Work fluctuation theorem for a classical circuit coupled to a quantum conductor

We propose a setup for a quantitative test of the quantum fluctuation theorem. It consists of a quantum conductor, driven by an external voltage source, and a classical inductor-capacitor circuit. The work done on the system by the voltage source can be expressed by the classical degrees of freedom of the $LC$ circuit, which are measurable by conventional techniques. In this way, the circuit acts as a classical detector to perform measurements of the quantum conductor. We prove that this definition is consistent with the work fluctuation theorem. The system under consideration is effectively described by a Langevin equation with non-Gaussian white noise. Our analysis extends the proof of the fluctuation theorem to this situation.

Figure 1

(a) Schematics of the system, consisting of a coherent quantum conductor (denoted by $G$) connected to an inductor and a capacitor. The voltage drop across the quantum conductor $V$ is measured by a voltmeter. (b) A Brownian particle in a driven harmonic potential.

Figure 2

Aharonov-Bohm interferometer with a quantum dot embedded in one arm. The magnetic flux $\Phi$ threads the ring, and the electron wave function acquires the AB phase $\phi$ for an electron traveling in clockwise direction. The energy parameters ${\Gamma }_{L/R}$ characterize the strength of the tunnel couplings between the quantum dot and the left/right leads. An electron can also be transmitted through the lower reference arm, characterized by the tunneling amplitude $t_{\mathrm{ref}}$. The left/right chemical potentials are ${\mu }_{L/R}$.

Figure 3

(a) Probability distributions of the work for $\phi =\pm \pi /4$. (b) Ratio between the positive and negative work probability distributions. When the direction of magnetic field is also reversed, the steady-state work fluctuation theorem is satisfied. The parameters are $U/\Gamma =4$, $V=\Gamma$, $k_{\mathrm{B}}T=0.2\Gamma$, ${\Gamma }_L=0.25\Gamma$, ${\Gamma }_R=0.75\Gamma$, and $t_{\mathrm{ref}}=0.25$ and ${\kappa }_L=-{\kappa }_R=0.5$. The average is $\langle \langle w(\phi =0)\rangle \rangle /\left(Me{V}_{\mathrm{ext}}\right)\approx 2.9\times {10}^{-2}\Gamma /\hslash$. The plots are not sensitive to the specific value of $\Gamma$ since it is canceled after the work $w$ is normalized to $\langle \langle w(\phi =0)\rangle \rangle$.