Structures of quantum Fokker-Planck equations are characterized with respect to the properties of complete positivity, covariance under symmetry transformations and satisfaction of equipartition, referring to recent mathematical work on structures of unbounded generators of covariant quantum dynamical semigroups. In particular the quantum optical master-equation and the quantum Brownian motion master-equation are shown to be associated to $\mathrm{U(1)}$ and $\mathrm{R}$ symmetry respectively. Considering the motion of a Brownian particle, where the expression of the quantum Fokker-Planck equation is not completely fixed by the aforementioned requirements, a recently introduced microphysical kinetic model is briefly recalled, where a quantum generalization of the linear Boltzmann equation in the small energy and momentum transfer limit straightforwardly leads to quantum Brownian motion.
Hu, Paz and Zhang [ B.L. Hu, J.P. Paz and Y. Zhang, Phys. Rev. D {\bf 45} (1992) 2843] have derived an exact master equation for quantum Brownian motion in a general environment via path integral techniques. Their master equation provides a very useful tool to study the decoherence of a quantum system due to the interaction with its environment. In this paper, we give an alternative and elementary derivation of the Hu-Paz-Zhang master equation, which involves tracing the evolution equation for the Wigner function. We also discuss the master equation in some special cases.
The "standard" Brownian motion master equation, used to describe thermal damping, is not completely positive, and does not admit a Monte Carlo method, important in numerical simulations. To eliminate both these problems one must add a term that generates additional position diffusion. He we show that one can obtain a completely positive simple quantum Brownian motion, efficiently solvable, without any extra diffusion. This is achieved by using a stochastic Schroedinger equation (SSE), closely analogous to Langevin's equation, that has no equivalent Markovian master equation. Considering a spec…
It is shown that the standard quantum Brownian equation (QBE) can violate positivity not only past the thermal correlation time, but at arbitrarily long times at high system frequencies. In an effort to improve the standard QBE, exact operator solutions are provided for a class of non-autonomous master equations. These exact solutions are used to derive sufficient positivity conditions for the coefficients of the master equations.
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…
A many particle Hamiltonian, where the interaction term conserves the number of particles, is considered. A master equation for the populations of the different levels is derived in an exact way. It results in a local equation with time-dependent coefficients, which can be identified with the transition probabilities in the golden rule approximation. A reinterpretation of the model as a set of coupled harmonic oscillators enables one to obtain for one of them an exact local Langevin equation, with time-dependent coefficients.
We revisit the model of a quantum Brownian oscillator linearly coupled to an environment of quantum oscillators at finite temperature. By introducing a compact and particularly well-suited formulation, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. The flexibility of our approach allows for an immediate generalization to cases with an external force and with an arbitrary number of Brownian oscillators. More importantly, we point out an important mathematical subtlety concerning boundary-value prob…
An original method to exactly solve the non-Markovian Master Equation describing the interaction of a single harmonic oscillator with a quantum environment in the weak coupling limit is reported. By using a superoperatorial approach we succeed in deriving the operatorial solution for the density matrix of the system. Our method is independent of the physical properties of the environment. We show the usefulness of our solution deriving explicit expressions for the dissipative time evolution of some observables of physical interest for the system, such as, for example, its mean energy.
Central to many discussion of decoherence is a master equation for the reduced density matrix of a massive particle experiencing scattering from its surrounding environment, such as that of Joos and Zeh. Such master equations enjoy a close relationship with spontaneous localization models, like the GRW model. This aim of this paper is to present two derivations of the master equation. The first derivation is a pedagogical model designed to illustrate the origins of the master equation as simply as possible, focusing on physical principles and without the complications of S-matrix theory. This …
Quantum brownian motion is a fundamental model for a proper understanding of open quantum systems in different contexts such as chemistry, condensed matter physics, bio-physics and opto- mechamics. In this paper we propose a novel approach to describe this model. We provide an exact and analytic equation for the time evolution of the operators, and we show that the corresponding equation for the states is equivalent to well-known results in literature. The dynamics is expressed in terms of the spectral density, regardless the strength of the coupling between the system and the bath. Our allows…
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…
In this paper, we generalize the quantum Brownian motion to include momentum-dependent system-environment couplings. The conventional QBM model corresponds to the spacial case $W_k = V_k$. The generalized QBM is more complicated but the generalization is necessary. This is because the particle transition and the pair production between the system and the environment represent two very different physical processes, and usually cannot have the same coupling strengths. Thus, the conventional QBM model, which is well-defined at classical level, is hardly realized in real quantum physical world. We…
We compare the thermodynamic entropy of a quantum Brownian oscillator derived from the partition function of the subsystem with the von Neumann entropy of its reduced density matrix. At low temperatures we find deviations between these two entropies which are due to the fact that the Brownian particle and its environment are entangled. We give an explanation for these findings and point out that these deviations become important in cases where statements about the information capacity of the subsystem are associated with thermodynamic properties, as it is the case for the Landauer principle.
