Amplified response of cavity-coupled quantum-critical systems

TL;DR

We show that coupling a cavity mode to a quantum-critical degree of freedom dramatically enhances photon–matter responses and readily accesses a superradiant state. Directly coupling to the critical mode yields a vanishing threshold for superradiance as the quantum critical point is approached, with $g_c\to 0$ at $h\to h_{\text{TFIM}}$; enhanced intrinsic squeezing occurs with $\Delta X_{\text{min}}^2$ vanishing near SRPT and $\Delta X_{\text{max}}^2$ diverging as $1/(g-g_c)$. When the cavity couples to a noncritical mode, the light–matter competition yields a minimum SRPT boundary near the TFIM QCP and reveals a tricritical point with anomalous scaling; the optical order parameter scales as $\langle a\rangle \sim (g-g_c)^{\beta}$ with $\beta\approx 0.25$. The results identify quantum criticality as a general resource for enhanced entanglement and metrological performance in cavity–matter platforms and are supported by large-$S$ theory, 1D DMRG, and exact analyses of related XY models.

Abstract

A quantum critical point develops when matter undergoes a continuous transformation between distinct ground states at absolute zero. It hosts pronounced quantum fluctuations, which render the system highly susceptible to external perturbations. While light-matter coupling has rapidly moved forward as a means to probe and control quantum materials, the capacity of quantum critical fluctuations in the photon-mediated responses has been largely unexplored. Here we advance the notion that directly coupling a quantum critical mode to a quantized cavity field dramatically facilitates the onset of superradiance. When the coupling between the two fields is bilinear, the transition is found to occur at vanishingly small light-matter coupling and is accompanied by strongly enhanced intrinsic squeezing. Our results identify a particularly favorable setting for realizing the elusive superradiant state, and point to a general principle by which quantum criticality amplifies photon-matter entanglement and enhances the associated metrological performance.

Figures (6)

• Figure 1: Schematic of a cavity mode coupled to matter degrees of freedom near a quantum critical point (QCP). (a) Phase diagram as a function of a non-thermal tuning parameter $x$. The coupling strength ($g_c$) required to induce a superradiant quantum phase transition (SRPT) is minimized at the matter QCP ($x_c$). The minimum $g_c$ vanishes when the cavity model couples directly to the critical mode (solid curve) but remains nonzero otherwise (dashed curve). (b) Schematic of the cavity magnetic field aligned parallel to the Ising spin-coupling direction. The red sinusoidal curve with arrows represents the cavity magnetic field mode. (c) Same setup as in (b), but with the cavity magnetic field oriented perpendicular to the Ising spin-coupling direction.
• Figure 2: Phase diagram, squeezing and entanglement for a cavity mode directly coupled to quantum critical mode. (a) Phase diagram in the large-$S$ limit, depicting $\expval{\hat{a}}$; black curve denotes continuous superradiant phase transitions that terminates at the TFIM QCP (red dot); blue bar refers to the ferromagnetic (FM) order. We set $zS=1$.
• Figure 3: Phase diagram for a cavity mode coupled to non-critical degrees of freedom. (a) Analytically obtained exact phase diagram where the color bar indicates $\expval{\hat{a}}$; red solid line marks the quantum critical point of the one-dimensional transverse-field Ising model; dashed (solid) black lines denote discontinuous (continuous) superradiant phase transitions; red dot represents the tricritical point. (b) Scaling of $\expval{\hat{a}}$ with $(g - g_c)^{\beta}$ in the vicinity of the tricritical point at $h/J \approx 0.55$. The crossover behavior is dictated by Eq. \ref{['eq:N']}.
• Figure S1: Expansion coefficients of the Landau free energy. Dependence of the quadratic $f(x)$, quartic $u(x)$, and sixth-order $v(x)$ coefficients on the scaled magnetic field $x=h/J$. The vertical dashed line marks the tricritical point at $h/J \approx 0.55$, separating first-order and second-order transitions.
• Figure S2: Quantum phase transitions in the anisotropic XY model coupled to cavity photons. (a) Zero-temperature phase diagram of the anisotropic XY model coupled to cavity photons for $\boldsymbol n = \hat{z}$. The vertical solid line at $\Delta=0$ marks the quantum critical point of the anisotropic XY model where the excitations beomes gapless. Dashed lines denote the first order transition from ferromagnetically ordered normal phase to the superradiant phase. (b) Superradiant order parameter $\phi=\langle a+a^{\dagger} \rangle/\sqrt{N}$ with respect to light-matter coupling $g$. (c) Zero-temperature phase diagram of the anisotropic XY model coupled to cavity photons for $\boldsymbol n = \hat{x}$ [cf. Eq. \ref{['eq:fullH']}] and $\Delta < 0$ [cf. Eq. \ref{['eq:XY']}] as the $XY$ quantum critical point at $\Delta = 0$ is approached. The color bar indicates the value of $\widetilde{\Theta}(\mathcal{N}) = \Theta(\mathcal{N} - 1) + \mathcal{N} \Theta(1 - \mathcal{N})$, where $\Theta$ is the Heaviside theta function and $\mathcal{N}$ is obtained from DMRG simulations.