We extend non-Hermitian topological quantum walks on a Su-Schrieffer-Heeger (SSH) lattice [M. S. Rudner and L. Levitov, Phys. Rev. Lett. 102, 065703 (2009)] to the case of non-Markovian evolution. This non-Markovian model is established by coupling each unit cell in the SSH lattice to a reservoir formed by a quasi-continuum of levels. We find a topological transition in this model even in the case of non-Markovian evolution, where the walker may visit the reservoir and return to the SSH lattice at a later time. The existence of a topological transition does, however, depend on the low-frequency properties of the reservoir, characterized by a spectral density $J(ε)\propto |ε|^α$. In particular, we find a robust topological transition for a sub-Ohmic ($α<1$) and Ohmic ($α=1$) reservoir, but no topological transition for a super-Ohmic ($α>1$) reservoir. This behavior is directly related to the well-known localization transition for the spin-boson model. We confirm the presence of non-Markovian dynamics by explicitly evaluating a measure of Markovianity for this model.
It has been found that Markovian quantum dissipative processes, described by the Lindblad equation, may have attractive steady-state manifolds, in which dissipation and decoherence can play a positive role to quantum information processing. In this article, we show that such attractive steady-state manifolds with the same positive features as in Markovian dissipative processes can exist in the non-Markovian dissipative processes. Our finding indicates that the dissipation-assisted schemes implemented in the Markovian systems can be directly generalized to the non-Markovian systems.
We show that two, non interacting 2-level systems, immersed in a common bath, can become mutually entangled when evolving according to a Markovian, completely positive reduced dynamics.
Work, moments of work and heat flux are studied for the generic case of a strongly driven twolevel system immersed in a bosonic heat bath in domains of parameter space where perturbative treatments fail. This includes particularly the interplay between non-Markovian dynamics and moderate to strong external driving. Exact data are compared with predictions from weak coupling approaches. Further, the role of system-bath correlations in the initial thermal state and their impact on the heat flux are addressed. The relevance of these results for current experimental activities on solid state devic…
The so-called Maryland model is a linear version of the quantum kicked rotor; it exhibits Anderson localization in momentum space. By turning the kicks into a Markovian stochastic process, the dynamics becomes a dissipative quantum process described by a discrete family of completely positive maps that allows to explicitly study the relation between divisibility of the maps and the degree of memory of the process.
We provide the exact analytic solution of the stochastic Schrödinger equation describing an harmonic oscillator interacting with a non-Markovian and dissipative environment. This result represents an arrival point in the study of non-Markovian dynamics via stochastic differential equations. It is also one of the few exactly solvable models, for infinite dimensional systems. We compute the Green's function; in the case of a free particle, and with an exponentially correlated noise, we discuss the evolution of Gaussian wave functions.
We propose a procedure to fully reconstruct the time-dependent coefficients of convolutionless non-Markovian dissipative generators via a finite number of experimental measurements. By combining a tomography based approach with a proper data sampling, our proposal allows to relate the time-dependent coefficients governing the dissipative evolution of a quantum system to experimentally accessible quantities. The proposed scheme not only provides a way to retrieve full information about potentially unknown dissipative coefficients but also, most valuably, can be employed as a reliable consistenc…
We investigate the non-Markovian quantum dynamics of a hybrid open system consisting of one qubit and one qutrit by employing a stochastic Schrödinger equation to generate diffusive quantum trajectories. We have established an exact quantum state diffusion (QSD) equation for the dissipative qubit-qutrit system coupled to a bosonic heat bath at zero temperature. As an important application, the non-Markovian QSD equation is employed to simulate the entanglement decay and generation measured by negativity. Finally, some steady state properties of the hybrid system are also discussed.
We study a stochastic Schroedinger equation that generates a family of Gaussian dynamical maps in one dimension permitting a detailed exam of two different definitions of non-Markovianity: one related to the explicit dependence of the generator on the starting time, the other to the non-divisibility of the time-evolution maps. The model shows instances where one has non-Markovianity in both senses and cases when one has Markovianity in the second sense but not in the first one.
