Dicke model: Entanglement as a finite size effect

We analyze the Dicke model at zero temperature by matrix diagonalization to determine the entanglement in the ground state. In the infinite system limit the mean field approximation predicts a quantum phase transition from a non-interacting state to a BoseEinstein condensate at a threshold coupling. We show that in a finite system the spin part of the ground state is a bipartite entangled state, which can be tested by probing two parts of the spin system separately, but only in a narrow regime around the threshold coupling. Around the resonance, the size of this regime is inversely proportional to the number of spins and shrinks down to zero for infinite systems. This spin entanglement is a non-perturbative effect and is also missed by the mean-field approximation. Coherent interaction between electromagnetic photon fields and matter attracted interest a long time ago with renewed attention gained in the last decade due to significant developments in the experimental techniques in various areas of physics. Achievement of Bose Einstein condensation of cold atomic gases in electromagnetic traps enabled the coherent coupling of hyperfine states of 105 atoms to a single photon mode of an optical resonator [1]. Advances in the semiconductor technology allowed to obtain optical microcavities where electron-hole excitations inside the semiconductor quantum well are strongly coupled to an eigenmode of the optical resonator [2]. Strong coupling of a single mode of a transmission line resonator to a Cooper pair box and a quantized mode of an optical crystal cavity to several semiconductor quantum dots [3] have been demonstrated as a possible way to a quantum computing device [4]. Theoretical understanding of all these systems is based on a model proposed by Dicke [5] which describes N spins 1/2 (identical two-level systems) with splitting energy 2! coupled to a single mode of electromagnetic field ω. It was shown that this model is exactly diagaonalisable [6]. At zero temperature it undergoes a quantum phase transition from a non interacting state with unpopulated bosonic mode to a condensed state with with a highly populated bosonic mode [7] if coupling between the boson and a single spin g is greater than a threshold value. In the thermodynamic limit a phase transition occurs in the region of strong coupling if temperature is less than a critical temperature which can be described by a Bogolyubov Hamiltonian similarly to the pairing model of superconductivity [8]. Recently, a variational wave function approach to the generalised Dicke model was used [9] to describe Bose Einstein condensation of excitonpolaritons in a semiconductor optical cavity. In this paper we analyse the Dicke model at zero temperature for a finite N using matrix diagonalisation methods. We find that for the particular coupling strength,