat
(5.251) (saturation parameter) for the saturation parameter. The saturation effect here represents the nonlinear response of the two-level atom to the field, which is not predicted by the classical (linear) Lorentz model.
5.7 Spectrum of Resonance Fluorescence¶
Chapter 5. Two-Level Atom Interacting with a Classical Field But now, are the two saturation intensities from Eqs. (5.244) and (5.249) equivalent? Consider the
\Gamma = \omega 3
(5.252) to eliminate the dipole matrix element in Eq. (5.249), we find
0 \Gamma 4\pic2 . (5.253) In the case of the rate equations, the resonant cross section for linearly polarized light and random orientation of the atom from Eq. (3.21) is
2\pi . (5.254)
(5.253) for the saturation intensity. 5.6.2.2 Validity of the Rate-Equation Limit As we discussed above, the optical Bloch equations can be accurately represented by the rate equations when the collisional damping rate is large, because the coherences are quickly damped away. The rate equations are also valid in the case of incoherent (broadband) excitation, since there the dipole is driven by many frequencies, and thus the dephasing of the different frequency components mimics fast damping. 5.7 Spectrum of Resonance Fluorescence We will now consider the radiation (resonance fluorescence) due to a single, isolated atom driven by a monochromatic field. The optical Wiener–Khinchin theorem that we wrote down before in Eq. (2.21) was Z \infty
\eta D
E . (5.255) Inverting this relation, keeping in mind that I(\omega) will be centered near a large, positive frequency (of the driving laser), gives the intensity spectrum in terms of the field autocorrelation function:
\pi\eta Z \infty −\infty D
E
(5.256) Note that the spectrum is real due to the time-inversion symmetry of the correlation function. Note also that we are maintaining a particular ordering of the positive- and negative-frequency components of the electric field, a point we will return to below. Applying this to the scattered light from a driven, two-level atom, recall that we use \omega to refer to the frequency of the driving field and \omega0 to refer to the atomic resonance frequency. Thus, we will use \omegas to refer to the frequency of the scattered light. Also, recall from Eq. (1.43) that in the radiation zone, the electric field of an oscillating dipole is
¨d(+)(tr) r . (5.257) This classical expression is still appropriate for radiation from a quantum dipole, although obviously we will need to remember that d(t) here is a Heisenberg-picture operator. We will work in the near-resonant regime, and thus we assume that any scattered light will be close to \omega0 in frequency. Thus, we make the approximation
0 d(+). (5.258)
5.7 Spectrum of Resonance Fluorescence We also note that¶
(5.259) which allows us to use the angular distribution function
8\pi
. (5.260) from before. Putting all this together, we can write down the scattered spectrum as
\omega 4
Z \infty −\infty
E , (5.261) where we used \eta = 1/ϵ0c. The angle brackets here imply both a time average and an expectation value, since we are now dealing with operators instead of classical quantities (Heisenberg-picture operators, that is, since they now carry the explicit time dependence). We can now separate out the dipole matrix elements from the dipole product to obtain
Z \infty −\infty
. (5.262) Now we see the importance of the so called normal ordering, where E(+) appears to the right of E(−) in the expectation value in Eq. (5.256): it implies that the atomic lowering operator is to the right in the expectation value, which thus vanishes if the atom is in the ground state. With any other ordering, the expectation value would be nonzero for an atom in the ground state, and would thus need to be compensated by another explicit term. Finally, we can use the result (5.242) from our rate-equation analysis to write this expression in terms of the spontaneous decay rate:
Z \infty −\infty
. (5.263) The spectral content is entirely in the integral factor, and thus we may define the (unnormalized) spectral function
2\pi Z \infty −\infty
, (5.264) (radiation spectrum) in terms of which the intensity spectral density becomes
r2
(fluorescence spectral density) (5.265) Note that the expectation value here is in the standard frame, whereas the optical Bloch equations are in the rotating frame. To use the rotating-frame solutions, we transform as follows:
(5.266) so that we may use the spectral function
2\pi Z \infty −\infty
(radiation spectrum, rotating frame) (5.267) if we take the expectation value with respect to rotating-frame solutions. We will do this henceforth. Thus, to find the spectrum of the scattered light, we need to compute the two-time atomic correlation function [which is of course proportional to g(1)(\tau)] and then compute its Fourier transform. We will do so shortly, but first we will make a few comments about the total scattered radiation.
5.7.1 Scattering Cross Section, Line Shape, and Power Broadening¶
Chapter 5. Two-Level Atom Interacting with a Classical Field 5.7.1 Scattering Cross Section, Line Shape, and Power Broadening To compute the total scattered intensity, we can integrate the spectrum over all frequencies, keeping in mind that this is a one-sided spectrum: Z \infty
(5.268) Here, the steady-state population is appropriate in view of the time average. Thus, the scattered intensity is proportional to the excited-state population, as we would expect from the rate-equation model. We can, of course, define a normalized spectrum by
S(\omegas) \rhoee(t −\rightarrow \infty). (5.269) (normalized spectrum) The total scattered intensity is thus
r2
(5.270) and we obtain the total scattered power by integrating the intensity over a spherical shell of radius r:
(5.271) The photon scattering rate is given by dividing the scattered power by the photon energy ¯h\omega0 (again, assuming that scattering occurs near resonance),
(5.272) (photon scattering rate) and we see that this is simply the excited-state decay rate multiplied by the excited-state population. A common way to describe the total scattered power is the scattering cross section \sigmasc, which we define as the power radiated by the atom divided by the incident energy flux. That is, the scattered power is \sigmascI, where I is the intensity of the driving field. This is, of course, the same quantity that we defined in Section 1.2.1 in our treatment of the Lorentz atom. Recalling from Eq. (5.250) that we can write the excited-state population in steady state as
1 I/Isat
, (5.273) we see that we can write the scattering cross section as
\sigma0
, (5.274) (scattering cross section) where the on-resonance, small-signal cross section is given by
2Isat . (5.275) (on-resonance, small-signal cross section) Obviously, the cross section falls to zero as the driving laser is tuned away from resonance, or due to saturation as the incident intensity becomes large. We will now examine the absorption line shape, which is just the frequency dependence of the cross sec- tion. It is somewhat more convenient to examine the excited-state population, which we saw is proportional to the cross section. We can write the population in the form of Eq. (5.137) as
Ω2/\Gamma2 1 + 2∆ \Gamma 2 + 2Ω2 \Gamma2 . (5.276) (absorption line shape)
5.7.2 Coherent and Incoherent Scattering¶
5.7 Spectrum of Resonance Fluorescence For a weak driving field, this reduces to
Ω2/4
(5.277) (absorption line shape, weak drive)
is also referred to as the transition linewidth (representing the angular frequency width of the transition). In the strong-field limit, the excited-state population becomes
Ω2/4
(5.278) (absorption line shape, strong drive) This line shape is also Lorentzian, but now with a much larger width (FWHM) of \sqrt 2 Ω, and a maximum value of 1/2 as we expect for a saturated transition. -10 D/G 0.5 Pe W = 0.1 G W = 0.3 G W = 10 G W = 3 G W = G Thus, the line shape of the transition is effectively larger due to the strong coupling to the field. This phenomenon is called power broadening of the transition. 5.7.2 Coherent and Incoherent Scattering Before grinding through the full solution of the g(1)(\tau) coherence function, we can gain some insight by first considering the asymptotic limit. Given that \sigma(t) is a stochastically fluctuating operator, due to the random nature of spontaneous emission, it should become uncorrelated with itself at very different times. Thus, lim \tau−\rightarrow \infty
=
(5.279) Thus, the coherence function decays to a possibly nonzero constant. The Fourier transform of this dc component leads to a delta function in the radiated spectrum, which we can refer to as the ‘‘coherent’’ component of the spectrum (even though it turns out not to be coherent to all orders). The decaying part of the correlation function leads to a broadened or ‘‘incoherent’’ component of the spectrum, which we will evaluate below. Formally, we decompose the scattering rate as
sc + R(inc) sc
. (coherent and incoherent scattering rates) (5.280)
Chapter 5. Two-Level Atom Interacting with a Classical Field That is, the coherent part is due to the square of the mean of the dipole moment, which corresponds to what we found in the classical analysis (recall that the classical electron ‘‘position’’ was in fact the mean electron position). The incoherent part includes the mean square dipole moment, and thus accounts for the fluctuations of the dipole moment within the (possibly fictitious) ensemble. As for the actual spectral content of the coherent part, a constant value for the coherence function in Eq. (5.267) gives a perfectly defined spectral peak
\pi Z \infty −\infty
(5.281) (elastic spectral peak) That is, the coherent part of the spectrum is exactly at the driving laser frequency, and thus represents elastically scattered light. (In the regime of far red detuning, the elastic scattering is often referred to as Rayleigh scattering.) We recall that this was also the classical prediction for the scattered light in steady state, since the steady state of the Lorentz atom is sinusoidal oscillation at the driving frequency. The incoherent part will be spread over a range of frequencies and thus will represent inelastic scattering. What fraction of the scattered light is coherent? We can calculate this simply using the steady-state solutions of the optical Bloch equations in the forms (5.135): R(coh) sc Rsc
\rhoee(t −\rightarrow \infty) = \Gamma 4\gamma⊥ s (1 + s)2 s/2 1 + s = \Gamma 2\gamma⊥ (1 + s). (5.282) Here, s is again the saturation parameter, given by s =
⊥ . (5.283)
R(coh) sc Rsc = 1 + s, (5.284) (fraction of elastic scattering) where the saturation parameter becomes s = 2Ω2/\Gamma2
(5.285) In this case, for small saturation parameter (small driving intensity), the scattered light is completely elastic. As the driving laser intensity increases, the elastic component vanishes, and the light is all inelastically scattered. In the presence of collisional damping (\gamma⊥> \Gamma/2), there is some inelastic scattering even for a vanishingly small driving field. In the homogeneously broadened case, the steady-state solutions again give explicit expressions for the components. The total scattering rate is
s 1 + s, (5.286) (total scattering rate) while the coherent scattering rate is R(coh) sc
s (1 + s)2 . (5.287) (coherent scattering rate)
5.7.3 Quantum Regression Theorem¶
5.7 Spectrum of Resonance Fluorescence The incoherent scattering rate is the difference of these two expressions: R(inc) sc
sc = \Gamma s2 (1 + s)2 . (5.288) (incoherent scattering rate) Thus, since the saturation parameter is proportional to the driving intensity, for small drive intensities the coherent component ‘‘turns on’’ linearly with the intensity, whereas the incoherent component turns on quadratically with intensity. s 0.5 R /G sc R /G sc (coh) R /G sc R /G sc (inc) We thus see that the coherent component represents the linear response of the atom to the field, and thus the classical component. The incoherent part is the nonlinear part of the response of the two-level atom to the field, and is manifestly quantum. 5.7.3 Quantum Regression Theorem In order to evaluate the frequency dependence of the scattered radiation, we must calculate the correlation functions \langle \sigma\dagger(t)\sigma(t + \tau)\rangle . One method of calculating these correlation functions comes from the quantum regression theorem,20 which we now describe. The upshot is that according to the quantum regression theorem, two-time correlation functions obey the same equations of motion as one-time averages, which considerably simplifies their calculation. In the spirit of the Church of the Larger Hilbert Space, we will regard the evolution of the atomic system according to a master equation as unitary evolution of the system coupled to an external ‘‘reservoir.’’ Before we consider the correlation functions, we will briefly review the calculation of single-time aver- ages. Recall that if A is a system operator (i.e., it does not operate on the reservoir coupled to the system), then its time-averaged value is given in the Heisenberg picture by
(5.289) where Tr is a trace over both the system and reservoir variables, \rhoSR is the composite density operator, \rho = TrR[\rhoSR] is the reduced density operator for the system, and \rhoR = TrS[\rhoSR] is the reduced density operator for the reservoir. Also, TrS and TrR are partial traces over the system and reservoir, respectively. 20Melvin Lax, ‘‘Quantum Noise. IV. Quantum Theory of Noise Sources,’’ Physical Review 145, 110 (1966) (doi: 10.1103/Phys- Rev.145.110); S. Swain, ‘‘Master equation derivation of quantum regression theorem,’’ J. Phys. A 14, 2577 (1981) (doi:
Chapter 5. Two-Level Atom Interacting with a Classical Field
the unitary time-evolution operator from 0 to t, with the result
(5.290) where we have used the invariance of the trace operation under cyclic permutations and we have carried out the trace over the reservoir by setting as before
(5.291) Then, if \rho(t) satisfies the master equation
(5.292) with Liouvillian operator L, the evolution of \langle A(t)\rangle can be computed by solving Eq. (5.292) for the time evolution of \rho(t). To calculate the correlation function (or two-time average) \langle A(t)B(t + \tau)\rangle , where A(t) and B(t) are arbitrary Heisenberg operators, we proceed in a similar manner. Factoring out the explicit time dependence and using the composition and inversion properties of the evolution operator,
(5.293) we find
(5.294) Then if we define the two-time operator
(5.295) the two-time correlation function becomes
(5.296) This last expression looks much like an evolving expectation value, as in Eq. (5.290), with \Lambda(t+\tau, t) replacing the reduced density operator \rho(t). Similarly, comparing to Eq. (5.291), the definition (5.295) of \Lambda looks much like a density operator, which \rhoSR replaced by \rhoSR(t) A. Thus, we see that \Lambda(t+\tau, t) obeys the same equation of motion as \rho(t), but as a function of \tau,
(5.297) because the time evolution is governed by the same evolution operators in each case. This evolution is subject to the boundary condition
(5.298) Hence, in the long-time limit, we may summarize the quantum regression theorem as lim
(quantum regression theorem) (5.299) where
(quantum regression theorem: evolution) (5.300)
5.7 Spectrum of Resonance Fluorescence and¶
(quantum regression theorem: initial condition) (5.301) is the initial condition for the evolution. If we apply the quantum regression theorem to the emission correlation operator \langle \sigma\dagger(t)\sigma(t + \tau)\rangle , we
˜\rho(t \rightarrow \infty)\sigma\dagger. Writing out the matrix elements of the initial condition explicitly, these conditions become
(5.302)
in terms of the solution \Lambda(\tau) of the optical Bloch equations with these initial conditions, the correlation function that we need is given by
(5.303) Using these two relations, we can now use the solutions of the optical Bloch equations that we already obtained to evaluate the two-time correlation function for the resonance fluorescence spectrum. 5.7.3.1 Alternate Form The quantum regression theorem can be written in a useful alternate form as follows. Suppose that the one-time average of an operator A can be written in the form
X j
(5.304) where gj(t) are functions representing the solution in terms of initial conditions \langle Aj(0)\rangle of some set of operators, then the two-time average may be written
X j
(quantum regression theorem) (5.305) To show this, we can see that
B\Lambda(\tau) = X j gj(\tau) TrS Bj\Lambda(0) = X j gj(\tau) TrS Bj\rho(t −\rightarrow \infty)A = X j
(5.306) where we used Eq. (5.304) in the second step, recalling that \Lambda(\tau) is formally equivalent to a density operator,
This form of the quantum regression theorem can also be generalized a bit to read
X j
(quantum regression theorem) (5.307) This form is useful in computing the second-order coherence g(2)(\tau) with normally-ordered operators. We leave the proof of this form as an exercise (Problem 5.22).
5.7.4 Mollow Triplet¶
Chapter 5. Two-Level Atom Interacting with a Classical Field 5.7.4 Mollow Triplet The above recipe for computing the spectrum is good for a numerical computation. However, we can make our lives a bit easier by modifying the equations for the analytical calculation. The resulting spectrum was first computed by Mollow,21 and is now called the Mollow spectrum or Mollow triplet. Given that the initial conditions for \Lambda(\tau) are zero for the (ge) component but not for the (eg) com- ponent, it is not so convenient to use Torrey’s solutions from Section 5.5.2.2, since they are best when real values are expected for \langle \sigmax\rangle and \langle \sigmay\rangle . Instead, let us cast the optical Bloch equations in the form
, (5.308) which is similar to the Bloch-vector form of Eqs. (5.127), but keeps the complex coherences instead of the transverse Bloch-vector components. Since we need only compute the incoherent part of the spectrum, we need only treat the fluctutation parts of the atomic operators. Thus, we wish to compute
t\rightarrow \infty, (5.309) where
(5.310) When we subtract off the steady-state components of the Bloch equations (5.308), we obtain the matrix form \partial t
= −\Gamma 2 + i∆ iΩ −\Gamma 2 −i∆−iΩ iΩ −iΩ −\Gamma
=: P
, (5.311)
than the form (5.127) that we used for Torrey’s solutions, since there is no extra constant component. We thus have a purely linear system to solve. We then need to work out a modified form of the quantum regression theorem, since we want to compute the fluctuation part of the correlation function. From Eq. (5.302), the initial conditions are given by subtracting the steady-state values from the previous initial condition:
(5.312) 21B. R. Mollow, ‘‘Power Spectrum of Light Scattered by Two-Level Systems,’’ Physical Review 188, 1969 (1969) (doi: 10.1103/PhysRev.188.1969); B. R. Mollow, ‘‘Absorption and Emission Line-Shape Functions for Driven Atoms,’’ Physical Re- view A, 5, 1522 (1972) (doi: 10.1103/PhysRevA.5.1522); B. R. Mollow, ‘‘Stimulated Emission and Absorption near Resonance for Driven Systems,’’ Physical Review A, 5, 2217 (1972) (doi: 10.1103/PhysRevA.5.2217). The calculation presented here is sim- ilar to that of Howard Carmichael, An Open System Approach to Quantum Optics (Springer-Verlag, 1993), section 3.3. For the experimental observation of the Mollow triplet, see F. Schuda, C. R. Stroud, Jr., and M. Hercher, ‘‘Observation of the resonant Stark effect at optical frequencies,’’ Journal of Physics B: Atomic and Molecular Physics 7, L198 (1974) (doi: 10.1088/0022- 3700/7/7/002); F. Y. Wu, R. E. Grove, and S. Ezekiel, ‘‘Investigation of the Spectrum of Resonance Fluorescence Induced by a Monochromatic Field,’’ Physical Review Letters 35, 1426 (1975) (doi: 10.1103/PhysRevLett.35.1426); R. E. Grove, F. Y. Wu, and S. Ezekiel, ‘‘Measurement of the spectrum of resonance fluorescence from a two-level atom in an intense monochromatic field,’’ Physical Review A 15, 227 (1977) (doi: 10.1103/PhysRevA.15.227); W. Hartig, W. Rassmussen, R. Schieder, and H. Walther, ‘‘Study of the frequency distribution of the fluorescent light induced by monochromatic radiation,’’ Zeitschrift für Physik A 278, 205 (1976) (doi: 10.1007/BF01409169); J. L. Carlsten, A. Szöke, and M. G. Raymer, ‘‘Collisional redistribution and saturation of near-resonance scattered light,’’ Physical Review A 15, 1029 (1977) (doi: 10.1103/PhysRevA.15.1029); and J. Hunnekens and A. Gallagher, ‘‘Self-broadening of the sodium resonance lines and excitation transfer between the 3P3/2 and 3P1/2 levels,’’ Physical Review A 27, 1851 (1983) (doi: 10.1103/PhysRevA.27.1851).
