1. Classical Atom-Field Interactions¶
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1.1 Polarizability¶
Chapter 1 Classical Atom–Field Interactions We will now model the interaction between light and atoms, using a classical model of the atom. This will allow us to treat a variety of phenomena from the refractive index of atomic vapors to the conductivity of metals to laser cooling and trapping of atoms. We will model the atom as a classical harmonic oscillator, an electron bound to the nucleus by a harmonic force (linear spring):
Here, \(x\) represents the average position of the electron, since quantum-mechanically, the electron is not localized, and \(\omega_0\) is the resonant frequency of the harmonic potential. The above equation is also in center-of-mass coordinates, so that we can ignore the motion of the nucleus. Thus, \(m\) is the reduced mass of the electron, given by
where \(m_e\) is the electron mass, and \(m_n\) is the nuclear mass. Generally \(m_e \ll m_n\), so
and generally, it is a good approximation to use \(m \approx m_e\). Why use a classical calculation, when an atom is a manifestly quantum-mechanical object? It turns out that the classical calculation gets many phenomena correct, and these results can be justified by quantum calculations. Essentially, the classical calculation is good for weak atomic excitations, when the harmonic potential, the lowest-order approximation to an arbitrary potential, is an accurate model. (It is even a good approximation to treat the quantum electromagnetic field classically as long as many photons are present, since the field turns out to be a set of harmonic oscillators, which are “not very quantum-mechanical.” Then our requirement of weak excitation of the atom implies an atom–field coupling that is in some sense very weak; we will see that this is true when discussing the atomic cross section in Section 1.2.1.) In particular, the classical model does not predict any saturation effects, and as we will see, it requires a bit of patching to make it quantitatively correct, even in the limit of small intensity.
1.1 Polarizability We will now consider the interaction of the atom with a monochromatic field of the form¶
$$ E^{(+)}(t) = \hat{\epsilon} E^{(+)-i\omega t}, \qquad (1.4)
Chapter 1. Classical Atom–Field Interactions where ˆ\epsilon is the unit polarization vector. Here, we are using the complex notation for the field, where we separate according to the positive- and negative-frequency components:
2 ei\omegat
(1.5) Recall that we are defining E(\pm) to go with e∓i\omegat, since by convention e−i\omegat corresponds to the positive
sum of the positive- and negative-frequency parts. But notice that these parts are complex conjugates, as is required to get a real (physical) field. Thus, we can always write the physical field as E(+) with its conjugate:
n
. (1.6) Of course, we apply this notation to all other quantities driven by the field, such as the displacement of the electron that we consider below. Mathematically, it is simpler to keep only one part of the solution, but to obtain the physical result, you always need to add the complex conjugate (assuming that all the calculations are linear). Note that classically, this decomposition arises as a mathematical convenience. As we will see much later, in the quantum treatment of the field this decomposition is more fundamental and significant, since the two components will play the roles of photon creation and annihilation operators. In writing down the expression (1.4), we are making the dipole approximation: we are assuming that the size of the atom is much smaller than the optical wavelength, so that the electron only sees the field at the nuclear position. Thus, we need not consider the spatial dependence or propagation direction of the field. The force on the electron due to the field is
(1.7) where e is the fundamental charge, the magnitude of the electron charge (so that the electron charge is −e). Then the equation of motion for the electron becomes
e−i\omegat. (1.8) We need only worry about the electron motion in the direction of the electric field; we will ignore any motion except that induced by the field, as we will justify when considering the damped version of the harmonic oscillator. We will now make the ansatz that the solution has the same time dependence as the field:
e−i\omegat. (1.9) With this solution, Eq. (1.8) becomes
0 x(+)
, (1.10) which we can solve for x(+) to obtain the solution x(+)
0 /m
. (1.11)
x(−)(t). The dipole moment of the atom is
(1.12)
1.1.1 Connection to Dielectric Media¶
1.2 Damping: Lorentz Model where x = ˆ\epsilonx. Since the dipole moment is induced by the field (the electron displacement is zero in equilibrium), we can define the polarizability \alpha to describe how easily the field induces the dipole moment by
(1.13) (polarizability definition) From Eqs. (1.11) and (1.12), we can write the polarizability as
e2/m \omega 2 0 −\omega2 . (1.14) (classical polarizability) The polarizability completely characterizes the response of the atom to the applied field. Of course, this is the frequency-space response function, which we have obtained via an implicit Fourier transform of the applied field. 1.1.1 Connection to Dielectric Media Recall that the polarization density P is the dipole moment per unit volume. Thus, for an atomic vapor of number density N,
\omega 2
e−i\omegat. (1.15) This expression is valid for a rarefied medium, where the interactions between the atoms are negligible. In dense media, correlations between dipoles cause deviations from these results. We can thus write the susceptibility for the vapor as
\omega 2 0 −\omega2 , (1.16) (classical susceptibility) in view of the defining relation P = ϵ0\chiE. Keeping the polarizability as the fundamental microscopic quantity, we can of course also write
ϵ0
(1.17) (susceptibility–polarizability relation) for the susceptibility of a vapor of number density N in terms of the polarizability. 1.2 Damping: Lorentz Model A better model of the atom is a damped harmonic oscillator. This improved model is known as the Lorentz model of the atom, and the equation of motion is
e−i\omegat. (1.18) (Lorentz model) The damping (‘‘friction’’) term models radiation reaction due to the charge acceleration (the classical ana- logue of spontaneous emission) and collisions with other atoms. A quantum-mechanical calculation shows that for an isolated atom, the damping rate is the same as the Einstein A coefficient (spontaneous emission
e−i\omegat. Following the method above, the solution is x(+) = eE(+) 0 /m
(1.19) Now the displacement is complex, reflecting a phase lag of the displacement behind the field, with phase angle
\gamma\omega \omega 2 0 −\omega2 . (1.20)
Chapter 1. Classical Atom–Field Interactions
case, the polarizability becomes
e2/m \omega 2
(1.21) (polarizability with damping) The susceptibility likewise becomes
Ne2/mϵ0 \omega 2
(1.22) (susceptibility with damping) It is worth reiterating here that \alpha and \chi are complex quantities defined for the positive-rotating fields via
If \chi is small (as for a dilute vapor), the complex refractive index is
p
2mϵ0 (\omega 2 0 −\omega2) (\omega 2
2mϵ0 \gamma\omega (\omega 2
(1.23) The real and imaginary parts of the complex index have distinct interpretations. Recall that a plane wave propagates with a phase factor according to
(1.24) and so we can define the phase index and absorption coefficient respectively as
(1.25) (phase index and absorption coefficient) where k is the wave number in the medium and k0 is the vacuum wave number, so that n(\omega) represents the phase shift of the propagating wave, while a(\omega) represents attenuation of the field due to absorption. Note that the absorption coefficient is defined such that the intensity I of a wave propagating in the z direction decays according to dI dz = −aI =\Rightarrow
(1.26) which explains the factors of 2 and k0. We can thus read off the phase index as the real part,
2mϵ0 (\omega 2 0 −\omega2) (\omega 2
(1.27) (classical refractive index) while the (intensity) absorption coefficient becomes
mϵ0c \gamma (\omega 2
(classical absorption coefficient) (1.28) Significant absorption occurs in the region of small detunings of the field from the atomic resonance, |\omega −
\omega 2
(1.29) This is effectively equivalent to the rotating-wave approximation that we discuss later (Section 5.1.2). With this approximation, the phase index and absorption become
2mϵ0
Ne2 mϵ0c\gamma (\gamma/2)2
(1.30) (Lorentzian absorption profile)
1.2.1 Oscillator Strength¶
1.2 Damping: Lorentz Model Thus, we recover the Lorentzian absorption profile (hence the name) with full width at half maximum \gamma and
\gamma Ne2
(\gamma/2)2
\gamma Im[˜n(\omega)], (1.31) as required by the Kramers–Kronig relations (Section 14.1.4.2). Im[n~(w)] Re[n~(w)]o-o1 w This gives the dispersive form for the phase index, as shown here. In general, we can have atoms with multiple electrons that we need to sum over. Then the polarizability and susceptibility become
X j e2 m f0j \omega 2
X j Ne2 mϵ0 f0j \omega 2
