Stabilizing Arrays of Photonic Cat States via Spontaneous Symmetry Breaking

The controlled generation and the protection of entanglement is key to quantum simulation and quantum computation. At the single-mode level, protocols based on photonic cat states hold strong promise as they present unprecedentedly long-lived coherence and may be combined with powerful error correction schemes. Here, we demonstrate that robust ensembles of “many-body photonic cat states” can be generated in a Bose-Hubbard model with pair hopping via a spontaneous $U(1)$ symmetry-breaking mechanism. We identify a parameter region where the ground state is a massively degenerate manifold consisting of local cat states which are factorized throughout the lattice and whose conserved individual parities can be used to make a register of qubits. This phenomenology occurs for arbitrary system sizes or geometries, as soon as long-range order is established, and it extends to driven-dissipative conditions. In the thermodynamic limit, it is related to a Mott insulator to pair-superfluid phase transition.

Figure 1

(a) Zero-temperature phase diagram in the even-parity sector as a function of the pair hopping $J$ and the chemical potential $\mu$ in units of the interaction $U$. The fidelity $F_{\text{cat}}=\underset{\alpha }{\max }(|⟨{\psi }_{\mathrm{GW}}|{\mathcal{C}}^{+(\alpha )}⟩{|}^2)$ between the Gutzwiller wave function and the closest cat state is represented in color plot. (b) Wigner quasiprobability distribution [39] of the Gutzwiller wave function $|{\psi }_{\mathrm{GW}}⟩$ for three points in the phase diagram, respectively located in the Mott Insulating phase ($P_1$) and in the pair-superfluid phase in the weak ($P_2$) and strong ($P_3$) PSF regimes. (c) Pair-superfluid order parameter as a function of $J$ at fixed chemical potential $\mu /U=1.2$. The green solid line and the orange circles represent the predictions of the Gutzwiller analysis and the semiclassical result of Eq. (5), respectively. In the inset, the formation of a Mexican hat potential sketches the spontaneous symmetry-breaking mechanism generating the many-body photonic cat states.

Figure 2

(a),(b) Phase diagram in the driven-dissipative scenario, truncated to the even-parity sector, as a function of the pair hopping $J$ and the detuning $\delta ={\omega }_{\text{at}}/2-{\omega }_c$ in units of the interaction $U$. The two-photon loss rate is set to ${\mathrm{\Gamma }}_l={10}^{-2}{\mathrm{\Gamma }}_p$. The pair-superfluid order parameter ${\psi }_0^{(2)}=⟨a_i^2⟩$ and the fidelity $F_{\text{cat}}=\underset{\alpha }{\max }⟨{\mathcal{C}}^{+(\alpha )}|{\rho }_{\mathrm{GW}}|{\mathcal{C}}^{+(\alpha )}⟩$ between the Gutzwiller density matrix ${\rho }_{\mathrm{GW}}$ and the closest cat state are represented in color plot in (a) and (b), respectively. (c),(d) ${\psi }_0^{(2)}$ and $F_{\text{cat}}$ are respectively represented for various dissipative rates ${\mathrm{\Gamma }}_{\mathrm{em}}^0$, ${\mathrm{\Gamma }}_l$ as a function of $J$ and at fixed detuning $\delta =2U$. The results of the Gutzwiller analysis are displayed in solid lines, and compared for ${\psi }_0^{(2)}$ to the semiclassical results with (without) the effect of saturation, displayed in crosses (black dashed line). All panels feature an interaction strength $U/{\mathrm{\Gamma }}_p=0.7$, and the Rabi coupling ${\mathrm{\Omega }}_R$ is chosen to maintain a fixed ratio ${\mathrm{\Gamma }}_{\mathrm{em}}^0/{\mathrm{\Gamma }}_l=9$ between pumping and losses. The various curves of (c) and (d) correspond to ${\mathrm{\Gamma }}_l/{\mathrm{\Gamma }}_p=3\times {10}^{-2}$ (blue), $1\times {10}^{-2}$ (orange), $3\times {10}^{-3}$ (yellow), and $1\times {10}^{-3}$ (purple).