5.7 Spectrum of Resonance Fluorescence In component form, this reads¶
(5.313) which we can write out even more explicitly as
t\rightarrow \infty =
−˜\rho2 ge
t\rightarrow \infty , (5.314) where all the atomic expectation values here refer to the steady-state values.
s 1 + s
1 + s
rs 1 + s
s (1 + s)2
s (1 + s)2 , (5.315) where s = 2Ω2 \Gamma2 . (5.316) Thus, we can write the initial conditions as
s (1 + s)2 s −i \sqrt 2s . (5.317) We also note that the on-resonance evolution matrix P = −\Gamma iΩ −\Gamma −iΩ iΩ −iΩ −\Gamma (5.318) has eigenvalues −\Gamma 2 , −3\Gamma 4 \pm iΩ\Gamma, (5.319) where as before Ω2 \Gamma := Ω2 − \Gamma 2 . (5.320) The corresponding eigenvectors are , −1 2Ω i\Gamma 4 ∓Ω\Gamma 2Ω i\Gamma 4 ∓Ω\Gamma . (5.321)
Chapter 5. Two-Level Atom Interacting with a Classical Field If we write these as the columns of the matrix S = −1 2Ω i\Gamma 4 −Ω\Gamma −1 2Ω i\Gamma 4 + Ω\Gamma 2Ω i\Gamma 4 −Ω\Gamma 2Ω i\Gamma 4 + Ω\Gamma , (5.322) then this matrix diagonalizes the evolution matrix:
−\Gamma −3\Gamma 4 + iΩ\Gamma −3\Gamma 4 −iΩ\Gamma . (5.323) Now that we have an evolution equation of the form
(5.324) we can use P = SDS−1 to obtain the solution
(5.325) The element of the operator \Lambda that we need is
t\rightarrow \infty, (5.326) which is equivalent to the desired correlation function according to the quantum regression theorem. In the vector form of the solution (5.325), this is simply the first (topmost) component in the ordering we have used here. After multiplying everything out (a symbolic algebra package helps a great deal here), we obtain the result
= s
- s 8(1 + s)2 s −1 −i \Gamma 4Ω\Gamma (5s −1)
- s 8(1 + s)2 s −1 + i \Gamma 4Ω\Gamma (5s −1)
(atomic dipole correlation function) (5.327) We can see that there are three components here: the first is a simple damped exponential and thus corre- sponds to a Lorentzian of width \Gamma (FWHM) centered on the resonant frequency; the other two are Lorentzian peaks shifted by \pmΩ\Gamma from the resonance frequency (if Ωis large enough that Ω\Gamma is real), each of width 3\Gamma/2 (FWHM). In the weak-field limit, all component are centered at the resonance frequency, since Ω\Gamma is
= 1
(dipole correlation, strong-field limit) (5.328) Noting that the coherent part of the spectrum is negligible in this regime, we see that the spectrum consists of three well-separated Lorentzian peaks. Accounting for the extra factor of 2 that we pick up when calculating the one-sided spectrum, we note that the total integrated spectrum is given by the correlation function evaluated at \tau = 0, so we conclude that 1/2 of the total power is in the central lobe, and 1/4 of the total power is in each of the side lobes.
5.7 Spectrum of Resonance Fluorescence Computing the Fourier transform in Eq. (5.264) to find the explicit spectrum, we find [including the elastic component from Eq. (5.281)]
s
s
\Gamma
- s
h
i + s
h
i (Mollow triplet, strong-field limit) (5.329) in the strong-field case where Ω\Gamma is real (Ω> \Gamma/4), and
s
s
\Gamma
- s
(Mollow spectrum, weak-field limit) (5.330) in the weak-field case where Ω\Gamma is imaginary (Ω< \Gamma/4). Several resonant spectra are plotted below. -20
W = 10G W = 3G W = 0.3G W = G D = 0 s 0.2 spectral density The elastic delta function of varying heights is schematically included. The spectrum is always symmetric about the laser (and hence atomic) frequency. 5.7.4.1 Off Resonance In the off-resonant case, the eigenvalues do not have a simple form, and it is difficult to treat the problem analytically. However, in the case of very large detuning or driving, such that \Gamma is negligible, the eigenvalues of P are 0 and \pmi˜Ω, so that the splitting reduces to the generalized Rabi frequency. In the case of large
at the laser frequency \omega, a side peak at the atomic resonance \omega0, and the other side peak at \omega+∆= 2\omega−\omega0. The line shapes for detuning ∆= 1 are quite different for small drive, but similar to the resonant line shapes for large drive, with the outer lobes being slightly larger and farther out.
Chapter 5. Two-Level Atom Interacting with a Classical Field
-20 0.16 spectral density W = 10G W = 3G W = 0.3G (¥20) W = G D = G s For larger detunings, the central peak is suppressed compared to the resonant case, especially for small driving intensities. Again, the spectrum is always centered about the laser frequency, not the atomic resonance frequency.
-20 0.012 spectral density W = 10G W = 3G W = G (¥150) (¥30) D = 10G s 5.7.4.2 Interpretations One nice physical picture of the Mollow triplet is an amplitude-modulated radiator. A source at frequency \omega modulated in amplitude at frequency \omegam can be modeled as
Amplitude modulation of a wave thus produces two sidebands with the original ‘‘carrier,’’ where the sideband splitting is just the modulation frequency. In the atom, the emission probability is proportional to \rhoee, which is modulated by the Rabi oscillations, and thus we expect a similar triplet spectrum based on the modulation of the emission probability. The Mollow spectrum is thus a direct signature of Rabi oscillations. In the resonant case, the Rabi oscillations happen at the damped Rabi frequency Ω\Gamma, and thus we expect the Mollow splitting to occur at this frequency. This argument extends to the far-off-resonant case, where
5.7 Spectrum of Resonance Fluorescence Rabi oscillations occur at the generalized Rabi frequency ˜Ω. However, this model is too simple to account for the spectral widths or relative heights of the peaks in the off-resonant case. The other nice intepretation of the Mollow triplet comes from the dressed-atom picture. We worked out the dressed states for the atom interacting with a classical field before in Section 5.3, where we found that the new eigenstates of the combined atom-field system are split by ˜Ω, as opposed to the uncoupled states, which are split by the detuning ∆. Although we have not explicitly quantized the field, we have discussed the idea of photons, and so it is easy to extend the dressed-atom picture to the case of the atom coupled to the quantized field. The only difference from the previous analysis is that the two atomic levels are repeated in energy corresponding to the presence of different numbers of field quanta. The repetition occurs every ¯h\omega in energy. w = wº |g, no+o1Ò, |e, nÒ fi |g, nÒ, |e, no-o1Ò
|+, (n)Ò |-, (n)Ò
|g, no+o2Ò, |e, no+o1Ò w = wº W As we see, for the resonant case, the rotating-frame states |g, n + 1\rangle and |e, n\rangle are degenerate, but they are coupled to each other by the field. They are thus split into a doublet, with the splitting again given by the Rabi frequency Ω. On the right-hand side of the diagram are shown four possible decay paths. Note
corresponds to the incoherent spectrum. The dressed-state splitting in this regime is likewise Ω\approx Ω\Gamma. We see that two possible paths give rise to the central peak, centered about the resonant frequency, while the two other decay paths are shifted in energy by \pmΩfrom the resonance. Also, on resonance, each dressed state is an equal superposition of the atomic excited and ground states, and thus we expect each decay path to occur at the same rate. Thus, the central band has twice the integrated area as each of the side bands, as we have already seen in the Ω≫\Gamma limit. Of course, in the off-resonant case, the uncoupled states are split by ∆, and the dressed states are split by ˜Ω, so the splittings of the Mollow triplet are just given by the generalized Rabi frequency ˜Ω. In addition to this simple physical picture, it is possible to compute quantitatively the general line widths and weights of the lines in the secular limit (of large driving or detuning).22 5.7.4.3 Energy Conservation These scalings have a nice interpretation in terms of scattering processes in perturbation theory.23 Consider the case of small saturation s (i.e., we take s to be the expansion parameter). Then the diagrammatic representation of the first-order atom-field interaction is as follows. w w |og‚ |oe‚ |og‚ 22Claude Cohen-Tannoudji, Jacques Dupont-Roc, and Gilbert Grynberg, Atom–Photon Interactions: Basic Processes and Applications (Wiley, 1992), Section VI.E, p. 437. 23Claude Cohen-Tannoudji, ‘‘Atoms in Strong Resonant Fields,’’ in Frontiers in laser spectroscopy: Proceedings of the Les Houches Summer School, Session XXVII, 1975, R. Balian, S. Haroche, and S. Liberman, Eds. (North-Holland, 1977), p. 1.
Chapter 5. Two-Level Atom Interacting with a Classical Field That is, we associate a factor of s with each absorption/emission cycle. Since there is only one emitted photon, and the atom ends in its initial state, the emitted photon must have the same frequency \omega as the incident photon. This scattering process represents the elastic peak, the coherent delta-function spectrum from Section 5.7.2. The second-order diagram, on the other hand, is as follows. w w1 w w2 |og‚ |oe‚ |og‚ |oe‚ |og‚ Energy needs only to be conserved for the entire process here, so energy conservation imposes the requirement
(5.332) on the two emitted photons. This is how photons may be inelastically scattered, say into the Mollow side bands, while maintaining energy conservation: any inelastically scattered photon must be balanced by another one (or possibly more, in higher order) that maintains overall energy balance. From before, we saw that for small s, the amplitude of the elastic peak scales as s, while the power of the inelastic component scales as s2. This matches our intuition here from perturbation theory. At higher orders, perturbation theory becomes rapidly more complicated, and not so useful to consider, but we can get some powerful intuition in the weak-excitation limit, as the next example shows. 5.7.4.4 Nonclassical Correlations The above analysis implies that for weak excitation, any photon emitted in one Mollow side band will be strongly correlated with another photon emitted in the other side band. This effect was observed in a beautiful experiment24 on the fluorescence of a strontium beam. The atoms were excited very far off
Again, the outer side bands are located approximately at \omega0 and 2\omega −\omega0, and photon coincidences in these two bands were monitored as a function of time delay. The data showed a large correlation over what would be expected for random coincidences, with the peak correlation occuring for a time delay of about one lifetime. Again, this is not explained by the linear Rayleigh process, |eÒ |gÒ w w but rather by the nonlinear, second-order multiphoton process in the following diagram (shown for the blue detuning of the experiment). |oeo‚ |ogo‚ w wº 2wo-owº w 24A. Aspect, G. Roger, S. Reynaud, J. Dalibard, and C. Cohen-Tannoudji, Physical Review Letters 45, 617 (1980) (doi: 10.1103/PhysRevLett.45.617); Jean Dalibard and Serge Reynaud, ‘‘Non-Classical Properties of Resonance Fluorescence Light,’’ in New trends in atomic physics: Proceedings of the Les Houches Summer School, Session XXXVIII, 1982, G. Grynberg and R. Stora, Eds. (Elsevier, 1984), Seminar 2, p. 181.
5.7.5 Antibunching of Resonance Fluorescence¶
5.7 Spectrum of Resonance Fluorescence Note the ordering here, as the photon of frequency \omega0 nearly always follows the emission of the photon of frequency 2\omega−\omega0. This is because the diagram for emission in the reverse order does not have an intermediate resonance with the excited state, as shown here. |oeo‚ |ogo‚ w wº 2wo-owº w Thus, the reverse process proceeds at a much smaller rate. (By contrast, for a resonantly driven atom, the photons in the two side bands can come in either order.) It is important to realize that this effect is manifestly quantum: it represents a correlation between the photons in two frequency modes, while no such correlation exists between photons in the same mode. The atom thus emits nonclassically correlated pairs of photons into the two frequency bands. 5.7.5 Antibunching of Resonance Fluorescence The elastic part of the spectrum is first-order coherent, because it is monochromatic. It turns out not to be coherent at second order, as we will now show, and in fact it turns out to be antibunched.25 To see this antibunching, we will need to examine the second-order coherence function. From our discussion of coherence in Section 2.6, Eq. (2.68), the normalized, second-order coherence function is
. (5.333) Again, we can replace the scattered-field operators with the atomic operators, with the result
. (5.334)
form
(5.335) where we can see where things will become nonclassical: at \tau = 0,
(5.336) since \sigma2\psi = 0. This is impossible for a classical field, so resonance fluorescence from a two-level atom is manifestly quantum. The correlation function vanishes at \tau = 0 because just after a photon is detected, the atom is known to be in the ground state, and cannot emit again until a time of order 2\pi/Ωelapses. In other words, two photons cannot be detected simultaneously in the resonance fluorescence of a single atom. 25For the first theoretical proposals, see H. J. Carmichael and D. F. Walls, ‘‘Proposal for the measurement of the resonant Stark effect by photon correlation techniques,’’ Journal of Physics B: Atomic and Molecular Physics 9, L43 (1976) (doi: 10.1088/0022-3700/9/4/00); H. J. Kimble and L. Mandel, ‘‘Theory of resonance fluorescence,’’ Physical Review A 13, 2123 (1976) (doi: 10.1103/PhysRevA.13.2123); H. J. Carmichael and D. F. Walls, ‘‘A quantum-mechanical master equation treatment of the dynamical Stark effect,’’ Journal of Physics B: Atomic and Molecular Physics 9, 1199 (1976) (doi: 10.1088/0022- 3700/9/8/007). For the experimental observation, see H. J. Kimble, M. Dagenais, and L. Mandel, ‘‘Photon Antibunching in Resonance Fluorescence,’’ Physical Review Letters 39, 691 (1977) (doi: 10.1103/PhysRevLett.39.691); M. Dagenais and L. Mandel, ‘‘Investigation of two-time correlations in photon emissions from a single atom,’’ Physical Review A 18, 2217 (1978) (doi: 10.1103/PhysRevA.18.2217).
Chapter 5. Two-Level Atom Interacting with a Classical Field To evaluate the correlation function, we will use a variation on the quantum regression theorem. Following along the lines of the above derivation,
= Tr
= Tr
= TrS
, (5.337) where
. (5.338) Then in the t −\rightarrow \inftylimit, \Lambda(\tau) satisfies the optical Bloch equations
(5.339) with initial condition
(5.340)
of the solution that we want is,
(5.341) the excited-state ‘‘population.’’ Since the optical Bloch equations in terms of ˜\rho\alpha\beta form a linear system, any scalar multiple of this solution is also a solution. In particular, rescaling the solution by \rhoee(t −\rightarrow \infty) matches the initial conditions for the quantum regression theorem, and thus
(5.342)
is intially in the ground state. Thus, for arbitrary excitation, the correlation function G(2)(\tau) starts off at zero, and for sufficiently large excitation shows damped oscillations towards steady state.
and all other components are initially zero from Eq. (5.176):
= Ω2/2
1 −e−(3\Gamma/4)t cos Ω\Gammat + 3\Gamma 4Ω\Gamma sin Ω\Gammat . (5.343) Thus, on resonance, the correlation function is
Ω2/2
2
4Ω\Gamma sin Ω\Gamma\tau . (5.344) We can find the normalized correlation function by dividing by \rho2 ee(t −\rightarrow \infty) (or by requiring that the correlation function settle to unity), with the result
4Ω\Gamma sin Ω\Gamma\tau . (second-order coherence for resonance fluorescence) (5.345) Again, for small excitation, the correlation function decays smoothly to unity. For larger excitations, the behavior is similar, but with Rabi oscillations along the way.
5.7.6 Probe Absorption¶
5.7 Spectrum of Resonance Fluorescence Gt W = 10oG W = G W = 0.1oG g (t) (2) 5.7.6 Probe Absorption Some interesting features come up when we consider the effect of a driven, two-level atom on a second, weak probe field. Namely, we will now derive the probe-absorption spectrum for the auxiliary probe field.26 Following Mollow’s argument, we start by computing the lowest-order perturbation to the atomic density operator due to the coupling to the probe field. Let Hp(t) denote the atom–probe coupling Hamiltonian. Then in time-dependent perturbation theory in the interaction picture, the perturbation to the state is
¯h Z t −\infty dt′[Hp(t′), \rho]. (5.346) Here, \rho is the Heisenberg-picture density operator with respect to the probe-free evolution, and thus has no time dependence. The effect of the Rabi oscillations must be incorporated into the time-dependence of Hp(t). The rate at which energy is absorbed from the probe is the rate at which the probe field does work on the system, which is given by
= −i ¯h Z t −\infty dt′ Tr {[\partial tHp(t), Hp(t′)] \rho} . (5.347) Note that there is no contribution from the unperturbed state \rho, since it does not include the effect of the probe. Assuming a monochromatic probe of frequency \omegap, we can write the atom–probe coupling Hamiltonian in the usual form
, (5.348) 26B. R. Mollow, op. cit.; see also N. Bloembergen and Y. R. Shen, ‘‘Quantum-Theoretical Comparison of Nonlinear Suscepti- bilities in Parametric Media, Lasers, and Raman Lasers,’’ Physical Review 133, A37 (1964) (doi: 10.1103/PhysRev.133.A37 ); Murray Sargent III, ‘‘Spectroscopic Techniques Based on Lamb’s Laser Theory,’’ Physics Reports (Section C of Physics Letters) 43, 223 (1978) (doi: 10.1016/0370-1573(78)90163-1); Donald J. Harter and Robert W. Boyd, ‘‘Nearly Degenerate Four-Wave Mixing Enhanced by the ac Stark Effect,’’ IEEE Journal of Quantum Electronics, QE-16, 1126 (1980); Robert W. Boyd, Michael G. Raymer, Paul Narum, and Donald J. Harter, ‘‘Four-wave parametric interactions in a strongly driven two-level system,’’ Physical Review A 24, 411 (1981) (doi: 10.1103/PhysRevA.24.411). For the experimental observation of the Mollow absorption spectra, see F. Y. Wu, S. Ezekiel, M. Ducloy, and B. R. Mollow, ‘‘Observation of Amplification in a Strongly Driven Two-Level Atomic System at Optical Frequencies,’’ Physical Review Letters 38, 1077 (1977) (doi: 10.1103/PhysRevLett.38.1077).
