Abstract By virtue of the superoperator technique, we solve analytically the non-Markovian master equation that describes single-mode continuous variable system interacting with a structure reservoir. Our analytical solution allows us to investigate the non-Markovian effects on the dynamics of quantum coherence and non-classicality of Schrödinger cat states embedded in various types of structured reservoirs. We show that the dynamical behaviors of the fidelity, entanglement potentials, and higher-order antibunching of such states can be divided into two phases: they decay rapidly within a short time, but would either vanish completely or tend to a nonzero steady-state value in the long-time limit depending on the initial Schrödinger cat states and system parameters. We also show that by examining the Wigner function and higher-order sub-Poissonian statistics, the super-Ohmic environment can better help to protect the non-classicality, whereas the sub-Ohmic environment causes the disappearance of the non-classicality. Our approach is concise and can generalize to other open quantum system coupled to non-Markovian environment at finite-temperature or to linear non-Markovian Gaussian system.
Abstract By virtue of the superoperator technique, we solve analytically the non-Markovian master equation that describes single-mode continuous variable system interacting with a structure reservoir. Our analytical solution allows us to investigate the non-Markovian effects on the dynamics of quantum coherence and non-classicality of Schrödinger cat states embedded in various types of structured reservoirs. We show that the dynamical behaviors of the fidelity, entanglement potentials, and higher-order antibunching of such states can be divided into two phases: they decay rapidly within a short time, but would either vanish completely or tend to a nonzero steady-state value in the long-time limit depending on the initial Schrödinger cat states and system parameters. We also show that by examining the Wigner function and higher-order sub-Poissonian statistics, the super-Ohmic environment can better help to protect the non-classicality, whereas the sub-Ohmic environment causes the disappearance of the non-classicality. Our approach is concise and can generalize to other open quantum system coupled to non-Markovian environment at finite-temperature or to linear non-Markovian Gaussian system.
Understanding the non-Markovian dynamics of dissipative processes induced by memory effects of the environment is a fundamental subject of open quantum systems. Because of the complexity of open quantum systems, e.g., the multiple energy scales involving that of the system, the environment, and their mutual coupling, it is generally a challenging task to characterize the relationship among the parameters of the system dynamics and the reservoir spectra. For the two-level spontaneous emission model within structured environments, it was shown in a recent literature (Opt. Lett. 38, 3650) that a functional relation could be established between the asymptotically non-decaying population and the spectral density of the reservoir as the system undergoes a long-time evolution. It hence renders a distinct perspective to look into the character of long-lived quantum coherence in the corresponding non-Markovian process. This article is devoted to further investigate the phenomena of limit cycle oscillations possibly occurring in such non-Markovian dissipative systems in a long-time evolution. For a two-level system subjected to an environment with Ohmic class spectra, due to the presence of a unique bound-state mode of the system, the evolution trajectory of the given initial states will converge to a limit cycle in the Bloch space. The dependence of the radius and the location of the limit cycle on the spectral density function of the reservoir are manifested by virtue of the described functional relation. For the model subjected to a photonic crystal environment with multiple bands, our studies reveal that, owing to the presence of two or more bound states, the evolution trajectory of the system will converge to a toric curve of a paraboloid in the Bloch space and the phenomena of periodic or quasi-periodic oscillations could exhibit. While the equation of the parabolic curve is fully determined by the initial values of the state vector in the Bloch space, our results reveal that the scope of the evolution trajectory inside the toric curve is related to the spectral density of the reservoir and their quantified relation is distinctly characterized. Finally, the asymptotic dynamics of the correlations of a two-qubit system is discussed when it is subjected locally to the non-Markovian dissipative process.
The precise characterization of dynamics in open quantum systems often presents significant challenges, leading to the introduction of various approximations to simplify a model. One commonly used strategy involves Markovian approximations, assuming a memoryless environment. In this study, such approximations are not used and an analytical dynamical depiction of an open quantum system is provided. The system under consideration is an oscillator that is surrounded by a bath of oscillators. The resulting dynamics are characterized by a second-order complex coefficient linear differential equation, which may be either homogeneous or inhomogeneous. Moreover, distinct dynamical regions emerge, depending on certain parameter values. Notably, the steady-state average excitation number (AEN) of the system shows rapid escalation with increasing non-Markovianity, reflecting the intricacies of real-world dynamics. In cases where there is detuning between the system frequency and the environmental central frequency within a non-Markovian regime, the AEN maintains its initial value for an extended period. Furthermore, the application of pulse control can effectively protect the quantum system from decoherence effects without using approximations. The pulse control can not only prolong the relaxation time of the oscillator, but can also be used to speed up the relaxation process, depending on the specifications of the pulse. By employing a kick pulse, the Mpemba effect can be observed in the non-Markovian regime in a surprisingly super-cooling-like effect.