Quantum coherence and non-Markovianity of an atom in dissipative cavity under weak measurement are investigated in this work. We find that, the quantum coherence obviously depends on the atomic initial state, the strength of the weak measurement and its reversal, the atom-cavity coupling constant and the non-Markovian effecct. The more obvious the weak measurement effect is, the better the protection of coherence is. The quantum coherence is preserved more efficiently for lager the atom-cavity coupling. The stronger the non-Markovian effect, the more slowly the coherence reduces. This is, the …
We study the dissipative dynamics of $N$ quantum spins with Lindblad generator consisting of operators scaling as fluctuations, namely with the inverse square-root of $N$. In the large $N$ limit, the microscopic dissipative time-evolution converges to a non-Markovian unitary dynamics on strictly local operators, while at the mesoscopic level of fluctuations it gives rise to a dissipative non-Markovian dynamics. The mesoscopic time-evolution is Gaussian and exhibits either a stable or an unstable asymptotic character; furthermore, the mesoscopic dynamics builds correlations among fluctuations t…
We investigate the optimal control problem for non-Markovian open, dissipative quantum system. Optimal control using Pontryagin maximum principle is specifically derived. The influences of Ohmic reservoir with Lorentz-Drude regularization are numerically studied in a two-level system under the following three conditions: ω_0\llω_c, ω_0\approxω_c or ω_0\ggω_c, where ω_0 is the characteristic frequency of the quantum system of interest, and ω_c the cut-off frequency of Ohmic reservoir. The optimal control process shows its remarkable influences on the decoherence dynamics. The temperature is a k…
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 …
The dynamics of quantum Fisher information (QFI) of the phase parameter in a driven two-state system is studied within the framework of non-Markovian dissipative process. The influences of memory effects, classical driving and detunings on the parameter-estimation precision are demonstrated by exactly solving the Hamiltonian under rotating-wave approximation. In sharp contrast with the results obtained in the presence of Markovian dissipation, we find that classical driving can drastically enhance the QFI, namely, the precision of parameter estimation in the non-Markovian regime. Moreover, the…
Non-Markovian quantum effects are typically observed in systems interacting with structured reservoirs. Discrete-time quantum walks are prime example of such systems in which, quantum memory arises due to the controlled interaction between the coin and position degrees of freedom. Here we show that the information backflow that quantifies memory effects can be enhanced when the particle is subjected to uncorrelated static or dynamic disorder. The presence of disorder in the system leads to localization effects in 1-dimensional quantum walks. We shown that it is possible to infer about the natu…
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.
The nonequilibrium dynamics of a small quantum system coupled to a dissipative environment is studied. We show that (1) the oscillatory dynamics close to a coherent-to-incoherent transition is surprisingly different from the one of the classical damped harmonic oscillator and that (2) non-markovian memory plays a prominent role in the time evolution after a quantum quench.
We investigate the geometric phase of a two-level atom (qubit) coupled to a bosonic reservoir with Lorentzian spectral density, and find that for the non-Markovian dynamics in which rotating-wave approximation (RWA) is performed, geometric phase has a $π$-phase jump at the nodal point. However, the exact result without RWA given by hierarchy equation of motion method shows that there is no such a phase jump or nodal structure in geometric phase. Thus our results demonstrate that the counter-rotating terms significantly contribute to the geometric phase in multi-mode Hamiltonian under certain c…
We consider the non-Markovian Langevin evolution of a dissipative dynamical system in quantum mechanics in the path integral formalism. After discussing the role of the frequency cutoff for the interaction of the system with the heat bath and the kernel and noise correlator that follow from the most common choices, we derive an analytic expansion for the exact non-Markovian dissipation kernel and the corresponding colored noise in the general case that is consistent with the fluctuation-dissipation theorem and incorporates systematically non-local corrections. We illustrate the modifications t…
We give the map representing the evolution of a qubit under the action of non-dissipative random external fields. From this map we construct the corresponding master equation that in turn allows us to phenomenologically introduce population damping of the qubit system. We then compare, in this system, the time-regions when non-Markovianity is present on the basis of different criteria both for the non-dissipative and dissipative case. We show that the adopted criteria agree both in the non-dissipative case and in the presence of population damping.