Chapter 5. Two-Level Atom Interacting with a Classical Field where the probe Rabi frequency is given by
¯h , (5.349) where E0p is the real amplitude of the probe field. Here, \sigma(t) is the Heisenberg-picture (with respect to the probeless atom) atomic lowering operator, which we assume to be slowly varying on the scale of \omegap if the pump and probe are nearly resonant. Then we can evaluate the derivative
. (5.350) Putting these Hamiltonian expressions into Eq. (5.347), we find
p Z t −\infty dt′ Tr nh
\rho o + c.c., (5.351) where we have dropped terms of the form \sigma(t)\sigma(t′)ei\omegap(t+t′), which will vanish under the subsequent time average. Notice that this spectrum is automatically one-sided since \omegap is the frequency of the real probe
so that we can drop the explicit t dependence and perform a time average,
p Z 0 −\infty d\tau
- c.c.
p Z 0 −\infty d\tau
p Z 0 −\infty d\tau
p Z \infty d\tau
p Z \infty −\infty d\tau
(5.352) We may thus write the absorption spectrum in terms of a new correlation function,
p Z \infty −\infty
(5.353) (probe absorption) where
. (probe absorption correlation function) (5.354) The t −\rightarrow \inftylimit is implied here. This correlation function allows us to define the (unnormalized) probe- absorption spectrum
Z \infty −\infty
(5.355) [Note that there is an arbitrary but different normalization coefficient here compared to the emission spectrum of Eq. (5.264).] Again, the Heisenberg operator \sigma(t) only has slow time dependence in the rotating frame of the laser field, and if we evaluate the correlation function in this frame, the spectrum becomes
Z \infty −\infty
(probe-absorption spectrum, rotating frame) (5.356) and is implicitly centered about the laser frequency.
5.7 Spectrum of Resonance Fluorescence When we integrate over all frequencies, we find the total absorbed power, Pabs = Z \infty d\omegap Pabs(\omegap) \approx ¯h\omega0Ω2 p Z \infty
p Z \infty −\infty
Z \infty
p Z \infty −\infty
p ga(0), (5.357) where we have assumed that the absorption spectrum is peaked near the atomic resonance. The undelayed correlation function is
(5.358) and is thus related only to the population inversion. This absorbed power implies an absorption coefficient for a vapor of number density N of a = PabsN I
p 2I
p 2I (Ng −Ne). (5.359) Noting that the saturation intensity is defined such that Ω2
¯h\omega3 0\Gamma/4\pic2 [from Eq. (5.253)], the integrated absorption coefficient becomes
\omega 2 (Ng −Ne). (5.360) This turns out to match the result we get directly from the rate equations: recall that the rate-equation absorption coefficient from Eq. (3.25) is
(5.361) where the laser cross section is
4 s(\omega). (5.362) Integration over all frequencies gives Z \infty
\omega 2 , (5.363) which gives precisely the same integrated absorption coefficient. We can thus be confident that we are more or less on the right track. The absorption correlation function can be written as
−
(5.364) where ge(\tau) is the correlation function that we already evaluated in deriving the emission spectrum. Hence, it remains to compute the correlation function
. (5.365) The usual business of the quantum regression theorem (see Problem 5.21) tells us to solve the optical Bloch equations, with initial condition
(5.366)
Chapter 5. Two-Level Atom Interacting with a Classical Field
then use the component
(5.367) of the solution, whose Fourier transform then gives the probe absorption spectrum. This problem can also be solved by explicitly including the coupling to the weak probe field in the optical Bloch equations, essentially by making the substitution
(5.368) in the laboratory (nonrotating) frame, and then finding the (oscillating) equilibrium solution. Using this procedure, Mollow27 gave an analytic form for the absorption spectrum (Problem 5.20):
h
i
iΩ2∆p
- c.c. (Mollow probe-absorption spectrum) (5.369) Here,
(5.370)
or without collisions). Some on-resonance absorption line shapes are plotted below in the absence of collisions. -20 -0.12 relative probe absorption D = 0
p W = 10G W = 3G W = 0.3G W = G The absorption line is strongest and Lorentzian for small pump intensities. As the pump intensity gets larger, the absorption line shape becomes more complicated, showing multiple peaks. A vertically zoomed version of the same plot is shown below. 27B. R. Mollow, ‘‘Stimulated Emission and Absorption near Resonance for Driven Systems,’’ Physical Review A, 5, 2217 (1972) (doi: 10.1103/PhysRevA.5.2217). See in particular Eqs. (3.8) and (3.11a).
5.7 Spectrum of Resonance Fluorescence -20 0.12 -0.12 relative probe absorption
p W = 10G W = 3G W = 0.3G, G D = 0 For saturating pump intensities, the absorption line shape crosses through zero and is negative for certain frequency ranges. These regions of negative absorption correspond, of course, to stimulated emission of the probe. For large pump intensities, the outer zero crossings occur at \pmΩfrom the pump laser frequency. Right on resonance, the probe is absorbed, but just off of resonance stimulated emission dominates. For larger detunings (and still with no collisions), the line shape is pretty much the same for small pump intensities: a single Lorentzian peak at the atomic resonance. But as the pump intensity gets larger, the line shape becomes dramatically different from the on-resonance case. -20 -0.2 relative probe absorption
p W = 10G W = 3G W = 0.3G W = G Side peaks develop near \pm˜Ωfrom the pump laser frequency, one positive and one negative. Also, as we can see from the zoomed version of the same plot, there is also a central dispersive structure in the line shape.
Chapter 5. Two-Level Atom Interacting with a Classical Field -20 1.2 -0.2 relative probe absorption
p W = 10G W = 3G W = 0.3G W = G While the center structure can be understood qualitatively,28 its interpretation is relatively subtle. The side peaks, however, are reasonably easy to understand in the dressed-state picture (which again applies if
|+, (n)Ò |-, (n)Ò
W ~ The absorption spectrum is essentially a probe of the dressed levels, which off resonance are split approx- imately by the generalized Rabi frequency ˜Ω, and hence the location of the side peaks. Recall also that for large negative detuning, the |−\rangle dressed state is essentially the same as |g\rangle , and hence should be more populated in steady state. Then the more energetic (blue) side peak probes a transition where the ground state is more populated than the excited state, hence giving rise to an absorptive peak. The less energetic (red) side peak probes a transition with a population inversion, and thus gives rise to a negative-absorption peak. The central structure is due to two transitions that probe levels with no population difference, and again has a more subtle interpretation. The absorption line shapes in the regime of strong collisional damping are also qualitatively quite
intensities, the line shape is a broadened Lorentzian, corresponding to the collisional line width. Interestingly, as the pump intensity is increased, a narrow dip appears in the line shape. 28Gilbert Grynberg and Claude Cohen-Tannoudji, ‘‘Central resonance of the Mollow absorption spectrum: physical origin of gain without population inversion,’’ Optics Communications 96, 150 (1993) (doi: 10.1016/0030-4018(93)90538-G).
5.7 Spectrum of Resonance Fluorescence -100 0.21 -0.002 relative probe absorption D = 0 g = 10G ^
p W = 0 W = G W = 3G W = 10G W = 30G As the pump intensity is increased to large values, the dip becomes a large ‘‘hole’’ in the line shape, and the line shape eventually takes on a form much like the homogeneously broadened case, with regions of stimulated emission. -100 0.005 -0.002 relative probe absorption
p W = 10G W = 30G W = 3G W = G W = 0 D = 0 g = 10G ^ Notice that the collisional damping overall suppresses the absorption, as the absorption lines are much weaker than in the homogeneously broadened cases. 5.7.6.1 Autler–Townes Doublet We saw above how the peaks in the emission and probe absorption spectra of the driven two-level atom can be explained in the strongly driven limit in terms of the splitting of the dressed states of the atom. The Autler–Townes doublet29 is an even more direct manifestation of the dressed-state splittings. Consider the usual two-level atom, driven by a resonant field of Rabi frequency Ω. Now consider a third, auxiliary level |e′\rangle , an energy ¯h\omega′ 0 above the usual excited state |e\rangle . We will assume this state decays to |e\rangle at a rate \Gamma′. We will also assume a weak probe field of frequency \omegap coupling |e\rangle −\rightarrow |e′\rangle (such that
0) with Rabi frequency Ωp. 29S. H. Autler and C. H. Townes, ‘‘Stark Effect in Rapidly Varying Fields,’’ Physical Review 100, 703 (1955) (doi: 10.1103/PhysRev.100.703).
Chapter 5. Two-Level Atom Interacting with a Classical Field G W W G¢ Wp |gÒ |eÒ |e¢Ò fi In the presence of a strong drive (large Ω), the excited state splits into a doublet of splitting Ωdue to mixing with the ground state. Thus, we expect the probe-absorption spectrum to have two peaks, corresponding to resonance of |e′\rangle with each of the dressed states. In the limit of large Ω, we expect the absorption spectrum to be a sum of two Lorentzian peaks. However, we need a formalism for dealing with this more quantitatively. Fortunately, it is easy to extend the formalism we have already developed to handle this problem. To compute the probe-absorption spectrum, we will treat the probe perturbatively. Thus, we will again need the correlation function
, (5.371) where now \sigma′ = |e\rangle \langle e′| is the lowering operator for the probe transition. The first term in this correlation function is
. (5.372) Working out the quantum regression theorem, we solve the master equation for this three-level atom, with initial condition
(5.373) which becomes in components
(5.374) Then the correlation function corresponds to the component
(5.375) The other part of the absorption correlation function is the emission correlation function
. (5.376) This correlation function satisfies the atomic master equation with initial condition
(5.377) which becomes in components
(5.378) This the correlation function corresponds to the same component as gd(\tau):
(5.379)
because there is no field coupling any state to |e′\rangle , these components remain zero for all time. Thus, we can
we expect since it is not pumped by any strong field. The master equation for the atom is the same as for the usual optical Bloch equations, with an extra dissipation term for the second decay channel,
¯h h
i
(5.380)
5.7 Spectrum of Resonance Fluorescence where the atomic Hamiltonian is¶
(5.381) in the rotating frame where the |e′\rangle state is degenerate with the |e\rangle state. The interaction is still ˜HAF = ¯hΩ
, (5.382) since we are neglecting the effect of the probe field on the atom. Written out as a set of coupled equations for the density-matrix elements, we find
2 ˜\rhoee′
2 ˜\rhoge′
2 ˜\rhoe′e
2 ˜\rhoe′g
(5.383) These equations are easy to solve numerically. The normalized spectrum is then again the Fourier transform of the correlation function
Z \infty −\infty
(5.384) which is also readily computed numerically. However, we can actually get a closed-form expression for the spectrum: the correlation function becomes
2\xi
1 + i \xi \Gamma 2 + i∆
1 −i \xi \Gamma 2 + i∆
, (5.385) where
r Ω2 + ∆2 −\Gamma2 4 −i∆\Gamma. (5.386)
p Ω2 −\Gamma2/4, and
2Ω2\Gamma
\rhoee(t −\rightarrow \infty)
1 + i\Gamma 2Ω2\Gamma
1 − i\Gamma 2Ω2\Gamma
. (5.387)
Chapter 5. Two-Level Atom Interacting with a Classical Field In this resonant case, for Ω2 \ge \Gamma2/4, the absorption spectrum consists of something like a pair of lines of width \Gamma/2 + \Gamma′, split by Ω2\Gamma (but with dispersive components),
Ω2\Gamma n
o
"
2 + 1 \Gamma 2 + \Gamma′ 2# + (Ω2\Gamma −\rightarrow −Ω2\Gamma) , (Autler–Townes absorption spectrum) (5.388) assuming Ωis large enough that Ω2\Gamma is real. As the pump intensity becomes large, the dispersive components become unimportant near each resonance, and at the same time Ω2\Gamma −\rightarrow Ω. The dispersive components thus cease to shift the spectral peaks in this limit—although they are still important in principle for the wings, since they fall off more slowly than the absorptive part—and the peaks are then separated by Ω, as we expected from the dressed-atom picture. Of course, we can get the absorption spectrum in the general case, but it’s complicated enough that it’s not very illuminating. Why the line width of \Gamma/2 + \Gamma′? In general, the weak-probe absorption width of two states of total decay rates \Gamma1 and \Gamma2 is simply \Gamma1 + \Gamma2, because the convolution of two Lorentzians of these widths is a Lorentzian of width \Gamma1 + \Gamma2. The dressed states on resonance are equal superpositions of |g\rangle and |e\rangle , which are states of respective decay rates 0 and \Gamma. Thus, each dressed state should only decay at rate \Gamma/2, which sets the width of each state. In general, from Eq. (5.385), we can see that the line widths are given by the total exponential decaying part, and so
(5.389) Again, in the limit of large Ω, we have ∆\omega = \Gamma/2 + \Gamma′. On the other hand, in the limit of small Ωand
absorption on |g\rangle −\rightarrow |e′\rangle , and \Gamma + \Gamma′, which is what we expect for probe absorption on |e\rangle −\rightarrow |e′\rangle . For
Interestingly, the on-resonance Autler–Townes spectrum has a minimum width for Ω= \Gamma/2, when Im[\xi] vanishes. The absorption spectra for \Gamma′ = \Gamma for several different values of the pumping rate Ωare shown below. -10 relative probe absorption
p ¢ D = 0 G¢ = G W = 0.3G W = G W = 3G W = 10G As we expected, we see two absorption lines, which become resolved as the pumping rate becomes large. In the case where the pump field is detuned from resonance, the doublet lines have asymmetric am- plitudes. The dressed states are an asymmetric superposition of the bare states |g\rangle and |e\rangle , which turns out
5.7 Spectrum of Resonance Fluorescence to be more important than the asymmetric steady-state populations of the dressed states. Also, the doublet center shifts, as we expect from the shift of the center of the bare states. It is a useful exercise to understand the placement and weights of the two lines here. -10 0.8 relative probe absorption
p ¢
G¢ = G W = 10G W = 10G W = 3G W = G W = 0.3G In a magnified view of the same spectrum, we can see the behavior of the cases of small pump intensity, where the displaced line is dominant, because there is little mixing of the pump-transition states. -10 0.04 relative probe absorption
p ¢
G¢ = G W = 10G W = 3G W = G W = 0.3G Minor secondary absorption peaks are also barely visible in the spectra in this magnified plot. 5.7.6.2 Lamb Dip One difficulty in precision spectroscopy of an atomic vapor is the Doppler broadening of the atomic transi- tions. For example, to use 87Rb as an absoute frequency reference, the natural linewidth is about 5 MHz, and the center of the transition can be determined far better than this width. However, the Doppler-broadened width at room temperature is about 500 MHz, making the precision of measurements much worse (and in fact blending several hyperfine levels into a single Doppler line). Of course, one can now use laser-cooled atoms, where the Doppler effect is unimportant. However, there is a much easier trick for sub-Doppler spec- troscopy: the Lamb dip or saturated-absorption spectroscopy. The basic idea is as follows. Consider a vapor of atoms where the Doppler width is larger than the natural line width. Now illuminate the atoms with two counterpropagating lasers.
Chapter 5. Two-Level Atom Interacting with a Classical Field pump W probe (weak) v The pump laser functions simply to saturate the atoms, while we monitor the absorption of the probe laser due to the atomic vapor. The two fields are generally produced from the same source, and thus their frequencies are swept together. There may be a frequency offset between them, but to simplify our discussion for the moment, assume they have the same frequency. Atoms at a particular velocity v experience the two fields with equal but opposite Doppler shifts, and thus a moving atom experiences them as having different frequencies. Moving atoms can thus only be resonant with at most one of the fields. Atoms at rest, however, see two fields at the same frequency, and can be resonant with both fields simultaneously. Because these atoms are effectively pumped with higher intensity than other atoms, the saturation reduces the resonant absorption coefficient. This reduction happens again only for atoms (nearly) at rest, and thus this effect is not Doppler broadened. The saturation produces a ‘‘dip’’ (the Lamb dip) in the Doppler absorption profile, centered at the atomic resonance, which has a width that can be on the order of the natural linewidth. Of course, if the pump and probe are not degenerate, the dip still occurs, but is displaced from the atomic resonance by an amount proportional to the pump-probe detuning. To analyze this problem quantitatively, we first note that in the interest of precision spectroscopy, the pump and especially the probe will have low intensity to avoid power broadening of the transition. We will thus treat the probe field perturbatively, and only explicitly include the effect of the pump field on the atom. We can then use Mollow’s formula (5.369) for the probe-absorption spectrum, writing \rhoee(t −\rightarrow \infty) explicitly using Eq. (5.132):
* 1 −
iΩ2∆p
- v . (saturated-absorption spectrum) (5.390) The angle brackets denote an average over the atomic velocity distribution, and recall that ∆= \omega −\omega0 is the pump detuning from resonance, and ∆p = \omegap −\omega is the probe detuning from the pump. Both of these frequencies now depend on the atomic velocity due to the Doppler shift, and the average ‘‘smears out’’ these frequencies. Recall that the Doppler shift of a field of wave vector k, as seen by an atom of velocity v, is \delta\omega = −k\cdotv. Why is this? A plane wave has the form E0 cos(k \cdot x −\omegat), (5.391) and the position
Z t dt′ v(t′) (5.392) of the atom is time-dependent. Thus, we can write the field as E0 cos k \cdot x0 + Z t dt′ v(t′) −\omegat = E0 cos k \cdot x0 −
Z t dt′ v(t′) . (5.393) In the moving frame, the effective frequency is the time derivative of the plane-wave phase:
(5.394) This establishes the Doppler shift, even for a time-varying velocity. We can thus implement the Doppler shift of the pump field by the replacement
(5.395)
5.7 Spectrum of Resonance Fluorescence In the case of the probe field, recall that the detuning ∆p is relative to the pump field, and thus we need to include the Doppler shift twice, once for the pump and once for the probe: ∆p −\rightarrow ∆p + kpump \cdot v −kprobe \cdot v \approx ∆p + 2k \cdot v. (5.396)
exactly counterpropagating pump and probe fields. Again, we will want the probe absorption as a function of the probe frequency, but the pump and probe are scanned together. Thus, ∆p (after the velocity replacments) is a fixed parameter, while the spectrum is as a function of ∆. Also, in general, since we want the spectrum location relative to the atomic resonance, we will make the plot of ∆p + ∆= \omegap −\omega0. Thus, in principle we have our result in Eq. (5.390), after making the replacements (5.395) and (5.396). Unfortunately, this result is rather complicated to interpret. Thus, we will expand the spectrum to lowest order in Ω2 as
(5.397) since it is reasonable to assume a weak pump. The zeroth-order spectrum, S(0)
2\gamma⊥ \gamma 2
v =
2\gamma⊥ \gamma 2
v , (5.398) is simply the Doppler-broadened absorption line (the Doppler velocity profile convolved with the atomic Lorentzian line). The next order is much more complicated, but includes the Lamb dip: S(2)
−
2\gamma2 ⊥ \Gamma[\gamma2
v +
v , −
v . (5.399) To interpret this equation, we can focus on the resonant factor in the denominators. There are several different factors to discuss here: 1. [\gamma2 ⊥+(∆−k\cdotv)2] or [\gamma⊥+i(∆−k\cdotv)]: simply says that the pump beam at detuning ∆is on resonance with atoms of velocity ∆/k along the direction of the pump beam. This factor is simply the overall Doppler profile. Of course, a Lamb dip can only occur inside the Doppler-broadened line. 2. [\gamma2
This is the same overall Doppler profile, but for probe resonance. Thus, the first term in S(2) a (\omegap) does not contribute to the Lamb dip, since it contains this and the former factor. This term simply represents lowest-order overall saturation of the Doppler transition, without any coherence-type effects.
atoms at rest. For nondegenerate pump-probe pairs, a moving velocity class is selected, effectively moving the Lamb dip. Notice that the last two terms seems to compete; but while the second term only has dispersive behavior, the last term has the only absorptive-type structure, which we identify as the dip.
