Universal Phase Transitions of Matter in Optically Driven Cavities

Abstract

Optical cavities have been widely applied to manipulate the properties of solid state materials inside them. We propose that in systems embedded within optical cavities driven by incident pump light, the pump induces generic phase transitions into new nonequilibrium steady states. This effect arises from the ponderomotive potential, the effective static potential exerted by the pump on the low energy degrees of freedom, which exhibits a universal steplike structure that pushes the matter degrees of freedom in the direction that redshifts the cavity photon modes. For a two dimensional electron liquid in a driven cavity, this steplike potential pushes the electron density to jump to a smaller value so that a hybrid cavity photon mode is redshifted to slightly below the pump frequency. Similarly, for a dirty superconductor in such a driven cavity, this potential acts on the superconducting order parameter and causes a first order phase transition to a new steady state with a smaller gap. By realistic electromagnetic modeling of the cavity that includes all cavity modes, we construct the nonequilibrium phase diagrams for experimentally relevant devices.

Figures (4)

• Figure 1: (a) Schematic of a Fabry–Pérot cavity of thickness $h$ with the sample (an ultra-thin film or a strictly two-dimensional material) inside it. The mirror has the reflection and transmission coefficients $R(\omega)$ and $T(\omega)$. The red color scale represents the in-plane electric field of an anti-node cavity photon mode. (b) The dressed cavity photon frequencies $\omega_{\text{N}}$ at zero in-plane momentum as functions of $\phi$, the slow degree of freedom on the sample. The red color scale represents the ponderomotive force on $\phi$, which peaks at $\phi_{\text{c}}$ when the cavity photon is in resonance with the pump $\omega$. (c) The ponderomotive potential $V_{\text{P}}$ and ponderomotive force $F_{\text{P}}$ felt by $\phi$. (d) The typical total potential felt by $\phi$ for $E_{\text{p}}=0$ (black dashed line), $E_{\text{p}}= E_{\text{c}}$ (red dashed line), and $E_{\text{p}}>E_{\text{c}}$ (solid red line).
• Figure 2: (a) Schematic of the 2D electron gas in the driven cavity. The 2DEG is electrically connected to a gate outside of the cavity. (b) Phase diagram of the device on the plane of pump frequency $\omega$ and electric field $E_{\text{P}}$, where the color scale corresponds to the calculated $\Delta n$, showing a clear phase boundary. The red dashed line shows the analytical prediction for the phase boundary from Eq. \ref{['eq:Ec']} with the damping correction. The parameters are $m=0.01 \,\mathrm{m_e}$, $n_0=3\times10^{14} \,\mathrm{\mathrm{cm}^{-2}}$, $h=1.2 \,\mathrm{\mu m}$, $C=0.1 \,\mathrm{\mu \mathrm{F}/\mathrm{cm}^{-2}}$, $T^2=1-R^2= 2 \times 10^{-4}$, and $\gamma = 1 \,\mathrm{meV}$ for the Drude optical conductivity. (c) The blue curve is $n_c$ as a function of $\omega$. The black dashed line represents $n_0$ and the red zone highlights its difference from the possible new phase.
• Figure 3: (a) The total potential felt by the order parameter $\Delta$ of a superconductor in the device in Fig. \ref{['fig:FPcavity']}(a) for $E_{\text{p}}=0$ (black dashed line), $E_{\text{p}}\approx E_{\text{c}}$ (blue dashed line), and $E_{\text{p}}>E_{\text{c}}$ (red solid line). (b) Numerically exact phase diagram of the device where the color scale represents $\delta\Delta = \Delta_0-\Delta_{\text{s}}$, the (negative) change of gap relative to its equilibrium value. A clear phase boundary is marked by the red solid curve, while the red dashed line is the analytical approximation to it. The parameters are $m=0.3 \,\mathrm{m_e}$, $\tau=0.03 \,\mathrm{ps}$, $\Delta_0=12\,\mathrm{K}$, $g\nu=0.5$, $n_0=1\times10^{13}\mathrm{cm}^{-2}$, $h=5\,\mathrm{mm}$ and $R =0.99$.
• Figure 4: (a) The three Feynman diagrams contributing to the ponderomotive force $F_{\text{P}}$ at second order in $A$. Solid lines are the $2\times2$ Green functions of quasi-particles in Keldysh notation with their arrows pointing from particle creation to annihilation. (b) The top panel shows the equilibrium superconducting gap as a function of temperature (blue curve). The dashed line shows half of the driving frequency. The bottom panel shows the $f_1$ (red curves) and $f_2+f_3$ (blue/cyan curves) components of the ponderomotive force normalized by $|F_{\text{P}}|$ at zero temperature. The solid lines correspond to $\eta/\Delta_0=1/10$ and dashed lines correspond to $\eta/\Delta_0=1/100$. The parameters are $g\nu =0.5$, $\Delta_0 = 12\,\mathrm{K}$ and $\Omega = 1.7 \,\mathrm{K}$ ($0.035\,\mathrm{THz}$).