6. Three-Level Atom Interacting with a Classical Field¶
PDF pages 269–300
6.1 Stimulated Raman Transitions¶
Chapter 6 Three-Level Atom Interacting with a Classical Field Now having developed the theory of two-level atoms quite thoroughly, we will spend a bit more time ex- amining slightly more complicated atoms, those with three levels. Some dramatic effects can take place in such systems due to quantum coherence and interference, much more so than in the two-level atom, due to multiple ‘‘pathways’’ between the different levels. 6.1 Stimulated Raman Transitions One effect that we considered briefly in the context of Bragg scattering in an optical standing wave is the stimulated Raman effect. We consider the atomic energy level structure in the \Lambda-configuration shown below, where two ground states |g1,2\rangle are coupled to an excited state |e\rangle by two optical fields. Our goal is to show that under suitable conditions, the atomic population can be driven between the ground states as in an effective, two-level system. In classical, nonlinear optics, you can think of the effect this way: two waves hitting the same atoms are mixed together by the nonlinearity of the atom, leading to an effective polarization wave at the beat (difference) frequency of the two waves, which drives the atomic transition between the ground states. Quantum mechanically, the coherence between the dipole moments associated with each transition causes them to work together, transferring population between the ground states without significantly populating the excited state. Of course, this is only possible with far-detuned excitations, so that spontaneous emission does not ‘‘scramble’’ the phases of the dipoles and ruin the quantum coherence. Thus, in this section we will stick to a Schrödinger-equation model, explicitly assuming far-off-resonant excitation to the excited state and thus ignoring spontaneous emission. The combined optical field has the form
(6.1) where E(\pm)(r, t) are the positive and negative rotating components of the field, given by
ˆϵ1E01e\pmik1\cdotre∓i\omega1t + ˆϵ2E02e\pmik2\cdotre∓i\omega2t , (6.2) and ˆϵ1,2 are the unit polarization vectors of the two fields.
Chapter 6. Three-Level Atom Interacting with a Classical Field w¡ wº¡ D¡ w™ wº™ D™ |g™Ò |g¡oÒ |eoÒ The free atomic Hamiltonian can then be written HA = p2
(6.3) where we have taken the excited state to have zero energy. In the dipole and rotating-wave approximations, the atom-field interaction Hamiltonian is
(6.4) We have assumed that the ground-state splitting is much smaller than the optical transition frequencies:
have decomposed the dipole operator d into its positive- and negative-rotating components,
= h
i + h
i , (6.5)
principle, we should include a relative phase for this to be generally true, but this disappears anyway in the transformation to the rotating frame). Substituting (6.5) into (6.4), we find HAF = −1
1eik1\cdotre−i\omega1t −1
2eik2\cdotre−i\omega2t . (6.6) We will assume the detunings ∆\alpha := \omega\alpha −\omega0\alpha are nearly equal; hence, to make this problem more tractable, we assume that the field E\alpha couples only |g\alpha\rangle to |e\rangle . After solving this problem, we can treat the cross- couplings as a perturbation to our solutions. If we define the Rabi frequencies
¯h , (6.7) which describe the strength of the coupling from level |g\alpha\rangle through field E\alpha to the excited level |e\rangle , we arrive at HAF = ¯hΩ1
1eik1\cdotre−i\omega1t + ¯hΩ2
2eik2\cdotre−i\omega2t (6.8) as a slightly more compact form for the interaction Hamiltonian. Now, before examining the equations of motion, we transform the ground states into the rotating frame of the laser field. This is like our transformation for the two-level atom, as in Section 5.1.5 on p. 154, but
6.1.1 Effective Two-Level Dynamics¶
6.1 Stimulated Raman Transitions here it is most convenient to use the excited state as the energy reference. Thus, writing the internal part of the state vector as
(6.9) we can define the rotating-frame state vector by
(6.10) where the slowly varying ground-state amplitudes are
(6.11) Since the extra phase factors effectively boost the energies of the |g\alpha\rangle states by ¯h\omega\alpha, the dynamics in the rotating frame are generated by the rotating-frame, free-atom Hamiltonian, given by ˜HA = p2
(6.12) The interaction Hamiltonian in the rotating frame is
= ¯hΩ1
1eik1\cdotr + ¯hΩ2
2eik2\cdotr , (6.13) where the slowly varying field amplitudes are given by ˜E(+)
6.1.1 Effective Two-Level Dynamics Turning to the equations of motion, we will manifestly neglect spontaneous emission, since ∆\alpha ≫\Gamma, where \Gamma is the decay rate of |e\rangle , by using a Schrödinger-equation description of the atomic evolution. Then we have
(6.14) where the state vector can be factored into external and internal components as
(6.15)
(6.16)
we identify the bracketed quantity [\psi\alphaei∆t] as the new state, which amounts to the replacement \psi\alpha −\rightarrow
frequencies of order |∆| ≫\Gamma. We are interested in motion on timescales slow compared to 1/\Gamma, and the fast oscillations are damped by coupling to the vacuum on timescales of 1/\Gamma, so we can adiabatically eliminate
p2/2m ≪¯h|∆|, with the result,
(6.17)
Chapter 6. Three-Level Atom Interacting with a Classical Field Notice that in deriving this relation, it was important to choose the proper energy shift −¯h∆to minimize the natural rotation of the states that remain after the adiabatic elimination; indeed, if the resonance condition that we will derive is satisfied, the two ground states have no natural oscillatory time dependence. This procedure would be much more clear in a density-matrix treatment (as in Section 5.8.3.1), where the oscillating coherences would be eliminated, but this description is cumbersome due to the number of energy levels in the problem. Using this relation in the remaining equations of motion, we obtain two coupled equations of motion for the ground states,
(6.18) where we have removed the energy shift of −¯h∆. These equations are formally equivalent to the equations of motion for a two level atom, with Rabi frequency (Raman or two-photon Rabi frequency) ΩR := Ω1Ω2 2∆ (6.19) (Raman Rabi frequency) and Stark shifts
\alpha 4∆. (6.20) (ac Stark shifts) These equations of motion are just the equations generated by the effective Raman Hamiltonian HR = p2
- ¯hΩR
Rei(k1−k2)\cdotr , (effective, two-level Hamiltonian) (6.21)
|g1\rangle is accompanied by a kick of up to two photon-recoil momenta, and the reverse transition is accompanied by the opposite kick of up to two photon recoils. We can write out the coupled equations of motion due to the Hamiltonian (6.21) more explicitly as
p2
2m
2 \psig1(p), (6.22) where 2\deltak := k1 −k2. The resonance condition for this transition |p\rangle |g1\rangle −\rightarrow |p + 2¯h\deltak\rangle |g2\rangle is
2m¯h
− p2
= 0, (6.23) which can be rewritten as
¯h\deltak
(Raman resonance condition) (6.24)
6.1.3 Multiple Excited States¶
6.1 Stimulated Raman Transitions Here, p∥is the component of p along the direction of \deltak, and we have defined the Raman recoil energy by
Doppler shift of the two optical fields due to motion at the average of the upper and lower state momenta. Thus, we see that the stimulated Raman problem reduces to an effective two-level system of splitting ¯h∆R, coupled by a dc interaction of strength ¯hΩR/2. 6.1.1.1 Cross-Couplings Finally, we account for the effects of the cross-couplings that we previously ignored. The lifetimes of the two ground states are in practice extremely long, so that the line width of the Raman transition is quite narrow, being limited only by the finite interaction time. Since it is assumed that the Raman resonance condition (6.23) is approximately true, the Raman cross-coupling is much further away from resonance than the intended coupling (typically several orders of magnitude), so this extra Raman coupling can be neglected in a secondary rotating-wave approximation. However, the cross-couplings can induce additional ac Stark shifts of the ground levels. So, we simply modify (6.20) to include these extra shifts:
4∆+ Ω2 1(2) 4(∆−\omega21)
4∆+ Ω2 2(1)
(ac Stark shifts with cross-couplings) (6.25) Here, Ω\alpha(\beta) is the cross-coupling Rabi frequency for field \beta on transition |g\alpha\rangle −\rightarrow |e\rangle ,
¯h , (6.26) and we have assumed \omega21 ≫|∆|. These additional Stark shifts may not in general be negligible compared to the original Stark shifts. 6.1.2 Spontaneous Emission We can also obtain an estimate of the spontaneous emission rate, which gives us a measure of how accurate our treatment is (since we have explicitly neglected it), by using (6.17) to write the total excited state population in terms of the density matrix elements:
= \GammaΩ2
4∆2 ei(k1−k2)\cdotr\rhog2g1. (6.27) Here, \rho\alpha\alpha is the population in state |\alpha\rangle , with \rhog1g1 + \rhog2g2 ≃1, and \Gamma is the total decay rate from the excited state. Note that this result assumes implicitly that ∆1 \approx ∆2. The second two terms represent an enhancement or suppression of spontaneous scattering due to atomic coherences; for example, the state
