Origin of the near-zero-momentum behavior of the momentum distribution in subrecoil laser cooling: the weak deviation from Lévy-statistics predictions at $\zeta\simeq5$ in the nonergodic cooling of metastable helium

Velocity-selective coherent population trapping (VSCPT) cools atoms below the recoil limit because the fluorescence rate $R(p)\propto(p/p_0)^2$ vanishes at momentum $p=0$ , so trapping times are Lévy distributed with a diverging mean and the cooling is nonergodic. In the 1999 experiment of Saubaméa, Leduc and Cohen-Tannoudji on metastable helium, the momentum distribution was probed by two-wave-packet interferometry: averaged over 12 data sets, the rescaled noncoupled fraction collapses onto a single curve versus $\zeta=\tilde t_D\tilde\theta^{-1/2}$ , in excellent agreement with the Lévy-statistics prediction ( $\chi^2=0.0049$ ), while an ergodic (exponential/Lorentzian) model fails at $\zeta<4$ . Around $\zeta\simeq5$ , however, at delay times much larger than the decay time of the signal, the data deviate weakly but statistically significantly from the Lévy prediction; the authors related this to details of the momentum distribution $P(p)$ in the close vicinity of $p=0$ that their model does not identify. The open problem is the physical identification and quantitative modeling of the near-zero-momentum behavior of $P(p)$ — including possible unmodeled microscopic processes — that accounts for the $\zeta\simeq5$ deviation without degrading the established agreement at smaller $\zeta$ .

Background

Subrecoil laser cooling cools atoms below the recoil limit, the lowest momentum scale $\hbar k$ set by the momentum $\hbar k$ of one photon of the cooling light (with $k=2\pi/\lambda$ its wavenumber). Ordinary Doppler or polarization-gradient cooling cannot cross this scale because every scattered photon gives the atom a random recoil kick. Velocity-selective coherent population trapping (VSCPT) bypasses it: atoms are optically pumped into a noncoupled ("dark") state — a coherent superposition of ground-state sublevels that is decoupled from the light exactly when the atomic momentum vanishes. In the quantum-jump picture used in the theoretical description of the same group, an atom of momentum $p$ scatters photons at a rate $R(p)$ that vanishes at $p=0$ ; for the 1D helium geometry used here, $R(p)\propto(p/p_0)^2$ for small $p$ (i.e. the rate exponent is $\alpha=2$ in $R\propto(p/p_0)^\alpha$ ). Atoms thus perform an inhomogeneous random walk in momentum space and accumulate near $p=0$ , where they remain trapped for very long times.

Because the mean trapping time diverges — the trapping-time distribution has a power-law tail $\propto\tau^{-3/2}$ for $\alpha=2$ — the accumulated dark time obeys Lévy statistics, the generalized central-limit regime for broad power-law distributions, and the cooling is nonergodic: the momentum distribution never reaches a time-independent steady state, and single-atom time averages do not converge to ensemble averages. The Lévy-statistical treatment of the same group predicts that after an interaction time $\theta$ the momentum distribution $P(p)$ of the atoms develops a narrow peak of half-width $p_u$ , defined by $R(p_u)\theta=1$ , narrowing as $p_u\propto\theta^{-1/2}$ , with algebraic tails $P(p)\propto1/p^2$ for $|p|\gg p_u$ . In contrast, any model with a nonzero residual rate $R(0)\neq0$ crosses over, in the regime $\theta\gg R(0)^{-1}$ , to an ergodic steady state — a Lorentzian momentum distribution (a curve of the form $1/(p^2+p_u^2)$ ) whose width saturates — so the shape of $P(p)$ near $p=0$ is the fingerprint that discriminates the nonergodic (Lévy) regime from the ergodic one: the Lévy prediction flattens near $p=0$ less abruptly than a Lorentzian.

