Non-Gaussian superradiant transition via three-body ultrastrong coupling

We introduce a class of quantum optical Hamiltonians characterized by three-body couplings and propose a circuit-QED scheme based on state-of-the-art technology that implements the considered model. Unlike two-body light-matter interactions, this three-body coupling Hamiltonian is exclusively composed of terms which do not conserve the particle number. We explore the three-body ultrastrong-coupling regime, showing the emergence of a superradiant phase transition which is of first order, is characterized by the breaking of ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ symmetry, and has a strongly non-Gaussian nature. Indeed, in contrast to what is observed in any two-body-coupling model, in proximity of the transition the ground state exhibits a divergent coskewness, i.e., quantum correlations that cannot be captured within semiclassical and Gaussian approximations. Furthermore, we demonstrate the robustness of our findings by including dissipative processes in the model, showing that the steady state of the system inherits from the ground states the most prominent features of the transition.

Figure 1

(a) Schematic representation of a system characterized by three-body USC realized, e.g., by the superconducting circuit in (b). (c) Rescaled photon number $\langle {\widehat{a}}^{†}\widehat{a}\rangle /\eta$ in the ground state [cf. Eq. (2)] vs the three-body coupling strength $g_0$. We go towards the thermodynamic limit by increasing $\eta$. At the transition, the population shows a discontinuity indicating a first-order transition. Also shown is the energy of (d) the first excited state $E_1$ and (e) the fourth excited state $E_4$. Note that the first three excited states are degenerate, $E_1=E_2=E_3$. At the transition, these energies go to zero and we obtain a fourfold-degenerate ground state, signaling spontaneous symmetry breaking. The next excited state $E_4$ shows the first-order discontinuity. The first-order transition point predicted by the semiclassical theory is indicated by a vertical dashed line in (c)–(e). We set $U_0/\omega =1$.

Figure 2

(a) Rescaled equal-time second-order correlation function $g^{\left(2\right)}\left(0\right)$ and (b) coskewness of the ground state vs the three-body coupling strength $g_0$. The inset shows a close-up of the coskewness for $\eta =10$. The first-order transition point predicted by the semiclassical theory is indicated by a vertical dashed line. The legend and parameters are the same as in Fig. 1.

Figure 3

(a) Rescaled photon number and (b) coskewness of the steady state vs the three-body coupling strength $g_0$. The parameters are the same as in Fig. 1 and $\kappa =\omega$. The number of trajectories at each point ensures that observables reached convergence within an error of $5\%$. Even for these values of noise, we still achieve a coskewness below $-1$.