Emergent thermal fluctuations and non-Hermitian phase transitions in open photon condensates

Abstract

We investigate the nonequilibrium dynamics of an open photon Bose-Einstein condensate in a dye-filled microcavity using a Lindblad master-equation approach, treating the condensate and the noncondensed fluctuations on the same footing. The driven-dissipative condensate exhibits a long-lived, metastable plateau stabilized by a ghost attractor, a fixed point that lies outside the physical domain in configuration space, yet stalls the condensate dynamics for exceedingly long times before it dephases to zero [Phys. Rev. Lett. 135, 053402 (2025)]. Despite the nonequilibrium origin of this dynamical stabilization, the condensate exhibits quasithermal fluctuations in the plateau in that the relative order-parameter fluctuations scale as the inverse square root of the system size. A linear stability analysis further reveals the presence of exceptional points, resulting in multiple non-Hermitian phase transitions associated with the relaxation dynamics into and out of the metastable condensate.

Figures (8)

• Figure 1: Sketch of the setup. The microcavity with single-mode frequency $\Omega$ is filled with dye molecules of electronic transition frequency $\Delta$. The inset shows fast relaxation of the vibrational modes of molecules (green arrows). The dissipative processes include the phonon-assisted absorption (emission) rates $\Gamma_{\mathrm{a}}$$(\Gamma_{\mathrm{e}}$), the cavity photon loss $\kappa$, the external pumping and nonradiative decay of the excited molecules $\gamma_{\pm}$, respectively.
• Figure 2: Relaxation dynamics. The time evolution of the photon condensate density $|\psi|^2$ is shown for three different values of the absorption and emission ratio, $\log_{10}(\Gamma_{\mathrm{a}}/\Gamma_{\mathrm{e}})=$1 -4.5, 2 -3.3 and 3 -0.1, as marked in \ref{['fig:stability map']}. Correspondingly, the dynamics of the excited-molecule fraction $m_{\mathrm{e}}$ are shown in the inset, where the dashed line indicates $m_{\mathrm{e}} = 0.5$. The other parameters are set to be $\gamma_+/\kappa=1.3\times10^{-4}$, $\gamma_- / \kappa=0$, $M=5.17\times10^9$, $g_\beta/\kappa=10^{-6}$, and $\Gamma_{\mathrm{a}}/\kappa=10^{-9}$Kroha21TimBode19. Unless stated otherwise, these values are used throughout the paper.
• Figure 3: Flow diagram. The projection of the phase-space trajectory corresponding to the time trace 1 of \ref{['fig:time-evol']} is shown in the plane of condensate fraction $\nu$ versus total photon number $n$. The slow dynamics near the unphysical fixed point $\overline{X}^{\mathrm{G}}$ (blue diamond) are demonstrated in the inset by the times of the evolution. The dashed line represents the physical boundary of $\nu=1$. The incoming and outgoing arrows from $\overline{X}^{\mathrm{G}}$ indicate the slowest attractive and repulsive eigenmodes. $\overline{X}^0$ (orange diamond) denotes the steady-state fixed point with photon number $\overline{n}^0$.
• Figure 4: Colormap of condensate lifetime. The lifetime of the photon condensate $\tau^G$ is plotted as a color map in the the ratio $\Gamma_{\mathrm{a}}/\Gamma_{\mathrm{e}}$ versus pumping rate $\gamma_+/\kappa$ plane. The vertical, white lines represent constant photon densities $\overline{n}^0$. The white area at the top corresponds to the lasing regime, with infinitely-lived condensate ($\tau\cdot \kappa \rightarrow \infty$) and molecular population inversion, $m_{\mathrm{e}}>1/2$. The condensate time-evolution at the marked points $\Circled{1}$, $\Circled{2}$ and $\Circled{3}$ are shown in \ref{['fig:time-evol']}.
• Figure 5: Order-parameter fluctuations. Normalized fluctuations $\sigma$ of the condensate order parameter $\psi$ are shown as a function of molecule number $M$ for $\gamma_+/\kappa = 0.1$: (a) at different times indicated in the legend and marked in the time evolution (inset); (b) at a plateau time ($t\cdot \kappa=10^2$) for different photon-molecule coupling strengths. The dashed lines indicate a slope of $-1/2$, corresponding to the scaling $\sigma \sim 1/\sqrt{M}$. The inset in (a) shows the time evolution of total photon number $n$ and condensate photon number $|\psi|^2$ at $M=5.17\times 10^9$ (arrow). In (b) the effective photon-molecule coupling is controlled by the bare Jaynes-Cummings coupling $g$, where $g_\beta \propto g$ and $\Gamma_{\mathrm{a,e}} \propto g^2$Pelster18. The inset shows the saturation values of $\sigma$ versus $g$ in the limit of large system size, $M=3 \times 10^{14}$.