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

Generation of nonclassical states is a necessary requirement for all quantum science and technology protocols. In this regard, Schrödinger cat states, as quantum superpositions of mocroscopically distinguishable coherent states with equal amplitude and opposite phases, have been of great importance in recent years. In this paper, using some appropriate nonlinear optical media, a theoretical scheme is employed to generate various classes of cat-like states with the help of the connection between quantum and nonlinear optics. In fact, we have exploited a single simple method to produce even, odd and also Yurke-Stoler cat-like states by considering different input states such as coherent squeezed, squeezed coherent, photon-added coherent, pair coherent and trio coherent states. To achieve the purpose, we extensively use cross-Kerr interaction and its further generalizations to higher-orders. Projective measurement is the original conception behind the represented approach by which distinct types of nonclassical states are obtained. At last, we pay our attention to the unavoidable effect of field dissipation and show that in most of the above-studied states we may still achieve the cat-like states, however, with attenuated amplitudes.

The Wehrl information entropy and its phase density, the so-called Wehrl phase distribution, are applied to describe Schr\"odinger cat and cat-like (kitten) states. The advantages of the Wehrl phase distribution over the Wehrl entropy in a description of the superposition principle are presented. The entropic measures are compared with a conventional phase distribution from the Husimi Q-function. Compact-form formulae for the entropic measures are found for superpositions of well-separated states. Examples of Schr\"odinger cats (including even, odd and Yurke-Stoler coherent states), as well as the cat-like states generated in Kerr medium are analyzed in detail. It is shown that, in contrast to the Wehrl entropy, the Wehrl phase distribution properly distinguishes between different superpositions of unequally-weighted states in respect to their number and phase-space configuration.

We study the improvement in the phase sensitivity of a Mach–Zehnder interferometer using the superposition of Schrödinger’s cat-like state with the Fock state (SCFS) and the coherent state as inputs. With this setup, we examine the effect on phase sensitivity of the interferometer using a two-channel detection (TCD) scheme [Opt. Express 29, 95 (2021)OPEXFF1094-408710.1364/OE.413391] in which we take the optimal combination of the intensities of both output ports. We find better phase sensitivity under some conditions for this setup as compared to other known combinations of inputs such as squeezed vacuum and coherent states, vacuum and coherent states, etc. Therefore, we expect that the SCFS may be an alternative nonclassical resource for improvement in the phase sensitivity of a Mach–Zehnder interferometer, having potential application in quantum sensing.

A two-step interaction scheme involving (2) and (3) nonlinear media is suggested for the generation of Schrodinger cat-like states of a single-mode optical field. In the first step, a weak coherent signal undergoes a self-Kerr phase modulation in a (3) crystal, leading to a Kerr kitten, namely a microscopic superposition of two coherent states with opposite phases. In the second step, such a Kerr kitten enters a (2) crystal and, in turn, plays the role of a quantum seed for stimulated phase-sensitive amplification. The output state in the above-threshold regime consists in a quantum superposition of mesoscopically distinguishable squeezed states, i.e. an optical cat-like state. The whole setup does not rely on conditional measurements, and is robust against decoherence, as only weak signals interact with the Kerr medium.

Quantum sub-phase sensitivity of a Mach–Zehnder interferometer with the superposition of Schrödinger’s cat-like state with vacuum state as an input under product detection scheme

We derive sampling functions for estimation of quantum state fidelity with Schr\"odinger cat-like states, which are defined as superpositions of two coherent states with opposite amplitudes. We also provide sampling functions for fidelity with squeezed Fock states that can approximate the cat-like states and can be generated from Gaussian squeezed states by conditional photon subtraction. The fidelities can be determined by averaging the sampling functions over quadrature statistics measured by homodyne detection. The sampling functions are designed such that they can compensate for losses and inefficient homodyning provided that the overall efficiency exceeds certain threshold. The fidelity with an odd coherent state and the fidelity with a squeezed odd Fock state provide convenient witnesses of negativity of Wigner function of the measured state. The negativity of Wigner function at the origin of phase space is certified if any of these fidelities exceeds 0.5. Finally, we discuss the possibility of reducing the statistical uncertainty of the fidelity estimates by a suitable choice of the dependence of the number of quadrature samples on the relative phase shift between local oscillator and signal beam.

Metabolic overflow (enhanced uptake of substrate and secretion of intermediates) is a phenomenon often observed for cells grown under substrate excess. Growth inhibition by substrate and/or product is also normally found for this kind of culture. An effort is made in this work to analyze the dynamic behavior of a continuous culture subject to metabolic overflow and growth inhibition by substrate and/or product. Analysis of a model system shows that in a certain range of operating conditions three nonwashout steady state solutions are possible. Local stability analysis indicates that only two of them are stable thus leading to multiplicity and hysteresis. Further analysis of the intrinsic effects of different terms describing the metabolic overflow and growth inhibitions reveals that for the model system and the parameters considered, the combined effects of product inhibition and an enhanced formation rate of product under substrate excess cause the multiplicity and hysteresis. Growth inhibition by substrate and/or an enhanced substrate uptake appear not to be necessary conditions. The combined effects of enhanced product formation and product inhibition can also lead to unusual dynamic behavior such as a prolonged time period to reach a steady state, oscillatory transition from one steady state to another, and sustained oscillations. Using the occurrence of multiplicity and oscillation as criteria, the operating regime of a continuous culture can be divided into four domains: one with multiplicity and oscillation, one with unique steady state but possible oscillatory behavior, the other two with unique and stable steady state. The model predictions are in accordance with recent experimental results. The results presented in this work may be used as guidelines for choosing proper operating conditions of similar culture systems to avoid undesired instability and multiplicity. Copyright 1998 John Wiley & Sons, Inc.

