Fluctuation theorem for heat transport probed by a thermal probe electrode

We analyze the full-counting statistics of the electric heat current flowing in a two-terminal quantum conductor whose temperature is probed by a third electrode (“probe electrode”). In particular we demonstrate that the cumulant-generating function obeys the fluctuation theorem in the presence of a constant magnetic field. The analysis is based on the scattering matrix of the three-terminal junction (comprising the two electronic terminals and the probe electrode), and a separation of time scales: it is assumed that the rapid charge transfer across the conductor and the rapid relaxation of the electrons inside the probe electrode give rise to much slower energy fluctuations in the latter. This separation allows for a stochastic treatment of the probe dynamics, and the reduction of the three-terminal setup to an effective two-terminal one. Expressions for the lowest nonlinear transport coefficients, e.g., the linear-response heat-current noise and the second nonlinear thermal conductance, are obtained and explicitly shown to preserve the symmetry of the fluctuation theorem for the two-terminal conductor. The derivation of our expressions, which is based on the transport coefficients of the three-terminal system explicitly satisfying the fluctuation theorem, requires full calculations of vertex corrections.

Figure 1

A three-terminal setup. The figure depicts a quantum conductor (the elliptical area) connected to two reservoirs denoted $L$ and $R$, which are held at two different temperatures; those are expressed in terms of the affinities ${\mathcal{A}}_{L,R}\equiv \beta -{\beta }_{L,R}$ (${\beta }^{-1}$ is the temperature of the entire system when at equilibrium); see the text. The thermal probe (the upper square) is specified by its own affinity ${\mathcal{A}}_P\equiv \beta -{\beta }_P$, which fluctuates in time. Also shown are the three auxiliary “counting” fields ${\lambda }_r$ ($r=L$, $R$, and $P$) which measure the energy flowing in and out of electrode $r$. The quantum conductor is threaded by a perpendicularly oriented magnetic field $B$. Once the stochastic dynamics of the energy current in and out of the probe electrode is taken care of, the junction becomes an effective two-terminal one.

Figure 2

(a) A bare vertex (empty circle) $L_{j\ell }^{ik}$; (b1) and (b2) corrected vertices (solid triangles).

Figure 3

Diagrams for the linear-response conductance (a1) and the equilibrium noise (a2). Only the “bare” diagrams, indicated by dotted squares, are included in the minimal-correlation coordinate approach explained in Appendix pp3.

Figure 4

The six diagrams comprising the second nonlinear conductance. The three diagrams enclosed in the dotted squares are the bare one and its cascade corrections, which are accounted for in the minimal-correlation coordinate approach (Appendix pp3). The second and fifth diagrams on the right-hand side contain $L_{11}^{10}$ and $L_{11}^{01}$, respectively, and thus vanish.

Figure 5

The 14 diagrams contributing to the expression for the linear-response noise. The four diagrams enclosed in the dotted squares are accounted for in the minimal-correlation coordinate approach (Appendix pp3). The third and fifth diagrams on the right-hand side contain $L_{11}^{10}$ and the 10th and the 11th ones contain $L_{11}^{01}$. These diagrams therefore vanish.

Figure 6

(a) Schematic picture of three-terminal triple quantum dot. The flux threads the triangular region. (b) The second nonlinear thermal conductance (solid line) and the linear-response thermal noise (dotted line) as functions of the flux. $L_2^1\left(0\right)=1.39\times {10}^{-4}{\Gamma }^3/\left(e^2{R}_K\right)$, where $R_K=h/e^2$ is the resistance quantum. (c) and (d) show the symmetric (solid lines) and antisymmetric (dashed lines) components in the flux. Solid and dashed lines in each panel overlap each other. Parameters: $t=0.1\Gamma$, ${\epsilon }_0=-0.5\Gamma$, $\beta \Gamma =10$, and $x=0.25$.

Figure 7

(a) The rate function and (b) the fluctuation theorem for $\varphi =0$. Parameters: $\mathcal{A}\Gamma =0.5$, $\beta \Gamma =10$.