. (1.32) (corrected response functions) Here, f0j is the absorption oscillator strength, which acts as a weighting factor for each electron, or possibly something like a probability for an electron to behave as different possible harmonic oscillators. The quantum- mechanical (and correct) interpretation of these expressions is that each term in the sum represents a transition from the ground level 0 to excited level j. The oscillator strength can only be obtained from a quantum calculation, and is necessary to make the classical calculation quantitatively correct. Because of the quantitative importance of this factor, we will explore it in more detail. 1.2.1 Oscillator Strength Since the absorption coefficient scales as the susceptibility and thus the oscillator strength (for a dilute gas), the oscillator strength also scales with the cross section. On resonance, the cross section for absorption is defined by
(1.33) Thus, using Eq. (1.28), we can write the classical absorption cross section as
mϵ0c \gamma (\omega 2
\omega=\omega0 = e2 mϵ0c\gamma . (1.34) (classical cross section) This cross section is not quantitatively correct, as the correct quantum-mechanical expression for the cross section for the transition to level j [see Eq. (3.21)] is
j0
\omega 2 j0 . (1.35) (quantum cross section)
1.3 Dipole Radiation¶
Chapter 1. Classical Atom–Field Interactions Note that this cross section assumes an orientational average (and thus a factor of 1/3) that is generally appropriate for our purposes. We can then define the absorption oscillator strength to be the ‘‘fudge factor’’ to fix the classical cross section: f0j := \sigma0j \sigma0, classical
e2\omega 2 j0 . (1.36) (oscillator strength) We can also write the cross section as
e2 mϵ0c\gammaj , (1.37) which will be useful later. More commonly, the absorption oscillator strength is defined to include the degeneracy of the level structure,1
e2\omega 2 j0 gj g0 , (1.38) where g\alpha is the degeneracy of level \alpha (i.e., the number of ways to have energy E\alpha), with a separate expression defined for the emission oscillator strength f0j (which just flips the degeneracy ratio). Also, in the limit of large frequency, the susceptibility of Eqs. (1.32) becomes
mϵ0\omega2 X j f0j. (1.39) In this limit, the induced electron displacements are small, and thus the damping and harmonic-potential forces are not important. We thus expect to recover the behavior of the free-electron plasma in the high- frequency limit (i.e., the conductor without damping). This corresponds to Eq. (1.16) in the limit \omega0 −\rightarrow 0, since the electrons are not bound. Thus, we can write
mϵ0\omega2 . (1.40) Comparing these two expressions for the susceptibility, we find the Thomas–Reiche–Kuhn sum rule for the oscillator strength: X j f0j = 1. (1.41) (Thomas–Reiche–Kuhn sum rule) Since f0j > 0, the sum rule tells us that f0j < 1. The interpretation is that the classical cross section represents the maximum possible cross section, which turns out to be distributed over all the possible transi- tions from the ground level. Note that transitions to unbound (ionized) states are also included in this sum, making it difficult to verify this with atomic transition data.2 1.3 Dipole Radiation The electric and magnetic fields for an oscillating dipole are3
4\piϵ0
" d(+)(tr) r3 + ˙d(+)(tr) cr2 # + 4\piϵ0
¨d(+)(tr) c2r
" ˙d(+)(tr) cr2 + ¨d(+)(tr) c2r # , (dipole radiation fields) (1.42) 1Alan Corney, Atomic and Laser Spectroscopy (Oxford, 1987). 2See Peter W. Milonni and Joseph H. Eberly, Lasers (Wiley, 1988), p. 239. 3See John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), p. 411 or Peter W. Milonni and Joseph H. Eberly, Lasers (Wiley, 1988), p. 44.
1.3 Dipole Radiation¶
dipole orientation vector). Only the 1/r terms actually transport energy to infinity (i.e., they correspond to radiation), so we can drop the rest to obtain
¨d(+)(tr) r
¨d(+)(tr) r . (1.43) The energy transport is governed by the Poynting vector, which we can write as
= 16\pi2ϵ0c3 | ¨d(+)|2 r2
= ˆr 16\pi2ϵ0c3 | ¨d(+)|2 r2
- c.c., (1.44) where we have used
ˆr (1.45) for the angular dependence. There are two main possibilities for the polarization vector: the incident light can be linearly or circularly polarized.
pattern for an oscillating dipole.
\sqrt
shaped’’ radiation pattern for a rotating dipole. Here, \theta is the angle from the z-axis, while \phi is the angle around the azimuth. Note that any arbitrary polar- ization can be represented as a superposition of these three basis vectors. The (intensity/power) radiation patterns for the linear and circular dipole cases are shown here. circular dipole linear dipole x z
1.3.1 Damping Coefficient¶
Chapter 1. Classical Atom–Field Interactions The three-dimensional distributions are generated by sweeping these patterns around the z-axis. The corresponding electric fields for the dipole radiation are polarized. From Eq. (1.43), we can see
\sqrt 2), the polarization vector is proportional
\sqrt 2. Now let’s define the angular-distribution function via
8\pi
. (1.46) (radiative angular distribution) For linear and circular polarization, this takes the form
16\pi
. (1.47) This function has the nice property that it is normalized, and thus represents a probability distribution for photon emission in quantum mechanics: Z
(1.48) Here, dΩ= sin \theta d\theta d\phi is the usual solid-angle element. Now we can write the Poynting vector in terms of the angular-distribution function as
ˆr 3\piϵ0c3 | ¨d(+)|2 r2
(1.49) The power radiated per unit solid angle is then dPrad dΩ
(1.50) and the total radiated power is Prad = Z dΩdPrad dΩ
3\piϵ0c3 . (1.51) Of course, the incident intensity is contained implicitly in the electron acceleration ¨x. 1.3.1 Damping Coefficient Now we can connect the radiated power to the damping term in the Lorentz model,4 Eq. (1.8). Note that the radiated power in Eq. (1.51) is the time-averaged power, since we used the complex representation. In terms of the real displacement, we can make the replacement
2 , (1.52) where the angle brackets denote the time average. Then the average work done by radiation reaction must balance the energy emitted into the field: Z x x0
Z t t0
e2 6\piϵ0c3 Z t t0 (¨x)2 dt′ = − e2 6\piϵ0c3 " ˙x¨x
t t0 − Z t t0 ˙x...x dt′ # . (1.53) 4This argument follows Alan Corney, Atomic and Laser Spectroscopy (Oxford, 1987), p. 230.
1.4 Atom Optics: Mechanical Effects of Light on Atoms¶
1.4 Atom Optics: Mechanical Effects of Light on Atoms Here Frr refers to the radiation-reaction force. If we pick t−t0 to be an integer multiple of the optical period, the boundary term vanishes (it is also negligible for large t −t0). Then the radiation-reaction force is Frr = e2 6\piϵ0c3 ...x. (1.54) This is, in fact, the Abraham–Lorentz model of radiation reaction, which has problems involving un- physical runaway solutions.5 We can avoid such problems by going back to the complex representation and noting that the displacement is a harmonic function, so we can make the approximation F(+) rr = e2 6\piϵ0c3
(1.55) which assumes the atom is driven close to resonance. If we define
e2\omega 2 6\pimϵ0c3 , (1.56) (classical damping rate) then we recover the damping term in the harmonic oscillator: F(+) rr
(1.57)
where re \approx 2.8 \times 10−15 m is the classical electron radius and \lambda is the optical wavelength. For example, 122 nm is the lowest-lying hydrogen line, which gives \gamma/\omega0 ∼10−7. For real quantum transitions, this ratio is slightly smaller (due to the addition of the oscillator strength as we mention below), on the order of 10−8. We now have the classical result for the spontaneous emission rate, which isn’t quite correct. Again, we can patch this with the substitution e2/m −\rightarrow (e2/m)f0j, with the result
0jf0j 6\pimϵ0c3 . (1.58) (quantum damping rate)
j0/2\pi (i.e., no orientational average for the dipole). Again, there are some subtleties here regarding the cross sections and orientational averages that are better handled by angular-momentum algebra. 1.4 Atom Optics: Mechanical Effects of Light on Atoms Now we will have a brief look at the field of atom optics, or optics with matter (de Broglie) waves. We will only be looking here at how to trap and cool atoms with laser light using the classical Lorentz model of the atom, so in a sense we will be doing ‘‘geometrical atom optics.’’ Broadly speaking, there are two types of mechanical forces that light can have on atoms.6 The first, the dipole force, is related to the potential energy of the induced dipole in the electric field, and is thus related to the real part of \alpha(\omega) [see Eq. (1.66)]. The second is radiation pressure due to absorption and rescattering of the incident light, which is thus related to the imaginary part of \alpha(\omega) [see Eq. (1.85)]. 5See David J. Griffiths, Introduction to Electrodynamics, 4th ed. (Prentice-Hall, 2013), p. 489, and John David Jackson, Classical Electrodynamics, 3rd ed. (Wiley, 1999), Chapter 16, p. 748. 6Strictly speaking, this decomposition into two forces is only true for scalar atoms—atoms with no orientation. This is appropriate for the form of the polarizability we have assumed here, but in the case of atoms with nonvanishing vector or tensor polarizabilities, as in Eq. (7.472), other forces associated with polarization gradients arise. See, for example, G. Nienhuis, P. van der Straten, and S-Q. Shang, Physical Review A 44, 462 (1991) (doi: 10.1103/PhysRevA.44.462 ).
