# Fate of dynamical phases of a BCS superconductor beyond the dissipationless regimen

## TL;DR

This paper investigates how dissipation affects the dynamical phases of a BCS superconductor, showing that environmental coupling leads to eventual equilibrium and recovery of long-range order, with implications for quantum technologies.

## Contribution

It introduces a self-consistent approach using the Keldysh formalism to study dissipative effects on superconductor dynamics beyond idealized models.

## Key findings

- All dynamical phases reach equilibrium order parameter over time.
- Overdamped phase exhibits a rapid crossover to long-range order.
- Damped Higgs mode oscillations are observed near equilibrium.

## Abstract

The BCS model of an isolated superconductor initially prepared in a nonequilibrium state, predicts the existence of interesting dynamical phenomena in the time-dependent order parameter as decaying oscillations, persistent oscillations and overdamped dynamics. To make contact with real systems remains an open challenge as one needs to introduce dissipation due to the environment in a self-consistent computation. Here, we reach this goal with the use of the Keldysh formalism to treat the effect of a thermal bath. We show that, contrary to the dissipationless case, all dynamical phases reach the equilibrium order parameter in a characteristic time that depends on the coupling with the bath. Remarkably, as time evolves, the overdamped phase shows a fast crossover where the superconducting order parameter recovers to reach a state with a well-developed long range order that tends towards equilibrium with the damped Higgs mode oscillations. Our results provide a benchmark for the description of the dynamics of real out-of-equilibrium superconductors relevant for quantum technological applications.