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 drivendissipative resonators subject to engineered twophoton processes. Here we present an exact analytical solution of the steadystate density matrix for this class of systems, including onephoton losses, which are considered detrimental for the achievement of cat states. We demonstrate that the unique steady state is a statistical mixture of two catlike states with opposite parity, in spite of significant onephoton 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 photoncounting 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 catlike steady state.
Introduction
Quantum nonlinear optical systems are an invaluable tool to explore the quantum world and its striking features^{1}. Generally, these systems are outofequilibrium: photons must be continuously pumped into the system to replace those inevitably dissipated. Effective photonphoton interactions can be mediated by an active medium, such as atoms or excitons in cavity QED or Josephson junctions in circuit QED resonators^{2}. The concepts of reservoir engineering^{3,4,5,6,7,8} and feedback^{9,10,11,12,13,14} emerged to stabilize nonclassical steady states in photonic and optomechanical resonators. In particular, new opportunities arise via engineering of twophoton pumping and/or twophoton dissipation^{15,16}.
Since their theoretical conception^{17}, Schrödinger’s cats have captured the collective imagination, because they are nonclassical states at the macroscopic level. In quantum optics, the states of the electromagnetic field closest to the classical ones are the coherent states , having a welldefined mean amplitude α and phase (being n〉 the nphoton Fock state). Photonic Schrödinger cat states are a quantum superposition of coherent states α〉 and −α〉^{1,18,19}:
Contrarily to the coherent states, are eigenstates of the photonparity operator , with the photon number operator. In fact, they are a superposition of only even (odd) number states. Twophoton processes are known to drive the system towards this kind of photonic states^{18,19}. However, onephoton losses are unavoidable even in the best resonators. As a result, the presence of both one and twophoton dissipations makes the life of the cat states more intriguing^{15,19,20,21}.
In this work, we provide an exact analytical solution for the steady state of this class of systems. We show that the rich transient dynamics depends dramatically on the initial state. It can exhibit metastable plateaux lasting several orders of magnitude longer than the singlephoton lifetime. We demonstrate that, for a wide range of parameters around typical experimental ones^{21}, the unique steadystate density matrix has as eigenstates two catlike states even for significant onephoton losses, with all the other eigenstates having negligible probability. The study of individual trajectories reveals that, under photon counting on the environment, the system jumps between the two cat states, a property suggesting a feedback scheme to create pure catlike steady states.
Results
Theoretical framework and analytic solution
To start our treatment, we consider the master equation for the density matrix. For a system interacting with a Markovian reservoir, the time evolution of the reduced density matrix is captured by the Lindblad master equation . The operator is the Hamiltonian, while is the Lindblad dissipation superoperator^{1,22}. In the frame rotating at the pump frequency,
where Δ is the pumpcavity detuning and U the photonphoton interaction strength. G is the amplitude of the twophoton pump, while γ and η represent, respectively, the one and twophoton dissipation rates (see Fig. 1). The Lindblad dissipator is the sum of one and twophoton loss contributions. This model has been investigated theoretically^{16,19,23,24,25} and implemented experimentally^{21}.
In order to find a general and analytic solution for the steadystate density matrix , we used the formalism of the complex Prepresentation^{26}. Details about the derivation of our solution are in the Methods section. In spite of the several parameters in the model, our solution depends only on two dimensionless quantities, namely c = (Δ + iħγ/2)/(U − iħη) and g = G/(U − iħη). The former can be seen as a complex singleparticle detuning Δ + iħγ/2 divided by a complex interaction energy U − iħη; g is instead the twophoton pump intensity normalized by the same quantity. Hence, our exact solution for the steadystate density matrix elements in the Fock basis reads
where is the normalization factor, chosen such that . , _{2}F_{1} being the Gaussian hypergeometric function. Notably, for odd, meaning that, for any finite value of the system parameters, there will be no evenodd coherences in the steady state. In what follows, all the quantities marked with a tilde will refer to steadystate values.
