Non-Markovian dissipative dynamics of two coupled qubits in independent reservoirs: a comparison between exact solutions and master equation approaches
The reduced dynamics of two interacting qubits coupled to two independent bosonic baths is investigated. The one-excitation dynamics is derived and compared with that based on the resolution of appropriate non-Markovian master equations. The Nakajima-Zwanzig and the time-convolutionless projection operator techniques are exploited to provide a description of the non-Markovian features of the dynamics of the two-qubits system. The validity of such approximate methods and their range of validity in correspondence to different choices of the parameters describing the system are brought to light.
Non-Markovian master equations describe general open quantum systems when no approximation is made. We provide the exact closed master equation for the class of Gaussian, completely positive, trace preserving, non-Markovian dynamics. This very general result allows to investigate a vast variety of physical systems. We show that the master equation for non-Markovian quantum Brownian motion is a particular case of our general result. Furthermore, we derive the master equation unraveled by a non-Markovian, dissipative stochastic Schrödinger equation, paving the way for the analysis of dissipative…
We establish a quantum functional central limit for the dynamics of a system coupled to a Fermionic bath with a general interaction linear in the creation, annihilation and scattering of the bath reservoir. Following a quantum Markovian limit, we realize the open dynamical evolution of the system as an adapted quantum stochastic process driven by Fermionic Noise.
We extend the non-Markovian quantum state diffusion (QSD) equation to open quantum systems which exhibit multi-channel coupling to a harmonic oscillator reservoir. Open quantum systems which have multi-channel reservoir coupling are those in which canonical transformation of reservoir modes cannot reduce the number of reservoir operators appearing in the interaction Hamiltonian to one. We show that the non-Markovian QSD equation for multi-channel reservoir coupling can, in some cases, lead to an exact master equation which we derive. We then derive the exact master equation for the three-level…
The non-Markovianity is a recently proposed characterization of the non-Markovian behavior in an open quantum system, based on which we first present a practical idea for directly measuring the non-Markovian character of a single qubit coupled to a zero-temperature bosonic reservoir, and then extend to investigate the dynamics of two noninteracting qubits subject to two reservoirs respectively with a lower bound of non-Markovianity. Our scheme, with no need of optimization procedures and quantum state tomography, is helpful for experimental implementation.
The reservoir memory effects can lead to information backflow and recurrence of the previously lost quantum correlations. We establish connections between the direction of information flow and variation of the geometric quantum discords (GQDs) measured respectively by the trace distance, the Hellinger distance, and the Bures distance for two qubits subjecting to the bosonic structured reservoirs, and unveil their dependence on a factor whose derivative signifies the (non-)Markovianity of the dynamics. By considering the reservoirs with Lorentzian and Ohmic-like spectra, we further demonstrated…
A theoretical investigation on slow light propagation based on eletromagnetically induced transparency in a three-level quantum-dot system is performed including non-Markovian effects and correlated dephasing reservoirs. It is demonstraonated that the non-Markovian nature of the process is quite essential even for conventional dephasing typical of quantum dots leading to significant enhancement or inhibition of the group velocity slow-down factor as well as to the shifting and distortion of the transmission window. Furthermore, the correlation between dephasing reservoirs may also either enhan…
An easily solvable quantum master equation has long been sought that takes into account memory effects induced on the system by the bath, i.e., non-Markovian effects. We briefly review the Post-Markovian master equation (PMME), which is relatively easy to solve, and analyze a simple example where solutions obtained exhibit non-Markovianity. We apply the distinguishability measure introduced by Breuer et al., and we also explicitly analyze the divisibility of the associated quantum dynamical maps. We give a mathematical condition on the memory kernel used in the PMME that guarantees non-CP-divi…
We study two distinct multi-level atomic models in which one transition is coupled to a Markovian reservoir, while another linked transition is coupled to a non-Markovian reservoir. We show that by choosing appropriately the density of modes of the non-Markovian reservoir the spontaneous emission to the Markovian reservoir is greatly altered. The existence of `dark lines' in the spontaneous emission spectrum in the Markovian reservoir due to the coupling to specific density of modes of the non-Markovian reservoir is also predicted.