1.4.1 Dipole Force¶
Chapter 1. Classical Atom–Field Interactions 1.4.1 Dipole Force The dipole moment of the atom is induced by the external field, so we can write the potential energy of the
Vdipole = −d \cdot E = −d E 2 . (1.59) The extra factor of 1/2 compared to the usual dipole energy is because the dipole is induced, and thus Vdipole = − Z E
Z E
2\alphaE2. (1.60) Since we found the solution for the positive-frequency component of the field, we should write out the potential in terms of the same components: Vdipole = −1
\cdot
. (1.61) Noting that d(\pm) ∼e∓i\omegat, E(\pm) ∼e∓i\omegat, (1.62) we can see that the terms of the form
(1.63) rotate at twice the optical frequency, which is too fast for the atoms to respond mechanically. So we will drop these terms in the time average (the same average that leads to the intensity). The terms of the form d(\pm) \cdot E(∓) ∼1 (1.64) are dc, so we can keep these. Thus, Vdipole = −1
= −1 h
\cdot E(−) −1 h
E(+) = −\eta0
(1.65) and in terms of the intensity, Vdipole = −\eta0
(1.66) (dipole potential) Here, \eta0 is the vacuum wave impedance
rµ0 ϵ0
ϵ0c \approx 377 Ω, (1.67) (wave impedance of vacuum) and we are regarding the electric-field envelope E(+)(r) to be a slowly varying function of position. Recall that the intensity in vacuum is given in terms of the real and complex field amplitudes E0 and E(+) by
(1.68) (intensity related to field amplitude) Putting in the explicit form for the polarizability, we can write the dipole potential as Vdipole = −e2 2mϵ0c \omega 2 0 −\omega2 (\omega 2
(1.69)
1.4 Atom Optics: Mechanical Effects of Light on Atoms Thus, the atom sees a spatial potential proportional to I(r) and to (n −1). This potential-shift effect (also known as the ac Stark shift) is the atomic counterpart of the phase shift (due to n −1) of a beam propagating through a vapor. Both effects follow from the coupling of the field to the atom. The corresponding force is given by the potential gradient
(1.70) Thus, the dipole force responds to intensity gradients. If the dipole is viewed as two slightly separated, opposite charges, there is only a net force if the two charges see a different electric field, which is only possible in the ideal dipole limit if the field has a gradient. The sign of the dipole potential is set solely by the detuning of the field from the atomic resonance. Defining the detuning ∆:= \omega −\omega0, we can write the dipole potential as Vdipole = e2 2mϵ0c
(1.71) Everything in this expression is positive except for the factor of ∆in the numerator. Thus, for positive ∆ (\omega > \omega0, or blue detuning), Vdipole > 0, while for negative ∆(\omega < \omega0, or red detuning), Vdipole < 0. That is, a bright spot in space (e.g., due to a tightly focused Gaussian beam) will repel an atom for blue detunings, forming a potential barrier, while for red detunings, the spot attracts atoms and forms a potential well. The sign dependence of Vdipole makes sense in terms of the phase lag (1.20). Recall that for small frequencies (∆< 0), the phase lag of the dipole behind the field is smaller than \pi/2, while for large frequencies (∆> 0), the phase lag is between \pi/2 and \pi. Since Vdipole ∝−d \cdot E, the phase lag is important because then d and E are mostly aligned or mostly opposed for ∆< 0 and ∆> 0, respectively. Thus, Vdipole ≷0 for ∆≷0. 1.4.1.1 Dipole Potential: Standard Form By writing the dipole potential in a more standard form, we can see that it matches the result of a quantum calculation, at least in the limit of low intensity. To do this, we first need to patch the classical result of Eq. (1.69) as before by including the oscillator strength and summing over all transitions: Vdipole = − X j e2f0j 2mϵ0c \omega 2 j0 −\omega2 (\omega 2
j \omega2 I(r). (1.72) (corrected dipole potential) Now to put this in more standard form, we need to define the saturation intensity for the atom. When we encounter rate equations in the next chapter, we will see that it is sensible to define an intensity scale known as the saturation intensity, given by
2\sigma0 . (1.73) (saturation intensity) (The damping rate \gamma here will correspond to the Einstein A coefficient in the rate-equation treatment.) Briefly, the saturation intensity is relevant here in that this classical model is valid—that is, it agrees with
the D2 transition (852 nm), while for 87Rb on the same transition (780 nm), the saturation intensity is Isat = 1.67 mW/cm2. We can also write the saturation intensity in terms of the oscillator strength by using Eq. (1.37), with the result
j 2e2f0j . (1.74) Even though the above numerical values are often quoted for the saturation intensity, this is actually a context-dependent quantity. A safe but cumbersome approach is to use the quantum-mechanical formalism for angular momentum to directly calculate the cross section and thus saturation intensity.7 7Daniel A. Steck, ‘‘Cesium D Line Data,’’ 2003. Available online at http://steck.us/alkalidata.
Chapter 1. Classical Atom–Field Interactions Using Eq. (1.74), we can write the dipole potential (1.72) as Vdipole = − X j
j \omega 2 j0 −\omega2 (\omega 2
j \omega2 I(r) Isat,j . (quantum dipole potential, small intensity) (1.75) This is the general expression for any frequency, so long as the intensity is small. To simplify this, we can
Vdipole = X j
j
j0 I(r) Isat,j = X j ¯h\gamma 2 j
−
I(r) Isat,j . (1.76) (far off resonance) The first term in the parentheses is the inverse of the detuning, and represents the Stark shift due to the atomic resonances. The second term can be interpreted as the weak, additional Stark shift due to resonances at the corresponding negative frequencies. This secondary shift is always negative (like a red detuning), and accounts for part of the Bloch–Siegert shift (Section 5.9), as well as other effects such as the Lamb shift (Section 13.12) and the Casimir–Polder effect (Chapters 13-14). Note that this expression also recovers
terms contribute equal, negative energy shifts. If one resonance is dominant (that is, the laser is tuned far away from resonance, but much closer to one than all the others), then we can make the rotating-wave approximation and neglect the second term in the parentheses of Eq. (1.76) to obtain Vdipole = ¯h\gamma 2 8∆ I(r) Isat , (1.77) (far off single dominant resonance) where again ∆= \omega −\omega0 is the detuning from resonance. Note that for a far-detuned, linearly polarized laser creating this potential, it turns out that \sigma0 = \lambda 2 0 /2\pi is the appropriate resonant cross section, so the above saturation intensity values should be multiplied by 3 before being used in this formula. Typically, a focused, red-detuned, Gaussian laser beam is used to make a dipole trap or far-off resonance trap (FORT)8 for atoms via the dipole force.9 Below is an example image of about 105 87Rb atoms confined in a dipole trap formed by a 10 W, 1090 nm Gaussian laser beam (far below the 780 and 794 nm main resonances) focused to a 31 µm beam waist (1/e2 radius), implying a Rayleigh length (depth of focus along the beam direction) of 2.8 mm. The dipole trap clearly runs from left to right with a slight downward angle; the dimensions of the image are 270 \times 29 CCD pixels (6.59 \times 0.71 mm). This is an absorption image, where the shadow cast by the atoms in a brief, resonant laser probe is imaged and recorded on a CCD camera. (The image greyscale is inverted so the atoms appear bright rather than dark.) But now the important question to address is under what conditions the trap is stable, since as the atom scatters photons, it heats up until it boils out of the trap. So we will need to take a closer look at the radiation of the Lorentz atom. 8Steven L. Rolston, Christoph Gerz, Kristian Helmerson, P. S. Jessen, Paul D. Lett, William D. Phillips, R. J. Spreeuw, and C. I. Westbrook, ‘‘Trapping atoms with optical potentials,’’ Proceedings of SPIE 1726, 205 (1992) (doi: 10.1117/12.130392). 9The first observation of atoms trapped in a dipole-force potential was Steven Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, ‘‘Experimental Observation of Optically Trapped Atoms,’’ Physical Review Letters 57, 314 (1986) (doi: 10.1103/Phys- RevLett.57.314).