Chapter 5. Two-Level Atom Interacting with a Classical Field
lowest-order ‘‘Doppleron’’ resonance (a higher-order treatment reveals more resonances of this form).30 Since the velocity class is swept with the probe frequency, we do not see structures due to this term in the spectra.
atomic resonance, and it has a width of order \Gamma (though in reality somewhat larger than \Gamma). As the pump becomes stronger, the dip becomes more pronounced, but for high pump intensities, the dip power broadens. -20
0.3 relative probe absorption Dpo=o0 Wo=o0 Wo=o0.3G Wo=oG Wo=o3G In the nondegenerate pump-probe case, we see essentially the same behavior, but now the Lamb dip is shifted by half the pump-probe detuning ∆p = 20\Gamma. When the probe is at +10\Gamma, the pump is at −10\Gamma, so the average frequency matches the atomic resonance. In this case, the dip is due to atoms moving at v = −10\Gamma/2k along the pump direction, which sees both beams as resonant. -20
0.3 relative probe absorption Dpo=o20G Wo=o0 Wo=o0.3G Wo=oG Wo=o3G 30Stig Stenhom, Foundations of Laser Spectroscopy (Wiley, 1984), section 4.2, p. 149.
5.7 Spectrum of Resonance Fluorescence¶
As we have mentioned, one of the main applications for saturation spectroscopy is the creation of precision absolute frequency references, such as for active frequency stabilization and absolute referencing of lasers. Saturation spectroscopy is the primary method used to stabilize lasers employed in laser cooling and trapping systems; it can also be utilized passively (automatically) for stabilizing a laser system itself.
A notable example involves methane-stabilization of a He–Ne laser, where an \(89 \mathrm{CH}_4\) cell is placed within the linear resonator cavity. This approach depends on coincidence between a particular He–Ne laser line and a \(\Lambda = 3\mbox{-}39~µ\mathrm{m}\) methane absorption transition. With its lifetime of approximately \(10^{-2} \mathrm{s}\) (around 10 ms), this generates an extremely narrow Lamb dip—roughly 400 kHz wide—which sits near the Doppler-broadened center line of stronger transitions in \(\mathrm{CH}_4\) and produces minimal extra loss for that laser. Consequently, the laser naturally oscillates within this narrowly defined frequency band without active locking (only using passive stabilization mechanisms inherent to this setup). In early experiments, two He–Ne lasers were simultaneously stabilized by such methods with sub-kHz-level precision of a few kHz between them; absolute reproducibility reached about \(10^{-11}\). Modern measurements report the methane-stabilized frequency as \(88~376~182~599~976.1(10) \text{ Hz }\)—very well-characterized to roughly 0.1 part per quadrillion relative uncertainty or so!
An additional phenomenon appearing in saturation spectroscopy is the crossover resonance. When multiple hyperfine/microscopic transitions lie within one Doppler-broadened absorption profile (i.e., frequency separation smaller than atomic thermal motion-induced linewidth), you naturally expect a Lamb dip corresponding to each individual transition. However, extra resonances also appear for certain pairs of these nearby levels: specifically when degenerate pump–probe fields couple separately via different Zeeman/hyperfine sublevels at first then become phase-matching conditions where atoms moving transversely with velocity component between them can simultaneously absorb both—effectively "averaging out" atomic kinetic effects. These crossover dips appear approximately halfway in frequency space (i.e., midway) between the individual Lamb dips of separate transitions and exhibit somewhat different spectral structures: they are typically broader but still visible due to interference contributions from multiple pathways contributing weakly near these crossing frequencies relative background signal strength alone, etc…
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5.7 Spectrum of Resonance Fluorescence sorption lines, two lines for each isotope of rubidium. Each pair of absorption lines corresponds to the two hyperfine ground states of each rubidium isotope, and each line represents a triplet of hyperfine transitions merged together within the Doppler width. Thus, we expect three Lamb dips and three crossover dips in each absorption line. For a sense of scale, the splitting between the two 87Rb multiplets is 6.8 GHz, the splitting between the two 85Rb multiplets is 3.0 GHz, the Doppler width is about 500 MHz at room temperature, and the natural line widths are about 5 MHz. Along with the saturation spectrum, the unsaturated spectrum (measured from the same probe beam, but with the pump beam blocked). Also shown here are zoomed versions of each of the absorption lines, with each of the saturation and crossover dips marked explicitly by the excited-state hyperfine number F ′. For the laser cooling transition of 87Rb (the top right plot), the dips are well-resolved, though not necessarily strong. For the repumping transition of 87Rb (the top left plot), the dips are more closely spaced, but still resolved. This is because the hyperfine splittings between the excited states is relatively large: 267 MHz between F ′ = 2 and 3, 157 MHz between F ′ = 1 and 2, and 72 MHz between F ′ = 1 and 2, compared to the 6 MHz line width. (Only 3 excited states can couple to each ground state, because F, being an angular momentum interacting with a spin-1 photon, can only change by 1, at least to leading
are sometimes possible,33 and reflect the complicated degenerate Zeeman-level substructure of the hyperfine levels. In 85Rb, the states are much less well-resolved, because the hyperfine splittings are smaller: 39 MHz between F ′ = 2 and 3, 21 MHz between F ′ = 1 and 2, and 10 MHz between F ′ = 1 and 2, compared to the 6 MHz line width. (It is precisely this reason that 85Rb is a difficult isotope to use for laser cooling and trapping.) In particular, for the ‘‘repumping’’ transition (lower left plot), four of the lines are merged together, and the other two lines are only marginally well resolved. Especially in the case of the laser cooling transitions (right two plots), the dips are centered to the right of the Doppler line center. This is not because of a frequency offset between the pump and probe lasers. This is because the state with the largest hyperfine number is the strongest transition, contributing the most to the Doppler absorption profile. The Doppler line center is thus pulled towards these states. probe wavelength probe transmission 87Rb, Fo=o1 á Fo ' Fo 'o=o2 Fo 'o=o1,2 Fo 'o=o0,2 Fo 'o=o1 Fo 'o=o0,1 Fo 'o=o0 probe wavelength probe transmission 87Rb, Fo=o2 á Fo ' Fo 'o=o3 Fo 'o=o2,3 Fo 'o=o1,3 Fo 'o=o2 Fo 'o=o1,2 Fo 'o=o1 probe wavelength probe transmission 85Rb, Fo=o2 á Fo ' Fo 'o=o3 Fo 'o=o1 Fo 'o=o2,3 Fo 'o=o1,3 Fo 'o=o2 Fo 'o=o1,2 probe wavelength probe transmission 85Rb, Fo=o3 á Fo ' Fo 'o=o4 Fo 'o=o3,4 Fo 'o=o2,4 Fo 'o=o3 Fo 'o=o2,3 Fo 'o=o2 33O. Schmidt, K.-M. Knaak, R. Wynands, and D. Meschede, ‘‘Cesium saturation spectroscopy revisited: How to reverse peaks and observe narrow resonances,’’ Applied Physics B: Lasers and Optics 59, 167 (1994) (doi: 10.1007/BF01081167).
5.8.1 Atom-Field Interaction¶
Chapter 5. Two-Level Atom Interacting with a Classical Field The following four plots are the same as the previous four zoomed plots, but here the Doppler back- ground is (mostly) subtracted away by using the reference beam in the experimental setup. The dips are somewhat easier to see here, although their placement within the Doppler lines is of course not apparent probe wavelength probe transmission 87Rb, Fo=o1 á Fo ' Fo 'o=o2 Fo 'o=o1,2 Fo 'o=o0,2 Fo 'o=o1 Fo 'o=o0,1 Fo 'o=o0 probe wavelength probe transmission 87Rb, Fo=o2 á Fo ' Fo 'o=o3 Fo 'o=o2,3 Fo 'o=o1,3 Fo 'o=o2 Fo 'o=o1,2 Fo 'o=o1 probe wavelength probe transmission 85Rb, Fo=o2 á Fo ' Fo 'o=o3 Fo 'o=o1 Fo 'o=o2,3 Fo 'o=o1,3 Fo 'o=o2 Fo 'o=o1,2 probe wavelength probe transmission 85Rb, Fo=o3 á Fo ' Fo 'o=o4 Fo 'o=o3,4 Fo 'o=o2,4 Fo 'o=o3 Fo 'o=o2,3 Fo 'o=o2 5.8 Mechanical Effects of Light on Two-Level Atoms We considered the dipole force in our classical treatment of the atom, but now it is time for a proper quantum derivation. In particular, the internal and external atomic dynamics are coupled and we will show that under suitable conditions, they decouple to good approximation. We will begin with a perturbative treatment, the basic conclusion being that under the proper con- ditions, it is possible to ignore the internal electronic structure of the atom, and treat the atom as a point particle. Furthermore, the ‘‘reduced’’ atom moves under the influence of the effective center-of-mass Hamil- tonian Heff = p2 2m + Veff(x), (5.400) where m is the atomic mass and the potential Veff is proportional to the laser intensity and inversely pro- portional to the detuning from the (nearest) atomic resonance. We will then examine things more generally in terms of dressed states and look at corrections to the simple dipole-force picture. 5.8.1 Atom-Field Interaction We must now redo the atom-field interaction, this time including the center-of-mass motion of the atom. Considering a linearly polarized field,
(5.401)
5.8.2 Schrödinger Equation¶
5.8 Mechanical Effects of Light on Two-Level Atoms where again E(+) and E(−) are the positive- and negative-rotating components of the field, respectively, and E(\pm)(x) is the space-dependent amplitude of the field. The atomic free-evolution Hamiltonian is then given by HA = p2
(5.402) which is the same as before except for the inclusion of the kinetic energy. The atom-field interaction Hamiltonian is still given (in the dipole approximation) by
(5.403) where d is the atomic dipole operator. In the rotating-wave approximation, this becomes HAF = ¯h
, (5.404) where
¯h (5.405) is the space-dependent Rabi frequency, which is no longer a real number in general (due to the eikx-type dependence of the field). In the rotating frame, the interaction becomes ˜HAF = ¯h
, (5.406) (atom–field interaction) and the free Hamiltonian becomes ˜HA = p2
(5.407) (free atomic evolution) so that the electronic states are nearly degenerate. 5.8.2 Schrödinger Equation
use the Schrödinger equation
(5.408) to describe the atomic evolution. It is convenient to decompose the state vector |\psi\rangle into a product of internal and external states,
(5.409) where the |\psi\alpha(t)\rangle are state vectors in the center-of-mass space of the atom. In the following, we will associate all time dependence of the atomic state with the center-of-mass components of the state vector. Defining
(5.410) Separating the coefficients of |e\rangle and |g\rangle , we obtain the coupled pair of equations
\psie. (5.411) for the wave functions \psi\alpha(x, t).
5.8.3 Adiabatic Approximation¶
Chapter 5. Two-Level Atom Interacting with a Classical Field 5.8.3 Adiabatic Approximation The equations of motion (5.411) are greatly simplified by using the adiabatic approximation, which we have seen a couple of times thus far. We can motivate this approximation by examining the various time scales in the evolution of \psie and \psig. The kinetic-energy terms in Eqs. (5.411) induce variations on time scales corresponding to kHz frequencies for ultracold atoms. However, the pump-field terms induce motion on a time scale corresponding to the Rabi frequency—typically from zero to several hundred MHz—and the free evolution term induces motion of \psie on a time scale corresponding to ∆, typically several to many
p
atomic motion lies the damping time scale due to coupling with the vacuum, which corresponds to the natural decay rate \Gamma, and \Gamma/2\pi is typically on the order of a few MHz for alkali atoms. Because we are primarily interested in the slow center-of-mass atomic motion, and the internal atomic dynamics take place over times much shorter than the damping time, it is a good approximation to assume that the internal motion is damped instantaneously to equilibrium, when compared to the external motion. Thus, \partial t\psie = 0, because \psie is the variable that carries the natural internal free-evolution time dependence at frequency ∆, whereas \psig has no natural internal oscillation, because the state |g\rangle is at zero energy. This approximation then gives a relation between \psie and \psig: ¯h∆−p2 2m \psie \approx ¯hΩ(x) \psig. (5.412) Noting that the kinetic energy p2/2m is negligible compared to ¯h∆, we can then use this constraint to eliminate \psie in the second of Eqs. (5.411), with the result
p2 2m
4∆ \psig. (5.413) Since the detuning is large, nearly all the population is contained in |g\rangle , so the excited state completely drops out of the problem. Hence, the atom obeys the Schrödinger equation with the effective center-of-mass Hamiltonian Heff = p2 2m + Veff(x), (5.414) where
4∆ , (5.415) (effective optical potential) and the atom behaves like a point particle in an effective potential, where the strength of the potential is given by (5.415). 5.8.3.1 Master-Equation Approach It is also instructive to make the adiabatic approximation from the viewpoint of a master equation, where we can more explicitly see the effects of damping on the atomic motion. The idea will follow that of the adiabatic approximation for obtaining the rate equations from the optical Bloch equations from before (Section 5.6.2). The master equation for the atomic evolution (i.e., the optical Bloch equations generalized to include center-of-mass motion) has the general form in the absence of collisions
Z
(5.416) where kL is the wave vector of the emitted photon (and dΩis the angular integration element, not to be confused with the Rabi frequency). This is the same master equation as for the optical Bloch equations, except for three modifications: (1) the atomic kinetic energy is now included in ˜HA, (2) the spatial de- pendence of the field is now included in ˜HAF, and (3) we have made the replacement \sigma −\rightarrow \sigmae−ikL\cdotr in
5.8 Mechanical Effects of Light on Two-Level Atoms the Lindblad superoperator, since any spontaneous emission must be accompanied by a photon recoil to conserve total momentum, and then we have integrated over all possible emission directions, weighted by the dipole radiation pattern fˆ\epsilon(\theta, \phi). We can write out the effect of the dissipation operator more explicitly, with the result
¯h[ ˜HA + ˜HAF, ˜\rho(t)] −\Gamma
Z
(5.417) where for simplicity we now restrict out attention to one dimension. Note that in writing down the master equation (5.417), we have assumed purely radiative damping. We can then write out the equations of motion
¯h p2 2m, \rhogg −i
Z dΩfˆ\epsilon(\theta, \phi)eikLx sin \theta cos \phi\rhoeee−ikLx sin \theta cos \phi
¯h p2 2m, \rhoee + i
¯h p2 2m, ˜\rhoge − \Gamma 2 + i∆ ˜\rhoge −i
¯h p2 2m, ˜\rhoeg − \Gamma 2 −i∆ ˜\rhoeg −i
(5.418)
comparable to or greater than \Gamma) and slow center-of-mass terms; this time, however, the equations of motion for the coherences (which are responsible for the population oscillations) have explicit damping terms. Since we are interested in the slow external motion, we can use the fact that the steady-state solution for ˜\rhoee is of order (\Gamma/∆)2, whereas the steady state solutions for the coherences ˜\rhoeg and ˜\rhoge are of order |\Gamma/∆|, so that we can neglect the ˜\rhoee terms on the right-hand sides of these equations. Now, we will assume that the quickly rotating coherences are damped to equilibrium on a time scale short compared to the external motion of interest, and hence set \partial t˜\rhoge \approx \partial t˜\rhoeg \approx 0. Doing so leads to the adiabatic relations
Ω∗(x) 2∆
2∆\rhogg, (5.419) where we have neglected the momentum and \Gamma terms in comparison to the ∆term. Substituting Eqs. (5.419) into the equation of motion for ˜\rhogg (and neglecting the ˜\rhoee term), we find
¯h p2 2m + ¯h|Ω(x)|2 4∆ , \rhogg . (5.420) This equation is simply the equation of motion for ˜\rhogg under the Hamiltonian
2m + ¯h|Ω(x)|2 4∆ , (5.421) which is just the effective Hamiltonian (5.414). Note that we have ultimately discarded the spontaneous emission effects, which lead to extra diffusive terms in the reduced evolution equations here, as we discuss below. From this approach, it is clear that the adiabatic approximation is good after a time on the order of 1/\Gamma, when the coherences have damped away. After this initial transient, the adiabatic approximation remains good as long as any modulations of the optical potential take place over a time long compared to 1/˜Ω. This is clear from the dressed-state analysis below, because such modulations will not excite transitions between the dressed states and thus cause the adiabatic approximation to break down.
Chapter 5. Two-Level Atom Interacting with a Classical Field 5.8.3.2 Bragg Scattering in an Optical Standing Wave As an example to gain some physical insight into the nature of the dipole force, we will consider the problem of Bragg scattering of a two-level atom in a weak, optical standing wave. The potential is, of course, sinusoidal in space with period \lambda/2, and so this setup is equivalent to the quantum pendulum, but in a regime where the atoms have enough energy that they are not bound to the optical lattice of potential wells (rotational pendulum motion). The quantum-pendulum dynamics show a feature that is distinctly nonclassical: the momentum transferred from the potential to the atoms is quantized. To see this directly, we consider The following argument. For a standing wave composed of two equal but counterpropagating traveling waves, with a field of the form
(5.422)
constant)
(5.423) where the potential amplitude is V0 = ¯h|Ω0|2 8∆ . (5.424) The Schrödinger equation is
p2 2m + V0 cos(2kx)
= p2 2m + V0 ei2kx + e−i2kx
(5.425) which can be written in the momentum representation as
(5.426)
erator, or by carrying out an explicit Fourier transform of the equation from the position to the momentum representation, and the explicit proof is left as an exercise. So, the evolution in the standing wave imposes a ‘‘ladder’’ structure in momentum, such that an atom beginning in a plane-wave state |p\rangle can only subse- quently occupy the states |p + n2¯hk\rangle for integer n. This momentum quantization has a clear interpretation in terms of the stimulated scattering of lattice photons: if the atom absorbs a photon that was traveling in one direction and then re-emits it into the counterpropagating mode, the atom will recoil, changing its momentum by twice the photon momentum, or by 2¯hk. Of course, the argument that we just considered was based on a classical treatment of the field, so it is the spatial periodicity of the potential that imposes the ladder structure in this model. However, the momentum transfer to the atoms can be viewed as a stimulated Raman transition—a two-photon transition from one ground state to an excited state and back to another ground state—between different motional states, say |g, p\rangle and |g, p + 2¯hk\rangle .
