Dissipation-Induced Instabilities of a Spinor Bose-Einstein Condensate Inside an Optical Cavity

We investigate the dynamics of a spinor Bose-Einstein condensate inside an optical cavity, driven transversely by a laser with a controllable polarization angle. We focus on a two-component Dicke model with complex light-matter couplings, in the presence of photon losses. We calculate the steady-state phase diagram and find dynamical instabilities in the form of limit cycles, heralded by the presence of exceptional points and level attraction. We show that the instabilities are induced by dissipative processes that generate nonreciprocal couplings between the two collective spins. Our predictions can be readily tested in state-of-the-art experiments and open up the study of nonreciprocal many-body dynamics out of equilibrium.

Figure 1

Spinor Bose-Einstein condensate composed of two hyperfine states $m_F=\pm 1$, coupled to a single-mode optical cavity with photon loss rate $\kappa$ and transversely driven by a laser whose polarization vector ${\overrightarrow{\epsilon }}_p$ is at an angle $\phi$ with respect to the cavity field polarization vector ${\overrightarrow{\epsilon }}_c$. This leads to a finite contribution from the vectorial polarizability of the atoms, resulting in complex light-matter couplings, of equal strength but opposite phase, between the hyperfine states and the cavity field [29].

Figure 2

(a) Ground-state phase diagram, obtained from mean-field theory, and (b) steady-state phase diagram, determined by the semiclassical equations of motion and a linear stability analysis (2), as a function of the light-matter coupling $\lambda$ and phase $\varphi$, for $\mathrm{\Delta }=-400{\omega }_0$ and $\kappa =250{\omega }_0$. Going beyond adiabatic elimination and including the cavity field fluctuations renders the NP unstable (light orange shading) for $\varphi \ne 0,\pm (\pi /2)$.

Figure 3

(a) The imaginary and real part of the eigenvalues ${\eta }_{\pm ,\pm }$ of the dynamical matrix (2), resulting from adiabatic elimination of the cavity field, for $\lambda =2{\omega }_0$, $\mathrm{\Delta }=-400{\omega }_0$, and $\kappa =250{\omega }_0$. We observe level attraction in the spectrum, consequence of the emergence of exceptional points. (b) The real part of the pair of eigenvalues ${ϵ}_{+}^{\pm }$ (solid lines) responsible for antidamping in the NP, obtained from the full dynamical matrix including cavity field fluctuations, for $|\mathrm{\Delta }|/{\omega }_0=10,25,50,100,1000,10000$, $\kappa =0.625|\mathrm{\Delta }|$ and $\lambda =2{\omega }_0$. As the bad-cavity limit is approached, the eigenvalues reduce to ${\eta }_{-,+}$ (dashed lines) given by Eq. (3). All quantities are in units of ${\omega }_0$.

Figure 4

Dynamics in the unstable regime, with $\mathrm{\Delta }=-40{\omega }_0$, $\kappa =25{\omega }_0$, $\lambda =5{\omega }_0$, and $\varphi =\pi /4$. (a) Time evolution of the average photon number, displaying limit-cycle oscillations in the long-time limit. (b) Bloch spheres depicting the long-time dynamics (blue) of the collective spins preceding the start of the steady-state limit cycle (thick red line). Insets: (i) Magnified picture of the limit-cycle oscillations. (ii) Time evolution for $\varphi =0.4$, corresponding to the NP within adiabatic elimination, but unstable when including the cavity field dynamics.