1.4 Atom Optics: Mechanical Effects of Light on Atoms 1.4.1.2 Photon Scattering Rate Now we can compute the rate of photon scattering as a way to get to the rate of momentum transfer. We can write the total radiated power from Eq. (1.51) in terms of the polarizability as
6\piϵ 2 0 c4 I, (1.78) (total radiated power)
e−i\omegat. As a brief aside, though, we can write down the scattering cross section from the total radiated power, given the defining relation Prad = \sigmaI:
6\piϵ 2 0 c4 . (1.79) (Rayleigh scattering cross section) The overall scaling is as \omega4 (neglecting the small modification due to the polarizability). This is the usual explanation for why the sky is blue and sunsets are red: blue wavelengths are preferentially scattered by the atmosphere, while red wavelengths are preferentially transmitted. We can continue by writing out explicitly the polarizability in Eq. (1.78), using Eq. (1.32): Prad = e2\omega4 6\pim2ϵ 2 0 c2
X j f0j \omega 2
I. (1.80) Using Eq. (1.58) to eliminate the oscillator strengths,
X j \omega2 \omega 2 j0 \gammaj \omega 2
I = ¯h
X j \omega2 \sqrt\omegaj0 \gamma3/2 j \omega 2
s I Isat,j
, (1.81) where we used \sigma0j = 3\lambda 2 0 /2\pi to write the saturation intensity as
2\sigma0j
j0\gammaj 4\pic2 . (1.82) The photon scattering rate Rsc is the radiated power divided by the photon energy ¯h\omega: Rsc = Prad
X j \omega3/2 p2\omegaj0 \gamma3/2 j \omega 2
s I Isat,j
. (1.83) (photon scattering rate) Again, this expression simplifies greatly for certain detunings. Near one resonance, we can ignore the contribution of the others: Rsc \approx \omega3 2\omega0 \gamma3 |\omega 2
I Isat = \omega3 2\omega0 \gamma3 (\omega 2
I Isat (1.84) Using Eq. (1.32) restricted to a single resonance, we find
¯h \omega2 \omega 2 Im[\alpha]I(r), (1.85) (single dominant resonance) which shows the connection of the scattering rate (and hence the radiation pressure force below) to the absorptive part of the polarizability. But as we have seen, this is only true near a single resonance.
Chapter 1. Classical Atom–Field Interactions 1.4.1.3 Optical Theorem Actually, what we just said isn’t quite true: the scattering rate can be generally written in terms of the imaginary part of the polarizability. The problem is actually with the Lorentz model, when we tried to mock it up to model multiple transitions. To see this, we start with the susceptibility expression (1.17),
write
2ϵ0
(1.86) Then the absorption coefficient from (1.28) for the vapor is
(1.87) The power absorbed from an incident, monochromatic field of intensity I(\omega) by the atomic vapor is given
absorbed is
N
(1.88) For energy to be conserved, the power emitted from Eq. (1.78) must match this expression for the power absorbed. Equating these, we arrive at the optical theorem
4\piϵ0 2\omega3
(1.89) (optical theorem) Of course, this statement of energy conservation in principle only applies in steady state (since we neglect energy stored in the atom), but we have already implied we are considering steady-state behavior just by writing down the polarizability. Thus, the result above from Eq. (1.85),
¯h \omega2 \omega 2 Im[\alpha]I(r), (1.90) (general scattering rate) is generally true, and it emphasizes that the imaginary part of the polarizability represents absorption or loss, and equivalently scattering. We will return to this statement later in Section 14.1.4.1 and see that the imaginary parts of generalized response functions always represent loss. However, as we have already noted, the Lorentz polarizability for multiple transitions (‘‘electrons’’) does not satisfy the optical theorem.10 We can see this by looking at Eq. (1.83), where the terms in the sum [representing the same sum in \alpha(\omega)] are combined and then squared, so that there are cross terms involving different resonances. But when considering the equivalent expression in terms of Im[\alpha(\omega)], this scattering rate involves a sum over terms of the form of Eq. (1.84), which always involve a single resonance. There are no cross terms in this case. In the first case, the cross terms reflect the fact that the fields radiated on each transition can interfere, which is why field amplitudes are added and then squared in Eq. (1.83). This is apparently not captured in the imaginary part of \alpha(\omega), which indicates a defect in our polarizability expression. This problem is somewhat academic, however, since optical atomic resonances have separations much larger than their widths, and the difference between the two expressions is usually negligible. For practical purposes, the Lorentz polarizability (1.32) is just fine. 10For a discussion of quantum-mechanical polarizabilities and their compatibility with the optical theorem, see Paul R. Berman, Robert W. Boyd, and Peter W. Milonni, ‘‘Polarizability and the optical theorem for a two-level atom with radiative broadening,’’ Physical Review A 74, 053816 (2006) (doi: 10.1103/PhysRevA.74.053816).
1.4.2 Radiation Pressure¶
1.4 Atom Optics: Mechanical Effects of Light on Atoms 1.4.1.4 Scaling¶
we find that Rsc \approx \gamma3 8∆2 I Isat = \gamma ¯h∆Vdipole, (1.91) (far off single dominant resonance) where in the last formula we have used Eq. (1.77) for the dipole potential. This is a fundamentally important scaling law for making productive use of dipole forces, as we will now discuss. The result (1.91) for Rsc, along with (1.77) for Vdipole, are of prime importance in the design of an optical dipole trap. The photon scattering rate represents heating of the atoms, as we will discuss, because of the random nature of photon emission. But the scattering rate and dipole potential scale as Rsc ∝I ∆2 ; Vdipole ∝I ∆. (1.92) (far off single dominant resonance) These scaling laws are actually interrelated by a nice qualitative argument. From the scaling of Rsc, we conclude that the radiated field is proportional to \sqrt I/∆. Now recall that the dipole potential scales as the phase refractive index n −1, which at a microscopic level arises due to the interference of the radiated field with the forward field (thus changing the phase velocity of the transmitted field), whose amplitude is \sqrt I. Hence n −1 and thus the dipole potential scale as ( \sqrt I/∆) \sqrt I, or as I/∆. So, for a given desired potential depth, the scattering (heating) rate can be made very small by making the detuning ∆large. The resulting decrease in trap depth is compensated by increasing the intensity. Thus, dipole traps with small heating rates and hence long lifetimes (up to minutes for dipole traps created by CO2 laser light) can be created in this way. For example, a CO2-laser dipole trap has been realized for Cs atoms,11 where a beam waist of 110 µm with 25 W of power gives a trap 120 µK deep, deep enough to confine laser-cooled atoms. The intensities here are incredibly high compared to the saturation intensity
In this regime the above scaling laws are invalid, and in fact the laser field can be treated as static to good approximation. Note that for a linearly polarized, far-detuned dipole trap where these scalings are valid, the same saturation intensities are to be used to calculate the dipole force and scattering rate, as discussed above. 1.4.2 Radiation Pressure Now we will examine the forces due to absorption and reemission of the incident light. Each photon carries a momentum of ¯hk. Thus, the photon scattering rate (1.84) implies a rate of momentum transfer and thus a force due to radiation pressure of Frad = ¯hkRsc, (1.93) Note that even though we have invoked the concept of the photon here, which will be convenient when discussing the heating rate, everything here is really classical, since the classical field momentum is related to the absorbed beam power: F = dp/dt = Pabs/c = \sigmaI/c. To get a sense of scale of the momenta involved, we can compute the recoil velocity vr, defined as the velocity corresponding to one photon recoil momentum ¯hk: vr = ¯hk m . (1.94) For 133Cs at 852 nm, vr = 3.5 mm/s, and for 87Rb at 780 nm, vr = 5.9 mm/s, so the recoil velocity is orders of magnitude smaller than typical room-temperature velocities. Close to a single resonance, the scattering rate from Eq. (1.84) is Rsc = (\gamma/2)3
I Isat , (1.95) 11H. Engler, T. Weber, M. Mudrich, R. Grimm, and M. Weidemüller, ‘‘Very long storage times and evaporative cooling of cesium atoms in a quasielectrostatic dipole trap,’’ Physical Review A 62, 031402 (2000) (doi: 10.1103/PhysRevA.62.031402).
