Quantum oscillator in a blackbody radiation field

Quantum Work Statistics at Strong Reservoir Coupling

Determining the statistics of work done on a quantum system while strongly coupled to a reservoir is a formidable task, requiring the calculation of the full eigenspectrum of the combined system and reservoir. Here, we show that this issue can be circumvented by using a polaron transformation that maps the system into a new frame where weak-coupling theory can be applied. Crucially, this polaron approach reproduces the Jarzynski fluctuation theorem, thus ensuring consistency with the laws of stochastic thermodynamics. We apply our formalism to a system driven across the Landau-Zener transition, where we identify clear signatures in the work distribution arising from a non-negligible coupling to the environment. Our results provide a new method for studying the stochastic thermodynamics of driven quantum systems beyond Markovian, weak-coupling regimes.

Imaginary-time hierarchical equations of motion for thermodynamic variables

The partition function (PF) plays a key role in the calculation of quantum thermodynamic properties of a system that interacts with a heat bath. The imaginary-time hierarchical equations of motion (imHEOM) approach was developed to evaluate in a rigorous manner the PF of a system strongly coupled to a non-Markovian bath. In this paper, we present a numerically efficient scheme to evaluate the imHEOM utilizing the β-differentiated imHEOM (BD-imHEOM) that are obtained by differentiating the elements of the imHEOM with respect to the inverse temperature. This approach allows us to evaluate the system, system-bath interaction, and heat-bath parts of the PF efficiently. Moreover, we employ a polyharmonic decomposition method to construct a concise hierarchical structure with better convergence, thus reducing the cost of numerical integrations. We demonstrate the proposed approach by compute thermodynamic quantities of a spin-boson system and a 2 × 2 antiferromagnetic triangular spin lattice system with an Ohmic spectral distribution.

Consistent Hamiltonian models for space-momentum diffusion

We develop a unified Hamiltonian approach to the diffusion of a particle coupled to a dissipative environment, an archetypal model widely invoked to interpret condensed phase phenomena, such as polymerization and cold-atom diffusion in optical lattices. By appropriate choices of the coupling functions, we reformulate phenomenological diffusion models by adding otherwise ignored space-momentum terms. We thus numerically predict a variety of diffusion regimes, from diffusion saturation to superballistic diffusion. With reference to ultracold atoms in optical lattices, we also show that time correlated external noises prevent superdiffusion from exceeding Richardson's law. Some of these results are unexpected and call for experimental validation.

Quantum counterpart of energy equipartition theorem for a dissipative charged magneto-oscillator: Effect of dissipation, memory, and magnetic field

In this paper, we formulate and study the quantum counterpart of the energy equipartition theorem for a charged quantum particle moving in a harmonic potential in the presence of a uniform external magnetic field and linearly coupled to a passive quantum heat bath through coordinate variables. The bath is modeled as a collection of independent quantum harmonic oscillators. We derive closed form expressions for the mean kinetic and potential energies of the charged dissipative magneto-oscillator in the forms E_{k}=〈E_{k}〉 and E_{p}=〈E_{p}〉, respectively, where E_{k} and E_{p} denote the average kinetic and potential energies of individual thermostat oscillators. The net averaging is twofold; the first one is over the Gibbs canonical state for the thermostat, giving E_{k} and E_{p}, and the second one, denoted by 〈·〉, is over the frequencies ω of the bath oscillators which contribute to E_{k} and E_{p} according to probability distributions P_{k}(ω) and P_{p}(ω), respectively. The relationship of the present quantum version of the equipartition theorem with that of the fluctuation-dissipation theorem (within the linear-response theory framework) is also explored. Further, we investigate the influence of the external magnetic field and the effect of different dissipation processes through Gaussian decay and Drude and radiation bath spectral density functions on the typical properties of P_{k}(ω) and P_{p}(ω). Finally, the role of system-bath coupling strength and the memory effect is analyzed in the context of average kinetic and potential energies of the dissipative charged magneto-oscillator.

Heat capacities of thermally manipulated mechanical oscillator at strong coupling

Coherent quantum oscillators are basic physical systems both in quantum statistical physics and quantum thermodynamics. Their realizations in lab often involve solid-state devices sensitive to changes in ambient temperature. We represent states of the solid-state optomechanical oscillator with temperature-dependent frequency by equivalent states of the mechanical oscillator with temperature-dependent energy levels. We interpret the temperature dependence as a consequence of strong coupling between the oscillator and the heat bath. We explore parameter regimes corresponding to anomalous behavior of mechanical and thermodynamic characteristics as a consequence of the strong coupling: (i) The localization and the purification induced by heating, and (ii) the negativity of two generalized heat capacities. The capacities can be used to witness non-linearity in the temperature dependency of the energy levels. Our phenomenological experimentally-oriented approach can stimulate development of new optomechanical and thermomechanical experiments exploring basic concepts of strong coupling thermodynamics.

