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arXiv:1601.02861 · Scientific Reports

Exact results for Schrödinger cats in driven-dissipative systems and their feedback control

Fabrizio Minganti, Nicola Bartolo, Jared Lolli, Wim Casteels, Cristiano Ciuti

Published May 31, 2016• 94 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1038/srep26987

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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.
Since their conception, Schr\odinger's cats have captured the collective imagination. Photonic cat states 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 a driven-dissipative resonator subject to engineered two-photon processes. Here we present an exact analytical solution of the steady-state density matrix for this class of systems by including one-photon losses, that 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. We propose a feedback protocol based on this behaviour generating a pure cat-like steady-state.
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.
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.
A system of two-level noninteracting atoms driven by superposition of two Glauber coherent photonic states (a cat state) is studied. The field state is continuously restored by a source explicitly incorporated into the model. Due to its nature, the cat state changes phase by $\ensuremath{\pi}$ upon stimulated excitation of any atom, a peculiar kind of coherent quantum feedback. That results in correlations between photonic and atomic subsystems. In the limit of a strong field, the ansatz for the system's density matrix is proposed and an approximate analytical solution to the master equation is obtained in the case of a large number of atoms and slow spontaneous emission. Based on this solution, the steady-state second-order correlation function of atomic photoemissions is evaluated and investigated. The results demonstrate a remarkable difference from the case of the classical field (i.e., the field in a Glauber coherent state).
We show that by engineering the interaction with the environment there exists a large class of systems that can evolve irreversibly to a cat state. To be precise we show that it is possible to engineer an irreversible process so that the steady state is close to a pure Schrödinger’s cat state by using double well systems and an environment comprising two-photon (or phonon) absorbers. We also show that it should be possible to prolong the lifetime of a Schrödinger’s cat state exposed to the destructive effects of a conventional single-photon decohering environment. In addition to our general analysis we present a concrete circuit realisation of both system and environment that should be fabricatable with current technologies. Our protocol should make it easier to prepare and maintain Schrödinger cat states which would be useful in applications of quantum metrology and information processing as well as being of interest to those probing the quantum to classical transition.
We present a method to implement two-phonon interactions between mechanical resonators and spin qubits in hybrid setups, and show that these systems can be applied for the dissipative generation of nonclassical mechanical states. In particular, we demonstrate that the implementation a two-phonon Jaynes-Cummings Hamiltonian under coherent driving of the qubit yields a dissipative phase transition with similarities to the one predicted in the model of the degenerate parametric oscillator: beyond a certain threshold in the driving amplitude, the driven-dissipative system sustains a steady state consisting of a jumping cat, i.e., a Schr\odinger cat undergoing random jumps between two phases. We consider realistic setups and show that, in samples within reach of current technology, the low rate at which jumps between Schr\odinger cat states occur allows these to survive with fidelities >0.99>0.99>0.99 for longer than one milisecond, without the need of any particular protocol or preparation.