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Edited by: Andrew D. Greentree, RMIT University, Australia

Reviewed by: Jonas Maziero, Universidade Federal de Santa Maria, Brazil; Zoltán Zimborás, University College London, UK

^{†}Present address: Richard D. Wilson, Sonobex Limited, Warwick, UK

This article was submitted to Quantum Computing, a section of the journal Frontiers in ICT.

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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 realization 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.

The development of many quantum technologies depends on an ability to engineer strongly non-classical states. Such states take the form of either highly entangled states of distinct degrees of freedom or a quantum coherent superposition of macroscopically distinct states in a single degree of freedom (Sanders,

In this work, we present a possible realization of a protocol for double well system [in this case a Superconducting Quantum-Interference Device (SQUID)] to create Schrödinger cat states using the interaction of the system with a special kind of environment. To be specific, we engineer an environment comprising a bath of two-photon absorbers, for certain initial states, such that the system relaxes to a steady state, which is close to a pure Schrödinger cat state. The use of open systems as well as the measurement process was proposed by Yurke, Schleich, and Walls (Yurke and Stoler,

We note that our scheme is simpler than and different from other driven dissipative bistable systems [for example, the coherently driven optical cavity containing a Kerr medium (Walls and Milburn,

For the results presented in this paper, we have used as an example system a superconducting quantum-interference device (SQUID) ring. Our reason for choosing SQUIDs is that these devices are routinely fabricated and their theory is very well understood. We note that we have investigated a number of other systems (but do not include results here) and our analysis indicates that the key feature of the ring is that it can be made to form a double well potential. Moreover, non-linear systems derived from the Josephson junction in circuit QED exhibit multi photon resonance when driven by an external field (Deppe et al., _{0} = ^{−10} H for the ring’s inductance and _{c}^{−15} F). We set the externally applied magnetic flux Φ_{x}_{0}, so that the ring’s potential forms a degenerate double well. It is also convenient to introduce the bosonic annihilation ^{†} operators where

^{−10} H, capacitance ^{−15} F, critical current of the weak link _{c}_{x}_{0}. Note that, we have exaggerated the energy difference between the ground and first excited states as well as stationary states two and three in order to make the different energies visible on this plot.

We model the effect of the environment on the system using the master equation in the Lindblad form (Viola et al., _{j}

We find very different behavior if one chooses a different environment comprising two-photon absorbers, described by a Lindblad operator proportional to the square of the annihilation operator. In Figure

In order to demonstrate that the system does indeed decay to a Schrödinger cat state, we will make use of the Wigner function. These pseudo probability density functions in phase space have been of great utility in demonstrating that some quantum states are Schrödinger cats (Deléglise et al., ^{2} will only couple even states to even states and odd states to odd states. Thus, the action of ^{2} on any initial state must preserve its parity. Hence, an environment comprising only two-photon absorbers would ensure that the system will relax to a steady state with same parity as the initial state. It is this symmetry property of the environment together with the symmetry in the Hamiltonian and initial condition that leads to steady state solutions that are Schrödinger cat states.

In order to examine quantitatively the emergence of this cat from the initial coherent state we introduce, following (Nogues et al.,

In absolute terms, this is a useful measure, but when we know (by inspecting the Wigner function) that the states we are examining are cat-like, a more useful measure may well be a relative cattiness to some reference Schrödinger cat state.

Hence, we define:
_{ref}) > 1], less [Cat(_{ref}) < 1], or just as [Cat(_{ref}) = 1] catty than the other. We have chosen to introduce this measure over using existing metrics such as the fidelity as it does not contain any contributions of the type that occur from, for example, correlations between a cat and its related mixed state (which might be thought of as the overlap of the “classical” like parts of the state). Computing relative measures such as the fidelity is further complicated by the fact that the final states, in terms of the size and orientation, for the different environments are very different from each other (having very little overlap). Hence, performing meaningful estimates of fidelity would be quite difficult, perhaps even impossible as we would have to provide different reference states for each environment against which to measure the fidelity. We note that while there are some limitations with the Cat measure and it should be applied with care, for the problem we study here, it suits our purposes very well. In Figure _{ref} the final cat state shown in Figure

_{ref}) for the dynamics leading to Figure _{ref} the final cat state shown in Figure _{ref}) for an environment of two-photon absorbers and damping.