1.4.3 Laser Cooling: Optical Molasses¶
Chapter 1. Classical Atom–Field Interactions so that the force due to radiation pressure becomes Frad =
I Isat . (1.96) (radiation pressure) Again, depending on the polarization and exactly how close the detuning is (i.e., whether or not the hyperfine structure is resolved), the appropriate value of the saturation intensity might be very different, so some caution is necessary in applying these formulae. 1.4.3 Laser Cooling: Optical Molasses Now let’s explore how we can use the radiation-pressure force to cool atoms. The simplest setup we can consider is an atom moving with velocity v, exposed to identical but counterpropagating laser fields along the velocity direction. laser 1 laser 2 v The radiation-pressure force on the atom due to the two fields from Eq. (1.96) is
∆2
∆2
I Isat , (1.97) where ∆1,2 are the effective detunings of the two lasers. The detunings of the two lasers are the same in the laboratory frame, but the idea behind Doppler cooling is to tune the lasers below the atomic resonance, so that the beam that opposes the atomic velocity is Doppler-shifted into resonance, thus tending to stop the atom. With the pictured setup, the frequency of laser 1 is Doppler shifted (red shifted) by −kv, while the frequency
can write ∆1 = ∆−kv
(1.98) where ∆= \omega −\omega0 is the detuning in the laboratory frame. Then the force is
I Isat . (1.99) Regarded as a function of velocity, this expression is the difference of two Lorentzians, each displaced by |∆|/k from zero velocity. This force is plotted for ∆= −\gamma/2, and the two offset Lorentzians are shown as dashed lines. F \pm|D|/kº rad v
1.4 Atom Optics: Mechanical Effects of Light on Atoms¶
(‘‘friction’’) force:
∆
I Isat v. (1.100) (optical molasses, small v) Because this force is damping for ∆< 0, typically leading to heavily overdamped motion for trapped atoms, this light configuration is called optical molasses.12 The maximum damping rate occurs for ∆= −\gamma/2 \sqrt 3, although it turns out that the optimal detuning is actually something else for reasons we will soon discuss. The velocity capture range is the range in velocity for which the force is appreciable. Thus, the capture range is on the order of \pm|∆| k ∼\pm \gamma
4\pi , (1.101) (capture velocity range)
at all with this method. Traditionally to load atomic traps, atoms were slowed by other methods from hot atomic beams to below the capture velocity and then trapped. However, it is possible to load a trap from room-temperature vapor with this method by capturing only the small fraction of the atoms with small enough velocity. 1.4.3.1 Doppler Cooling Limit For laser cooling in three dimensions, it is sufficient to simply combine three of the above one-dimensional setups, one along each axis.13 Then we can write the force vector for small velocities as,
∆
I Isat v. (1.102) where I is still the intensity of a single beam. Our treatment so far makes it appear as though the atomic velocity may be damped completely away. However, we have only considered the average cooling force. There are also fluctuations of the cooling force that lead to a temperature limit. We will now derive this temperature limit, the Doppler limit, for the cooling mechanism presented here. Let’s look at the variance of the velocity distribution: d dt
v2 = 2
v \cdot dv dt
= mA
v \cdot dp dt
= mA
(1.103) 12Steven Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, ‘‘Experimental Observation of Optically Trapped Atoms,’’ Physical Review Letters 57, 314 (1986) (doi: 10.1103/PhysRevLett.57.314). 13Graphics by Windell Oskay.
Chapter 1. Classical Atom–Field Interactions Here, mA is the atomic mass, and the angle brackets denote an ensemble average. With the small-velocity expression (1.100) for the average cooling force, this equation of motion becomes d
mA ∆
I Isat
(1.104) Again, according to this differential equation, the velocity damps to zero for ∆< 0. Now we will include the force fluctuations heuristically, since the fluctuations are quantum–mechan- ical in origin (although there is a more general connection between damping and fluctuations known as the fluctuation–dissipation relation; see Problem 5.26 and Section 14.3.8.1). In the course of scattering a photon from one of the laser beams, there is a photon absorption and a photon emission. Each absorption leads to a momentum ‘‘kick’’ of magnitude ¯hk, and the direction is random but along one of the six beams. The emission is also in a random direction (not in a dipole-radiation pattern if we assume all polarizations to be equally present), leading to a second kick of magnitude ¯hk in a random direction. Thus, a scattering event is effectively equivalent to two steps in a random walk in velocity space, where the step size is ¯hk/mA. These scattering events happen at the scattering rate Rsc = (\gamma/2)3
6I Isat , (1.105) since there are six beams present. Recall that for a random walk (see Section 17.1, p. 697), each step increases \langle v2\rangle by a fixed amount, given by the variance after one step starting from the origin. The three-dimensional probability distribution for a single scattering event is confined to a shell of radius ¯hk/mA in velocity space for either the absorption or emission event. The probability distribution is also inversion symmetric in either case. Thus if the atom is initially at rest, then after one step in the random walk, we can write \langle v2 initial\rangle = 0 −\rightarrow \langle v2
¯hk mA 2 , (1.106) so that \langle v2\rangle increases at the rate 2Rsc ¯hk mA 2 . (1.107) Including the heating rate in Eq. (1.104), we find d
mA ∆
I Isat
I Isat ¯hk mA 2 . (1.108) In steady state, we can set the right-hand side to zero, with the result
4mA
. (1.109) This is an expression for the equilibrium kinetic energy, which we can convert to a temperature via
2kBT, (1.110) where kB is the Boltzmann constant. This gives
. (1.111) The temperature is minimized for the detuning ∆= −\gamma/2, giving the Doppler temperature TD:
2 . (1.112) (Doppler limit)
1.4 Atom Optics: Mechanical Effects of Light on Atoms This temperature is the best expected for Doppler cooling. For 133Cs at 852 nm, TD = 125 µK, and for 87Rb at 780 nm, TD = 146 µK. These temperatures are extremely low. We can compare these temperatures to the recoil temperature Tr, which is the temperature corresponding to atoms with an average momentum of one photon recoil ¯hk (i.e., a one-dimensional rms momentum of one photon recoil):
mA . (1.113) (recoil temperature) For 133Cs, Tr = 198 nK, and for 87Rb, Tr = 362 nK, so the Doppler limit is TD = 631 Tr for 133Cs and
vrms = r TD Tr ¯hk mA , (1.114) which is 8.8 cm/s for 133Cs and 12 cm/s for 87Rb. These velocities are about three orders of magnitude slower than room-temperature rms velocities. It turns out that for alkali vapors, typical laser-cooled samples exhibit temperatures well below the Doppler limit. Such ‘‘sub-Doppler’’ cooling is due to the degenerate level structure of alkali atoms.14 For example, 133Cs can be laser cooled with the same general setup described above to about 2.5 µK.15 1.4.3.2 Magneto-Optical Trap Optical molasses tends to stop atoms, making them ‘‘stuck,’’ but it does not confine atoms to a particular place. A slight modification to the three-dimensional optical molasses is to impose the magnetic field due to two opposed current loops in the ‘‘anti-Helmholtz’’ configuration. This arrangement is called the magneto- optical trap (MOT).16 The magnetic field vanishes at the center point of the trap, thus defining a point for atoms to accumulate. We will not go into the operation of the trap in detail yet, but essentially the idea is very similar to laser cooling. The additional complication is that the laser beams must all be correctly (circularly) polarized to address magnetic substates in the degenerate excited level. The magnetic field gives a position-dependent ‘‘Zeeman’’ shift of the transition frequency, such that if the atom is away from the center of the trap, the appropriate beam comes into resonance and pushes it towards the trap center.17 14Sub-Doppler temperature were observed in some of the first laser cooling experiments. The first reported observation is P. D. Lett, W. D. Phillips, S. L. Rolston, C. E. Tanner, R. N. Watts, and C. I. Westbrook, ‘‘Optical molasses,’’ Journal of the Optical Society of America B 6, 2084 (1989). A classic treatment of sub-Doppler cooling mechanisms is J. Dalibard and C. Cohen-Tannoudji, ‘‘Laser cooling below the Doppler limit by polarization gradients: simple theoretical models,’’ Journal of the Optical Society of America B 6, 2023 (1989). 15C. Salomon, J. Dalibard, W. D. Phillips, A. Clairon, and S. Guellati, ‘‘Laser Cooling of Cesium Atoms below 3 µK,’’ Europhysics Letters 12, 683 (1990). 16Graphics by Windell Oskay. 17Jean Dalibard proposed the idea for the magneto-optical trap, and the MOT was first demonstrated by E. L. Raab, M. Prentiss, Alex Cable, Steven Chu, and D. E. Pritchard, ‘‘Trapping of Neutral Sodium Atoms with Radiation Pressure,’’ Physical Review Letters 59, 2631 (1987) (doi: 10.1103/PhysRevLett.59.2631).
