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arXiv:quant-ph/0309027 · Physical Review Letters

Entanglement and the Phase Transition in Single-Mode Superradiance

Neill Lambert, Clive Emary, Tobias Brandes

Published Feb 19, 2004• 482 citations
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Abstract

We consider the entanglement properties of the quantum phase transition in the single-mode superradiance model, involving the interaction of a boson mode and an ensemble of atoms. For an infinite size system, the atom-field entanglement diverges logarithmically with the correlation length exponent. Using a continuous variable representation, we compare this to the divergence of the entropy in conformal field theories and derive an exact expression for the scaled concurrence and the cusplike nonanalyticity of the momentum squeezing.

DOI
10.1103/physrevlett.92.073602

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We consider the entanglement properties of the quantum phase transition in the single-mode superradiance model, involving the interaction of a boson mode and an ensemble of atoms. For an infinite size system, the atom-field entanglement diverges logarithmically with the correlation length exponent. Using a continuous variable representation, we compare this to the divergence of the entropy in conformal field theories and derive an exact expression for the scaled concurrence and the cusplike nonanalyticity of the momentum squeezing.
We investigate the entanglement properties of an ensemble of atoms interacting with a single bosonic field mode via the Dicke (superradiance) Hamiltonian. The model exhibits a quantum phase transition and a well-understood thermodynamic limit, allowing the identification of both quantum and semiclassical many-body features in the behavior of the entanglement. We consider the entanglement between the atoms and the field, an investigation initiated in Lambert et al. [Phys. Rev. Lett. 92, 073602 (2004)]. In the thermodynamic limit, we give exact results for all entanglement partitions and observe a logarithmic divergence of the atom-field entanglement, and discontinuities in the average linear entropy.
We investigate the entanglement properties of an ensemble of atoms interacting with a single bosonic field mode via the Dicke (superradiance) Hamiltonian. The model exhibits a quantum phase transition and a well-understood thermodynamic limit, allowing the identification of both quantum and semi-classical many-body features in the behavior of the entanglement. We consider entanglement between the atoms and field, as well both bi- and multi- partite entanglement between atoms. In the thermodynamic limit, we give exact results for all entanglement partitions and observe a logarithmic divergence of the atom-field entanglement, and discontinuities in the pairwise and multipartite entanglement.
Entanglement and quantum correlations between atoms are not usually considered key ingredients of the superradiant phase transition. Here we consider the Tavis-Cummings model, a solvable system of two-levels atoms, coupled with a single-mode quantized electromagnetic field. This system undergoes a superradiant phase transition, even in a finite-size framework, accompanied by a spontaneous symmetry breaking, and an infinite sequence of energy level crossings. We find approximated expressions for the ground state, its energy, and the position of the level crossings, valid in the limit of a very large number of photons with respect to that of the atoms. In that same limit, we find that the number of photons scales quadratically with the coupling strength, and linearly with the system size, providing a new insight into the superradiance phenomenon. Resorting to novel multipartite measures, we then demonstrate that this quantum phase transition is accompanied by a crossover in the quantum correlations and entanglement between the atoms (qubits). The latters therefore represent suited order parameters for this transition. Finally, we show that these properties of the quantum phase transition persist in the thermodynamic limit.
Entanglement and quantum correlations between atoms are not usually considered key ingredients of the superradiant phase transition. Here we consider the Tavis−Cummings model, a solvable system of two-levels atoms, coupled with a single-mode quantized electromagnetic field. This system undergoes a superradiant phase transition, even in a finite-size framework, accompanied by a spontaneous symmetry breaking, and an infinite sequence of energy level crossings. We find approximated expressions for the ground state, its energy, and the position of the level crossings, valid in the limit of a very large number of photons with respect to that of the atoms. In that same limit, we find that the number of photons scales quadratically with the coupling strength, and linearly with the system size, providing a new insight into the superradiance phenomenon. Resorting to novel multipartite measures, we then demonstrate that this quantum phase transition is accompanied by a crossover in the quantum correlations and entanglement between the atoms (qubits). The latters therefore represent suited order parameters for this transition. Finally, we show that these properties of the quantum phase transition persist in the thermodynamic limit.
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.