On the Bosonic Phase Operator Realization for Josephson Hamiltonian Model

Hong-Yi Fan, Ji‐Suo Wang, Shu-guang Liu

Fan Hong-Yi, Fan Yue

On the assumption that a Cooper pair acts as a Bose particle and based on the newly established representation, which is the common eigenvector of two particles' relative position and total momentum, we introduce a mesoscopic Josephson junction Hamiltonian constituted by two-mode Bose phase operator and number-difference operator. The number-difference-phase uncertainty relation can then be set up, which implies the existence of Josephson current.

Hong-Yi Fan, Ji‐Suo Wang, Xiang‐Guo Meng

Based on Feynman's explanation about Cooper pair that "a bound pair acts as a Bose particle" and the bosonic operator Hamiltonian of the Josephson junction, we realize that the quantum state of the Josephson junction is a Cooper pair number-phase entangled state constructed by the phase operator across the junction. Its Schmidt decomposition is derived. The Cooper pair number-phase squeezed state's projection onto this entangled state leads to a geometric distribution.

Luca Salasnich

Michael Martin Nieto

J. R. Anglin, P. D. Drummond, Augusto Smerzi

The quantum Josephson Hamiltonian of two weakly linked Bose-Einstein condensates is written in an overcomplete phase representation, thus avoiding the problem of defining a Hermitian phase operator. We discuss the limit of validity of the standard, non-rigorous Mathieu equation, due to the onset of a higher order $\cos 2 ϕ$ term in the Josephson potential, and also to the overcompleteness of the representation (the phase $ϕ$ being the relative phase between the two condensates). We thereby unify the Boson Hubbard and Quantum Phase models.