Quantum Thermodynamics: Inside-Outside Perspective

Introduction. Construction of quantum molecular devices became a reality due to the advancement of experimental techniques at nanoscale Jezouin et al. (2013); Pekola (2015); Hartman et al. (2018); Klatzow et al. (2019). This development poses a challenge to theory making thermodynamic formulation at nanoscale important from both fundamental and applicational perspectives. Indeed, besides purely academic interest, such formulation is at the heart of any efficiency estimate of thermoelectric nano-devices Reddy et al. (2007); Lee et al. (2013); Kim et al. (2014); Zotti et al. (2014); Cui et al. (2017a, b, 2018, 2019). Existing theories rely on figure of merit concept which stems from classical macroscopic formulation. It is restricted to equilibrium considerations (linear response) and completely disregards quantum fluctuations.

Significant progress was achieved in theoretical formulations of quantum thermodynamics for systems weakly coupled to their baths. Analogs of traditional thermodynamics which focuses on average system characteristics as well as considerations of quantum fluctuations (thermodynamic fluctuation theorems) are available in the literature Esposito et al. (2009, 2014).

In nanoscale devices, molecule usually forms a covalent bond with contacts on at least one of its interfaces which results in hybridization of molecular states with those of the contacts. This appears as system-bath interaction of a strength comparable to energy of the isolated system. Therefore, thermodynamic formulation of systems strongly coupled to their baths (i.e. situation where energy of system-bath interaction cannot be disregarded) becomes a practical necessity. Contrary to weakly coupled situation, thermodynamic formulations for strongly coupled systems are still at their infancy with most of discussion focused on formulations involving system averages. Only a few publications on steady-state regime considered quantum fluctuations in such systems Esposito et al. (2015a), while fluctuation theorems formulated so far were shown to be violated in strongly coupled systems Brandner et al. (2018).

Here, we focus on thermodynamic theory of averages for systems strongly coupled to their baths keeping in mind that for future attempts of stochastic thermodynamic formulation for strongly coupled systems one of the guiding principles of the formulation should be consistency between thermodynamic and microscopic dynamical descriptions Kosloff (2013). As far as we know, only one consistent thermodynamic formulation is available in the literature today. It postulates the von Neumann entropy expression for reduced density matrix of the system as proper thermodynamics entropy. The approach was originally proposed in Refs. Lindblad, 1983; Peres, 2002; Esposito et al., 2010 and later used in a number of studies Sagawa (2012); Kato and Tanimura (2016); Strasberg et al. (2017); Seshadri and Galperin (2021). The formulation guarantees that in a thermodynamic process which starts from decoupled system and baths entropy production is positive (i.e. integrated form of the second law of thermodynamics is satisfied) while entropy production rate is non-monotonic (i.e. differential form of the second law is not guaranteed) Esposito et al. (2010). When a thermodynamic process does not start from the decoupled state, the second law is not guaranteed in any form.

We argued Seshadri and Galperin (2021) that the deficiency of this von Neumann formulation is due to its neglect of energy resolution in system entropy: the von Neumann expression operates with reduced density matrix, which is a time-local (integrated in energy) object. Here, we introduce a general energy-resolved thermodynamic formulation for systems strongly coupled to their baths which is consistent with underlying microscopic dynamics. This is done by employing nonequilibrium Green’s function method in reformulating the inside-outside approach in Ref. Bruch et al., 2018 originally developed for noninteracting systems under adiabatic driving. We extend the theory to interacting systems under arbitrary driving. The resulting formulation is capable of overcoming limitations of the von Neumann formulation in Ref. Esposito et al., 2010.

Model. First, we introduce a generic model of open quantum system and mention several concepts from its microscopic dynamics (quantum transport) description which will be necessary for thermodynamic formulation.

We consider a system $S$ strongly coupled to a number of baths $\{B\}$ and subjected to arbitrary external driving applied to the system and system-baths couplings. Hamiltonian of the model is

where Hamiltonian of the system $\hat{H}_{S}(t)$ contains any intra-system interactions, and

describe bath $B$ and its coupling to the system. Here, $\hat{d}_{m}^{\dagger}$ ($\hat{d}_{m}$) and $\hat{c}_{k\alpha}^{\dagger}$ ($\hat{c}_{k\alpha}$) creates (annihilates) an electron in orbital $m$ of the system $S$ and state $k$ in channel $\alpha$ of the bath $B$, respectively.

