The Reaction Coordinate Mapping in Quantum Thermodynamics

Abstract

We present an overview of the reaction coordinate approach to handling strong system-reservoir interactions in quantum thermodynamics. This technique is based on incorporating a collective degree of freedom of the reservoir (the reaction coordinate) into an enlarged system Hamiltonian (the supersystem), which is then treated explicitly. The remaining residual reservoir degrees of freedom are traced out in the usual perturbative manner. The resulting description accurately accounts for strong system-reservoir coupling and/or non-Markovian effects over a wide range of parameters, including regimes in which there is a substantial generation of system-reservoir correlations. We discuss applications to both discrete stroke and continuously operating heat engines, as well as perspectives for additional developments. In particular, we find narrow regimes where strong coupling is not detrimental to the performance of continuously operating heat engines.

Appendices

Appendix A: Heisenberg Equations for the Phonon Mapping

The Heisenberg equations of motion for a system observable $A=A^\dagger$ read in the original representation

$$\begin{aligned} \dot{A}&= \mathrm {i}S_1(t) + \mathrm {i}S_2(t) \sum _k \left( h_k a_k + h_k^* a_k^\dagger \right) \,,\qquad S_1(t) = [H_S, A]\,,\qquad S_2(t) = [S, A]\,,\nonumber \\\dot{a}_k&= -\mathrm {i}\omega _k a_k -\mathrm {i}h_k^* S\,,\qquad \dot{a}_k^\dagger = +\mathrm {i}\omega _k a_k^\dagger + \mathrm {i}h_k S\,. \end{aligned}$$

(A1)

We now Fourier-transform these equations according to $\int [\ldots ] e^{+\mathrm {i}z t} dt$ with the convention $\mathfrak {I}z > 0$. In z-space, the creation and annihilation operators are no longer adjoint to each other, but we will keep the $\dagger$-notation. This yields the algebraic equations (convolution theorem)

$$\begin{aligned} \mathrm {i}z A(z)&= \mathrm {i}S_1(z) + \frac{\mathrm {i}}{2\pi } \int S_2(z') \sum _k \left[ h_k a_k(z-z') + h_k^* a_k^\dagger (z-z')\right] dz'\,,\nonumber \\\mathrm {i}z a_k(z)&= -\mathrm {i}\omega _k a_k(z) -\mathrm {i}h_k^* S(z)\,,\qquad \mathrm {i}z a_k^\dagger (z) = +\mathrm {i}\omega _k a_k^\dagger (z) + \mathrm {i}h_k S(z)\,. \end{aligned}$$

(A2)

We can solve the last two equations $a_k(z) = -\frac{h_k^*}{z+\omega _k} S(z)$ and $a_k^\dagger (z) = +\frac{h_k}{z-\omega _k} S(z)$, and insert them into the first

$$\begin{aligned} z A(z)&= S_1(z)+ \frac{1}{2\pi } \int S_2(z') \left[ \sum _k \frac{-{\left| h_k \right| }^2}{z-z'+\omega _k} + \sum _k \frac{+{\left| h_k \right| }^2}{z-z'-\omega _k}\right] S(z-z') dz'\nonumber \\&= S_1(z) + \frac{1}{2\pi } \int S_2(z') \left[ \frac{1}{\pi } \int _0^\infty \frac{\omega \Gamma ^{(0)}(\omega )}{(z-z')^2-\omega ^2} d\omega \right] S(z-z') dz'\nonumber \\&= S_1(z) - \frac{1}{2\pi } \int S_2(z') \frac{1}{2} W^{(0)}(z-z') S(z-z') dz'\,. \end{aligned}$$

(A3)

Here, we have in the first step used the fact that the harmonic oscillator frequencies $\omega _k$ are by construction all positive and we have introduced the Cauchy transform

$$\begin{aligned} W^{(n)}(z) = \frac{2}{\pi } \int _0^\infty \frac{\omega \Gamma ^{(n)}(\omega )}{\omega ^2-z^2} d\omega = \frac{1}{\pi } \int _{-\infty }^{+\infty } \frac{\Gamma ^{(n)}(\omega )}{\omega -z} d\omega \,, \end{aligned}$$

