In this paper, we discuss how effective environments incorporating periodic measurements can be used to prepare a two-level system (TLS) in almost arbitrary thermal states: Concretely, we study a TLS coupled to a spin environment, the magnetization of which is measured periodically. In ensemble average these measurements cause a relaxation of the TLS into a thermal (diagonal) state. By adjusting the time between the measurements and the detuning of the environmental spins, the creation of very low temperatures as well as inversion becomes possible. Our analytical results derived for large environments are numerically shown to be valid even for quite small environments, down to only a few spins.
In this paper, we discuss how effective environments incorporating periodic measurements can be used to prepare a two-level system (TLS) in almost arbitrary thermal states: Concretely, we study a TLS coupled to a spin environment, the magnetization of which is measured periodically. In ensemble average these measurements cause a relaxation of the TLS into a thermal (diagonal) state. By adjusting the time between the measurements and the detuning of the environmental spins, the creation of very low temperatures as well as inversion becomes possible. Our analytical results derived for large environments are numerically shown to be valid even for quite small environments, down to only a few spins.
Within a quantum-thermodynamical approach a single subsystem can be shown to relax to a thermal state if appropriately embedded in a quantum environment. This picture appears to be at variance with the foundations of statistical mechanics, according to which thermal properties emerge either as a long-time or ensemble average. Here we argue that periodic measurements on the environment may establish the missing link: we consider a two-level system (TLS) coupled to an environment consisting of many spins, the magnetization of which is measured periodically. These measurements destroy system-environment correlations and influence the state of the TLS due to so-called co-jumps, which lead to quasi-classical trajectories for the TLS. We derive an analytical solution for the ensemble-averaged attractor state reached after many measurements. Reinterpreting the setup as an indirect measurement process for the TLS also sheds new light on the properties of such periodic indirect measurements.
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.
We discuss the long-time relaxation of a qubit linearly coupled to a finite bath of [Formula: see text] spins (two-level systems, TLSs), with the interaction Hamiltonian in rotating wave approximation. We focus on the regime [Formula: see text], assuming that the qubit–bath coupling is weak, that the range of spin frequencies is sufficiently broad, and that all the spins are initialized in the ground state. Despite the model being perfectly integrable, we make two interesting observations about the effective system relaxation. First, as one would expect, the qubit relaxes exponentially towards its zero-temperature state at a well characterized rate. Second, the bath spins, even when mutually coupled, do not relax towards a thermal distribution, but rather form a Lorentzian distribution peaked at the frequency of the initially excited qubit. This behaviour is captured by an analytical approximation that makes use of the property [Formula: see text] to treat the TLS frequencies as a continuum and is confirmed by our numerical simulations.
We investigate a two-level system in contact with a larger quantum environment, often consisting of many two-level systems itself (spin environment) which may or may not interact. The total system is considered to be closed. The environment typically is in a canonical state with a given temperature initially. Depending on the precise spectral structure of the environment and the type of coupling between both systems, the smaller part may relax to a canonical state with the same temperature as the environment (i.e., thermal relaxation) or to some other quasiequilibrium state (nonthermal relaxation). The type of (quasi)equilibrium state can be related to the distribution of certain properties of the energy eigenvectors of the total system. We examine these distributions for several abstract and concrete (spin environment) Hamiltonian systems; the significant aspect of these distributions can be related to the relative strength of the local and interaction parts of the Hamiltonian.
Interactions between a quantum system and its environment at low temperatures can lead to apparent violations of thermal laws for the system. The source of these violations is the coupling between the system and environment, which prevents the system from entering into a thermal state, a state of Gibb's form. On the other hand, for two-state systems, we show that one can define an effective temperature, placing the system into a 'pseudo-thermal' state where effective thermal laws can be applied. We then numerically explore these assertions for an n -state environment inspired by the spin-boson environment.
The coupling of two-level quantum systems to the thermal environment is a fundamental problem with applications of such usually single-particle or fermionic systems ranging from qubit state preparation to spin models. The present Letter studies the elementary problem of the thermodynamics of an ensemble of bosons populating a two-level system. Using an optical dye microcavity platform, we thermalize photons in a two-mode system at conditions of tunable chemical potential, demonstrating the statistical mechanical problem of N bosons populating a two-level system, coupled to a heat bath. Under pulsed excitation, we observe Josephson oscillations between the two quantum states, which verifies the possibility for coherent manipulation. In contrast, under stationary conditions, the thermalization of the two-mode system is observed, arising from radiative coupling of photons to the dye. As the energetic splitting between eigenstates is 2 orders of magnitude smaller than thermal energy, at low occupations, an almost equal distribution of the occupation of the modes is observed, as expected from Boltzmann statistics. For larger occupation, we observe efficient population of the ground state and saturation of the upper level at high filling, expected from quantum statistics. Our experiment holds promise for state preparation in quantum technologies as well as for quantum thermodynamics studies.