Linear response as a singular limit for a periodically driven closed quantum system

TL;DR

This paper investigates the limits of linear response theory in periodically driven closed quantum systems, showing it fails after a certain time for extensive perturbations but remains valid for local perturbations, with implications for experiments.

Contribution

The study demonstrates a singular limit for linear response validity in many-body quantum systems under periodic driving, highlighting the role of perturbation extent and providing detailed analysis with a quantum Ising chain.

Findings

Linear response predicts energy absorption only up to a finite time t* for extensive perturbations.

For local perturbations, energy absorption persists indefinitely, and LRT remains valid.

Experimental relevance discussed for cold atoms and ultrafast spectroscopy.

Abstract

We address the issue of the validity of linear response theory for a closed quantum system subject to a periodic external driving. Linear response theory (LRT) predicts energy absorption at frequencies of the external driving where the imaginary part of the appropriate response function is different from zero. Here we show that, for a fairly general non-linear many-body system on a lattice subject to an extensive perturbation, this approximation should be expected to be valid only up to a time $t^*$ depending on the strength of the driving, beyond which the true coherent Schr\"odinger evolution departs from the linear response prediction and the system stops absorbing energy form the driving. We exemplify this phenomenon in detail with the example of a quantum Ising chain subject to a time-periodic modulation of the transverse field, by comparing an exact Floquet analysis with the standard results of LRT. In this context, we also show that if the perturbation is just local, the system is expected in the thermodynamic limit to keep absorbing energy, and LRT works at all times. We finally argue more generally the validity of the scenario presented for closed quantum many-body lattice systems with a bound on the energy-per-site spectrum, discussing the experimental relevance of our findings in the context of cold atoms in optical lattices and ultra-fast spectroscopy experiments.