(6.28) (where \eta is the appropriate normalization factor) is dark, since Rsc vanishes for this state. However, this state is only dark if the cross-couplings can be ignored. More realistically, the scattering rate can be modeled as an incoherent sum over all the couplings of the form (\GammaΩ2/4∆2)\rhog\alphag\alpha. This ‘‘dark’’ phenomenon is coherent population trapping, which we will treat in more detail below. 6.1.3 Multiple Excited States It turns out we can work out this problem in the case where the ground states are coupled to multiple excited states |en\rangle , as essentially always happens in real atoms. The idea is the same as above, except now the excited states have energies \deltan with respect to some arbitrary reference in the excited-state manifold
6.1.4 Velocity Selectivity¶
Chapter 6. Three-Level Atom Interacting with a Classical Field
states, and n enumerates the excited states). Generalizing Eq. (6.7), we have Rabi frequencies
¯h , (6.29) for each possible transition. Then in the rotating frame the free atomic Hamiltonian is ˜HA = p2
X n
(6.30) and the interaction Hamiltonian in the rotating frame is ˜HAF = X n ¯hΩ1n
1neik1\cdotr + X n ¯hΩ2n
2neik2\cdotr , (6.31) where the lowering operators are now given by \sigma\alphan := |g\alpha\rangle \langle en|. This setup is essentially the same as before, except for the summation of the excited states and the dependence of the detunings from the excited state on n. Following the same procedure as above, we find that the effective Raman Hamiltonian (6.21) is still valid, but where the Raman Rabi frequency is given by ΩR = X n Ω1nΩ2n 2(∆−\deltan), (Raman Rabi frequency, multiple excited states) (6.32) and the Stark shifts are given by
X n Ω2 \alphan 4(∆−\deltan), (ac Stark shifts, multiple excited states) (6.33) which can be generalized as above to include cross-couplings and couplings to other levels. Ignoring any interference effects, the spontaneous-emission rate is a sum over terms of the form \GammaΩ2 n/4∆2 n for every transition coupled by the fields. 6.1.4 Velocity Selectivity From Eq. (6.24), the resonance condition for the stimulated Raman transition is
¯h\deltak
(6.34) If we choose to ignore the atomic motion, we can do this by letting m −\rightarrow \infty, in which case \omegaR −\rightarrow 0, and
(6.35) Thus, the transition is resonant if the detunings of the two fields to the excited states are equal, including any ac Stark shifts. However, in general, the resonance condition involves the atomic momentum. Noting that \omegaR ∼(¯h\deltak)2, we can see that the atomic-velocity contribution is largest when \deltak is maximum. In
have \deltak \approx k, and the momentum change in the Raman transition is 2¯hk [this is, in fact, exact if we define
case, we find the resonance momentum
4\omegaR −¯hk (6.36) (resonant momentum)
6.1.5 Pulse-Shape Considerations¶
6.1 Stimulated Raman Transitions for atoms in the |g1\rangle state, where in this case the Raman recoil frequency reduces to the usual recoil frequency: ¯h\omegaR = ¯h2k2/2m = ¯h\omegar. After making the transition to the |g2\rangle state, the atoms have the momentum
4\omegaR + ¯hk. (6.37) (resonant momentum) In the copropagating case, when the ground-state splitting is small (say, a microwave transition as in the alkali atoms), the momentum recoil ¯h\deltak is several orders of magnitude smaller than the optical recoil ¯hk1,2. In this case, the momentum-dependent term makes an essentially negligible contribution to the resonance
21/2mc2. For a microwave ground-state splitting \omega21/2\pi of 1 GHz, and a mass of 10−25 kg, we find \omegaR/2\pi = 40 nHz, compared to a recoil frequency of \omegar/2\pi = 4 kHz if the optical transition is \omega01 \approx \omega02 = 5 \times 1014 Hz. Thus, for typical ‘‘long’’ Raman pulses of ms durations, the velocity selectivity can be on the order of the atomic recoil or better for the counterpropagating case, but has essentially no selectivity on the scale of many recoils for the copropagating case. Thus, in the counterpropagating configuration, the velocity dependence makes stimulated Raman transitions a valuable tool for atomic velocity selection.1 The idea is fairly simple: to select atoms with a particular velocity, we simply drive a \pi-pulse between the atomic ground states, tuning the frequency difference between the two laser fields according to the resonance condition to select the desired velocity group. Since the frequency difference must be stable to typically better than the kHz level, the two fields are often generated from the same laser source by acousto-optic or electro-optic modulation, or they are generated by two separate lasers that are actively phase-locked (by detecting the beat note on a photodiode and using a phase-locked loop to feed back to the ‘‘slave’’ laser frequency). The line widths of the ground states are typically quite narrow compared to any laser line width, and so the width of the velocity selection is dominated by power-broadening of the two-photon transition. That is, there is an effective range of detunings on the order of ΩR, so that the width of the selected momentum distribution is \deltap∥\approx ΩR/4\omegar in the counterpropagating case. We will be more quantitative about the distribution below, but for now note that recoil-level resolution requires \pi-pulses of ms or longer durations. After the Raman pulse, the atoms near the desired momentum are ‘‘tagged,’’ by their internal state: if all the atoms start in |g1\rangle , the atoms with the desired momentum end up in |g2\rangle . This may be sufficient for some purposes, or the undesired atoms may be ‘‘blown away’’ by a resonant beam. For example, a beam that couples |g1\rangle to another excited state (that decays only to |g1\rangle and not to |g2\rangle ) can push the atoms away via radiation pressure. One problem with this technique is extreme sensitivity to stray fields. Magnetic fields cause Zeeman shifts in otherwise degenerate levels of real atoms on the order of 0.1 MHz/G, where the Earth’s magnetic field is around 0.7 G. But recoil-level velocity selection requires frequency precisions of kHz or better. Experi- mentally, stray magnetic fields must therefore be eliminated with compensation coils, high-µ metal shielding, and elimination of ferromagnetic materials in the vicinity of the atoms. 6.1.5 Pulse-Shape Considerations 6.1.5.1 Square Pulse Since the velocity-selective Raman pulses (in the counterpropagating configuration) are generally used to ‘‘tag’’ a subset of an atomic distribution according to their momentum, it is important to consider the impact of the temporal pulse profile on the tagged distribution. The simplest pulse profile is the square profile, where the light is turned on at a constant intensity for some duration. Assuming that the atoms are all initially in the same internal atomic state, the tagging process is described by the solution of the optical Bloch equations for the excited state population of a two-level atom with Rabi frequency ΩR, Raman detuning ∆R (given by 1Mark Kasevich, David S. Weiss, Erling Riis, Kathryn Moler, Steven Kasapi, and Steven Chu, ‘‘Atomic velocity selection using stimulated Raman transitions,’’ Physical Review Letters 66, 2297 (1991) (doi: 10.1103/PhysRevLett.66.2297); Kathryn Moler, David S. Weiss, Mark Kasevich, and Steven Chu, ‘‘Theoretical analysis of velocity-selective Raman transitions,’’ Physical Review A 45, 342 (1992) (doi: 10.1103/PhysRevA.45.342).
Chapter 6. Three-Level Atom Interacting with a Classical Field the left-hand side of Eq. (6.24)), and with all initial population in the ground Raman state |g1\rangle :
Ω2 R Ω2 R + ∆2 R sin2 1 p Ω2 R + ∆2 R t . (6.38) The dynamics here are just the familiar generalized Rabi oscillations from the two-level atom, Eq. (5.60).
shape has wings that decay relatively slowly, with a series of locations where the line shape goes to zero. The locations of the zeros for an interaction time of \deltat is given by ∆R = s 4n2\pi2 (\deltat)2 −Ω2 R (6.39) for positive integer n. This relation simplifies for specific interaction times; for example, for a ‘‘\pi-pulse’’ of duration \deltat = \pi/ΩR, the locations are at ∆R = ΩR \sqrt
the locations are ∆R = ΩR \sqrt 16n2 −1. These zeros were important in a previous implementation of Raman cooling,2 where the first zero of the profile (6.38) was placed at zero momentum to form a dark interval where atoms would accumulate. The square-pulse excitation line shape is plotted in below for a \pi/2-pulse, a \pi-pulse, and a 2\pi-pulse. -10 Doooo/WR R Transferred population p-pulse 2p-pulse
Note that for the important case of the \pi-pulse, the central population lobe is characterized by a half width at half maximum of 0.799 \cdot Ω. It is also important to note that because one typically excites a range of detunings with a velocity- selective Raman pulse, the transferred population does not undergo simple sinusoidal Rabi oscillations. For a square pulse, the excitation profile (6.38) must be averaged over the atomic velocity distribution. In the limit of a broad velocity distribution, the excited population is proportional to Z \infty −\infty
Ji0(ΩRt) = \piΩ2 Rt n
2 [J1(ΩRt)H0(ΩRt) −J0(ΩRt)H1(ΩRt)] o , (6.40)
R x 0 Jn(x′)dx′. The population in this case still oscillates as a function of time, but with some effective damping due to dephasing of the different momenta. 2J. Reichel, F. Bardou, M. Ben Dahan, E. Peik, S. Rand, C. Salomon, and C. Cohen-Tannoudji, ‘‘Raman Cooling of Cesium below 3 nK: New Approach Inspired by Lévy Flight Statistics,’’ Physical Review Letters 75, 4575 (1995) (doi: 10.1103/Phys- RevLett.75.4575); Jakob Reichel, Refroidissement Raman et vols de Lévy: atomes de césium au nanokelvin, Thése de Doctorat, École Normale Supérieure (1996).