(A4)

where the last equality sign holds for analytic continuation as an odd function $\Gamma (-\omega )=-\Gamma (+\omega )$. In particular, we note the important property

$$\begin{aligned} \Gamma ^{(n)}(\omega ) = \lim _{\epsilon \rightarrow 0^+} \mathfrak {I}W^{(n)}(\omega +\mathrm {i}\epsilon )\,. \end{aligned}$$

(A5)

Similarly, we can derive the Heisenberg equations of motion in the mapped representation, and Fourier-transform them according to the same prescription, yielding

$$\begin{aligned} z A(z)&= S_1(z) + \frac{\lambda }{2\pi } \int S_2(z') \left[ b(z-z') + b^\dagger (z-z')\right] dz'\,,\nonumber \\z b(z)&= -\lambda S(z) - \Omega b(z) - \sum _k \left[ H_k b_k(z) + H_k^* b_k^\dagger (z)\right] \,,\nonumber \\z b^\dagger (z)&= +\lambda S(z) + \Omega b^\dagger (z) + \sum _k \left[ H_k b_k(z) + H_k^* b_k^\dagger (z)\right] \,,\nonumber \\z b_k(z)&= -\Omega _k b_k(z) - H_k^* \left[ b(z) + b^\dagger (z)\right] \,,\qquad z b_k^\dagger (z) = +\Omega _k b_k^\dagger (z) + H_k \left[ b(z) + b^\dagger (z)\right] \,. \end{aligned}$$

(A6)

Again, we follow the approach of successively eliminating the $b_k(z)$, $b_k^\dagger (z)$, and then the b(z), $b^\dagger (z)$ variables, yielding for the remaining equation

$$\begin{aligned} z A(z) = S_1(z) + \frac{1}{2\pi } \int S_2(z') \frac{2 \lambda ^2 \Omega }{(z-z')^2 - \Omega ^2 + \Omega W^{(1)}(z-z')} S(z-z') dz'\,. \end{aligned}$$

(A7)

Comparing this with the original representation, we can infer a relation between $W^{(0)}(z)$ and $W^{(1)}(z)$, which can be used to obtain the transformed spectral density

$$\begin{aligned} \Gamma ^{(1)}(\omega )&= -\lim _{\epsilon \rightarrow 0^+} \mathfrak {I}\frac{4\lambda ^2}{W^{(0)}(\omega +\mathrm {i}\epsilon )} = \frac{+4 \lambda ^2 \Gamma ^{(0)}(\omega )}{\left[ \frac{1}{\pi }\mathcal{P} \int \frac{\Gamma ^{(0)}(\omega ')}{\omega -\omega '} d\omega '\right] ^2 + \left[ \Gamma ^{(0)}(\omega )\right] ^2}\,. \end{aligned}$$

(A8)

Appendix B: Heisenberg Equations for the Particle Mapping

Now, the Heisenberg equations of motion for a system observable $A=A^\dagger$ read in the original representation

$$\begin{aligned} \dot{A}&= \mathrm {i}S_1(t) + \mathrm {i}S_2(t) \sum _k h_k^* a_k^\dagger - \mathrm {i}S_2^\dagger (t) \sum _k h_k a_k\,,\quad S_1(t) = [H_S, A]\,,\quad S_2(t) = [S, A]\,,\nonumber \\\dot{a}_k&= -\mathrm {i}\omega _k a_k -\mathrm {i}h_k^* S\,,\qquad \dot{a}_k^\dagger = +\mathrm {i}\omega _k a_k^\dagger + \mathrm {i}h_k S^\dagger \,. \end{aligned}$$

(B1)