To further characterise the steadystate, we consider the spectral decomposition of the density matrix , with Ψ_{κ}〉 the κ^{th} eigenstate of with eigenvalue p_{κ}. The latter corresponds to the probability of finding the system in Ψ_{κ}〉. The eigenstates are sorted in such a way that p_{κ} ≥ p_{κ+1}. For a pure state, p_{1} = 1 and all the other probabilities p_{κ} are zero. We numerically diagonalised the density matrix in a truncated Fock basis, choosing a cutoff ensuring a precision of 10^{−14}. For our calculations, we chose a set of parameters around typical experimental ones^{21}, i.e. , , and . In this regime, for the steady state (5) only two eigenstates dominate the density matrix. As shown in Fig. 2(a), typically and . The aforementioned absence of evenodd coherences implies that is composed of only even (odd) Fock states. Furthermore, we find that and for an appropriate choice of α. For the parameters of Fig. 2(d), for . We have varied Δ/ħη between −0.2 and 0.2, G/ħη between 0 and 15, γ/η between 0 and 5, U/ħη between 1 and 10, always finding that . Moreover, in these ranges, we verified that there exists a value of α such that . Hence, we can conclude that for a broad range of parameters the eigenstates of are two catlike states of opposite parity.
Using the linearity of the trace, for any operator one can write , where . In Fig. 2(b) we plot, as a function of the pump amplitude G, the steadystate mean density , together with its contributions . For weak pumping one has and , in agreement with what one would obtain for the even and the odd cat by taking the limit α → 0 of Eq. (1). These two contributions become equal in the limit of a very large number of photons. As shown in Fig. 2(c), the two contributions to the mean parity always stay clearly different, being orthogonal eigenstates of with eigenvalues ±1. A valuable tool to visualise the nonclassicality of a state is the Wigner function , defined in terms of the displacement operator^{27} . Indeed, negative values of W(β) indicate strong nonclassicality^{1}. The Wigner function corresponding to the density matrix (5) is always positive [cf. Eq. (12) in Methods], while the separate contributions and exhibit an interference pattern with negative regions, typical of cat states [cf. Fig. 2(d–f)].
We emphasize that for finite γ the considered system has always a unique steady state. However, the temporal relaxation towards the steady state depends dramatically on the initial condition. This is revealed by the timedependent fidelity with respect to the steady state, presented in Fig. 3, obtained by numerical integration of the master equation. In particular, initialising the system in one of the coherent states ±α〉 composing the steadystate cats, it persists nearby for a time several orders of magnitude longer than 1/γ and 1/η. Hence, the “multiple stable steady states” in^{21} are actually metastable.
Quantum trajectories
We now examine the quantum trajectories of the system, which give an insight on the pure states that the system explores during its dynamics^{28,29,30}. In principle, keeping track of all the photons escaping the cavity allows to follow the system wave function (cf. Methods)^{28,31}. A photoncounting trajectory is presented in Fig. 4, where in panels (a,b) we follow, respectively, the time evolution of the photon number and of the parity , starting from the vacuum state as initial condition. On a single trajectory, twophoton processes initially dominate, driving the system towards . Indeed, and . Twophoton losses do not affect a state parity, indeed . This is why the system persists nearby the even cat until a onephoton loss occurs. At this point, the state abruptly jumps to the odd manifold^{24}, since . After the jump, twophoton processes stabilise , so that and . When another onephoton jump takes place, the system is brought back to the even manifold and so on. Hence, if the quantum trajectory is monitored via photon counting^{31}, the system can only be found nearby or . The probability of being in each cat is given by the corresponding eigenvalue of , namely and . Since , it is impossible to discern the cats’ jumps by tracking the photon density. A parity measurement, instead, would be suitable^{32}. In Fig. 4(a,b) we also present the average over 100 trajectories, which, as expected, converges to the master equation solution (also shown). The latter corresponds to the full average over an infinite number of realizations^{22}. The fullyaveraged and singletrajectory evolutions of the Wigner function are shown in Fig. 4(c). In the averaged one, an evencat transient appears, but negativities are eventually washed out for ηt, ^{19,21,24}. By following a single quantum trajectory, instead, we see that W_{t}(β) quickly tends to the one of . Then, it abruptly switches to that of , then back at each onephoton jump.
