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.
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.
Contrary to the conventional wisdom that deviations from standard thermodynamics originate from the strong coupling to the bath, it is shown that in quantum mechanics, these deviations originate from the uncertainty principle and are supported by the non-Markovian character of the dynamics. Specifically, it is shown that the lower bound of the dispersion of the total energy of the system, imposed by the uncertainty principle, is dominated by the bath power spectrum; therefore, quantum mechanics inhibits the system thermal-equilibrium-state from being described by the canonical Boltzmann's distribution. We show for a wide class of systems, systems interacting via central forces with pairwise-self-interacting environments; this general observation is in sharp contrast to the classical case, for which the thermal equilibrium distribution, irrespective of the interaction strength, is exactly characterized by the canonical Boltzmann distribution; therefore, no dependence on the bath power spectrum is present. We define an effective coupling to the environment that depends on all energy scales in the system and reservoir interaction. Sample computations in regimes predicted by this effective coupling are demonstrated. For example, for the case of strong effective coupling, deviations from standard thermodynamics are present and for the case of weak effective coupling, quantum features such as stationary entanglement are possible at high temperatures.
Chemical Physics Theory Group, Department of Chemistry and Center for Quantum Information and Quantum Control,University of Toronto, Toronto, Canada M5S 3H6(Dated: January 8, 2014)Contrary to the conventional wisdom that deviations from standard thermodynamics originatefrom the strong coupling to the bath, it is shown that these deviations are intimately linked to thepower spectrum of the thermal bath. Speci cally, it is shown that the lower bound of the dispersionof the total energy of the system, imposed by the uncertainty principle, is dominated by the bathpower spectrum and therefore, quantum mechanics inhibits the system thermal-equilibrium-statefrom being described by the canonical Boltzmann’s distribution. This is in sharp contrast to theclassical case, for which the thermal equilibrium distribution of a system interacting via centralforces with pairwise-self-interacting environment, irrespective of the interaction strength, is shown tobe exactly characterized by the canonical Boltzmann distribution. As a consequence of this analysis,we de ne an e ective coupling to the environment that depends on all energy scales in the systemand reservoir interaction. Sample computations in regimes predicted by this e ective coupling aredemonstrated. For example, for the case of strong e ective coupling, deviations from standardthermodynamics are present and, for the case of weak e ective coupling, quantum features such asstationary entanglement are possible at high temperatures.
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.
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.
The focus is on understanding the quantum thermodynamics of strongly coupled non-Markovian quantum systems. To this end, a non-trivial, non-Markovian model of a central spin surrounded by a spin bath is taken up, and its exact evolution is derived for arbitrary system-bath couplings. The fundamental quantum thermodynamic quantities, such as system and bath internal energies, work, heat, entropy production, and ergotropy, are calculated using the dynamics and the original system (bath) Hamiltonian. An explicit expression for the work, a mismatch between the system and bath internal energies, is derived. The thermodynamic entropy of the system at thermal equilibrium is studied using the Hamiltonian of mean force in the strong coupling regime. The role of a canonical Hamiltonian in calculating the above thermodynamic quantities, a recently developed technique, is also investigated. Furthermore, an interesting observation relevant to the spin bath acting as a charger is made in a scenario where the central spin is envisaged as a quantum battery.
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.