Fourier-transformation yields

$$\begin{aligned} z A(z)&= S_1(z) + \frac{1}{2\pi } \int \left[ S_2(z') \sum _k h_k^* a_k^\dagger (z-z') - S_2^\dagger (z') \sum _k h_k a_k(z-z')\right] dz'\,,\nonumber \\z a_k(z)&= -\omega _k a_k(z) - h_k^* S(z)\,,\qquad z a_k^\dagger (z) = +\omega _k a_k^\dagger (z) + h_k S^\dagger (z)\,. \end{aligned}$$

(B2)

Inserting the solutions of the last two equations into the first we get

$$\begin{aligned} z A(z) = S_1(z) + \frac{1}{2\pi } \int \left[ S_2(z') \sum _k \frac{{\left| h_k \right| }^2}{z-z'-\omega _k} S^\dagger (z-z') + S_2^\dagger (z') \sum _k \frac{{\left| h_k \right| }^2}{z-z'+\omega _k} S(z-z')\right] \,. \end{aligned}$$

(B3)

In the mapped representation, we have

$$\begin{aligned} \dot{A}&= \mathrm {i}S_1(t) + \mathrm {i}\lambda S_2(t) b^\dagger - \mathrm {i}\lambda S_2^\dagger (t) b\,,\nonumber \\\dot{b}&= -\mathrm {i}\lambda S - \mathrm {i}\Omega b - \mathrm {i}\sum _k H_k b_k\,,\qquad \dot{b}^\dagger = +\mathrm {i}\lambda S^\dagger + \mathrm {i}\Omega b^\dagger + \mathrm {i}\sum _k H_k^* b_k^\dagger \,,\nonumber \\\dot{b}_k&= -\mathrm {i}H_k^* b-\mathrm {i}\Omega _k b_k\,,\qquad \dot{b}_k^\dagger = +\mathrm {i}H_k b^\dagger + \mathrm {i}\Omega _k b_k^\dagger \,, \end{aligned}$$

(B4)

such that Fourier transformation yields

$$\begin{aligned} z A(z)&= S_1(z) + \frac{\lambda }{2\pi } \int \left[ S_2(z') b^\dagger (z-z') - S_2^\dagger (z') b(z-z')\right] dz'\,,\nonumber \\z b(z)&= - \lambda S(z) - \Omega b(z) - \sum _k H_k b_k(z)\,,\qquad z b^\dagger (z) = +\lambda S^\dagger (z) + \Omega b^\dagger (z) + \sum _k H_k^* b_k^\dagger (z)\,,\nonumber \\z b_k(z)&= -H_k^* b(z) - \Omega _k b_k(z)\,,\qquad z b_k^\dagger (z) = +H_k b^\dagger (z) + \Omega _k b_k^\dagger (z)\,. \end{aligned}$$

(B5)

Successive elimination of the last four equations yields for the remaining one

$$\begin{aligned} z A(z) =&S_1(z)\\&+ \frac{\lambda }{2\pi } \int \left[ S_2(z') \frac{+\lambda }{z-z'-\Omega -\sum _k \frac{{\left| H_k \right| }^2}{z-z' -\Omega _k}} S^\dagger (z-z')\right. \nonumber \\&+ \left. S_2^\dagger (z') \frac{\lambda }{z-z'+\Omega -\sum _k \frac{{\left| H_k \right| }^2}{z-z' +\Omega _k}} S(z-z')\right] \,.\nonumber \end{aligned}$$

(B6)

From comparison with the first representation, we conclude

$$\begin{aligned} \sum _k \frac{{\left| h_k \right| }^2}{z-\omega _k} = \frac{\lambda ^2}{z-\Omega -\sum _k \frac{{\left| H_k \right| }^2}{z-\Omega _k}}\,,\qquad \sum _k \frac{{\left| h_k \right| }^2}{z+\omega _k} = \frac{\lambda ^2}{z+\Omega -\sum _k \frac{{\left| H_k \right| }^2}{z+\Omega _k}}\,, \end{aligned}$$

(B7)

where the second equation just encodes the first at $-z$ and is therefore not independent.

