Self-Ordered Limit Cycles, Chaos, and Phase Slippage with a Superfluid inside an Optical Resonator

We study dynamical phases of a driven Bose-Einstein condensate coupled to the light field of a high-$Q$ optical cavity. For high field seeking atoms at red detuning the system is known to show a transition from a spatially homogeneous steady state to a self-ordered regular lattice exhibiting superradiant scattering into the cavity. For blue atom pump detuning the particles are repelled from the maxima of the light-induced optical potential suppressing scattering. We show that this generates a new dynamical instability of the self-ordered phase, leading to the appearance of self-ordered stable limit cycles characterized by large amplitude self-sustained oscillations of both the condensate density and cavity field. The limit cycles evolve into chaotic behavior by period doubling. Large amplitude oscillations of the condensate are accompanied by phase slippage through soliton nucleation at a rate that increases in the chaotic regime. Different from a superfluid in a closed setup, this driven dissipative superfluid is not destroyed by the proliferation of solitons since kinetic energy is removed through cavity losses.

Figure 1

A Bose-Einstein condensate (blue surface) trapped inside an optical resonator is laser driven with a Rabi frequency $\mathrm{\Omega }$. The laser frequency is blue detuned by ${\mathrm{\Delta }}_{\mathrm{a}}$ with respect to an internal atomic transition $g\leftrightarrow e$ and by ${\mathrm{\Delta }}_c$ with respect a standing-wave cavity mode $\sim \cos (k_{c}x)$ defining the characteristic recoil frequency ${\omega }_R=\hslash k_c/2m$ with atomic mass $m$. The coupling between a single atom and the cavity mode has a strength $g_0$. For large ${\mathrm{\Delta }}_{\mathrm{a}}\gg ({\mathrm{\Delta }}_c,\kappa ,g_0,{\omega }_R)$, with $\kappa$ being the cavity linewidth due to leakage out of the mirrors, the dispersive regime is reached. Atoms experience negligible spontaneous emission and an optical potential (orange surface) arising from the interfering cavity and pump fields [see Eq. (1)].

Figure 2

(a) Nonequilibrium phase diagram as a function of the effective pump strength $\eta =\sqrt{N}{g}_0\mathrm{\Omega }/{\mathrm{\Delta }}_{\mathrm{a}}$ and the detuning ${\mathrm{\Delta }}_c$, for $\kappa =10{\omega }_R$, $U_{0}N=12.1{\omega }_R$, and $g_{\text{aa}}=0$. The color scale indicates the growth rate $\mathrm{Re}\omega$ of the most unstable collective excitation mode above the steady state. The time evolution of the cavity mode amplitude $\alpha$ is shown in (b), where panels I, II, and III correspond to the points marked in the phase diagram of (a). Four different phases emerge. In the normal phase ($N$) the steady state is an empty cavity with a homogeneous condensate. In the self-ordered phase (S-O) the steady state is stable and has a finite $\alpha$ accompanied by the corresponding condensate density modulation in space. In the self-ordered limit cycle (S-O-L-C) phase the steady state is unstable and evolves into periodic self-sustained oscillations of large amplitude about a finite value of $\alpha$. At every time during the oscillation the condensate is self-ordered, i.e., has chosen the density modulation giving rise to the instantaneous cavity amplitude. In the self-ordered chaotic phase (S-O-Chaos) the collective oscillations lose their periodicity. The transition from limit cycles to chaos takes place by period doubling, as illustrated in Fig. 3.

Figure 3

Transition from limit cycles to chaos. The Fourier transform of the cavity amplitude time oscillations (left column) is shown together with the condensate wave function recurrence [29] $R(t_1,t_2)=\int dx|\psi (x,t_1)-\psi (x,t_2){|}^2$ (middle column). For the same parameters as in Fig. 2, results are shown for three different pump strengths: (a) $\eta =9.5{\omega }_R$, (b) $\eta =10{\omega }_R$, and (c) $\eta =11{\omega }_R$ at ${\mathrm{\Delta }}_c=4{\omega }_R$. The spectra clearly show period doubling [between (a) and (b)] before the onset of the noisy background in (c). This indication for the appearance of chaos is confirmed by the large-scale structures in the recurrence $R$, as opposed to the stripes characterising a periodic behavior, visible in (a) and (b). In the latter, period doubling shows up as further small-scale structures perpendicular to the stripes in $R$.

Figure 4

Phase slippage through soliton nucleation for a finite value of the atom-atom interaction $g_{\mathrm{aa}}=1.0\hslash {\omega }_R$. The condensate density (color scale) as a function of position and time is shown in panels (a) and (b). The time evolution of the cavity amplitude is shown in panels (f) and (g). Panels (a) and (f) correspond to limit cycle dynamics at $\eta =12.0{\omega }_R$ while panels (b) and (g) correspond to chaotic dynamics at $\eta =22.0{\omega }_R$. In panels (a) and (b), white circles indicate the space-time coordinate of phase slips, i.e., phase singularities in the form of solitons having a zero-density core together with a $\pi$ phase difference across it. The dynamics of a single phase slip is illustrated in panels (c)–(e) for $\eta =22.0{\omega }_{\mathrm{R}}$. The phase singularity is present in (d) as a $\pi$-phase jump localized at the dark [$n(x)=0$] soliton position. The latter has a size of the order of the healing length $\xi =1/\sqrt{2m{g}_{c}n}\sim 0.1\lambda$. The phase slip consists in changing the sign of the phase jump $\pi \to -\pi$ [they are equivalent at the point $n(x)=0$]. Subsequently, in (e) the soliton increases its minimum density from zero and moves away, carrying the energy subtracted from the initial phase gradient in (c). Panel (h) shows the total number of phase slips $N_s(t)$ (solid line) generated up to time $t$ together with the condensate kinetic energy $E_k$ (dotted line) as a function of time, for $\eta =12.0{\omega }_R$ (in blue) and $\eta =22.0{\omega }_R$ (in red).