5.8 Mechanical Effects of Light on Two-Level Atoms w w |g, po+o2h—kÒ |g, poÒ We can use Eq. (5.426) to write down coupled equations for the two states. Assuming that couplings to other states are negligible, we have
2m
2 \psi(p)., (5.427) These are the equations of motion for a two-level system, coupled with Rabi frequency ΩR = V0 ¯h = |Ω0|2 8∆, (5.428) with a dc interaction between states of energy difference
2m −p2 2m = 2¯hkp m
(5.429) where the recoil energy
2m (5.430) (recoil energy) is the atomic kinetic energy associated with a single photon recoil. Thus, atomic population oscillates between the two momentum states at the Raman Rabi frequency ΩR: since we have adiabatically eliminated the intermediate (excited) state, the three-level system behaves approximately as an effective two-level system. More generally speaking, the coupling between these two levels is described by a Raman Rabi frequency (as in the two-level atom), given by ΩR = Ω1Ω2 2∆, (5.431) (Raman Rabi frequency) where Ω1,2 are the Rabi frequencies associated separately with each traveling-wave component of the stand- ing wave, and ∆is the mutual detuning to the atomic excited state (the relative frequency difference is constrained by energy conservation to be the splitting between the motional states). To connect with the notation that we have already used, Ω1 = Ω2 = Ω0/2 for the case of identical traveling waves, so that ¯hΩR = V0, and thus again V0 also represents the strength of the Raman couplings. The two-photon, stimulated Raman transition is an example of a Bragg scattering process.34 In fact, it is the simplest (‘‘first-order’’) form of Bragg scattering; in general, nth-order Bragg scattering is a 34For experiments and more details about Bragg scattering in atom optics, see Peter J. Martin, Bruce G. Oldaker, Andrew H. Miklich, and David E. Pritchard, ‘‘Bragg scattering of atoms from a standing light wave,’’ Physical Review Letters 60, 515 (1988) (doi: 10.1103/PhysRevLett.60.515); David M. Giltner, Roger W. McGowan, and Siu Au Lee, ‘‘Theoretical and experimental study of the Bragg scattering of atoms from a standing light wave,’’ Physical Review A 52, 3966 (1995) (doi: 10.1103/PhysRevA.52.3966), and M. Kozuma, L. Deng, E. W. Hagley, J. Wen, R. Lutwak, K. Helmerson, S. L. Rolston, and W. D. Phillips, ‘‘Coherent Splitting of Bose-Einstein Condensed Atoms with Optically Induced Bragg Diffraction,’’ Physical Review Letters 82, 871 (1999) (doi: 10.1103/PhysRevLett.82.871).
Chapter 5. Two-Level Atom Interacting with a Classical Field 2n-photon transition spanning an interval of 2n¯hk in momentum between the |\pmn¯hk\rangle states.. The term ‘‘Bragg scattering’’ applies to the weakly coupled regime, where the intermediate states are not appreciably populated, and so the transition between the two distant momentum states can be treated as a two-level problem. In this regime, classical transport between these distinct momentum regions is forbidden, as the classical potential is not sufficiently strong to cause a correspondingly large change in the classical momentum. As such, Bragg scattering is an example of dynamical tunneling, which is quantum tunneling between regions in phase space between which classical transport is forbidden, but by the dynamics (here, the nature of asymptotically free-particle motion) rather than by a potential barrier. Although the potential has a small amplitude, quantum coherence can build up as the atoms sample the potential and cause the atoms to significantly change their motion. We will illustrate this process by considering the relatively simple case of second-order Bragg scattering, and then we will generalize our results to the nth-order case. We consider the case where the standing wave is stationary, so that only the states |−2¯hk\rangle and |2¯hk\rangle are resonantly coupled in the limit of small ΩR. No other states will be substantially coupled by these fields, unless the Raman Rabi frequency is large enough to power-broaden the off-resonant transitions, which would not correspond to the Bragg regime. The relevant energy-level diagram is shown below, which shows that the detuning from the |p = 0\rangle motional state is simply the kinetic-energy shift. w 4wr p = 0 p = 2h—k
equation for the three coupled momentum states then becomes
2m
2 \psi(0, t)
2m
2 \psi(0, t). (5.432) Adding an energy offset of −4¯h\omegar, the equations become
2 \psi(0, t)
(5.433)
R/\omega2 r ) and hence negligible. Additionally, we can make an adiabatic approximation for the evolution of the |p = 0\rangle state, by formally
matrix picture and replacing the rapidly-varying coherences with their locally average value (although this procedure is a result of coarse-graining here, rather than radiative damping as in the previous treatment). Doing so leads to the adiabatic relation
, (5.434)
5.8.4 Nonperturbative Analysis¶
5.8 Mechanical Effects of Light on Two-Level Atoms which can be used to eliminate the intermediate state, resulting in a two-level evolution:
R 16\omegar
. (5.435) Hence, we see explicitly the Raman-Rabi oscillations between the two motional states (which is not the
Ω2 R/8\omegar. The first term represents a Stark shift of ΩB,2/2, due to scattering processes where the absorbed and emitted photons have the same k, while the second term represents the Rabi-type coupling, where the absorbed and emitted photons have opposite k. Comparing this expression to the form (5.431) for the two- photon Rabi frequency, we see that this second-order Bragg process can be viewed also as a Raman process of two Raman transitions, where the detuning to the intermediate state ∆is identified as 4\omegar. Continuing in this manner, the Bragg rate for nth-order scattering from n¯hk to −n¯hk is given by ΩB,n = Ωn R 2n−1 n−1 Y k=1 \deltak , (5.436) where \deltak is the detuning of the kth intermediate motional state. Notice that the intermediate detunings are given by [n2 −(n−2)2]\omegar, [n2 −(n−4)2]\omegar, …, [n2 −(2−n)2]\omegar, so that this Bragg frequency can be written as ΩB,n = Ωn R (8\omegar)n−1[(n −1)!]2 (5.437) (Bragg transition rate) The transition frequency obviously becomes small for high-order Bragg processes, as the Rabi frequency decreases exponentially with the order. Nevertheless, Bragg oscillations of up to sixth35 and eighth36 order have been observed experimentally for an atomic beam crossing an optical standing wave. 5.8.4 Nonperturbative Analysis The above analysis of the dipole force was a perturbative treatment in Ω/∆for the ground-state energy shift. We will now perform a better analysis that gets the (adiabatic) potential correct even for strong excitation, as well as the radiation-pressure force.37 We start with the Heisenberg-picture force,
(5.438) where the last equality follows because p = −i¯h\nabla . Here, the atomic position r is in principle an operator, but we will take on the semiclassical view that it refers to the mean atomic position to simplify this treatment, in contrast the preceding perturbative treatment. Again, the rotating-frame interaction Hamiltonian is given by ˜HAF = ¯h
. (5.439) Here, we have written the Rabi frequency again as
(r) ¯h
(5.440) where E(+) (r) is the positive-rotating part of the field. The spatial dependence includes both any phase rotation as well as slow envelope variations. 35David M. Giltner et al., op. cit. 36Armand Eugéne Albert Koolen, Dissipative Atom Optics with Cold Metastable Helium Atoms, Ph.D. thesis, Technische Universiteit Eindhoven (2000). 37J. P. Gordon and A. Ashkin, ‘‘Motion of atoms in a radiation trap,’’ Physical Review A 21, 1606 (1980) (doi: 10.1103/Phys- RevA.21.1606).
Chapter 5. Two-Level Atom Interacting with a Classical Field The force then depends on the gradient of the Rabi frequency according to F = −¯h
. (5.441) The gradient is given by
\nabla |Ω|
, (5.442) so we can write
Ω(r)
(5.443) for the gradient. The mean force then becomes
(5.444) There are two terms here. Both go as the interaction energy
D
(5.445) for the dipole in the external field, but only the first depends on gradients of the field amplitude, and this term corresponds to the dipole force. The second term is due to absorption, since it is 90◦out of phase with respect to to the dipole force. This is the radiation-pressure force. In the case where the atom is at rest or moves slowly on time scales of order \Gamma−1, we can use the steady-
for the fact that Ωis no longer necessarily real, the steady-state coherences are
\Gamma 1 + i2∆ \Gamma 1 + 2∆ \Gamma 2 + 2|Ω|2 \Gamma2 = − iΩ
iΩ∗
(5.446) where the saturation parameter is
|Ω(r)|2
(5.447) With these relations, the adiabatic mean force is
i¯h|Ω(r)|2
+ c.c. = ¯hs(r) 1 + s(r)
(5.448) We can write the second term, the mean radiation pressure, as
(5.449) (radiation-pressure force)
force becomes
(5.450)
5.8 Mechanical Effects of Light on Two-Level Atoms The radiation pressure thus has the physically reasonable interpretation of being the photon scattering rate multiplied by the photon recoil momentum. The force is in the direction of the wave vector k. On the other hand, for a standing wave composed of two equal but counterpropagating traveling waves, with a field of the form
, (5.451) the phase \phi(r) is a constant, and the (mean) radiation-pressure force vanishes. The first term, the mean dipole force, is
1 + s(r)Re \nabla log |Ω(r)| . (5.452) This force depends on ∆and the field intensity (via s), and thus on the phase between the applied and dipole radiated fields, giving a dispersive frequency dependence. The force represents a change in the field momentum due to the interference of the radiated and (outgoing) applied fields. The quantum-mechanical interpretation is that the force occurs via coherent scattering processes of absorption and stimulated emission, where the absorbed and emitted photons have different k vector orientations. The atom therefore recoils to conserve the total atom-field momentum. In a plane wave, there is only one k, and so there is no possibility for changing k on scattering. There is thus no dipole force in a plane wave: intensity gradients, which are connected to uncertainty in the direction of k, are necessary to produce a dipole force. Since the gradient of the saturation parameter is
2|Ω|\nabla |Ω|
|Ω|
(5.453) we can write the dipole force in the form
\nabla s
(5.454) where Vdip = ¯h∆
2 log 1 + |Ω(r)|2
= ¯h∆ 2 log 1 + I(r)/Isat
. (dipole potential) (5.455) This is the main result of this section. This gives the dipole potential for any field intensity, for a stationary atom, or for an atom moving very slowly (as we have used the steady-state solutions of the optical Bloch equations). Notice that while the radiation pressure force saturates for large s, the dipole force can continue to increase without bound, though for large intensities it only does so logarithmically. Also, the dipole potential is a negative shift for red detuning and a positive shift for blue detuning. Exactly on resonance, the dipole force vanishes. Additionally, note that there is no counter-rotating term of the form (\omega + \omega0)−1 as there was in the classical result of Eq. (1.76). The solution here is based on the optical Bloch equations, which assumed the rotating-wave approximation. Far off resonance, we can expand the logarithm to first order in s(r): Vdip \approx ¯h∆
4∆ . (5.456) (dipole potential, far off resonance) Thus, we recover our effective dipole potential from our previous perturbative treatment. In terms of the saturation intensity, using I/Isat = 2Ω2/\Gamma2 we find the perturbative result in standard form Vdip \approx ¯h\Gamma2 8∆ I(r) Isat , (5.457) (dipole potential, far off resonance) in agreement with the classical expression (1.77) in the same regime.
5.8.6 Fluctuations of the Optical Force¶
Chapter 5. Two-Level Atom Interacting with a Classical Field 5.8.5 Dressed-State Interpretation Using the dressed-atom picture, we can obtain more generally valid results for the dipole potential.38 The dressed-state energies from Eq. (5.64) are E\pm = −¯h∆ 2 \pm ¯h˜Ω 2 , (5.458) where the generalized Rabi frequency is ˜Ω= p |Ω|2 + ∆2 (5.459) in the case of a complex Rabi frequency. The energies of the dressed states relative to the mean level energy −¯h∆/2 are thus \pm¯h˜Ω/2. For a Gaussian laser beam, the dressed states shift in opposite directions. For
ground-state atom sees a potential well, while an excited-state atom sees a potential barrier. |eÒ |gÒ |eÒ |+Ò |-Ò |gÒ h—|D| h—|D| position h—W~ The ground-state shift is then Vgnd = −¯h˜Ω 2 = −¯h 2 |∆| r 1 + |Ω|2 ∆2 \approx −¯h 2 |∆| 1 + |Ω|2 2∆2 \approx ¯h∆ 2 + ¯h|Ω|2 4∆. (5.460) The first term is the bare-state energy, while the second term is the lowest-order dipole potential. Of course,
in the ground state. For larger excitation, the atom is in a mixture of the two dressed states in steady state, and since the shifts are opposite, the total potential shift is less than this perturbative result indicates. This motivates the logarithmic saturation of the dipole potential with intensity (but see the homework for a more precise interpretation of the dipole potential in terms of dressed states). The sign of the dipole force, which again depends only on the sign of the detuning, is thus explained by which dressed state has the most population. Far to the red of resonance, |g\rangle \approx |−\rangle , and so the shift is negative, while far to the blue, |g\rangle \approx |+\rangle , and so the dipole potential shift is positive. 5.8.6 Fluctuations of the Optical Force The optical force on the atoms in the standing wave can also lead to momentum diffusion. Part of this diffusion is due to spontaneous emission. The dipole moment of the atom fluctuates due to spontaneous emission, and this fluctuating dipole interacts with the field gradients in the standing wave to produce momentum diffusion. Alternately, you can think of it this way: the atom occasionally jumps to the ground state whenever a photon is emitted (seen by a fictitious photodetector), and then relaxes towards equilibrium. This means that the atom is changing its weight stochastically between the dressed states, which have shifts of opposite signs. Thus, the dipole force the atoms experience is also stochastic. 38J. Dalibard and C. Cohen-Tannoudji, ‘‘Dressed-atom approach to atomic motion in laser light: the dipole force revisited,’’ Journal of the Optical Society of America B 2, 1707 (1985).
5.8 Mechanical Effects of Light on Two-Level Atoms 5.8.6.1 Fokker–Planck Equation To handle the effect of fluctuations on the atoms, we will take the semiclassical view of atoms as localized particles on the scale of the potential, and treat the momentum probability density f(p, t) for an ensemble of atoms. In treating the mean force, what we have derived is the drift coefficient A for the advection equation
(5.461) The solution to this equation is simply f(p −At), and thus this equation simply represents translation of the momentum distribution by At. Thus, the drift coefficient A is the mean force on the atoms. We can see the effect of the right-hand-side on the distribution by visualizing its derivatives on a simple distribution. On the left-hand side, where the derivative is positive, the advection term causes the function to decrease, assuming A > 0. On the right-hand side, the opposite is true. The net effect is motion of the distribution to the right. To treat diffusion, we use a second-order term to obtain the diffusion equation
2 \partial 2 p f(p, t) (5.462) with diffusion coefficient D. The diffusion term causes the distribution to spread, which we can again see by visualizing the derivatives for a smooth, single-peaked distribution. For a positive diffusion coefficient, the second derivative is negative at the peak but positive in the wings, resulting in a net spreading. The diffusion equation has the Gaussian solution, assuming an initial condition of \delta(x −x0), of
\sqrt 2\piDt exp −1 (x −x0)2 Dt , (5.463) which has variance Dt and width \sqrt Dt. Note that under this evolution, even non-Gaussian initial conditions become asymptotically Gaussian, as we expect for a random-walk process. This is because the Gaussian solution is the solution for the delta-function initial condition, and thus the solution at time t for a general initial condition is the convolution of the initial condition with the above Gaussian. At late times, when the Gaussian solution is much broader than the initial condition, the contribution of the initial condition to the solution is negligible. Combining these two effects, we arrive at a simple advection-diffusion equation
2 \partial 2 p f(p, t). (5.464) In the more general, one-dimensional case, the advection and diffusion coefficients can depend on the mo- mentum itself
2\partial 2 p D(p)f(p, t). (5.465) (Fokker–Planck equation) This equation is the one-dimensional Fokker–Planck equation. Again, note that we have computed the advection coefficient for the optical force, although thus far we have ignored its momentum dependence. In
Chapter 5. Two-Level Atom Interacting with a Classical Field the following sections, we will be concerned with computing the diffusion coefficient, also ignoring its velocity dependence. The simplest velocity dependent case that we would like to consider for laser cooling is the
p
exp −1 (p −p0eAt)2 (D/2A)(e2At −1) . (5.466) For A > 0, the system is unstable and runs away, while for A < 0, the solution settles down to the steady state Gaussian centered at p = 0 and width p D/2|A|:
p
exp −1 p2 D/2|A| . (steady-state, linear solution) (5.467) This, of course, is the problem of laser cooling with intrinsic noise, as we discussed when deriving the Doppler limit of Section 1.4.3.1. In the most general case, the Fokker–Planck equation in three dimensions is (note the implied sum- mations)
\partial 2
(5.468) However, we will only be concerned with the total diffusion rate D\alpha\alpha, rather than with the anisotropic components of the diffusion tensor D\alpha\beta. 5.8.6.2 Diffusion Coefficient For the calculation of the diffusion coefficient due to optical forces, we will again assume an atom at rest (or slowly moving), and we will also assume that the atom is spatially localized. Based on our above discussion, we may take the diffusion coefficient (henceforth denoted by Dp) to be defined by the rate at which the momentum variance increases:
p2
(5.469) where we have used F = \partial tp. Then expressing p as the time integral of F, Dp = Z t −\infty
= Z 0 −\infty
(5.470) where we have set t′ = t + \tau and assumed stationarity of the force. Thus, we can write Dp = Z \infty −\infty d\tau h
. (5.471) (diffusion coefficient) Now recalling from Eq. (5.441) that F = −¯h
, (5.472) 39H. J. Carmichael, Statistical Methods in Quantum Optics 1: Master Equations and Fokker–Planck Equations (Springer, 1999), p. 148.
5.8 Mechanical Effects of Light on Two-Level Atoms we can write
|\nabla Ω(r)|2h
+
i
. (5.473) Had we treated the field quantum-mechanically here, there would be an extra term describing diffusion due to spontaneous emission.40 However, this is missing in our semiclassical treatment, and we will treat that effect separately below. 5.8.6.3 Quantum Regression Theorem Thus, we will need integrals of two-time averages of the form Z \infty −\infty d\tau
. (5.474) To do this, we will use the alternate form of the quantum regression theorem (Section 5.7.3.1): given that one-time average \langle \sigma(t)\rangle has a solution of the form
\sigma\dagger(0)
(5.475) which it must have to be completely determined by the initial quantum state, it follows from the quantum regression theorem that the two-time average has the similar solution
\sigma\dagger
, (5.476)
(5.477) Similarly, we will need the correlation function
, which should have a similar solution as
= g∗ 0(t) + g∗ 1(t)
\sigma\dagger(0)
- g∗
(5.478) in which case we note that to get the right steady state, we replace g∗ 0(t) −\rightarrow g∗ 0(\tau) \langle \sigma\rangle , and for the other coefficients we replace \langle C(0)\rangle −\rightarrow \langle \sigmaC\rangle to obtain
= g∗ 0(\tau)
\sigma\dagger + g∗ 1(\tau)
- g∗
= [g∗
(5.479) Finally, for the two remaining correlation functions, we have
= g∗ 0(\tau)
\sigma\dagger + g∗ 1(\tau)
- g∗ 2(\tau)
- g∗ 3(\tau)
= [g∗ 0(\tau) −g∗
(5.480) and
(5.481) 40J. P. Gordon and A. Ashkin, op. cit.
