Phase Diagram and dynamical phases of self organization of a Bose-Einstein condensate in a transversely pumped red-detuned cavity

TL;DR

This work advances the understanding of a Bose-Einstein condensate coupled to a red-detuned single-mode cavity by solving the full mean-field dynamics with an extended momentum-state basis. The authors map the steady-state phase diagram as a function of pump strength and cavity detuning, revealing rich behavior beyond the Dicke model, including bistability, chaotic dynamics, polariton-resonance instabilities, and stable atomic superpositions with zero cavity field. They develop and apply a fixed-point and Floquet analysis to classify phases, identify mechanisms for instabilities, and show how higher momentum states crucially modify the phase structure. The results have direct implications for experiments in cavity QED with BECs, suggesting new dynamical regimes to probe and guiding future beyond-mean-field treatments in driven-dissipative quantum systems.

Abstract

We study a transversely pumped atomic Bose-Einstein Condensate coupled to a single-mode optical cavity, where effective atom-atom interactions are mediated by pump and cavity photons. A number of experiments and theoretical works have shown the formation of a superradiant state in this setup, where interference of pump and cavity light leads to an optical lattice in which atoms self-consistently organize. This self-organization has been extensively studied using the approximate Dicke model (truncating to two momentum states), as well as through numerical Gross-Pitaevskii simulations in one and two dimensions. Here, we perform a full mean-field analysis of the system, including all relevant atomic momentum states and the cavity field. We map out the steady-state phase diagram vs pump strength and cavity detuning, and provide an in-depth understanding of the instabilities that are linked to the emergence of spatio-temporal patterns. We find and describe parameter regimes where mean-field predicts bistability, regimes where the dynamics form chaotic trajectories, instabilities caused by resonances between normal mode excitations, and states with atomic dynamics but vanishing cavity field.

Figures (12)

• Figure 1: A cartoon of the experiment and the self-organization transition. The atom cloud (a) below and (b) above the critical pumping threshold. Due to the intracavity light field which builds up in the superradiant phase, the atoms organize in a checkerboard pattern. Adapted with permission from Ref. bhaseen_dynamics_2012, Copyright (2012) by the American Physical Society.
• Figure 2: Energy level scheme, showing the possible photon-mediated transitions starting from the $|0,0\rangle$ state. Blue lines denote interaction with a pump photon, red lines interaction with a cavity photon. The energy difference between ground and excited atomic states is $\omega_a$. The pump frequency $\omega_p$ is red detuned from this energy by $\Delta_a$. The detuning of the pump from the cavity frequency, $\omega_c$, is denoted by $\omega$. The atomic momentum states are multiples of the cavity recoil momentum $q$ and their energies are thus set by the corresponding recoil energy $\omega_r={q^2}/{2m}$.
• Figure 3: (c) The phase diagram of the model as a function of dimensionless pump strength $P$ and pump-cavity detuning $\omega$. (a)--(b): the cavity field amplitude $|\lambda|^2$ of the fixed points and their stability along two cuts marked by solid black lines in the phase diagram. Additional cuts corresponding to subsequent figures are marked as gray lines, and specific points at which dynamics is shown later are marked as yellow diamonds.
• Figure 4: Linear stability mode analysis along the cuts at (a)--(c) $P=0.55$ and (d)--(f) $P=0.75$ indicated in Fig. \ref{['fig:full_diagram']}. (a)/(d): Energies $\omega_\gamma$ of the modes while varying $\omega$. (b)/(e): The corresponding growth/decay rates ($\Gamma_\gamma$, same color), zoomed out in inset. (c)/(f): The photonic participation factor (see Eq. \ref{['eq:participation_factor']}) of the modes. At mode crossings, mixing of the eigenvectors can modify the photonic part of otherwise mostly density-wave like modes, changing their decay rates. The modes' directions are identified by comparing their energies to the eigenenergies of 1D cosine potentials, see App. \ref{['app:mode_class']}.
• Figure 5: A fit of the matrix eigenvalues of Eq. \ref{['eq:fitmatrix']} to the energies and growth/decay rates of the resonance at $\omega/E_0 \approx 2.25$, $P=0.55$ (see Fig. \ref{['fig:modes']}. Fit parameters found are $\alpha_1=(7.03+0.0053i)\times 10^{-3}$, $\beta_1=2\pi\times(-25.4 - 0.0094 i)$ MHz,$\alpha_2=(1.96+0.0303i)\times10^{-2}$, $\beta_2=2\pi\times(-18.5 + 0.045 i)$ MHz, $\delta=2\pi\times(0.0466 + 0.423 i)\times10^{-3}$ MHz.