For description of the system microscopic dynamics we employ the nonequilibrium Green’s function (NEGF) method Haug and Jauho (2008); Stefanucci and van Leeuwen (2013), which is routinely employed in quantum transport formulations for open nanoscale systems. Thermodynamic formulation below requires several basic concepts of the quantum transport theory. In particular, we utilize the single-particle Green’s function defined on the Keldysh contour,

(here and below $e=k_{B}=\hbar=1$), and expressions for particle $I_{B}^{N}(t)$ and energy $I_{B}^{E}(t)$ fluxes at the interface with bath $B$ Jauho et al. (1994); Galperin et al. (2007)

Here,

is the unitless energy-resolved particle flux in channel $\alpha$, $T_{c}$ is the contour ordering operator, $f_{\alpha}(E)$ is the Fermi-Dirac distribution in channel $\alpha$, $<$ and $r$ are the lesser and retarded projections of the system Green’s function (3), $\rho_{\alpha}(E)$ is the density of states in channel $\alpha$, and $V_{m,k\alpha}(t)=V_{m,\alpha}(\varepsilon_{k\alpha},t)$. Finally, we use expression for heat flux $\dot{Q}_{B}(t)$ from the quantum transport theory Esposito et al. (2015a)

With these definitions we are ready to discuss thermodynamic formulation for systems strongly coupled to their baths.

Quantum thermodynamics. For systems strongly coupled to their baths the main problem is formulation of the second law of thermodynamics in terms of quantities used in dynamical description (fluxes and populations of states). Differential form of the second law is

where $S(t)$ is system entropy, $\dot{Q}_{B}(t)$ is heat flux at system interface with bath $B$, and $\dot{S}_{i}(t)$ is entropy production. In this expression only heat flux is clearly defined by the microscopic theory, Eq.(6).

To define entropy and entropy production we employ the following observations from the quantum transport theory:

1. 1.

   Overall (system plus baths) dynamics is unitary. Thus, entropy of the universe given by the von Neumann expression for the total density operator $\hat{\rho}(t)$, $S_{tot}(t)\equiv-\mbox{Tr}\left[\hat{\rho}(t)\,\ln\hat{\rho}(t)\right]$, does not change during evolution: $\frac{d}{dt}S_{tot}(t)=0$.

2. 2.

   Particle fluxes from baths into system are thermal. Non-thermal fluxes from system into baths do not return into the system. Indeed, within the NEGF dynamics of the system is governed by the Dyson equation - equation-of-motion for the Green’s function (3). This dynamical law is exact. Effect of the baths enters the Dyson equation through corresponding self-energies which only contain information on thermal distribution in the baths.

3. 3.

   Thermalization processes take place far away from the system. They do not affect the system dynamics.

Below we use these observation to develop thermodynamic formulation for systems strongly coupled to their baths.

The first observation allows to define system entropy $S(t)$ from total baths entropy $S_{B,tot}(t)$ as ¹¹1We note that Eq.(8) is not a manifestation of additivity of system and baths contributions to the total entropy (due to entanglement the contributions are clearly not additive); rather it is definition of the system entropy. That is, one considers evolution of the total universe (system plus baths) and separately evolution of the baths alone; difference between results of the two evolutions is defined as a quantity characterizing system entropy. Please note that such way of defining properties of systems strongly coupled to baths goes back to works by Kirkwood Kirkwood (2004) and was used in many previous thermodynamic considerations Talkner and Hänggi (2020); Bruch et al. (2016); Ochoa et al. (2016); Bruch et al. (2018).

Defining system quantity from baths characteristics is at the heart of the inside-outside approach first introduced in Ref. Bruch et al., 2018. Original (Landauer-Buttiker) formulation of Ref. Bruch et al., 2018 is restricted to noninteracting systems under adiabatic driving. Here, we generalize it to interacting systems and to arbitrary driving, thus providing a general thermodynamic formulation applicable in any transport regime and for any system. Technically, the generalization requires employing the NEGF method in place of single-particle scattering theory, utilizing independent current-carrying states in the baths in place of scattering states, and accounting for correlations between states of different energies in addition to inter-channel correlations.

