Quantum Thermodynamics: A Nonequilibrium Green’s Function Approach

We establish the foundations of a nonequilibrium theory of quantum thermodynamics for noninteracting open quantum systems strongly coupled to their reservoirs within the framework of the nonequilibrium Green’s functions. The energy of the system and its coupling to the reservoirs are controlled by a slow external time-dependent force treated to first order beyond the quasistatic limit. We derive the four basic laws of thermodynamics and characterize reversible transformations. Stochastic thermodynamics is recovered in the weak coupling limit.

Figure 1

Sketch of a fermionic single quantum level junction. The level is broadened by the strong coupling to the reservoirs and is driven by a time-dependent force.

Figure 2

Heat flux [Eq. (14)] and entropy production [Eq. (19)], for the quantum level in contact with a single reservoir at $T=300\mathrm{K}$. The external force drives the level energy as $ϵ(t)={ϵ}_0+\mathrm{\Delta }(1-\cos {\omega }_{0}t)/2$ from ${ϵ}_0$ at $t=0$ to ${ϵ}_0+\mathrm{\Delta }$ at $t=\pi /{\omega }_0$, where ${ϵ}_0=-0.02\mathrm{eV}$ and $\mathrm{\Delta }=0.02\mathrm{eV}$. Entropy production (solid line, red) and heat flux (dashed line, blue) are depicted in (a) as functions of time for $\mathrm{\Gamma }=0.01\mathrm{eV}$ and ${\omega }_0=0.01\mathrm{eV}$. The ratio of their time-integrated values is depicted in (b) as a function of the driving rate ${\omega }_0$.

Figure 3

Entropy production for the quantum level at steady state between two reservoirs $\nu =L,R$ (with $T_L=300\mathrm{K}$, $T_R=10\mathrm{K}$, ${\mu }_L=-0.05\mathrm{eV}$, ${\mu }_R=E_F=0$) as a function of: (a) the interaction strength to the reservoirs $\mathrm{\Gamma }=2{\mathrm{\Gamma }}_L=2{\mathrm{\Gamma }}_R$, (b) the position of the level $ϵ$. The strong coupling entropy production [Eq. (19)] is depicted for $ϵ=-0.05\mathrm{eV}$ (dotted line, blue) and $ϵ=0.05\mathrm{eV}$ (solid line, blue), and its weak coupling counterpart for $ϵ=-0.05\mathrm{eV}$ (dash-dotted line, red) and $ϵ=0.05\mathrm{eV}$ (dashed line, red). The energy grid used spans from $-3$ to 3 eV with step ${10}^{-6}\mathrm{eV}$.