6.1 Stimulated Raman Transitions 2.5 Transferred population (relative) Wooooot/2p R Notice that for short times, the function (6.40) reduces to (\pi/2)Ω2 Rt + O(t2), so that one can associate a nonzero transition rate, proportional to Ω2 R (which is in turn proportional to the product of the laser
6.1.5.2 Blackman Pulse An alternative pulse profile, the Blackman pulse profile, is useful for suppressing the side lobes of the tagged distribution.3 This profile, when normalized to have unit area, can be written as
(6.41)
-0.5 1.5 t/t 2.381 Pulse amplitude (normalized) The Blackman profile has compact support and also, because it is continuous, has the property that the tails in the Fourier spectrum are suppressed relative to the square pulse. Hence, the Raman excitation spectrum of the Blackman pulse falls off much more sharply than the corresponding square-pulse spectrum. 3Mark Kasevich and Steven Chu, ‘‘Laser Cooling below a Photon Recoil with Three-Level Atoms,’’ Physical Review Letters 69, 1741 (1992) (doi: 10.1103/PhysRevLett.69.1741); Nir Davidson, Heun Jin Lee, Mark Kasevich, and Steven Chu, ‘‘Raman Cooling of Atoms in Two and Three Dimensions,’’ Physical Review Letters 72, 3158 (1994) (doi: 10.1103/PhysRevLett.72.3158).
6.1.6 Stimulated Raman Cooling¶
Chapter 6. Three-Level Atom Interacting with a Classical Field -10 Transferred population Doooo/WR R Blackman pulse square pulse However, the implementation of Blackman pulses is more complicated if the Raman beams induce an ac Stark shift of the transition, since the Raman frequency must be chirped to match the Stark shift in order to get good frequency resolution. 6.1.6 Stimulated Raman Cooling 6.1.6.1 Free Space The velocity selectivity of stimulated Raman transitions makes them very useful for cooling atoms, and stimulated Raman cooling for neutral atoms has been successfully implemented.4 The method, while difficult to implement in practice, has the advantage of very cold (subrecoil) temperatures without substantial losses of atoms (as in forced evaporation). The important preliminary conceptual step is to define a ‘‘target zone’’ near p = 0, where the atoms will accumulate. Then we proceed with a cycle of steps. For simplicity, we’ll consider only one dimension for the moment. 1. Start with all atoms in one state, say |g1\rangle . 2. Tag all atoms outside the target zone by transferring them to the |g2\rangle state, using stimulated Raman pulses in the counterpropagating configuration. |g™Ò |g¡oÒ |eoÒ When tagging a particular velocity group, the orientation of the two beams should be such that the recoil of 2¯hk moves the atoms towards the target zone near p = 0. 4Mark Kasevich and Steven Chu, ‘‘Laser Cooling below a Photon Recoil with Three-Level Atoms,’’ Physical Review Let- ters 69, 1741 (1992) (doi: 10.1103/PhysRevLett.69.1741); J. Reichel, O. Morice, G. M. Tino, and C. Salomon, ‘‘Subrecoil Raman Cooling of Cesium Atoms,’’ Europhysics Letters 28, 477 (1994); J. Reichel, F. Bardou, M. Ben Dahan, E. Peik, S. Rand, C. Salomon, and C. Cohen-Tannoudji, ‘‘Raman Cooling of Cesium below 3 nK: New Approach Inspired by Lévy Flight Statistics,’’ Physical Review Letters 75, 4575 (1995) (doi: 10.1103/PhysRevLett.75.4575); H. J. Lee, C. S. Adams, M. Ka- sevich, and S. Chu, ‘‘Raman Cooling of Atoms in an Optical Dipole Trap,’’ Physical Review Letters 76, 2658 (1996) (doi: 10.1103/PhysRevLett.76.2658).
6.1 Stimulated Raman Transitions p po=o0 2h—k target zone Raman tag In general, a number of pulses are required to tag all of the necessary atoms. At higher momenta, shorter pulses (tagging wider momentum distributions) may be used, while near the target zone, long, high-resolution pulses are necessary. 3. Now ‘‘reset’’ the tagged atoms by applying light resonant with the |g2\rangle −\rightarrow |e\rangle transition, so that the atoms eventually decay back to the dark |g1\rangle state. |g™Ò |g¡oÒ |eoÒ This can in general be done by beams from all directions, such as the optical molasses beams that are likely to be present anyway. In this case, the tagged atom distribution from the last step will be broadened in momentum by an amount on the order of ¯hk. p This spontaneous Raman step provides the dissipation or ‘‘exit channel’’ for entropy necessary for any cooling scheme to work. 4. Repeat the above sequence many times. Why does this work so effectively? With the above sequence, all the atoms are essentially making a biased random walk towards the target zone. Ideally, once the atoms reach the target zone, they never leave it, because the velocity selectivity of the simulated Raman transitions. Even though the Raman pulses transfer momentum 2¯hk at a time, the spontaneous emission allows the atom to move by fractions of a momentum recoil ¯hk, and thus the target zone can be narrower than ¯hk, and cooling below the recoil limit has been demonstrated with this method. Of course, the above idealization where the atoms are permanently stuck in the target zone is not quite true: the tails of the tagging distributions as well as off-resonant excitations determine a limited lifetime for atoms in the target zone. So long as this lifetime is much longer than the
Chapter 6. Three-Level Atom Interacting with a Classical Field time to iterate the above cycle, the cooling still works, and can be understood in terms of Lévy flights,5 which amounts to diffusive behavior in momentum where every so often, the atoms ‘‘stick’’ to the region near p = 0 before diffusing again. In three dimensions, the tagging must take place in all three dimensions on each iteration, so the target region is a small box in the three-dimensional momentum space. The much smaller target region (relative to the initial distribution) implies a much longer cooling time, but the method can still be made to work. Obviously this requires more juggling of laser beams, which makes the method quite challenging to implement. This is especially true considering the sensitivity of Raman transitions to magnetic fields that we discussed above, and subrecoil cooling requires extensive measures against stray fields. 6.1.6.2 Resolved-Sideband Raman Cooling If atoms are bound in a tightly confining potential, another cooling method becomes possible if the splittings between the vibrational levels becomes much larger than the line width of the relevant atomic transition. In this case, the spectralsidebands are well-resolved, and the cooling method is known as resolved-sideband Raman cooling. The basic idea is as follows. Assuming a nearly harmonic trapping potential of frequency \omegatrap, we note that the bound atom oscillates mechanically at this frequency. If a monochromatic laser field impinges on the atom in the direction of motion, the atom thus sees a time-varying Doppler shift (phase-modulated wave) of the form
(6.42) The instantaneous frequency is simply given by the time derivative of the phase (up to a minus sign), or \omega −\delta\phi \omegatrap cos(\omegatrapt), but in view of the decomposition
eikxe−i\omegat \infty X j=−\infty Jj(\delta\phi)eij\omegatrapt, (6.43) we see that the spectrum is the ‘‘carrier’’ at frequency \omega plus a sequence of sidebands at frequencies \omegaj = \omega −j\omegatrap, where j is any nonzero integer. The above decomposition follows from the generating function for the Bessel functions: exp x t −1 t = \infty X j=−\infty Jj(x)tj. (6.44) The point is that the absorption spectrum of the bound atom consists of the usual atomic resonance \omega0, plus sidebands \omega0 + j\omegatrap spaced at the trap frequency, assuming that the sidebands are well resolved (in the limit where \omegatrap is much larger than any decay rates for the ground states). When absorbing on one of the sidebands, energy conservation demands that along with the electronic transition, the vibrational state change by the appropriate number of quanta. We can write down a recipe similar to that of free-space Raman cooling as follows. 1. Begin with all atoms in the same electronic state, say in |g1\rangle . 2. Drive a stimulated Raman transition on the \omega0 −\omegatrap sideband. This implies transitions of the form |g1, n\rangle −\rightarrow |g2, n −1\rangle , where the integer labels the vibrational quantum number. 5J. Reichel, F. Bardou, M. Ben Dahan, E. Peik, S. Rand, C. Salomon, and C. Cohen-Tannoudji, ‘‘Raman Cooling of Cesium below 3 nK: New Approach Inspired by Lévy Flight Statistics,’’ Physical Review Letters 75, 4575 (1995) (doi: 10.1103/Phys- RevLett.75.4575).
