Dicke-model quantum spin and photon glass in optical cavities: Nonequilibrium theory and experimental signatures

In the context of ultracold atoms in multimode optical cavities, the appearance of a quantum-critical glass phase of atomic spins has been predicted recently. Due to the long-range nature of the cavity-mediated interactions, but also the presence of a driving laser and dissipative processes such as cavity photon loss, the quantum optical realization of glassy physics has no analog in condensed matter and could evolve into a “cavity glass microscope” for frustrated quantum systems out of equilibrium. Here we develop the nonequilibrium theory of the multimode Dicke model with quenched disorder and Markovian dissipation. Using a unified Keldysh path integral approach, we show that the defining features of a low-temperature glass, representing a critical phase of matter with algebraically decaying temporal correlation functions, are seen to be robust against the presence of dissipation due to cavity loss. The universality class, however, is modified due to the Markovian bath. The presence of strong disorder leads to an enhanced equilibration of atomic and photonic degrees of freedom, including the emergence of a common low-frequency effective temperature. The imprint of the atomic spin-glass physics onto the photon subsystem realizes a “photon glass” state and makes it possible to detect the glass state by standard experimental techniques of quantum optics. We provide an unambiguous characterization of the superradiant and glassy phases in terms of fluorescence spectroscopy, homodyne detection, and the temporal photon correlation function $g^{\left(2\right)}\left(\tau \right)$.

Figure 1

Nonequilibrium steady-state phase diagram of the open multimode Dicke model (in units of the cavity detuning ${\omega }_0=1$), as a function of averaged atom-photon coupling $J$ ($y$ axis) and disorder variance $K$ ($x$ axis) and for ${\omega }_z=0.5$ (effective atom detuning) for different photon decay rates $\kappa$. QG is the quantum spin and photon glass; SR is the superradiant phase. The $T=0$ equilibrium phase diagram of Ref. [10] is recovered as $\kappa \to 0$. The SR-QG transition is not affected by $\kappa$.

Figure 2

Illustration of the dissipative spectral properties and universality class. As a function of probe frequency $\omega$ ($y$ axis) and the disorder variance $K$ ($x$ axis), we illustrate the different regimes in the phase diagram ($J=0$ for simplicity). In the normal phase, for frequencies $\omega <\alpha$ the system is represented by a dissipative Ising model, described by Eq. (2.2), while for frequencies $\omega >\alpha ,{\omega }_c$ it is described by nonuniversal behavior of a disordered spin fluid. In the glass phase ($K>K_c$), there exist two qualitatitvely distinct frequency regimes, separated by the crossover scale ${\omega }_c$; cf. Eq. (2.1). At the lowest frequencies, $\omega <{\omega }_c$ the system is described by the universality class of dissipative spin glasses. For $\omega >{\omega }_c$, we find that the system behaves quantitatively as an equilibrium spin glass. For $\alpha <{\omega }_c$ and $K<K_c$, there exists a dissipative crossover region (D-C in the figure), which is a precursor of the dissipative spin glass. It shows dissipative Ising behavior for the smallest frequencies and resembles the dissipative glass for frequencies ${\omega }_c>\omega >\alpha$.

Figure 3

Dissipative spectral properties and universality class of the single-atom spectral density $\mathcal{A}\left(\omega \right)$ (response signal of RF spectroscopy) in the quantum glass phase for parameters $K=0.01,J=0.1,{\omega }_z=2,\kappa =0.1,$ ${\omega }_0=0.7$. For frequencies $\omega <{\omega }_c$ below the crossover scale, the spectral density is overdamped and proportional to $\sqrt{\omega }$. For intermediate frequencies $\omega >{\omega }_c$, $\mathcal{A}$ is linear in the frequency, as for the nondissipative case [10], which is recovered in the limit $\kappa \to 0$.

Figure 4

Thermalization into quantum-critical regime of the atomic (red dashed line) and photonic (blue lines) distribution functions $F\left(\omega \right)$ when approaching the glass transition at a critical disorder variance $K_c$ for ${\omega }_0=1.3,{\omega }_z=0.5,\kappa =0.01,K_c=0.01$, $J=0.1$, and varying parameter $\delta =K_c-K$. For larger values of $J$, i.e., larger distance from the glass transition, the LET $2T_{\mathrm{eff}}={\lim }_{\omega \to 0}\omega F\left(\omega \right)$ of the photons is much lower than the LET of the atoms and the frequency interval for which atoms and photons are not equilibrated is larger. When the glass transition is approached, atoms and photons attain the same LET.

Figure 5

Emergent photon glass phase with algebraically decaying photon correlation function $g^{\left(2\right)}\left(\tau \right)$ at long times, for parameters ${\omega }_0=1,\kappa =0.4,{\omega }_z=6,J=0.4,K=0.16$. The time scale for which algebraic decay sets in is determined by the inverse crossover frequency ${\omega }_c$, given by Eq. (5.11). For comparison, we have also plotted the envelope of the exponential decay of the cor-relation function in the normal and superradiant phase. The short-time behavior of the correlation function is nonuniversal and not shown in the figure; however, $g^{\left(2\right)}\left(0\right)=3$ due to the effective thermal distribution for low frequencies. The parameter ${\tau }_0=O\left(\frac{1}{{\omega }_0}\right)$ was determined numerically.

Figure 6

Cavity glass microscope setup. Atoms are placed in a multimode cavity subject to a transversal laser drive with pump frequency ${\omega }_p$. The atoms are fixed at random positions by an external speckle trapping potential over regions inside the cavity, wherein mode functions $g({\mathbf{k}}_i,{\mathbf{x}}_l)$ randomly change sign as a function of the atomic positions, in order to provide frustration, as well as vary in magnitude. The more cavity modes, the better, and in particular the regime where the ratio of the number of cavity modes ($M$) over the number of atoms ($N$), $\alpha =M/N$ is kept sizable is a promising regime for glassy behavior [9, 15]. Photons leaking from the cavity with rate $\kappa$ give rise to additional dissipative dynamics and allow for output detection measurements.