Thermal power of heat flow through a qubit

In this paper we consider the thermal power of a heat flow through a qubit between two baths. The baths are modeled as a set of harmonic oscillators initially at equilibrium, at two temperatures. Heat is defined as the change of energy of the cold bath, and thermal power is defined as expected heat per unit time, in the long-time limit. The qubit and the baths interact as in the spin-boson model, i.e., through qubit operator σ_{z}. We compute thermal power in an approximation analogous to a "noninteracting blip" (NIBA) and express it in the polaron picture as products of correlation functions of the two baths, and a time derivative of a correlation function of the cold bath. In the limit of weak interaction we recover known results in terms of a sum of correlation functions of the two baths, a correlation functions of the cold bath only, and the energy split.

Steady-State Coherences by Composite System-Bath Interactions

We identify sufficient conditions on the structure of the interaction Hamiltonian between a two-level quantum system and a thermal bath that, without any external drive or coherent measurement, guarantee the generation of steady-state coherences (SSC). The SSC obtained this way, remarkably, turn out to be independent of the initial state of the system, which could therefore be taken as initially incoherent. We characterize in detail this phenomenon, first analytically in the weak coupling regime for two paradigmatic models, and then numerically in more complex systems without any assumption on the coupling strength. In all of these cases, we find that SSC become increasingly significant as the bath is cooled down. These results can be directly verified in many experimental platforms.

Energy-temperature uncertainty relation in quantum thermodynamics

It is known that temperature estimates of macroscopic systems in equilibrium are most precise when their energy fluctuations are large. However, for nanoscale systems deviations from standard thermodynamics arise due to their interactions with the environment. Here we include such interactions and, using quantum estimation theory, derive a generalised thermodynamic uncertainty relation valid for classical and quantum systems at all coupling strengths. We show that the non-commutativity between the system's state and its effective energy operator gives rise to quantum fluctuations that increase the temperature uncertainty. Surprisingly, these additional fluctuations are described by the average Wigner-Yanase-Dyson skew information. We demonstrate that the temperature's signal-to-noise ratio is constrained by the heat capacity plus a dissipative term arising from the non-negligible interactions. These findings shed light on the interplay between classical and non-classical fluctuations in quantum thermodynamics and will inform the design of optimal nanoscale thermometers.

Nonperturbative theory of atom-surface interaction: corrections at short separations

The nonperturbative expressions for the free energy and force of interaction between a ground-state atom and a real-material surface at any temperature are presented. The transition to the Matsubara representation is performed, whereupon the comparison is made with the commonly used perturbative results based on the standard Lifshitz theory. It is shown that the Lifshitz formulas for the free energy and force of an atom-surface interaction follow from the nonperturbative ones in the lowest order of the small parameter. Numerical computations of the free energy and force for the atoms of He[Formula: see text] and Na interacting with a surface of an Au plate have been performed using the frequency-dependent dielectric permittivity of Au and highly accurate dynamic atomic polarizabilities in the framework of both the nonperturbative and perturbative theories. According to our results, the maximum deviations between the two theories are reached at the shortest atom-surface separations of about 1 nm. Simple analytic expressions for the atom-surface free energy are derived in the classical limit and for an ideal-metal plane. In the lowest order of the small parameter, they are found in agreement with the perturbative ones following from the standard Lifshitz theory. Possible applications of the obtained results in the theory of van der Waals adsorption are discussed.

Entropy production and time asymmetry in the presence of strong interactions

It is known that the equilibrium properties of open classical systems that are strongly coupled to a heat bath are described by a set of thermodynamic potentials related to the system's Hamiltonian of mean force. By adapting this framework to a more general class of nonequilibrium states, we show that the equilibrium properties of the bath can be well defined, even when the system is arbitrarily far from equilibrium and correlated with the bath. These states, which retain a notion of temperature, take the form of conditional equilibrium distributions. For out-of-equilibrium processes we show that the average entropy production quantifies the extent to which the system and bath state is driven away from the conditional equilibrium distribution. In addition, we show that the stochastic entropy production satisfies a generalized Crooks relation and can be used to quantify time asymmetry of correlated nonequilibrium processes. These results naturally extend the familiar properties of entropy production in weakly coupled systems to the strong coupling regime. Experimental measurements of the entropy production at strong coupling could be pursued using optomechanics or trapped-ion systems, which allow strong coupling to be engineered.