It is interesting to consider what would happen to a ring that was initially in a Schrödinger cat state under the influence of a bath of two-photon absorbers. For systems with deep enough double well potentials, such as the one considered here the ground and first excited energy eigenstates are both Schrödinger cats. The ground state is, to good approximation, an even superposition of two macroscopically distinct coherent states while the first excited state is an odd superposition as can be seen from their Wigner functions in Figures

This approach seems all very well and good, but an environment of two-photon absorbers is very special. It would be hard to construct such an environment without having any other source of decoherence present. We therefore need to verify that the effects of a two-photon absorbing environment cannot be completely destroyed by the presence of a more traditional environment such as a lossy bath. In Figure _{ref}) using the initial stationary state as shown in Figure _{ref}) for these two initial conditions, we now note that the dashed line shown in Figure

_{ref}) for these three environments as a function of time (we have used the initial stationary state as shown in Figure

In order to make our above discussion a reality, we need to engineer a dissipative quantum channel that acts as a two-photon absorber. Here, we suggest a concrete realization that, whilst not perfect, still retains the key feature of environmentally induced “decoherence” to a Schrödinger cat state. Our proposal makes use of non-linearly coupled electromagnetic fields and SQUIDs. Such quantum electrodynamic circuits have already been investigated in the context of weak non-demolition measurement (Deng et al., ^{†}

There are a number of models (Deng et al., _{α}_{α}_{1} defined as the coefficient of the leading non-linear current term of the SQUID inductance. We will set cos^{2} ^{2}

The system can be quantized in the usual way in terms of the bosonic annihilation and creation operators ^{†} for the probe and ^{†} and for the signal cavity defined by (Wallquist et al.,

The Hamiltonian may then be written as

Unlike Kumar and DiVincenzo (^{2}^{†2} as we will choose _{p}_{s}^{1}

We now include the dissipative channels for this model in the usual way. The density operator for the total system, in the interaction picture, satisfies
_{a}_{b}_{b}^{2} is the photon flux of the driving field. We have also assumed that each cavity sees a zero temperature environment.

In the absence of the SQUID mediated interactions, the probe cavity will relax to a coherent state with the steady state amplitude

We will choose the phase of the probe driving as a reference phase and set _{0} to be real. If we make a canonical transformation to the displaced picture by
^{†}). The last term is equivalent to the quantum description of sub/second harmonic generation considered by Drummond et al. (

We now assume that _{b}_{2} and dephasing rate Γ_{⊥} are given by

A peculiar feature of using SQUID coupled cavities is that the price paid for two-photon decay is an additional dephasing term on the signal cavity field. Using the strong dependence on the steady state amplitude _{0} in the two-photon rate, we can make the two-photon decay term that dominate over the single photon decay of the signal cavity over the time scales of interest. In Figure _{a}_{⊥} to Γ_{2} has not been preserved. Our reason for presenting this data is to indicate that alternative circuit realizations, where the beneficial effects of the two-photon absorbing environment are reduced and the damaging effects of the dephasing term increased, might still be used to engineer a steady state cat. We therefore believe that the discussion in Section

_{2} = 0.2 (or _{⊥} = 0.05 (or _{a}

Finally, in Figure

There are two phenomena that embody quantum mechanics, namely entanglement and the Schrödinger’s cat thought experiment for making macroscopic superposition states (Schrödinger,

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Mark J. Everitt, Richard D. Wilson, and Richard D. Wilson thank the Templeton Foundation for their generous support. Gerard J. Milburn acknowledges the support of the National Science Foundation under Grant No. NSF PHY11-25915 and the Australian Research Council Centre of Excellence for Engineered Quantum Systems grant CE110001013. Mark J. Everitt would like to thank Andrew Archer, Gerry Swallowe, and Richard Giles for their help with the preparation of our manuscript.

^{1}Note that the coefficients in this equation are