Joshua Wilkie
A mean field approximation is employed to derive a master equation suitable for self-interacting baths and strong system-bath coupling. Solutions of the master equation are compared with exact solutions for a central spin interacting with a spin-bath.
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A mean field argument is used to derive a master equation for systems simultaneously interacting with external fields and coupled environmental degrees of freedom. We prove that this master equation preserves positivity of the reduced density matrix. Solutions of the master equation are compared with exact solutions for a system consisting of three spins which is manipulated with a sequence of laser pulses while interacting with a spin-bath. Exact solutions appear to converge to the master equation result as the number of bath spins increases.
Time evolution of a harmonic oscillator linearly coupled to a heat bath is compared for three classes of initial states for the bath modes - grand canonical ensemble, number states and coherent states. It is shown that for a wide class of number states the behavior of the oscillator is similar to the case of the equilibrium bath. If the bath modes are initially in coherent states, then the variances of the oscillator coordinate and momentum, as well as its entanglement to the bath, asymptotically approach the same values as for the oscillator at zero temperature and the average coordinate and …
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 …
We solve analytically master equations that describe a cavity filled with a kerr medium, taking into account the dissipation induced by the environment, and parametric down conversion processes. We use superoperator techniques.
We study the analytically solvable Ising model of a single qubit system coupled to a spin bath. The purpose of this study is to analyze and elucidate the performance of Markovian and non-Markovian master equations describing the dynamics of the system qubit, in comparison to the exact solution. We find that the time-convolutionless master equation performs particularly well up to fourth order in the system-bath coupling constant, in comparison to the Nakajima-Zwanzig master equation. Markovian approaches fare poorly due to the infinite bath correlation time in this model. A recently proposed p…
Open systems of coupled qubits are ubiquitous in quantum physics. Finding a suitable master equation to describe their dynamics is therefore a crucial task that must be addressed with utmost attention. In the recent past, many efforts have been made toward the possibility of employing local master equations, which compute the interaction with the environment neglecting the direct coupling between the qubits, and for this reason may be easier to solve. Here, we provide a detailed derivation of the Markovian master equation for two coupled qubits interacting with common and separate baths, consi…
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 thermodynamic consistency of the master equation description of heat transport through an optomechanical system attached to two heat baths, one optical and one mechanical. We employ three different master equations to describe this scenario: (i) The standard master equation used in optomechanics, where each bath acts only on the resonator that it is physically connected to; (ii) the so-called dressed-state master equation, where the mechanical bath acts on the global system; and (iii) what we call the global master equation, where both baths are treated non-locally and affec…
Dynamics of a quantum system interacting with white non-Gaussian baths: Poisson noise master equation
本文针对浴由两态系统或自旋等构成的非高斯环境，提出基于泊松噪声的量子浴模型并推导主方程，揭示白噪声区域中的非高斯浴效应，为描述通用浴影响下的开放量子动力学奠定理论基础。
Fermi's golden rule applies to a situation in which a single quantum state $|ψ\rangle$ is coupled to a near-continuum. This "quasi-continuum coupling" structure results in a rate equation for the population of $|ψ\rangle$. Here we show that the coupling of a quantum system to the standard model of a thermal environment, a bath of harmonic oscillators, can be decomposed into a "cascade" made up of the quasi-continuum coupling structures of Fermi's golden rule. This clarifies the connection between the physics of the golden rule and that of a thermal bath, and provides a non-rigorous but physica…
This paper presents an application of white noise functional approach to derive the quantum propagator and the evolution of the reduced density matrix for an open quantum system consisting of coupled harmonic oscillators which are coupled to a bath consisting of a multimode harmonic oscillator. It is shown that the full quantum propagator is a product of three individual propagators. These propagators were obtained after two successive transformations of the coordinates for the system, and the coordinates for the coupling between the bath and one of the two harmonic oscillators in the system. …
We present an exact master equation for a central spin-$1/2$ system coupled to a spin-$1/2$ bath in terms of hyperfine interaction, which provides a unified formalism for both free evolution and controlled dynamics of the central spin. The equation enables us to study controllability of dynamics processes with various degrees of non-Markovianity. We investigate the Overhauser's effect on decoherence dynamics of the central spin under different bath spectra and the system-bath coupling strengths. Nonperturbative leakage elimination approach is applied to the system to suppress decoherence in so…
We construct a quantum Markovian Master equation for a driven system coupled to a thermal bath. The derivation utilizes an explicit solution of the propagator of the driven system. This enables the validity of the Master equation to be extended beyond the adiabatic limit. The Non-Adiabatic Master Equation (NAME) is derived employing the weak system-bath coupling limit. The NAME is valid when a separation of timescales exists between the bath dynamics and the external driving. In contrast to the adiabatic Master equation, the NAME leads to coupled equations of motion for the population and cohe…
Starting from the Liouville-von Neumann equation, under a weak coupling limit we derive the Lindblad master equation for the one-dimensional quantum Ising model in a Markov approximation and a rotating wave approximation. We also prove that the steady solution of the Lindblad equation is the canonical distribution independent of the dissipation rate.