Open system trajectories specify fluctuating work but not heat

Based on the explicit knowledge of a Hamiltonian of mean force, the classical statistical mechanics and equilibrium thermodynamics of open systems in contact with a thermal environment at arbitrary interaction strength can be formulated. Yet, even though the Hamiltonian of mean force uniquely determines the equilibrium phase space probability density of a strongly coupled open system, the knowledge of this probability density alone is insufficient to determine the Hamiltonian of mean force, needed in constructing the underlying statistical mechanics and thermodynamics. We demonstrate that under the assumption that the Hamiltonian of mean force is known, an extension of thermodynamic structures from the level of averaged quantities to fluctuating objects (i.e., a stochastic thermodynamics) is possible. However, such a construction undesirably also involves a vast ambiguity. This situation is rooted in the eminent lack of a physical guiding principle allowing us to distinguish a physically meaningful theory out of a multitude of other equally conceivable ones.

First and Second Law of Thermodynamics at Strong Coupling

For a small driven system coupled strongly to a heat bath, internal energy and exchanged heat are identified such that they obey the usual additive form of the first law. By identifying this exchanged heat with the entropy change of the bath, the total entropy production is shown to obey an integral fluctuation theorem on the trajectory level implying the second law in the form of a Clausius inequalilty on the ensemble level. In this Hamiltonian approach, the assumption of an initially uncorrelated state is not required. The conditions under which the proposed identification of heat is unique and experimentally accessible are clarified.

Particle coupled to a heat bath in non-Abelian gauge potentials

We derive the quantum Langevin equation (QLE) for a harmonically single trapped cold atom subjected to artificial non-Abelian gauge potentials and linearly coupled to a heat bath. The independent-oscillator (IO) and the momentum-momenta coupling models are studied. In each case, the non-Abelian effect on the QLE is pointed out for a U(2) gauge transformation. For the IO model, only the generalized Lorentz force is modified by the appearance of an additive term. For the momentum-momenta coupling model, the generalized Lorentz force as well as the friction force are subjected to modifications. The dependence of the system on the magnetic field is explicit even if the gauge potential is uniform in space.

Coupled-oscillator theory of dispersion and Casimir-Polder interactions

We address the question of the applicability of the argument theorem (of complex variable theory) to the calculation of two distinct energies: (i) the first-order dispersion interaction energy of two separated oscillators, when one of the oscillators is excited initially and (ii) the Casimir-Polder interaction of a ground-state quantum oscillator near a perfectly conducting plane. We show that the argument theorem can be used to obtain the generally accepted equation for the first-order dispersion interaction energy, which is oscillatory and varies as the inverse power of the separation r of the oscillators for separations much greater than an optical wavelength. However, for such separations, the interaction energy cannot be transformed into an integral over the positive imaginary axis. If the argument theorem is used incorrectly to relate the interaction energy to an integral over the positive imaginary axis, the interaction energy is non-oscillatory and varies as r(-4), a result found by several authors. Rather remarkably, this incorrect expression for the dispersion energy actually corresponds to the nonperturbative Casimir-Polder energy for a ground-state quantum oscillator near a perfectly conducting wall, as we show using the so-called "remarkable formula" for the free energy of an oscillator coupled to a heat bath [G. W. Ford, J. T. Lewis, and R. F. O'Connell, Phys. Rev. Lett. 55, 2273 (1985)]. A derivation of that formula from basic results of statistical mechanics and the independent oscillator model of a heat bath is presented.

Fractional entropy decay and the third law of thermodynamics

We report in this paper a theoretical study on the quantum thermodynamic properties of a fractional damping system. Through the analysis, few nontrivial characteristics are revealed, which include (1) a fractional power-law decay entropy function, which provides an evidence for the validity of the third law of thermodynamics in the quantum dissipative region and (2) the varying of the entropy from a nonlinear divergent function to a semilinear decay function with a fractional exponent as the temperature approaches absolute zero.

Fluctuation corrections to thermodynamic functions: finite-size effects

The explicit thermodynamic functions, in particular, the specific heat of a spin system interacting with a spin bath which exerts finite dissipation on the system are determined. We show that the specific heat is a sum of the products of a thermal equilibration factor that carries the temperature dependence and a dynamical correction factor, characteristic of the dissipative energy flow under steady state from the system. The variation of specific heat with temperature is accompanied by an abrupt transition that depends on these dynamical factors characteristic of the finite system size.

