non-Markovian Master Equation Research Articles

Non-Markovian master equation for quantum transport of fermionic carriers.

We propose a simple, yet feasible, model for quantum transport of fermionic carriers across tight-binding chain connecting two reservoirs maintained at arbitrary temperatures and chemical potentials. The model allows for elementary derivation of the master equation for the reduced single particle density matrix in a closed form in both Markov and Born approximations. In the Markov approximation the master equation is solved analytically, whereas in the Born approximation the problem is reduced to an algebraic equation for the single particle density matrix in the Redfield form. The non-Markovian equation is shown to lead to resonant transport similar to Landauer's conductance. It is shown that in the deep non-Markovian regime the transport current can be matched with that obtained by the non-equilibrium Green's function method.

Quantum Trajectories for Time-Local Non-Lindblad Master Equations.

For the efficient simulation of open quantum systems, we often use quantum jump trajectories given by pure states that evolve stochastically to unravel the dynamics of the underlying master equation. In the Markovian regime, when the dynamics is described by a Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) master equation, this procedure is known as MonteCarlo wave function approach. However, beyond ultraweak system-bath coupling, the dynamics of the system is not described by an equation of GKSL type, but rather by the Redfield equation, which can be brought into pseudo-Lindblad form. Here, negative dissipation strengths prohibit the conventional approach. To overcome this problem, we propose a pseudo-Lindblad quantum trajectory (PLQT) unraveling. It does not require an effective extension of the state space, like other approaches, except for the addition of a single classical bit. We test the PLQT for the eternal non-Markovian master equation for a single qubit and an interacting Fermi-Hubbard chain coupled to a thermal bath and discuss its computational effort compared to solving the full master equation.

Effects of non-Markovian squeezed bath on the dynamics of open systems

Control of the dynamics of an open quantum system is crucial in quantum information processing. Basically there are two ways: one is the control on the system and the other is tuning the bath parameters. In this paper, we use the latter to analyze the non-Markovian dynamics of the open system. The model is that the system is immersed in non-Markovian squeezed baths. For the dynamics, a non-Markovian master equation is obtained using the quantum state diffusion (QSD) equation technique for the weak system–bath couplings. We take the adiabatic evolution or quantum state transmission as examples to analyze the effects of the bath parameters: non-Markovianity γ, the squeezed direction θ and squeezed strength r. For the adiabatic or state transmission fidelity, the calculation results show that they both can be enhanced by a smaller γ or bigger p-quadrature. Interestingly, when 0<θ<π/2, the squeezed quadrature is determined by the combination of r and θ, and by numerical simulation we find that the fidelity peak occurs at r=1−2θ/π. The fidelities increase with increasing r when r∈(0,1−2θ/π]. When θ≥π/2, lower fidelities are obtained due to the squeezed baths. Our results show that the dynamics of the open systems can be effectively controlled by reservoir engineering.

Relaxation dynamics of nonresonant excitation transfer processes assisted by coherent phonon environment

In this study, the steepest-entropy-ascent quantum thermodynamics (SEAQT) framework is used to investigate the excitation transfer (ET) dynamics of two-level nanosystems (TLSs), focusing on non-resonant processes that involve a dynamic local phonon system. In contrast to other methods based on Markovian or non-Markovian quantum master equations, SEAQT analysis always guarantees the positivity of the density operators, thus enabling the discussion of both transient and long-term dynamics of nanosystems. The findings of this study demonstrate that the relaxation time of coherent phonons, relative to the Rabi oscillation period in two TLSs, significantly affects the relaxation process of non-resonant ETs. Moreover, the degree of mutual synchronization between ET dynamics of TLSs and local (coherent) phonons can either prolong or shorten the decoherence time, presenting a way to control the coherence in nanosystems and stimulate quantum device applications.

Quantum recoherence in the early universe

Despite being created through a fundamentally quantum-mechanical process, cosmological structures have not yet revealed any sign of genuine quantum correlations. Among the obstructions to the direct detection of quantum signatures in cosmology, environmental-induced decoherence is arguably one of the most inevitable. Yet, we discover a mechanism of quantum recoherence for the adiabatic perturbations when they couple to an entropic sector. After a transient phase of decoherence, a turning point is reached, recoherence proceeds and adiabatic perturbations exhibit a large amount of self-coherence at late-time. This result is also understood by means of a non-Markovian master equation, which reduces to Wilsonian effective-field theory in the unitary limit. This allows us to critically assess the validity of open-quantum-system methods in cosmology and to highlight that re(de)coherence from linear interactions has no flat-space analogue.