Chapter 5. Two-Level Atom Interacting with a Classical Field Now, to carry out the appropriate integrals over the correlation functions, we can write Z \infty −\infty d\tau
Z \infty −\infty d\tau
0 + G∗
Z \infty −\infty d\tau h
−
0 −G∗
Z \infty −\infty d\tau h
(5.482) where
Z \infty
(5.483)
\alpha(\tau), so that the G\alpha are real. Note that we subtract the dc amplitude in G0, which implements the dc subtraction in Eqs. (5.474). In terms of these integrals, we can use Eqs. (5.482) in Eqs. (5.471) and (5.473) to write Dp = ¯h2 |\nabla Ω(r)|2h
i
(G∗ 0 −G∗
2\rhoee i
i + c.c. (5.484) But Dp is real by construction, so we can explicitly drop any imaginary terms with the result Dp = ¯h2 2 Re
, (5.485) so that all that remains is to evaluate the integrals G\alpha. To do these integrals, note that given a function f(t) and its Laplace transform L f, we have Z \infty
(5.486)
variable of the Laplace transform, not the saturation parameter). Now our equations of motion can be written \partial t
= −\Gamma 2 + i∆ iΩ −\Gamma 2 −i∆−iΩ∗ iΩ∗ −iΩ −\Gamma
=: P
. (5.487) Computing the Laplace transform, we find
= s −P
. (5.488)
5.8 Mechanical Effects of Light on Two-Level Atoms We only need the Laplace transform at s = 0, so it suffices to compute s −P
s=0 = −P =
2Ω2 −Ω(2∆−i\Gamma) 2[Ω∗]2
−2Ω∗(2∆−i\Gamma) −2Ω(2∆+ i\Gamma) \Gamma2 + 4∆2 . (5.489) Then, since the zero-frequency component of the Laplace transform has the form
(5.490)
(5.491) We thus obtain G1 =
G2 = Ω2
G3 = iΩ
(5.492) and using the steady-state values
−iΩ
1 + s(r), (5.493) we find G0 = − iΩ
\Gamma/2 −i∆ \Gamma − \Gamma \Gamma/2 −i∆−i2s∆ \Gamma . (5.494) Then, putting these integrals into Eq. (5.485), we find our main result Dp = ¯h2\Gamma \nabla s 2s 2 s (1 + s)3 1 + \Gamma2
\Gamma2 s3 +¯h2\Gamma (\nabla \phi)2 s (1 + s)3 1 + 3 − \Gamma2
s + s2 +¯h2∆
s s2 (1 + s)3 \Gamma2
. (diffusion coefficient) (5.495) This expression is somewhat cumbersome, and so we will examine it in the limits of low and high intensity. 5.8.6.4 Interpretation of the Diffusion Rate In the low-intensity limit, we find that only the first two terms contribute: Dp \approx ¯h2\Gammas "\nabla s 2s 2
+ O(s2). (diffusion coefficient, low-intensity limit) (5.496)¶
Chapter 5. Two-Level Atom Interacting with a Classical Field The first responds to the gradient of the field, and we can see explicitly now that the effect here is due to the fluctuations of the atomic dipole interacting with the field gradients. Hence, this effect is referred to as the stochastic-dipole force. The other term, which depends on the phase gradient is due to photon absorption; we will interpret this effect more carefully below when we add spontaneous emission. In the high-intensity limit, the stochastic-dipole force dominates so long as \nabla s̸ = 0, in which case the diffusion rate becomes Dp \approx ¯h2|Ω|2 \Gamma \nabla s 2s 2 . (diffusion coefficient, high-intensity limit) (5.497) As was the case for the mean force, the absorption contribution saturates, whereas the dipole contribution does not. Note that Dp increases like s for very large intensities, while Vdip increases only as log s, so that for a nearly conservative and long-lived trap it is not wise to use a very large saturation parameter. 5.8.6.5 Dressed-State Model In the high-intensity limit, we can understand the diffusion simply in terms of the dressed states.41 In this limit, the dressed states |\pm\rangle are approximately equal superpositions of |g\rangle and |e\rangle , and vice versa. Each spontaneous-emission event projects the atom into the ground state, and thus into an equal superposition of the dressed states. The atom therefore sees both dressed-state shifts \pm¯h˜Ω/2 \approx \pm¯h|Ω|/2. We can interpret this as follows: after each spontaneous-emission event, the atom sees a force
= ∓¯h|Ω| \nabla s 2s , (5.498) where the sign is chosen randomly, but with equal probability for the two possibilities. Assuming the atom is moving slowly, even after accumulating momentum, the momentum change associated with a single spontaneous-emission event is ∆p = ∓¯h|Ω| \nabla s 2s \xi (5.499) where \xi is the time until the next spontaneous-emission event, which is a random variable (\xi > 0) of mean 2/\Gamma and exponential probability density
2 exp −\Gamma 2 \xi . (5.500) To take into account the randomness of the sign, we can write ∆p = ¯h|Ω| \nabla s 2s \xi′ (5.501) where \xi′ \in R has a two-sided exponential probability density
4 exp −\Gamma 2 |\xi′| . (5.502) Then the mean-square kick is
(∆p)2 = ¯h2|Ω|2 \nabla s 2s 2 \xi′2 = 2¯h2|Ω|2 \Gamma2 \nabla s 2s 2 , (5.503) where
\xi′2 = \Gamma Z \infty −\infty d\xi′ exp −\Gamma 2 |\xi′|
\Gamma2 . (5.504) 41J. P. Gordon and A. Ashkin, op. cit.; see also J. Dalibard and C. Cohen-Tannoudji, op. cit.
5.8 Mechanical Effects of Light on Two-Level Atoms The diffusion rate is the mean-square step divided by the average step time ∆t = 2/\Gamma, so Dp =
(∆p)2 ∆t = ¯h2|Ω|2 \Gamma \nabla s 2s 2 , (5.505) which is precisely what we obtained from the full calculation in this limit. 5.8.6.6 Examples: Plane and Standing Waves To gain further insight, let’s also consider a couple of concrete examples. For a plane wave, \nabla s = 0 and \nabla \phi = k, so that only the absorption contribution remains: Dp = ¯h2k2\Gamma s (1 + s)3 1 + 3 − \Gamma2
s + s2 . (diffusion coefficient, plane wave) (5.506) For small intensity, this expression becomes Dp \approx ¯h2k2\Gammas
(diffusion coefficient, plane wave, low intensity) (5.507) As we will see below, this is what we expect for the diffusion due to photon recoils of ¯hk at an average rate \Gamma\rhoee(t −\rightarrow \infty). However, this is not due to spontaneous emission itself, which we have not yet accounted for, but rather the absorption of photons that later result in spontaneous emission events. This conclusion also applies in the high-intensity limit: Dp \approx ¯h2k2\Gamma
(diffusion coefficient, plane wave, high intensity) (5.508) For intermediate intensities, we can see that there are other correction factors in Eq. (5.506). These cor- rections have been shown to be related to the non-Poissonian character of spontaneous emission.42 This is related to the antibunching that we already examined; spontaneous emission becomes Poissonian, such that
The other example we will consider is a standing wave of light, where we can take \nabla \phi = 0 and Ω= Ω0 cos kx. Note that we are taking |Ω| −\rightarrow |Ω0| cos kx, so we do not have to deal with the sign of the Rabi frequency using the phase \phi. Thus, \nabla s
|Ω| = −k tan kx. (5.509) Only the dipole part contributes, so that the diffusion rate becomes Dp = 2¯h2k2\Gamma Ω2 0 sin2 kx
0 cos2 kx]3 \times " ∆2 + \Gamma2 2 + 3 4\Gamma2 −∆2 Ω2 0 cos2 kx + 3 4Ω4 0 cos4 kx + Ω6 2\Gamma2 cos6 kx # . (diffusion coefficient, standing wave) (5.510) For low intensities, this becomes Dp \approx ¯h2k2\Gammas0 sin2 kx, (diffusion coefficient, standing wave, low intensity) (5.511) 42Richard J. Cook, ‘‘Photon number statistics in resonance fluorescence,’’ Physical Review A 23, 1243 (1981) (doi: 10.1103/PhysRevA.23.1243); Stig Stenholm, ‘‘Distribution of photons and atomic momentum in resonance fluorescence,’’ Phys- ical Review A 27, 2513 (1981) (doi: 10.1103/PhysRevA.27.2513).
Chapter 5. Two-Level Atom Interacting with a Classical Field where s0 = Ω2
(5.512) We thus see that in this regime, the stochastic-dipole force is maximum where the gradients are maximum, which is where the intensity is minimum. This force is largest precisely where we expect spontaneous- emission noise to be smallest, and vice versa. It turns out that we can interpret the diffusion in this regime as due to stimulated absorption, as we will show below. In the large-intensity limit, the diffusion rate reduces to Dp = ¯h2k2Ω2 \Gamma sin2 kx, (diffusion coefficient, standing wave, high intensity) (5.513) which has the same spatial dependence as the small-field case. We should reiterate here that in writing down these diffusion rates, we have assumed nearly zero atomic velocity and ignored the velocity dependences of the diffusion rates. This is because we have used local values for the internal atomic variables, which is only valid if the atom really is localized, or if we are in the perturbative regime. 5.8.6.7 Spontaneous Emission In our discussion of the force fluctuations, we have ignored spontaneous emission because we have used a semiclassical field, and we did not use the Bloch equations in the form (5.418) that include the recoil kick in the dissipation terms. Thus, we will need to put it in by hand. Note that there are two effects on the atomic motion that one might consider: the absorption of the photon and the emission. As we noted above, fluctuations due to absorption are already included in the preceding analysis, but it is worth examining this in a bit more depth. In the perturbative regime, recall that the excited- and ground-state amplitudes are related from Eq. (5.412) by
2∆\psig. (5.514) Thus, the excited state has an ‘‘imprint’’ of the field profile when compared to the ground state. For
2∆cos kx \psig. (5.515) On a spontaneous-emission event, the atomic annihilation operator \sigma is applied to the atomic state vector (along with the recoil operator for the emission, which we will not consider for the moment). Then the post-emission state is
∝cos kx \psig(x)|g\rangle ∝ eikx + e−ikx
(5.516) Thus, the atom is in a superposition of having recoiled by one photon momentum in each direction along the standing wave, due to the indistinguishable possibilities of having absorbed a photon from either traveling wave. Notice also that the disturbance to the wave function due to absorption (multiplication by cos kx) is minimal at the extrema of the cosine function (where the cosine is approximately constant), but there is most disturbance on the gradients of the cosine function: the gradients induced by the absorption represent the added momentum. This explains the sin2 kx dependence of the diffusion rate (5.511) in the low-intensity limit, since sin2 kx is maximum precisely where the derivatives of cos kx are most extreme. In the general case, it is somewhat difficult to pin down the absorption effects that contribute to force fluctuations, but in principle we have already accounted for them.
5.8.7 Velocity Dependence¶
5.8 Mechanical Effects of Light on Two-Level Atoms It is easy to account for the diffusion due to spontaneous emission. The photon scattering rate is Rsc = \Gamma s(r)
(\Gamma/2)Ω2(r)
(5.517) The momentum recoils from the emitted photons, as we saw in the Lorentz atom treatment, result in atomic momentum diffusion at the rate D(se) p = ¯h2k2Rsc = ¯h2k2\Gamma s
Ω2(r)
(diffusion coefficient, spontaneous emission) (5.518) which is simply the mean-square momentum kick for one photon recoil, ¯h2k2, multiplied by the scattering
as Dpt. The form here follows from the fact that photons are emitted into random directions, and thus the resulting photon recoils cause the atom to execute a random walk in momentum space. The specific example of the plane wave is trivial here, since Ωis just a constant. For the standing wave, the diffusion rate becomes D(se) p = ¯h2k2\Gamma Ω2 0 cos2 kx
0 cos2 kx, (diffusion coefficient, spontaneous emission, standing wave) (5.519) which for small intensities becomes D(se) p \approx ¯h2k2\Gamma s0 cos2 kx, (diffusion coefficient, spontaneous emission, standing wave, low intensity) (5.520) which has the same form as Eq. (5.511), but with cos2 kx instead of sin2 kx. When added together, we see that the total diffusion rate (including both absorption and stimulated-emission contributions) becomes independent of position. If we restrict our attention to a single dimension, then the one-dimensional diffusion rate is D(se,1) p
Ω2(x)
(5.521) where \zeta2 is the mean-square projection of the photon recoil along the direction of the standing wave:
Z dΩsin2 \theta cos2 \phi. (5.522) For radiation from a pure linearly oscillating dipole oriented across the axis of interest, \zeta2 = 2/5. 5.8.7 Velocity Dependence When we take into account the atom’s motion, the Rabi frequency Ωbecomes time-dependent. Thus, for example,
\nabla |Ω|
\nabla s
. (5.523) For the saturation parameter, we can thus also write
2|Ω|\partial t|Ω|
|Ω| , (5.524) so that \partial ts
|Ω|
|Ω|
\nabla s 2s . (5.525)
Chapter 5. Two-Level Atom Interacting with a Classical Field Now we can obtain the atomic density matrix to lowest order in v as follows. First, differentiate the at-rest steady-state solutions
1 + s
iΩ
(5.526) and keep terms to first order in v to obtain
2s
\nabla s 2s \approx −2s
\nabla s 2s
(5.527) and
\partial tΩ Ω−\partial ts 1 + s = v \cdot \nabla s
− 2s
\nabla s 2s ˜\rhoeg = 1 −s 1 + s v \cdot \nabla s 2s
˜\rhoeg. (5.528) The optical Bloch equations are
−\Gamma 2 + i∆
(5.529) and equating these relations with the above expressions for the velocity-dependent time derivatives gives
(5.530) where \Gammav := \Gamma − 2s
\nabla s 2s
2 −i∆+ 1 −s 1 + s v \cdot \nabla s 2s
(5.531) Eliminating the coherences in the first of Eqs. (5.530) gives the solution
\Gamma
|\gammav|2 , (5.532) so that the second of Eqs. (5.530) gives the steady-state coherence
2\gammav
|\gammav|2 = − iΩ\Gamma\gamma∗ v/2
(velocity-dependent coherence) (5.533) The mean force from Eq. (5.441) as usual is then
(5.534)
5.8 Mechanical Effects of Light on Two-Level Atoms This is again somewhat complicated to interpret in general, so we will work out this expression in a couple of special cases.
Note that this is exactly the same as the static analysis for this case, except for the Doppler-shift replacement ∆−\rightarrow ∆−k \cdot v. Thus,
iΩ
(5.535) and following the steps leading up to Eq. (5.450), the force becomes
s(v) 1 + s(v), (velocity-dependent force, plane wave) (5.536) where
|Ω|2
(5.537) This force is only valid to first order in v, and so
1 + 2∆k \cdot v
s, (velocity-dependent saturation parameter) (5.538)
1 − s2∆k \cdot v
1 + s, (5.539) so that
s 1 + s 1 + 2∆k \cdot v
1 + 2∆k \cdot v
. (velocity-dependent force, plane wave) (5.540) The velocity-dependent part of this expression is the Doppler force, which, as we saw from the classical analysis of Section (1.4.2), is a damping, friction-like force for red detunings (∆< 0), which is what gives rise to optical molasses when multiple beams are present. (Note that from this expression, one beam is sufficient to cool an atom, provided that it is trapped by some other force that cancels the mean radiation pressure.) For a standing wave in the x-direction, we have Ω= Ω0 cos kx as usual, so that \nabla s/2s = −k tan kx
2s(r) 1 + s(r)(v \cdot k) tan kx
2 −i∆− 1 −s(r) 1 + s(r) (v \cdot k) tan kx. (5.541) The algebra here is more complicated, but the velocity-dependent part of the mean force reduces to
0 cos2 kx] −Ω6 0 cos4 kx
0 cos2 kx]3 . (velocity-dependent force, standing wave) (5.542) For small intensities, this expression becomes
Ω2
(velocity-dependent force, standing wave, small intensity) (5.543)
5.8.8 Doppler Cooling Limit¶
Chapter 5. Two-Level Atom Interacting with a Classical Field where
2 = Ω2
(5.544) is the zero-velocity equilibrium population of the atomic excited state, where the intensity is taken to be spatially averaged over the standing wave. Comparison of this expression to Eq. (5.540) makes it clear that, for small intensities, the velocity-dependent force in a standing wave is explained by optical molasses, but where the strength is modulated sinusoidally with period \lambda/2. The more interesting feature here is that the sign of the force changes as the intensity becomes very large. Thus, the velocity-dependent force is damping for red detuning and small intensities, but becomes a heating force for large detunings. The interpretation in the large-intensity regime is that the local steady- state, dressed-level populations lag behind the atomic position.43 |eÒ |gÒ |eÒ |+Ò |-Ò |gÒ h—|D| h—|D| position v h—W~ To visualize this, let’s consider an atom moving in a Gaussian beam, with ∆< 0. Thus, the atom is primarily in the |−\rangle state, which is more like the ground state than the excited state. Suppose that it is moving to the right. In an instant, when it moves a small distance to the right, the equilibrium population for the atom is less in |−\rangle and more in |+\rangle . (The populations are equal in the limit of large intensity.) However, the atom doesn’t adjust its populations instantaneously, so when it arrives at the new position, it has ‘‘too much’’ population in |−\rangle compared to equilibrium. Thus, the dressed-state shifts don’t cancel as much as they would otherwise, and so the force is larger than it would otherwise be. The velocity-dependent ‘‘correction’’ is thus in the same direction as the velocity, as we expect for ∆< 0. This argument works, of course, when the energy gradient has the opposite sign or when the atom has the opposite velocity. 5.8.8 Doppler Cooling Limit Now that we have the velocity-dependent force and the diffusion coefficient, we can treat the problem of laser cooling from the quantum-mechanical viewpoint. Recall from Eq. (5.467), the steady-state solution of
Gaussian (‘‘thermal’’) distribution in momentum of variance
D 2|A|. (5.545) We will consider the case of two traveling waves, which form a standing wave. The diffusion rate is given by the sum of the diffusion rate from the fluctuation force in Eq. (5.510) and the spontaneous-emission diffusion rate from Eq. (5.518). To avoid a cumbersome expression (and to focus on the cooling regime), we will consider the small-intensity limit, where the diffusion coefficients are given by Eqs. (5.511) and (5.520). The sum is independent of position, and is given by Dp = ¯h2k2\Gammas0 , (5.546) 43J. Dalibard and C. Cohen–Tannoudji, op. cit.
