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A system–bath (SB) model is considered to examine the Jarzynski equality in the fully quantum regime. In our previous paper [J. Chem. Phys. **153**, 234107 (2020)], we carried out “exact” numerical experiments using hierarchical equations of motion (HEOM) in which we demonstrated that the SB model describes behavior that is consistent with the first and second laws of thermodynamics and that the dynamics of the total system are time irreversible. The distinctive quantity in the Jarzynski equality is the “work characteristic function (WCF)”, 〈exp[−*βW*]〉, where *W* is the work performed on the system and *β* is the inverse temperature. In the present investigation, we consider the definitions based on the partition function (PF) and on the path, and numerically evaluate the WCF using the HEOM to determine a method for extending the Jarzynski equality to the fully quantum regime. We show that using the PF-based definition of the WCF, we obtain a result that is entirely inconsistent with the Jarzynski equality, while if we use the path-based definition, we obtain a result that approximates the Jarzynski equality, but may not be consistent with it.

In thermodynamics, work, *W*, and heat, ^{1}^{–}^{14}^{)} *τ*. Although investigating this equality in the classical regime is straightforward in both theoretical and experimental contexts, doing so in the quantum regime remains challenging,^{15}^{–}^{23}^{)} because the dynamics of a small quantum system itself are reversible in time and therefore the system cannot reach thermal equilibrium on its own without a system–bath (SB) interaction: We cannot assume a canonical distribution as the equilibrium state for the system itslef and the heat-bath, due to the presence of the SB interaction. In the present paper, we numerically evaluate the work characteristic function (WCF),

A commonly employed model for this kind of investigation is described by a SB Hamiltonian, in which a small quantum system *A* is coupled to a bath *B* modeled by an infinite number of harmonic oscillators. We found that the behavior described by this model is consistent with the first and second laws of thermodynamics and provides an ideal platform to examine various fundamental propositions of thermodynamics in the fully quantum regime.^{24}^{,}^{25}^{)} In this model, the Hamiltonian of the total system is given by *j*th bath oscillator, respectively. The SB interaction *j*th bath oscillator. The effect of the bath is characterized by the noise correlation function, *β* is the inverse temperature of the bath.

When we apply the SB model to problems of thermodynamics, because the main system is microscopic and because the quantum coherence between the system and bath characterizes the quantum nature of the system dynamics, the role of the SB interaction has to be examined carefully. For example, although the factorized thermal equilibrium state, ^{26}^{,}^{27}^{)}

In Refs. 24 and 25, we presented a scheme for calculating thermodynamic variables in the SB model on the basis of simulations including an external perturbation using the hierarchical equations of motion (HEOM).^{26}^{–}^{32}^{)} The key quantity in this investigation is the change of the “quasi-static Helmholtz energy” at time *τ*, which is defined as^{25}^{)} ^{25}^{)} Moreover, we have numerically confirmed that the total entropy production is alway positive. With these results strongly supporting the validity of our approach, in this paper, we evaluate

In what follows, we examine two definitions of the WCF: (i) a definition based on the partition function (PF) (the PF-WCF)^{33}^{)} and (ii) a definition based on trajectory (path) (the path-WCF). For an isolated quantum system, the PF-WCF has been defined as^{34}^{)} ^{25}^{)}

Alternatively, we can use the path-WCF on the basis of the non-equilibrium trajectories (paths) as ^{34}^{)} we use the symmetric form in Eq. (7^{,}), because otherwise ^{,}): This violates the condition to satisfy the Jarzynski equality. Nevertheless, we use Eq. (7), because it is natural to assume that the measurement of the work cannot be carried out without disturbing the dynamics of the main system, because it is regarded as small.

For an open quantum system, we can derive the HEOM for Eq. (9^{,}) using the same procedure as that used to obtain the HEOM for Eq. (1^{,}),^{27}^{–}^{32}^{)} because the only difference between the quantum Liouville equation and Eq. (8^{,}) is the presence of the work operator *η* is the SB coupling strength, and *γ* is the inverse correlation time of the bath-induced noise. Then, the noise correlation function takes the form of a linear combination of exponential functions and a delta function: ^{,}) is expressed as *k*th direction. Each hierarchical density matrix is specified by the index ^{,}) with the fixed Hamiltonian

We now report the results of our numerical computations of the PF-WCF, ^{,}), and the path-WCF, ^{,}) [or Eq. (9^{,})], under the periodic external force described by ^{,}), the procedures for computing ^{24}^{,}^{25}^{,}^{35}^{)} Throughout this investigation, we fix

In Fig. 1(a), we display the time dependences of the PF-WCF, path-WCF, and

Figure 1. (Color online) The quantity

While the time evolution of the PF-WCF differs significantly from ^{,}) is taken after the time integration of

In Figs. 1(b) and 1(c), we present the results for the intermediate and strong SB coupling cases. As pointed out previously,^{35}^{,}^{36}^{)} the efficiency of the heat current is suppressed when the SB coupling becomes strong. Because the effective SB coupling depends on the characteristic time scale of the system, the increase of the PF-WCF is suppressed in the case ^{37}^{)} In Fig. 1(b), due to the effect of the moderately strong SB coupling, the system closely follows its instantaneous equilibrium state, as the entropy production,

For the path-WCF results represented by the solid curves, the deviation becomes larger as the strength of the SB coupling increases. This is because the calculated ^{25}^{)} whereas

In the weak coupling case considered in Fig. 1(a), the difference between the path-WCF results and the free energy results is large in the resonant excitation case,

In this paper, we demonstrated a method for extending the Jarzynski equality to the fully quantum regime. We evaluated the WCF defined in two ways, the PF-WCF and path-WCF, using the numerally rigorous HEOM formalism. Although the path-WCF agrees with the free energy reasonably well, in particular in a weak SB coupling case or the fast excitation cases, while the PF-WCF exhibits very different time-dependence due to the heat production, the result is not equality but approximation. This discrepancy arises from the contribution of the SB interaction, which should also play a role in the classical case if the SB coupling strength is comparable to the system energy. Indeed, if we employ quantum hierarchical Fokker–Planck equations (QHFPEs) for a system described by Wigner distribution functions, we can investigate not only the quantum case but also the classical case by taking the classical limit: We can easily identify purely quantum mechanical effects by comparing the classical and quantum results for the Wigner distribution.^{37}^{,}^{38}^{)}

It should be mention that, although here we introduced the path-WCF, this is not physical observable,^{21}^{,}^{34}^{)} as seen from Eq. (8^{,}). Moreover, we cannot determined the paths in the functional formalism of Eq. (9^{,}), due to the limitation introduced by the uncertainty principle. Thus, in order to evaluate the free energy in the fully quantum regime, the path-WCF is not practical. Instead, Eq. (4) should be used to evaluate the free energy.

Although the present investigation is limited to spin-Boson systems for the specific definitions of the WCF, the applicability of our approach based on the HEOM formalism is in fact more general. Indeed, the same approach can be applied to all of the systems to which the HEOM formalism has been previously applied.^{26}^{,}^{27}^{)} Different definitions of the WCF should also be examined. We leave such extensions to future studies to be carried out in the context of the fluctuation theorem.

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