The second observation allows to separate total particle flux into incoming thermal, $\phi^{in}(E)$, and outgoing non-thermal, $\phi^{out}(t,E)$, contributions

Here, $i_{\alpha}(t,E)$ is energy-resolved particle flux defined in Eq.(5), $\phi^{in}_{\alpha}(E)$ is thermal population of incoming state in channel $\alpha$ of the baths, and $\phi^{out}_{\alpha}(t,E)$ is non-thermal population of outgoing state in channel $\alpha$ of the baths. To account for system-induced coherences between states of different energies and different channels in the baths, in addition to populations one has to consider also coherences in the energy and channel spaces: $\phi^{out}_{\alpha\beta}(t;E_{\alpha},E_{\beta})$. Their explicit form can be obtained from equation-of-motion for the baths density matrix (see Appendix A for derivation)

Here, $\rho_{\alpha}(E_{\alpha})$ and $\rho_{\beta}(E_{\beta})$ are densities of states of channels $\alpha$ and $\beta$ of the baths, respectively. While our approach is general, at steady-state Eq.(10) reduces to the Landauer-Buttiker scattering theory result.

Having expressions for populations and coherences, Eq.(10), one can define rate of entropy change in the baths as difference between entropy flux outgoing from the system and entropy flux incoming to the system

Here, $S_{B,tot}$ is the total entropy of all the baths, $\mbox{Tr}_{c,E}\{\ldots\}$ is trace over channels and energies in the baths, and

is the von Neumann expression for entropy in the baths.

The third observation indicates that neither thermalization process affects physics of the system, nor system participates in the process. Thus, entropy production, which takes place during thermalization, can be modeled as a process of reducing non-thermal distribution over bath states $\phi^{out}$ to thermal distribution $\phi^{in}$. Following Ref. Vedral, 2002 entropy production can be rationalized as information erasure due to measurement of non-thermal baths by set of thermal super-baths weakly coupled to the baths. The process leads to entropy production in the universe which consists of entropy change in the baths, $\mbox{Tr}_{c,E}\left\{\sigma\left[\phi^{in}\right]-\sigma\left[\phi^{out}(t)\right]\right\}$, and heat flux into the super-baths, $-\sum_{B}\beta_{B}\dot{Q}_{B}(t)$. Adding the two contributions leads to expression for entropy production

Note that contrary to formulation of Ref. Esposito et al., 2010 entropy production rate in the inside-outside approach is always positive Sagawa (2012); Esposito et al. (2015b).

Finally, using (11) and (13) in (8) leads to the differential form of the second law of thermodynamics for entropy of the system $S(t)$, Eq.(7) (see Appendix B for derivation).

This completes thermodynamic formulation for system strongly coupled to its baths. Note that the introduced formulation readily yields access to energy-resolved version of the second law as discussed in Ref Seshadri and Galperin, 2021.

Numerical results. We now illustrate the inside-outside approach for the resonant level model of phonon assisted tunneling where a single molecular level $\varepsilon_{0}$ represents a junction connecting two electron reservoirs ($L$ and $R$) while being also coupled to a single harmonic mode $\omega_{0}$. The mode represents molecular vibration and is coupled to a thermal (phonon) bath ($P$). Hamiltonian of the system is given in Eq.(1) with

and $B\in\{L,R,P\}$ where

Driving is performed in the position of the level, $\varepsilon_{0}(t)$, and in system-fermionic baths coupling strengths, $V_{k}(t)\equiv u(t)V_{k}$. To simplify simulations we assume the wide-band approximation (WBA) which allows to reduce the general expressions (10) to energy diagonal form (see Appendix C). Also, incorporation of the bosonic bath $P$ requires slight generalization of the thermodynamic formulation (see Appendix D). Electron-phonon interaction $M$ is treated within the self-consistent Born approximation (SCBA) Park and Galperin (2011).

