We discuss general properties of the equilibrium state of parametric down-conversion in superconducting quantum circuits with detunings and Kerr anharmonicities, in the strongly nonlinear regime. By comparing moments of the steady state and those of a Schr"odinger cat, we show that true Schr"odinger cats cannot survive in the steady state if there is any single-photon loss. A delta-function ``catlike'' steady-state distribution can be formed, but this only exists in the limit of an extremely large nonlinearity. The steady state is a mixed state, which is more complex than a mixture or linear combination of delta functions, and the purity of which is reduced by driving. We expect this general behavior to occur in other driven, dissipative quantum subharmonic nonequilibrium open systems.
We discuss general properties of the equilibrium state in superconducting quantum circuits with anharmonic nonlinearities and detunings, in the nonlinear regime. By comparing moments of the steady state and those of a Schroinger cat, we show that true Schroinger cats cannot survive in the steady state if there is a single-photon loss. A delta-function `cat-like' steady-state distribution can be formed, but this only exists in the limit of extremely large nonlinearity. In general, the steady state is a mixed state whose purity is reduced by driving.
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.
Out-of-equilibrium driven nonlinear oscillators have attracted interest as a platform for tailoring Floquet Hamiltonian spectra and stabilizing quantum states for quantum information. However, it is usually ignored that the necessary parametric drive can amplify environmental fluctuations limiting applications. In this work, we capture this effect by computing the effective Lindbladian of a Kerr parametric oscillator beyond the rotating wave approximation. We find that the relevant dissipators depend strongly on the nonlinearity and the mean-photon number in the oscillator. In particular, we address recent measurements of the logical lifetime on a Schr\"odinger's-cat qubit stabilized by a parametric drive, thereby providing an explanation for the observed three-order-of-magnitude discrepancy between experimental data and an ordinary Lindblad treatment. Our systematic approach paves the path to treat dissipative effects in Floquet-engineered nonlinear quantum systems.
Abstract The quantum to classical transition is determined by the interplay of a trio of parameters: dissipation, nonlinearity, and macroscopicity. Why is nonlinearity needed to see quantum effects? And, is not an ordinary pendulum quite nonlinear already? In this manuscript, we discuss the parameter regime where the dynamics of a massive oscillator should be quantum mechanical in the presence of dissipation. We review the outstanding challenge of the dynamical generation of highly quantum mechanical cat states of a massive ‘pendulum’, known as Kerr-cats. We argue that state-of-the-art cold atom experiments may be in a position to reach such a nonlinear regime, which today singles out superconducting quantum circuits. A way to stabilize Schrödinger cat superpositions of a mechanical atomic oscillator via parametric squeezing and further protected by an unusual form of quantum interference is discussed. The encoding of a neutral atom Kerr-cat qubit is proposed.
In quantum optics, photonic Schrödinger cats are superpositions of two coherent states with opposite phases and with a significant number of photons. Recently, these states have been observed in the transient dynamics of driven-dissipative resonators subject to engineered two-photon processes. Here we present an exact analytical solution of the steady-state density matrix for this class of systems, including one-photon losses, which are considered detrimental for the achievement of cat states. We demonstrate that the unique steady state is a statistical mixture of two cat-like states with opposite parity, in spite of significant one-photon losses. The transient dynamics to the steady state depends dramatically on the initial state and can pass through a metastable regime lasting orders of magnitudes longer than the photon lifetime. By considering individual quantum trajectories in photon-counting configuration, we find that the system intermittently jumps between two cats. Finally, we propose and study a feedback protocol based on this behaviour to generate a pure cat-like steady state.
We propose a fast scheme to generate Schr\"odinger cat states in a superconducting resonator using a continuously driven qubit without resorting to the dispersive regime, two-photon drives, or engineered two-photon dissipation. We provide analysis for when the qubit is on and off resonance from the drive. We extend our analysis to account for a third level in a weakly anharmonic qutrit. We also discuss the case of a strongly anharmonic qutrit. Throughout the paper, we corroborate our analytical results with numerical simulations in the presence of energy relaxation and dephasing of the qubit and resonator using realistic experimental parameters.