# Charge-vibration interaction effects in normal-superconductor quantum dots

## TL;DR

This paper investigates charge-vibration interactions in a normal-superconductor quantum dot system, revealing how inelastic tunneling processes influence current, vibrational states, and enable vibrational spectroscopy at high temperatures.

## Contribution

It provides a detailed analysis of inelastic tunneling effects and vibrational state control in quantum dots coupled to superconductors using Keldysh Green's functions.

## Key findings

- Inelastic tunneling peaks are sharp and temperature-independent, enabling vibrational spectroscopy.
- Inelastic processes induce diverse vibrational states, including cooling, heating, and instabilities.
- Ground-state cooling can occur for multiple vibrational modes simultaneously.

## Abstract

We study the quantum transport and the nonequilibrium vibrational states of a quantum dot embedded between a normal and a superconducting lead with the charge on the quantum dot linearly coupled to a harmonic oscillator of frequency $\omega$. To the leading order in the charge-vibration interaction, we calculate the current and the nonequilibrium phonon occupation by the Keldsyh Green's function technique. We analyze the inelastic, vibration-assisted tunneling processes in the regime $\omega <\Delta$, with the superconducting energy gap $\Delta$, and for sharp resonant transmission through the dot. When the energy $\varepsilon_0$ of the dot's level is close to the Fermi energy $\mu$, i.e. $|\varepsilon_0-\mu|\ll \Delta$, inelastic Andreev reflections dominate up to voltage $eV\gtrsim\Delta$. The inelastic quasiparticle tunneling becomes the leading process when the dot's level is close to the gap $|\varepsilon_0-\mu|\sim \Delta \pm \omega$. In both cases, the inelastic tunneling processes appear as sharp and prominent peaks - not broadened by temperature - in the $I$-$V$ characteristic and pave the way for inelastic spectroscopy of vibrational modes even at temperatures $T \gg \omega$. We also found that inelastic Andreev reflections as well as quasiparticle tunneling induce a strong nonequilibrium state of the oscillator. In different ranges on the dot's level, we found that the current produces: (i) ground-state cooling of the oscillator with phonon occupation $n\ll 1$, (ii) accumulation of energy in the oscillator with $n\gg 1$ and (iii) a mechanical instability which is a precursor of self-sustained oscillations. We show that ground-state cooling is achieved simultaneously for several modes of different frequencies. Finally, we discuss how the nonequilibrium vibrational state can be detected by the asymmetric behavior of the inelastic current peaks respect to the gate voltage.