Incoherent Control of Open Quantum Systems

This work reviews various topics in the control of open quantum systems interacting with the environment. The topics include the formulation of coherent and incoherent quantum control, analysis of control landscapes and their critical points for typical objective functionals, controllability properties, and the relation to the optimization over complex Stiefel manifolds.     

Repeated and Continuous Interactions in Open Quantum Systems

We consider a finite quantum system S{\mathcal {S}} coupled to two environments of different nature. One is a heat reservoir R{\mathcal {R}} (continuous interaction) and the other one is a chain C{\mathcal {C}} of independent quantum systems E{\mathcal {E}} (repeated interaction). The interactions of S{\mathcal {S}} with R{\mathcal {R}} and C{\mathcal {C}} lead to two simultaneous dynamical processes. We show that for generic such systems, any initial state approaches an asymptotic state in the limit of large times. We express the latter in terms of the resonance data of a reduced propagator of S+R{\mathcal {S}+\mathcal {R}} and show that it satisfies a second law of thermodynamics. We analyze a model where both S{\mathcal {S}} and E{\mathcal {E}} are two-level systems and obtain the asymptotic state explicitly (at lowest order in the interaction strength). Even though R{\mathcal {R}} and C{\mathcal {C}} are not directly coupled, we show that they exchange energy, and we find the dependence of this exchange in terms of the thermodynamic parameters. We formulate the problem in the framework of W ^*-dynamical systems and base the analysis on a combination of spectral deformation methods and repeated interaction model techniques. We analyze the full system via rigorous perturbation theory in the coupling strength, and do not resort to any scaling limit, like e.g. weak coupling limits, or any other approximations in order to derive some master equation.     

Inchworm Monte Carlo Method for Open Quantum Systems

We investigate in this work a recently proposed diagrammatic quantum Monte Carlo method—the inchworm Monte Carlo method—for open quantum systems. We establish its validity rigorously based on resummation of Dyson series. Moreover, we introduce an integro-differential equation formulation for open quantum systems, which illuminates the mathematical structure of the inchworm algorithm. This new formulation leads to an improvement of the inchworm algorithm by introducing classical deterministic time-integration schemes. The numerical method is validated by applications to the spin-boson model. © 2020 Wiley Periodicals, Inc.     

Algebraic Aspects of the Dynamics of Quantum Multilevel Systems in the Projection Operator Technique

Using the projection operator method, we obtain approximate time-local and time-nonlocal master equations for the reduced statistical operator of a multilevel quantum system with a finite number N of quantum eigenstates coupled simultaneously to arbitrary classical fields and a dissipative environment. We show that the structure of the obtained equations is significantly simplified if the free Hamiltonian dynamics of the multilevel system under the action of external fields and also its Markovian and non-Markovian evolutions due to coupling to the environment are described via the representation of the multilevel system in terms of the SU(N) algebra, which allows realizing effective numerical algorithms for solving the obtained equations when studying real problems in various fields of theoretical and applied physics.     

Dynamics of Thermally-Insulated Nonequilibrium Stefan Problem

We study a two-phase Stefan problem with kinetics. Here we prove existence of a finite-dimensional attractor for the problem without heat losses. Fot the most part we use a more elegant technique of energetic type estimates in appropriately defined weighted Sobolev spaces as opposite to the parabolic potentials of [9]. We demonstrate existence of compact attractors in the Sobolev spaces and prove that the attractor consists of sufficiently regular functions. This allows us to show that the Hausdorff dimension of the attractor is finite.     

Decoherence and Coherence Preservation in the Solutions of the GKSL Equation in the Theory of Open Quantum Systems

Mathematical Notes - The properties of solutions of the Gorini–Kossakowski-Sudarshan–Lindblad (GKSL) equation for the density operator (matrix) of a system that has nondegenerate energy...     

On Hyperbolic Pencils for the Grad Moment Systems in Nonequilibrium Thermodynamics

In this paper, we derive sufficient conditions for the stability of a solution to the Cauchy problem of systems of moments of nonequilibrium thermodynamics linearized in the neighborhood of the equilibrium state that are very close to the necessary conditions. The stability conditions are presented in the form of a parametric Hermite theorem for polynomial hyperbolic pencils of any order.__________Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 24, pp. 67–94, 2004.     

On the Global Stability of Solutions of Moment Systems in Nonequilibrium Thermodynamics

In this paper, we study the linearization of the Cauchy problem and the mixed problem for the system of Grad--Hermite moments in nonequilibrium thermodynamics in the neighborhood of the equilibrium state. Stability conditions for solutions of the Cauchy problem are proved as a generalization of the classical Hermite--Biller theorem on stable polynomials. For the mixed problem, we prove an analog of the Vishik--Lyusternik theorem on small singular perturbations of general elliptic problems. The last observation allows us to introduce the Shapiro--Lopatinskii condition, which implies the well-posedness of the mixed problem.     