Caldeira and Leggett (CL) in a seminal paper derived a master equation describing Markovian Quantum Brownian motion. Such an equation suffered of not being completely positive, and many efforts have been made to solve this issue. We show that, when a careful mathematical analysis is performed, the model considered by CL leads to a non dissipative master equation. We argue that the correct way to understand the master equation derived in the CL regime is to consider it non-Markovian. Moreover, we show that if one wants to provide a microscopic description of Quantum Brownian motion with the CL …
Quantum Brownian motion of a harmonic oscillator in the Markovian approximation is described by the respective Caldeira-Leggett master equation. This master equation can be brought into Lindblad form by adding a position diffusion term to it. The coefficient of this term is either customarily taken to be the lower bound dictated by the Dekker inequality or determined by more detailed derivations on the linearly damped quantum harmonic oscillator. In this paper, we explore the theoretical possibilities of determining the position diffusion term's coefficient by analyzing the entropy production …
We derive and solve analytically the non-Markovian master equation for harmonic quantum Brownian motion proving that, for weak system-reservoir couplings and high temperatures, it can be recast in the form of the master equation for a harmonic oscillator interacting with a squeezed thermal bath. This equivalence guarantees preservation of positivity of the density operator during the time evolution and allows one to establish a connection between the dynamics of Schrödinger cat states in squeezed environments and environment-induced decoherence in quantum Brownian motion.
We characterize the pointer states generated by the master equation of quantum Brownian motion and derive stochastic equations for the dynamics of their trajectories in phase space. Our method is based on a Poissonian unraveling of the master equation whose deterministic part exhibits soliton-like solutions that can be identified with the pointer states. In the semiclassical limit, their phase space trajectories turn into those of classical diffusion, yielding a clear picture of the induced quantum- classical transition.
The equation for the quantum motion of a Brownian particle in a gaseous environment is derived by means of S-matrix theory. This quantum version of the linear Boltzmann equation accounts non-perturbatively for the quantum effects of the scattering dynamics and describes decoherence and dissipation in a unified framework. As a completely positive master equation it incorporates both the known equation for an infinitely massive Brownian particle and the classical linear Boltzmann equation as limiting cases.
We revisit the model of a system made up of a Brownian quantum oscillator under the influence of an external classical force and linearly coupled to an environment made up of many quantum oscillators at zero or finite temperature. We show that the HPZ master equation for the reduced density matrix derived earlier [B.L. Hu, J.P. Paz, Y. Zhang, Phys. Rev. D 45, 2843 (1992)] with coefficients obtained from solutions of integro-differential equations can assume closed functional forms for a fairly general class of spectral densities of the environment at arbitrary temperature and coupling strength…
We discuss the description of full counting statistics in quantum transport with a nonMarkovian master equation. We focus on differences arising from whether charge is considered as a classical or a quantum degree of freedom. These differences manifest themselves in the inhomogeneous term of the master equation which describes initial correlations. We describe the influence on current and in particular, the finite-frequency shotnoise. We illustrate these ideas by studying transport through a quantum dot and give results that include both sequential and cotunneling processes. Importantly, the n…
We introduce a classification scheme for the generators of open fermionic Gaussian dynamics. We simultaneously partition the dynamics along the following four lines: (1) unitary versus non-unitary, (2) active versus passive, (3) state-dependent versus state-independent, and (4) single-mode versus multi-mode. We find that only nine of these 16 types of dynamics are possible. Using this partition we discuss the consequences of imposing complete positivity on fermionic Gaussian dynamics. In particular, we show that completely positive dynamics must be either unitary (and so can be implemented wit…
We consider the exact reduced dynamics of a two-level system coupled to a bosonic reservoir, further obtaining the exact time-convolutionless and Nakajima-Zwanzig non-Markovian equations of motion. The considered system includes the damped and undamped Jaynes-Cummings model. The result is obtained by exploiting an expression of quantum maps in terms of matrices and a simple relation between the time evolution map and time-convolutionless generator as well as Nakajima-Zwanzig memory kernel. This non-perturbative treatment shows that each operator contribution in Lindblad form appearing in the e…