5.9 Bloch-Siegert Shift¶
5.9 Bloch–Siegert Shift where again s0 = Ω2 0/2[(\Gamma/2)2 + ∆2]. The linear drift coefficient is given from Eq. (5.543) by
m = ¯h\Gammak2s0 2m ∆
(5.547) where we have replaced the sin2 kx dependence by the spatially averaged value of 1/2, since we assume the atoms are not so cold that they are stationary on wavelength distance scales. Now remember that Dp represents diffusion in three dimensions, so to have three-dimensional cooling we must have three sets of standing waves. This means that the cooling force is three-dimensional, with the same coefficient A, but the diffusion rate is larger by a factor of three. Thus, the steady-state variance for negative detunings is
2|A| = 3¯hm
|∆| . (5.548) We can translate this into a kinetic energy via
2m = 3¯h
|∆| . (5.549)
kBT = ¯h\Gamma
2|∆|/\Gamma . (5.550) (Doppler limit to laser cooling) This is precisely the same result that we obtained from our Lorentz-model treatment of Section 1.4.3.1. Again, all the ingredients are the same: we have included the cooling force (optical molasses) via the velocity-dependent force of Eq. (5.543), the diffusion due to absorption from a random beam in the diffusion rate of Eq. (5.510), and the diffusion rate due to spontaneous emission from Eq. (5.518). In principle, though the more general expressions can also treat more general situations than the weak-field case, and we have now shown more explicitly that we expect (to lowest order in velocity) a thermal steady-state distribution. 5.9 Bloch–Siegert Shift All of the results that we have developed for the two-level atom have involved the rotating-wave approxima- tion. So, then, what is the effect of the neglected term? One well-known effect that is closely related to the dipole shift is the Bloch–Siegert shift of the atomic resonance.44 Recall that, within the rotating-wave approximation, a monochromatic field of frequency \omega induces an ac Stark shift of the ground state, which we can see from Eq. (5.460) is given by ∆Eg = ¯hΩ2 4∆, (5.551) to lowest order in Ω2/∆2, where as usual ∆= \omega −\omega0. The excited-state shift is exactly opposite the ground-state shift: ∆Ee = −¯hΩ2 4∆. (5.552) We would now like to treat the counterrotating term as another monochromatic field, but of frequency −\omega. To lowest order, the shift is just an additive shift of the same form, but with \omega −\rightarrow −\omega. This treatment is corroborated by the classical treatment of the dipole potential, specifically Eq. (1.76), where the counterrotating field gave an additional dipole shift of the same form, but with the replacement ∆= \omega −\omega0 −\rightarrow −
. (5.553) 44After F. Bloch and A. Siegert, ‘‘Magnetic Resonance for Nonrotating Fields,’’ Physical Review 57, 522 (1940) (doi: 10.1103/PhysRev.57.522).
5.9.1 Magic Wavelength¶
Chapter 5. Two-Level Atom Interacting with a Classical Field Thus, we expect a ground-state shift due to the counterrotating term of ∆Eg,c = − ¯hΩ2
(5.554) while the excited state experiences an equal but opposite shift. The shift of the transition frequency due to the counterrotating field is thus ∆\omegac = ∆Ee,c −∆Eg,c ¯h = Ω2
(5.555) The shift in the atomic resonance is given to lowest order by setting \omega = \omega0 in this expression, so that
4\omega0 . (5.556) (Bloch–Siegert shift) This is the lowest-order expression for the Bloch–Siegert shift.45 The shift is typically quite weak: a relatively large Rabi frequency of Ω/2\pi = 1 GHz is large enough to drive a 2\pi-pulse in only 1 ns, but at optical frequencies (say for 1 µm light), the Bloch–Siegert shift is only 1 kHz, or a fractional shift of about 10−12. This is well within the power-broadened line shape, and quite difficult to detect.46 Thus, the first-order result is quite adequate for most optical situations. An important point regarding the Bloch–Siegert shift is that it is due to a nonresonant interaction. Nonresonance is the justification for treating an atom as a two-state system: the other levels are not reso- nantly coupled and are therefore ignorable. However, if the detuning is large enough or the desired accuracy of the calculation is high enough that the effects of the counterrotating term are important, then so are the effects of couplings to other levels. That is, the two-level and rotating-wave approximations are at the same level of accuracy. For consistency of approximations either both or neither should be made. (Of course, in magnetic resonance it is possible to have an exact two-state system where the Bloch–Siegert shift is significant.) 5.9.1 Magic Wavelength In the two-level atom, the ac Stark shift always results in opposite shifts between the excited and ground states, and thus always leads to a shift of the transition frequency. Of course, the fact that couplings to other states exist can work to our advantage. In particular, when accounting for the counterrotating interactions and couplings to other levels, it may be possible to find situations where the excited- and ground-state shifts are equal, with the same sign. This situation happens when the atom is driven at particular wavelengths, called magic wavelengths. To illustrate this, consider the level shifts of 87Rb of the two levels in the D1 transition at 794 nm due to a monochromatic laser field. 45It is possible to obtain higher-order expressions for the Bloch–Siegert shift. See Jon H. Shirley, ‘‘Solution of the Schrödinger Equation with a Hamiltonian Periodic in Time,’’ Physical Review, 138, B979 (1965) (doi: 10.1103/PhysRev.138.B979); Stig Stenholm, ‘‘Quantum theory of RF resonances: The semiclassical limit,’’ Journal of Physics B: Atomic and Molecular Physics, 6, 1650 (1973) (doi: 10.1088/0022-3700/6/8/042); C. Cohen-Tannoudji, J. Dupont-Roc and C. Fabre, ‘‘A quantum calculation of the higher order terms in the Block Siegert shift,’’ Journal of Physics B: Atomic and Molecular Physics, 6, L214 (1973) (doi: 10.1088/0022-3700/6/8/007); P. Hannaford, D. T. Peg, and G. W. Series, ‘‘Analytical expressions for the Bloch-Siegert shift,’’ Journal of Physics B: Atomic and Molecular Physics, 6, L222 (1973) (doi: 10.1088/0022-3700/6/8/009). 46Note, however, that the shift, including higher order corrections, have been studied experimentally for microwave transitions. See C. Cohen-Tannoudji, J. Dupont-Roc and C. Fabre, ‘‘An experimental check of higher order terms in the radiative shift of a coherence resonance,’’ Journal of Physics B: Atomic and Molecular Physics, 6, L218 (1973) (doi: 10.1088/0022-3700/6/8/008).
5.9 Bloch–Siegert Shift ground state excited state wavelength (nm) −200 level shift (MHz) −100 The plotted energy shifts correspond to those experienced by an atom at the center of a 1 W Gaussian beam of radius w0 = 10 µm. We will defer the details of the calculation until we develop more sophisticated technology for dealing with atomic structure. For now, we will note that we have done the calculation accounting for 24 total atomic transitions, some of which are visible as dispersive resonances in the level- shift curves. Magic wavelengths occur at the locations where the two curves intersect. For example, a laser spot at 1350.39(65) nm can be used as a dipole trap for 87Rb, where the frequency of the D1 transition is not affected. Why is this interesting? There are two well-known application. The first is, suppose you have an atom in a dipole trap, and you want to probe a transition with a resonant laser. Well, recall that the stochastic dipole force is due precisely to having different dipole potentials in different states, and then stochastically jumping between them due to spontaneous emission. Of course, the same thing will happen in this situation: when probed, the atom will jump between the ground and excited states, where it will experience different trapping potentials. This can be a large source of heating and decoherence. But trapping at the magic wavelength suppresses this form of heating. This approach is used to great advantage in cavity quantum electrodynamics to obtain long trap lifetimes of a single atom in an ultrahigh-finesse cavity.47 The other application is to atomic frequency standards. The current frequency standard is based on a microwave hyperfine transition in 133Cs. But increasing the quality factor \omega0/\delta\omega of the oscillator standard requires both decreasing the line width \delta\omega as well as pushing the resonance frequency \omega0 into the optical. To allow for long interrogation times, the atoms must be trapped. One promising approach is the single-ion atomic clock, where the oscillating trap fields average to zero to lowest order. Another promising approach is to trap atoms in a magic-wavelength, three-dimensional optical lattice.48 This allows the atoms to be trapped with little shift of the optical transition. Also, the atoms do not suffer collisions because they are isolated at different lattice sites. So the clock does not suffer from a collisional shift, but retains the advantage of averaging over many atoms. 47J. McKeever, J. R. Buck, A. D. Boozer, A. Kuzmich, H.-C. Nägerl, D. M. Stamper-Kurn, and H. J. Kimble, ‘‘State- Insensitive Cooling and Trapping of Single Atoms in an Optical Cavity,’’ Physical Review Letters 90, 133602 (2003) (doi: 10.1103/PhysRevLett.90.133602). 48Masao Takamoto and Hidetoshi Katori, ‘‘Spectroscopy of the 1S0-3P0 Clock Transition of 87Sr in an Optical Lattice,’’ Physical Review Letters 91, 223001 (2003) (doi: 10.1103/PhysRevLett.91.223001); Carsten Degenhardt, Hardo Stoehr, Uwe Sterr, Fritz Riehle, and Christian Lisdat, ‘‘Wavelength-dependent ac Stark shift of the 1S0-3P1 transition at 657 nm in Ca,’’ Physical Review A 70, 023414 (2004) (doi: 10.1103/PhysRevA.70.023414); Masao Takamoto, Feng-Lei Hong, Ryoichi Higashi, and Hidetoshi Katori, ‘‘An optical lattice clock,’’ Nature 435, 321 (2005) (doi: 10.1038/nature03541).
Chapter 5. Two-Level Atom Interacting with a Classical Field 5.10 Exercises Problem 5.1 In the classical treatment of the dipole potential, we used the expression −(1/2)d \cdot E for the dipole energy, where the 1/2 accounts for the fact that the dipole is induced. However, for the two-level atom, we used the interaction Hamiltonian HAF = −d \cdot E, which has no such factor. What gives? Hint: the interaction Hamiltonian is also valid classically, so argue that the above two expressions are consistent in the classical model. In particular, show that the Lorentz model assumed an interaction of the form of HAF. Then, modeling the atom as a classical harmonic oscillator, show that a perturbation of the form HAF leads to an energy shift −(1/2)d \cdot E (it suffices to assume a static perturbation). Problem 5.2 (a) Derive the equations of motion for the amplitudes of the quantum state
(5.557) under the evolution of the Hamiltonian HA + HAF. (b) Then make the transformation to the rotating frame by defining ˜ce := ceei\omegat, (5.558) and rewrite the equations of motion in terms of ˜ce. (c) Finally, define the rotating-frame quantum state
(5.559) and show that the equations of motion for this state under the rotating-frame Hamiltonian ˜HA + ˜HAF are equivalent to your results for part (b). Problem 5.3 Consider a quantum-mechanical particle in a double-well potential, not necessarily symmetric. For simplicity, we will make a two-state approximation for the particle, restricting our Hilbert space to the lowest two energy levels. (a) In the uncoupled (i.e., no tunneling through the barrier) limit, we can write the state in the left- hand well as |L\rangle , and the state in the right-hand well is |R\rangle . Using this basis, write down the most general Hamiltonian that describes this system, including the tunneling interaction. Introduce new parameters as necessary but explain what they represent and any necessary constraints.
Show that the state will oscillate periodically in time between the two wells. Rewrite the Hamiltonian from part (a) for this case in terms of the period T of oscillation. (c) For the case in part (b), what are the eigenstates and eigenenergies of the Hamiltonian? How does the structure of the eigenstates and eigenenergies explain the tunneling process? (You need not derive them if you know them, but give precise answers and explain.) Problem 5.4 Show that the eigenstates (dressed states) of the Hamiltonian ˜HA + ˜HAF can be written
(5.560) where
∆
. (5.561)
5.10 Exercises Problem 5.5 Go through the derivation of the Landau–Zener transition probability in Section 5.3.2 carefully, filling in the missing steps, to derive the loss probability
−\piΩ2 2|\partial t∆| . (5.562) Problem 5.6 (a) Show that the transition probability for a Ramsey-type interference experiment can be written Pe = 4Ω2 ˜Ω2 sin2 ˜Ω\tau ! " cos ∆T cos ˜Ω\tau ! −∆ ˜Ω sin ∆T sin ˜Ω\tau !#2 , (5.563) where \tau is the interaction of each of the two laser pulses of Rabi frequency Ωand detuning ∆, and T is the ‘‘drift’’ time between pulses. (b) Make a plot of the excitation vs. the microwave detuning. Use parameters appropriate for the NIST-7 cesium beam clock. (c) Make another excitation plot, but this time average the fringes over the velocity distribution of the cesium beam. Assume a temperature of 100◦C, corresponding to a velocity width of 240 m/s. (I suggest doing the averaging numerically, not analytically.) Problem 5.7 Solve the optical Bloch equations to obtain the population inversion \langle \sigmaz(t)\rangle in the limit of weak
Assume that the atom is initially in the ground state. Problem 5.8
arbitrary detuning ∆. Keep only lowest-order contributions in Ω−2 in your solution. Do not assume Ω≫|∆|. Problem 5.9 (a) Summarize the effects of the nonlinear response of a two-level atom to an externally applied, classical, monochromatic field. (Treat all aspects we covered, but ignore atomic motion.) (b) In Problem 1.5, you found that adding nonlinearities to the Lorentz model resulted in new frequen- cies being generated from the original, monochromatic driving field. Explain qualitatively why the nonlinear response of the two-level atom does not similarly generate harmonics of the original driving field. How would the two-level atom need to be generalized to model harmonic generation (assuming that we keep the dipole approximation)? Problem 5.10 Use the solutions of the optical Bloch equations to derive an expression for the frequency-dependent polarizability \alpha(\omega) for the two-level atom. Show that your results are consistent with the result from the Lorentz atom in the appropriate regime. Discuss the significance of any differences. Problem 5.11 If the density operator of the two-level atom evolves according to
(5.564) derive an equation of motion for the purity in terms of the elements \rhoee, \rhoeg, \rhoge, and \rhogg of the density matrix.
Chapter 5. Two-Level Atom Interacting with a Classical Field Problem 5.12 Go through the derivation of the Mollow triplet on resonance, including the details of the quantum regression theorem, filling in the missing steps. Problem 5.13 Use the quantum regression theorem to derive the inelastic spectrum for a two level atom driven far
Problem 5.14 Derive expressions for the elastic and inelastic emission spectra for a two-level atom with collisional
inelastic component is a single peak at the atomic resonance frequency. Problem 5.15 Go through the derivation of the second-order coherence function g(2)(\tau) on resonance, filling in the missing steps. Problem 5.16 Derive the expression for the Autler–Townes absorption correlation function ga(\tau) in the case of homo- geneous broadening but arbitrary pump intensity and detuning, Eq. (5.373) in the notes. To do this, work out the details of the quantum regression theorem for this system, and then solve the resulting system of equations. Hint: the quantum regression theorem gives the correlation function in terms of a solution of 9 coupled differential equations. However, only two of them are nonzero. Start off by eliminating the unnecessary components. Problem 5.17 Give an alternate derivation of the general dipole potential Vdip = ¯h∆ 2 log[1 + s(r)] (5.565) in the dressed-state picture, and thus interpret the potential as a combination of the Stark shifts of the two dressed levels, plus a contribution from the field due to scattering into the Mollow sidebands. Use the following outline for your derivation.49 (a) Show that when the Rabi frequency Ωis complex, the dressed-state energies are E\pm = −¯h∆ 2 \pm ¯h˜Ω 2 , (5.566) where the generalized Rabi frequency is ˜Ω= p |Ω|2 + ∆2, and the dressed states are given by
(5.567) where
∆
, (5.568)
49J. Dalibard and C. Cohen-Tannoudji, ‘‘Dressed-atom approach to atomic motion in laser light: the dipole force revisited,’’ Journal of the Optical Society of America B 2, 1707 (1985).
5.10 Exercises Hint: argue that the Hamiltonian has the form ˜H = ¯h −∆ Ω/2 Ω∗/2 , (5.569) and can be diagonalized by a unitary matrix. What is the most general form of a unitary matrix? (b) Derive the steady-state populations \rho\pm\pm and coherences \rho\pm∓for the dressed levels in the limit of
are well resolved and thus give a valid description of the system. Hint: first find the density-matrix elements in the bare basis, and make appropriate approximations in this limit. Then use the appropriate matrix transformation to switch to the dressed basis. Your answers should be
" 1 + ˜Ω ∆(1 + s) #
(5.570)
(c) Starting with the expression from Eq. (5.441),
- c.c., (5.571) we drop the radiation-pressure contribution (the term with \nabla \phi) to obtain the dipole force
(5.572) Show that the force can be written in terms of the dressed states as
(5.573) (d) The work to move the atom a distance dr is
= ¯h
(5.574) For a static or slowly moving atom, only the first term contributes to the dipole force; the second term involves the dressed-state coherences, and represents nonadiabatic transitions between the dressed states due to the displacement. Show explicitly based on your answer from (b) that the second term vanishes for adiabatic displacements. (e) The displacement work can then be written as dW = ¯hd˜Ω
(5.575) Use your results of part (b) to evaluate this expression, and then integrate it with respect to position to obtain the dipole potential (valid in the limit of well-resolved dressed states).
Chapter 5. Two-Level Atom Interacting with a Classical Field (f) The position-dependent level shift of the dressed states, weighted by the dressed-state population, is
(5.576) where ∆E\pm = \pm¯h˜Ω (5.577) are the dressed-state energy shifts relative to the mean level energy. Show that UA is not the correct dipole potential.