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 …
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…
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 open quantum system in contact with an infinite bath approaches equilibrium, while the state of the bath remains unchanged. If the bath is finite, the open system still relaxes to equilibrium, but it induces a dynamical evolution of the bath state. In this work, we study the dynamics of open quantum systems in contact with finite baths. We obtain a hierarchy of master equations that improve their accuracy by including more dynamical information of the bath. For instance, as the least accurate but simplest description in the hierarchy we obtain the conventional Born-Markov-secular master equ…
A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipa…
This paper considers the extension of the non-Markovian stochastic approach for quantum open systems strongly coupled to a fermionic bath, to the models in which the system operators commute with the fermion bath. This technique can also be a useful tool for studying open quantum systems coupled to a spin-chain environment, which can be further transformed into an effective fermionic bath. We derive an exact stochastic Schrödinger equation (SSE), called fermionic quantum state diffusion (QSD) equation, from the first principle by using the fermionic coherent state representation. The reduced d…
An exact quantum master equation formalism is constructed for the efficient evaluation of quantum non-Markovian dissipation beyond the weak system-bath interaction regime in the presence of time-dependent external field. A novel truncation scheme is further proposed and compared with other approaches to close the resulting hierarchically coupled equations of motion. The interplay between system-bath interaction strength, non-Markovian property, and required level of hierarchy is also demonstrated with the aid of simple spin-boson systems.
We consider a molecular aggregate consisting of $N$ identical monomers. Each monomer comprises two electronic levels and a single harmonic mode. The monomers interact with each other via dipole-dipole forces. The monomer vibrational modes are bilinearly coupled to a bath of harmonic oscillators. This is a prototypical model for the description of coherent exciton transport, from quantum dots to photosynthetic antennae. We derive an exact quantum master equation for such systems. Computationally, the master equation may be useful for the testing of various approximations employed in theories of…
In this paper, we apply the polaron master equation, which offers the possibilities to interpolate between weak and strong system-bath coupling, to study how system-bath couplings affect charge transfer processes in Photosystem II reaction center (PSII RC) inspired quantum heat engine (QHE) model in a wide parameter range. The effects of bath correlation and temperature, together with the combined effects of these factors are also discussed in details. The results show a variety of dynamical behaviours. We interpret these results in terms of noise-assisted transport effect and dynamical locali…
The effect of a thermal reservoir is investigated on a bipartite Gaussian state. We derive a pre-Lindblad master equation in the non-rotating wave approximation for the system. We then solve the master equation for a bipartite harmonic oscillator Hamiltonian with entangled initial state. We show that for strong damping the loss of entanglement is the same as for freely evolving particles. However, if the damping is small, the entanglement is shown to oscillate and eventually tend to a constant nonzero value.
The Hu-Paz-Zhang equation is a master equation for an oscillator coupled to a linear passive bath. It is exact within the assumption that the oscillator and bath are initially uncoupled . Here an exact general solution is obtained in the form of an expression for the Wigner function at time t in terms of the initial Wigner function. The result is applied to the motion of a Gaussian wave packet and to that of a pair of such wave packets. A serious divergence arising from the assumption of an initially uncoupled state is found to be due to the zero-point oscillations of the bath and not removed …
Canonically consistent quantum master equation for proton-transfer reactions
本文基于CCQME方法研究TAA分子内质子转移，对比HEOM和Redfield理论，揭示Redfield在强耦合下偏离HEOM，而CCQME保持一致性。
The non-Markovian master equation for open quantum systems is obtained by generalization of the ordinary Zwanzig-Nakajima (ZN) projection technique. To this end, a coupled chain of equations for the reduced density matrices of the bath $\varrho_{B}(t)$ and the system $\varrho_{S}(t)$ are written down. A formal solution of the equation for $\varrho_{B}(t)$, having been inserted in the equation for the reduced density matrix of the system, in the 2-nd approximation in interaction yields a very specific extra term in the generalized master equation. This term, being nonlinear in $\varrho_{S}(t)$,…
We show that the influence of the shared phonon bath considered in H. Hossein-Nejad and G. D. Scholes, New J. Phys. 12, 065045 (2010) on the exciton transfer in a two-molecule system can be reproduced by that of an independent bath model.