We show how the quantum analog of the Fokker-Planck equation for describing Brownian motion can be obtained as the diffusive limit of the quantum linear Boltzmann equation. The latter describes the quantum dynamics of a tracer particle in a dilute, ideal gas by means of a translation-covariant master equation. We discuss the type of approximations required to obtain the generalized form of the Caldeira-Leggett master equation, along with their physical justification. Microscopic expressions for the diffusion and relaxation coefficients are obtained by analyzing the limiting form of the equatio…
A nonlinear master equation is derived, reflecting properly the entropy of open quantum systems. In contrast to linear alternatives, its equilibrium solution is exactly the canonical Gibbs density matrix. The corresponding nonlinear equation for the Wigner function accounts rigorously for the thermo-quantum entropy. It reduces at large friction to the Smoluchowski-Bohm equation in the coordinate subspace, which reflects the stochastic Langevin-Bohm dynamics. The previously derived Maxwell-Heisenberg relation for the nonequilibrium momentum dispersion of quantum Brownian particles is confirmed …
We analyze the microscopic model of quantum Brownian motion, describing a Brownian particle interacting with a bosonic bath through a coupling which is linear in the creation and annihilation operators of the bath, but may be a nonlinear function of the position of the particle. Physically, this corresponds to a configuration in which damping and diffusion are spatially inhomogeneous. We derive systematically the quantum master equation for the Brownian particle in the Born-Markov approximation and we discuss the appearance of novel terms, for various polynomials forms of the coupling. We disc…
Understanding the behaviour 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…
We study the dynamics of a Brownian quantum particle hopping on an infinite lattice with a spin degree of freedom. This particle is coupled to free boson gases via a translation-invariant Hamiltonian which is linear in the creation and annihilation operators of the bosons. We derive the time evolution of the reduced density matrix of the particle in the van Hove limit in which we also rescale the hopping rate. This corresponds to a situation in which both the system-bath interactions and the hopping between neighboring sites are small and they are effective on the same time scale. The reduced …
Adiabatic elimination and Wigner function approach in microscopic derivation of Open Quantum Brownian Motion
University of KwaZulu-Natal（夸祖鲁-纳塔尔大学）；North-West University（西北大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）；Stellenbosch University（斯泰伦博斯大学）
We consider a model of non-Markovian Quantum Brownian motion that consists of an harmonic oscillator bilinearly coupled to a thermal bath, both via its position and momentum operators. We derive the master equation for such a model and we solve the equations of motion for a generic Gaussian system state. We then investigate the resulting evolution of the first and second moments for both an Ohmic and a super-Ohmic spectral density. In particular, we show that, irrespective of the specific form of the spectral density, the coupling with the momentum enhances the dissipation experienced by the s…
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…
A quantum-mechanical version of Einstein's 1905 theory of Brownian motion is presented. Starting from the Hamiltonian dynamics of an isolated composite of objective and environmental systems, subdynamics for the objective system is derived in the spirit of Einstein. The resulting master equation is found to have the Lindblad structure.
We apply the restricted-path-integral (RPI) theory of non-minimally disturbing continuous measurements for correct description of frictional Brownian motion. The resulting master equation is automatically of the Lindblad form, so that the difficulties typical of other approaches do not exist. In the special case of harmonic oscillator the known familiar master equation describing its frictionally driven Brownian motion is obtained. A thermal reservoir as a measuring environment is considered.
We perform a quantum theoretical calculation of the noise power spectrum for a phase measurement of the light output from a coherently driven optical cavity with a freely moving rear mirror. We examine how the noise resulting from the quantum back action appears among the various contributions from other noise sources. We do not assume an ideal (homodyne) phase measurement, but rather consider phase modulation detection, which we show has a different shot noise level. We also take into account the effects of thermal damping of the mirror, losses within the cavity, and classical laser noise. We…
We stress the relevance of the two features of translational invariance and atomic nature of the gas in the quantum description of the motion of a massive test particle in a gas, corresponding to the original picture of Einstein used in the characterization of Brownian motion. The master equation describing the reduced dynamics of the test particle is of Lindblad form and complies with the requirement of covariance under translations.
