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.
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.
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.
Taking the decoherence into account, we investigate nonclassical features of the optical Schr\"odinger cat states in one-mode and two-coupled-modes systems with two-photon driving. In the one-mode system, the relationship between the Schr\"odinger cat states and the system parameters is derived. We observe that in the presence of single-photon decay the steady states would be a mixture of Schr\"odinger cat states. The dynamics and steady states of such a cat versus single-photon decay are examined. In the two-coupled-modes cases with linear and nonlinear couplings, the dynamics of entanglement and mutual information are examined with two different initial states and single-photon decay. Compared to the linear coupling case, more complicated structure appears in the Wigner function in the nonlinear coupling case. The joint quadrature distributions are also explored. Such nonclassical states can be used not only in exploring the boundary between the classical and the quantum worlds but also in quantum metrology and quantum information processing.
Non-Gaussian quantum states have been deterministically prepared and autonomously stabilized in single-and two-mode circuit quantum electrodynamics architectures via engineered dissipation.However, it is currently unknown how to scale up this technique to multimode non-Gaussian systems.Here, we upgrade dissipation engineering to collective (normal) modes of nonlinear resonator arrays and show how to stabilize multimode Schrödinger cat states.These states are multiphoton and multimode quantum superpositions of coherent states in a single normal mode delocalized over an arbitrary number of cavities.We consider tailored dissipative coupling between resonators that are parametrically driven and feature an on-site nonlinearity, which is either a Kerr-type nonlinearity or an engineered two-photon loss.For both types of nonlinearity, we find the same exact closed-form solutions for the two-dimensional steady-state manifold spanned by superpositions of multimode Schrödinger cat states.We further show that, in the Zeno limit of strong dissipative coupling, the even-parity multimode cat state can be deterministically prepared from the vacuum.Remarkably, engineered two-photon loss gives rise to a fast relaxation toward the steady state, protecting the state preparation against decoherence due to intrinsic single-photon losses and imperfections in tailored dissipative coupling, which sets in at longer times.The relaxation time is independent of system size making the state preparation scalable.Multimode cat states are naturally endowed with a noise bias that increases exponentially with system size and can thus be exploited for enhanced robust encoding of quantum information.
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.