Energy entanglement in normal metal-superconducting forks

The possibility for detecting energy entanglement in normal metal-superconductor junctions is examined. We prove that two electrons in a NS structure originating from the same Cooper pair are entangled in the energy subspace. This work follows previous works where spin entanglement was studied in similar circuits. The device consists of a superconducting beam splitter connected to two electronic Mach-Zehnder interferometers. In each arm of the interferometers, energies are filtered with coherent quantum dots. In contrast to previous studies of zero-frequency cross-correlations of electrical currents for this system, attention is drawn to finite-time measurements. This allows one to observe two-particle interference for particles with different energies above and below Fermi level. Entanglement is first characterized via the concurrence for the two-particle spatial density matrix. Next, we formulate the Bell inequality, which is written in terms of finite-time noise correlators, and thereby we find a specific set of parameters for which entanglement can be detected.