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Cambridge University Press eBooks

Correspondence between parameters of the statistical models and atomic and laser parameters

François Bardou, Jean‐Philippe Bouchaud, Alain Aspect, Claude N. Cohen-Tannoudji

Published Jul 2, 2010• 0 citations
Publisher

Abstract

We establish here the correspondence between the statistical models introduced in Chapter 3 and the quantum evolution of atoms undergoing subrecoil laser cooling. This enables us to establish analytical expressions connecting the parameters of the statistical models (τ 0 , p 0 , p D , Δ p , p max , τ b and) to atomic and laser parameters relevant to subrecoil laser cooling. Such a ‘dictionary’ is useful for the numerical estimation of the results derived in this book (see Chapter 8). It also leads to analytical relations between τ b and, which are used for cooling optimization (see Chapter 9). We first treat in detail Velocity Selective Coherent Population Trapping in Section A.1. Analytical expressions are given for the statistical parameters. Special attention is given to the p -dependences of the jump rates both for small p and for large p , because they control the asymptotic behaviours of the trapping and recycling times. It is thus important to include these p -dependences correctly in the simplified jump rates in order to ensure the validity of the statistical model. Raman cooling is then briefly treated in Section A.2. We only consider here the limit of small laser intensities (and a null detuning for VSCPT) but it is clear that the calculations can easily be generalized if needed. Velocity Selective Coherent Population Trapping We first present the quantum optics treatment of one-dimensional σ + /σ − VSCPT (Section A.1.1).

DOI
10.1017/cbo9780511755668.012

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Motivation As explained in Chapter 2, the Lévy statistics treatment of subrecoil laser cooling has been introduced after a series of simplifications, where we have dropped details of the quantum microscopic description to only keep the main features of the physical process. Such a way of reasoning is standard in statistical physics. It is difficult, however, to be sure a priori that one has not dropped important features, and the validity of the statistical approach needs to be checked. An important step of this verification, although not a rigorous proof, is to compare a posteriori the predictions of the statistical approach with experimental results as well as with the predictions of microscopic theoretical approaches. This chapter presents such comparisons. We present in Section 8.2 the approaches (theoretical and experimental) to which our statistical approach can be compared. We then proceed to compare the results obtained by the different approaches. First, in Section 8.3, we treat in detail the predictions for a global quantity, the proportion of trapped atoms. This is done for the three recycling models introduced in this work, in the one-dimensional case. Then, in Section 8.4, we study another physical quantity with a richer content, the momentum distribution of cooled atoms. In Section 8.5, we investigate the influence of the dimensionality of the problem, and the role of friction during the recycling periods – which are crucial predictions of the Lévy statistics analysis.

Cambridge University Press eBooks•Jul 2, 2010•0 citations

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Introduction The statistical approach presented in this book provides not only a deeper physical understanding of subrecoil cooling, but also analytical expressions for the various characteristics of the momentum distribution of the cooled atoms. A great confidence in the validity of these predictions has been obtained in the previous chapter, by comparing them with experimental and numerical results. Therefore, we are now entitled to apply the approach developed in this work to specific problems, such as the optimization of one particular feature of the cooling process, namely the height of the peak of cooled atoms. This is the subject of this chapter. Finding empirically the optimum conditions for a subrecoil cooling experiment is a difficult task. There are a priori many parameters to be explored and each experiment with a given set of parameters is in itself lengthy. The same can also be said of numerical simulations. One needs guidelines such as those provided by the present statistical approach to reduce the size of the parameter space to be explored. There is a variety of optimization problems that can be considered. Following usual motivations of laser cooling, like the increase of atomic beam brightness or the search for quantum degeneracy, we will concentrate here on optimizing the height h (θ) of the peak of the momentum distribution of the cooled atoms, which corresponds also to the gain in phase space provided by the cooling (see Section 6.2.3).

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Several asymptotic subrecoil laser-cooling behaviors in arbitrary degenerate Λ systems

Hong Yuan Ling

Subrecoil cooling can be achieved in three-level \ensuremath{\Lambda}-type atoms subject to counterpropagating laser beams via velocity-selective coherent population trapping [A. Aspect et al., J. Opt. Soc. Am. B 6, 2112 (1989)]. This process leads to narrow peaks in the ground but a flat structure in the excited atomic momentum distributions. By taking advantage of this difference, we construct several analytical functions from which the main cooling features (with the exception of the scaling law about the fraction of subrecoil cooled atoms) can be approximated under arbitrary parameters. This method allows us to generalize the peak-momentum width from symmetric to arbitrary degenerate \ensuremath{\Lambda} systems, and to discuss analytically the small dependence of the momentum distributions on the laser detuning near the two-photon resonance.