Decoherence and the ultraviolet cutoff: non-Markovian dissipative dynamics of a charged particle in a magnetic field

We derive a non-Markovian master equation for a charged particle in a magnetic field coupled to a bath and study decoherence by analyzing the temporal decay of the off-diagonal elements of the reduced density matrix in the position basis. The coherent oscillations characterized by the cyclotron frequency get suppressed as a result of decoherence due to coupling with the environment. We consider an Ohmic bath with three distinct models for the high-frequency cutoff for the spectral density of the bath and compare the three cases. As expected, the three cutoff models converge in the limit of the uppermost frequency of the bath tending to infinity. We notice a dramatic slowing down of loss of coherence in the low-temperature limit dominated by zero point quantum fluctuations compared to the high-temperature classical limit dominated by thermal fluctuations. We also go beyond the Ohmic model and study super-Ohmic and sub-Ohmic baths with the spectral densities deviating from a linear dependence on the frequency. Our results are testable in a state of the art cold atom laboratory.

Macroscopic entanglement generation in optomechanical system embedded in non-Markovian environment

In this paper, we analyze the macroscopic entanglement generation between two movable mirrors in an optomechanical system beyond the Markov approximation. We derive a non-Markovian master equation for the optomechanical system and analyze the entanglement generation based on the numerical simulation. It is found that the maximum entanglement can be achieved by utilizing both the optomechanical coupling and the non-Markovian properties of the bath. In particular, we show the contribution from the system-bath coupling has a positive impact only if the bath is non-Markovian. Besides, the symmetric properties of the optomechanical system as well as the initial state are crucial to the entanglement generation. The results presented in the paper illustrate that the impact of the non-Markovian bath is an important factor in the entanglement generation in the optomechanical system.

Non-Markovian dynamics under time-translation symmetry

A dynamical symmetry is employed to determine the structure of the quantum non-Markovian time-local master equation. Such a structure is composed from two components: scalar kinetic coefficients and the standard quantum Markovian operator form. The kinetic coefficients are generally time-dependent and incorporate information on the kinematics and memory effects, while the operators manifest the dynamical symmetry. Specifically, we focus on time-translation symmetric dynamics, where the Lindblad jump operators constitute the eigenoperators of the free dynamics. This symmetry is motivated by thermodynamic microscopic considerations, where strict energy conservation between system and environment imposes the time-translation symmetry. The construction is generalized to other symmetries, and to driven quantum systems. The formalism is illustrated by three exactly solvable non-Markovian models, where the exact reduced description exhibits a dynamical symmetric structure. The formal structure of the master equation leads to a first principle calculation of the exact kinetic coefficients. This opens the possibility to simulate in a modular fashion non-Markovian dynamics.

Dynamics of quantum Fisher information from a time-local non-Markovian master equation with decoherence rates and operators depending on the estimated parameter

We analyze the dynamics of the quantum Fisher information (QFI) in the case of a two-dimensional open quantum system obeying a time-local non-Markovian master equation in the canonical form, characterized by canonical decoherence rates and operators that depend on the parameter to be estimated. This last condition brings a framework in which the information about the parameter is structurally encoded not only in the open system, but also in the system-environment interaction or/and correlations, and in some environment properties. We derive analytical formulas for the decompositions of the QFI flow and of the purity dynamics in terms associated to the canonical decoherence rates. In contrast to the results presented in X.-M. Lu et al. [Phys. Rev. A 82, 042103 (2010)], we show that, in this extended framework, the QFI flow contains not only the subflows corresponding to the decoherence rates, but also supplementary terms which originate in the state dependence on the estimated parameter; moreover, the signs of the QFI subflows associated to the decoherence rates are not correlated to the signs of the rates. We show that a pertinent connection between the QFI flow and non-Markovianity can be realized using the canonical measures defined from the negative canonical decoherence rates. We employ this theoretical framework to explore the QFI flow and its subflows in two cases of quantum evolutions directed by master equations with decoherence rates or/and operators depending on the estimated parameter: (i) the Markovian nonunital time evolution of a qubit under the generalized amplitude damping channel; (ii) the non-Markovian time evolution of a two-dimensional electronic subsystem entangled with its vibrational environment in a molecule.

Field-theoretical approach to open quantum systems and the Lindblad equation

We develop a systematic field-theoretical approach to open quantum systems based on condensed-matter many-body methods. The time evolution of the reduced density matrix for the open quantum system is determined by a transmission matrix. Developing diagrammatic perturbation theory, invoking Wick's theorem in connection with a Caldeira-Leggett quantum oscillator environment in thermal equilibrium, the transmission matrix satisfies a Dyson equation characterized by an irreducible kernel. Unlike the Nakajima-Zwanzig and standard approaches, the Dyson equation is equivalent to a general non-Markovian master equation for the reduced density matrix, incorporating secular effects and independent of the initial preparation. The kernel is determined by a systematic diagrammatic expansion in powers of the interaction. We consider the Born approximation for the kernel. Applying a condensed-matter pole or, equivalently, a quasiparticle-type approximation, equivalent to the usual assumption of a timescale separation, we derive a master equation of the Markov type. Furthermore, imposing the rotating-wave approximation, we obtain a Markov master equation of the Lindblad form. To illustrate the method, we consider the standard example of a single qubit coupled to a thermal heat bath.