Equilibration and thermalization of the dissipative quantum harmonic oscillator in a nonthermal environment

We study the dissipative quantum harmonic oscillator with general nonthermal preparations of the harmonic oscillator bath. The focus is on equilibration of the oscillator in the long-time limit and the additional requirements for thermalization. Our study is based on the exact solution of the microscopic model obtained by means of operator equations of motion, which provides us with the time evolution of the central oscillator density matrix in terms of the propagating function. We find a hierarchy of conditions for thermalization, together with the relation of the asymptotic temperature to the energy distribution in the initial bath state. We discuss the presence and absence of equilibration for the example of an inhomogeneous chain of harmonic oscillators, and we illustrate the general findings about thermalization for the nonthermal environment that results from a quench.

Role of quantum heat bath and confinement in the low-temperature thermodynamics of cyclotron motion

In this Brief Report we show how the low-temperature thermodynamics of the dissipative motion of an electron in a magnetic field is sensitive to the nature of the spectral density function, J(omega), of the quantum heat bath. In all cases of couplings considered here the free energy and the entropy of the cyclotron motion of the electron fall off to zero as power law in conformity with the third law of thermodynamics. The power of the power law however depends on the nature of J(omega). We also separately discuss the influence of confinement.

Clausius inequality beyond the weak-coupling limit: the quantum Brownian oscillator

We consider a quantum linear oscillator coupled at an arbitrary strength to a bath at an arbitrary temperature. We find an exact closed expression for the oscillator density operator. This state is noncanonical but can be shown to be equivalent to that of an uncoupled linear oscillator at an effective temperature T*(eff) with an effective mass and an effective spring constant. We derive an effective Clausius inequality deltaQ*(eff)< or =T*(eff)dS , where deltaQ*(eff) is the heat exchanged between the effective (weakly coupled) oscillator and the bath, and S represents a thermal entropy of the effective oscillator, being identical to the von-Neumann entropy of the coupled oscillator. Using this inequality (for a cyclic process in terms of a variation of the coupling strength) we confirm the validity of the second law. For a fixed coupling strength this inequality can also be tested for a process in terms of a variation of either the oscillator mass or its spring constant. Then it is never violated. The properly defined Clausius inequality is thus more robust than assumed previously.

Quantum dissipative Brownian motion and the Casimir effect

We explore an analogy between the thermodynamics of a free dissipative quantum particle in one dimension and that of an electromagnetic field between two mirrors of finite conductivity. While a free particle isolated from its environment will effectively be in the high-temperature limit for any nonvanishing temperature, a finite coupling to the environment leads to quantum effects ensuring the correct low-temperature behavior. Even then, it is found that under appropriate circumstances the entropy can be a nonmonotonic function of the temperature. Such a scenario with its specific dependence on the ratio of temperature and damping constant also appears for the transverse electric mode in the Casimir effect. The limits of vanishing dissipation for the quantum particle and of infinite conductivity of the mirrors in the Casimir effect both turn out to be noncontinuous.

Fluctuation theorem for arbitrary open quantum systems

Based on the observation that the thermodynamic equilibrium free energy of an open quantum system in contact with a thermal environment is the difference between the free energy of the total system and that of the bare environment, the validity of the Crooks theorem and of the Jarzynski equality is extended to open quantum systems. No restrictions on the nature of the environment or on the strength of the coupling between system and environment need to be imposed. This free energy entering the Crooks theorem and the Jarzynski equality is closely related to the Hamiltonian of mean force that generalizes the classical statistical mechanical concept of the potential of mean force.

Thermodynamics of a subensemble of a canonical ensemble

Two approaches to describing the thermodynamics of a subsystem that interacts with a thermal bath are considered. Within the first approach, the mean system energy E_{S} is identified with the expectation value of the system Hamiltonian, which is evaluated with respect to the overall (system+bath) equilibrium distribution. Within the second approach, the system partition function Z_{S} is considered as the fundamental quantity, which is postulated to be the ratio of the overall (system+bath) and the bath partition functions, and the standard thermodynamic relation E_{S}=-d(ln Z_{S})/dbeta is used to obtain the mean system energy. Employing both classical and quantum-mechanical treatments, the advantages and shortcomings of the two approaches are analyzed in detail for various different systems. It is shown that already within classical mechanics both approaches predict significantly different results for thermodynamic quantities provided the system-bath interaction is not bilinear or the system of interest consists of more than a single particle. Based on the results, it is concluded that the first approach is superior.

