Colloquium: Quantum fluctuation relations: Foundations and applications

Two fundamental ingredients play a decisive role in the foundation of fluctuation relations: the principle of microreversibility and the fact that thermal equilibrium is described by the Gibbs canonical ensemble. Building on these two pillars the reader is guided through a self-contained exposition of the theory and applications of quantum fluctuation relations. These are exact results that constitute the fulcrum of the recent development of nonequilibrium thermodynamics beyond the linear response regime. The material is organized in a way that emphasizes the historical connection between quantum fluctuation relations and (non)linear response theory. A number of fundamental issues are clarified which were not completely settled in the prior literature. The main focus is on (i) work fluctuation relations for transiently driven closed or open quantum systems, and (ii) on fluctuation relations for heat and matter exchange in quantum transport settings. Recently performed and proposed experimental applications are presented and discussed.

Figure 1

Example of statistics obeying the fluctuation relation, Eq. (1). Left panel: Probability distribution $p(q)$ of number $q$ of electrons, transported through a nanojunction subject to an electrical potential difference. Right panel: The linear behavior of $\ln [p(q)/p(-q)]$ evidences that $p(q)$ obeys the fluctuation relation, Eq. (1). In this example forward and backward protocols coincide yielding $p_B=p_F\equiv p$, and consequently $b=0$ in Eq. (1). From 126.

Figure 2

Microreversibility for nonautonomous classical (Hamiltonian) systems. The initial condition ${\mathbf{z}}_0$ evolves, under the protocol $\lambda$, from time $t=0$ until time $t$ to ${\mathbf{z}}_t={\phi }_{t,0}[{\mathbf{z}}_0;\lambda ]$ and until time $t=\tau$ to ${\mathbf{z}}_{\tau }$. The time-reversed final condition $\epsilon {\mathbf{z}}_{\tau }$ evolves, under the protocol ${\epsilon }_Q\tilde{\lambda }$ from time $t=0$ until $\tau -t$ to ${\phi }_{\tau -t,0}[\epsilon {\mathbf{z}}_{\tau };{\epsilon }_Q\tilde{\lambda }]=\epsilon {\phi }_{t,0}[{\mathbf{z}}_0;\lambda ]$, Eq. (10), and until time $t=\tau$ to the time-reversed initial condition $\epsilon {\mathbf{z}}_0$.

Figure 3

Microreversibility for nonautonomous quantum systems. The normalized initial condition $|i⟩$ evolves, under the unitary time evolution generated by $\mathcal{H}({\lambda }_t)$, from time $t=0$ until $t$ to $|{\psi }_t⟩=U_{t,0}[\lambda ]|i⟩$ and until time $t=\tau$ to $|f⟩$. The time-reversed final condition $\Theta |f⟩$ evolves, under the unitary evolution generated by $\tilde{\lambda }$ from time $t=0$ until $\tau -t$ to $U_{\tau -t,0}[\tilde{\lambda }]\Theta |f⟩=\Theta |{\psi }_t⟩$, Eq. (42), and until time $t=\tau$ to the time-reversed initial condition $\Theta |i⟩$. The motion occurs on the hypersphere of unitary radius in the Hilbert space.

Figure 4

Driven open system. A driven system [represented by its Hamiltonian ${\mathcal{H}}_S({\lambda }_t)$] is coupled to a bath (represented by ${\mathcal{H}}_B$) via the interaction Hamiltonian ${\mathcal{H}}_{SB}$. The compound system is in vanishingly weak contact with a superbath that provides the initial canonical state at inverse temperature $\beta$, Eq. (60).

Figure 5

Exchange fluctuation relation setup. Several reservoirs (large semicircles) interact via the coupling ${\mathcal{V}}_t$ (symbolized by the small circle), which is switched on at time $t=0$ and switched off at $t=\tau$. During the on period (0, $\tau$) the reservoirs exchange energy and matter with each other. The resulting net energy change of the $i$th reservoir is $\Delta E_i$ and its particle content changes by $\Delta N_i$. Initially the reservoirs prescribed temperatures $T_i=(k_B{\beta }_i{)}^{-1}$ , and chemical potentials ${\mu }_i$ of the particle species that are exchanged, respectively.

Figure 6

Scheme of a bidirectional counting statistics experiment. Two leads (large semicircles) with different electronic chemical potentials (${\mu }_1\ne {\mu }_2$) and same temperature (${\beta }_1={\beta }_2$) are connected through a double quantum dot (small circles), whose quantum state is continuously monitored. The state (1,0), i.e., one electron in the left dot, no electrons in the right dot, is depicted. The transition from this state to the state (0,1) signals the exchange of one electron from subsystem 1 to subsystem 2. ${\mathcal{H}}_{1,2}$ and ${\mathcal{N}}_{1,2}$ denote the Hamiltonian and electron number operators of the subsystems, respectively.