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arXiv:2308.07996

Quantum systems in Markovian environments

Henryk Gzyl

Published Aug 15, 2023• 0 citations
arXivHTMLPDF

Abstract

In this work, we develop a mathematical framework to model a quantum system whose Hamiltonian may depend on the state of changing environment, that evolves according to a Markovian process. When the environment changes its state, the quantum system may suffer a shock that produces an instantaneous transition among its states. The model that we propose can be readily adapted to more general settings.\\ To avoid collateral analytical issues, we consider the case of quantum systems with finite dimensional state space, in which case the observables are described by Hermitian matrices. We show how to average over the environment to predict the expected values of observables.

DOI
10.48550/arxiv.2308.07996

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In this work, we develop a mathematical framework to model a quantum system whose Hamiltonian may depend on the state of changing environment, that evolves according to a Markovian process. When the environment changes its state, the quantum system may suffer a shock that produces an instantaneous transition among its states. The model that we propose can be readily adapted to more general settings.\ To avoid collateral analytical issues, we consider the case of quantum systems with finite dimensional state space, in which case the observables are described by Hermitian matrices. We show how to average over the environment to predict the expected values of observables.
A weak value of an observable is studied for a quantum system which is placed under the influence of an environment, where a quantum system irreversibly evolves from a pre-selected state to a post-selected state. A general expression for a weak value influenced by an environment is provided. For a Markovian environment, the weak value is calculated in terms of the predictive and retrodictive density matrices, or by means of the quantum regression theorem. For a non-Markovian environment, a weak value is examined by making use of exactly solvable models. It is found that although the anomalous property is significantly suppressed by a Markovian environment, it can survive a non-Markovian environment.
In order to model realistic quantum devices it is necessary to simulate quantum systems strongly coupled to their environment. To date, most understanding of open quantum systems is restricted either to weak system-bath couplings or to special cases where specific numerical techniques become effective. Here we present a general and yet exact numerical approach that efficiently describes the time evolution of a quantum system coupled to a non-Markovian harmonic environment. Our method relies on expressing the system state and its propagator as a matrix product state and operator, respectively, and using a singular value decomposition to compress the description of the state as time evolves. We demonstrate the power and flexibility of our approach by numerically identifying the localisation transition of the Ohmic spin-boson model, and considering a model with widely separated environmental timescales arising for a pair of spins embedded in a common environment.
A novel framework for the analysis of observation statistics on time discrete linear evolutions in Banach space is presented. The model differs from traditional models for stochastic processes and, in particular, clearly distinguishes between the deterministic evolution of a system and the stochastic nature of observations on the evolving system. General Markov chains are defined in this context and it is shown how typical traditional models of classical or quantum random walks and Markov processes fit into the framework and how a theory of quantum statistics ({\it sensu} Barndorff-Nielsen, Gill and Jupp) may be developed from it. The framework permits a general theory of joint observability of two or more observation variables which may be viewed as an extension of the Heisenberg uncertainty principle and, in particular, offers a novel mathematical perspective on the violation of Bell's inequalities in quantum models. Main results include a general sampling theorem relative to Riesz evolution operators in the spirit of von Neumann's mean ergodic theorem for normal operators in Hilbert space.
We know that the quantum system evolves to different physical phenomena according to initial states. However, the situation is very different in the open system. We consider the system and the environment governed by the Markovian process. The density matrix (the reduced density matrix) of the system satisfies the Kossakowski-Lindblad equation. We calculate the expectation value of the z-direction-magnetization in the system for various types of 1) initial states and 2) interactions between the system and the environment. In all the cases, the values of the magnetization approached asymptotically to the same point even though the different initial states and different interaction types between the system and the environment are applied. The paths to asymptotically approach the final values are different according to the initial states and the interactions. These facts show that the physical results of expectation value are independent of the initial states.
Quantum mechanics must be regarded as open systems. On one hand, this is due to the fact that, like in classical physics, any realistic system is subjected to a coupling to an uncontrollable environment which influences it in a non-negligible way. The theory of open quantum systems thus plays a major role in many applications of quantum physics since perfect isolation of quantum system is not possible and since a complete microscopic description or control of the environment degrees of freedom is not feasible or only partially so [1]. Practical considerations therefore force one to seek for a simpler, effectively probabilistic description in terms of an open system. There is a close physical and mathematical connection between the evolution of an open system, the state changes induced by quantum measurements, and the classical notion of a stochastic process. The paper provides a bibliographic review of this interrelations, it shows the mathematical equivalence between markovian master equation and generalized piecewise deterministic processes [1] and it introduces the open system in an open observed environment model.
In this article we define and build the one method of description of stochastic evolution of a physical quantum system.For each quantum state U E ω ∈ we construct the probability measure ω µ in the space ( ) , U P S , where U P is the space of the pure states of the quantum system, S the Borel σ -algebra in U P .Farther, for any Hermit's positive element with norm 1 u = , in the C * -algebra of observables U , we define the probability measure u µ on the set of states U E .If strongly conti- nuous group { } t α of * automorphisms on U describes the evolution of struc- ture of observables, according to this, we have a picture of evolution of distribution of states of quantum system relatively to each observable u .