In this work we investigate the late-time steady states of open quantum systems coupled to a thermal reservoir in the strong coupling regime. In general such systems do not necessarily relax to a Boltzmann distribution if the coupling to the thermal reservoir is nonvanishing or equivalently if the relaxation time scales are finite. Using a variety of nonequilibrium formalisms valid for non-Markovian processes, we show that starting from a product state of the closed system = system+environment, with the environment in its thermal state, the open system which results from coarse graining the environment will evolve towards an equilibrium state at late times. This state can be expressed as the reduced state of the closed system thermal state at the temperature of the environment. For a linear (harmonic) system and environment, which is exactly solvable, we are able to show in a rigorous way that all multitime correlations of the open system evolve towards those of the closed system thermal state. Multitime correlations are especially relevant in the non-Markovian regime, since they cannot be generated by the dynamics of the single-time correlations. For more general systems, which cannot be exactly solved, we are able to provide a general proof that all single-time correlations of the open system evolve to those of the closed system thermal state, to first order in the relaxation rates. For the special case of a zero-temperature reservoir, we are able to explicitly construct the reduced closed system thermal state in terms of the environmental correlations.
In this work we investigate the late-time steady states of open quantum systems coupled to a thermal reservoir in the strong coupling regime. In general such systems do not necessarily relax to a Boltzmann distribution if the coupling to the thermal reservoir is nonvanishing or equivalently if the relaxation time scales are finite. Using a variety of nonequilibrium formalisms valid for non-Markovian processes, we show that starting from a product state of the closed system = system+environment, with the environment in its thermal state, the open system which results from coarse graining the environment will evolve towards an equilibrium state at late times. This state can be expressed as the reduced state of the closed system thermal state at the temperature of the environment. For a linear (harmonic) system and environment, which is exactly solvable, we are able to show in a rigorous way that all multitime correlations of the open system evolve towards those of the closed system thermal state. Multitime correlations are especially relevant in the non-Markovian regime, since they cannot be generated by the dynamics of the single-time correlations. For more general systems, which cannot be exactly solved, we are able to provide a general proof that all single-time correlations of the open system evolve to those of the closed system thermal state, to first order in the relaxation rates. For the special case of a zero-temperature reservoir, we are able to explicitly construct the reduced closed system thermal state in terms of the environmental correlations.
It is known that the origin of the deviations from standard thermodynamics proceed from the strong coupling to the bath. Here, it is shown that these deviations are related to the power spectrum of the bath. Specifically, it is shown that the system thermal-equilibrium-state cannot be characterized by the canonical Boltzmann's distribution in quantum mechanics. This is because the uncertainty principle imposed a lower bound of the dispersion of the total energy of the system that is dominated by the spectral density of the bath. However, in the classical case, for a wide class of systems that interact via central forces with pairwise-self-interacting environment, the system thermal equilibrium state is exactly characterized by the canonical Boltzmann distribution. As a consequence of this analysis and taking into account all energy scales in the system and reservoir interaction, an effective coupling to the environment is introduced. Sample computations in different regimes predicted by this effective coupling are shown. Specifically, in the strong coupling effective regime, the system exhibits deviations from standard thermodynamics and in the weak coupling effective regime, quantum features such as stationary entanglement are possible at high temperatures. Moreover, it is known that the spectrum of thermal baths depends on the non-Markovian character. Hence, non-Markovian interactions have a important role in the thermal equilibrium state of physical systems. For example, in quantum optomechanics is looked up to cool the mechanical system through an auxiliary system which generally is a cavity. This cooling process takes into account the non-Markovian interaction and as it is shown here, it is more effective than if we use only the Markovian approximation in the equation of motion for the different modes.
Abstract We introduce an exact open system method to describe the dynamics of quantum systems that are strongly coupled to specific types of environments comprising of spins, such as central spin systems. Our theory is similar to the established non-Markovian quantum state diffusion theory, but for a spin bath instead of a Gaussian bath. The method allows us to represent the time-evolved reduced state of the system as an ensemble average of stochastically evolving pure states. We present a comprehensive theory for arbitrary linear spin environments at both zero and finite temperatures. Furthermore, we introduce a hierarchical expansion method that enables the numerical computation of the time evolution of the stochastic pure states, facilitating a numerical solution of the open system problem in relevant strong coupling regimes.
Abstract Environments in quantum thermodynamics usually take the role of heat baths. These baths are Markovian, weakly coupled to the system, and initialized in a thermal state. Whenever one of these properties is missing, standard quantum thermodynamics is no longer suitable to treat the thermodynamic properties of the system that result from the interaction with the environment. Using a recently proposed framework for open system quantum thermodynamics which is valid for arbitrary couplings and non-Markovian effects, we show that within the very same model, described by a Fano–Anderson Hamiltonian, the environment can take three different thermodynamic roles: a standard heat bath, exchanging only heat with the system, a work reservoir, exchanging only work, and a hybrid environment, providing both types of energy exchange. The exact role of the environment is determined by the strength and structure of the coupling, and by its initial state. The latter also dictates the long time behaviour of the open system, leading to thermal equilibrium for an initial thermal state and to a nonequilibrium steady state when there are displaced environmental modes.
Quantum systems are inherently open to their environment and thus they evolve under the influence of dissipation, decoherence and driving from external fields. This poses a significant theoretical challenge: how to track the dynamics of the system of interest while tracing out environmental degrees of freedom. Markovian master equations offer a longstanding solution to this problem, where under a series of assumptions the reduced density matrix of the system can be obtained. These flexible approaches provide a conceptually and numerically simple framework for describing complex many-body systems. However, the assumptions enforced by Markovian master equations create a gap in the availability of generic, simple, and analytical techniques capable of treating open quantum systems beyond the regimes of standard Markovian master equations. In this thesis, we introduce three studies addressing three regimes that are inaccessible to standard Markovian master equations and investigate their consequences in quantum thermodynamics-inspired applications. We begin by developing a Markovian embedding technique to tackle non-Markovian dynamics induced by the long correlation times of a qubit coupled to a structured environment. Subsequently, we propose an "effective" Hamiltonian theory enabling the study of equilibrium properties of strong coupling to a thermal reservoir through a dressed version of the Gibbs state. This framework can also be applied to address problems of equilibration, transport, quantum thermal machines, and quantum dynamics. Lastly, we showcase the importance of including non-secular terms in the quantum master equation, which results in a drastic separation of timescales leading to rich metastable dynamics. We apply this quantum master equation approach to temperature sensing in a nonequilibrium setting offering an alternative to standard equilibrium probes. Overall, this thesis contributes novel theoretical techniques for tackling open quantum systems beyond standard approximations with applications in quantum thermodynamics.
We present a numerically exact method to compute the full counting statistics of heat transfer in non-Markovian open quantum systems, which is based on the time-evolving matrix product operator (TEMPO) algorithm. This approach is applied to the paradigmatic spin-boson model in order to calculate the mean and fluctuations of the heat transferred to the environment during thermal equilibration. We show that system-reservoir correlations make a significant contribution to the heat statistics at low temperature and present a variational theory that quantitatively explains our numerical results. We also demonstrate a fluctuation-dissipation relation connecting the mean and variance of the heat distribution at high temperature. Our results reveal that system-bath interactions make a significant contribution to heat dissipation even when the dynamics of the open system is effectively Markovian. The method presented here provides a flexible and general tool to predict the fluctuations of heat transfer in open quantum systems in non-perturbative regimes.
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.