In recent years, there has been an increased interest in the generation of superposition of coherent states with opposite phases, the so-called photonic Schrodinger-cat states. These experiments are very challenging and so far, cats involving small photon numbers only have been implemented. Here, we propose to consider two-mode squeezed states as examples of a Schrodinger-cat-like state. In particular, we are interested in several criteria aiming to identify quantum states that are macroscopic superpositions in a more general sense. We show how these criteria can be extended to continuous variable entangled states. We apply them to various squeezed states, argue that two-mode squeezed vacuum states belong to a class of general Schrodinger-cat states and compare the size of states obtained in several experiments. Our results not only promote two-mode squeezed states for exploring quantum effects at the macroscopic level but also provide direct measures to evaluate their usefulness for quantum metrology.

We study the dynamics of a catlike superposition of f-deformed coherent states under dissipative decoherence. For this purpose, we investigate two important categories of f-deformed coherent states: Gazeau–Klauder and displacement-type coherent states. In addition, we consider two deformation functions; one of them describes a harmonic oscillator in an infinite well and another corresponds to a harmonic oscillator in a quantum well with finite depth. The decoherence effects appeared through a dissipative interaction of the environment with the catlike states. In this study, we first show that the Gazeau–Klauder coherent state is more resistant under the decoherence process, in contrast to the displacement-type one, and second, that the potential range of the infinite well and the depth of potential possess a remarkable role in the decoherence process.

We propose a scheme to prepare a macroscopic mechanical oscillator in a catlike state, close to a coherent state superposition. The mechanical oscillator, coupled by radiation-pressure interaction to a field in an optical cavity, is first prepared close to a squeezed vacuum state using a reservoir engineering technique. The system is then probed using a short optical pulse tuned to the lower motional sideband of the cavity resonance, realizing a photon-phonon swap interaction. A photon number measurement of the photons emerging from the cavity then conditions a phonon-subtracted catlike state with a negative Wigner distribution exhibiting separated peaks and multiple interference fringes. We show that this scheme is feasible using state-of-the-art photonic crystal optomechanical system.

It well known that in multisector models of optimal growth, optimal paths converge to a unique steady state when future utilities are not discounted. Sutherland [1970], Kurz [1968], and Liviatan and Samuelson [1969] gave examples of multiple steady states when future utilities are discounted. Beals and Koopmans [1969] and Iwai [1972] used intertemporal utility functions, which also yield multiple steady states. The uniqueness of steady states with multiple consumption goods when future utility discounted has been studied by Brock [1973]. He showed tlhat if we assume that none of the goods are inferior in consumption (he calls this the normality condition for the utility function) the uniqueness of the steady state assured. Brock did not allow for pure consumption goods. Later Brock and Burmeister [1976] generalized Brock's result to allow pure consumption goods as well (Morishima [1974] type). Brock also formulated an alternative approach where uniqueness assured under the of a non-vanishing Jacobian for every non-negative discount rate. He writes however that the non-singularity of the Jacobian is an obscure assumption and that it would be worthwhile to relate it to the normality condition of the utility function. We first propose to weaken Brock's of non-vanishing Jacobian for every discount rate. Then in Section 3, Theorem 2, we show that the normality condition on the utility function implies a non-vanishing Jacobian. In Theorem 3 we weaken the normality condition for the uniqueness of the steady state by investigating the conditions for a non-vanishing Jacobian. We then clarify the economic content of our weaker conditions. In the final section we observe that the normality theorem can be proved for the joint production case (Mirrlees [1969] type) using a technique due to McKenzie [1963, 1973].

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Schroedinger cat-like neutron states and Wigner function formulation of interferometry and spin-echo experiments

The dynamic generation of Schrödinger cats and their detection

We study theoretically the dynamics of entangled states created in a beam splitter with a nonlinear Kerr medium placed into one input arm. Entanglement dynamics of initial classical and nonclassical states are studied and compared. Signatures of revival and fractional revival phenomena exhibited during the time evolution of states in the Kerr medium are captured in the entangled states produced by the beam splitter. Dynamics of entanglement shows local minima at the instants of fractional revivals. These minima correspond to the generation of two-component Schrödinger cat states or multi-component Schrödinger cat-like states if the initial state considered is a coherent state. Maximum entanglement is obtained at the instants of collapses of wave packets in the medium. Our analysis shows increase in entanglement with increase in the degree of nonclassicality of the initial states considered. We show that the states generated at the output of the beam splitter using initial nonclassical states are more robust against decoherence due to photon absorption by an environment than those formed by an initial classical state.