Digital-Analog Simulations of Schrödinger Cat States in the Dicke-Ising Model

TL;DR

The paper develops a digital-analog quantum simulator for the Dicke-Ising model to study its superradiant quantum phase transition, circumventing the no-go constraint via circuit-QED implementations. It introduces a parity-based protocol that disentangles the photonic condensate from the qubits to realize a Schrödinger cat–like photonic state, observable through Wigner tomography. A path-integral free-energy framework and a quasi-classical spin representation illuminate the QPT structure, including instanton-driven tunneling and Kibble-Zurek–type dynamics in imaginary time. The proposed circuit relies on Jaynes-Cummings– and Rabi-based gates arranged in Dicke and Dicke-Ising sequences, with a quench protocol to prepare the superradiant state and a parity projection to extract the cat state, robust to realistic dissipation in superconducting hardware. This framework enables controlled exploration of macroscopic quantum coherence and critical fluctuations in finite-size spin-boson systems with potential for broader quantum simulation of light-matter critical phenomena.

Abstract

The Dicke-Ising model, one of the few paradigmatic models of matter-light interaction, exhibits a superradiant quantum phase transition above a critical coupling strength. However, in natural optical systems, its experimental validation is hindered by a "no-go theorem''. Here, we propose a digital-analog quantum simulator for this model based on an ensemble of interacting qubits coupled to a single-mode photonic resonator. We analyze the system's free energy landscape using field-theoretical methods and develop a digital-analog quantum algorithm that disentangles qubit and photon degrees of freedom through a parity-measurement protocol. This disentangling enables the emulation of a photonic Schrödinger cat state, which is a hallmark of the superradiant ground state in finite-size systems and can be unambiguously probed through the Wigner tomography of the resonator's field.

Figures (10)

• Figure 1: Reduced density matrices (a) for the mixed state $\hat{\rho}_{\rm mix} = {\rm tr}_\sigma [\hat{\rho}_{\rm SR}]$ and (b) when projected to the positive-parity subspace $\hat{\rho}_+={\rm tr}_\sigma[\hat{\rho}_{\rm SR}\hat{P}_+]$. (c) Wigner function of the mixed state, and (d) of the projected state showing non-classical features indicative of a cat state. The photon Hilbert space has a cutoff of 20 photons. The coupling $g=0.9\sqrt{\omega_0J}$ is near the critical value $\tilde{g}_{\rm c}$, the chain has open ends and comprises $N=7$ qubits. The other parameters are $J=\omega_0$ and $\omega_z=0.05\omega_0$.
• Figure 2: Sketch of the free energies as functions of the superradiant order parameter for (a-c) the Dicke and (d-f) the Dicke-Ising model. (a, d) Normal phases. Critical points (b) of the second-order and (e) first-order QPTs. (d, f) Superradiant phases. The values $u=\pm J/g$ in (f) correspond to the critical Ising chain.
• Figure 3: (a) Angular representation of qubit states. The $xz$-plane contributing most to the mean-field solution is shown in blue. (b) The effective potential in the mean-field approximation. The red dots are the two minima representing the superradiant states $|\Psi_{R,L}\rangle$. The dashed curve is an instanton trajectory. (c) Schematic representation of the instanton trajectory $u_{\rm inst}(\tau)$.
• Figure 4: Sketch illustrating the quench dynamics of the photon probability distribution $w(x,t)$ in the potential formed by the free energy $\mathcal{F}_{\rm DI}(u)$. (a) Gaussian $w(x,t)$ at $t=0$ when $g=0$. (b) Evolution of $w(x,t)$ after the quench of $g$ from 0 to $g\approx g_c$. (c) End of the evolution at $t=t_{\rm f}$ shows two maxima of $w(x,t_{\rm f})$ at $u=\pm u_0$ corresponding to superradiant condensates.
• Figure 5: (a) Qubit-resonator architecture for the Rabi model. (b) Resonant pulse for the Jaynes-Cummings gate. (c) Pulse sequence representing the Rabi gate.