The dynamics of two interacting spins coupled to separate bosonic baths is studied. An analytical solution in Born approximation for arbitrary spectral density functions of the bosonic environments is found. It is shown that in the non-Markovian cases concurrence "lives" longer or reaches greater values.
We solve the long-standing central spin problem for a general set of inhomogeneous bath couplings and a large class of initial bath states. We compute the time evolution of the coherence of a central spin coupled to a spin bath by resumming all orders of the time-convolutionless master equation, thus avoiding the need to assume weak coupling to the bath. The fully quantum, non-Markovian solution is obtained in the large-bath limit and is valid up to a timescale set by the largest coupling constant. Our result captures the full decoherence of an electron spin qubit coupled to a nuclear spin bat…
We discuss a model of a closed quantum evolution of two-qubits where the joint Hamiltonian is so chosen that one of the qubits acts as a bath and thermalize the other qubit which is acting as the system. The corresponding exact master equation for the system is derived. Interestingly, for a specific choice of parameters the master equation takes the Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) form with constant coefficients, representing pumping and damping of a single qubit system. Based on this model we construct an Otto cycle connected to a single qubit bath and study its thermodynamic pro…
The equation for the Wigner function describing the reduced dynamics of a single harmonic oscillator, coupled to an oscillator bath, was obtained by Karrlein and Grabert [Phys. Rev. E, vol. 55, 153 (1997)]. It was shown that for some special correlated initial conditions the equation reduces, in the classical limit, to the corresponding classical Fokker-Planck equation obtained by Adelman [J. Chem Phys., vol. 64, 124 (1976)]. However for separable initial conditions the Adelman equations were not recovered. We resolve this problem by showing that, for separable initial conditions, the classica…
In the framework of the Lindblad theory for open quantum systems, a master equation for the quantum harmonic oscillator interacting with a dissipative environment, in particular with a thermal bath, is derived for the case when the interaction is based on deformed algebra. The equations of motion for observables strongly depend on the deformation function. The expectation values of the number operator and squared number operator are calculated in the limit of a small deformation parameter for the case of zero temperature of the thermal bath. The steady state solution of the equation for the de…
We consider a system subject to a quantum optical master equation at finite temperature and study a class of conditional dynamics obtained by monitoring its totally or partially purified environment. More specifically, drawing from the notion that the thermal state of the environment may be regarded as the local state of a lossy and noisy two-mode squeezed state, we consider conditional dynamics ("unravellings") resulting from the homodyne detection of the two modes of such a state. Thus, we identify a class of unravellings parametrised by the loss rate suffered by the environmental two-mode s…
We derive an exact non-Markovian master equation that generalizes the previous work [Hu, Paz and Zhang, Phys. Rev. D {\bf 45}, 2843 (1992)] to damped harmonic oscillators with time-varying parameters. This is achieved by exploiting the linearity of the system and operator solution in Heisenberg picture. Our equation governs the non-Markovian quantum dynamics when the system is modulated by external devices. As an application, we apply our equation to parity kick decoupling problems. The time-dependent dissipative coefficients in the master equation are shown to be modified drastically when the…
A model of an arbitrary spin coupled to a bath of spins 1/2 in a star configuration is considered. The exact reduced dynamics of the central spin is found for the case of non-correlated initial conditions of the system and the bath. The exact solution is used to test two approximation techniques, namely, the Nakajima-Zwanzig projection operator technique and the time-convolutionless projection operator technique corresponding to the second order of the coupling constant. Two types of projection operators are used for deriving the master equations and the results are compared with the exact sol…
In a recent Letter, Guarnieri et al. [1] reported on the possibility of generating steady-state coherence (SSC) in the energy basis of a two-level system by the sole action of a single thermal bath also for weak system-bath coupling and due to a composite form of this interaction. The results are based on a time-local Bloch-Redfield master equation without secular approximation and the associated set of Bloch equations, all derived in the Supplemental Material (SM) of Ref. [1]. We comment that the results reported by the authors present both technical flaws and conceptual limitations that subs…
By exact numerical and master equation approaches, we show that a central spin-1/2 can be configured to probe internal bath dynamics. System-bath interactions cause Rabi oscillations in the detector and periodic behavior of fidelity. This period is highly sensitive to the strength of the bath self-interactions, and can be used to calculate the intra-bath coupling.
