Full counting statistics of electron-photon hybrid systems: Joint statistics and fluctuation symmetry

TL;DR

This work develops a joint full counting statistics framework for electron-photon hybrids, specifically a double quantum dot embedded in an optical cavity, by augmenting the quantum Lindblad equation with electronic and photonic counting fields. It shows that while the photonic current grows with coupling, the ratio to the electronic current scales quadratically in the weak-coupling limit but deviates at larger $g$, and the cross-correlation can switch sign depending on the electronic splitting. A central finding is that the conventional zero-temperature photonic dissipation fails to satisfy the fluctuation theorem, necessitating a finite-temperature photon bath realized via a photon-gain channel, which restores the symmetry while leaving weak-coupling results qualitatively unchanged. The study provides a thermodynamically consistent blueprint for joint electron-photon statistics in quantum devices and clarifies when simpler, conventional master equations are sufficient.

Abstract

Electron-photon hybrid systems serve as ideal light-matter interfaces with broad applications in quantum technologies. These systems are typically operated dynamically under nonequilibrium conditions, giving rise to coupled electronic and photonic currents. Understanding the joint fluctuation behavior of these currents is essential for assessing the performance of light-matter interfaces that rely on electron-photon correlations. Here, we investigate the full counting statistics of coupled electronic and photonic currents in an experimentally feasible hybrid system composed of a double quantum dot coupled to an optical cavity. We employ the framework of quantum Lindblad master equation which is augmented with both electronic and photonic counting fields to derive their joint cumulant generating function--a treatment that differs significantly from existing studies, which typically focus on either electron or photon statistics separately. We reveal that the ratio between photonic and electronic currents, as well as their variances, can deviate from an expected quadratic scaling law in the large electron-photon coupling regime. Furthermore, we demonstrate that conventional modelings of photonic dissipation channels in quantum master equations must be modified to ensure that the joint cumulant generating function satisfies the fluctuation symmetry enforced by the fluctuation theorem. Our results advance the understanding of joint fluctuation behaviors in electron-photon hybrid systems and may inform the design of efficient quantum light-matter interfaces.

Figures (6)

• Figure 1: First-order and second-order scaled cumulants as functions of (a) chemical potential $\mu_c$ and (b) photon loss rate $\kappa$. In both plots, certain quantities are scaled by the factors indicated in the legends for a better illustration. All other parameters are set to their default values as specified in \ref{['tab:parameter_values']}.
• Figure 2: (a) Photonic ($\mathcal{I}_2$) and electronic ($\mathcal{I}_1$) currents and (b) their respective ratio $\mathcal{I}_2/\mathcal{I}_1$ as functions of the electron-photon coupling strength $g$ for three values of energy splitting $\epsilon$. The red dashed line in (b) serves as a guide to the eye for the power-law behavior $\mathcal{I}_2/\mathcal{I}_1\sim g^2$. All other parameters are set to their default values as specified in \ref{['tab:parameter_values']}.
• Figure 3: Second-order scaled cumulants versus coupling strength $g$ for three values of energy splitting $\epsilon$: (a) electronic current noise, (b) photonic current noise, and (c) electron-photon cross-correlation. All other parameters are set to their default values as specified in \ref{['tab:parameter_values']}.
• Figure 4: (a) $\epsilon$-dependence of first-order scale cumulants: Electronic current $\mathcal{I}_1$ (blue solid curve) and photonic current $\mathcal{I}_2$ (orange solid curve). We also verify the validity of Eq. (\ref{['eq:current-relation-deterministic']}) (green circles). (b) second-order scaled cumulants $\mathcal{M}_{2,0}$ (blue solid line), $\mathcal{M}_{0,2}$ (orange solid line) and $\mathcal{M}_{1,1}$ (green solid line) as functions of energy splitting $\epsilon$. All other parameters are set to their default values as specified in \ref{['tab:parameter_values']}.
• Figure 5: Real and imaginary part of the scaled cumulant generating functions on a two-dimensional plane passing through the fluctuation symmetric point $i\boldsymbol\sigma/2$ of $\boldsymbol\chi$ and $-\boldsymbol\chi+i\boldsymbol\sigma$ appeared in \ref{['eq:symmetry_scgf']}, using a modified quantum Lindblad master equation \ref{['eq:modified_master']}. Left panel: Real parts of (a) $\mathcal{C}_{\boldsymbol\chi}$ and (c) $\mathcal{C}_{-\boldsymbol\chi+i\boldsymbol\sigma}$. Right panel: Imaginary parts of (b) $\mathcal{C}_{\boldsymbol\chi}$ and (d) $\mathcal{C}_{-\boldsymbol\chi+i\boldsymbol\sigma}$. $x$ and $y$ are introduced as parameters such that the first four counting fields are parameterized as $(0.1+i\boldsymbol\sigma_{1-4}/2)x$, while the last counting field is given by $(0.1+i\sigma_{5}/2)y$. We set the chemical potential of the photonic reservoir $\mu_p=32.02\,\mu\text{eV}$, other parameters are set to their default values as specified in \ref{['tab:parameter_values']}.

Theorems & Definitions (1)

• Proposition