6.1.7 Atom Interferometry¶
6.1 Stimulated Raman Transitions |g™Ò |g¡oÒ The vibrational energy is thus reduced by one quantum. Note also that the |g1, 0\rangle state is dark, because the laser does not resonantly drive it to any other state. 3. Recycle the atoms to |g1\rangle by resonantly exciting it to the excited state. On average, the vibrational state does not change during the transition, particularly if the vibrational splitting is larger than the transition line width. Thus, on average, the atoms have reduced their vibrational energies by about one quantum. 4. Repeat. At the end of many iterations, it is possible to find the atom in the ground state with near-unit probability. In three dimensions, all three relevant sidebands must be driven sequentially, assuming nondegenerate trap frequencies, and the beams must not be along a principle axis of the trap. This method has been successfully implemented in ion traps6 as well as with neutral atoms in optical lattices.7 6.1.7 Atom Interferometry One other application of stimulated Raman transitions is in the realization of atom interferometers, where atoms are split and recombined to effect sensitive physical measurements. The first atom interferometers were realized with thermal atomic beams, with the ‘‘beam splitters’’ realized by passing atoms through indi- vidual slits in physical aperture masks8 or through microfabricated (absorptive) diffraction-grating masks.9 Ultracold-atom interferometers lend themselves naturally to measurements of increased sensitivity due to the high degree of available control and potentially long interaction times. 6C. Monroe, D. M. Meekhof, B. E. King, S. R. Jefferts, W. M. Itano, D. J. Wineland, and P. Gould, ‘‘Resolved-Sideband Raman Cooling of a Bound Atom to the 3D Zero-Point Energy,’’ Physical Review Letters 75, 4011 (1995) (doi: 10.1103/Phys- RevLett.75.4011). 7S. E. Hamann, D. L. Haycock, G. Klose, P. H. Pax, I. H. Deutsch, and P. S. Jessen, ‘‘Resolved-Sideband Raman Cooling to the Ground State of an Optical Lattice,’’ Physical Review Letters 80 4149 (1998) (doi: 10.1103/PhysRevLett.80.4149); Vladan Vuletić , Cheng Chin, Andrew J. Kerman, and Steven Chu, ‘‘Degenerate Raman Sideband Cooling of Trapped Cesium Atoms at Very High Atomic Densities,’’ Physical Review Letters 81 5768 (1998) (doi: 10.1103/PhysRevLett.81.5768); Andrew J. Kerman, Vladan Vuletić, Cheng Chin, and Steven Chu, ‘‘Beyond Optical Molasses: 3D Raman Sideband Cooling of Atomic Ceium to High Phase-Space Density,’’ Physical Review Letters 84 439 (2000) (doi: 10.1103/PhysRevLett.84.439); Dian-Jiun Han, Steffen Wolf, Steven Oliver, Colin McCormick, Marshall T. DePue, and David S. Weiss, ‘‘3D Raman Sideband Cooling of Cesium Atoms at High Density,’’ Physical Review Letters 85 724 (2000) (doi: 10.1103/PhysRevLett.85.724). 8O. Carnal and J. Mlynek, ‘‘Young’s Double-Slit Experiment with Atoms: A Simple Atom Interferometer,’’ Physical Review Letters 66, 2689 (1991) (doi: 10.1103/PhysRevLett.66.2689). 9David W. Keith, Christopher R. Ekstrom, Quentin A. Turchette, and David E. Pritchard, ‘‘An Interferometer for Atoms,’’ Physical Review Letters 66, 2693 (1991) (doi: 10.1103/PhysRevLett.66.2693).
Chapter 6. Three-Level Atom Interacting with a Classical Field An atom interferometer based on stimulated Raman transitions might work as follows.10 As atoms move slowly along (say, in free fall after being launched in an atomic fountain), they are exposed to a set of pulses from stimulated Raman lasers in the counterpropagating configuration. atomic trajectory
p-pulse If the atoms all start in one state |g1\rangle , a \pi/2 Raman pulse puts them in a superposition of |g1\rangle and |g2\rangle . The atoms in |g2\rangle have also suffered a momentum recoil of 2¯hk in this configuration, and if the Raman lasers are oriented normally to the atoms’ path, the atoms in the two states begin to separate transversely. Later, the atoms are exposed to a \pi Raman pulse, which exchanges the ground states as well as the transverse velocities of the two atomic groups, bringing them back together. When they again overlap, a final \pi/2 pulse mixes them and produces interference fringes. Thinking of this interferometer as analogous to the optical Mach–Zehnder interferometer, the \pi/2 pulses are analogous to (50/50) beam splitters, while the \pi pulse is analogous to a set of high reflectors. Of course, any interaction that induces a relative phase between the two groups of atoms can be sensitively measured with this technique. One of the more successful applications is to the measurement of gravity. In the above figure, we can imagine that gravity points towards the bottom of the page. In this case, the phases accumulated by the two atomic groups during the respective ‘‘tilted segments’’ of their journeys should be the same. However, during the ‘‘horizontal segments,’’ the two atoms travel along paths with different gravitational potentials, and thus there is a phase shift given by mg∆z ∆t/2¯h, where m is the atomic mass, g is the local acceleration of gravity, ∆z is the spatial separation of the two interferometer arms, and ∆t is the time between the \pi/2-pulses. The local gravitational acceleration g has been measured with this system with a resolution of \deltag/g ∼10−8 for ∼1 s integration times and ∼10−10 for integration times of ∼1 day,11 which begins to rival the current method of a falling corner-cube optical interferometer. Further, implementing a simultaneous pair of these measurements at different locations enables measurement of gravity gradients, which are otherwise quite difficult to measure.12 Further, the sensitivity of these measurements may be greatly enhanced by using Bose–Einstein condensates in place of ordinary cold-atom clouds.13 On the fundamental side, gravity and gravity gradient measurements enable measurements of the gravitational constant G and tests of general relativity, while on the applied side, interferometers are of technological interest for the detection of underground structures and reservoirs of oil and water, as well as completely passive navigation. An alternate, but substantially equivalent, method of atom interferometry uses Bragg scattering from optical lattices as atomic beam splitters and mirrors.14 As we discussed before, Bragg scattering can be viewed as a stimulated Raman process among momentum states, and thus the Bragg scatterings must be set to the equivalents of \pi/2- and \pi-pulses for beam splitters and high reflectors, respectively. Because the recoil energy enters the resonance condition, a variation on the above interferometer en- ables the measurement of the fine-structure constant \alpha, which is interesting from a fundamental perspective, since past measurements have had statistically significant discrepancies, and there is some speculation that the fine-structure constant may be time-dependent. The rough idea is that the resonance condition (6.24) for stimulated Raman transitions involves the recoil energy ¯h\omegar for Raman beams in the counterpropagating 10Mark Kasevich and Steven Chu, ‘‘Atomic interferometry using stimulated Raman transitions,’’ Physical Review Letters 67, 181 (1991) (doi: 10.1103/PhysRevLett.67.181). 11A. Peters, K. Y. Chung, and S. Chu, ‘‘High-precision gravity measurements using atom interferometry,’’ Metrologia 38, 25
12M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, ‘‘Measurement of the Earth’s Gravity Gra- dient with an Atom Interferometer-Based Gravity Gradiometer,’’ Physical Review Letters 81, 971 (1998) (doi: 10.1103/Phys- RevLett.81.971). 13P. Bouyer and M. A. Kasevich, ‘‘Heisenberg-limited spectroscopy with degenerate Bose-Einstein gases,’’ Physical Review A 56, R1083 (1997) (doi: 10.1103/PhysRevA.56.R1083 ). 14David M. Giltner, Roger W. McGowan, and Siu Au Lee, ‘‘Atom Interferometer Based on Bragg Scattering from Standing Light Waves,’’ Physical Review Letters 75, 2638 (1995) (doi: 10.1103/PhysRevLett.75.2638).