The decoherence of two initially entangled qubits in a non-Markovian reservoir has been investigated exactly without Born Markovian approximation and rotating-wave approximation(RWA). The non-perturbative quantum master equation is derived and its exact solution is obtained. The decoherence behaviors of two qubits, initially entangled in Bell states, has been investigated in three different cases of parameters. The results show that the counter-rotating wave terms have great influence on the decoherence behavior, and there are differences between the exact solution of the Hamiltonian with RWA …
The interaction of quantum emitters with structured baths modifies both their individual and collective dynamics. In González-Tudela \emph{et al} we show how exotic quantum dynamics emerge when QEs are spectrally tuned around the middle of the band of a two-dimensional structured reservoir, where we predict the failure of perturbative treatments, anisotropic interactions and novel super and subradiant behaviour. In this work, we provide further analysis of that situation, together with a complete analysis for the quantum emitter dynamics in spectral regions different from the center of the ban…
The interrelationship between the non-Markovian stochastic Schrödinger equations and the corresponding non-Markovian master equations is investigated in the finite temperature regimes. We show that the general finite temperature non-Markovian trajectories can be used to derive the corresponding non-Markovian master equations. A simple, yet important solvable example is the well-known damped harmonic oscillator model in which a harmonic oscillator is coupled to a finite temperature reservoir in the rotating wave approximation. The exact convolutionless master equation for the damped harmonic os…
We investigate the non-Markovian dynamics of two-state systems in structured reservoirs. We establish a connection between two theoretical quantum approaches, the pseudomodes [B. M. Garraway, Phys. Rev. A 55, 2290 (1997)] and the recently developed non-Markovian quantum jump method [J. Piilo et al., Phys. Rev. Lett. 100, 180402 (2008)]. This connection provides a clear physical picture of how the structured reservoir affects the system dynamics, indicating the role of the pseudomodes as an effective description of the environmental memory.
By using the effective non-Markovian measure [H.P. Breuer, E.M. Laine, J. Piilo, Phys. Rev. Lett. 103, 210401 (2009)], we investigate non-Markovian dynamics of a pair of two-level atoms (TLAs) system, each of which interacting with a local reservoir. We show that subsystem dynamics can be controlled by manipulating the coupling between TLAs, temperature and relaxation rate of the atoms. Moreover, the correlation between non-Markovianity of subsystem and entanglement between the subsystem and the structured bath is investigated, the results show that the emergence of non-Markovianity has a nega…
In this paper, we investigate preservation of quantum coherence of a single-qubit interacting with a zero-temperature thermal reservoir through the addition of noninteracting qubits in the reservoir. Moreover, we extend this scheme to preserve quantum entanglement between two and three distant qubits, each of which interacts with a dissipative reservoir independently. At the long time limit, we obtained analytical expressions for the coherence measure and the concurrence of two and three qubits in terms of the number of additional qubits. It is observed that, by increasing the number of additi…
Non-Markovian evolution of an open quantum system can be induced by the memory effects of a reservoir. Although a reservoir with stronger memory effects may seem like it should cause stronger non-Markovian effects on the system of interest, this seemingly intuitive thinking may not always be correct. We illustrate this by investigating a qubit (a two-level atom) that is coupled to a hierarchical environment, which contains a single-mode cavity and a reservoir consisting of infinite numbers of modes. We show how the non-Markovian character of the system is influenced by the coupling strength be…
The squeezing dynamics of a damped harmonic oscillator are studied for different types of environment without making the Markovian approximation. The squeezing dynamics of a coherent state depend on the reservoir spectrum in a unique way that can, in the weak coupling approximation, be analyzed analytically. Comparison of squeezing dynamics for Ohmic, sub-Ohmic and super-Ohmic environments is done showing a clear connection between the squeezing--non-squeezing oscillations and reservoir structure. Understanding the effects occurring due to structured reservoirs is important both from a purely …
For a bosonic (fermionic) open system in a bath with many bosons (fermions) modes, we derive the exact non-Markovian master equation in which the memory effect of the bath is reflected in the time dependent decay rates. In this approach, the reduced density operator is constructed from the formal solution of the corresponding Heisenberg equations. As an application of the exact master equation, we study the active probing of non-Markovianity of the quantum dissipation of a single boson mode of electromagnetic (EM) field in a cavity QED system. The non-Markovianity of the bath of the cavity is …
We present a detailed microscopic derivation for a non-Markovian master equation for a driven two-state system interacting with a general structured reservoir. The master equation is derived using the time-convolutionless projection operator technique in the limit of weak coupling between the two-state quantum system and its environment. We briefly discuss the Markov approximation, the secular approximation and their validity.
