Evolution of a self-consistent joint system (JS), i.e., a quantum system (QS) + thermal bath (TB), is considered within the framework of the Langevin–Schrödinger (L-Sch) type equation. As a tested QS, we considered two linearly coupled quantum oscillators that interact with TB. The influence of TB on QS is described by the white noise type autocorrelation function. Using the reference differential equation, the original L-Sch equation is reduced to an autonomous form on a random space–time continuum, which reflects the fact of the existence of a hidden symmetry of JS. It is proven that, as a result of JS relaxation, a two-dimensional quantized small environment is formed, which is an integral part of QS. The possibility of constructing quantum thermodynamics from the first principles of non-Hermitian quantum mechanics without using any additional axioms has been proven. A numerical algorithm has been developed for modeling various properties and parameters of the QS and its environment.
Evolution of a self-consistent joint system (JS), i.e., a quantum system (QS) + thermal bath (TB), is considered within the framework of the Langevin–Schrödinger (L-Sch) type equation. As a tested QS, we considered two linearly coupled quantum oscillators that interact with TB. The influence of TB on QS is described by the white noise type autocorrelation function. Using the reference differential equation, the original L-Sch equation is reduced to an autonomous form on a random space–time continuum, which reflects the fact of the existence of a hidden symmetry of JS. It is proven that, as a result of JS relaxation, a two-dimensional quantized small environment is formed, which is an integral part of QS. The possibility of constructing quantum thermodynamics from the first principles of non-Hermitian quantum mechanics without using any additional axioms has been proven. A numerical algorithm has been developed for modeling various properties and parameters of the QS and its environment.
The evolution of the joint system (JS) - ``quantum system (QS)+thermal bath (TB)" is considered in the framework of a complex probabilistic processes that satisfies the stochastic differential equation of the Langevin-Schrödinger type. Two linearly coupled oscillators that randomly interact with the environment and with each other are selected as QS. In the case when the interactions obey the law of a white random process, all the construction of the statistical parameters of the QS and its environment are performed analytically in the form of double integrals and solutions of second-order partial differential equations. Expressions of time-dependent von Neumann entropy and its generalization are obtained, taking into account the self-organization and entanglement processes occurring in the JS. It is mathematically proved that as a result of the relaxation of JS in the TB, a small quantized environment is formed, which can be interpreted as a continuation of QS or its halo. Bell states formed as a result of the decay of coupled two linear oscillators are constructed taking into account the influence of the environment. The transitions between (in) and (out) asymptotic states of QS are studied in detail taking into account the influence of TB. Within the framework of the model problem, the possibility of constructing quantum thermodynamics from the first principle is proved without using any additional conditions.
Spontaneous transitions between bound states of an atomic system, the “Lamb Shift” of energy levels and many other phenomena in real nonrelativistic quantum systems are connected with the influence of the quantum vacuum fluctuations (fundamental environment (FE)), which are impossible to consider in the framework of standard quantum-mechanical approaches. The joint system quantum system (QS) and FE is described in the framework of the stochastic differential equation (SDE) of Langevin-Schrödinger type and is defined on the extended space ℝ3⊗Ξ n , where ℝ3 and Ξ n are the Euclidean and functional spaces, respectively. The method of stochastic density matrix is developed and the von Neumann equation for reduced density matrix of QS with FE is generalized. The entropy of QS entangled with FE is defined and investigated. It is proved that the interaction of QS with the environment leads to emerging structures of various topologies which present new quantum-field properties of QS. It is shown that when the physical system (irrelatively to its being micro ormacro) breaks up into two fragments by means of FE, there arises between these fragments a nonpotential interaction which does not disappear at large distances.
The rather new field of quantum thermodynamics intends to derive thermodynamics from quantum mechanics, as well as to study thermodynamical behavior of quantum systems. It turns out that, indeed, thermalization can be obtained even for very small quantum systems due to the interaction with an appropriate environment. The consideration of an ensemble, as in statistical mechanics, is not necessary. However, the total system - consisting of system and environment - in quantum thermodynamics is usually considered to be closed. In particular, there exists no interaction with an external observer. Since quantum measurements typically influence the measured system, one may wonder which influence such an observation will have in the quantum thermodynamical setting. In this thesis, we study this influence of measurements in quantum thermodynamical systems. It turns out that various concepts known from statistical mechanics arise from such a "quantum thermodynamics under observation", although they do not have any meaning in the isolated quantum thermodynamical setting. The model discussed in the first part consists of a two-level system coupled to a large spin environment, the magnetization of which is measured periodically. This leads to quasi-classical trajectories for the system and - as in statistical mechanics - thermalization and temperature only have a meaning in terms of an ensemble average or a long-time average. Calculating these averages, we prove the ergodicity of the system. Interestingly, the entropy of the obtained attractor state depends on the time between the measurements and can be shown to be connected with the subjective lack of knowledge of the observer. In case of optimal measurements, the ensemble averaged state of the system under periodic measurements equals the state of the system in the quantum thermodynamical model without measurements. Introducing a detuning between system and environment, we find that the system can be prepared in almost arbitrary thermal states due to the periodic measurements. Even the creation of inversion becomes possible. As shown in numerical simulations, our results, which were analytically obtained for large environments, are even applicable to describe the behavior of the system coupled to small environments consisting of only a few spins. Besides measurements of the environment, we also study direct projective measurements of the system, as well as measurements with finite duration, obtaining that the dynamics of the system is almost independent of the concrete realization of the measurements. In the second part of the thesis, we discuss the meaning of temperature fluctuations in quantum thermodynamical systems. In statistical mechanics, there exist different approaches to the problem of temperature fluctuations, which lead to the well-known temperature fluctuation formula. The nature and operational accessibility of these fluctuations, however, remains rather unclear. We argue that according to quantum thermodynamics a system relaxes into a quasi-stationary thermal state and therefore the temperature does not fluctuate. However, since temperature is not an observable, it can only be determined indirectly. For modular systems, we show that the fluctuations of the temperature estimate obtained from a single energy measurement, indeed, are given by the temperature fluctuation formula derived in statistical mechanics, but only within a certain temperature range. The validity range of this formula increases with increasing size of the system. Thus, one can conclude that these "temperature fluctuations" can be interpreted as representing a fundamental limit for the instantaneous determination of temperature.