6.2 Coherent Population Trapping¶
6.2 Coherent Population Trapping configuration. The recoil energy gives a measurement of h/m, which can be combined with an atomic-mass measurement to give the ratio ¯h/me where me is the electron mass. This ratio can then be converted to a measurement of \alpha.15 6.2 Coherent Population Trapping Another important effect, coherent population trapping16 dramatically shows the influence of quantum interference in \Lambda atoms. Consider again the \Lambda atom from our discussion of stimulated Raman transitions of Section 6.1. w¡ wº¡ D¡ w™ wº™ D™ |g™Ò |g¡oÒ |eoÒ Following our previous treatment, but now ignoring center-of-mass motion of the atom, we can write the free-atomic Hamiltonian in the rotating frame as
(6.45) Similarly, the atom-field interaction Hamiltonian in the rotating frame is ˜HAF = ¯hΩ1
+ ¯hΩ2
. (6.46) It turns out that if we make a change of basis for the ground states, one of the new states decouples from the excited state, which of course simplifies things. In particular, motivated by the spontaneous-emission results in the context of stimulated Raman transitions from Section 6.1.2, we can make the transformation
p Ω2 1 + Ω2
=
p Ω2 1 + Ω2
(6.47) where the excited state is unchanged, and the rotation angle is defined by
Ω1 . (6.48) (decoupling rotation angle) 15David S. Weiss, Brenton C. Young, and Steven Chu, ‘‘Precision measurement of the photon recoil of an atom using atomic interferometry,’’ Physical Review Letters 70, 2706 (1993) (doi: 10.1103/PhysRevLett.70.2706); Brenton Christopher Young, ‘‘A Measurement of the Fine-Structure Constant using Atom Interferometry,’’ Ph.D. dissertation, Stanford University (1997). 16E. Arimondo and G. Orriols, ‘‘Nonabsorbing atomic coherences by coherent two-photon transitions in a three-level optical pumping,’’ Lettere al Nuovo Cimento della Societa Italiana di Fisica 17, 333 (1976) (doi: 10.1007/BF02746514); H. R. Gray, R. M. Whitley, and C. R. Stroud, Jr., ‘‘Coherent trapping of atomic populations,’’ Optics Letters 3, 218 (1978) (doi: 10.1364/OL.3.000218).
Chapter 6. Three-Level Atom Interacting with a Classical Field Clearly, the new states |g+\rangle and |g−\rangle are still normalized and orthogonal. Of course, the opposite basis change is given by reversing the rotation angle:
(6.49) Then, we can put these into Eq. (6.45) to find the atomic Hamiltonian in the new basis,
, (transformed free-atom Hamiltonian) (6.50) where the rotated detunings are
(6.51) and the coupling rate between the new states is
Ω1Ω2 Ω2 1 + Ω2 . (6.52) Similarly, the transformations (6.49) in Eq. (6.46) give the interaction Hamiltonian in the new basis
- + ¯hΩ−
− , (6.53) where \sigma\pm := |g\pm\rangle \langle e|, and the new Rabi frequencies are
Ω2 1 + Ω2 p Ω2 1 + Ω2 = q Ω2 1 + Ω2
p Ω2 1 + Ω2 = 0. (6.54) Thus, we may write the interaction Hamiltonian as ˜HAF = ¯h p Ω2 1 + Ω2
- , (transformed interaction Hamiltonian) (6.55) and thus we see that the coupling between |g−\rangle and |e\rangle vanishes, while the Rabi frequency for the coupling
dramatically. This is because Ωg = 0, and ∆+ = ∆−= ∆: ˜HA = ¯h∆
. (6.56) (free evolution, Raman resonance) Thus, we see that the free-atomic Hamiltonian becomes diagonal at Raman resonance (while ˜HAF is inde- pendent of detuning). We see in this case that under Hamiltonian evolution, |g−\rangle is completely uncoupled
Now, what about spontaneous emission? The operator form of the master equation reads
¯h h
i
(6.57)
6.2 Coherent Population Trapping where the \Gamma\alpha are the decay rates of |e\rangle to |g\alpha\rangle , so that the total decay rate of the excited state is \Gamma = \Gamma1+\Gamma2. In the new basis, the master equation becomes
¯h h
i
− , (6.58) where
(6.59) We see that the dissipation terms have a similar form in the new basis, but the last term is a correction term to handle asymmetric decay to |g1\rangle and |g2\rangle . The point here is that |e\rangle decays as usual to both |g+\rangle and |g−\rangle . Our basic conclusions will thus not be affected by the simplification \Gamma1 = \Gamma2 = \Gamma/2 = \Gamma+ = \Gamma−, so that
¯h h
i
(6.60) We have thus arrived at a new effective \Lambda atom, where the excited state decays to both ground states, but only one ground state, |g+\rangle , is pumped by the external fields to the excited state. |go–oÒ |goòoÒ |eoÒ G G Wò Again, at Raman resonance, there is no coupling between |g+\rangle and |g−\rangle , but away from resonance there is a coupling at rate Ωg. Thus, at Raman resonance, |g−\rangle is a dark state, and for Ω+̸ = 0, all the population will eventually end up in |g−\rangle . Thus, in steady state, the atom scatters no light. In the original basis, this is because the dipole moments for the two transitions either constructively or destructively interfere. If they destructively interfere, then the atom scatters no light, and effectively decouples from the field. If there is constructive interference, spontaneous emission scrambles the phases of the dipoles until the interference is purely destructive. This effect is coherent population trapping, because the population is ‘‘trapped’’ in |g−\rangle due to quantum interference. If we look at an absorption spectrum where one frequency, say \omega2, is swept, while the other is held fixed, we should expect to get the usual Lorentzian line shape for absorption, with width \Gamma1 + \Gamma2. However, we expect to see a dip in the absorption spectrum when ∆1 = ∆2.
Chapter 6. Three-Level Atom Interacting with a Classical Field D2o/oG -5 ree 0.022 D1oo=o0 G1o=ooG2o=oG W1o=ooW2o=oo0.3G The dip can be quite narrow, and by solving for the steady state of the three-level optical Bloch equations it is possible to show17 that the line shape is approximately the difference of two Lorentzians, one associated with the natural width of the excited state, and a narrow Lorentzian responsible for the coherent dip, of width (FWHM)
1 + Ω2 \Gamma1 + \Gamma2 (6.61) (width of coherent dip) in the weak-driving limit. Since this is a coherent effect, the dip becomes suppressed and wider if any dephasing process reduces the coherence of the ground states. If the first laser is detuned from resonance, and the second laser is made much weaker and swept in frequency across the first laser, a narrow, dispersive resonance occurs. D2o/oG -5 ree 4.8o!o10-6
G1oo=ooG2oo=oG W1oo=oo100oW2oo=o0.3G This shape has been shown18 to be a Fano profile19 (an asymmetric line shape that arises due to interference in ionization spectra). 17H. R. Gray et al., op. cit. 18B. Lounis and C. Cohen-Tannoudji, ‘‘Coherent population trapping and Fano profiles,’’ Journal de Physique II (France) 2, 579 (1992) (doi: 10.1051/jp2:1992153). 19U. Fano, ‘‘Effects of Configuration Interaction on Intensities and Phase Shifts,’’ Physical Review 124, 1866 (1961) (doi: 10.1103/PhysRev.124.1866 )
6.2.2 Electromagnetically Induced Transparency¶
6.2 Coherent Population Trapping 6.2.1 VSCPT¶
atomic momentum, and is essentially the same as the stimulated Raman resonance condition, Eq. (6.23). Doing so leads to a clever method for cooling atoms to extremely low temperatures, known as velocity- selective coherent population trapping, or VSCPT.20 The idea is as follows. Suppose an atom with two degenerate ground states has both levels coupled to the excited state by two counterpropagating lasers of equal optical frequency. (This level structure occurs for an angular-momentum transition of the form
the excited state (due, e.g., to different polarizations of the two beams). |g™Ò |g¡oÒ |eoÒ Then we can see that Doppler laser cooling works as usual, assuming that the common optical detuning is to the red of the atomic resonance: if the atom moves, it scatters photons preferentially from the opposing beam, which is shifted into resonance, thus tending to stop the atom. However, due to the three-level structure, there is a dark state, which for equal intensities is given by
\sqrt
, (6.62) once we have included the atomic motion. The frequency width of this dark state is set mostly by the common Rabi frequency as in Eq. (6.61), and for small intensities the dark-state width can be quite small, corresponding to a momentum width smaller than ¯hk. Thus, VSCPT gives rise to a sub-recoil cooling method, with the final momentum distribution consisting of two subrecoil peaks about \pm¯hk. Note also that the final momentum state (the dark state) is, in fact, an entangled state between the atomic internal and external states. In this one-dimensional configuration, this method works best for collimation of atomic beams. However it is possible to extend this method to three dimensions.21 It is also possible to apply this scheme to more complicated atoms with further degeneracy, so long as the lasers are pulsed to give a Ramsey-fringe-type effect.22 6.2.2 Electromagnetically Induced Transparency One way to view the phenomenon of coherent population trapping is that the absorption profile for a field is modified by the presence of another field. Thinking of field 1 as a (strong) pump field, and field 2 as a (weak) probe field, the absorption coefficient for the probe can drop to zero due to the presence of the pump. 20A. Aspect, E. Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, ‘‘Laser Cooling below the One-Photon Re- coil Energy by Velocity-Selective Coherent Population Trapping,’’ Physical Review Letters 61, 826 (1988) (doi: 10.1103/Phys- RevLett.61.826); A. Aspect, Ennio Arimondo, R. Kaiser, N. Vansteenkiste, and C. Cohen-Tannoudji, ‘‘Laser cooling below the one-photon recoil energy by velocity-selective coherent population trapping: theoretical analysis,’’ Journal of the Optical Society of America B 6, 2112 (1989); M. Widmer, M. R. Doery, M. J. Bellanca, W. F. Buell, T. H. Bergeman, and H. J. Metcalf, ‘‘High-velocity dark states in velocity-selective coherent population trapping,’’ Physical Review A 53, 946 (1996) (doi: 10.1103/PhysRevA.53.946). 21M. A. Ol’shanii and V. G. Minogin, ‘‘Three-dimensional velocity-selective coherent population trapping of a (3 + 3)-level atom,’’ Optics Communications 89, 393 (1992); J. Lawall, S. Kulin, B. Saubamea, N. Bigelow, M. Leduc, and C. Cohen- Tannoudji, ‘‘Three-Dimensional Laser Cooling of Helium Beyond the Single-Photon Recoil Limit,’’ Physical Review Letters 75, 4194 (1995) (doi: 10.1103/PhysRevLett.75.4194). 22Frank Sander, Thibaut Devolder, Tilman Esslinger, and Theodor W. Hänsch, ‘‘Ramsey-Type Subrecoil Cooling,’’ Physical Review Letters 78, 4023 (1997) (doi: 10.1103/PhysRevLett.78.4023).