The 1999 experiment tested this on a collimated beam of metastable helium atoms in 1D VSCPT. The observable was not $P(p)$ directly but its Fourier transform. The dark state is a superposition of two wave packets whose momenta differ by $2\hbar k$ ; after the cooling light is extinguished for a "dark" time $t_D$ , the two wave packets have drifted apart in position, and a short probe pulse (duration $t_A\simeq8\,\mu$ s) measures the fraction of atoms that have not been coupled out of the dark state. This noncoupled fraction, $P_{NC}(t_D)=\frac{1}{2}+\frac{1}{2}\int dp\,P(p)\cos(2kpt_D/M)$ with $M$ the atomic mass, is the momentum autocorrelation function, i.e. the Fourier transform of $P(p)$ . The finite probe duration itself perturbs the measurement — during the probe pulse, atoms with $R(p)t_A>1$ (i.e. large $|p|$ , since $R(p)$ grows with $p$ ) absorb light, so the observed $P(p)$ is truncated beyond a momentum much larger than $p_u$ , and the shape of $P_{NC}(t_D)$ is changed only in a small interval around $t_D=0$ , much smaller than its decay time — so the first data points ( $\zeta<0.8$ ) were discarded. Twelve data sets with different interaction times $\theta$ (dimensionless $\Gamma\theta$ of order 2000–15000, where $\Gamma$ is the excited-state linewidth) were averaged after rescaling the delay time to $\zeta=\tilde t_D\tilde\theta^{-1/2}$ , where tilded quantities are times multiplied by $\Gamma$ ; the measured interference contrast $F$ (the noncoupled fraction rescaled so that it equals the momentum autocorrelation) then collapses onto a single self-similar curve $F(\zeta)$ , as the Lévy theory predicts. The Rabi frequency ratio $\Omega/\Gamma$ (with $\Omega$ the light-atom coupling rate) was measured independently at the location of the atomic cloud, $\Omega/\Gamma=0.72(2)$ , and the 12 runs were taken at that value; in the fit of the averaged rescaled curve, $\Omega/\Gamma$ was the only adjustable parameter, giving $0.70$ in very good agreement with the independent measurement. The Lévy prediction fits the averaged curve with $\chi^2=0.0049$ , while the best exponential (ergodic/Lorentzian) fit fails ( $\chi^2=0.0285$ , with discrepancies 3–4 times the standard deviation at $\zeta<4$ ). Around $\zeta\simeq5$ , however, the averaged data deviate weakly but statistically significantly from the Lévy prediction — at delay times much longer than the decay time of $F$ — and the authors related this to details of $P(p)$ in the close vicinity of $p=0$ that their model does not identify. The model's simplifying assumptions include a single-exponential law for the time $\tau_S$ an atom entering the dark region at momentum $p$ stays dark, $P(\tau_S|p)=R(p)e^{-R(p)\tau_S}$ (the exact law is a sum of exponentials; the authors argue only prefactors change), a uniform distribution of trap-entry momenta over $|p|\lesssim p_{trap}$ with $p_{trap}\ll\hbar k$ , and neglect of the Doppler decrease of $R(p)$ at large $p$ .

Problem Statement

The problem left open by the 1999 experiment is the physical identification and quantitative modeling of the behavior of the momentum distribution $P(p)$ in the close vicinity of $p=0$ in 1D VSCPT of metastable helium that accounts for the weak but statistically significant deviation of the averaged, rescaled noncoupled fraction $F$ from the Lévy-statistics prediction around $\zeta=\tilde t_D\tilde\theta^{-1/2}\simeq5$ , a delay time much larger than the decay time of $F$ . The open problem is to (1) identify the responsible mechanism — for example a nonzero residual scattering rate at $p=0$ , the multi-exponential structure of the dark-state survival law, a finite trap-entry momentum $p_{trap}$ (the source argues that these two approximations change only prefactors of $P(p)$ , not its shape), the Doppler decrease of $R(p)$ , or an unmodeled microscopic process; (2) derive from it a corrected quantitative prediction for $F(\zeta)$ or for $P(p)$ ; and (3) show that this prediction accounts for the reported weak but statistically significant deviation around $\zeta\simeq5$ — removing the residual trend at large $\zeta$ — while maintaining the Lévy prediction's established agreement with the data over the rest of the observed range, including the demonstrated failure of the best exponential (ergodic/Lorentzian) fit at $\zeta<4$ , where its discrepancies reach 3–4 times the standard deviation. Any candidate mechanism invoking a nonzero residual rate $R_0$ near $p=0$ must keep the ergodic crossover beyond the observed interaction times ( $R_0^{-1}\gg\theta$ ), since for $\theta\gg R_0^{-1}$ the source shows the system would display ergodicity with a Lorentzian $P(p)$ . The authors left this point unidentified in their model.

References

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B. Saubaméa, T. W. Hijmans, S. Kulin, E. Rasel, E. Peik, M. Leduc, C. Cohen-Tannoudji, Direct Measurement of the Spatial Correlation Function of Ultracold Atoms, Physical Review Letters 79, 3146–3149 (1997). DOI: 10.1103/PhysRevLett.79.3146, https://doi.org/10.1103/PhysRevLett.79.3146 (the two-wave-packet correlation technique used in the primary source).
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