Mechanical Cooper Pair Transportation as a Source of Long-Distance Superconducting Phase Coherence

Transportation of Cooper pairs by a movable single Cooper-pair box placed between two remote superconductors is shown to establish coherent coupling between them. This coupling is due to entanglement of the movable box with the leads and is manifested in the suppression of quantum fluctuations of the relative phase of the order parameters of the leads. It can be probed by attaching a high resistance Josephson junction between the leads and measuring the current through this junction. The current is suppressed with increasing temperature.

Figure 1

Schematic diagram of the system discussed in the text. A movable nanoscale superconducting grain which may accommodate excess charge carried by either 0 or 1 extra Cooper pairs is placed between two finite isolated superconducting leads. Close to the leads, electrostatic gates ensure that the Coulomb blockade of tunneling is lifted. After a sequence of grain rotations, the grain moving periodically contacting the two leads sequentially, a probe (Josephson) junction is connected through which a current $I$ is measured.

Figure 2

Average current through a probe junction attached as in Fig. 1 after many grain rotations as a function of the phases ${\chi }_{\pm }$ (in units of $\pi$). The color in the plot indicates the magnitude of the current in units of the critical current $I_c$ of the probe junction. The brightest areas correspond to $|I/I_c|\approx 0.4$ while black areas correspond to zero current. The direction of the current is indicated by $+$ and $-$ signs.

Figure 3

Phase difference probability density $f(\Delta \Phi )$ as a function of the number of grain rotations. The graph shows how the system evolves from a state with maximum to a state with minimum quantum fluctuations as the number of rotations increases.

Figure 4

Probe junction current versus temperature. The probe junction current is here shown for fixed dynamical phases for a series of different temperatures.