Chapter 6. Three-Level Atom Interacting with a Classical Field w¡ (pump) (probe) D¡ w™ D™ |g™Ò |g¡oÒ |eoÒ This phenomenon is known as electromagnetically induced transparency, or EIT.23 Let’s analyze this situation in a bit more depth, taking advantage of the assumption of a weak probe field. We will assume essentially the same form of the master equation as in (6.57)
¯h h
i
(6.63)
between the ground states such as collisions in atomic vapors or dephasing due to local crystal fields. This master equation implies the equation for the probe-transition coherence
−\Gamma2 2 + i∆2
2 ˜\rhog1g2 \approx −\Gamma2 2 + i∆2 ˜\rhoeg2 −iΩ2 −iΩ1 2 ˜\rhog1g2. (6.64) Here, to lowest order in Ω2 we have made the replacements \rhog2g2 \approx 1 and \rhoee \approx 0 to arrive at the second expression, since both are multiplied by Ω2. We want this coherence in steady state, since it controls the photon-absorption rate of the probe, but it is coupled to ˜\rhog1g2. The equation of motion for this ground-state coherence is
2 ˜\rhog1e
2 ˜\rhoeg2. (6.65) Here, we have dropped the ˜\rhog1e term, since it is unpopulated to lowest order in Ω2, and it already involves a factor of Ω2. In steady state, \partial t˜\rhog1g2 = 0, and solving the above equation gives
Ω1˜\rhoeg2
(6.66) Then setting \partial t˜\rhoeg2 = 0 to find the steady-state coherence, we find −\Gamma2 2 + i∆2 ˜\rhoeg2 −iΩ1
2 . (6.67) Using Eq. (6.66) and solving for ˜\rhoeg2, we find
(6.68) 23The first proposals for EIT were by Surya P. Tewari and G. S. Agarwal, ‘‘Control of Phase Matching and Nonlinear Generation in Dense Media by Resonant Fields,’’ Physical Review Letters 56, 1811 (1986) (doi: 10.1103/PhysRevLett.56.1811); and S. E. Harris, J. E. Field, and A. Imamoğlu, ‘‘Nonlinear optical processes using electromagnetically induced transparency,’’ Physical Review Letters 64, 1107 (1990) (doi: 10.1103/PhysRevLett.64.1107). For a good review, see Robert W. Boyd and Daniel J. Gauthier, ‘‘‘Slow’ and ‘Fast’ Light,’’ in Progress in Optics, vol. 43, E. Wolf, ed. (Elsevier, Amsterdam, 2002), p. 497. Our treatment here parallels part of their discussion.
6.2 Coherent Population Trapping This coherence determines the optical properties of the medium, as far as the probe is concerned. The polarization is given by the dipole moment per unit volume, or
(6.69) where \chi is the linear susceptibility of the atomic vapor, N is the number density of the atomic vapor, and E(+) is the positive-rotating electric field amplitude for the probe. Thus, we can write the susceptibility as
ϵ0¯h
(EIT susceptibility) (6.70) Recall that the complex refractive index is the square root of the dielectric constant, so ˜n = p
2 , (6.71) since \chi is small, assuming a rarefied medium. Also, taking the real part of the refractive index,
. (6.72) The intensity absorption coefficient is related to the imaginary part of the refractive index by comparing the damping part of the plane-wave solution:
(6.73) so that a = 2k0Im[˜n] \approx k0 Im[\chi]. (6.74) Thus, in this regime, the absorption coefficient for the probe is given by the imaginary part of \chi. We can thus see the induced transparency by looking at the Raman resonance ∆1 = ∆2, and for simplicity we will also consider the resonant case ∆2 = 0. In this case the susceptibility
ϵ0¯h \gammag
(6.75) becomes purely imaginary (i.e., the phase index becomes unity), and \chi drops monotonically to zero with increasing pump intensity (Ω1). Thus, we see how (on resonance) transparency for the probe is induced by the pump field. The atomic medium is causal, and since the refractive index represents a causal response of the medium to the applied field, the real and imaginary parts of the complex refractive index obey the Kramers–Kronig relations24
\pi – Z \infty −\infty Im[˜n(\omega′)]
d\omega′
\pi – Z \infty −\infty Re[˜n(\omega′)] −1
d\omega′. (6.76) The integrals here are Hilbert transforms (the cut integration symbols denote Cauchy-principal-value integrals), which are effectively convolutions with the kernel 1/\omega. Since this kernel changes sign (and is
willing to be not-too-quantitative. Since the coherent dip in EIT can be very narrow, as we saw from our analysis of coherent population trapping, the imaginary part of ˜n has large derivatives, and hence the real 24See Section 14.1.4.2, or for an alternate treatment see also Daniel A. Steck, Classical and Modern Optics, available online at http://steck.us/teaching.
6.2.3 Stimulated Raman Adiabatic Passage¶
Chapter 6. Three-Level Atom Interacting with a Classical Field part of ˜n (the phase index) can have large values and steep gradients. For a resonant, arbitrarily strong
ϵ0¯h (i\gammag −∆2) (Ω1/2)2 , (6.77) so that the phase index n = Re[˜n] \approx Re[\chi]/2 becomes
ϵ0¯h ∆2 Ω2 . (6.78) The group index of refraction is given by25
d\omega , (6.79) and it measures the ratio of the vacuum speed of light to the propagation velocity (group velocity) of an optical pulse. Assuming the second term is the most important, for the EIT medium the group index becomes
ϵ0¯hΩ2 .
(6.80)
This also occurs where the medium is least absorbing, so long as the pulse spectrum is not too wide. Using this ‘‘slow light’’ technique, optical pulses have been slowed to 17 m/s.26 6.2.3 Stimulated Raman Adiabatic Passage Now suppose in a three-level atom, you want to move all the population from |g1\rangle to |g2\rangle , using two-photon stimulated Raman transitions. |go™oÒ |go¡oÒ |eoÒ W¡ W™ You can do the good old \pi-pulse, and of course you can chirp the Raman frequency to do adiabatic passage like in the two-level atom. However, because the dark states depend on the relative intensity, there is a different form of adiabatic passage, called stimulated Raman adiabatic passage, or STIRAP.27 The idea is that if you have two laser pulses, one for each optical transition, you should do something counterintuitive: you should first turn on the laser coupling |g2\rangle −\rightarrow |e\rangle , and then later turn on the laser coupling |g1\rangle −\rightarrow |e\rangle . 25Daniel A. Steck, op. cit. 26Lene Vestergaard Hau, S. E. Harris, Zachary Dutton, and Cyrus H. Behroozi, ‘‘Light speed reduction to 17 metres per second in an ultracold atomic gas,’’ Nature 397 (1999) (doi: 10.1038/17561). See also Michael M. Kash, Vladimir A. Sautenkov, Alexander S. Zibrov, L. Hollberg, George R. Welch, Mikhail D. Lukin, Yuri Rostovtsev, Edward S. Fry, and Marlan O. Scully, ‘‘Ultraslow Group Velocity and Enhanced Nonlinear Optical Effects in a Coherently Driven Hot Atomic Gas,’’ Physical Review Letters 82, 5229 (1999) (doi: 10.1103/PhysRevLett.82.5229). 27J. Oreg, F. T. Hioe, and J. H. Eberly, ‘‘Adiabatic following in multilevel systems,’’ Physical Review A 29, 690 (1984) (doi: 10.1103/PhysRevA.29.690); U. Gaubatz, P. Rudecki, M. Becker, S. Schiemann, M. Külz, and K. Bergmann, ‘‘Population switching between vibrational levels in molecular beams,’’ Chemical Physics Letters 149, 463 (1988) (doi: 10.1016/0009- 2614(88)80364-6); U. Gaubatz, P. Rudecki, S. Schiemann, and K. Bergmann, ‘‘Population transfer between molecular vibra- tional levels by stimulated Raman scattering with partially overlapping laser fields. A new concept and experimental results,’’ Journal of Chemical Physics 92, 5363 (1990) (doi: 10.1063/1.458514); Martin Weitz, Brenton C. Young, and Steven Chu, ‘‘Atomic Interferometer Based on Adiabatic Population Transfer,’’ Physical Review Letters 73, 2563 (1994) (doi: 10.1103/Phys- RevLett.73.2563).