1.5.1 Atom-Mirror Interaction¶
Chapter 1. Classical Atom–Field Interactions 1.5 Cooperative Radiation We will close our classical treatment with ‘‘cooperative effects,’’ where the radiation of an atom is influenced by other atoms. The two examples here serve as prototypes for the quantum-mechanical problems and to show how far one can get with classical arguments, as in the atom optics presentation above. 1.5.1 Atom–Mirror Interaction The first effect we consider is the influence of a perfectly conducting plane on a radiating dipole.18 This is a classical prototype for both cavity QED and the Casimir–Polder effect, which we will study much later. The setup is a dipole located a distance z from a mirror. The field due to the boundary is equivalent to that of an image dipole at position −z; due to the relative locations of the constituent charges, the image dipole is reflected in the direction transverse to the plane, but the orientation is the same in the orthogonal direction. z For a dipole oscillating at frequency \omega, the field amplitude is
4\piϵ0
1 r3 −i k r2
4\piϵ0
(monochromatic dipole field) (1.115) with \omega = ck. If we regard the image as the radiating dipole, we are interested in the field at the position of the atom, so that r = 2z and ˆr = ˆz. Also, it is useful to consider separately the cases of a dipole parallel to the surface,
(1.116) and a dipole perpendicular to the surface,
(1.117) where recall that ˆ\epsilon, the polarization vector of the applied field, is also the unit vector representing the dipole direction. Since we are concerned with the image and not the original dipole, we make the replacements ˆ\epsilon∥−\rightarrow −ˆ\epsilon∥and ˆ\epsilon⊥−\rightarrow ˆ\epsilon⊥, as we can see from the above diagram. Thus, the field due to the surface at the atom is E(+)
k3 4\piϵ0 (2kz)3 −i (2kz)2 h 2d(+)
∥ (\omega) i ei2kz − k3 4\piϵ0 2kz d(+) ∥ (\omega)ei2kz = −3 2\gamma k3 k 2 mc e 1 z′3 −i 1 z′2 h 2x(+)
∥ (\omega) i −1 z′ x(+) ∥ (\omega) eiz′, (1.118) 18See, e.g., H. Morawitz, ‘‘Self-Coupling of a Two-Level System by a Mirror,’’ Physical Review 187, 1792 (1969) (doi: 10.1103/PhysRev.187.1792).
1.5 Cooperative Radiation where recall that d = −ex, z′ := 2kz, (1.119)
Now we put this field back into the the Lorentz model (1.18) as the driving field:
mE(+) mirror(z, \omega)e−i\omegat. (1.120) We consider no other driving field, since we are simply interested in how the atom damps to equilibrium. Assuming again a solution of the form
e−i\omegat, (1.121) we have \omega 2
mirror(z, \omega0) mx(+) . (1.122)
can write \omega0 for \omega when it is isolated), we find
mirror(z, \omega0)
. (1.123) Since \omega is the rotation frequency of the dipole, the real part corresponds to the actual rotation (energy), while the imaginary part corresponds to damping. We can thus see that the mirror field induces shifts \delta\omega0 and \delta\gamma in the oscillator frequency and damping rate, respectively, according to
, (1.124) so that
"
mirror(z, \omega)
¶
"
mirror(z, \omega)
. (1.125) Evaluating these expressions for a perpendicular dipole, we find¶
2\gamma sin z′ z′2 + cos z′ z′3
cos z′ z′2 −sin z′ z′3 . (1.126) (dipole-mirror shifts) For the parallel dipole,
4\gamma 1 z′ −1 z′3 cos z′ −sin z′ z′2
2\gamma 1 z′ −1 z′3 sin z′ + cos z′ z′2 . (1.127) (dipole-mirror shifts) Note that here that in the near-resonant approximation, z′ = 2k0z. (1.128) The frequency shift (corresponding to a transition-frequency shift of a quantum-mechanical atom), is plotted here.
Chapter 1. Classical Atom–Field Interactions 2kºz 0.2 -0.4 dwº/g » ^ This effect can be interpreted as an ac Stark shift of the atom due to its own radiated field. Note that only the parallel component has a radiative (1/z′) component. In the near field, the frequency shift becomes very large until the dipole approximation breaks down. The decay rate has similar oscillatory behavior, but the prediction is not divergent as the atom approaches the wall. 2kºz » ^
Notice that the decay rate drops to zero as z −\rightarrow 0 for the parallel component; for this component, the dipole and its image are out of phase when the dipole is close to the mirror, and so there is complete destructive interference of the radiated field. For the perpendicular component, the decay rate increases to twice the free-space value as z −\rightarrow 0; for this component the dipole and image are in phase near the mirror surface, and there is constructive interference of the radiated and image fields, leading to a sort of superradiance. It is possible to combine the expressions (1.126) and (1.127) into a more compact form that will be useful for later comparison as follows. Noting that
4\gamma cos z′ z′
4\gamma 1 z′ −2 z′3 cos z′ −2sin z′ z′2 = −3 4\gamma\partial 2 z′ cos z′ z′ , (1.129) we see that we can write the total shift as
4\gamma h ˆ\epsilon 2
⊥ − ˆ\epsilon 2
⊥ \partial 2 z′ i cos z′ z′ . (transition frequency shift) (1.130)
1.5.2 Two-Atom Radiation¶
1.5 Cooperative Radiation Similarly, since¶
2\gamma sin z′ z′
2\gamma 1 z′ −2 z′3 sin z′ −2cos z′ z′2 = 3 2\gamma\partial 2 z′ sin z′ z′ , (1.131) we have
2\gamma h ˆ\epsilon 2
⊥ − ˆ\epsilon 2
⊥ \partial 2 z′ i sin z′ z′ . (1.132) (damping-rate shift) The polarization combinations have the following interpretation. Note that the combination ˆ\epsilon 2
⊥van- ishes if the dipole has all three components equally excited (i.e., isotropic excitation), since it is proportional
Thus, ˆ\epsilon 2
⊥gives the anisotropic part of the dipole excitation, while ˆ\epsilon 2
⊥gives the isotropic con- tribution to the dipole excitation (isotropic here is with respect to the parallel and perpendicular parts). The above forms (1.130) and (1.132) are somewhat nonsensical in the classical framework, since the different shifts in the different directions lead to precession of the dipole vector; however, the interpretation is more sensible when we go over to quantum mechanics and interpret the polarization combinations as dipole matrix elements (Chapter 14). But for now, we can use these forms to arrive at the compact expressions
8\gamma 1 −\partial 2 z′ cos z′ z′
4\gamma
z′ cos z′ z′
4\gamma 1 −\partial 2 z′ sin z′ z′
2\gamma
z′ sin z′ z′ (1.133) for the separate shifts. 1.5.2 Two-Atom Radiation A similar problem to the atom-wall interaction is the problem of two coupled atoms. (See Problem 6.7 for the quantum version of this problem.)
