The extent to which a temperature can be appropriately assigned to a small quantum system, as an internal property but not as a property of any large environment, is still an open problem. In this paper, a method is proposed for solving this problem, by which a studied small system is coupled to a two-level system as a probe, the latter of which can be measured by measurement devices. A main difficulty in the determination of possible temperature of the studied system comes from the back-action of the probe-system coupling to the system. For small quantum chaotic systems, we show that a temperature can be determined, the value of which is sensitive to neither the form, location, and strength of the probe-system coupling, nor the Hamiltonian and initial state of the probe. The temperature thus obtained turns out to have the form of Boltzmann temperature.
The extent to which a temperature can be appropriately assigned to a small quantum system, as an internal property but not as a property of any large environment, is still an open problem. In this paper, a method is proposed for solving this problem, by which a studied small system is coupled to a two-level system as a probe, the latter of which can be measured by measurement devices. A main difficulty in the determination of possible temperature of the studied system comes from the back-action of the probe-system coupling to the system. For small quantum chaotic systems, we show that a temperature can be determined, the value of which is sensitive to neither the form, location, and strength of the probe-system coupling, nor the Hamiltonian and initial state of the probe. The temperature thus obtained turns out to have the form of Boltzmann temperature.
Abstract Thermalization of isolated quantum systems has been studied intensively in recent years and significant progresses have been achieved. Here, we study thermalization of small quantum systems that interact with large chaotic environments under the consideration of Schrödinger evolution of composite systems, from the perspective of the zeroth law of thermodynamics. Namely, we consider a small quantum system that is brought into contact with a large environmental system; after they have relaxed, they are separated and their temperatures are studied. Our question is under what conditions the small system may have a detectable temperature that is identical with the environmental temperature. This should be a necessary condition for the small quantum system to be thermalized and to have a well-defined temperature. By using a two-level probe quantum system that plays the role of a thermometer, we find that the zeroth law is applicable to quantum chaotic systems, but not to integrable systems.
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.
Recent work has shown that the entanglement of finite-temperature eigenstates in chaotic quantum many-body local Hamiltonians can be accurately described by an ensemble of random states with an internal U(1) symmetry. We build upon this result to investigate the universal symmetry-breaking properties of such eigenstates. As a probe of symmetry breaking, we employ the entanglement asymmetry, a quantum information observable that quantifies the extent to which symmetry is broken in a subsystem. This measure enables us to explore the finer structure of finite-temperature eigenstates in terms of the U(1)-symmetric random state ensemble; in particular, the relation between the Hamiltonian and the effective conserved charge in the ensemble. Our analysis is supported by analytical calculations for the symmetric random states, as well as exact numerical results for the Mixed-Field Ising spin-1/2 chain, a paradigmatic model of quantum chaoticity.
Potential realization of a quantum thermometer operating in the nanokelvin regime, formed by a few-fermionic mixture confined in a one-dimensional harmonic trap, is proposed. Thermal states of the system are studied theoretically from the point of view of fundamental sensitivity to temperature changes. It is pointed out that the ability to control the interaction strength in such systems allows obtaining high-temperature sensitivity in the regime where the temperature is much lower than the characteristic temperature scale determined by a harmonic confinement. This sensitivity is very close to the fundamental bound that involves optimal engineering of level separations. The performance of practical measurement schemes and the possible experimental coupling of the thermometer to the probe are discussed.
We investigate the possibility to monitor the dynamics of an open quantum system with the help of a small probe system, coupled via dephasing coupling to the open system of interest. As an example, we consider a dissipative harmonic oscillator and a single qubit as probe system. Qubit plus oscillator are described by a finite temperature quantum master equation, where the dynamics of the whole system can be obtained analytically. We find that the short time behavior of the reduced qubit state (its coherence) provides exhaustive information on the dissipative dynamics of the oscillator. Observing this coherence for two initial states with different out-of-equilibrium temperatures, one can determine all coupling constants and the equilibrium temperature fixed by the external heat bath. In addition, the dephasing coupling to the qubit probe, may be considered as a perturbation of the dissipative oscillator. The corresponding quantum fidelity can be calculated analytically, also. Hence, we find the precise relation between the behavior of the reduced qubit state (its coherence) and that fidelity.
We use the decoherent histories approach to quantum theory to derive the form of an effective theory describing the coupling of classical and quantum variables. The derivation is carried out for a system consisting of a large particle coupled to a small particle with the important additional feature that the large particle is also coupled to a thermal environment producing the decoherence necessary for classicality. The effective theory is obtained by tracing out both the environment and the small particle variables. It consists of a formula for the probabilities of a set of histories of the large particle, and depends on the dynamics and initial quantum state of the small particle. It has the form of an almost classical particle coupled to a stochastic variable whose probabilities are determined by a formula very similar to that given by quantum measurement theory for continuous measurements of the small particle's position. The effective theory gives intuitively sensible answers when the small particle is in a superposition of localized states (unlike the simple mean field approach of coupling to the expectation values of the small system). The derived effective theory suggests a form of the semiclassical theory even when the quantum theory of the large system is not known, as is the case, for example, when a classical gravitational field is coupled to a quantized matter field, thus offering a new approach to the back reaction problem.