We show how a large family of master equations, describing quantum Brownian motion of a harmonic oscillator with translationally invariant damping, can be derived within a phenomenological approach, based on the assumption that an environment can be simulated by two classical stochastic forces. This family is determined by three time-dependent correlation functions (besides the frequency and damping coefficients), and it includes as special cases the known master equations, whose dissipative part is bilinear with respect to the operators of coordinate and momentum.
We show a completely analytical approach to the decoherence induced by a zero temperature environment on a Brownian test particle. We consider an Omhic environment bilinearly coupled to an oscillator and compute the master equation. From diffusive coefficients, we evaluate the decoherence time for the usual quantum Brownian motion and also for an upside-down oscillator, as a toy model of a quantum phase transition.
We derive a quantum master equation from first principles to describe friction in one dimensional, collisional Brownian motion. We are the first to avoid an ill-defined square of the Dirac delta function by using localized wave packets rather than plane waves. Solving the Schrödinger equation for two colliding particles, we discover that the previously found position diffusion is not a physical process, but an artifact of the approximation of a coarse grained time scale, which in turn is needed to find Markkovian dynamics.
Stochastic exclusion processes play an integral role in the physics of non-equilibrium statistical mechanics. These models are Markovian processes, described by a classical master equation. In this paper a quantum mechanical version of a stochastic hopping process in one dimension is formulated in terms of a quantum master equation. This allows the investigation of coherent and stochastic evolution in the same formal framework. The focus lies on the non-equilibrium steady state. Two stochastic model systems are considered, the totally asymmetric exclusion process and the fully symmetric exclus…
A systematic first-order correction to the standard Markov master equation for open quantum systems interacting with a bosonic bath is presented. It extends the Markov Lindblad master equation to the more general case of non-Markovian evolution. The meaning and applications of our `post'-Markov master equation are illustrated with several examples, including a damped two-level atom, the spin-boson model and the quantum Brownian motion model. Limitations of the Markov approximation, the problem of positivity violation and initial slips are also discussed.
We generalize the classical theory of Brownian motion so as to reckon with non-Markovian effects on both Klein-Kramers and Smoluchowski equations. For a free particle and a harmonic oscillator, it is shown that such non-Markovian effects account for the differentiability of the Brownian trajectories as well as the breakdown of the energy equipartition of statistical mechanics at short times in some physical situations. This non-Markovian approach is also extended to look at anomalous diffusion. Next, we bring in the dynamical-quantization method for investigating open quantum systems, which do…
We analyze the evolution of a quantum Brownian particle starting from an initial state that contains correlations between this system and its environment. Using a path integral approach, we obtain a master equation for the reduced density matrix of the system finding relatively simple expressions for its time dependent coefficients. We examine the evolution of delocalized initial states (Schrödinger's cats) investigating the effectiveness of the decoherence process. Analytic results are obtained for an ohmic environment (Drude's model) at zero temperature.
A recently proposed master equation in the Lindblad form is studied with respect to covariance properties and existence of a stationary solution. The master equation describes the interaction of a test particle with a quantum fluid, the so-called Rayleigh gas, and is characterized by the appearance of a two-point correlation function known as dynamic structure factor, which reflects symmetry and statistical mechanics properties of the fluid. In the case of a free gas all relevant physical parameters, such as fugacity, ratio between the masses, momentum transfer and energy transfer are put into…
A master equation with a Lindblad structure is derived, which describes the interaction of a test particle with a macroscopic system and is expressed in terms of the operator valued dynamic structure factor of the system. In the case of a free Fermi or Bose gas the result is evaluated in the Brownian limit, thus obtaining a single generator master equation for the description of quantum Brownian motion in which the correction due to quantum statistics is explicitly calculated. The friction coefficients for Boltzmann and Bose or Fermi statistics are compared.
