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arXiv:1202.2992 · Physical Review A

Rate-equation approach to cavity-mediated laser cooling

Tony Blake, Andreas Kurcz, Almut Beige

Published Jul 23, 2012• 7 citations
arXivHTMLPDF

Abstract

The cooling rate for cavity-mediated laser cooling scales as the Lamb-Dicke parameter $\ensuremath{\eta}$ squared. A proper analysis of the cooling process hence needs to take terms up to ${\ensuremath{\eta}}^{2}$ in the system dynamics into account. In this paper, we present such an analysis for a standard scenario of cavity-mediated laser cooling with $\ensuremath{\eta}\ensuremath{\ll}1$. Our results confirm that there are many similarities between ordinary and cavity-mediated laser cooling. However, for a weakly confined particle inside a strongly coupled cavity, which is the most interesting case for the cooling of molecules, numerical results indicate that more detailed calculations are needed to model the cooling process accurately.

DOI
10.1103/physreva.86.013419

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The cooling rate for cavity-mediated laser cooling scales as the Lamb-Dicke parameter \ensuremathη\ensuremath{\eta}\ensuremathη squared. A proper analysis of the cooling process hence needs to take terms up to \ensuremathη2{\ensuremath{\eta}}^{2}\ensuremathη2 in the system dynamics into account. In this paper, we present such an analysis for a standard scenario of cavity-mediated laser cooling with \ensuremathη\ensuremath≪1\ensuremath{\eta}\ensuremath{\ll}1\ensuremathη\ensuremath≪1. Our results confirm that there are many similarities between ordinary and cavity-mediated laser cooling. However, for a weakly confined particle inside a strongly coupled cavity, which is the most interesting case for the cooling of molecules, numerical results indicate that more detailed calculations are needed to model the cooling process accurately.
Cavity-mediated cooling has the potential to become one of the most efficient techniques to cool molecular species down to very low temperatures. In this paper we analyse cavity cooling with single-laser driving for relatively large cavity decay rates κ and relatively large phonon frequencies ν. It is shown that cavity cooling and ordinary laser cooling are essentially the same within the validity range of the Lamb–Dicke approximation. This is done by deriving a closed set of rate equations and calculating the corresponding stationary state phonon number and cooling rate. For example, when ν is either much larger or much smaller than κ, the minimum stationary state phonon number scales as κ2/16ν2 (strong confinement regime) and as κ/4ν (weak confinement regime), respectively.
Cavity-mediated cooling has the potential to become one of the most efficient techniques to cool molecular species down to very low temperatures. This thesis studies the use of rate equations to analyse the cooling process in such systems. In particular the master equation is used to find rate equations that can determine the rate of change of phonons in the system. The general idea behind cavity cooling is the continuous conversion of phonons into cavity photons. While there is no spontaneous emission and decay rate associated with the concept of phonons, photons are created after a change in the phonon number has occurred and can then leak out through the cavity mirrors easily. Hence the conversion of phonons into photons can result in the constant removal of phonon energy from the system. In this thesis we compare cavity mediated cooling with single particle laser cooling. It is shown that cavity cooling is essentially the same as ordinary laser cooling. This is done by calculating the stationary state phonon number mss and the cooling rate y as a function of the system parameters. For example, when the trap phonon frequency υ is either much larger or much smaller than the cavity decay rate k , the minimum stationary state phonon number scales as k²/16v² (strong confinement regime) and as k/4 (weak confinement regime), respectively. Replacing k with Ѓ yields the steady states associated with ordinary laser cooling.
Cavity-mediated cooling has the potential to become one of the most efficient techniques to cool molecular species down to very low temperatures. In this paper we analyse cavity cooling with single-laser driving for relatively large cavity decay rates kappa and relatively large phonon frequencies nu. It is shown that cavity cooling and ordinary laser cooling are essentially the same within the validity range of the Lamb-Dicke approximation. This is done by deriving a closed set of rate equations and calculating the corresponding stationary state phonon number and cooling rate. For example, when nu is either much larger or much smaller than kappa, the minimum stationary state phonon number scales as kappa^2/16 nu^2 (strong confinement regime) and as kappa / 4 nu (weak confinement regime), respectively.
We analyze theoretically a one-dimensional model of laser cooling of an atom or ion trapped in a cavity. We assume that the cavity loss rate is much larger than the atom-cavity coupling (bad-cavity limit) and that the atomic excited state is weakly occupied (low saturation limit). After elimination of the cavity mode and the atomic excited state, we derive rate equations for the populations of the trap states. We find that in the Lamb-Dicke limit the atom can be cooled to the ground state of the trap even in the strong confinement limit. This result is interpreted in terms of quantum interferences between different cooling and heating processes involving spontaneous emission in the cavity.
This paper analyzes the cooling of a single particle in a harmonic trap with red-detuned laser light with fewer approximations than previously done in the literature. We avoid the adiabatic elimination of the excited atomic state but are still interested in Lamb-Dicke parameters $\ensuremath{\eta}\ensuremath{\ll}1$. Our results show that the Rabi frequency of the cooling laser can be chosen higher than previously assumed, thereby increasing the effective cooling rate but not affecting the final outcome of the cooling process. Since laser cooling is already a well-established experimental technique, the main aim of this paper is to present a model which can be extended to more complex scenarios, like cavity-mediated laser cooling.
We extend an earlier semiclassical model to describe the dissipative motion of N atoms coupled to M modes inside a coherently driven high-finesse cavity. The description includes momentum diffusion via spontaneous emission and cavity decay. Simple analytical formulas for the steady-state temperature and the cooling-time for a single atom are derived and show surprisingly good agreement with direct stochastic simulations of the semiclassical equations for N atoms with properly scaled parameters. A thorough comparison with standard free-space Doppler cooling is performed and yields a lower temperature and a cooling-time enhancement by a factor of M times the square of the ratio of the atom-field coupling constant to the cavity decay rate. Finally, it is shown that laser cooling with negligible spontaneous emission should indeed be possible, especially for relatively light particles in a strongly coupled field configuration.