Parameters of the simulation are given in terms of arbitrary unit of energy $E_{0}$ and corresponding unit of time $t_{0}=2\pi/E_{0}$. Unless stated otherwise, the parameters are as follows. Vibrational frequency $\omega_{0}=0.05$, electron-phonon interaction $M=0.01$, temperature $T=0.05$, electron escape rate $\Gamma_{0}\equiv 2\pi\sum_{k\in K}\lvert V_{k}\rvert^{2}\delta(E-\varepsilon_{k})=0.1$, and energy dissipation rate $\gamma(\omega)\equiv 2\pi\sum_{p\in P}\lvert U_{\alpha}\rvert^{2}\delta(\omega-\omega_{\alpha})=\theta(\omega)\,\gamma_{0}\,\frac{\omega^{2}}{\omega_{0}^{2}}exp(1-\omega/\omega_{0})$, where $\gamma_{0}=0.1$. The junction is not biased, the Fermi energy is taken as origin, $E_{F}=0$, and the temperatures in the fermionic and bosonic baths are assumed to be the same. Simulations are performed on energy grid with $4001$ points spanning a range from $-2$ to $2$ with step size $10^{-3}$. FFTW fast Fourier transform library Frigo and Johnson (2005) was employed in the simulations (see Appendix E for details).

Figure 2 shows entropy production rate for the resonant level coupled to a reservoir and driven from $0.2$ to $-0.2$ position with the constant rate $\dot{\varepsilon_{0}}=1.6\times 10^{-3}\,E_{0}/t_{0}$. The simulation is performed in the absence of electron-phonon coupling, $M=0$. We compare our results with the von Neumann approach in Refs. Lindblad, 1983; Peres, 2002; Esposito et al., 2010. One can see that the lack of energy resolution in the von Neumann approach results in appearance of negative entropy production rate. As a result, integrating over part of the thermodynamic process may yield negative entropy production which contradicts the second law of thermodynamics. Note that our inside-outside approach yields positive entropy production for an arbitrary initial state.

We now consider the Carnot cycle in the resonant junction model (see Fig. 1 for a sketch). In step 1 (isothermal part of the cycle with a constant coupling strength to the hot reservoir) the resonant level is driven from $0.2$ to $0.1$ with a variety of driving rates. For simplicity, step 2 (decoupling from the hot and subsequent coupling to the cold reservoir) is performed adiabatically slow. This allows to find an analytical connection between rates $\dot{u}$ and $\dot{\varepsilon}_{0}$ from the requirement that $\dot{Q}=0$ during the process (see Appendix F for details). Also, this guarantees zero entropy production during the step. Step 3 is performed under the same rate $\dot{\varepsilon}_{0}$ as step 1. Finally, step 4 is again performed adiabatically slow. Temperature of the hot reservoir is $T_{H}=0.1$, and the cold reservoir has temperature $T_{C}=0.05$. Figure 3 shows results for the Carnot cycle vs. resonant level driving rate with (filled markers) and without (empty markers) electron-phonon interaction. Panel (a) shows entropy productions during the isothermal parts of the cycle. As anticipated, entropy production grows with driving rate, and the entropy production for step 3 is higher than that of step 1. Also, electron-phonon interaction increases entropy production due to the presence of additional (bosonic) bath. Panel (b) shows that the efficiency of the cycle is expectedly lower for faster driving and in the presence of interaction. Note that its dependence on the driving rate is non-monotonic which is expected for the employed model (see Ref. Migliore et al., 2012 for discussion on the relevance of the model to phonon assisted electron transport in junctions). Indeed, within the model, strength of electron-phonon interaction depends on the level population which at intermediate rates drops fast during step 3 of the cycle thus limiting the amount of heat which can be transferred to the cold bosonic bath. At even higher rates, the amount of heat from the hot bosonic baths also becomes affected which eventually leads to decreasing efficiency.

Conclusion. We established a connection between the inside-outside approach originally suggested in Ref. Bruch et al., 2018 and the nonequilibrum Green’s function formulation which allowed us to extend the method to interacting systems with arbitrary drivings in the system and system-baths couplings. This generalized thermodynamic formulation applies to strong system-bath coupling and is consistent with the underlying microscopic dynamics (quantum transport). It overcomes limitations of the only other consistent thermodynamic formulation Esposito et al. (2010) by satisfying any form of the second law of thermodynamics for any initial state of a thermodynamic process and for any driving protocol.