Entropy consumption in primary photosynthesis

Knox and Parson have objected to our previous conclusion on possible negative entropy production during primary photochemistry, i.e., from photon absorption to primary charge separation, by considering a pigment system in which primary photochemistry is not specifically considered. This approach does not address our proposal. They suggest that when a pigment absorbs light and passes to an excited state, its entropy increases by hnu/T. This point is discussed in two ways: (i) from considerations based on the energy gap law for excited state relaxation; (ii) using classical thermodynamics, in which free energy is introduced into the pigment (antenna) system by photon absorption. Both approaches lead us to conclude that the excited state and the ground state are isoentropic, in disagreement with Knox and Parson. A discussion on total entropy changes specifically during the charge separation process itself indicates that this process may be almost isoentropic and thus our conclusions on possible negentropy production associated with the sequence of reactions which go from light absorption to the first primary charge separation event, due to its very high thermodynamic efficiency, remain unchanged.

Quantum master equations from classical Lagrangians with two stochastic forces

We show how a large family of master equations, describing quantum Brownian motion of a harmonic oscillator with translationally invariant damping, can be derived within a phenomenological approach, based on the assumption that an environment can be simulated by two classical stochastic forces. This family is determined by three time-dependent correlation functions (besides the frequency and damping coefficients), and it includes as special cases the known master equations, whose dissipative part is bilinear with respect to the operators of coordinate and momentum.

A quantum violation of the second law?

An apparent violation of the second law of thermodynamics occurs when an atom coupled to a zero-temperature bath, being necessarily in an excited state, is used to extract work from the bath. Here the fallacy is that it takes work to couple the atom to the bath and this work must exceed that obtained from the atom. For the example of an oscillator coupled to a bath described by the single relaxation time model, the mean oscillator energy and the minimum work required to couple the oscillator to the bath are both calculated explicitly and in closed form. It is shown that the minimum work always exceeds the mean oscillator energy, so there is no violation of the second law.

Classical and quantum diffusion in the presence of velocity-dependent coupling

A generalized system-plus-reservoir model is introduced, which includes four kinds of couplings between the coordinates and velocities of a system and its environment. It is found that the velocity-dependent coupling is not equivalent to a coordinate coupling due to the different power spectra of thermal noise. Harmonic velocity and acceleration noises are proposed which correspond to the coordinate-velocities and velocity-velocities couplings, respectively, if the environmental oscillators are assumed to have a harmonic spectral distribution. Indeed, the velocity-dependent coupling can induce ballistic diffusion of a force-free particle and the mean square velocity depends on the initial preparation. Quantum ballistic diffusion is also presented and its velocity correlation function is found to be unstable at any time. One of real examples of velocity-coordinates coupling is a one-electron atom interacting with the radiation field. A particle moving in a periodic potential shows a nonergodic behavior.

Non-Markovian Brownian dynamics and nonergodicity

We report the breaking of ergodicity for a class of generalized, Brownian motion obeying a non-Markovian dynamics being driven by a generalized Langevin equation (GLE). This very feature originates from a vanishing of the effective friction. A novel quantity b (being uniquely determined from the corresponding memory friction kernel gamma(t)of the GLE) is introduced as a parameter that is capable of measuring the strength of ergodicity breaking. The ergodicity breaking is accompanied by a nonunique stationary probability density for the corresponding embedded Markovian dynamics. Differing physical situations for a Brownian, non-Markovian particle dynamics occurring either in free Brownian motion, in a periodic potential, or in a confining potential are elucidated.

Harmonic velocity noise: non-Markovian features of noise-driven systems at long times

We propose a harmonic velocity noise with a broadband feature, which is the time derivative of the harmonic noise. If this noise is regarded as a thermal one, the system has a vanishing effective friction and it should induce ballistic diffusion of a free particle at long times. The effective temperature of the system coupled to such a structured heat bath represented by the harmonic velocity noise is introduced. This means that any initial preparation will approach asymptotically a preparation-dependent variance and mean value for velocity variable. Thus the fluctuation-dissipation theorem does not hold as there is no unique stationary state being connected with a breakdown of ergodicity. This noise can show greenness when it is taken as an external noise source to drive a correlation ratchet.

Anomalous dissipation: strong non-Markovian effect and its dynamical origin

We report the effects of anomalous dissipation with a vanishing effective friction, which can induce ballistic diffusion and dissipative acceleration, where the fluctuation-dissipation theorem is fulfilled. An influence factor is introduced in order to describe the role of non-Markovian friction and the force-folded effect on the long-time results. The velocity-dependent coupling and force might be a dynamical origin of this dissipation. The steady acceleration of a particle moving in periodic and magnetic-force potentials are calculated.