Lifshitz Quantum Mechanics and Anisotropic Josephson Junction

TL;DR

The paper investigates Lifshitz quantum mechanics with time anisotropy and fractional derivatives, formulating a Hamiltonian $H = -\frac{\hbar^z}{M^{z-2}}\frac{\Delta_z}{2m} + V$ and proving unitary evolution with a local conservation law $\partial_t \rho + \nabla \cdot \mathbf{J} = 0$, where the anisotropic current $\mathbf{J}$ is defined in momentum space and is not unique up to curl terms. It clarifies that previous claims of probability non-conservation in fractional quantum mechanics stem from incomplete current identification, and shows that probability is conserved with the correct current. As an application, it analyzes an anisotropic SIS Josephson Junction, deriving a penetration length $\zeta = \zeta_0 (\zeta_0/\lambda_M)^{2/z-1}$ and a rescaled critical current $\mathrm{J}_c = \gamma \mathrm{J}_{c0}$ with $\gamma = \frac{z \zeta}{2 \zeta_0} \frac{\sinh(2a/\zeta_0)}{\sinh(2a/\zeta)}$, illustrating how anisotropy can substantially boost tunneling. These results point to practical implications for tuning quantum transport in nanoscale devices and potential benefits for quantum computing architectures.

Abstract

We consider quantum mechanics in spacetime with anisotropy in time. Such Lifshitz quantum mechanics is characterized by a kinetic term with fractional derivatives. We show that, contrary to a common claim in the literature, local conservation of probability is respected when the probability current is properly identified. As an application we consider a Josephson Junction with an insulating layer exhibiting Lifshitz anisotropy. We show that anisotropy modifies the tunneling rate and can significantly enhance the performance of the Josephson Junction.

Figures (2)

• Figure 1: Two superconductors separated by a thin insulator
• Figure 2: Tunneling through a potential