The eventoodd jumps in a photoncounting quantum trajectory deserve a more detailed discussion. Each trajectory corresponds to the behaviour of our quantum system on a singleshot experiment^{22,28}. Indeed, to simulate a quantum trajectory it is necessary to model how an observer measures the environment to probe the system (in the presented case, by a photoncounting measure). The same Lindblad equation can be described via different measurement protocols, resulting in different single trajectories. Their average result reproduces the master equation solution^{1,29,30}. Our steady state, given by in Eq. (5), is typically a mixture of an even and an odd cat state. To unveil such a mixture at the singletrajectory level, a photoncounting measurement is suitable. But this does not mean that any physical system described by the same Lindblad equation would always be in a paritydefined state. We emphasise that this behaviour is not exclusively caused by the chosen measurement process: other systems under the same photoncounting trajectory do not show eventoodd jumps. For example, if one considers a onephoton pump and no twophoton driving, at any given time the trajectories do not show a defined parity (the state of the system in not an eigenstate of ).
A feedback protocol
The results presented above suggest that, in order to have a catlike steady state (e.g. keep interference fringes in the fully averaged Wigner function), one may try to unbalance the even and odd contributions to . This effect can be envisioned through a paritytriggered feedback mechanism^{11,12,33,34} opening a onephoton loss channel. In practice, this can be implemented via nondestructive parity measurements^{32,35}, whose rate must be larger than any other rate in the system. Note that, in general, a parity measurement projects the system into the even or oddparity manifold, affecting the dynamics by destroying the evenodd coherences. In the present case, however, those coherences are proven to be always zero in the steady state by the analytic solution (5). Thus, a highrate and nondestructive parity measure does not alter significantly the system dynamics and allows to continuously monitor . When the undesired value is measured, an auxiliary qubit is put into resonance with the cavity, inducing the absorption of a photon. After the desired parity is restored, the qubit is brought out of resonance, closing the additional dissipation channel. Such a qubit acts as a nonMarkovian bath for the system and in principle its effects can not be simply assimilated to those of a Markovian environment. However, if one assumes that the excitedstate lifetime of the qubit is shorter than the inverse of the qubitcavity coupling rate, one can safely treat it as an additional Markovian dissipator^{34,36}. In other words, the coupled qubit must be engineered to easily lose the photon to the environment, which seems a reasonable task for the present experimental techniques^{9,11,12,25}. Under these assumptions, the proposed feedback protocol can be effectively described by the additional jump operator and the corresponding dissipator
Qualitatively, leaves the even cat undisturbed, while it enhances the dissipation for the odd one.
In Fig. 5(a) we show the time evolution of for three different values of γ_{f}. These results have been obtained via numerical integration of the Lindblad master equation with a total dissipator . At the steady state, as γ_{f} increases so does , indicating that the positive cat has a larger weight in . In Fig. 5(b) we show the corresponding steadystate Wigner functions . For finite γ_{f}, negative fringes appear in the Wigner function. They are more pronounced as γ_{f} is increased, revealing a highly nonclassical state. In the limit , . By using, instead, the jump operator , one can similarly stabilize the odd cat state. Note that the Wignerfunction negativities in Fig. 5 are those of the full steadystate density matrix. Hence, the quantum state of the system is on average nonclassical.
Discussion
In conclusion, we presented the exact steadystate solution for a general photonic resonator subject to onephoton losses and twophoton drive and dissipation. Remarkably, the unique steady state appears to be a mixture of two orthogonal cat states of opposite parity. We have also shown that the transient dynamics to the unique steady state can depend dramatically on the initial condition, revealing the existence of metastable states. Furthermore, by monitoring the quantum trajectory of the system via photon counting, we found that it explores the two cat states composing the steadystate statistical mixture. On this ground, we proposed to engineer a paritydependent dissipation which allows to stabilize a catlike steady state.