Figure 7

Cavity glass microscope output of a typical fluorescence spectrum $S\left(\omega \right)$ (not normalized), decomposed in coherent $S_{\mathrm{c}}$ and incoherent part $S_{\mathrm{inc}}$ for the three distinct phases in the multimode Dicke model. The parameters $J,K$ are varied, while ${\omega }_0=1,\kappa =0.1,{\omega }_z=0.5$ are kept fixed for each panel. (Top) Normal phase, $(J,K)=(0.13,0.008)$. Central and outer doublets are visible but broadened by the disorder; only the incoherent contribution is nonzero. (Middle) Superradiant phase, $(J,K)=(0.4,0.008)$. The central doublets have merged due to the presence of a critical mode at $\omega =0$. At zero frequency there is a coherent $\delta$ contribution indicated by the arrow (dashed). (Bottom) Glass phase, $(J,K)=(0.13,0.017)$. There is a characteristic $\frac{1}{\sqrt{\omega }}$ divergence for small $\omega <{\omega }_c$ due to the nonclassical critical modes at zero frequency. The peak at $\omega =0$ is incoherent and can therefore easily be discriminated from the coherent peak in the middle panel. The inset in the top panel shows the behavior of the peak distance of $S\left(\omega \right)$ in the normal phase when approaching the glass phase. The two peaks approach each other and merge at the glass transition. The distance follows the dominant coherent exponent ${\alpha }_{\delta }\propto {\delta }^{\frac{3}{2}}$; cf. Sec. 2b.

Figure 8

Internal level scheme to generate tunable Dicke couplings between the ground-state levels $|1\rangle$, $|0\rangle$ and the cavity. Adapted from Dimer et al. [35].

Figure 9

Regular part of the spectral density $\mathcal{A}\left(\omega \right)$ in the SR phase for parameters $K=0.05$ and $J=0.4$ and varying $\kappa$ and ${\omega }_0$. For small frequencies $\omega <{\alpha }_{\delta }$ the spectral density is linear in $\omega$ and $\kappa$ and behaves as a square root for intermediate frequencies $\omega >{\alpha }_{\delta }$. For the nondissipative case ($\kappa \to 0$), the spectral weight develops a gap at low frequencies, which is indicated for $\kappa ={10}^{-3}$ (solid line). The lower panel depicts the low-frequency behavior of $\mathcal{A}$ (red dash-dotted line) for values $\kappa =0.03$ and ${\omega }_0=0.9$. The green (solid) and the black (dashed) line indicate the linear and square-root behaviors, respectively. Approaching the glass transition, ${\alpha }_{\delta }$ scales to zero $\propto {\delta }^{\frac{3}{2}}$.

Figure 10

Schematic illustration of the pole structure and critical dynamics in the present model: (a) the normal-to-superradiance transition in the dissipative Dicke model, (b) the normal-to-glass transition in the zero-temperature equilibrium model, (c) the normal-to-glass transition in the dissipative model. (a) When approaching the superradiance transition, two of the polaritonic modes advance to the imaginary axis and become purely imaginary before the transition point. This leads to the effective classical relaxational dynamics close to the transition. At the transition point, the ${\mathbb{Z}}_2$ symmetry is broken by only a single mode approaching zero and becoming critical for $J\to J_c$. (b) For moderate disorder $K$, the poles are located on the real axis away from zero. For increasing $K$, the poles approach zero, with the closest pole scaling proportional to $|K-K_c{|}^{\frac{1}{2}}$. At $K=K_c$ the modes form a continuum which reaches zero and becomes quantum critical. No dissipative dynamics is involved. (c) For moderate disorder $K$, the set of modes is located in the complex plane, away from zero. For increasing variance $K$, the modes get shifted closer to the origin, however, due to the scaling of real ($\propto$$|K-K_c{|}^{\frac{3}{2}}$) and imaginary part ($\propto$$|K-K_c{|}^2$), they become neither purely real nor purely imaginary. At $K=K_c$ a continuum of modes reaches zero.

Figure 11

Spectral equilibration. Photon $x$-$x$ spectral response ${\mathcal{A}}_{xx}\left(\omega \right)=-2\text{Im}\left[G_{xx}^R\left(\omega \right)\right]$ in the glass phase for parameters $K=0.04,J=0.12,{\omega }_z=2,{\omega }_0=1,\kappa =0.02$. As for the SR phase, it shows the same low-frequency behavior as the atomic spectral response $-2\text{Im}\left[Q^R\left(\omega \right)\right]$ (multiplied with a constant $\frac{{\omega }_z{\omega }_0}{2\lambda }$). As for the atomic spectral density, one can clearly identify the overdamped regime with the unusual square-root behavior and the linear regime, separated by the frequency ${\omega }_c$.

Figure 12

Illustration of homodyne detection of a weakly driven cavity. The cavity is driven via a weak coherent input field $\eta \left(t\right)$ entering the cavity through one of the mirrors. Then a homodyne measurement is performed on the output signal of the driven cavity. For this, the output signal is superimposed with a reference laser $\beta \left(t\right)$ via a $50/50$ beam splitter and the difference current of the two outgoing channels is measured. From this, the response function of the photons in the cavity can be measured by tuning the relative phases and frequencies of $\beta \left(t\right)$ and $\eta \left(t\right)$, as explained in the text.