中国股票市场的非对称动态相关性研究

基于沪深A、B股1994.1.3到2012.7.31的四组日回报率数据,本文研究了中国股票市场间条件波动率和相关系数的动态性与非对称性,分别建立了EGARCH模型和非对称形式的动态条件相关模型(DCC)进行分析。研究表明,我国股票市场的条件波动率在利空消息冲击时普遍表现出很强的非对称性,其中深B股市场的波动率非对称性表现为市场受利好信息冲击时的反应更强而不同于其他三个股票市场;尽管股票市场间条件相关系数存在着不同的非对称表现形式,但是无论是A股市场还是B股市场,其条件相关系数都表现出显著的动态非对称性。     

On Linear Dynamics in Quantum Calculus

In this paper, we are concerned with establishing some new elements of q-harmonic analysis. Namely, we prove a generalized q-Paley-Wiener theorem. As a consequence, we obtain some results of linear dynamics related to the symmetric q-derivative.     

Quantum Transport in Degenerate Systems

Transport in nonequilibrium degenerate quantum systems is investigated. The transfer rate depends on the parameters of the system. In this paper we investigate the dependence of the flow (transfer rate) on the angle between “bright” vectors (which define the interaction of the system with the environment). We show that in some approximation for the system under investigation the flow is proportional to the cosine squared of the angle between the “bright” vectors. Earlier the author has shown that in this degenerate quantum system excitation of nondecaying quantum “dark” states is possible; moreover, the effectiveness of this process is proportional to the sine squared of the angle between the “bright” vectors (this phenomenon was discussed as a possible model of excitation of quantum coherence in quantum photosynthesis). Thus quantum transport and excitation of dark states are competing processes; “dark” states can be considered as a result of leakage of quantum states in a quantum thermodynamic machine which performs the quantum transport.     

Conditional Entropy and Information in Quantum Systems

The concepts of conditional entropy of a physical system given the state of another system and of information in a physical system about another one are generalized for quantum systems. The fundamental difference between the classical case and the quantum one is that the entropy and information in quantum systems depend on the choice of measurements performed over the systems. It is shown that some equalities of the classical information theory turn into inequalities for the generalized quantities. Specific quantum phenomena such as EPR pairs and superdense coding are described and explained in terms of the generalized conditional entropy and information.     

Systems Dynamics Applications to Industrial and Other Systems

This paper describes the results of ideas derived from the work of the System Dynamics Research Group at the University of Bradford. The basic underlying ideas of System Dynamics are outlined and examples of their application to a number of practical problems given. The major results obtained and the lines of research suggested by these studies are discussed.     

Polynomial Conservation Laws in Quantum Systems

We consider systems with a finite number of degrees of freedom and potential energy that is a finite sum of exponentials with purely imaginary or real exponents. Such systems include the generalized Toda chains and systems with a toric configuration space. We consider the problem of describing all the quantum conservation laws, i.e., the differential operators that are polynomial in the derivatives and commute with the Hamiltonian operator. We prove that in the case where the potential energy spectrum is invariant under reflection with respect to the origin, such nontrivial operators exist only if the system under consideration decomposes into a direct sum of decoupled subsystems. In the general case (without the spectrum symmetry assumption), we prove that the existence of a complete set of independent conservation laws implies the complete integrability of the corresponding classical system.     

Markov Approximations of the Evolution of Quantum Systems

Doklady Mathematics - The convergence in probability of a sequence of iterations of independent random quantum dynamical semigroups to a Markov process describing the evolution of an open quantum...     

A Nonequilibrium Statistical Model of Spectrally Truncated Burgers‐Hopf Dynamics

Exact spectral truncations of the unforced, inviscid Burgers‐Hopf equation are Hamiltonian systems with many degrees of freedom that exhibit intrinsic stochasticity and coherent scaling behavior. For this reason recent studies have employed these systems as prototypes to test stochastic mode reduction strategies. In the present paper the Burgers‐Hopf dynamics truncated to n Fourier modes is treated by a new statistical model reduction technique, and a closed system of evolution equations for the mean values of the m lowest modes is derived for m ? n. In the reduced model the m‐mode macrostates are associated with trial probability densities on the phase space of the n‐mode microstates, and a cost functional is introduced to quantify the lack of fit of paths of these densities to the Liouville equation. The best‐fit macrodynamics is obtained by minimizing the cost functional over paths, and the equations governing the closure are then derived from Hamilton‐Jacobi theory. The resulting reduced equations have a fractional diffusion and modified nonlinear interactions, and the explicit form of both are determined up to a single closure parameter. The accuracy and range of validity of this nonequilibrium closure is assessed by comparison against direct numerical simulations of statistical ensembles, and the predicted behavior is found to be well represented by the reduced equations. © 2014 Wiley Periodicals, Inc.