A cat state is formed as the steady-state solution for the signal mode of an ideal degenerate parametric oscillator, in the limit of negligible single-photon signal loss. In the presence of signal loss, this is no longer true over timescales much longer than the damping time. However, for sufficient parametric nonlinearity, a cat state can still exist as a transient state. In this paper we study the dynamics of the creation and decoherence of cat states in degenerate parametric oscillation, both with and without the Kerr nonlinearity found in recent superconducting-circuit experiments that generate cat states in microwave cavities. We determine the time of formation and the lifetime of a cat state of fixed amplitude in terms of three dimensionless parameters $\ensuremath{\lambda}$,
g, and $\ensuremath{\chi}$. These relate to the driving strength, the parametric nonlinearity relative to signal damping, and the Kerr nonlinearity, respectively. We find that the Kerr nonlinearity has little effect on the threshold parametric nonlinearity (
g>1) required for the formation of cat states and does not significantly alter the decoherence time of the cat state, but can reduce the time of formation. The quality of the cat state increases with the value of
g. To verify the existence of the cat state, we consider several signatures, including interference fringes and negativity. We emphasize the importance of taking into account more than one of these signatures. We simulate a superconducting-circuit experiment using published experimental parameters and find good agreement with experimental results, indicating that a nonclassical catlike state with a small Wigner negativity is generated in the experiment. Interference fringes, however, are absent, requiring higher
g values. Finally, we explore the feasibility of creating large cat states with a coherent amplitude of 20, corresponding to 400 photons, and study finite-temperature reservoir effects.
Non-Gaussian continuous-variable states play a central role both in the foundations of quantum theory and for emergent quantum technologies. In particular, ``cat states,'' i.e., two-component macroscopic quantum superpositions, embody quantum coherence in an accessible way and can be harnessed for fundamental tests and quantum information tasks alike. Degenerate optical parametric oscillators can naturally produce single-mode cat states and thus represent a promising platform for their realization and harnessing. We show that a dissipative coupling between degenerate optical parametric oscillators extends this to two-mode entangled cat states, i.e., two-mode entangled cat states are naturally produced under such dissipative coupling. While overcoming single-photon loss still represents a major challenge towards the realization of sufficiently pure single-mode cat states in degenerate optical parametric oscillators, we show that the generation of two-mode entangled cat states under such dissipative coupling can then be achieved without additional hurdles. We numerically explore the parameter regime for the successful generation of transient two-mode entangled cat states in two dissipatively coupled degenerate optical parametric oscillators. To certify the cat-state entanglement, we employ a tailored, variance-based entanglement criterion, which can robustly detect cat-state entanglement under realistic conditions.
The degenerate parametric oscillator is an important source of squeezed light. If the pump mode is adiabatically eliminated, its operation is governed by two dimensionless parameters: the pump parameter λ and the coupling constant g. The pump parameter controls the threshold behavior predicted by semiclassical theory (threshold is reached for λ = 1). The coupling constant g determines the strength of the quantum fluctuations. The optical parametric oscillators used to generate squeezed light have g ~ 10–4 ≪ 1.1 Therefore, in these systems the quantum fluctuations only produce very small perturbations about semiclassical states. We have recently shown how the steady state of the degenerate parameter oscillator changes as a function of coupling strength.2 In the strong coupling limit, major departures from semiclassical states occur; for g2 ≫ 1 and 2λ2/g2 ≪ 1 the steady state approches the coherent state superposition | ψ + 〉 = | λ / g 〉 + | − λ / g .
Nondegenerate parametric oscillation and four-wave mixing generate correlated twin-beams that exhibit squeezing in the intensity difference fluctuations. Using a simple theoretical model for parametric amplification we predict that the twin-beam devices generate macroscopic states that violate Bell’s inequality. In four-wave mixing, or parametric oscillation with an amplitude-squeezed pump, the twin-beams may also exhibit squeezing in the intensity of each beam. We predict that the use of these intensity-squeezed devices is advantageous in the preparation of the macroscopic quantum states. It is also possible to use nondegenerate systems to generate macroscopic superposition (Schrodinger cat) states.
We theoretically investigate the generation of non-Gaussian quantum states, specifically Schrödinger cat-like states (SCLSs), via degenerate dual-pump spontaneous four-wave mixing in a
χ(3)-based microring resonator. By introducing a unitary transformation that exactly decouples the self-phase modulation (SPM) and cross-phase modulation (XPM) terms, we reduce the full nonlinear Hamiltonian to an effective three-mode interaction. The resulting dynamics (decoupled and full Hamiltonians) are studied using the Lindblad master equation, accounting for cavity losses. Unlike semiclassical or parametric approximations, our full quantum mechanical approach explicitly includes quantum pump depletion, which enables the emergence and observation of non-Gaussian features. We compute the Wigner function, photon number distributions, quadrature variances, Fano factor, Schmidt number, and fidelity to characterize the generated states. For the non-dissipative case, we find that the signal mode
b^3 or
a^3 exhibits clear non-Gaussian features with a structured Wigner function and even-dominated photon number distribution, characteristic of an even coherent state. In the presence of dissipation (
γj=0.2), the interference fringes become faint, odd photon numbers appear, and the fidelity with the ideal state remains high (
>0.9), indicating robustness. The pump mode
b^1 or
a^1 remains Gaussian, while both modes display super-Poissonian statistics and entanglement (
>2). Our results demonstrate that degenerate dual-pump spontaneous four-wave mixing in microring resonators is a promising platform for generating and controlling cat-like states under dissipative conditions.