Quantitative three-dimensional treatment of Sisyphus-enhanced velocity-selective coherent population trapping: Doppler-force losses, the critical speed, and the temperature-momentum-density trade-off

The 1995 experiment of Lawall, Kulin, Saubamea, Bigelow, Leduc and Cohen-Tannoudji achieved the first subrecoil laser cooling in three dimensions, cooling metastable helium to 180 nK — 22 times below the single-photon recoil temperature — with a 16-fold momentum-space-density enhancement, by combining velocity-selective coherent population trapping (VSCPT) with Sisyphus-type frictional forces in a six-beam geometry. The authors stated that no quantitative study of VSCPT cooling including coherent population trapping, Sisyphus and Doppler forces existed, that the available Lévy-flight and random-walk models lacked a loss term for atoms Doppler-accelerated out of the cooling random walk, and left open the dependence of the 'critical' speed (where Sisyphus and Doppler forces compensate, about 16 recoil velocities in that experiment) on laser parameters, the possible anisotropy of the supplementary forces in velocity space ('channels' of reduced friction), and the optimization of the temperature-momentum-density trade-off. The intervening literature contains one-dimensional force-assisted VSCPT theory (Wu et al. 1995; Morigi et al. 1996; Zambon and Nienhuis 2006), a completed friction-free Lévy-statistics theory (Bardou et al. 2001), and the practical goals achieved by other techniques (3D Raman sideband cooling, gray molasses, evaporative cooling) — but no published unified three-dimensional treatment with a Doppler loss term, no critical-speed parameter characterization, and no resolution of the anisotropy question. The problem remains open.

Background

Laser cooling of neutral atoms uses the momentum transferred in resonant absorption and spontaneous-emission cycles to reduce the atoms' velocity spread. Two classical mechanisms set the scales: Doppler cooling, in which the radiation-pressure force of light detuned to the red of resonance damps atomic motion, and Sisyphus cooling (also called polarization-gradient cooling), in which spatially modulated light shifts and optical pumping (light-driven transfer among the atom's internal sublevels) make atoms climb internal-state 'potential hills' more often than they descend, giving strong friction at low velocity (detuning is the laser frequency offset from atomic resonance; blue/red means above/below resonance). Each absorbed or spontaneously emitted photon also delivers a random recoil kick of momentum $\hbar k$ ($\hbar$ is the reduced Planck constant and $k$ the optical wave number), which defines the recoil velocity $v_R=\hbar k/m$ (about 9.2 cm/s on the 1.083 μm helium transition used below) and the recoil temperature $T_R$, fixed by $k_B T_R = m v_R^2$ (about 4 μK on the same transition). Reaching velocity widths below $v_R$ ('subrecoil' cooling) is impossible in any scheme where fluorescence cycles continue at low velocity, because the unceasing random recoils set a floor of order $\hbar k$ on the momentum spread.

Two techniques evade this floor by making the photon absorption rate itself vanish at a selected velocity. In velocity-selective coherent population trapping (VSCPT), optical pumping by quantum interference between excitation amplitudes drives atoms into a 'dark' coherent superposition of ground-state sublevels and momenta that no longer absorbs the laser light; slow atoms stay trapped there while faster atoms keep executing a random walk in velocity space, until they fall into the dark state. VSCPT was demonstrated in one dimension on metastable helium — helium atoms held in the long-lived $2^3S_1$ excited state, which serves as an effective ground state for laser manipulation (Aspect et al., 1988) — extended to two dimensions (Lawall et al., 1994), and the existence of velocity-selective dark states for an arbitrary set of equal-modulus wave vectors on a $J=1\to J=1$ transition was established by Ol'shanii and Minogin (1992). The companion technique, Raman cooling (Kasevich and Chu, 1992), uses sequences of Raman pulses (two-photon stimulated transitions between ground states) to the same end. Because the jump rate vanishes at the target velocity, sojourn times near zero velocity follow broad power-law statistics (Lévy flights), making subrecoil cooling a canonical physical realization of Lévy statistics (Bardou et al., 1994).

Three dimensions bring a qualitative difficulty: a free random walk in 3D velocity space is transient (a nonzero fraction of walkers never returns to any bounded region), so the accumulation efficiency into the dark state falls rapidly with dimension. The remedy is to add 'supplementary' frictional forces that confine the walk; several groups had shown, in one and two dimensions, that combining coherent population trapping with polarization-gradient or Doppler forces speeds up accumulation into dark states (Mauri and Arimondo, 1991; Shahriar et al., 1993; Marte et al., 1994; Weidemüller et al., 1994).