A closed, uncountably infinite dimensional system of “quantum object + thermostat” is mathematically stated in terms of a complex probabilistic process for the wave function, i.e. the solution of Langevin-Schrödinger (L-Sch)-type stochastic differential equation (SDE). The L-Sch SDE has been studied for nonstationary 1D random potential with quadratic space form. It was proved that when the random processes in potential determined by δ-shaped correlators, then closed, exactly solvable nonrelativistic quantum mechanics may be constructed. In this case all physically measured parameters of the system are built as multiple integrals of the fundamental solution of some second-order partial differential equations (etalon equations). In the present work we obtain expressions for transition probabilities in a quantum subsystem. It is shown that depending on the coupling constant of the thermostat with the parametric quantum harmonic oscillator (PQHO), some phase transitions of second kind may occur in the expressions for microscopic quantum transitions. A method of stochastic density matrix is developed for calculation of thermodynamic potentials of the quantum harmonic oscillator (QHO) immersed into a thermostat. The analytic expressions for the ground state energy level widening and shift of QHO are obtained. In other words the possibility of violating the second law of thermodynamics due to quantum fluctuations, (i.e., spontaneous transitions in QHO from vacuum to the excited states) is shown.
We explore some aspects of nonequilibrium statistical mechanics of classical and quantum systems. Two chapters are devoted to fluctuation theorems which were originally derived for classical systems. The main challenge in formulating them in quantum mechanics is the fact that fundamental quantities of interest, like work, are defined via the classical concept of a phase space trajectory. We utilize the decoherent histories conceptual framework, in which classical trajectories emerge in quantum mechanics as a result of coarse graining, and provide a first-principles analysis of the nonequilibrium work relation of Jarzynski and Crooks's fluctuation theorem for a quantum system interacting with a general environment based on the quantum Brownian motion (QBM) model. We indicate a parameter range at low temperatures where the theorems might fail in their original form. Fluctuation theorems of Jarzynski and Crooks for systems obeying classical Hamiltonian dynamics are derived under the assumption that the initial conditions are sampled from a canonical ensemble, even though the equilibrium state of an isolated system is typically associated with the microcanonical ensemble. We address this issue through an exact analysis of the classical Brownian motion model. We argue that a stronger form of ensemble equivalence than usually discussed in equilibrium statistical mechanics is required for these theorems to hold in the infinite environment limit irrespective of the ensemble used, and proceed to prove it for this model. An exact expression for the probability distribution of work is obtained for finite environments. Intuitively one expects a system to relax to an equilibrium state when brought into contact with a thermal environment. Yet it is important to have rigorous results that provide conditions for equilibration and characterize the equilibrium state. We consider the dynamics of open quantum systems using the Langevin and master equations and rigorously show that under fairly general conditions quantum systems interacting with a heat bath relax to the equilibrium state defined as the reduced thermal state of the system plus environment, even in the strong coupling regime. Our proof is valid to second-order in interaction strength for general systems and exact for the linear QBM model, for which we also show the equivalence of multi-time correlations. In the final chapter we give a sampling of our investigations into macroscopic quantum phenomena. We work out in detail a specific example of how and under what conditions the center of mass (CoM) coordinate of a macroscopic object emerges as the relevant degree of freedom. Interaction patterns are studied in terms of the couplings they induce between the CoM and relative coordinates of two macroscopic objects. We discuss the implications of these interaction patterns on macroscopic entanglement.
Why is thermalisation a universal phenomenon? How does a quantum system reach thermodynamical equilibrium? These questions are not new, dating even from the very birth of quantum theory and have been the subject of a renewed interest over the two last decades (see for instance the review in Eisert et al, Nature Physics 2015). In this work, we propose a universal model demonstrating that thermalisation of a small quantum system is an emergent property of the unitary evolution under a Schrödinger equation of a larger composite system, whose initial state can be arbitrary. We show that the origin of universality lies in the phenomenon of 'measure concentration', which provides self-averaging properties for the reduced density matrix characterizing the state of the small subsystem. Using our framework, we focus on the asymptotic state at long times and consider its stationary properties. In typical macroscopic conditions, we recover the canonical state and the Boltzmann distribution well known from statistical thermodynamics. This findings lead us to propose an alternative and more general definition of the canonical partition function which also allow us to describe non thermal stationary states.