Quantum metrology utilizes quantum effects to reach higher precision measurements of physical quantities compared with their classical counterparts. However the ubiquitous decoherence obstructs its application. Recently, non-Markovian effects are shown to be effective in performing quantum optical metrology under locally dissipative environments\cite{PhysRevLett.123.040402}. However, the mechanism is still rather hazy. Here, we uncover the reason why forming a bound state can protect the quantumness against a dissipative ambient via the quantum Fisher information of entangled coherent states. …
In the case of the discrete time coined quantum walk the reduced dynamics of the coin shows non-Markovian recurrence features due to information back-flow from the position degree of freedom. Here we study how this non-Markovian behavior is modified in the presence of open system dynamics. In the process, we obtain useful insights into the nature of non-Markovian physics. In particular, we show that in the case of (non-Markovian) random telegraph noise (RTN), a further discernbile recurrence feature is present in the dynamics. Moreover, this feature is correlated with the localization of the w…
Quantum walk acts obviously different from its classical counterpart, but decoherence will lessen and close the gap between them. To understand this process, it is necessary to investigate the evolution of quantum walk under different situation of decoherence. In this article, we study a non-Markovian decoherent quantum walk on a line. In the short time regime, the behavior of the walk deviates from both idea quantum walks and classical random walks. The position variance as a measure of quantum walk starts oscillating from the first several steps and tends to be linear on time and showing a d…
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…
Quantum non-Markovianity of a quantum noisy channel manifests typically as information backflow, characterized by the departure of the intermediate map from complete positivity, though we indicate certain noisy channels that don't exhibit this behavior. In complex systems, non-Markovianity becomes more involved on account of subsystem dynamics. Here we study various facets of non-Markovian evolution, in the context of coined quantum walks, with particular stress on disambiguating the internal vs. environmental contributions to non-Markovian backflow. For the above problem of disambiguation, we…
We investigate the dynamical information exchange between a two-state system and its environment which is measured by von Neumann entropy. It is found that in the underdamping regime, the entropy dynamics exhibits an extremely non-Markovian oscillation-hump feature, in which oscillations manifest quantum coherence and a hump of envelop demonstrates temporal memory of bath. It indicates that the process of entropy exchange is bidirectional. When the coupling strength increases a certain threshold, the hump along with ripple disappears, which is indicative of the coherent-incoherent dynamical cr…
Understanding whether dissipation in an open quantum system is truly quantum is a question of both fundamental and practical interest. We consider n qubits subject to correlated Markovian dephasing and present a sufficient condition for when bath-induced dissipation can generate system entanglement and hence must be considered quantum. Surprisingly, we find that the presence or absence of time-reversal symmetry plays a crucial role: broken time-reversal symmetry is required for dissipative entanglement generation. Further, simply having nonzero bath susceptibilities is not enough for the dissi…
We study the geometric phase of a two-level atom coupled to an environment with Lorentzian spectral density. The non-Markovian effect on the geometric phase is explored analytically and numerically. In the weak coupling limit the lowest-order correction to the geometric phase is derived analytically and the general case is calculated numerically. It is found that the correction to the geometric phase is significantly large if the spectral width is small and in this case the non-Markovian dynamics has a significant impact to the geometric phase. When the spectral width increases, the correction…
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…
The quantum channels with memory, known as non-Markovian channels, are of crucial importance for a realistic description of a variety of physical systems, and pave ways for new methods of decoherence control by manipulating the properties of environment such as its frequency spectrum. In this work, the reduced dynamics of coin in a discrete-time quantum walk is characterized as a non-Markovian quantum channel. A general formalism is sketched to extract the Kraus operators for a $t$-step quantum walk. Non-Markovianity, in the sense of P-indivisibility of the reduced coin dynamics, is inferred f…
We consider the dynamical properties of dissipative continuous-time quantum walks on directed graphs. Using a large-deviation approach we construct a thermodynamic formalism allowing us to define a dynamical order parameter, and to identify transitions between dynamical regimes. For a particular class of dissipative quantum walks we propose a quantum generalization of the the classical PageRank vector, used to rank the importance of nodes in a directed graph. We also provide an example where one can characterize the dynamical transition from an effective classical random walk to a dissipative …
