# Time dependent local potential in a Tomonaga-Luttinger liquid

## TL;DR

This paper investigates how a time-dependent local potential affects energy absorption in a one-dimensional Tomonaga-Luttinger liquid, revealing power-law behaviors and interaction-dependent crossovers relevant for cold atom experiments.

## Contribution

It introduces a detailed analysis of energy deposition due to oscillating local potentials in TLLs, highlighting interaction-dependent crossover phenomena and calculating the associated frequency scale.

## Key findings

- Energy deposition follows a power-law in frequency reflecting TLL properties.
- A crossover between weak and strong backscattering regimes depends on the TLL parameter K.
- Different exponents for energy absorption are found for K>1 and K<1.

## Abstract

We study the energy deposition in a one dimensional interacting quantum system with a point like potential modulated in amplitude. The point like potential at position $x=0$ has a constant part and a small oscillation in time with a frequency $\omega$. We use bosonization, renormalization group and linear response theory to calculate the corresponding energy deposition. It exhibits a power law behavior as a function of the frequency that reflects the Tomonaga-Luttinger liquid (TLL) nature of the system. Depending on the interactions in the system, characterized by the TLL parameter $K$ of the system, a crossover between week and strong coupling for the backscattering due to the potential is possible. We compute the frequency scale $\omega_\ast$, at which such crossover exists. We find that the energy deposition due to the backscattering shows different exponent for $K>1$ and $K<1$. We discuss possible experimental consequences, in the context of cold atomic gases, of our theoretical results.