Analytical Bounce Solution in a Dissipative Quantum Tunneling

TL;DR

This paper derives an analytical bounce solution for dissipative quantum tunneling using the polygamma function, providing insights into how dissipation affects the tunneling process, and offers a basis for approximate calculations of the prefactor.

Contribution

It presents a new analytical bounce solution in a dissipative quantum tunneling model, expressed with the polygamma function, and discusses its properties and limitations.

Findings

Bounce peak point increases from 1 to 4/3 with dissipation coefficient

Classical action bounds are established across the dissipation range

Solution serves as a starting point for approximate prefactor calculations

Abstract

The analytical bounce solution is derived in terms of the polygamma function in the Caldeira-Leggett's dissipative quantum tunneling model. The classical action for the bounce solution lies between the upper and lower bounds in the full range of $\alpha$, where $\alpha$ is a dissipation coefficient. The bounce peak point increases from 1 to 4/3 with increase of $\alpha$. In spite of various nice features we have shown that the solution we have derived is not exact one by making use of the zero mode argument in the linearized fluctuation equation. However, our solution can be a starting point for approximate computation of the prefactor in this model.