The general nature and richness of the results predicted here paves the way to a wide variety of experimental and theoretical investigations. As a future perspective, a challenging but intriguing problem is the study of other photonic catlike states in the transient and steadystate regime for arrays of coupled resonators. The generation and stabilization of orthogonal catlike states is of great interest for quantum computation, since they can be used as qubits logic states^{16,37,38}. Besides the implications for quantum information, our results are also relevant for the study of exotic phases based on the manybody physics of light^{2}.
Methods
Complex Prepresentation
The complex Prepresentation of the density matrix reads^{26}
where the complex amplitudes α and β define the corresponding (nonorthogonal) coherent states. The two independent and closed integration paths and must encircle all the singularities of P(α, β) in the complex plane. A Lindblad master equation for the density matrix translates into a FokkerPlanklike differential equation for the function P(α, β)^{39}. In the case defined by Eqs (2, 3, 4), one has
where . The corresponding steadystate equation, defined by , is satisfied by
where c and g were introduced in the main text. Taylor expanding the exponential and projecting on the number states 〈n and m〉, we obtain a formal expression for the matrix elements of :
The appropriate choice of the integration paths is a central issue. A suitable one is the Pochhammer contour^{39}, which leads to Eq. (5). Similarly, it is also possible to calculate the general steadystate expectation value of the correlation functions
and the steadystate Wigner function (without feedback)
Determination of quantum trajectories
The Lindblad master equation defined by Eqs (2, 3, 4) describes the evolution of the density matrix if we do not collect any information about the system or the environment. Hence, the analytic and numerical solution of a Linbdblad master equation predicts the average outcomes of an experimental realisation. Detecting the photons escaping the system would provide more information about the time evolution of the system itself^{29,30,31}. That is, the evolution of the wave function of an open quantum system can be followed gathering information on its exchanges with the environment. The operator describes the time evolution of the system state between two detections of a photon emission. For our system, the nonhermitian effective Hamiltonian reads:
At t = t_{i}, the emission of v_{i} photons is detected (v_{i} = 1, 2) and the state abruptly changes according to . This is a quantum jump, which can be simulated stochastically: in the interval [t, t + δt] the probability of onephoton emission is , while that of twophoton emission is . After a jump, the time evolution continues under the action of until the next photon emission. On this ground, a single trajectory is simulated by randomly determining if, at each time step, the state jumps or evolves under the action of . We stress that for perfect detection (all the emitted photons are gathered) and a pure initial condition the state ψ(t)〉 stays pure at any time. For the same Lindblad equation, other quantum trajectories than the photoncounting ones can be modelled and simulated^{1,22,31}. However, the average over an infinite number of trajectories will always give the solution of the Lindblad master equation.
Additional Information
How to cite this article: Minganti, F. et al. Exact results for Schrödinger cats in drivendissipative systems and their feedback control. Sci. Rep. 6, 26987; doi: 10.1038/srep26987 (2016).
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Acknowledgements
We thank B. Huard, Z. Leghtas and G. Rembado for discussions. We acknowledge support from ERC (via the Consolidator Grant “CORPHO” No. 616233) and from ANR (Project QUANDYDE No. ANR11BS100001).
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F.M., N.B. and W.C. obtained the exact solution for the steadystate density matrix. N.B. and F.M. performed the timedependent solutions of the master equation, while F.M. and J.L. studied the individual quantum trajectories. N.B. composed the figures. All authors contributed to the critical analysis of the results and writing of the manuscript. C.C. proposed and supervised the project.
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Minganti, F., Bartolo, N., Lolli, J. et al. Exact results for Schrödinger cats in drivendissipative systems and their feedback control. Sci Rep 6, 26987 (2016). https://doi.org/10.1038/srep26987
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