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Europhysics Letters (EPL)

Entropic uncertainty and the quantum phase transition in the Dicke model

E. Romera, Manuel Calixto, Á. Nagy

Published Jan 1, 2012• 32 citations
Publisher

Abstract

We show that the description of the quantum phase transition in terms of the entropic uncertainty relation turns out to be more suitable than in terms of the standard variance-based uncertainty relation. The entropic uncertainty relation detects the quantum phase transition in the Dicke model and it provides a correct description of the quantum fluctuations or quantum uncertainty of the system.

DOI
10.1209/0295-5075/97/20011

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We show that the description of the quantum phase transition in terms of the entropic uncertainty relation turns out to be more suitable than in terms of the standard variance-based uncertainty relation. The entropic uncertainty relation detects the quantum phase transition in the Dicke model and it provides a correct description of the quantum fluctuations or quantum uncertainty of the system.
Our knowledge of quantum mechanics can satisfactorily describe simple microscopic systems, but is yet to explain the macroscopic everyday phenomena we observe. Here we aim to shed some light on the quantum-to-classical transition as seen through the analysis of uncertainty relations. We employ entropic uncertainty relations to show that it is only by the inclusion of imprecision in our model of macroscopic measurements that we can prepare a system with two simultaneously well-defined quantities, even if their associated observables do not commute. We also establish how the precision of measurements must increase in order to keep quantum properties, a desirable feature for large quantum computers.
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 NNN of two-level systems: as a reminiscence of the entanglement properties between light and the two-level systems, the product $\ensuremath{\Delta}x\ensuremath{\Delta}p$ diverges at the quantum critical point as N1/6{N}^{1/6}N1/6. 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.
We present a general way of quantifying the entropic uncertainty of quantum field configurations in phase space in terms of entropic distinguishability with respect to the vacuum. Our approach is based on the functional Husimi Q -distribution and a suitably chosen relative entropy, which we show to be non-trivially bounded from above by the uncertainty principle. The resulting relative entropic uncertainty relation is as general as the concept of coherent states and thus holds for quantum fields of bosonic and fermionic type. Its simple form enables diverse applications, among which we present a complete characterization of the uncertainty surplus of arbitrary states in terms of the total particle number for a scalar field and the fermionic description of the Ising model. Moreover, we provide a quantitative interpretation of the role of the uncertainty principle for quantum phase transitions.
Abstract We investigate the effectiveness of entropic uncertainty, entanglement and steering in discerning quantum phase transitions (QPTs). Specifically, we observe significant fluctuations in entropic uncertainty as the driving parameter traverses the phase transition point. It is observed that the entropic uncertainty, entanglement and quantum steering, based on the electron distribution probability, can serve as indicators for detecting QPTs. Notably, we reveal an intriguing anticorrelation relationship between entropic uncertainty and entanglement in the Aubry–André model. Moreover, we explore the feasibility of detecting a QPT when the period parameter is a rational number. These observations open up new and efficient avenues for probing QPTs.