Abstract
Impurities coupled to superconductors offer a controlled platform to understand the interplay between superconductivity, many-body interactions, and nonequilibrium physics. In the equilibrium situation, local interactions at the impurity induce a transition from the spin-singlet to the spin-doublet ground state, resulting in a supercurrent sign reversal ($0\text{\ensuremath{-}}\ensuremath{\pi}$ transition). In this work, we apply the exact time-dependent density matrix renormalization group method to simulate the transient dynamics of such superconducting systems. We also use a perturbative approximation to analyze their properties at longer times. These two methods agree for a wide range of parameters. In a phase-biased situation, the system gets trapped in a metastable state characterized by a lower supercurrent compared to the equilibrium case. We show that local Coulomb interactions do not provide an effective relaxation mechanism for the initially trapped quasiparticles. In contrast, other relaxation mechanisms, such as coupling to a third normal lead, make the impurity spin relax for parameter values corresponding to the equilibrium 0 phase. For parameters corresponding to the equilibrium $\ensuremath{\pi}$ phase the impurity converges to a spin-polarized stationary state. Similar qualitative behavior is found for a voltage-biased junction, which provides an effective relaxation mechanism for the trapped quasiparticles in the junction.
Highlights
Superconductors are macroscopic coherent materials, described by a complex order parameter, measuring the pairing amplitude between electrons
We develop an exact numerical method that simulates the real-time evolution of the system, based on the time-dependent density matrix renormalization group
Perturbation theory on the interaction strength, U, provides accurate results in the U/ 1 regime. This approximation has been shown to describe accurately physical observables in the stationary regime and the nonmagnetic phase [22,53,54]. We extend this approximation to the time domain, allowing us to describe the nonstationary and nonequilibrium situations
Summary
Superconductors are macroscopic coherent materials, described by a complex order parameter , measuring the pairing amplitude between electrons. The time dynamics of impurities coupled to superconductors can answer fundamental questions, like the time onset of electron correlations, formation of subgap states, dependence on the initial conditions, and relaxation of quasiparticles. The inclusion of a third normal lead or a finite bias allows for quasiparticle relaxation due to transitions to excited states with energies above | | [80] In this last case, the superconducting phase difference evolves linearly in time, giving rise to a time-oscillating current due to multiple Andreev reflections (MARs) [59,81,82,83,84,85,86,87,88,89,90].
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