Physical Review A•May 1, 1999•5 citations
We establish here the correspondence between the statistical models introduced in Chapter 3 and the quantum evolution of atoms undergoing subrecoil laser cooling. This enables us to establish analytical expressions connecting the parameters of the statistical models (τ 0 , p 0 , p D , Δ p , p max , τ b and) to atomic and laser parameters relevant to subrecoil laser cooling. Such a ‘dictionary’ is useful for the numerical estimation of the results derived in this book (see Chapter 8). It also leads to analytical relations between τ b and, which are used for cooling optimization (see Chapter 9). We first treat in detail Velocity Selective Coherent Population Trapping in Section A.1. Analytical expressions are given for the statistical parameters. Special attention is given to the p -dependences of the jump rates both for small p and for large p , because they control the asymptotic behaviours of the trapping and recycling times. It is thus important to include these p -dependences correctly in the simplified jump rates in order to ensure the validity of the statistical model. Raman cooling is then briefly treated in Section A.2. We only consider here the limit of small laser intensities (and a null detuning for VSCPT) but it is clear that the calculations can easily be generalized if needed. Velocity Selective Coherent Population Trapping We first present the quantum optics treatment of one-dimensional σ + /σ − VSCPT (Section A.1.1).
Motivation As explained in Chapter 2, the Lévy statistics treatment of subrecoil laser cooling has been introduced after a series of simplifications, where we have dropped details of the quantum microscopic description to only keep the main features of the physical process. Such a way of reasoning is standard in statistical physics. It is difficult, however, to be sure a priori that one has not dropped important features, and the validity of the statistical approach needs to be checked. An important step of this verification, although not a rigorous proof, is to compare a posteriori the predictions of the statistical approach with experimental results as well as with the predictions of microscopic theoretical approaches. This chapter presents such comparisons. We present in Section 8.2 the approaches (theoretical and experimental) to which our statistical approach can be compared. We then proceed to compare the results obtained by the different approaches. First, in Section 8.3, we treat in detail the predictions for a global quantity, the proportion of trapped atoms. This is done for the three recycling models introduced in this work, in the one-dimensional case. Then, in Section 8.4, we study another physical quantity with a richer content, the momentum distribution of cooled atoms. In Section 8.5, we investigate the influence of the dimensionality of the problem, and the role of friction during the recycling periods – which are crucial predictions of the Lévy statistics analysis.
One-dimensional laser cooling based on velocity-selective coherent population trapping (VSCPT) has been investigated numerically through the solution of the optical Bloch equations and through a Monte Carlo analysis. The 1\ensuremath{\rightarrow}1 and 2\ensuremath{\rightarrow}2 transitions have been examined as a function of the atomic recoil frequency, the spontaneous-emission decay rate, and the Rabi frequency of the cooling laser. It has been found that for a large set of those parameters, the VSCPT cooling process may be described through scaling-law relations. The scaling laws are not valid at long atom-laser interaction times or large Rabi frequencies, where the atomic Doppler shift plays a significant role in the atomic motion evolution. Similar results for two atomic transitions suggest the validity of the scaling law for any one-dimensional VSCPT process. \textcopyright{} 1996 The American Physical Society.
We represent a new solution of the set of equations describing laser cooling of atoms by velocity selective coherent population tapping. Our solution is valid at large interaction time t in a full range of atomic kinetic momenta. Referring to this analytic expression we obtain the formula for lowest achievable temperature of cooled atoms for given experimental conditions. We demonstrate that the cooling efficiency decreases as t -3/4, when t is greater than or equal to (Gamma) 12-1, where (Gamma) 12 is the low-frequency atomic coherence decay rate. Also the case of subrecoil laser cooling in vertical direction is examined and the gravitational force influence is estimated.
Introduction The statistical approach presented in this book provides not only a deeper physical understanding of subrecoil cooling, but also analytical expressions for the various characteristics of the momentum distribution of the cooled atoms. A great confidence in the validity of these predictions has been obtained in the previous chapter, by comparing them with experimental and numerical results. Therefore, we are now entitled to apply the approach developed in this work to specific problems, such as the optimization of one particular feature of the cooling process, namely the height of the peak of cooled atoms. This is the subject of this chapter. Finding empirically the optimum conditions for a subrecoil cooling experiment is a difficult task. There are a priori many parameters to be explored and each experiment with a given set of parameters is in itself lengthy. The same can also be said of numerical simulations. One needs guidelines such as those provided by the present statistical approach to reduce the size of the parameter space to be explored. There is a variety of optimization problems that can be considered. Following usual motivations of laser cooling, like the increase of atomic beam brightness or the search for quantum degeneracy, we will concentrate here on optimizing the height h (θ) of the peak of the momentum distribution of the cooled atoms, which corresponds also to the gain in phase space provided by the cooling (see Section 6.2.3).
We present a fully quantum-mechanical analysis of laser cooling of an angular momentum Jg{\mathit{J}}_{\mathit{g}}Jg​=1 to Je{\mathit{J}}_{\mathit{e}}Je​=1 transition in a laser configuration consisting of two counterpropagating linearly polarized laser beams. The essential feature of this configuration is the coexistence of velocity-selective coherent population trapping (VSCPT) and polarization-gradient cooling. The role of polarization-gradient cooling is to provide (i) for short interaction times ``precooling'' of the initial momentum distribution and (ii) in the long-time limit ``confinement of velocities.'' This eventually leads to a larger number of atoms being captured in the dark state when compared with the schme of Aspect et al. [Phys. Rev. Lett. 61, 826 (1988)]. We find that the optimum parameter values for polarization-gradient cooling and VSCPT are in a completely different parameter regime: polarization-gradient cooling works best off resonance and for low intensities, while VSCPT works best on resonance. We can combine the advantages of polarization-gradient cooling and VSCPT in a scheme where we cycle in time between the optimum cooling parameters for both cooling mechanisms.
Subrecoil cooling can be achieved in three-level \ensuremath{\Lambda}-type atoms subject to counterpropagating laser beams via velocity-selective coherent population trapping [A. Aspect et al., J. Opt. Soc. Am. B 6, 2112 (1989)]. This process leads to narrow peaks in the ground but a flat structure in the excited atomic momentum distributions. By taking advantage of this difference, we construct several analytical functions from which the main cooling features (with the exception of the scaling law about the fraction of subrecoil cooled atoms) can be approximated under arbitrary parameters. This method allows us to generalize the peak-momentum width from symmetric to arbitrary degenerate \ensuremath{\Lambda} systems, and to discuss analytically the small dependence of the momentum distributions on the laser detuning near the two-photon resonance.