6.2.4 Quantum Beats¶
6.2 Coherent Population Trapping t W¡ W™ The key is the overlap of the pulses, and the form of the dark state. When Ω2 is large and Ω1 = 0, then clearly the dark state is |g1\rangle . This represents the initial configuration. Similarly, when Ω1 is large and Ω2 = 0, the dark state is |g2\rangle , the desired final state. We showed above that there is a dark state |−\rangle for any pair (Ω1, Ω2), and thus if we transform the field amplitudes adiabatically, slowly on time scales of (Ω2 1 + Ω2 2 )−1/2, then the atom will follow the dark state |−\rangle until it reaches the final state |g2\rangle . Since the atom is always in the dark state, there is no problem with spontaneous emission, even if the lasers are near resonance. Of course, with different pulse shapes it is possible to end in any superposition of the two ground states. 6.2.4 Quantum Beats Until now we have discussed only the three-level atom in the \Lambda-configuration, but how do atoms in other configurations differ? One of the most significant differences is the possibility of quantum beats in resonance fluorescence. The basic idea is fairly simple28 if we first consider the radiation from the \Lambda atom. The dipole moment is proportional to the annihilation operators for the two transitions,
(6.81) where \sigma\alpha := |g\alpha\rangle \langle e|. For simplicity we are dropping the dipole matrix elements, which may be different for the two transitions but do not affect our qualitative conclusions. The radiated field intensity thus scales as
E(−)E(+) ∝ D d(−)d(+)E ∝ D \sigma\dagger
E
(6.82) (radiated intensity, \Lambda-atom) where we have used \sigma\dagger \alpha\sigma\beta = |e\rangle \langle g\alpha|g\beta\rangle \langle e| = |e\rangle \langle e|\delta\alpha\beta. Thus, the total radiation rate is proportional to the excited-state population. The ‘‘vee’’ atom—where a single ground state |g\rangle is coupled to two excited states |e1\rangle and |e2\rangle —is more complicated, however. |e™Ò |e¡oÒ |goÒ If the two transitions decay into the same polarization mode, we can also write the dipole operator as
(6.83) 28P. W. Milonni, ‘‘Semiclassical and quantum-electrodynamical approaches in nonrelativistic radiation theory,’’ Physics Re- ports 25, 1 (1976).
Chapter 6. Three-Level Atom Interacting with a Classical Field where \sigma\alpha := |g\rangle \langle e\alpha|, and we have again dropped the dipole matrix elements. In this case, we have \sigma\dagger
(6.84) and thus
E(−)E(+) ∝ D d(−)d(+)E ∝ D \sigma\dagger
E
(6.85) (radiated intensity, V-atom) We see that the radiated intensity is proportional to the sum of the excited-state populations, which we might expect, but also the last two coherence terms represent interferences between the two populations. In the case where |e1\rangle and |e2\rangle have different energies, these coherences (transiently) rotate at the splitting frequency, thus leading to the quantum beats in the resonance fluorescence. This is the same beat note that we expect from any two radiating oscillators, but it goes to show that spontaneous emission isn’t entirely coherent—interference effects are manifest in spontaneous emission. The above argument leading to quantum beats rests on the assumption of decay into the same mode. If the radiation from the two transitions is distinguishable, say, if the two transitions radiated orthogonal polarizations, then the decay operators should not be added before taking the expectation value, and the quantum beats are not present. 6.2.4.1 Master Equations and Quantum Beats The above argument addresses an ambiguity that arises when writing down the master equation for the three-level atom. For the \Lambda atom, we assumed in Eq. (6.57) that the master equation takes the form
¯h h
i
(6.86) (distinguishable radiation) That is, we use separate dissipation terms for each operator, recalling that the Lindblad superoperator has the form
. (6.87) Of course, we could have used a single dissipation term, had we added the operators together, to arrive at the master equation
¯h h
i + D hp
p \Gamma2\sigma2 i ˜\rho. (indistinguishable radiation) (6.88) Which one is correct? It depends on the physical situation. The master equation (6.86) corresponds to the case where the radiation from the two transitions is distinguishable, while the master equation (6.88) corresponds to the case where the radiation from the two transitions is indistinguishable (and thus we add the ‘‘amplitudes,’’ or decay operators, for the two fields together before detection). This interpretation will be more clear when we study master equations in the context of continuous measurement, but the general rule is: we use separate decay terms when the decay processes can be monitored by separate detectors, while if the decay processes can’t be distinguished, there is no ‘‘which-way’’ information, and we model the resulting potential interference via a single decay term. To emphasize the difference between Eqs. (6.86) and (6.88), we note that we can rewrite Eq. (6.88) for the \Lambda atom as
¯h h
i
p \Gamma1\Gamma2
. (6.89) These last two terms involve the excited-state population and only couple to the ground-state coherences, and represent additional coherence induced between the ground states by spontaneous emission. Normally, the
6.2 Coherent Population Trapping indistinguishability is not such an important issue for \Lambda atoms, because even if the radiated polarizations are the same, the ‘‘which-way’’ information is provided by the atom itself, since we can in principle interrogate it to see which ground state it is in. However, we have already seen the coherent version of the master equation in the general form of Eq. (6.89) in the context of coherent population trapping, where the extra coherence terms popped up when we switched to the dark/bright-state basis in Eq. (6.58). In that case, the point was that each decay was to a superposition of the two ground states in the new basis, which was reflected by the additional coherence terms. 6.2.4.2 Steady-State Quantum Beats A dramatic manifestation of the above difference between distinguishable and indistinguishable master equa- tions occurs in a variant of the vee atom, resulting in something termed steady-state quantum beats.29 The configuration is the vee atom from above, but in the case where the excited states are nondegenerate with splitting \delta but both coupled from the ground state by the same field. |e™Ò |e¡oÒ w d |goÒ If we define the rotating-frame Hamiltonians, detunings ∆\alpha, Rabi frequencies Ω\alpha, and decay rates \Gamma\alpha (\alpha \in {1, 2}) in the usual way, then the Bloch equations for the excited-state populations have the form
2 (˜\rhoe1g −˜\rhoge1) + (dissipations terms)
2 (˜\rhoe2g −˜\rhoge2) + (dissipations terms). (6.90) We may thus interpret the photon absorption rate to be the rate at which atoms are being excited, and thus Rabs = iΩ1
(6.91) Again, there are two possible master equations that we can write down to describe this system. In the case where there is distinguishable emission from the two excited states (say, different polarizations), we use the master equation with separate decay terms:
¯h h
i
(6.92) A sample absorption spectrum (in steady state) for this system is shown below, plotted as a function of the mean detuning (∆1 + ∆2)/2, relative to the common decay rate \Gamma. 29D. A. Cardimona, M. G. Raymer, and C. R. Stroud, Jr., ‘‘Steady-state quantum interference in resonance fluorescence,’’ Journal of Physics B: Atomic and Molecular Physics 15, 55 (1982) (doi: 10.1088/0022-3700/15/1/012).
Chapter 6. Three-Level Atom Interacting with a Classical Field
-20 (absorption rate)/G 0.5 do=o10G G1o=ooG2oo=oG W1o=ooW2o=o3G The spectrum consists of two peaks, as one might expect the sum of two Lorentzian peaks if there is no ‘‘interaction’’ between the transitions. For indistinguishable emission from the excited states, we again allow for interference of the radiated fields, and we use the master equation
¯h h
i + D hp
p \Gamma2\sigma2 i ˜\rho. (6.93) In this case, for the same parameters, something remarkable happens: the absorption vanishes at the mid- point between the peaks. This is an effect that persists even at high intensity (note that saturation effects
excited states.
-20 (absorption rate)/G 0.5 do=o10G G1o=ooG2oo=oG W1o=ooW2o=o3G This is clearly an interference effect, similar to coherent population trapping, but in a sense more remarkable because there is still population in the excited state, even when the atom is dark, as we can see by examining the total excited-state population.