We analyze a system coupled to a bath of independent harmonic oscillators. We transform the bath in chain structure by solving an inverse eigenvalue problem. We solve the equations of motion for the collective variables defined by this transformation, and we derive the exact dynamics for an harmonic oscillator in terms of the microscopic motion of the environmental modes. We compare this approach to the well-known Generalized Langevin Equation and we show that our dynamics satisfies this equation.
Dynamics of quantum entanglement shared between system spins which are connected to thermal equilibrium baths is studied. Central spin system comprises of the entangled spins, and is connected to baths and one of the bath has strong intra-environmental coupling. The dynamics between the system and baths are studied and inferred that that intra-envirnomental coupling guards against the system from the effects of the spin baths.
We consider a macroscopic quantum system such as a qubit, interacting with a bath of fermions as in the Fröhlich polaron model. The interaction Hamiltonian is thus linear in the macroscopic system variable, and bilinear in the fermions. Using the recently developed extension of Feynman-Vernon theory to non-harmonic baths we evaluate quadratic and the quartic terms in the influence action. We find that for this model the quartic term vanish by symmetry arguments. Although the influence of the bath on the system is of the same form as from bosonic harmonic oscillators up to effects to sixth orde…
Quantum master equation from the eigenstate thermalization hypothesis
该研究基于本征态热化假设推导了弱耦合混沌纯态浴系统的量子主方程，揭示马尔可夫性与局域细致平衡的产生机制，经数值验证并对比可积浴差异，拓展了开放量子系统理论的适用范围。
Adiabatically-manipulated systems interacting with spin baths beyond the Rotating Wave Approximation
本文通过超越旋转波近似的微扰方法，研究了三态系统与自旋浴弱耦合时受激拉曼绝热通道的粒子数转移效率，发现环境效应通常可忽略，从而验证了该过程的鲁棒性。
Beyond the Quantum Regression Theorem in Variational Polaron Master Equations: from Ohmic to Super-Ohmic Environments
本文提出扩展量子回归定理以处理动态相关性，应用于极子主方程，展示在强耦合下对多时间观测的准确计算。
Given a multilevel system coupled to a bath, we use a Feshbach P,Q partitioning technique to derive an exact trace-nonpreserving master equation for a subspace $\mathcal{S}_{i}$ of the system. The resultant equation properly treats the leakage effect from $\mathcal{S}_{i}$ into the remainder of the system space. Focusing on a second-order approximation, we show that a one-dimensional master equation is sufficient to study problems of quantum state storage and is a good approximation, or exact, for several analytical models. It allows a natural definition of a leakage function and its control, …
Memory loss is contagious in open quantum systems
本文证明开放量子系统中马尔可夫性可通过有损耗系统从一个浴“传染”到另一个浴，并提出一种结合非厄米哈密顿量与主方程的 Bloch-Redfield 启发方法，以更低计算成本描述有损耗系统与非马尔可夫浴的相互作用。
We derive the exact time-evolution for a general quantum system under the influence of pure phase-noise and demonstrate that for a Gaussian initial state of the bath, the exact result can be obtained also within a perturbative time-local master equation approach already in second order of the system-bath coupling strength. We reveal that this equivalence holds if the initial state of the bath can be mapped to a Gaussian phase-space distribution function. Moreover, we discuss the relation to the standard Bloch-Redfield approach.
Understanding central spin decoherence due to interacting dissipative spin baths
本文提出基于簇关联展开的新方法模拟中心自旋在相互作用耗散自旋浴中的退相干，并通过一维、二维自旋浴及金刚石近表面NV色心验证了方法的准确性与耗散的关键作用。
In this paper we study the decoherence and purity of a driven solid-state qubit in the Ohmic bath by using the method based on the master equation. At first, instead of solving the master equation we investigate the coefficients of the equation which describe the shift in frequency, diffusive, decoherence, and so on. It is shown that one of the coefficients (we called it decoherence coefficient) is crucial to the decoherence of the qubit in the model. Then we investigate the evolution of the purity of the state in the model. From the analysis of the purity we see that the decoherence time of t…
The strong system-bath correlation is a typical initial condition in many condensed matter and some quantum optical systems. Here, the dynamics of a spin interacting with a spin bath through an intermediate spin are studied. Initial correlations between the spin and the intermediate spin are taken into account. The exact analytical expression for the evolution operator of the spin is found. Furthermore, correlated projection superoperator techniques are applied to the model and a time-convolutionless master equation to second order is derived. It is shown that the time-convolutionless master e…
The low temperature solution of the exact master equation for an oscillator coupled to a linear passive heat bath is known to give rise to serious divergences. We now show that, even in the high temperature regime, problems also exist, notably the fact that the density matrix is not necessarily positive.
