Dynamical phases of a BEC in a bad optical cavity at optomechanical resonance

TL;DR

The study addresses dynamical phases of a Bose-Einstein condensate coupled to a dissipative optical cavity in the bad-cavity limit, focusing on the regime near optomechanical resonance where the atoms’ refractive index tunes the cavity into resonance. Using second-order perturbation theory in the atom-cavity coupling, the authors derive a mean-field atomic Hamiltonian that includes a density-dependent dynamical Stark shift and compare full cavity dynamics to adiabatic and non-adiabatic eliminations. They find that away from resonance adiabatic elimination yields chaotic-like instabilities, while near resonance non-adiabatic corrections are essential and reveal stable limit-cycle density oscillations that are metastable under adiabatic dynamics. Retardation effects encoded in a minimal non-adiabatic elimination (Eq. (23)) reproduce relaxation and stabilize the limit cycle, providing a clear mechanism for coherent dynamical phases in systems with large timescale separation and guiding experimental observation of limit cycles in optomechanical atom-cavity setups.

Abstract

We study the emergence of dynamical phases of a Bose-Einstein condensate that is optomechanically coupled to a dissipative cavity mode and transversally driven by a laser. We focus on the regime close to the optomechanical resonance, where the atoms' refractive index shifts the cavity into resonance, assuming fast cavity relaxation. We derive an effective model for the atomic motion, where the cavity degrees of freedom are eliminated using perturbation theory in the atom-cavity coupling and benchmark its predictions using numerical simulations based on the full model. Away from the optomechanical resonance, perturbation theory in the lowest order (adiabatic elimination) reliably describes the dynamics and predicts chaotic phases with unstable oscillations. Interestingly, the dynamics close to the optomechanical resonance are qualitatively captured only by including the corrections to next order (non-adiabatic corrections). In this regime we find limit cycle phases that describe stable oscillations of the density with a well defined frequency. We further show that such limit cycle solutions are metastable configurations of the adiabatic model. Our work sheds light on the mechanisms that are required to observe dynamical phases and predict their existence in atom-cavity systems where a substantial timescale separation is present.

Figures (5)

• Figure 1: Schematic of a experimentally realizable setup. A BEC of $^{87}\text{Rb}$ in the dispersive regime is trapped inside an optical cavity and is transversely driven by a coherent pump laser with Rabi frequency $\Omega$ that is far red-detuned with respect to the atomic transition of interest. The BEC is coupled to a single dissipative mode with decay rate $\kappa$.
• Figure 2: Steady state imaginary time phase diagrams of the (a) rescaled cavity field intensity $\abs{\alpha_0}^2$, (b) order parameter $\abs{\Theta}$, and (c) effective cavity detuning $\delta_c/\kappa$ as a function of the bare cavity detuning $\Delta_c$ and effective cavity pump strength $\sqrt{N}\eta$. We vary $\sqrt{N}\eta$ and fix $N$ keeping $NU_0 = -2241.38\omega_r$ constant and $\kappa = 344.83\omega_r$. The solid white line denotes the phase boundary from the unorganized to self-organized phase [Eq. \ref{['last']}]. We choose a momentum cutoff of 20$\hbar k_c$.
• Figure 3: (a) Maximum imaginary eigenvalue Im$(\omega_{\mathrm{crit}})$ determined from Eq. \ref{['eq19']} as a function of the effective cavity pump strength $\sqrt{N}\eta$ and bare cavity detuning $\Delta_c$. (b) Maximum imaginary eigenvalue determined from Eq. \ref{['eq21']}. The light grey shaded regions represent the unorganized stable phase (US) and the darker grey shaded regions represent the organized stable phase (OS). The black dashed line is at $\Delta_c = NU_0/2$ and the black contour is determined by Eq. \ref{['last']}. We vary $\sqrt{N}\eta$ and keep $N$ constant and use $NU_0 = -2241.38\omega_r$ and $\kappa = 344.83\omega_r$. We choose a momentum cutoff of 20$\hbar k_c$.
• Figure 4: Real time quench dynamics of the rescaled cavity field intensity $\abs{\alpha}^2$. (a) Heating phase with $\Delta_c = -700\omega_r$ and $\sqrt{N}\eta = 80\omega_r$$[\square-$symbol in Fig. \ref{['fig4']}(a)], (b) stationary self-organized phase with $\Delta_c = -2150\omega_r$ and $\sqrt{N}\eta = 27\omega_r$ [$\boldsymbol{+}$-symbol in Fig. \ref{['fig4']}(b)], (c) limit cycle phase with $\Delta_c = 1900\omega_r$ and $\sqrt{N}\eta = 22\omega_r$$[\boldsymbol{\diamond}-$symbol in Fig. \ref{['fig4']}(a)], and (d) chaotic phase with $\Delta_c = 1450\omega_r$ and $\sqrt{N}\eta = 60\omega_r$$[\boldsymbol{\circ-}$symbol in Fig. \ref{['fig4']}(a)]. For the red lines we simulate the cavity field expression of \ref{['eq:mfcavity']}, the black lines we simulate Eq. \ref{['eq5']}, and for the blue lines we simulate Eq. \ref{['eq23']}. The other parameters are the same as in Figs. \ref{['fig3']}.
• Figure 5: (a) Real time quench dynamics of the rescaled cavity field intensity for the limit cycle phase, where we simulate Eqs. \ref{['eq3']} and \ref{['eq:mfcavity']}. (b) Real time dynamics with adiabatic cavity field Eq. \ref{['eq5']}, where we initialized (b) with the final point of (a). (c) Imaginary time dynamics using Eq. \ref{['eq5']} and Eq. \ref{['eq6']}. (d) Real time dynamics where we initialized (d) with the final point of (c). The parameters are $\Delta_c = -1900\omega_r$ and $\sqrt{N}\eta = 22\omega_r$, while the other parameters are the same as in Fig. \ref{['fig3']}