6.2 Coherent Population Trapping¶
-20 (excited-state population)/G 0.5 do=o10G G1o=ooG2oo=oG W1o=ooW2o=o3G re¡e¡o+oore™oe™ re¡e¡ re™oe™ Of course, the absorption rate must equal the emission rate in steady state, and we have seen that the emission rate is not just proportional to the total excited-state population, but rather to
(6.94) The coherences, or interference terms, prevent the atom from decaying even though the excited states are populated. Thus, coherent population trapping is due to interference in the Hamiltonian evolution of a \Lambda atom, while steady-state quantum beating is due to interference in the dissipative evolution of a vee atom. A final amusing thing to note is that steady-state quantum beating gives rise to an alternate interpre- tation of EIT. In the \Lambda atom, you can imagine that the pump laser dresses the excited state, splitting it into a doublet (as in the Autler–Townes doublet). The probe beam thus effectively couples to a vee atom, and with the proper detuning, steady-state quantum beating suppresses absorption of the probe.
6.3 Exercises¶
Chapter 6. Three-Level Atom Interacting with a Classical Field 6.3 Exercises Problem 6.1 A phase-modulated optical wave has the form
(6.95) where \omegamod is the modulation frequency. Such a wave could result, for example, by running the wave through an electro-optic crystal, whose refractive index is modulated by an applied ac signal with frequency \omegamod. (a) For a wave with time dependence exp[−i\phi(t)], we can define the instantaneous frequency as
dt . (6.96) Compute the instantaneous frequency of the phase-modulated wave and thus show that the frequency oscillates about \omega. That is, phase modulation is in some sense the same as frequency modulation. (b) Write the phase-modulated wave as a sum of plane waves, with the general form \infty X j=−\infty cjei(kx−\omegajt). (6.97) Hint: start by writing down a Bessel series for the function exp(iK sin x), using the generating function given in the notes. (c) From your answer to (b), argue that the intensity spectrum (as viewed through a Fabry–Perot spectrum analyzer) consists of a series of peaks with relative intensity J 2
You may assume the response of the Fabry–Perot analyzer is slow compared to the modulation frequency. This phase- modulation technique is commonly used in the laboratory to shift the frequency of a laser or to generate multiple laser frequencies. Problem 6.2 Consider the vee atom, where steady-state quantum beating can be observed, where both excited states are coupled by a single, monochromatic, electric field. |e™Ò |e¡oÒ w d |goÒ (a) Write down expressions for the free atomic Hamiltonian ˜HA and the interaction Hamiltonian ˜HAF in the rotating frame, in terms of the appropriate detunings and Rabi frequencies. (b) Assuming a master equation of the form
¯h h
i
(6.98) appropriate for distinguishable decay channels, write out the Bloch equations for the density-matrix
6.3 Exercises (c) In the case of indistinguishable decay channels, with master equation of the form
¯h h
i + D hp
p \Gamma2\sigma2 i ˜\rho, (6.99) what are the new terms in the Bloch equations compared to what you wrote in part (b)? Give specific interpretations to these extra terms where possible. Problem 6.3 For the vee atom in Problem 6.2, consider the case of indistinguishable decay channels, with \Gamma1 = \Gamma2 = \Gamma and Ω1 = Ω2, with the field tuned exactly halfway between the excited states. Solve for the steady state of the optical Bloch equations for this system to lowest nontrivial order in Ω1 and Ω2 (i.e., find the linear response for very weak fields), and thereby prove that steady-state quantum beats occur. The effect does not depend on the smallness of the fields, and the analytic solution can be worked out for arbitrary parameters, but this problem is much easier in the perturbative limit. Problem 6.4 Consider an atomic transition between states |g\rangle and |e\rangle , of resonance frequency \omega0, driven at nearly half the resonance frequency, so that 2\omega \approx \omega0. In this case, it is possible to have two-photon absorption and nonlinearly drive the transition. |eÒ |joÒ |gÒ wº w w Of course, this happens because of Raman-type transitions involving the other states |j\rangle as intermediate states. However, our stimulated-Raman analysis does not apply here, because we cannot make the usual rotating wave approximation, since \omega does not resonantly couple |g\rangle or |e\rangle to any intermediate level. Your goal is to work out the theory of two-photon transitions, and thus to show that this system effectively reduces to a two-level system for |g\rangle and |e\rangle . To do this, use the following outline. 1. Write down the free atomic Hamiltonian, using the following definitions: the energy of |g\rangle is zero, and the energy of the |g\rangle −\rightarrow |j\rangle is \omegaj. 2. Write down the atom–field interaction Hamiltonian, using Rabi frequencies Ωgj for the |g\rangle −\rightarrow |j\rangle transitions and Ωej for the |e\rangle −\rightarrow |j\rangle transitions. For the moment, ignore the direct coupling between |g\rangle and |e\rangle (assume, for example, that the transition is dipole-forbidden). Do not make any rotating-wave approximations at this stage. 3. Write the state vector as
X j cj|j\rangle , (6.100) and derive equations of motion for the coefficients. 4. Transform into a rotating frame by changing to the slowly varying coefficient ˜ce = ceei2\omegat, (6.101) which is appropriate for \omega0 \approx 2\omega, and rewrite the coefficient equations in terms of this new variable. 5. Integrate the equation for \partial tcj to obtain an approximate expression for cj(t), assuming that cg and ˜ce are slowly varying on the time scales of optical oscillations. This is justified since we are interested in the slow dynamics of these variables.
Chapter 6. Three-Level Atom Interacting with a Classical Field 6. Use your approximate result to eliminate cj from the equations of motion, and write the equations of motion in the form of a two-level system. Now you should make appropriate rotating-wave approximations to put the equations in the proper form. In your answer, give expressions for the Stark shifts of |g\rangle and |e\rangle , and also for the two-photon Rabi frequency. At the same level of approximation, how do your results change if the transition |g\rangle −\rightarrow |e\rangle is also coupled directly by the field with Rabi frequency Ω? Problem 6.5 Name as many approximations as you can that go into the result ΩR = Ω1Ω2 2∆. (6.102) Problem 6.6 Consider a transition between the two ground states of a \Lambda atom via the STIRAP procedure. |go™oÒ |go¡oÒ |eoÒ W¡ W™ Estimate the probability that a photon is scattered from the atom during the transition, which happens if the pulse sequence does not drive the atom adiabatically through the transition. To do this, set this problem up as an avoided-crossing problem, and use Landau–Zener theory to estimate the probability that the atom ‘‘tunnels’’ out of the dark state. Model the two laser pulses as Gaussian pulses, t W¡ W™ with pulse profile
−t2 2 \deltat2 . (6.103) Assume that the two laser pulses are identical, both exactly resonant with their respective transitions, and that the peak Rabi frequencies are the same for both transitions. Take the time separation between the two pulse peaks to be \tau. Note: strictly speaking, Landau–Zener theory only applies to an avoided crossing where the bare-state energies are linear functions of time. However, you may approximately apply it to any avoided crossing by noting that most of the tunneling across the gap occurs during the times when the gap is narrowest. Thus, you should set up the problem such that the Landau–Zener problem approximates the energy levels of this problem in the vicinity of the avoided crossing.
6.3 Exercises Problem 6.7 Consider two two-level atoms situated near each other. No optical fields are present other than those radiated by the atoms. Recall that the spatial profile of the atomic radiation field is given by the classical dipole pattern. (a) Consider the interaction of the two atoms as the usual dipole interaction of one atom with the radiated field of the other. Show that after making a suitable rotating-wave approximation the atom– atom interaction can be written Hint = ¯h
1\sigma2 , (6.104) where \sigma1,2 are the lowering operators for atoms 1 and 2, and the free-atom Hamiltonian is
\sigma\dagger
2\sigma2 . (6.105) Write down an expression for the coupling rate \Xi, which depends on the separation and orientation of the atoms. Assume the two atomic dipoles have the same orientation. (b) Argue that in a suitable rotating frame, the interaction Hamiltonian is unchanged but the free- evolution Hamiltonian becomes ˜H0 = 0. (c) Now consider the symmetric initial state
\sqrt h
i , (6.106) and the antisymmetric initial state
\sqrt h
i , (6.107) both corresponding to a single excitation. Show that in the limit of small atom separations, where \Xi \in R, both states are eigenstates of the rotating-frame Hamiltonian ˜Hint. (d) Assume that the atoms evolve according to a master equation of the form
(6.108) That is, we assume the dipole fields radiated by the two atoms to interfere perfectly, which is only true if the atoms have the same orientation, and they are very close together. Show that the decay of the atom pair starting in the symmetric state proceeds more quickly than for a single, isolated atom in the excited state. (You need only show this to be true at short times.) This effect is called Dicke superradiance,30 and arises physically due to the constructive interference of the two radiated fields. (e) Show also that the decay of the atom pair starting in the antisymmetric state proceeds more slowly than for a single, isolated atom in the excited state. This effect is called subradiance, and is due to the destructive interference of the two radiated fields. (f) Why is the description ‘‘two atoms playing photon ping-pong’’ appropriate to this problem, specif- ically to the form of Hint? 30R. H. Dicke, ‘‘Coherence in Spontaneous Radiation Processes,’’ Physical Review 93, 99 (1954) (doi: 10.1103/PhysRev.93.99).