From realizing that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \frac{1}{2\pi } \int _0^\infty \frac{\Gamma (\omega ')}{\omega -\omega '+\mathrm {i}\epsilon } d\omega ' {\mathop {=}\limits ^{\omega >0}} \frac{1}{2\pi } \mathcal{P} \int _0^\infty \frac{\Gamma (\omega ')}{\omega -\omega '} d\omega ' - \frac{\mathrm {i}}{2} \Gamma (\omega )\,, \end{aligned}$$

(B8)

we can use e.g. the first of these relations to infer a mapping relation between the spectral densities,

$$\begin{aligned} \Gamma ^{(1)}(\omega ) = \frac{4 \lambda ^2 \Gamma ^{(0)}(\omega )}{\left[ \frac{1}{\pi }\mathcal{P}\int _0^\infty \frac{\Gamma ^{(0)}(\omega ')}{\omega -\omega '} d\omega '\right] ^2 + \left[ \Gamma ^{(0)}(\omega )\right] ^2}\,, \end{aligned}$$

(B9)

where $\omega >0$ is assumed throughout.

Appendix C: Heisenberg Equations for Fermionic Reservoirs

To avoid case distinctions on whether the system operator A commutes or anti-commutes with the coupling operator, we just consider the Heisenberg equations for the creation and annihilation operators. In the original representation, they become

$$\begin{aligned} \dot{c} = \mathrm {i}[H_S, c] + \mathrm {i}\sum _k t_k c_k = \mathrm {i}S(t) + \mathrm {i}\sum _k t_k c_k\,,\qquad \dot{c}_k = \mathrm {i}t_k^* c - \mathrm {i}\epsilon _k c_k\,, \end{aligned}$$

(C1)

and similarly for the creation operators. Since at this level they do not mix, we consider only the annihilation operators. Fourier-transformation yields

$$\begin{aligned} z c(z) = S(z) + \sum _k t_k c_k(z)\,,\qquad z c_k(z) = t_k^* c(z) - \epsilon _k c_k(z)\,. \end{aligned}$$

(C2)

Eliminating the second equation then gives

$$\begin{aligned} z c(z) = S_1(z) + \sum _k \frac{{\left| t_k \right| }^2}{z+\epsilon _k} c(z)\,. \end{aligned}$$

(C3)

In contrast, the mapped representation yields

$$\begin{aligned} \dot{c} = \mathrm {i}S(t) + \mathrm {i}\lambda d\,,\qquad \dot{d} = -\mathrm {i}\lambda c - \mathrm {i}\epsilon d + \mathrm {i}\sum _k T_k d_k\,,\qquad \dot{d}_k = \mathrm {i}T_k^* d - \mathrm {i}\epsilon _k d_k\,. \end{aligned}$$

(C4)

Fourier-transforming and eliminating the non-system variables then gives

$$\begin{aligned} z c(z) = S(z) - \frac{\lambda ^2}{z+\epsilon -\sum _k \frac{{\left| T_k \right| }^2}{z+\epsilon _k}} c(z)\,, \end{aligned}$$

(C5)

and from comparison we get the relation

$$\begin{aligned} \sum _k \frac{{\left| t_k \right| }^2}{z+\epsilon _k} = - \frac{\lambda ^2}{z+\epsilon -\sum _k \frac{{\left| T_k \right| }^2}{z+\epsilon _k}}\,. \end{aligned}$$

(C6)

Converting the sums to integrals and evaluating at $z=-\omega +\mathrm {i}\delta$ when $\delta \rightarrow 0^+$ we obtain a mapping relation between the fermionic spectral densities:

$$\begin{aligned} \Gamma ^{(1)}(\omega ) = \frac{4 \lambda ^2 \Gamma ^{(0)}(\omega )}{\left[ \frac{1}{\pi } \mathcal{P}\int \frac{\Gamma ^{(0)}(\omega ')}{\omega -\omega '} d\omega '\right] ^2 + \left[ \Gamma ^{(0)}(\omega )\right] ^2}\,. \end{aligned}$$

(C7)