Opening a gap in the collective excitation modes of a driven-dissipative condensate in the presence of an external coherent drive

TL;DR

This work develops a minimal model for driven-dissipative condensates under an external coherent drive, using a generalized Gross-Pitaevskii framework and Bogoliubov theory extended to limit cycles via Floquet analysis. By solving for stationary phase-locked states and exploring their linear and Floquet excitations, the authors identify regimes where a gap opens in the Bogoliubov spectrum, which can be purely imaginary or acquire a finite real part depending on detuning Δ and drive strength. When phase locking fails, a gapless Goldstone mode reappears and Floquet-band folding introduces additional structure; in strong-drive regions finite-wavenumber instabilities emerge, suggesting a route to supersolid-like spatial modulation. The results connect to recent exciton-polariton experiments and offer a framework applicable to other spatially extended optical oscillators and lasers, elucidating how external injection controls collective modes in non-equilibrium condensates.

Abstract

We build a minimal theoretical model to describe the opening of a gap in the dispersion of the collective excitations of a driven-dissipative condensate when the condensate phase is fixed by an additional coherent phase-locking drive. We map out the phase diagram as a function of the frequency and the strength of the coherent drive. We identify regions where the gap is purely imaginary or has a finite real part. When the coherent drive is unable to lock the condensate phase, a gapless Goldstone mode is recovered in the Floquet-Bogoliubov dispersion of collective modes. We finally characterize regions of finite-wavevector dynamical instability, where the condensate tends to develop a supersolid-like spatial modulation. While our theoretical framework is directly related to recent experiments with exciton-polariton condensates, it can be applied to describe the effect of external injection also in a variety of spatially extended optical parametric oscillators or laser devices.

Figures (9)

• Figure 1: Examples of the steady-state intensity $I_{ss}$ vs. incident intensity $I_{inc}$ for $\tilde{P}= 2$ (left) and $\tilde{P} = 0.75$ (right). The solid lines indicate dynamically stable steady states at $k=0$, while the dashed lines indicate dynamically unstable spatially uniform steady states. Three cases - no detuning (blue: $\tilde{\Delta} = 0$), weak detuning (red: $\tilde{\Delta} = 0.1$), and strong detuning (green: $\tilde{\Delta} = 0.3$) - are shown. Gray dashed lines and colored points indicate the values of the incident field intensity used in the next figures.
• Figure 2: Left: flow lines of the dynamic evolution of $\bar{E}$ in the complex plane for a vanishing detuning $\tilde{\Delta}=0$ and a relatively weak value of the external field $E_{inc}$ as indicated by the gray dashed line on the blue curve of the left panel of Fig. \ref{['graph: qualitative graphs cases 1 and 2']}. Right: Bogoliubov spectrum corresponding to the high-intensity attractor solution indicated in the left panel as a blue point.
• Figure 3: Flow lines of the dynamic evolution of $\bar{E}$ in the complex plane for a fixed weak detuning $\tilde{\Delta}= 0.1$ and decreasing values of $E_{inc}$ as indicated by the gray dashed lines on the red curve in the left panel of Figure \ref{['graph: qualitative graphs cases 1 and 2']}. The panels show the formation of a limit-cycle from a multi-solution region as $E_{inc}$ is decreased.
• Figure 4: Flow lines of the dynamic evolution of $\bar{E}$ in the complex plane for a fixed larger detuning $\tilde{\Delta}=0.3$ and decreasing values of $E_{inc}$ as indicated by the gray dashed lines on the green curve in the left panel of Figure \ref{['graph: qualitative graphs cases 1 and 2']}. The panels show the formation of a limit-cycle from a single-solution region for decreasing $E_{inc}$.
• Figure 5: Left: absence of real gap in the Bogoliubov spectrum for the stable solution in the small $\Delta \neq 0$ for both ${\Delta}<0$ (top) and ${\Delta} > 0$ (bottom). The specific parameters are indicated by the green points inside Fig. \ref{['graph: phase diagram']}. Right: emergence of a real gap in the Bogoliubov spectrum for the stable solution in a larger $\Delta \neq 0$ case for both ${\Delta}<0$ (top) and ${\Delta} > 0$ (bottom). The specific parameters are indicated by the blue points inside Fig. \ref{['graph: phase diagram']}. Dashed curves in the imaginary part highlight the onset of instability at finite wavevectors $k \neq 0$ as the system approaches the transition to a limit cycle. The specific parameters are indicated by the yellow points inside Fig. \ref{['graph: phase diagram']}.