A wide class of exact master equations for a multiple qubit system can be explicitly constructed by using the corresponding exact non-Markovian quantum state diffusion equations. These exact master equations arise naturally from the quantum decoherence dynamics of qubit system as a quantum memory coupled to a collective colored noisy source. The exact master equations are also important in optimal quantum control, quantum dissipation and quantum thermodynamics. In this paper, we show that the exact non-Markovian master equation for a dissipative N-qubit system can be derived explicitly from th…
In this paper, we provide a mechanism of decoherence suppression for open quantum systems in general, and that for "Schrodinger cat-like" state in particular, through the strong couplings to non-Markovian reservoirs. Different from the usual strategies of suppressing decoherence by decoupling the system from the environment in the literatures, here the decoherence suppression employs the strong back-reaction from non-Markovian reservoirs. The mechanism relies on the existence of the singularities (bound states) of the nonequilibrium retarded Green function which completely determines the dissi…
We evaluate exactly the non-Markovian effect on the decoherence dynamics of a qubit interacting with a dissipative vacuum reservoir and find that the coherence of the qubit can be partially trapped in the steady state when the memory effect of the reservoir is considered. Our analysis shows that it is the formation of a bound state between the qubit and its reservoir that results in this residual coherence in the steady state under the non-Markovian dynamics. A physical condition for the decoherence suppression is given explicitly. Our results suggest a potential way to decoherence control by …
Based on tensor network realizations of path integrals reducing exponential memory scaling to polynomial efficiency and a Liouville space implementation of a time-discrete quantum memory, we investigate a quantum system simultaneously exposed to two structured reservoirs. For this purpose, we employ a numerically exact quasi-2D tensor network combining both diagonal and off-diagonal system-reservoir interactions with a twofold memory for continuous and discrete retardation effects. As a possible example, we study the non-Markovian dynamical interplay between discrete photonic feedback and stru…
The uncertainty relation is a fundamental limit in quantum mechanics and is of great importance to quantum information processing as it relates to quantum precision measurement. Due to interactions with the surrounding environment, a quantum system will unavoidably suffer from decoherence. Here, we investigate the dynamic behaviors of the entropic uncertainty relation of an atom-cavity interacting system under a bosonic reservoir during the crossover between Markovian and non-Markovian regimes. Specifically, we explore the dynamic behavior of the entropic uncertainty relation for a pair of inc…
The time-convolutionless (TCL) non-Markovian master equation was generally thought to break down at finite time due to its singularity and fail to produce the asymptotic behavior in strong coupling regime. However, in this paper, we show that the singularity is not an obstacle for validity of the TCL master equation. Further, we propose a multiscale perturbative method valid for solving the TCL master equation in strong coupling regime, though the ordinary perturbative method invalidates therein.
