Quantum-correlated photons generated by nonlocal electron transport

Since the realization of high-quality microwave cavities coupled to quantum dots, one can envisage the possibility to investigate the coherent interaction of light and matter in semiconductor quantum devices. Here we study a parallel double quantum dot device operating as single-electron splitter interferometer, with each dot coupled to a local photon cavity. We explore, how quantum correlation and entanglement between the two separated cavities are generated by the coherent transport of a single electron passing simultaneously through the two different dots. We calculate the covariance of the cavity occupations by using a diagrammatic perturbative expansion based on Keldysh Green's functions to fourth order in the dot-cavity interaction strength, taking into account vertex diagrams. Furthermore, we demonstrate the creation of entanglement by showing that the classical Cauchy-Schwarz inequality is violated if the energy levels of the two dots are almost degenerate. For large level detuning or a single dot coupled to two cavities, we show that the inequality is not violated.

Figure 1

Sketch of the studied systems and idea. (a) Main model: a parallel double quantum dot with a single electronic energy level in each quantum dot. Each dot is coupled to one of two separated microwave cavities and common left and right leads. (b) Basic idea: An electron travels through both branches of the parallel double quantum dot simultaneously. Its state is a coherent superposition of the states in the two dots. The delocalization of the single electron yields quantum correlations between the two cavities and remains even when the electron leaves the system. (c) Single quantum dot with a single electronic energy level coupled to two microwave cavities.

Figure 2

Examples of vertex diagrams, which contribute to the integrals of $C$ and $S$. Black dots represent the electron-photon interaction proportional to $\lambda$. Wiggly lines correspond to bosonic Green's functions of the microwave cavities with resonance frequency ${\omega }_{\text{a,b}}$, distinguished by the colors red and blue. Straight lines signify fermionic Green's functions of the double quantum dot system. Diagonal Green's functions are indicated by blue and red lines; off-diagonal Green's functions are represented in green.