Master equation for non-Markovian quantum Brownian motion: The emergence of lateral coherences

Understanding the behavior of a quantum system coupled to its environment is of fundamental interest in the general field of quantum technologies. It also has important repercussions on foundational problems in physics, such as the process of decoherence and the so-called quantum measurement problem. There have been many approaches to explore Markovian and non-Markovian dynamics within the framework of open quantum systems, but the richness of the ensuing dynamics is still not fully understood. In this paper we develop a non-Markovian extension of the standard Caldeira-Leggett model, based on expanding the dynamics of the reduced system at high temperature in inverse powers of the high-frequency cutoff of the Ohmic spectral density of the environment and derive a non-Markovian master equation for the reduced density matrix for the case of a general potential. We also obtain a fully analytical solution in the free particle case. While the short-time behavior of this solution does not diverge substantially from the Markovian behavior, at intermediate times we find a resurgence of coherence, which we name lateral coherence. We identify this with a corresponding transient negative entropy production rate, which is understood to be characteristic of non-Markovian dynamics. We also analyze the positivity of the reduced density matrix and derive the corresponding Fokker-Planck equation in the classical limit.

Trajectory tracking for non-Markovian quantum systems

We propose a systematic scheme to engineer quantum states of a quantum system governed by a time-convolutionless non-Markovian master equation. According to the idea of reverse engineering, the general algebraic equation to determine the control parameters, such as coherent and incoherent control fields, is presented. Without artificially engineering the time-dependent decay rates and retaining the environment-induced Lamb shifts, the quantum state can still be transferred into the target state in a finite period of time along an arbitrary designed trajectory strictly in Hilbert space. As an application, we apply our scheme to a driven two-level non-Markovian system and realize instantaneous-steady-state tracking and a complete population inversion with control parameters which are available in experimental settings.

Non-Markovian dynamics of open quantum system with memory

In this paper, non-Markovian generalization of master equation for open quantum system is proposed. Non-Markovian dynamics of two-level quantum system with memory and interaction with environment is described. To describe this system, the Gorini–Kossakowski–Sudarshan equation for open quantum states is generalized by taking into account power-law fading memory. The non-Markovian quantum processes with power-law memory are described by using integration and differentiation of non-integer orders. Complete positivity and bi-positivity in non-Markovian quantum dynamics with memory are described. An example of two-level quantum systems with power-law memory is suggested. Exact solution of the non-Markovian master equations of two-level open quantum systems with memory is derived.

Solvable class of non-Markovian quantum multipartite dynamics

We study a class of multipartite open quantum dynamics for systems of arbitrary number of qubits. The non-Markovian quantum master equation can involve arbitrary single or multipartite and time-dependent dissipative coupling mechanisms, expressed in terms of strings of Pauli operators. We formulate the general constraints that guarantee the complete positivity of this dynamics. We characterize in detail underlying mechanisms that lead to memory effects, together with properties of the dynamics encoded in the associated system rates. We specifically derive multipartite "eternal" non-Markovian master equations that we term hyperbolic and trigonometric due to the time dependence of their rates. For these models we identify a transition between positive and periodically divergent rates. We also study non-Markovian effects through an operational (measurement-based) memory witness approach.

Resonant transmission of fermionic carriers: Comparison between solid-state physics and quantum optics approaches

We revisit the phenomenon of the resonant transmission of fermionic carriers through a quantum device connected to two contacts with different chemical potentials. We show that, besides the traditional Landauer-B\"uttiker approach in solid-state physics, this phenomenon can also be described by the non-Markovian master equation for the reduced density matrix of the fermions in the quantum device. We identify validity regions for both approaches in the system parameter space and argue that for large relaxation rates the accuracy of the latter approach greatly exceeds the accuracy of the former.

Pumping current in a non-Markovian N -state model

A periodically modulated N-state model whose dynamics is governed by a time-convoluted generalized master equation is theoretically analyzed. It is shown that this non-Markovian master equation can be converted to a Markovian master equation having a larger transition matrix, which affords easier analysis. The behavior of this model is investigated by focusing on the cycle-averaged pumping current. In the adiabatic limit, the geometrical current is calculated analytically, and compared to numerical results which are available for a wide range of modulation frequencies.