equations give
(5.578) for the (position-dependent) dressed-state populations, where
(5.579) represent the decay rates between the dressed states. To do this, make the adiabatic, or secular approximation, where the populations and coherences evolve on very different time scales and thus can be decoupled by replacing the coherences by their steady-state (mean) values in the population equations. (Why do the populations vary slowly?) Then \Gamma+−\rho++ is the rate for |+\rangle −\rightarrow |−\rangle transitions, and thus the rate for producing photons in the \omega + ˜ΩMollow side band, and \Gamma−+\rho−−is the rate for |−\rangle −\rightarrow |+\rangle transitions, and thus the rate for producing photons in the \omega −˜ΩMollow side band. In equilibrium, the two rates are equal, implying a symmetric Mollow spectrum. Note that the optical Bloch equations in the dressed-state basis reduce to simple rate equations in the secular approximation. Otherwise, the dressed-state equations are rather more complicated than for the bare states. Thus, the dressed basis is most useful when the secular approximation holds. (h) Finally, show that dW = dUA + dUF, (5.580) where
(5.581) Interpret this relation as the energy transferred to the field in time dt due to the displacement of the atom. The dipole potential thus has contributions from both the atom and field energies. Problem 5.18 Recall the one-dimensional Fokker–Planck equation:
2\partial 2 p D(p)f(p, t). (5.582) Since the drift coefficient A(p) and diffusion coefficient D(p) depend on momentum, we can’t write down a general solution, and the interpretation of these coefficients is a bit tricky. However, computing the equations of motion for the first two moments helps with the interpretation. (a) Show that
(5.583) Here, the expectation value refers to an average with respect to the distribution f(p, t). Assume that f(p, t) falls off quickly enough with |p| that any boundary terms are negligible. Thus, ‘‘drift’’ of the distribution mean is caused both by the drift coefficient, averaged over the distribution. (b) Show that
(5.584)
5.10 Exercises¶
p2 −\langle p\rangle 2. Thus, the diffusion coefficient causes spreading of the variance, again weighted by the distribution f(p, t). Also, variations in the drift coefficient can cause the distribution to spread or contract, since different accelerations for parts of the distribution at different momenta can tear them apart or squish them together, depending on the relative accelerations. Problem 5.19 Consider an atom trapped in a potential well of an optical standing wave (one-dimensional optical lattice). (a) The standing wave is composed of two counterpropagating traveling waves. Suppose that the frequency of one of the traveling waves is shifted by a small amount \delta\omega. Show mathematically that the new configuration corresponds to a moving optical lattice. Intuitively, you can see this because there exists a moving reference frame where the Doppler-shifted frequencies of the two waves are equal, and thus the optical lattice is at rest in this frame. What is the velocity of the standing wave? What is the frequency shift that matches the velocity of an atom moving with momentum ¯hk, where k is the wave number of the optical lattice? Compute this last frequency shift for 87Rb, assuming that the optical lattice is tuned close to the 780 nm resonance. (b) If you add a linear chirp (frequency sweep) to the frequency of one of the traveling waves, then you end up with an optical lattice of constant acceleration. An atom bound to the optical lattice will accelerate with it, making transitions to higher momentum states. Consider the component of the atom in the p = 0 state. Use what you know about Bragg scattering and adiabatic passage to explain qualitatively how the atom in the accelerating optical lattice makes transitions to the states
argument), how the atom is ‘‘lost’’ from the lattice and stops accelerating if the acceleration is too large. (c) Given a particular potential depth V0, estimate the critical acceleration ac, above which the atom is lost from the lattice. Give this estimate using both a classical argument, assuming a classical potential of the form V0 cos 2kx, as well as a quantum-mechanical argument based on adiabatic passage. Problem 5.20 Derive the expression (5.369)
h
i
iΩ2∆p
- c.c. (5.585) for the Mollow probe-absorption spectrum, using the following outline. This problem is a good prototype for how to treat an atom interacting with a bichromatic field. By treating one of the fields as a perturbation, the solution is greatly simplified (relatively speaking). (a) Write down the interaction Hamiltonian for the probe field with the atom in terms of the probe Rabi frequency Ωp and the probe frequency \omegap. Then write down the same Hamiltonian in the rotating frame of the pump laser field (of Rabi frequency Ωand frequency \omega), in terms of the probe-pump detuning ∆p := \omegap −\omega. You may assume both Ωand Ωp to be real. (b) Write out the equations of motion for the atomic density-matrix elements, obtained from the master equation
¯h h
i
(5.586) (c) Now we will treat the probe-field interaction as a perturbation to the steady-state solution of ˜HA + ˜HAF by making the ansatz
(5.587)
Chapter 5. Two-Level Atom Interacting with a Classical Field for the long-time solution to the equations of motion. Here, the ˜\rho(0) \alpha\beta are the steady-state solutions under the evolution of ˜HA + ˜HAF; the \delta˜\rho(0) \alpha\beta are the dc corrections due to the perturbation; and the
\alpha\beta are (complex) constants giving the amplitudes of the oscillating corrections to the long-time solutions. Note that
\beta\alpha ∗
gg , (5.588) so that only nine of the coefficients are independent. Justify (qualitatively) this form of the ansatz as sufficient to describe the corrected solution to lowest order in the probe intensity. Then substitute these equations into the above equations, and use the ‘‘steady-state’’ condition \partial t\rho\alpha\beta = 0 to derive a set of nine(!) coupled, linear equations for the perturbation coefficients (use the above constraints to eliminate the coefficients \rho(\alpha) gg ). Check your answers: −iΩ
ge + iΩ
ee −iΩp
ge + iΩp
eg = 0 −iΩ
ge −iΩp 2 ˜\rho(0) ge + iΩ
ee = 0
ge −iΩ \delta\rho(0)
gg −iΩp
gg = 0
ge −iΩ
gg = 0
ge −iΩ
ee
gg −iΩp \rho(0) ee −\rho(0) gg = 0. (5.589) (d) Solve the set of equations to obtain an expression for \delta˜\rho(+) eg . A computer and a symbolic algebra program could save you both time and severe cramping in your writing hand. (e) Recall the Bloch equation for the atom interacting with the pump field (without the probe)
(5.590) Identify the function of these terms and argue that in steady state, the rate of photon absorption is Rabs = iΩ
(5.591) Then argue that the rate of photon absorption from the probe field is Rabs, probe = iΩp
ge , (5.592) and finally show that Rabs, probe is equivalent to the above absorption spectrum up to an overall factor. Problem 5.21 Prove the following form of the quantum regression theorem, for computing a correlation function of the form \langle A(t + \tau)B(t)\rangle . In this case, the correlation function may be written lim
(5.593) where \Lambda(\tau) obeys the master equation with initial conditions
(5.594)
5.10 Exercises Problem 5.22 Prove the following form of the quantum regression theorem. Suppose that the one-time average of an operator A can be written in the form
X j
(5.595) where gj(t) are functions representing the solution in terms of initial conditions \langle Bj(0)\rangle of some set of operators, then the two-time average \langle A(t)B(t + \tau)C(t)\rangle t\rightarrow \inftymay be written
X j
(5.596) Problem 5.23 Consider the quantum damped harmonic oscillator, with master equation
(5.597) with free Hamiltonian
(5.598) and driving Hamiltonian Hint = ¯hE
. (5.599) (a) Derive the equations of motion for the density matrix elements \rhonm in the energy basis. (b) Find a transformation to a rotating frame, where the equations of motion for the slow variables ˜\rhonm have no explicit time dependence. (c) Write out explicit expressions for the equations of motion for \rho11, ˜\rho10, ˜\rho01, and \rho00, under the assumption that ˜\rhonm = 0 if n > 1 or m > 1. Compare your results to the optical Bloch equations. Problem 5.24 In this problem we will construct and analyze a simple model for a laser by using the optical Bloch equations and the results of Problem 1.6 to describe the atom–field interaction. For our purposes here, we will think of a laser as comprised of (1) an optical cavity (resonator), which traps light and confines it in some region of space, as in a pair of parallel mirrors (Fabry–Perot cavity); (2) a gain medium, consisting of a vapor of two-level atoms uniformly filling the cavity; and (3) some pump source that promotes the atoms to the excited state. (a) Derive the Maxwell–Bloch equations for the interaction of a single cavity-field mode with an ensemble of quantum-mechanical two-level atoms of number density N that fill the cavity:
ϵ0
2 E0
(5.600) Here, E0 is the slowly varying field amplitude as defined in Problem 1.6; the field is exactly resonant
in the absence of coupling to the cavity field; \kappa is the energy decay rate of the cavity; and dge := \langle g|dz|e\rangle is the matrix element that appears in the Rabi frequency. To do the derivation:
Chapter 5. Two-Level Atom Interacting with a Classical Field 1. Start by writing down the optical Bloch equations on resonance. Assuming the atoms are initially unexcited, argue that \langle \sigmax\rangle can be removed from the problem. Note that the Rabi frequency here represents the coupling parameter of the atoms to the laser field. 2. Tack on an extra term of the form −R[\langle \sigmaz\rangle −1] to the \langle \sigmaz\rangle Bloch equation to model excitation due to a pump field. Justify why in fact such a term gives the appropriate behavior. 3. Use the results of Problem 1.6 to write down an equation of motion for the cavity-field amplitude E0, ignoring any spatial dependence of the field amplitude or the polarization amplitude. Also ignore any overlap of the circulating field with itself, as in a ring cavity. Use what you know about the polarization and the atomic dipoles to obtain the first term in the first Maxwell–Bloch equation above. Finally, tack on a damping term to model leakage of the cavity energy through the laser output port. (b) Let E0ss, \langle \sigmay\rangle ss, and \langle \sigmaz\rangle ss denote the steady-state values of E0, \langle \sigmay\rangle , and \langle \sigmaz\rangle , respectively. Now define the variables
;
\kappa 2\gamma⊥ ; b := \GammaR \gamma⊥ , (5.601) and define the scaled coordinates
x := p b(r −1) E0 E0ss ; y := p
;
. (5.602) Show that the Maxwell–Bloch equations may be written in the form50
(5.603) These equations are well-known in the area of nonlinear dynamics as the Lorenz model51 (not Lorentz!), essentially one of the simplest possible nontrivial models for turbulent fluid flow or for the global weather system. Depending on the parameter values, the Lorenz model—and thus the Maxwell–Bloch equations for the laser—display a complex variety of dynamical behaviors, from steady behavior to periodic oscillation to chaos.52 The fluid interpretation of the Lorenz model is briefly as follows: the Lorenz model describes a fluid (subject to gravity) confined between two horizontal plates, where the lower plate is maintained a temperature ∆T higher than the upper plate. Then the parameter \sigma is a sort of scaled fluid viscosity (the Prandtl number), r is a scaled temperature difference ∆T (the Rayleigh number), and b is a scaled plate separation. Physically, for small temperature differences, the fluid simply supports the temperature gradient by conduction, without any movement. For larger temperature differences, the stationary behavior becomes unstable due to the buoyancy of the warm fluid near the bottom plate, and a convection pattern forms: a periodic pattern of ‘‘rolls’’ forms, transporting the warm fluid quickly to the upper plate, where it cools and then falls back down. 50This correspondence was first shown by H. Haken, ‘‘Analogy Between Higher Instabilities in Fluids and Lasers,’’ Physics Letters A 53, 77 (1975). 51E. N. Lorenz, ‘‘Deterministic Nonperiodic Flow,’’ Journal of Atmospheric Science 20, 130 (1963). 52Lorenz–Haken-type instability was observed in an ammonia laser, see C. O. Weiss and J. Brock, ‘‘Evidence for Lorenz- Type Chaos in a Laser,’’ Physical Review Letters 57, 2804 (1986) (doi: 10.1103/PhysRevLett.57.2804). For a review, see Eugenio Roldán, G. J. de Valcárcel, R. Vilaseca, R. Corbalán, V. J. Martínez, and R. Gilmore, ‘‘The dynamics of optically pumped molecular lasers. On its relation with the Lorenz–Haken model,’’ Quantum and Semiclassical Optics 9, R1 (1997)
5.10 Exercises¶
To+o∆T To go For larger temperature differences, the flow becomes more complicated, but the Lorenz model doesn’t capture this: it assumes physical quantities to be periodic in the horizontal direction with a single spatial frequency (i.e., truncating the Fourier series for the velocity, temperature, etc. fields after the first term). The coordinates x, y, and z are respectively the amplitude of the velocity modulation (i.e., the maximum upward velocity), the amplitude of the temperature modulation (i.e., the maximum horizontal temperature difference), and the dc offset of the temperature field from the conductive value. This general fluid problem is called Rayleigh–Bénard convection, and the Lorenz model is a greatly simplified description of this system.
requiring \sigma > b + 1). (c) To analyze this laser model, begin by finding the fixed points or stationary solutions of the Lorenz
solutions, corresponding to conductive and convective fluid flow. Identify which is which. In terms of laser output, what is the (correct) interpretation of each fixed point? (d) Perform a linear stability analysis to find the ranges of the parameter r for which the conductive and convective solutions are stable. You may assume \sigma, b and r to be positive for the purposes of this problem. Make a plot (sketch) of the fixed-point solutions, plotting x∗vs. r, indicating their stability on your plot. To do the stability analysis, write each dynamical variable as a small perturbation to a fixed point,
(5.604) and substitute these into the Lorenz equations, keeping only first order terms in the perturbations. The result is a set of linearized equations for \deltax, \deltay, and \deltaz in the neighborhood of the fixed point (x∗, y∗, z∗). Now assume a solution of the form
(5.605) and use the linearized equations to find expressions for \lambda near each fixed point. If Re[\lambda] > 0, then the solution runs away, and the fixed point is linearly unstable; If Re[\lambda] \le 0, then the solution remains bounded, and the fixed point is linearly stable. Hint: you don’t necessarily need to find the values of \lambda (i.e., the eigenvalues of the evolution matrix) in each case, you just need to determine the signs of the real parts. (e) A universal feature in laser physics is threshold behavior. In a laser, what is a physical mechanism that prevents the unstable conductive solution from occuring? Viewing r as a ‘‘rescaled’’ version of the pumping rate R, interpret your solutions for the fixed points and their stability in terms of the steady-state laser output, and explain how these result predict a threshold in the laser output as a function of pump rate. Also, when a laser above threshold begins to oscillate (lase), when starting
breaking. Explain what this term means in the context of your plot from part (d). (f) Below is shown a bifurcation diagram for the x coordinate of the Lorenz model as a function of
Chapter 5. Two-Level Atom Interacting with a Classical Field
forward for a long time to get rid of any transient behavior. Then, integrate it for an even longer time; if the solution has settled to a stationary state, then plot the steady x∗value at coordinates (r, x∗); if it oscillates, plot a point on the graph at coordinates (r, x), each time the slope of the variable x changes sign. Thus, each trajectory will look like a bunch of points on your bifurcation diagram, and gives some idea of what happens at any value of r. For the purposes of this diagram, we have chosen the common values \sigma = 10 and b = 8/3. r -25 x
The plot above represents the parameter region that we’ll mostly focus on. Explain how the behavior shown here is consistent with your fixed-point stability analysis. Note that if you were to put a photodetector in front of your laser and measure the power as a function of the pump rate on your oscilloscope, you’d be looking at intensity, not the field amplitude, so you’d see something more like this: r x € soo=oo10, boo=oo8/3
5.10 Exercises Just for kicks, we can also look at a much wider range of r. Note that a large value of r corresponds to a very strong pump, and thus likely puts the laser into a highly nonlinear regime where this model breaks down. We would thus expect higher-order instabilities to make things more complicated. However, even within this model, the bifurcation diagram is quite intricate. You can see all sorts of behavior, such as an inverse period-doubling cascade, stability windows, and period-3 trajectories, which demonstrate that the model must exhibit chaotic behavior53 r -60 x
(g) Noting the symmetry of your steady-state sketch in (d), and the apparent reflection symmetry of the bifurcation diagrams, find the underlying symmetry in the Lorenz equations. Give a physical interpretation to the symmetry.
other parameters and initial conditions as noted in (f). Interpret the behavior you see in each case, and comment on what the dynamical plot tells you about each relevant region in phase space. Note that any noisy behavior you see is likely to be chaos (deterministic ‘‘randomness’’ in low-dimensional systems).
spiking or relaxation oscillation, a transient, oscillatory behavior in the laser output when a laser parameter (usually, the pump power) is changed suddenly. It can be explained as follows. When the pump source is suddenly ‘‘switched on,’’ the strong pump rapidly excites the gain medium, such that the medium polarization build up to well beyond the threshold value before the cavity field can respond. The intensity, as a result, builds up rapidly, quickly depleting (saturating) the gain medium as the laser oscillates. The cavity field amplitude becomes so large that the gain-medium polarization drops below threshold, and soon afterwards, the field amplitude begins to drop. This process repeats, producing the oscillations.
consistent with this explanation. Physically, why do these oscillations damp to a steady equilibrium? (j) 20 years from now, you find yourself in the position of Über-Manager at Acme Laser Corp., overseeing the Nifty New Lasers division. One day, one of your many peons comes up to you in a panic, saying that the new high-power, ultrastable laser prototype that your division is now finally testing after years of engineering work turns out to not be very stable at all. In fact, the output intensity appears to be 53Tien-Yien Li and James A. Yorke, ‘‘Period Three Implies Chaos,’’ The American Mathematical Monthly 82, 985 (1975) (doi: 10.2307/2318254).
Chapter 5. Two-Level Atom Interacting with a Classical Field very noisy whenever it is run at full power, even though the power supply is very quiet. Your first hunch, based on years of experience in laser building and tweaking, is that an inappropriate output coupler (output mirror) somehow made it into the design. But to fix things, should you try substituting an output coupler with greater or lesser reflectance than the one in the prototype now? Fortunately, ages ago, you worked out the above theory, which you can use as a semi-quantitative guide to laser stability. By working out how the changing output-coupler reflectance modifies the parameters in the Lorenz model, you make an intelligent guess that should put the laser back into a stable regime. What do you advise your employee, O Sage One? (Note that when you solved this problem years ago in a quantum optics class, you explained your reasoning.) Here are some observations that may help: 1. Changing the mirror reflectance, and hence the value of \kappa, will affect both the values of r and \sigma. How do these parameters change with the mirror reflectance? This will not be obvious for r, so you should work out an expression for it in terms of \kappa and the other physical parameters. 2. Which way should you go in terms of r and \sigma to get back into a stable region? 3. You may assume that R is not much larger than \Gamma, as is typical for high-density gain media (for three-level lasers, technically speaking). Also, assume nearly natural broadening, so that \gamma⊥∼\Gamma. Finally, assume the laser to be in the ‘‘bad cavity’’ regime of \kappa ≫\Gamma, \gamma⊥. This doesn’t necessarily mean that the cavity is bad (in fact a truly bad cavity would violate the slowly varying amplitude approximation for the electric-field envelope), if the atomic transition is very narrow. (k) You will get a reasonable answer with the above considerations, but you should keep the following caveat in mind. The Lorenz–Haken instability is unusual in practice, because lasers do not usually operate in the bad-cavity limit—usually, the atomic transition is broadened well beyond the width of the cavity resonance. Usually, though, other instabilities cause transitions to dynamical (though not necessarily chaotic) ‘‘steady’’ states. In the more typical limit of strong broadening (large \gamma⊥), when the atomic coherence can be adiabat- ically eliminated, show that the laser can then be modeled by the rate equations
\gamma⊥
\gamma⊥ \sigma0I ¯h\omega0
h
i , (5.606) which are written only in terms of the cavity intensity I and the atomic inversion. Here \sigma0 = \lambda 2
is the resonant cross section for homogeneous broadening, with \lambda0 the resonant wavelength. Bonus question: is chaotic instability possible in this regime? Why or why not? Problem 5.25 Consider again the forced, damped, quantum harmonic oscillator in the form
(5.607) with free-oscillator Hamiltonian
(5.608) and driving Hamiltonian Hint = ¯hE
. (5.609) (a) Show that the expectation values evolve under the master equation as
\partial t
a\daggera
= iE h
a\dagger i −\gamma
a\daggera
, (5.610)
5.10 Exercises once a transformation to a rotating frame has been made. Find the steady-state values for \langle a\rangle and
a\daggera
. (b) Use the quantum regression theorem to compute the first-order coherence function
t\rightarrow \infty
. (5.611) What is the emission spectrum S(\omegas) for the damped harmonic oscillator? Comment on how your result compares to the two-level atom modeled by the optical Bloch equations. Hint: Use the fact that \langle a(t)\rangle may be written in the form
a\dagger(0)
- g3(t)
a\dagger(0)a(0)
, (5.612) and find the functions g\alpha(t). Then use the quantum regression theorem to write down an expression for
. Problem 5.26 Consider the quantum damped harmonic oscillator, with master equation
(5.613) with Hamiltonian
. (5.614) The fact that damping occurs here implies that the oscillator also experiences a stochastic (heating) force. This is one way to view the zero-point energy: the damping tends to make the oscillator stop, but in the ground state the damping and stochastic heating exactly balance, producing a steady state of nonzero energy. This is also essentially the content of what is called the fluctuation–dissipation relation (see Section 14.3.8.1). To see this, recall that the momentum diffusion coefficient Dp is be the rate at which the momentum variance increases,
p2
(5.615) and we showed that it is given in terms of the force correlation function as Dp = Z \infty −\infty d\tau h
. (5.616)
value of the diffusion coefficient for the damped harmonic oscillator.
at arbitrary times and the quantum regression theorem to find the correlation function. (b) Show that in steady-state, the combined effect of damping and the stochastic force leads to the correct zero-point momentum variance (i.e., the momentum variance of the ground state) in steady state.