Tunable intertwining via collective excitations

TL;DR

This work demonstrates that a minimal driven-dissipative platform—namely a Bose-Einstein condensate at the intersection of two crossed optical cavities—hosts two competing $\ extbf{Z}_2$ density-wave orders that can be dynamically intertwined via periodic driving. By using asynchronous driving with a phase lag between cavities, the authors induce a dynamical breaking of U(1) symmetry and stabilize a rich landscape of intertwined phases, including intertwining of Landau and time-crystal orders as well as between Landau and time-crystal orders across both Higgs and Goldstone polariton branches. The approach combines mean-field theory, fluctuation analysis, Floquet-Magnus expansion, and Lindblad dynamics to map out a comprehensive phase diagram featuring Landau, time-crystal, and their intertwinings, all within experimentally accessible parameter regimes. These findings open a path to realizing and controlling out-of-equilibrium vestigial and intertwined phases in cavity-QED platforms with potential applications in quantum simulation and non-equilibrium materials science.

Abstract

The intertwining of multiple order parameters is a widespread phenomenon in equilibrium condensed matter systems, yet its exploration is often hindered by the complexity of real materials. Here, we present a controlled study of intertwined orders in a minimal and versatile driven-dissipative quantum-engineered platform. We consider a Bose-Einstein condensate at the intersection of two optical cavities, realizing two competing copies of a $\mathbb{Z}_2$ symmetry-breaking superradiant phase transition characterized by density wave orders. Using periodic drives that exploit dynamical symmetry reduction, we show that collective excitations can be harnessed to stabilize a variety of novel intertwined orders. Going beyond the conventional phenomenology involving Landau orders, we show the emergence of a larger class of out-of-equilibrium intertwined phases, including intertwining of purely time-crystalline orders, as well as between Landau and time crystal orders. These results should be observable in state of the art experimental setups.

Figures (9)

• Figure 1: (a) Schematic setup of the cross cavity. The BEC (turquoise patch) is illuminated by a transverse laser pump, from which photons scatter into cavity modes 1 and 2 (green and purple). The light-matter couplings $\lambda_{1,2}$ are modulated with time period $T$ and a phase difference $\phi$. (b) Static phase diagram of the model in Eq. (\ref{['eq:Hamiltonian']}) and schematic asynchronous driving scheme (red ellipse). White, green, and purple regions correspond to normal phase, and superradiant phase for cavity 1 and 2, respectively. Solid (dashed) lines indicate second (first) order phase transitions. The line $\lambda_1=\lambda_2$ is endowed with U(1) symmetry. (c) Polariton energies for $\lambda_1=\lambda_2$ , with Goldstone (Higgs) branches shown in cyan (black), and the higher-energy branches in gray. (d) Schematic depiction of intertwining, with $\alpha_i^{L,M}$ indicating Landau ($L$) and TC ($M$) order w.r.t. cavity $i$. Green (purple) diamonds indicate single-mode superradiance in cavity 1 (2) in the static system.
• Figure 2: (a) Phase diagram of the Higgs mode in the ($\phi$,$\epsilon$) plane, with $\kappa=0.1$, $\lambda_0=0.85$, and $\omega=0.92$, satisfying parametric resonance with the Higgs polariton branch. All frequencies and units are in terms of $\omega_a$ which has been set equal to one. Here $\phi=0$ is the U(1)-symmetry broken phase, shown in green. While the white region is devoid of a coherent steady state, gray, blue, and red regions indicate periodic symmetry-broken phases, see text. Representative trajectories in the ($\alpha_1$,$\alpha_2$) plane are shown in panels (b), (c), and (d), respectively, where gray circle indicates the static potential minimum. In (d) the two steady state trajectories are shown in dark and pale markers as a guide for the eye. For a complete set of steady states see supplementary.
• Figure 3: (a) Phase diagram associated with the GS mode in the ($\phi$,$\omega$) plane, with $\epsilon=0.1$, $\lambda_0=0.85$, and $\kappa=0.17$. The white, green and gray regions are the same as in Fig. \ref{['fig:Higgs']}. Blue, red, and orange regions indicate symmetry broken phases, with representative steady state trajectories shown in panels (b), (c), and (d), respectively, see text. Here gray circle shows the static potential minimum. The shaded region in (a) indicates coexistence of gray and blue. Highly fine tuned steady states occupying very narrow regions have been removed from the lowest-frequency regime. For a complete set of steady states see supplementary.
• Figure 4: Phase diagram with the GS mode in the $(\phi, \omega)$ plane, with cavity detuning $\omega_a=10$ an order of magntitude larger than the atomic detuning $\omega_b = 1$. The rest of the parameters in untis of $\omega_b$ are $\lambda_0 =2.5$, $\epsilon=0.1$, $\kappa=0.17$. Colors represent the same phases as in Fig. 3 of the main text. Highly fine tuned steady states occupying very narrow regions have been removed from the lowest-frequency regime.
• Figure 5: Four steady states corresponding to the L-TC phase for the Higgs resonance with $\lambda=0.85$, $\omega=0.92$, $\epsilon=0.1$, $\phi=0.8$, $\kappa=0.1$.