First-order transition into a topological superfluid state in an atom-cavity system

Abstract

We propose to combine Bose-Einstein condensation in higher Bloch bands and a driven-dissipative cavity-BEC system into a hybrid light-matter platform. Specifically, the condensate is trapped in a bipartite $s$-$p_x$-$p_y$-lattice, with a tunable energy offset. This enables a controlled population transfer from the $s$-orbital to the nearly degenerate $p_x$ and $p_y$ orbitals. The system forms a chiral ground state with $p_x \pm i p_y$ symmetry, with staggered orbital currents. By increasing the transverse pump strength, we drive the system into the superradiant phase, resulting in a self-organized, density checkerboard, which rectifies the staggered chiral order into a topological superfluid state. Using truncated Wigner simulations and complementary mean-field analysis, we determine the phase transition into this state as first order. Our results show that higher-band condensates coupled to a cavity provide a promising platform for engineering non-trivial orbital order and topological superfluid phases in quantum optical many-body systems.

Figures (6)

• Figure 1: Sketch of the cavity-BEC system. (a) A BEC trapped in an optical lattice with wavelength $\lambda_\mathrm{L}$, placed in a high-finesse cavity and transversely pumped by a laser beam with wavelength $\lambda_p$ and pump strength $\epsilon_p$ along the $y$-direction. Photons leaking out of the cavity with the loss rate $\kappa$ along the $x$-direction. (b) Lattice geometry consisting of the two sublattices $A$ and $B$, with tunable relative potential depth $\Delta V$. Sublattice $A$ contains $s$-orbitals (circles), while sublattice $B$ contains $p_x$-$p_y$-orbitals (dumbbells). The orbital phase convention is indicated by $\pm$ signs, which determine the sign of the tunneling amplitude $J$ between neighbouring sites. The unit cell is indicated by the gray shaded rectangle.
• Figure 2: Symmetry-breaking sequence in the $s$–$p_x$–$p_y$ lattice coupled to an optical cavity (gray regions indicate cavity mirrors). Reduced orbital visibility reflects lower occupation. (I) Normal Phase (NP): Atoms occupy $s$-orbitals on $A$-sites without phase coherence. Time-reversal and inversion symmetry are preserved. (II) Chiral Phase (CP): tuning the relative sublattice potential depth $\Delta V$ leading atoms to condense into chiral states $p_x \pm i p_y$, breaking time-reversal symmetry and generating local orbital currents (indicated by circular arrows). (III) Topological superfluid state, emerging due to chiral and superradiant ordering (CP + SP): Increasing pump strength $\epsilon_p$ induces a Dicke-type superradiant transition. The system condenses on even or odd $B$-sublattice sites, breaking lattice and time-reversal symmetry. The resulting states $\lvert \Phi^{\pm}_\mathrm{even} \rangle$ and $\lvert \Phi^{\pm}_\mathrm{odd} \rangle$ reflect this sublattice selectivity.
• Figure 3: (I) Dynamical emergence of the topological superfluid phase. We show the steady-state mean light field amplitude $|\alpha|$ as a function of the maximal pump strength $\epsilon_f$, in panel (I). The system evolves until it reaches a dynamical steady state. Each data point is the average over 100 trajectories using Truncated Wigner Approximation (TWA). (II) Individual TWA trajectories (blue) for three exemplary final pump strengths $\epsilon_f$ as a function of time $t$ in ms: (a,d,g) directly below the phase transition at $\epsilon_f=0.245$ , (b,e,h) directly above at $\epsilon_f=0.251$, and (c,f,i) far above at $\epsilon_f=0.450$. The pump beam protocol $\epsilon_p$ is indicated in red. We linearly increase $\epsilon_p$ within $5$ ms to its final values $\epsilon_f$ in each run. After reaching $\epsilon_f$, we hold it constant until the end of the simulation. Panel (a-c): We show the light field amplitude $|\alpha|$. Panel (d-f): We show the occupation imbalance between the two possible self-organized checkerboard patterns, normalised by $N_p$ the total number of particles in the $p_x$ and $p_y$-orbital. (g-i) We show the total angular momentum $L_z$ per particle in the $p$-orbitals in the lattice.
• Figure 4: Mean‐field energy landscape $E/E_0$ as a function mean $p$‐orbital occupancy $\bar{N}_p$ and imbalance $\Delta$. For coupling $g<g_c$, the single minimum at $(\bar{N}_p=N/2,\Delta=0)$ marks the normal phase (grey circle). For $g>g_c$, two degenerate minima (red dots) at finite $\pm\Delta$ emerge, indicating the superradiant phase transiton. $\bar{N}_p=N/2,\Delta=0$ remain a local minima (grey circle). Unphysical regions $|\Delta|>\bar{N}_p$ are shaded dark blue. (b) $\mathrm{min}_{\Bar{N}_p}E^\mathrm{MF}(\Delta)$ as a function of $\Delta$ for various $g$. The curves evolve from a parabola ($g = 0$, red) to a $\Phi^6$-type potential. At $g = g_c$, three degenerate minima appear. For $g > g_c$ (blue), two global minima emerge at finite $\Delta$ and a local minimum remains at $\Delta = 0$.
• Figure 5: Hysteresis of the intracavity field amplitude $|\alpha|$ as a function of the pump strength $\epsilon_p$. The pump strength is linearly increased from $\epsilon_p=0$ to $0.351$ over $300\,\mathrm{ms}$ (up-ramp) and, after a hold time of $100\,\mathrm{ms}$, decreased back to zero over the same duration (down-ramp).