x y z It is convenient again here to decompose the dipoles into parts that are perpendicular or parallel to the z = 0 plane that separates to the two atoms. This is because the field due to the parallel part of one dipole couples only to the parallel part of the other, and the same is true for the perpendicular parts. This follows from the polarization properties of the radiated field. The field of dipole 1 at the location of dipole 2 has a form similar to that of (1.118), but now without the reversal of ˆϵ∥, and with a separation of r. E(+)
e (k0r)3 −i (k0r)2 h 2˜x(+)
∥ (t) i − k0r ˜x(+) ∥ (t) eik0r, (1.134)
Chapter 1. Classical Atom–Field Interactions Here, we have assumed that the dipoles are oscillating at approximately their common resonance frequency \omega0, justifying the decomposition x(+)
⊥,∥(t)e−i\omega0t, (1.135) where ˜x(+) ⊥,∥(t) is a slowly varying amplitude, so that
⊥,∥(t)|. (1.136) Then we can note that the approximate derivatives have the form
(1.137) Using the Lorentz model (1.18) again, with the above expressions for the time derivatives, we can write the equations of motion for the components of the second dipole reacting to the first dipole as ˙˜z(+)
2 ˜z(+) = −i3 2\gamma (k0r)3 −i (k0r)2 eik0r˜z(+)
˙˜x(+)
2 ˜x(+) = −i3 2\gamma (k0r)3 −i (k0r)2 − k0r eik0r˜x(+)
. (1.138) Here, we are writing out the explicit displacement components in coordinates, so that z is the perpendicular component, and x and y are the parallel components. Of course, ˜y(+) satisfies an equation of the same form as ˜x(+) . Similarly, the equations of motion for the first dipole become ˙˜z(+)
2 ˜z(+)
˙˜x(+)
2 ˜x(+)
, (1.139) leading to pairs of coupled equations for each vector component of the displacement.19 For any component, we thus have a pair of coupled equations of the form
(1.140) We can solve these by the method of Laplace transforms. The Laplace transforms of the equations are
(1.141) which we can decouple and solve. For example, the solution for \alpha is
Ω\beta0
(1.142) 19Note that we are ignoring the time delay in the propagating waves between the two dipoles, which leads to ‘‘signaling’’ behavior between the two atoms. This amounts to a coarse-graining on time scales of order \gamma−1, and is valid as long as r ≪c/\gamma. For alkali atoms, \gamma−1 ∼30 ns, so this approximation is good as long as r ≪10 m. See P. W. Milonni and P. L. Knight, ‘‘Retardation in coupled dipole–oscillator systems, American Journal of Physics 44, 741 (1976) (doi: 10.1119/1.10122).
1.5 Cooperative Radiation The inverse transform gives the solution in terms of the initial values \alpha0 and \beta0:
(1.143) The first term here represents an antisymmetric or out-of-phase component in the initial excitation, while the second term represents the symmetric or in-phase component. In view of the decomposition (1.135), we can see that the real and imaginary parts of the exponential frequencies are significant as shifts in the frequency and damping rate, as in the atom–mirror problem. In particular,
(1.144) Here, the + subscript refers to the in-phase part, and the −subscript refers to the out-of-phase part. From Eq. (1.138), we can write the shifts as
2\gamma sin k0r (k0r)2 + cos k0r (k0r)3 = ∓3 4\gamma
r′ cos r′ r′
cos k0r (k0r)2 −sin k0r (k0r)3 = \pm3 2\gamma
r′ sin r′ r′ (1.145) for the perpendicular components, and
4\gamma 1 k0r − (k0r)3 cos k0r −sin k0r (k0r)2 = ∓3 8\gamma 1 −\partial 2 r′ cos r′ r′
2\gamma 1 k0r − (k0r)3 sin k0r + cos k0r (k0r)2 = \pm3 4\gamma 1 −\partial 2 r′ sin r′ r′ (1.146) for the parallel components. To shorten the notation, we have used r′ = k0r here. Also, the y component again satisfies an equation of this same form. These relations have the same form as the atom–mirror shifts of Eqs. (1.126) and (1.127), in the sense that for the perpendicular component, the + solution matches the atom–mirror result, while for the parallel component, the −solution matches the atom–mirror case. This is what we expect from the phases of the image-dipole components relative to the source dipole. The shifts for both components and both relative phases are plotted below. kºr 0.4 -0.4 dwº/g
»,+ »,- For the frequency (energy) shift, we can see that for small distances and in-phase dipoles, the perpendicular (z-z) orientation produces an attractive potential, while the parallel (x-x) orientation produces a repulsive potential, as we expect for static dipoles.
Chapter 1. Classical Atom–Field Interactions kºr
»,+ »,- For the decay-rate shift, we see superradiant behavior for either orientation when the dipoles are in phase and subradiant behavior when the dipoles are out of phase. This is what we expect from the respective constructive or destructive interference of the emitted waves. Note that again we can write all components together with both the symmetric and antisymmetric phases, here in coordinate-free form:
8\gamma
r′
1 −\partial 2 r′ cos r′ r′
4\gamma
r′
1 −\partial 2 r′ sin r′ r′ . (two-atom resonance and damping-rate shifts) (1.147) In the far-field (the radiation zone), these simplify to
r′
r′ , (1.148) since in this regime, only the 1/r′ terms are important.
1.6 Exercises¶
1.6 Exercises 1.6 Exercises Problem 1.1 Estimate the absorption coefficient of room-temperature rubidium (87Rb) on the D2 resonance at 780 nm as follows. (a) Write down an expression for the one-dimensional atomic velocity distribution in the direction of the pumping laser beam. What is the rms velocity? Recalling that the Doppler shift is given by ∆\omega = k0v, where v is the velocity component opposite to the pumping-beam direction, what is the corresponding full width at half maximum ∆\omegaD in frequency (the Doppler width)? (b) Now write down an expression for the absorption coefficient that accounts for the Doppler broad- ening effect. The result involves a convolution of the frequency distribution from (a) with the natural lineshape, but you only need the result at one frequency. Noting that 1/\gamma = 26.2 ns, you should argue that \gamma ≪∆\omegaD to simplify your calculation. (c) Use the above values for \gamma and \lambda0 to compute the absorption oscillator strength fD2 for this transition. Note that this is a J = 1/2 −\rightarrow J′ = 3/2 transition, so that the degeneracy ratio g′/g = 2 (the prime here refers to the excited level). (d) Now give a numerical value for the absorption coefficient. Assume a vapor pressure for Rb of 3 \times 10−7 torr at room temperature. Note that the relative abundance of 87Rb is about 28% (the rest being 85Rb). (e) The answer to (d) does not predict the absorption coefficient well, because it assumes that the Doppler width is much larger than the splittings of the various hyperfine levels. In fact, this is only marginally true for the excited states, which span an effective range of 400 MHz. The ground-state doublet is in fact resolved at room temperature, since the splitting is 6.8 GHz. For what temperature range would you expect the above treatment to become valid? We will return to the room-temperature case when we have the formalism to properly handle the hyperfine structure. Problem 1.2 The radiation of an atom is collected and collimated by a lens of radius a and focal length f, reflected by a planar mirror, and imaged back onto the atom by the lens. f
As in the direct atom–mirror interaction, the interaction of the atom with the distant mirror here causes a shift of the energy level that changes sinusoidally as the mirror is translated. This leads to potential wells for the atom due to its interaction with its own radiated field.20 (a) Give a qualitative explanation for why the sign of the potential varies with the distance to the mirror, and from your explanation predict the period of the variation. (b) Using the classical model of the atom, give an expression for the depth of the potential wells near the focus (the maximum energy shift, in this case). Also assume for simplicity that the atom radiates 20This mechanical effect of a distant mirror on an atom has been observed experimentally with a trapped Ba ion. See Pavel Bushev, Alex Wilson, Jürgen Eschner, Christoph Raab, Ferdinand Schmidt-Kaler, Christoph Becher, and Rainer Blatt, ‘‘Forces between a Single Atom and Its Distant Mirror Image,’’ Physical Review Letters 92 223602 (2004).