Non-Markovian perturbation theories for phonon effects in strong-coupling cavity quantum electrodynamics

Phonon interactions are inevitable in cavity quantum electrodynamical systems based on solid-state emitters or fluorescent molecules, where vibrations of the lattice or chemical bonds couple to the electronic degrees of freedom. Due to the non-Markovian response of the vibrational environment, it remains a significant theoretical challenge to describe such effects in a computationally efficient manner. This is particularly pronounced when the emitter-cavity coupling is comparable to or larger than the typical phonon energy range, and polariton formation coincides with vibrational dressing of the optical transitions. In this Article, we consider four non-Markovian perturbative master equation approaches to describe such dynamics over a broad range of light-matter coupling strengths and compare them to numerically exact reference calculations using a tensor network. The master equations are derived using different basis transformations and a perturbative expansion in the new basis is subsequently introduced and analyzed. We find that two approaches are particularly successful and robust. The first of these is suggested and developed in this Article and is based on a vibrational dressing of the exciton-cavity polaritons. This enables the description of distinct phonon-polariton sidebands that appear when the polariton splitting exceeds the typical phonon frequency scale in the environment. The second approach is based on a variationally optimized polaronic vibrational dressing of the electronic state. Both of these approaches demonstrate good qualitative and quantitative agreement with reference calculations of the emission spectrum and are numerically robust, even at elevated temperatures, where the thermal phonon population is significant.

Finite-time quantum Stirling heat engine

We study the thermodynamic performance of a finite-time non-regenerative quantum Stirling-like cycle used as a heat engine. We consider specifically the case in which the working substance (WS) is a two-level system (TLS). The Stirling cycle is made of two isochoric transformations separated by a compression and an expansion stroke during which the WS is in contact with a thermal reservoir. To describe these two strokes we derive a non-Markovian master equation which allows to study the real-time dynamics of a driven open quantum system with arbitrary fast driving. Following the real-time dynamics of the WS using this master equation, the endpoints of the isotherms can deviate from the equilibrium thermal states. The role of this deviation in the performance of the heat engine is addressed. We found that the finite-time dynamics and thermodynamics of the cycle depend non-trivially on the different time scales at play. In particular, driving the WS at a time scale comparable to the resonance time of the bath enhances the performance of the cycle and allows for an efficiency higher than the efficiency of the quasistatic cycle, but still below the Carnot bound. However, by adding thermalization of the WS with the baths at the end of compression/expansion processes one recovers the conventional scenario in which efficiency decreases by speeding up the processes. In addition, the performance of the cycle is dependent on the compression/expansion speeds asymmetrically, which suggests new freedom in optimizing quantum heat engines. The maximum output power and the maximum efficiency are obtained almost simultaneously when the real-time endpoints of the compression/expansion processes are considered instead of the equilibrium thermal endpoint states. However, the net extractable work always declines by speeding up the drive.

Range of applicability of the Hu-Paz-Zhang master equation

We investigate a case of the Hu-Paz-Zhang master equation of the Caldeira-Leggett model without Lindblad form obtained in the weak-coupling limit up to the second-order perturbation. In our study, we use Gaussian initial states to be able to employ a sufficient and necessary condition, which can expose positivity violations of the density operator during the time evolution. We demonstrate that the evolution of the non-Markovian master equation has problems when the stationary solution is not a positive operator, i.e., does not have physical interpretation. We also show that solutions always remain physical for small-times of evolution. Moreover, we identify a strong anomalous behavior, when the trace of the solution is diverging. We also provide results for the corresponding Markovian master equation and show that positivity violations occur for various types of initial conditions even when the stationary solution is a positive operator. Based on our numerical results, we conclude that this non-Markovian master equation is superior to the corresponding Markovian one.

Full-polaron master equation approach to dynamical steady states of a driven two-level system beyond the weak system-environment coupling

We apply a full-polaron master equation and a weak-coupling non-Markovian master equation to describe the steady-state time-averaged properties of a driven two-level system, an electron coherently tunneling between double quantum dots (DQDs), interacting with a bosonic phonon bath. Comparing the results obtained using these two master equations with those from a recent DQD experiment and its corresponding weak-coupling theoretical method, we find that the original parameter set used in the experiment and theoretical method is not in the weak-coupling parameter regime. By using the full-polaron master equation with a slight adjustment on only the value of the interdot separation in the original experimental parameter set, we find that a reasonable fit to the experimentally measured time-averaged steady-state population data can be achieved. The adjusted interdot separation is within the possible values allowed by the geometry of the surface gates that define the DQD in the experiment. Our full-polaron equation approach does not require the special renormalization scheme employed in their weak-coupling theoretical method, and can still describe the experimental results of driving-induced phonon-enhanced steplike shoulder behaviors in the experiment. This demonstrates that the full-polaron master equation approach is a correct and efficient tool to describe the steady-state properties of a driven spin-boson model in the case of strong system-environment coupling.