Several modifications of the Schr\"odinger equation that dynamically reduce the off-diagonal elements of the density operator in the position representation (localization) have received recent attention in the literature. Models of quantum open systems provide localization for the system of interest by considering the effects of an environment on the effective master equation that describes the dynamics of the reduced density operator for the system of interest. In contrast, localization is introduced as a fundamental modification to the Schr\"odinger equation in quantum mechanics with spontaneous localization (QMSL) [G. C. Ghirardi, A. Rhimini, and T. Weber, Phys. Rev. D 34, 470 (1986)]. We compare several environment models and discuss some of the anomalous features of the effective master equations noted by Ballentine [L. E. Ballentine, Phys. Rev. A 43, 9 (1991)] by examining the Ehrenfest relations for position, energy, and momentum. Insight is provided into the relative importance of various terms in the effective master equations for several environment models. We also briefly discuss the implications of similar anomalous features present in the QMSL master equation in light of our analysis of the open-system models.
Several modifications of the Schr"odinger equation that dynamically reduce the off-diagonal elements of the density operator in the position representation (localization) have received recent attention in the literature. Models of quantum open systems provide localization for the system of interest by considering the effects of an environment on the effective master equation that describes the dynamics of the reduced density operator for the system of interest. In contrast, localization is introduced as a fundamental modification to the Schr"odinger equation in quantum mechanics with spontaneous localization (QMSL) [G. C. Ghirardi, A. Rhimini, and T. Weber, Phys. Rev. D 34, 470 (1986)]. We compare several environment models and discuss some of the anomalous features of the effective master equations noted by Ballentine [L. E. Ballentine, Phys. Rev. A 43, 9 (1991)] by examining the Ehrenfest relations for position, energy, and momentum. Insight is provided into the relative importance of various terms in the effective master equations for several environment models. We also briefly discuss the implications of similar anomalous features present in the QMSL master equation in light of our analysis of the open-system models.
The dynamical elimination or reduction of macroscopic superpositions has long been of interest, particularly with regard to the quantum theory of measurement. A number of models have demonstrated this for the reduced density matrix of a system interacting with an environment. Alternatively, Ghirardi, Rhimini, and Weber [Phys. Rev. D 34, 470 (1986)] have proposed a fundamental modification of the Schr\"odinger equation, quantum mechanics with spontaneous localization (QMSL), which provides a master equation similar in mathematical form. In this paper we consider an isotropic environment that is elastically scattered by the system of interest with negligible momentum transfer, extending a previous result of Joos and Zeh [Z. Phys. B. 59, 223 (1985)] from small length scales to all length scales. We discuss the physical nature and relevance of the differences between our result and similar open systems calculations. We describe the mathematical similarity between our extended environmental model and the QMSL dynamics determining the QMSL parameters that allow calculations using QMSL to be used as a model for the effect of the environment. That gives us access to a number of interesting results obtained for the QMSL master equation. Finally, we discuss some experimental considerations for the purposes of detecting effective nonunitary evolution of this form.
Within the framework of the quantum-statistical approach, utilizing both non-Hermitian Hamiltonian and Lindblad’s jump operators, one can derive various generalizations of the von Neumann equation for reduced density operators, also known as hybrid master equations. If one considers the evolution of pure states only, i.e., disregarding the coherence between states and spontaneous transitions from pure to mixed states, then one can resort to quantum-mechanical equations of the Schrödinger type. We derive them from the hybrid master equations and study their main properties, which indicate that our equations have a larger range of applicability compared to other generalized Schrödinger equations proposed hitherto. Among other features, they can describe not only systems which remain in the stationary eigenstates of the Hamiltonian as time passes, but also those which evolve from those eigenstates. As an example, we consider a simple but important model, a quantum harmonic oscillator driven by both Hamiltonian and non-Hamiltonian terms, and derive its classical limit, which turns out to be the damped harmonic oscillator. Using this model, we demonstrate that the effects of dissipative environments of different types can cancel each other, thus resulting in an effectively dissipation-free classical system. Another discussed phenomenon is whether a non-trivial quantum system can reduce to a classical system in free motion, i.e., without experiencing any classical Newtonian forces. This uncovers a large class of quantum-mechanical non-Hamiltonian systems whose dynamics are not determined by conventional mechanics’ potentials and forces, but rather come about through quantum statistical effects caused by the system’s environment.