We solve the model of N quantum Brownian oscillators linearly coupled to an environment of quantum oscillators at finite temperature, with no extra assumptions about the structure of the system-environment coupling. Using a compact phase-space formalism, we give a rather quick and direct derivation of the master equation and its solutions for general spectral functions and arbitrary temperatures. Since our framework is intrinsically nonperturbative, we are able to analyze the entanglement dynamics of two oscillators coupled to a common scalar field in previously unexplored regimes, such as off…
Starting from a microscopic theory, we derive a master equation for a harmonic oscillator coupled to a bath of non-interacting oscillators. We follow a non-perturbative approach, proposed earlier by us for the free Brownian particle. The diffusion constants are calculated analytically and the positivity of the Master Equation is shown to hold above a critical temperature. We compare the long time behaviour of the average kinetic and potential energies with known thermodynamic results. In the limit of vainishing oscillator frequency of the system, we recover the results of the free Brownian par…
In this paper, we present a comprehensive account of quantum dissipation theories with the quadratic environment couplings. The theoretical development includes the Brownian solvation mode embedded hierarchical quantum master equations, a core-system hierarchy construction that verifies the extended dissipaton equation of motion (DEOM) formalism [R. X. Xu et al., J. Chem. Phys. 148, 114103 (2018)]. Developed are also the quadratic imaginary-time DEOM for equilibrium and the λ (t)-DEOM for nonequilibrium thermodynamics problems. Both the celebrated Jarzynski equality and Crooks relation are acc…
We present the stochastic Schroedinger equation for the dynamics of a quantum particle coupled to a high temperature environment and apply it the dynamics of a driven, damped, nonlinear quantum oscillator. Apart from an initial slip on the environmental memory time scale, in the mean, our result recovers the solution of the known non-Lindblad quantum Brownian motion master equation. A remarkable feature of our approach is its localization property: individual quantum trajectories remain localized wave packets for all times, even for the classically chaotic system considered here, the localizat…
Quantum resources, such as entanglement, steering, and Bell nonlocality, are evaluated for three coupled qubits in the steady-state configuration. We employ the phenomenological master equation and the microscopic master equation to probe such quantum resources, which provide very different results depending on the system configuration. In particular, steering and Bell nonlocality are null within the phenomenological model, while they reach considerable values within the microscopic model. These results show that the phenomenological approach is not able to capture all quantum resources of the…
In this paper, we study the dynamical properties of two coupled quantum harmonic oscillators coupled with bosonic non-Markovian environment both in position and momentum. We deduce the exact analytical master equation using Quantum State Diffusion method and give the quantum trajectory description when the control is added to the system by applying interaction between two harmonic oscillators. With numerical simulation, we compare the evolution of entanglement under different controlling effects. At last, we use nonlinear Quantum State Diffusion method to strengthen our above results by gettin…
Starting from the Caldeira-Leggett (CL) model, we derive the equation describing the Quantum Brownian motion, which has been originally proposed by Dekker purely from phenomenological basis containing extra anomalous diffusion terms. Explicit analytical expressions for the temperature dependence of the diffusion constants are derived. At high temperatures, additional momentum diffusion terms are suppressed and classical Langivin equation can be recovered and at the same time positivity of the density matrix(DM) is satisfied. At low temperatures, the diffusion constants have a finite positive v…
We study the fluctuation-induced dissipative dynamics of the quantized center of mass motion of a polarizable dielectric particle trapped near a surface. The particle's center of mass is treated as an open quantum system coupled to the electromagnetic field acting as its environment, with the resulting system dynamics described by a quantum Brownian motion master equation. The dissipation and decoherence of the particle's center of mass are characterized by the modified spectral density of the electromagnetic field that depends on surface losses and the strength of the classical trap field. Ou…
The theory of quantum Brownian motion describes the properties of a large class of open quantum systems. Nonetheless, its description in terms of a Born-Markov master equation, widely used in the literature, is known to violate the positivity of the density operator at very low temperatures. We study an extension of existing models, leading to an equation in the Lindblad form, which is free of this problem. We study the dynamics of the model, including the detailed properties of its stationary solution, for both constant and position-dependent coupling of the Brownian particle to the bath, foc…
A new method for stochastic unraveling of general time-local quantum master equations (QME) which involve the reduced density operator at time t only is proposed. The present kind of jump algorithm enables a numerically efficient treatment of QMEs which are not of Lindblad form. So it opens new large fields of application for stochastic methods. The unraveling can be achieved by allowing for trajectories with negative weight. We present results for the quantum Brownian motion and the Redfield QMEs as test examples. The algorithm can also unravel non-Markovian QMEs when they are in a time-local…