In this paper we derive the Lindblad and Redfield forms with and without secular approximation from the Born-Markov master equation for open quantum systems. The spectral correlation tensor of bath (the Fourier transform of the bath correlation function) and then the coefficients in the two forms of the master equation are reevaluated according to the scheme in Ref.[Phys. Rev. A 99, 022118 (2019)]. They are complex numbers rather than the real numbers getting from traditional simplified methods. The dynamics of two models [one is an open three-level quantum system model, and the other is the m…
The recently developed technique combining the weak coupling limit with the Floquet formalism is applied to a model of two-level atom driven by a strong laser field and weakly coupled to heat baths. Firstly, the case of a single electromagnetic bath at zero temperature is discussed and the formula for resonance fluorescence is derived. The expression describes the well-known Mollow triplet, but its details differ from the standard ones based on additional simplifying assumptions. The second example describes the case of two thermal reservoirs: an electromagnetic one at finite temperature and t…
The entanglement dynamics in a bipartite system consisting of a qubit and a harmonic oscillator interacting only through their coupling with the same bath is studied. The considered model assumes that the qubit is coupled to the bath via the Jaynes-Cummings interaction, whilst the position of the oscillator is coupled to the position of the bath via a dipole interaction. We give a microscopic derivation of the Gorini-Kossakowski-Sudarshan-Lindblad equation for the considered model. Based on the Kossakowski Matrix, we show that non-classical correlations including entanglement can be generated …
We study the Markovian dynamics of a collection of n quantum systems coupled to an irreversible environmental channel consisting of a stream of n entangled qubits. Within the framework of repeated quantum interactions, we derive the master equation that describes the dynamics of the composite quantum system. We investigate the evolution of the joint system for two-qubit environments and find that (1) the presence of antidiagonal coherences (in the local basis) in the environment is a necessary condition for entangling two remote systems, and (2) that maximally entangled two-qubit baths are an …
We generalize the method of third quantization to a unified exact treatment of Redfield and Lindblad master equations for open quadratic systems of n fermions in terms of diagonalization of 4n x 4n matrix. Non-equilibrium thermal driving in terms of the Redfield equation is analyzed in detail. We explain how to compute all physically relevant quantities, such as non-equilibrium expectation values of local observables, various entropies or information measures, or time evolution and properties of relaxation. We also discuss how to exactly treat explicitly time dependent problems. The general fo…
Invariant-based master equation applied to driven qutrit coupled to a bath and a leaky cavity
研究驱动开放N能级量子系统主方程，用基于Lewis-Riesenfeld不变量的广义方法，避免传统方法驱动强度限制，能统一多种主方程框架，还应用于量子点，为有耗散时的量子控制应用铺路。
We derive a Born-Markov master equation describing the dissipation induced by a bath of lossy but coherently driven two-level systems (TLS) coupled to a bosonic system via Jaynes-Cummings interaction. We analytically derive all the master equation rates. We characterize these rates for the particular case of a single-mode system coupled to identical TLS. We study the steady state of the system and its exotic properties stemming from the non-thermal stationary state of the driven TLS bath. These properties include dissipative amplification, bath-induced linear instability, and both coherent and…
We provide an explicit method for solving general markovian master equations for quadratic bosonic Hamiltonians with linear bath operators. As an example we consider a one-dimensional quantum harmonic oscillator chain coupled to thermal reservoirs at both ends of the chain. We derive an analytic solution of the Redfield master equation for homogeneous harmonic chain and recover classical results, namely, vanishing temperature gradient and constant heat current in the thermodynamic limit. In the case of the disordered gapped chains we observe universal heat current scaling independent of the ba…
Adiabatic manipulation of a system interacting with a spin-bath
本文分析了受激拉曼绝热通道在与自旋浴相互作用系统中的应用，利用旋转波近似简化处理并评估总幺正演化，揭示了不同耦合与失谐机制下布居数转移效率的变化及强阻尼下效率降低的Zeno子空间成因。
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