We present a formalism that enables the study of the non-Markovian dynamics of a three-level ladder system in a single structured reservoir. The three-level system is strongly coupled to a bath of reservoir modes and two quantum excitations of the reservoir are expected. We show that the dynamics only depends on reservoir structure functions, which are products of the mode density with the coupling constant squared. This result may enable pseudomode theory to treat multiple excitations of a structured reservoir. The treatment uses Laplace transforms and an elimination of variables to obtain a …
We discuss the derivation of master equations in the presence of initial correlations with the reservoir. In van Hove's limit, the total system behaves as if it started from a factorized initial condition. A proper choice of Nakajima-Zwanzig's projection operator is crucial and the reservoir should be endowed with the mixing property.
We study the non-Markovian quantum interference phenomenon of a multi-state atomic system coupled to a bosonic dissipative environment by using the exact master equations derived in this paper. Two examples involving four-level systems with or without external control fields are investigated. Our results show that non-Markovian master equations are capable of exhibiting quantum interference phenomena in a temporal domain that has not fully explored before. In particular, we show that the environmental memory is instrumental in the onset of non-Markovian quantum interference pattern in differen…
We provide the exact non-Markovian master equation for a two-level system interacting with a thermal bosonic bath, and we write the solution of such a master equation in terms of the Bloch vector. We show that previous approximated results are particular limits of our exact master equation. We generalize these results to more complex systems involving an arbitrary number of two-level systems coupled to different thermal baths, providing the exact master equations also for these systems. As an example of this general case we derive the master equation for the Jaynes-Cummings model.
An exactly solvable model for the multi-level system interacting with several reservoirs at zero temperatures is presented. Population decay rates and decoherence rates predicted by exact solution and several approximate master equations, which are widespread in physical literature, are compared. The space of parameters is classified with respect to different inequities between the exact and approximate rates.
We study the non-Markovianity and quantum speedup of a two-level atom (quantum system of interest) in a dissipative Jaynes-Cumming model, where the atom is embedded in a single-mode cavity, which is leaky being coupled to an external reservoir with Ohmic spectral density. We obtain the non-Markovianity characterized by using the probability of the atomic excited state and the negative decoherence rate in the time-local master equation. We also calculate the quantum speed limit time (QSLT) of the evolution process of the atom. The results show that, the atom-cavity coupling is the main physical…
The Monte Carlo wave function method or the quantum trajectory/jump approach is a powerful tool to study dissipative dynamics governed by the Markovian master equation, in particular for high-dimensional systems and when it is difficult to simulate directly. In this paper, we extend this method to the non-Markovian case described by the generalized Lindblad master equation. Two examples to illustrate the method are presented and discussed. The results show that the method can correctly reproduce the dissipative dynamics for the system. The difference between this method and the traditional Mar…
We present a derivation of the Markovian master equation by the renormalization group method. Starting from a naive perturbative solution of the von Neumann equation, the reduced density matrix with the coarse grained time steps is obtained using the assumption of short correlation time of the bath field. Then by applying the renormalization group method, we show that the dependence of the specific initial time on the perturbative solution can be removed and the Markovian semigroup master equation in the Gorini--Kossakowski--Lindblad--Sudarshan (GKLS) form is obtained in the weak coupling limi…
The exact non-Markovian dynamics of a microcavity strongly coupled to a general reservoir at arbitrary temperature is studied. With the exact master equation for the reduced density operator of the cavity system, we analytically solve the time evolution of the cavity state and the associated physical observables. We show that the non-Markovian dynamics is completely determined by the propagating (retarded) and correlation Green functions. Compare the non-Markovian behavior at finite temperature with those at zero-temperature limit or Born-Markov limit, we find that the non-Markovian memory eff…
We derive Markovian master equations of single and interacting harmonic systems in different scenarios, including strong internal coupling. By comparing the dynamics resulting from the corresponding Markovian master equations with exact numerical simulations of the evolution of the global system, we precisely delimit their validity regimes and assess the robustness of the assumptions usually made in the process of deriving the reduced dynamics. The proposed method is sufficiently general to suggest that the conclusions made here are widely applicable to a large class of settings involving inte…
Non-Markovian dynamics of two interacting two-level qubits coupled to a bosonic bath was previously studied using the quantum-state-diffusion (QSD) equation, where a stochastic state is used to describe the system. In this study, we provide another perspective on this system by deriving the analytic form of the master equation, which describes the system with a reduced density matrix. Then, we validate the master equation by examining entanglement generation and state purity. In addition, with the master equation, we observe the effects from first-order noise and notice that a good asymptotic …
The model of a multi-level system interacting with several reservoirs is considered. The exact reduced density matrix evolution could be obtained for this model without Markov approximation. Namely, this evolution is fully defined by the finite set of linear differential equations. In this work the results which were obtained previously for only one Lorentz peak in the spectral density are generalized to the case of an arbitrary number of such peaks. The case of the Ohmic contribution in the spectral density is also taken here into account.
