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Journal of Physics Conference Series

Quantum optical properties of the radiation field in the Dicke model

Timothy Jarrett, Alexandra Olaya-Castro, Neil F. Johnson

Published Oct 1, 2007• 1 citation
Publisher

Abstract

We study the optical signatures of the quantum critical behaviour associated with the quantum phase transition exhibited by the Dicke model in the thermodynamic limit. We obtain an effective Hamiltonian for the radiation field, which resembles a degenerate parametric amplifier and which reproduces the Dicke's model critical behaviour. We identify the state of the radiation field in the sub-radiant and super-radiant phases, and show that the optical squeezing and photon statistics present striking behaviour in the vicinity of the phase transition. Our results are both of theoretical interest and of relevance to recent experimental proposals that hope to realize the quantum phase transition in the Dicke model.

DOI
10.1088/1742-6596/84/1/012009

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We study the optical signatures of the quantum critical behaviour associated with the quantum phase transition exhibited by the Dicke model in the thermodynamic limit. We obtain an effective Hamiltonian for the radiation field, which resembles a degenerate parametric amplifier and which reproduces the Dicke's model critical behaviour. We identify the state of the radiation field in the sub-radiant and super-radiant phases, and show that the optical squeezing and photon statistics present striking behaviour in the vicinity of the phase transition. Our results are both of theoretical interest and of relevance to recent experimental proposals that hope to realize the quantum phase transition in the Dicke model.
Information about quantum phase transitions in conventional condensed matter systems, must be sought by probing the matter system itself. By contrast, we show that mixed matter-light systems offer a distinct advantage in that the photon field carries clear signatures of the associated quantum critical phenomena. Having derived an accurate, size-consistent Hamiltonian for the photonic field in the well-known Dicke model, we predict striking behavior of the optical squeezing and photon statistics near the phase transition. The corresponding dynamics resemble those of a degenerate parametric amplifier. Possible realizations include atom-cavity, nanostructure-cavity, and nanostructure-photonic-band-gap systems.
The single-mode Dicke model is well known to undergo a quantum phase transition from the so-called normal phase to the superradiant phase (hereinafter called the 'superradiant quantum phase transition'). Normally, quantum phase transitions can be identified by the critical behavior of quantities such as entanglement, quantum fluctuations, and fidelity. In this paper, we study the role of the quantum Fisher information (QFI) of both the field mode and the atoms in the ground state of the Dicke Hamiltonian. For a finite but large number of atoms, our numerical results show that near the critical atom-field coupling, the QFI of the atomic and the field subsystems can surpass their classical limits, due to the appearance of nonclassical quadrature squeezing. As the coupling increases far beyond the critical point, each subsystem becomes a highly mixed state, which degrades the QFI and hence the ultimate phase sensitivity. In the thermodynamic limit, we present the analytical results of the QFI and their relationship with the reduced variances of the field mode and the atoms. For each subsystem, we find that there is a singularity in the derivative of the QFI at the critical point, a clear signature of the quantum criticality in the Dicke model.
The Dicke model describing an ensemble of two-state atoms interacting with a single quantized mode of the electromagnetic field (with omission of the ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{A}}^{2}$ term) exhibits a zero-temperature phase transition at a critical value of the dipole coupling strength. We propose a scheme based on multilevel atoms and cavity-mediated Raman transitions to realize an effective Dicke model operating in the phase transition regime. Optical light from the cavity carries signatures of the critical behavior, which is analyzed for the thermodynamic limit where the number of atoms is very large.
The Dicke model was proposed by Dicke in 1954 to describe a single mode of photon coupled to an assembly of N atoms with the same strength. It was found that there is a quantum phase transition from a normal phase at weak coupling to a super-radiant phase at strong coupling at the thermodynamic limit N =∞. Here, we solve the model at N ≥ 1 and also discuss its possible experimental implementations inside a cavity. By studying the Dicke model by 1/N expansion, we identify an emergent quantum phase diffusion mode inside the super-radiant phase and also work out many remarkable experimental consequences of this mode such as its low frequency, photon number squeezing properties and photon statistics. The energy of the phase diffusion modecan be continuously tuned into many frequency ranges from Micro-wave to Terahertz (THz) to Infra-red. The photons from the super-radiant phase are in a number squeezed state with much enhanced signal/noise ratio which may have wide applications in quantum information processing. The photon statistics is strong sub-Poissonian. The effects of dissipations due to leaking photons out of the cavity are also discussed. The connections with the recent experiments of the strong coupling of a BEC of N ~ 105 87Rb atoms to the photons inside an ultrahigh-finess optical cavity are carefully analyzed. Several experimental schemes to detect the phase diffusion mode are proposed.
doi:10.1088/1367-2630/16/6/063039 The single-mode Dicke model is well known to undergo a quantum phase transition from the so-called normal phase to the superradiant phase (hereinafter called the ‘superradiant quantum phase transition’). Normally, quantum phase transitions can be identified by the critical behavior of quantities such as entanglement, quantum fluctuations, and fidelity. In this paper, we study the role of the quantum Fisher information (QFI) of both the field mode and the atoms in the ground state of the Dicke Hamiltonian. For a finite but large number of atoms, our numerical results show that near the critical atom-field coupling, the QFI of the atomic and the field subsystems can surpass their classical limits, due to the appearance of nonclassical quadrature squeezing. As the coupling increases far beyond the critical point, each subsystem becomes a highly mixed state, which degrades the QFI and hence the ultimate phase sensitivity. In the thermodynamic limit, we present the analytical results of the QFI and their relationship with the reduced variances of the field mode and the atoms. For each subsystem, we find that there is a singularity in the derivative of the QFI at the critical point, a clear signature of the quantum criticality in the Dicke model.
The transition to the super-radiant regime in the Dicke model of two-level atoms interacting with a one-mode radiation field has been argued to be impossible for many years. Recently, however, experimental results show that this transition is possible in an open system [1, 2]. Here we exhibit the power of the use of coherent states with restored symmetry to describe the superradiant regime of the Dicke Hamiltonian. These symmetry-adapted states allow for an analytic description of both the ground and first excited states. From these, analytical expressions for all expectation values of interest are obtained, valid for any number of atoms. All these results are exact in the thermodynamic limit.