We explore the dynamics of $N$ coupled atoms to a generic bosonic reservoir under specific system symmetries. In the regime of multiple atoms coupled to a single reservoir with identical couplings, we identify remarkable effects, notably that the initial configuration of the atomic excited state amplitudes strongly impacts the dynamics of the system and can even fully sever the system from its environment. Additionally, we find that steady state amplitudes of the excited states become independent of the choice of the reservoir. The framework introduced is applied to a structured photonic reser…
We address the dynamics of continuous-time quantum walks on one-dimensional disordered lattices inducing dynamical noise in the system. Noise is described as time-dependent fluctuations of the tunneling amplitudes between adjacent sites, and attention is focused on non-Gaussian telegraph noise, going beyond the usual assumption of fast Gaussian noise. We observe the emergence of two different dynamical behaviors for the walker, corresponding to two opposite noise regimes: slow noise (i.e. strong coupling with the environment) confines the walker into few lattice nodes, while fast noise (weak c…
We study a quantum walker on a one-dimensional lattice with a single defect site characterized by a phase. The spread and localization of discrete-time quantum walks starting at the impurity site are affected by the appearance of bound states and their reflection symmetry. We quantify the localization in terms of an effective localization length averaged over all eigenstates and an effective participation ratio after time evolution averaged over all initial states. We observe that the reduced coin system dynamics undergoes oscillations in the long-time limit, the frequencies of which are relat…
Floquet-Magnus (FM) expansion theory is a powerful tool in periodically driven (Floquet) systems under high-frequency drives. In closed systems, it dictates that their stroboscopic dynamics under a time-periodic Hamiltonian is well captured by the FM expansion, which gives a static effective Hamiltonian. On the other hand, in dissipative systems driven by a time-periodic Liouvillian, it remains an important and nontrivial problem whether the FM expansion gives a static Liouvillian describing continuous-time Markovian dynamics, which we refer to as the Liouvillianity of the FM expansion. We ans…
An open quantum walk formalism for dissipative quantum computing is presented. The approach is illustrated with the examples of the Toffoli gate and the Quantum Fourier Transform for 3 and 4 qubits. It is shown that the algorithms based on the open quantum walk formalism are more efficient than the canonical dissipative quantum computing approach. In particular, the open quantum walks can be designed to converge faster to the desired steady state and to increase the probability of detection of the outcome of the computation.
In this work, we obtain an analytical representation of the density operator of an atom in dissipative cavity when the reservoir is at zero temperature and the total number of excitation is N=1. We also investigated the quantum speed limit time(QSLT) of the atom and the non-Markovianity in the dynamics process. The results show that the QSLT and the non-Markovianity can be effectively manipulated by the atom-cavity coupling and the reservoir parameters. Both of the atom-cavity coupling and the detuning can induce a sudden transition from Markovian to non-Markovian dynamics and this transition …
We analyze in detail the discrete--time quantum walk on the line by separating the quantum evolution equation into Markovian and interference terms. As a result of this separation, it is possible to show analytically that the quadratic increase in the variance of the quantum walker's position with time is a direct consequence of the coherence of the quantum evolution. If the evolution is decoherent, as in the classical case, the variance is shown to increase linearly with time, as expected. Furthermore we show that this system has an evolution operator analogous to that of a resonant quantum k…
We investigate the time evolution of the chirality reduced density matrix for a discrete-time quantum walk on a one-dimensional lattice, which is obtained by tracing out the spatial degree of freedom. We analyze the standard case, without decoherence, and the situation where decoherence appears in the form of broken links in the lattice. By examining the trace distance for possible pairs of initial states as a function of time, we conclude that the evolution of the reduced density matrix is non-Markovian, in the sense defined in [H. P. Breuer, E. M. Laine, and J. Piilo, Phys. Rev. Lett. 103, 2…
We study the laser-driven Dicke model beyond the rotating-wave approximation. For weak coupling of the system to environmental degrees of freedom the dissipative dynamics of the emitter-cavity system is described by the Floquet master equation. Projection of the system evolution onto the emitter degrees of freedom results in non-Markovian behavior. We quantify the non-Markovianity of the resulting emitter dynamics and show that this quantity can be used as an indicator of the dissipative quantum phase transition occurring at high driving amplitudes.