In this paper, the exact dynamics of open quantum systems in the presence of initial system-reservoir correlations is investigated for a photonic cavity system coupled to a general non-Markovian reservoir. The exact time-convolutionless master equation incorporating with initial system-reservoir correlations is obtained. The non-Markovian dynamics of the reservoir and the effects of the initial correlations are embedded into the time-dependent coefficients in the master equation. We show that the effects induced by the initial correlations play an important role in the non-Markovian dynamics o…
Master equations govern the time evolution of a quantum system interacting with an environment, and may be written in a variety of forms. Time-independent or memoryless master equations, in particular, can be cast in the well-known Lindblad form. Any time-local master equation, Markovian or non-Markovian, may in fact also be written in a Lindblad-like form. A diagonalisation procedure results in a unique, and in this sense canonical, representation of the equation, which may be used to fully characterize the non-Markovianity of the time evolution. Recently, several different measures of non-Ma…
We study the dynamics of a macroscopic superconducting qubit coupled to two independent non-stationary reservoirs by using time-dependent perturbation theory. We show that an equilibrium environment surpasses the coherent evolution of the macroscopic qubit completely. When the qubit couples to two different reservoirs, exemplifying a non-equilibrium environment, the short-time dynamics is affected by the interference between two reservoirs, implying the non-additivity of effects of two reservoirs. The non-additivity can be traced back to a non-Markovian effect, even though two reservoirs are i…
In this paper we demonstrate that two commonly used phenomenological post-Markovian quantum master equations can be derived without using any perturbative approximation. A system coupled to an environment characterized by self-classical configurational fluctuations, the latter obeying a Markovian dynamics, defines the underlying physical model. Both Shabani-Lidar equation [A. Shabani and D. A. Lidar, Phys. Rev. A 71, 020101(R) (2005)] and its associated approximated integro-differential kernel master equation are obtained by tracing out two different bipartite Markovian Lindblad dynamics where…
We study the decoherence properties of an entangled bipartite qubit system, represented by two two-level atoms that are individually coupled to non-Markovian reservoirs. This coupling ensures that the dynamical equations of the atoms can be treated independently. The non-Markovian reservoirs are described by a model which leads to an exact non-Markovian master equation of the Nakajima-Zwanzig form [J. Salo, S. M. Barnett, and S. Stenholm, Optics Commun. 259, 772 (2006)]. We consider the evolution of the entanglement of a two-atom state that is initially completely entangled, quantified by its …
We study the non-Markovian dynamics of a two-mode bosonic system interacting with two uncorrelated thermal bosonic reservoirs. We present the solution to the exact microscopic Master equation in terms of the quantum characteristic function and study in details the dynamics of entanglement for bipartite Gaussian states. In particular, we analyze the effects of short-time system-reservoir correlations on the separability thresholds and show that the relevant parameter is the reservoir spectral density. If the frequencies of the involved modes are within the reservoir spectral density entanglemen…
Stochastic Schr{ö}dinger equations for quantum trajectories offer an alternative and sometimes superior approach to the study of open quantum system dynamics. Here we show that recently established convolutionless non-Markovian stochastic Schr{ö}dinger equations may serve as a powerful tool for the derivation of convolutionless master equations for non-Markovian open quantum systems. The most interesting example is quantum Brownian motion (QBM) of a harmonic oscillator coupled to a heat bath of oscillators, one of the most-employed exactly soluble models of open system dynamics. We show explic…
