Hierarchical Equations of Motion Approach to Quantum Thermodynamics

Abstract

We present a theoretical framework to investigate quantum thermodynamic processes under non-Markovian system-bath interactions on the basis of the hierarchical equations of motion (HEOM) approach, which is convenient to carry out numerically “exact” calculations. This formalism is valuable because it can be used to treat not only strong system-bath coupling but also system-bath correlation or entanglement, which will be essential to characterize the heat transport between the system and quantum heat baths. Using this formalism, we demonstrated an importance of the thermodynamic effect from the bath-system-bath tri-partite correlations (TPC) for a two-level heat transfer model and a three-level autonomous heat engine model under the conditions that the conventional quantum master equation approaches are failed. Our numerical calculations show that TPC contributions, which distinguish the heat current from the energy current, have to be take into account to satisfy the thermodynamic laws.

Appendices

Appendix A: Derivation of Equation (24.6)

The heat current is defined as the rate of decrease of the bath energy, $\dot{Q}_{\mathrm {HC},k} (t) = - d\langle { \hat{H}_\mathrm {bath}^{(k)} } (t)) \rangle /dt$, which can be rewritten by using the Heisenberg equations for ${ \hat{H}_\mathrm {int}^{(k)} }$ and ${ \hat{H}_\mathrm {bath}^{(k)} }$ as

$$\begin{aligned} \dot{Q}_{\mathrm {HC},k}(t)&= { \frac{i}{\hbar } } \left\langle \left[ { \hat{H}_\mathrm {bath}^{(k)} } (t), { \hat{H}_\mathrm {int}^{(k)} } (t) \right] \right\rangle \nonumber \\&= { \frac{i}{\hbar } } \left\langle \left[ { \hat{H}_\mathrm {int}^{(k)} } (t), { \hat{H}_\mathrm {sys} } (t) \right] \right\rangle + { \frac{d}{dt} } \left\langle { \hat{H}_\mathrm {int}^{(k)} } (t) \right\rangle + \sum _{k^\prime } { \frac{i}{\hbar } } \left\langle \left[ { \hat{H}_\mathrm {int}^{(k)} } (t), { \hat{H}_\mathrm {int}^{(k^\prime )} } (t) \right] \right\rangle . \end{aligned}$$

(24.16)

The first term of r.h.s. of Eq. (24.16) is related to the energy flow to the kth bath via the energy conservation equation for ${ \hat{H}_\mathrm {sys} }$ as ${ \frac{d}{dt} } \langle { \hat{H}_\mathrm {sys} } (t) \rangle = \dot{W}(t) + \sum _k \dot{Q}_{\mathrm {SEC},k}(t)$, where $\dot{Q}_{\mathrm {SEC},k}(t) = (i/\hbar ) \langle [ { \hat{H}_\mathrm {int}^{(k)} } (t), { \hat{H}_\mathrm {sys} } (t) ] \rangle$. Therefore, by using the above definition for $\dot{Q}_{\mathrm {SEC},k}(t)$ and Eqs. (24.8), (24.6) is derived.

Appendix B: Derivation of Equation (24.14)

To derive Eq. (24.14), we adapt a generating functional approach by adding the source term, $f_k(t)$, for the kth interaction Hamiltonian as

$$\begin{aligned} \hat{V}_k \hat{X}_k \rightarrow \hat{V}_{k,f}(t) \hat{X}_k \equiv \left( \hat{V}_k + f_k(t) \right) \hat{X}_k \end{aligned}$$

(24.17)

Here, in order to evaluate an expectation value, we add the source term to the ket (left) side of the density operator, which does not change a role of the system-bath interaction in the time-evolution operator. This source term enables us to have a collective bath coordinate with the functional derivative as

$$\begin{aligned} \tilde{X}_k(t) \tilde{\rho }_\mathrm {tot}(t) = i \hbar \frac{\delta }{\delta f_k(t)} \left. \tilde{\rho }_{\mathrm {tot},f}(t) \right| _{f \equiv 0}. \end{aligned}$$

(24.18)

Then, the expectation value of the operator $\hat{Z}_k \equiv \hat{A} \hat{X}_k$ for any system operator $\hat{A}$ reads

$$\begin{aligned} \left\langle \hat{Z}_k(t) \right\rangle&= \mathrm {Tr}_\mathrm {sys} \left[ \tilde{A}(t) i\hbar \left. \frac{\delta }{\delta f_k(t)} \tilde{\rho }_f(t) \right| _{f\equiv 0} \right] \nonumber \\&= \frac{2}{\hbar } \int _0^t d\tau \, \mathrm {Im} \left[ C_k(t-\tau ) \left\langle \hat{A}(t) \hat{V}_k(\tau ) \right\rangle \right] . \end{aligned}$$

(24.19)

Next, the kth HC, Eq. (24.6), is rewritten by using the Heisenberg equation for $\hat{V}_k$ as $\dot{Q}_k(t) = { \frac{d}{dt} } \langle \hat{H}_\mathrm {int}^{(k)}(t) \rangle - \langle ( { \frac{d}{dt} } \hat{V}_k(t) ) \hat{X}_k(t) \rangle$. The time derivatives, ${ \frac{d}{dt} } \langle \hat{H}_\mathrm {int}^{(k)}(t) \rangle$ and $\langle ( { \frac{d}{dt} } \hat{V}_k(t) ) \hat{X}_k(t) \rangle$, are given by the time differentiation of Eq. (24.19) for $\hat{A} = \hat{V}_k$ and Eq. (24.19) for $\hat{A} = { \frac{d}{dt} } \hat{V}_k$, respectively. This immediately leads to the expression for the kth HC in Eq. (24.14).