We investigate spin squeezing for a Lipkin-Meshkov-Glick (LMG) model coupled to a general non-Markovian environment in a finite temperature regime. Using the non-Markovian quantum state diffusion and master equation approach, we numerically study non-Markovian spin squeezing generation in LMG model. Our results show that the total spin number N, energy kBT, and certain coefficients in a LMG model can play a crucial role in generating spin squeezing. In particular, it shows that the maximum spin squeezing can be significantly enhanced when the participating environment has a relatively long mem…
We examine the non-Markovian dynamics in a multipartite system of two initially correlated atomic qubits, each located in a single-mode leaky cavity and interacting with its own bosonic reservoir. We show the dominance of non-Markovian features, as quantified by the difference in fidelity of the evolved system with its density matrix at an earlier time, in three specific two-qubit partitions associated with the cavity-cavity and atom-reservoir density matrices within the same subsystem, and the cavity-reservoir reduced matrix across the two subsystems. The non-Markovianity in the cavity-cavity…
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 study transitions between the Floquet states of a periodically driven oscillator caused by the coupling of the oscillator to a thermal reservoir. The analysis refers to the oscillator that is driven close to triple its eigenfrequency and displays resonant period tripling. The interstate transitions result in a random ``walk'' over the states. We find the transition rates and show that the walk is nonlocal in the state space: the stationary distribution over the states is formed by the transitions between remote states. This is to be contrasted with systems in thermal equilibrium, where the …
In this paper, a scheme for the generation of long-living entanglement between two distant Λ-type three-level atoms separately trapped in two dissipative cavities is proposed. In this scheme, two dissipative cavities are coupled to their own non-Markovian environments and two three-level atoms are driven by the classical fields. The entangled state between the two atoms is produced by performing Bell state measurement (BSM) on photons leaving the dissipative cavities. Using the time-dependent Schördinger equation, we obtain the analytical results for the evolution of the entanglement. It is re…
We investigate a `pseudo thermalization' effect, where an open quantum system coupled to several non-Markovian reservoirs presents an emergent thermal behaviour in spite of its coupling to a non-equilibrated environment. The thermal behaviour is visible at both static and dynamical levels and the system satisfies the fluctuation-dissipation theorem. Our analysis is focused on the exactly solvable model of a weakly interacting driven-dissipative Bose gas in presence of frequency-dependent particle pumping and losses, and is based on a quantum Langevin theory, which we derive starting from a mic…
We present a quantum algorithm to simulate general finite dimensional Lindblad master equations without the requirement of engineering the system-environment interactions. The proposed method is able to simulate both Markovian and non-Markovian quantum dynamics. It consists in the quantum computation of the dissipative corrections to the unitary evolution of the system of interest, via the reconstruction of the response functions associated with the Lindblad operators. Our approach is equally applicable to dynamics generated by effectively non-Hermitian Hamiltonians. We confirm the quality of …
Capturing non-Markovian dynamics of open quantum systems is generally a challenging problem, especially for strongly-interacting many-body systems. In this work, we combine recently developed non-Markovian quantum state diffusion techniques with tensor network methods to address this challenge. As a first example, we explore a Hubbard-Holstein model with dissipative phonon modes, where this new approach allows us to quantitatively assess how correlations spread in the presence of non-Markovian dissipation in a 1D many-body system. We find regimes where correlation growth can be enhanced by the…
Quantum Darwinism explains the emergence of a classical description of objects in terms of the creation of many redundant registers in an environment containing their classical information. This amplification phenomenon, where only classical information reaches the macroscopic observer and through which different observers can agree on the objective existence of such object, has been revived lately for several types of situations, successfully explaining classicality. We explore quantum Darwinism in the setting of an environment made of two level systems which are initially prepared in the gro…