Exact master equations describing the decay of a two-state system into a structured reservoir are constructed. Employing the exact solution for the model we determine analytical expressions for the memory kernel of the Nakajima-Zwanzig master equation and for the generator of the corresponding time-convolutionless master equation. This approach allows a detailed investigation and comparison of the convergence behavior of the corresponding perturbation expansions. Moreover, we find that the structure of widely used phenomenological master equations with memory kernel may be incompatible with a …
This paper deals with the dissipative dynamics of a quantum harmonic oscillator interacting with a bosonic reservoir. The Master Equations based on the Rotating Wave and on the Feynman-Vernon system--reservoir couplings are compared highlighting differences and analogies. We discuss quantitatively and qualitatively the conditions under which the counter rotating terms can be neglected. By comparing the analytic solution of the heating function relative to the two different coupling models we conclude that, even in the weak coupling limit, the counter rotating terms give rise to a significant c…
Within the f-deformed oscillator formalism, we derive a Markovian master equation for the description of the damped dynamics of nonlinear systems that interact with their environment. The applicability of this treatment to the particular case of a Morse-like oscillator interacting with a thermal field is illustrated, and the decay of quantum coherence in such a system is analyzed in terms of the evolution on phase space of its nonlinear coherent states via the Wigner function.
We investigate the entanglement dynamics of two initially entangled qubits interacting independently with two uncorrelated reservoirs beyond the Markovian approximation. Quite different from the Markovian reservoirs [C. E. Lopez et al., Phys. Rev. Lett. 101 (2008) 080503], we find that entanglement sudden birth (ESB) of the two reservoirs occurs without certain symmetry with respect to the entanglement sudden death (ESD) of the two qubits. A phenomenological interpretation of entanglement revival is also given.
Driven two-level system is a useful model to describe many quantum objects, particularly in quantum information processing. However, the exact master equation for such a system is barely explored. Making use of the Feynman-Vernon influence functional theory, we derive an exact non-Markovian master equation for the driven two-level system and show the lost feature in the perturbative treatment for this system. The perturbative treatment leads to the time-convolutionless (TCL) and the Nakajima-Zwanzig (NZ) master equations. So to this end, we derive the time-convolutionless (TCL) and the Nakajim…
Do phenomenological master equations with memory kernel always describe a non-Markovian quantum dynamics characterized by reverse flow of information? Is the integration over the past states of the system an unmistakable signature of non-Markovianity? We show by a counterexample that this is not always the case. We consider two commonly used phenomenological integro-differential master equations describing the dynamics of a spin 1/2 in a thermal bath. By using a recently introduced measure to quantify non-Markovianity [H.-P. Breuer, E.-M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 210401 (2009…
In this paper we investigate the non-Markovian dynamics of a qubit by comparing two generalized master equations with memory. In the case of a thermal bath, we derive the solution of the post-Markovian master equation recently proposed in Ref. [A. Shabani and D.A. Lidar, Phys. Rev. A {\bf 71}, 020101(R) (2005)] and we study the dynamics for an exponentially decaying memory kernel. We compare the solution of the post-Markovian master equation with the solution of the typical memory kernel master equation. Our results lead to a new physical interpretation of the reservoir correlation function an…
A new post-Markovian quantum master equation is derived, that includes bath memory effects via a phenomenologically introduced memory kernel k(t). The derivation uses as a formal tool a probabilistic single-shot bath-measurement process performed during the coupled system-bath evolution. The resulting analytically solvable master equation interpolates between the exact Nakajima-Zwanzig equation and the Markovian Lindblad equation. A necessary and sufficient condition for complete positivity in terms of properties of k(t) is presented, in addition to a prescription for the experimental determin…
