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.
A cat-state is a superposition of two coherent states with amplitudes
α0 and
−α0. Recent experiments create cat states in a microwave cavity field using superconducting circuits. As with degenerate parametric oscillation (DPO) in an adiabatic and highly nonlinear limit, the states are formed in a signal cavity mode via a two-photon dissipative process induced by the down conversion of a pump field to generate pairs of signal photons. The damping of the signal and the presence of thermal fluctuations rapidly decoheres the state, and the effect on the dynamics is to either destroy the possibility of a cat state, or else to sharply reduce the lifetime and size of the cat-states that can be formed. In this paper, we study the effect on both the DPO and microwave systems of a squeezed reservoir coupled to the cavity. While the threshold nonlinearity is not altered, we show that the use of squeezed states significantly lengthens the lifetime of the cat states. This improves the feasibility of generating cat states of large amplitude and with a greater degree of quantum macroscopic coherence, which is necessary for many quantum technology applications. Using current experimental parameters for the microwave set-up, which requires a modified Hamiltonian, we further demonstrate how squeezed states enhance the quality of the cat states that could be formed in this regime. Squeezing also combats the significant decoherence due to thermal noise, which is relevant for microwave fields at finite temperature. By modeling a thermal squeezed reservoir, we show that the thermal decoherence of the dynamical cat states can be inhibited by a careful control of the squeezing of the reservoir. To signify the quality of the cat state, we consider different signatures including fringes and negativity, and the
Cl1 measure of quantum coherence.
We explore in detail the possibility of generating a pair-coherent state in the nondegenerate parametric oscillator when decoherence is included. Such states are predicted in the transient regime in parametric oscillation where the pump mode is adiabatically eliminated. Two specific signatures are examined to indicate whether the state of interest has been generated, the Schrödinger cat state - like signatures, and the fidelity. Solutions in a transient regime reveal interference fringes which are indicative of the formation of a Schrödinger cat state. The fidelity indicates the purity of our prepared state compared with the ideal pair-coherent state.
We investigate the quantum properties of a nonlinear Kerr oscillator driven by a three-photon pump. We derive both exact and approximate analytical expressions for the ground state of this interacting model. The exact solution arises at an exact spectral degeneracy, while the approximate solution describes regimes of quasi-degeneracy of the energy spectrum. In both cases, the threefold (quasi)degenerate ground-state manifold consists of quantum superpositions of three macroscopically distinct states. These states differ qualitatively from conventional three-component Schrödinger's cat states due to the presence of squeezing with a distinctive parametric dependence. By varying the detuning between the oscillator and the three-photon pump, we show that the squeezing can be enhanced, suppressed, or even reversed, leading to a squeezing-to-anti-squeezing transition. We analyze the generation and stabilization of these superposition states, their robustness against perturbations and analytically quantify the leakage to excited states. Our analysis elucidates how the three-photon Kerr parametric oscillator can be used to encode a Kerr-cat qutrit protected against phase-flip errors.
A degenerate parametric oscillator operating above threshold is studied when the cavity is injected by a broadband squeezed light at the pump frequency. Using the solution of the quantum Langevin equations for the signal and pump modes, we have studied the quantum fluctuations in the intracavity and transmitted field quadratures for the signal and pump modes. For the intracavity modes we studied the variance in the field quadratures, and for the transmitted modes we studied the spectra for the variance in the squeezed quadratures. Our results have revealed that the same relative cavity damping rates between the signal and the pump modes that predict enhanced squeezing in the intracavity do not necessarily predict enhanced squeezing in the transmitted modes.