We apply the stochastic Schr\"odinger equation approach to study the non-Markovian dynamics of quantum systems coupled to a squeezed environment. We derive the non-Markovian quantum-state-diffusion equation using coherent-state unraveling and the associated zeroth-order master equation for general models in a microscopic quantum context. Focused on a dissipative optical cavity coupled with a squeezed vacuum, the numerical simulations demonstrate the time evolution of photon numbers in an optical cavity. We observe that the long-time limits of the non-Markovian dynamics are significantly different from those of the Markovian dynamics for various memory factors and squeezing factors. Additionally, non-Markovian dynamics exhibit a distinct pattern of behavior when parameters change. Our work provides a method to study the impact of multiple parameters on the non-Markovian dynamics and long-time limits in a joint manner. The method can be further extended to squeezed finite-temperature environments.
We discuss the generation of a Schrödinger cat in a nanocavity created by the coupling of an electromagnetic mode with an exciton in a quantum dot considering the dispersive limit of the Jaynes–Cummings model. More than the generation itself, we focus on the effects of the environment over the bosonic state in the nanocavity, which has losses simulated by coupling with two different kinds of reservoirs. In the first case, the interaction of the system with a finite reservoir shows that fragments of different sizes of the reservoir deliver the same amount of information about the physical system in the dynamics of the birth and death of the Schrödinger cat. The second case considers a structured reservoir, whose spectral density varies significantly with frequency. This situation becomes relevant in solid-state devices where quantum channels are embedded, as memory effects generally cannot be neglected. Under these circumstances, it is observed that the dynamics can differ substantially from the Markovian, presenting oscillations related to the average number of photons. These oscillations influence the information flow between the system and the environment, evidenced here by the measurement of non-Markovianity.
In this manuscript, we study the Schrödinger cat state transfer in a quantum optical version of Newton's cradle in non-Markovian environment. Based on a non-Markovian master equation, we show that the cat state can be transferred purely through the memory effect of the non-Markovian common environment, even without any direct couplings between neighbor cavities. The mechanism of the environment induced cat state transfer is analyzed both analytically and numerically to demonstrate that the transfer is a unique phenomenon in non-Markovian regime. From this example, the non-Markovian environment is shown to be qualitatively different from the Markovian environment reflected by the finite versus zero residue coherence. Besides, we also show the influence of environmental parameters are crucial for the transfer. We hope the cat state transfer studied in this work may shed more light on the fundamental difference between non-Markovian and Markovian environments.
The transient dynamics of quantum coherence of Gaussian states are investigated. The state is coupled to an external environment which can be described by a Fano-Anderson type Hamiltonian. Solving the quantum Langevin equation, we obtain the Greens functions which are used to compute the time evolved first and second moments of the quadrature operators. From the quadrature operator moments, we construct the covariance matrix which is used to measure the coherence in the system. The coherence is measured using the relative entropy of coherence measure. We consider three different classes of spectral densities in our analysis viz, the Ohmic, the sub-Ohmic, and the super-Ohmic densities. In our work, we study the dynamics of the coherent state, squeezed state, and displaced squeezed state. For all these states we observe that when the coupling with the system and the environment is weak, the coherence monotonically decreases and eventually vanishes in a long time. Thus all the states exhibit Markovian evolution in the weak coupling limit. In the strong coupling limit, the dynamics for the initial period is Markovian and after a certain period, it becomes non-Markovian where we observe an environmental backaction on the system. Thus in the strong coupling limit, we observe a dynamical crossover from Markovian nature to non-Markovian behavior. This crossover is very abrupt under some environmental conditions and for some parameters of the quantum state. Using a quantum master equation approach we verify the crossover from the dynamics of the dissipation and fluctuation parameters and the results endorse those obtained from coherence dynamics.