Heisenberg Uncertainty Principle as Probe of Entanglement Entropy: Application to Superradiant Quantum Phase Transitions

TL;DR

This paper establishes a scaling relation between entanglement entropy and the Heisenberg uncertainty principle, demonstrating that the latter can effectively probe superradiant quantum phase transitions in photon-matter systems.

Contribution

It introduces a novel approach linking the Heisenberg uncertainty principle to entanglement properties at quantum critical points, with exact results for the Dicke model.

Findings

The product ΔxΔp diverges as N^{1/6} at the critical point.

Heisenberg uncertainty can serve as a sensitive probe of quantum phase transitions.

Results are applicable to experimental platforms like Bose-Einstein condensates and circuit QED.

Abstract

Quantum phase transitions are often embodied by the critical behavior of purely quantum quantities such as entanglement or quantum fluctuations. In critical regions, we underline a general scaling relation between the entanglement entropy and one of the most fundamental and simplest measure of the quantum fluctuations, the Heisenberg uncertainty principle. Then, we show that the latter represents a sensitive probe of superradiant quantum phase transitions in standard models of photons such as the Dicke Hamiltonian, which embodies an ensemble of two-level systems interacting with one quadrature of a single and uniform bosonic field. We derive exact results in the thermodynamic limit and for a finite number N of two-level systems: as a reminiscence of the entanglement properties between light and the two-level systems, the product $\Delta x\Delta p$ diverges at the quantum critical point as $N^{1/6}$. We generalize our results to the double quadrature Dicke model where the two quadratures of the bosonic field are now coupled to two independent sets of two level systems. Our findings, which show that the entanglement properties between light and matter can be accessed through the Heisenberg uncertainty principle, can be tested using Bose-Einstein condensates in optical cavities and circuit quantum electrodynamics