Chapter 1. Classical Atom–Field Interactions in a spherically symmetric pattern, and that the lens is small enough that the radiation intensity is uniform in the plane of the lens. It will help to know the following result from diffraction theory: suppose a converging, spherical wave has radius of curvature f and a uniform intensity Iaperture over some aperture of radius a; then the maximum intensity is given by
\lambda2f 2 Iaperture (1.149) where the wave converges.21 (c) Use parameters appropriate to 87Rb and assume the lens has a numerical aperture of 0.4 to give a numerical value to your answer in (b). Report your answer as a temperature. Problem 1.3 Compute the frequency and decay-rate shifts for a Lorentz atom situated between two parallel mirrors separated by distance L and located a distance a from the closest mirror. You may assume that the dipole is oriented perpendicular to the mirrors. Make plots of your results for the case a = L/2. Hint: how many image dipoles do you need to use? (The answer isn’t two.) Problem 1.4 Consider a vapor of atoms (with number density N), where the atoms are described by the Lorentz polarizability
e2/m \omega 2
(1.150) We will use this model to explore the physics of free-electron gases, as in plasmas or metals, which corresponds to the limit \omega0 −\rightarrow 0. (Note that metals have density-induced correlations that are missed by the models here.) (a) First ignoring any damping (\gamma −\rightarrow 0), show that the complex refractive index for the free electron gas is
r 1 − \omegap \omega 2 , (1.151)
p Ne2/mϵ0 is the plasma frequency. This is the plasma model for the refractive index. (b) At an interface of the electron gas with vacuum (say, a metal–vacuum interface), the intensity (Fresnel) reflection coefficient for monochromatic light incident from the vacuum on the interface is
1 −˜n(\omega)
, (1.152) assuming normal incidence to the interface. This expression can be derived by imposing appropriate boundary conditions on the electromagnetic fields, but here you may take this expression to be given.
results imply for the reflectances of metal mirrors in the infrared and the deep UV, assuming a plasma frequency in the UV range? (c) Now put the damping back in, to account for radiation reaction, electron collisions, and so on, and show that the complex refractive index becomes
s 1 − \omega 2 p
(1.153) 21See Max Born and Emil Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge, 1999), Eq. (22), p. 489.
1.6 Exercises This is the Drude–Lorentz model for the refractive index. (d) Consider the limit of small frequency. In terms of the reflectance, how is the Drude–Lorentz model more physically sensible than the plasma model? Show as well that the Drude–Lorentz model reduces to the plasma model at high frequencies. (e) The current density induced by the field is j = −Ne ˙x. (1.154) Defining the conductivity of the free-electron gas by the relation j = \sigmaE, show that the conductivity is given according to the Drude–Lorentz model by
\sigma0
(1.155) where \sigma0 is the dc conductivity. What is \sigma0? Problem 1.5 We used the Lorentz model to describe the linear response of an atom to the field, but it can be modified to describe nonlinear optical media. Anharmonic terms added to the electron binding potential, for example, can describe basic nonlinear optics, but here we will explore nonlinear responses in free- electron gases.
However, the full Lorentz force, including the magnetic-field interaction, is F = −eE −ev \times B. The extra magnetic-field interaction is responsible for a nonlinear response at very large intensities. The classical equation of motion for a free electron (ignoring damping) is thus m ˙v = −eE −ev \times B. (1.156) Assume the electron is driven by a linearly polarized, monochromatic plane wave
(1.157) with B = ˆk \times E/c as required by Maxwell’s equations. Now make a Fourier-series ansatz of the form
\infty X j=1
(1.158) Put this form for v(t) into the electron equation of motion, ignoring the effect of the magnetic field, and obtain a zeroth-order perturbative solution for v(+)(t). (b) Put the zeroth-order solution back into the right-hand side of the equation of motion, and obtain the next-order solution for v(+)(t), this time including the effect of the magnetic-field coupling. Write down the corresponding induced dipole moment at frequency 2\omega. The nonlinear response thus allows second-harmonic generation from the driving field. In which direction is there no second-harmonic radiation? (Note that you should be able to interpret your solution as an electron tracing out a ‘‘figure eight.’’) (c) Estimate the intensity at which the second-harmonic dipole moment becomes comparable to the dipole moment at the fundamental frequency. (d) Now instead of the magnetic-field interaction, consider the relativistic interaction with the electric field, given by ˙p = d
dt mv r 1 −v2 c2 = −eE, (1.159)
Chapter 1. Classical Atom–Field Interactions where \gamma is the usual relativistic factor and the driving field is the same as before. Show that the second harmonic vanishes in this case. (e) Obtain a perturbative expression for the relativistic dipole moment at frequency 3\omega. (f) At what intensity is the third-harmonic relativistic dipole comparable to the fundamental dipole? Problem 1.6 We have already seen how the field responds to an atomic vapor, given the Lorentz model for the atoms: the real part of the polarizability results in a phase shift (dispersion), while the imaginary part leads to absorption. In this problem, you will explore a general classical model for how the electromagnetic field responds to the polarization field P(r, t) of a medium. The polarization, or the dipole moment per unit volume, is the macroscopic generalization of the atomic dipole moment. (a) Use Maxwell’s equations in a source-free, dielectric medium,
(1.160) where B = µ0H, (1.161) and the electric fields are related by
(1.162) to derive the polarization-driven wave equation \nabla 2E −1 c2 \partial 2 t E = ϵ0c2 \partial 2 t P. (1.163) You may assume the polarization field to be transverse, i.e., \nabla \cdot P = 0. Generally speaking, the polarization is induced by the field, but here we may view it as an independent object acting to modify the field. (b) Now assume the fields have the form
(1.164) where \phi(z, t) is a slowly varying phase, E0(z, t) is a slowly varying (real) field amplitude, and P (+) (t) is a slowly varying (complex, varying in time only) polarization amplitude. Of course, due to a possible phase lag of the medium, the polarization phase may have a phase different from that of the field. In this case, ‘‘slowly varying’’ means
|\partial zE0| ≪kE0, (1.165) with similar relations holding for \phi and |P (+) |. Put the fields into the above wave equation, making the approximation of slowly varying amplitude and phase, to derive the equations
ϵ0 Im h P (+) i E0
= k ϵ0 Re h P (+) i (1.166)
1.6 Exercises for the effect of the polarization on the field. We again see the imaginary part causes absorption (or gain), and the real part causes a phase shift. These relations are important, for example, in laser physics in treating the effect of the gain medium on the laser field. Note: in the slowly varying approximation, you should throw away second derivatives of E0 and \phi, as well as other second-derivative terms, e.g., of the form (\partial tE0)(\partial t\phi). This problem treats the lowest- order modification of the field due to the polarization field, so you should discard all derivatives of P (+) . Finally, since we are after a perturbative result, you may assume that E0 approximately satisfies the homogeneous equation
c2 E0 \approx 0. (1.167) That is, ignoring the variation, E0 represents a valid solution in the absence of the medium.