In this setting Lawall, Kulin, Saubamea, Bigelow, Leduc and Cohen-Tannoudji (1995) reported the first subrecoil laser cooling in three dimensions. Metastable helium atoms precooled in a magneto-optical trap — a magnetic-field-assisted version of optical molasses (the viscous damping of atomic motion by red-detuned light) that both confines and cools the atoms — were irradiated by six laser beams on the $2^3S_1\text{-}2^3P_1$ transition at 1.083 μm, a transition between levels of total angular momentum $J=1$ (the subscripts $g$ and $e$ below denote the lower and upper level): four beams lying in a vertical plane at 45 degrees to the vertical and two along the horizontal axis, each retro-reflected pair with counter-rotating circular $\sigma^+/\sigma^-$ polarizations. Atoms accumulated in six subrecoil peaks in velocity space of width $\delta v\approx v_R/4.8$ (180 nK, i.e. $T_R/22$, with an atomic coherence length near 5 μm) and a momentum-space density enhanced by a factor 16 per peak relative to the precooled trap, the temperature falling like $1/\theta$ with interaction time $\theta$ (the velocity width narrowing like $1/\sqrt{\theta}$). Two complications accompanied the cooling. With blue detuning, Sisyphus-type friction first compresses the velocity distribution, but atoms whose random walk takes them beyond a 'critical' speed are then accelerated by the Doppler radiation-pressure force and lost, showing up as a hot component in the time-of-flight distribution (arrival times of atoms ballistically falling onto a position- and time-resolved detector, which encode their velocities). With red detuning the roles interchange — Sisyphus heating at low velocity, Doppler friction at high velocity — producing a stable compensation sphere in velocity space at about 150 cm/s, roughly $16v_R$, in their parameters. Longer interaction times lower the temperature further but erode the momentum-space density. The authors state that no theoretical study of this combined process existed and that the available random-walk and Lévy-flight models lack a loss term for the Doppler-accelerated atoms; closing that gap is the subject of this problem.

Problem Statement

The problem, as posed at the close of the source experiment, is to develop the quantitative, unified theoretical treatment of three-dimensional subrecoil cooling by velocity-selective coherent population trapping in a multi-beam $J_g=1\to J_e=1$ geometry (realized with metastable helium on the $2^3S_1$-$2^3P_1$ transition), in which coherent population trapping, the accompanying Sisyphus-type frictional forces, and Doppler radiation-pressure forces are treated together, including a loss term describing atoms that are accelerated by Doppler forces to high velocities and thereby removed from the cooling random walk. Within this single objective, the source states four linked open questions:

(1) Construct such a model — microscopic or statistical — correcting the known deficiency of the existing trapping/recycling (Lévy-flight and random-walk) models, which contain no loss term for Doppler-accelerated atoms, and recovering the established friction-free asymptotics when the supplementary forces vanish (for example the temperature law $T\propto 1/\theta$, with the subrecoil velocity width $\delta v\propto 1/\sqrt{\theta}$).

(2) Characterize quantitatively how the 'critical' speed — the speed at which Sisyphus heating and Doppler cooling compensate (or, for blue detuning, beyond which Doppler acceleration overwhelms Sisyphus cooling) — depends on the laser parameters: detuning, intensity (Rabi frequency, the strength of the light-atom coupling, proportional to the laser field amplitude), and beam geometry. The source measured essentially one such point, about 150 cm/s or $16v_R$ at a red detuning of one natural linewidth and a Rabi frequency near 0.9 linewidths, and explored the dependence only empirically.

(3) Determine the actual anisotropy of the supplementary forces in three-dimensional velocity space — in particular whether preferential 'channels' with reduced friction exist — rather than assuming isotropy as a working hypothesis.

(4) Use the validated model to optimize three-dimensional VSCPT so as to maximize simultaneously very low temperature and high momentum-space density, mitigating the observed trade-off (the source reached $T_R/22$ at 16-fold momentum-space-density enhancement but $T_R/45$ at only about 4-fold enhancement).

The four questions are linked: the model of (1) is expected to yield the critical-speed characterization (2) and the anisotropy determination (3), which the validated model then supports in addressing the optimization (4). The open question concerns the three-dimensional multi-beam geometry itself, not only its one-dimensional force-assisted variants.

References

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