Master Equation for a Quantum Gas of Polarizable Particles in Cavities

Abstract

Quantum gases of atoms and molecules in optical cavities offer a formidable laboratory for studying the out-of-equilibrium dynamics of open quantum systems with long-range interactions. Long-range interactions are here mediated by multiple scattering of cavity photons and can induce the formation of quantum structures in space and time. Control of these dynamics requires a detailed understanding of all relevant mechanisms at play. Due to the strong correlations induced by light, however, perturbative theoretical models, which reduce the number of degrees of freedom, do not correctly capture the regime where the interplay of photon-mediated long-range forces and quantum fluctuations of light and matter become significant, such as across the transition to self-organization. In this work, we present the derivation of an effective Lindblad master equation for the dynamics of the sole motional variables of polarizable particles, such as atoms or molecules, that dispersively couple to cavity fields. The master equation is valid even for relatively large intracavity photon numbers, and is apt to study both the steady-state regime and the out-of-equilibrium dynamics where quantum fluctuations of the field seed the onset of macroscopic coherences. We validate the theoretical description by showing that it captures the dynamics across a wide temperature interval, from Doppler cooling down to the ultra-cold regime, and from weak to strong cavity-mediated interactions. Our theory provides a powerful framework for the description of cavity-induced dynamics of quantum matter. In doing so, it permits to connect models of statistical mechanics with cavity-QED experimental platforms, thus enabling quantum simulation of long-range interacting matter.

Figures (3)

• Figure 1: (a) Characteristic setup of many-body CQED that consists of polarizable particles (e.g., atoms or molecules) strongly coupled to one or multiple optical resonators (here two modes are illustrated). (b) In the optomechanical regime, the excited electronic states are only virtually populated: Photon absorption and emission coherently couple the motion of the particles (black circles) to the cavity field modes (large shaded circles), which in turn can dissipate photons via emission through the mirrors (wavy arrows). (c) The resulting atom-only dynamics is illustrated by a globally connected network, where the nodes represent the particles and the links correspond to photon-mediated interactions. (d) Regime of validity for different approaches to describe the optomechanical master equation. Mean-field approaches fail to describe atom-cavity correlations. Weak-coupling theories have limited validity for describing atom-cavity correlations and rely on a timescale separation. The focus of this work is to identify the regimes in which the dynamics can be described by atom-only master equations that can capture strong atom-cavity correlations.
• Figure 2: Evolution of the mean kinetic energy $E_\mathrm{kin}$ for an atom undergoing cavity cooling. Energy and time are in units of $\hbar \omega_\mathrm{R}$ and of $\omega_\mathrm{R}^{-1}$, respectively. The solid blue and the dashed orange curves are, respectively, obtained by numerically integrating the optomechanical, Eq. \ref{['eq:opto_master_equation_disp']}, and the atom-only master equation, Eq. \ref{['Eq:mu']}; see Appendix \ref{['App:Cavity cooling']} for their explicit expressions. The dotted green curve corresponds to the prediction of the Gaussian ansatz, Eq. \ref{['eq:eom_E_kin']}. The parameters are: a) $(\eta,\Delta,\kappa) = (1, -20, 20)\,\omega_\mathrm{R}$ and b) $(\eta,\Delta,\kappa) = (1, -20, 5)\,\omega_\mathrm{R}$. The initial atomic state is a thermal distribution $\exp(-\beta \hat{p}^2/(2m))$ with initial temperature $\beta^{-1} = 20\,\hbar\omega_\mathrm{R}$. In the optomechanical master equation, the cavity is initially in the vacuum state $\ket{\mathrm{vac}}$. The insets show the Kurtosis, Eq. \ref{['Eq:Kurtosis']}, extracted from the full simulations of the optomechanical master equation. All equations were implemented and numerically integrated using the library described in Ref. Kramer:2018.
• Figure 3: Benchmark of the atom-only master equation for the Bose-Hubbard model coupled to a dissipative cavity mode for a lattice with $N = 2$ bosons and $L = 4$ sites and open boundary conditions. The left panels correspond to the adiabatic regime, with $\varepsilon =0.04$. In the right panels, $\varepsilon =0.36$, such that diabatic corrections become relevant. Upper row: Eigenvalues $\lambda$ (in units of $J$) of the adiabatic atom-only Lindbladian $\mathcal{L}_{0,\mathrm{eff}}$, and of the atom-only Lindbladian $\mathcal{L}_{1,\mathrm{eff}}$ including the first diabatic correction. The eigenvalues of the optomechanical Lindbladian $\mathcal{L}_\mathrm{cav}$, Eq. \ref{['Eq:L:full']}, are presented up to a chosen energy cutoff. The insets zoom into the slowest-decaying bundle of eigenvalues. Lower row: Evolution of $\langle \hat{\Theta}_{\rm BH}^2 \rangle$ predicted using the optomechanical (blue solid line) and the atom-only Lindbladians, with (orange dotted line) and without (pink dashed line) the first diabatic correction. The black dotted-dashed line in subplot c) marks the value $\langle \hat{\Theta}_\mathrm{BH}^2\rangle/N^2 = 0.6$ for a fully mixed state. The inset of subplot d) shows the dynamics of the cavity population $\langle \hat{a}^\dagger\hat{a}\rangle$ using the optomechanical Lindbladian (blue solid line) and the expectation value $\langle \hat{\alpha}^\dagger\hat{\alpha}\rangle$ for $\mathcal{L}_{1,\mathrm{eff}}$ (orange dotted line). The atoms are initially in the ground state of the Bose-Hubbard model $\hat{H}_\mathrm{S}$, Eq. \ref{['Eq:H:BH']}. In the optomechanical simulations, the cavity mode is initially in the vacuum state. The parameters are $u = 2.5J$, $\Delta = -500J$, $\kappa = 500J$ and (a,c) $\eta=100J$, (b,d) $\eta=300J$. Numerical simulations were performed using the framework of Ref. Kramer:2018.