Several quantum open-system models have been examined recently that dynamically reduce the off-diagonal terms of the density operator in the position representation by incorporating the effects of interaction with external degrees of freedom. A recent model [M. R. Gallis and G. N. Fleming, Phys. Rev. A 42, 38 (1990)] includes explicit dependence on the spatial correlations of the decoherence rate of the off-diagonal elements (in the position representation) of the density operator. In this paper, we use influence-functional techniques to examine the effect of an external potential that randomly varies as function of position and time, which leads to a natural characterization of the decay rate of the coherence between spatially separated states in terms of the spatial correlations of the potential, and the corresponding fluctuating forces. We obtain the effective master equations for short environment relaxation time scales, and discuss the similarities and differences between the effects of random external potentials versus the effects of interactions with external degrees of freedom, such as that recently investigated by Stern, Aharonov, and Imry [Phys. Rev. A 41, 3436 (1990)].
The majority of quantum open system models in the literature are simplistic in the sense that they only explicitly account for that part of the environment that directly interacts with the system of interest. A quantum open system with an open environment is examined using the projection operator method in the weak coupling limit. The openness of environment is modelled by nonunitary evolution of the Lindblad form. Under certain conditions, the resulting master equation for the system is insensitive to the initial state of the environment and to initial entanglements between the system and environment for time scales greater than the environment relaxation timescales. For the particular case of an environment consisting of a harmonic oscillator bath, the resulting master equations are demonstrated to have the algebraic form for completely positive evolution. The open environment model is illustrated for the particular case of a system linearly coupled to an oscillator bath.
Positivity preservation is naturally guaranteed in exact non-Markovian master equations for open quantum system dynamics. However, in many approximated non-Markovian master equations, the positivity of the reduced density matrix is not guaranteed. In this paper we provide a general class of time-local, perturbative, and positivity-preserving non-Markovian master equations generated from stochastic Schr\"odinger equations, particularly quantum-state-diffusion equations. Our method has an expanded range of applicability for accommodating a variety of non-Markovian environments. We show the positivity-preserving master equation for a three-level system coupled to a dissipative bosonic environment as a particular example to exemplify our general approach. We illustrate the numerical simulations with an analysis explaining why the previous approximated non-Markovian master equations cannot guarantee positivity. Our work provides a consistent master equation for studying the non-Markovian dynamics in ultrafast quantum processes and strong-coupling systems.
Recent advances in quantum technologies and related experiments have created a need for highly accurate, versatile, and computationally efficient simulation techniques for the dynamics of open quantum systems. Long-lived correlation effects (non-Markovianity), system-environment hybridization, and the necessity for accuracy beyond the Born-Markov approximation form particular challenges. Approaches to meet these challenges have been introduced, originating from different fields, such as hierarchical equations of motion, Lindblad-pseudomode formulas, chain-mapping approaches, quantum Brownian motion master equations, stochastic unravelings, and refined quantum master equations. This diversity, while indicative of the field's relevance, has inadvertently led to a fragmentation that hinders cohesive advances and their effective cross-community application to current problems for complex systems. How are different approaches related to each other? What are their strengths and limitations? Here we give a systematic overview and concise discussion addressing these questions. We make use of a unified framework which very conveniently allows to link different schemes and, this way, may also catalyze further progress. In line with the state of the art, this framework is formulated not in a fully reduced space of the system but in an extended state space which in a minimal fashion includes effective reservoir modes. This in turn offers a comprehensive understanding of existing methods, elucidating their physical interpretations, interconnections, and applicability.