Quantum thermodynamic quantities, normally formulated with the assumption of weak system-bath coupling (SBC), can often be contested in physical circumstances with strong SBC. This work presents an alternative treatment that enables us to use standard concepts based on weak SBC to tackle with quantum thermodynamics with strong SBC. Specifically, via a physics-motivated mapping between strong and weak SBC, we show that it is possible to identify thermodynamic quantities with arbitrary SBC, including work and heat that shed light on the first law of thermodynamics with strong SBC. Quantum fluctuation theorems, such as the Tasaki-Crooks relation and the Jarzynski equality are also shown to be extendable to strong SBC cases. Our theoretical results are further illustrated with a working example.
Quantum thermodynamic quantities, normally formulated with the assumption of weak system-bath coupling (SBC), can often be contested in physical circumstances with strong SBC. This work presents an alternative treatment that enables us to use standard concepts based on weak SBC to tackle with quantum thermodynamics with strong SBC. Specifically, via a physics-motivated mapping between strong and weak SBC, we show that it is possible to identify thermodynamic quantities with arbitrary SBC, including work and heat that shed light on the first law of thermodynamics with strong SBC. Quantum fluctuation theorems, such as the Tasaki-Crooks relation and the Jarzynski equality are also shown to be extendable to strong SBC cases. Our theoretical results are further illustrated with a working example.
Identifying or constructing a fine-grained microscopic theory that will emerge under specific conditions to a known macroscopic theory is always a formidable challenge. Thermodynamics is perhaps one of the most powerful theories and best understood examples of emergence in physical sciences, which can be used for understanding the characteristics and mechanisms of emergent processes, both in terms of emergent structures and the emergent laws governing the effective or collective variables. Viewing quantum mechanics as an emergent theory requires a better understanding of all this. In this work we aim at a very modest goal, not quantum mechanics as thermodynamics, not yet, but the thermodynamics of quantum systems, or quantum thermodynamics. We will show why even with this minimal demand, there are many new issues which need be addressed and new rules formulated. The thermodynamics of small quantum many-body systems strongly coupled to a heat bath at low temperatures with non-Markovian behavior contains elements, such as quantum coherence, correlations, entanglement and fluctuations, that are not well recognized in traditional thermodynamics, built on large systems vanishingly weakly coupled to a non-dynamical reservoir. For quantum thermodynamics at strong coupling, one needs to reexamine the meaning of the thermodynamic functions, the viability of the thermodynamic relations and the validity of the thermodynamic laws anew. After a brief motivation, this paper starts with a short overview of the quantum formulation based on Gelin & Thoss and Seifert. We then provide a quantum formulation of Jarzynski's two representations. We show how to construct the operator thermodynamic potentials, the expectation values of which provide the familiar thermodynamic variables. Constructing the operator thermodynamic functions and verifying or modifying their relations is a necessary first step in the establishment of a viable thermodynamics theory for quantum systems. We mention noteworthy subtleties for quantum thermodynamics at strong coupling, such as in issues related to energy and entropy, and possible ambiguities of their operator forms. We end by indicating some fruitful pathways for further developments.
The nonequilibrium fluctuation relation is a cornerstone of quantum thermodynamics. It is widely believed that the system-bath heat exchange obeys the famous Jarzynski-Wójcik fluctuation theorem. However, this theorem is established in the Born-Markovian approximation under the weak-coupling condition. Via studying the energy exchange between a harmonic oscillator and its coupled bath in the non-Markovian dynamics, we establish a generalized quantum fluctuation theorem for energy exchange being valid for arbitrary coupling strength. The Jarzynski-Wójcik fluctuation theorem is recovered in the weak-coupling limit. We also find the average energy exchange exhibits rich nonequilibrium characteristics when different numbers of system-bath bound states are formed, which suggests a useful way to control the quantum heat. Deepening our understanding of the fluctuation relation in quantum thermodynamics, our result lays the foundation to design high-efficiency quantum heat engines.
We give a perturbative analysis of Crooks relation and Jarzynski equality in an arbitrary driven quantum system weakly coupled to a heat bath. Invoking no efficient Hamiltonian nor any restriction on the form of the coupling, we derive the first-order correction to Crooks relation and Jarzynski equality in terms of the interacting Hamiltonian. A Crooks type of relation about energy increment in the system and nonvanishing heat expenditure to the heat bath up to the second order of interaction would be given by the way. Our results tell us that deviation of the quantum fluctuation relations is mainly caused by energy trapped interaction rather than the heat flow to (or from) heat bath. We also show a way of defining work in weak coupling regime which is consistent to the work definition in closed systems and thus deepen our understanding of work in quantum realm.
Abstract The treatment of quantum thermodynamic systems beyond weak coupling is of increasing relevance, yet extremely challenging. The evaluation of thermodynamic quantities in strong-coupling regimes requires a nonperturbative knowledge of the bath dynamics, which in turn relies on heavy numerical simulations. To tame these difficulties, considering thermal bosonic baths linearly coupled to the open system, we derive expressions for heat, work, and average system-bath interaction energy that only involve the autocorrelation function of the bath and two-time expectation values of system operators. We then exploit the pseudomode approach, which replaces the physical continuous bosonic bath with a small finite number of damped, possibly interacting, modes, to numerically evaluate these relevant thermodynamic quantities. We show in particular that this method allows for an efficient numerical evaluation of thermodynamic quantities in terms of one-time expectation values of the open system and the pseudomodes. We apply this framework to the investigation of two paradigmatic situations. In the first instance, we study the entropy production for a two-level system (TLS) coupled to an ohmic bath, simulated via interacting pseudomodes, allowing for the presence of time-dependent driving. Secondly, we consider a quantum thermal machine composed of a TLS interacting with two thermal baths at different temperatures, showing that an appropriate sinusoidal modulation of the coupling with the cold bath only is enough to obtain work extraction.
We consider a time-dependent quantum linear oscillator coupled to a bath at an arbitrary strength. We then introduce a generalized Jarzynski equality (GJE) which includes the terms reflecting the system-bath coupling. This enables us to study systematically the coupling effect on the linear oscillator in a non-equilibrium process. This is also associated with the second law of thermodynamics beyond the weak-coupling limit. We next take into consideration the GJE in the classical limit. By this generalization we show that the Jarzynski equality in its original form can be associated with the second law, in both quantal and classical domains, only in the vanishingly small coupling regime.
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.