Scaling behavior of the adiabatic Dicke model

Abstract

We analyze the quantum phase transition for a set of N -two level systems interacting with a bosonic mode in the adiabatic regime. Through the Born-Oppenheimer approximation, we obtain the finite-size scaling expansion for many physical observables and, in particular, for the entanglement content of the system. PACS numbers: 64.60. Fr, 03.65.Ud, 05.70.Jk, 73.43.Nq Introduction. Many-body systems have become of central interest in the realm of quantum information theory as training grounds to study static, dynamical and sharing properties of quantum correlations. It has been natural, then, to employ the entanglement as a tool to analyze quantum phase transitions, one of the most striking consequences of quantum correlations in many body systems [1,2,3]. In this letter, we study the Dicke superradiant phase transition and obtain the finite size scaling behavior of the quantum correlations at criticality.The Dicke model (DM) describes the interaction of N two-level systems (qubits) with a single bosonic mode [4], and has become a paradigmatic example of collective quantum behavior. It exhibits a second-order phase transition [5], which has been studied extensively [6,7,8]. The continued interest in the DM arises from its broad application range [9] and from its rich dynamics, displaying many non-classical features [10,11,12,13]. The ground state entanglement of the DM has been recently analyzed [14,15,16], and some aspect of its finite size behavior has been obtained numerically. Finally, Vidal and Dusuel,[17], obtained the critical exponents by a modified Holstein-Primakoff approach.The exact treatment of the finite-size corrections to the Dicke transition is quite complicated and the study of some limiting cases can be useful. In this Letter we analyze the case of N qubits coupled to a slow oscillator. We discuss on equal footings both the finite size and the thermodynamic limit of the model, thus allowing to obtain the phase transition as well as its precursors at finite N . Indeed, we obtain the dominant scaling behavior for some entanglement measures and the entire 1/N expansion for all of the relevant physical observables, such as the order parameter. Concerning the quantum correlations, we argue that what is really relevant for the critical behavior is the bi-partite entanglement between the oscillator mode and the set of two level systems.Adiabatic limit. The interaction of N identical qubits with a bosonic mode is described by the Hamiltonian

Quantum phase transitions in coupled two-level atoms in a single-mode cavity

“…Quantum phase transitions (QPTs) in the original Dicke model has attracted considerable attentions recently [14][15][16][17][18][19][20]. With the consideration of additional interatomic interactions, one nature question is its effect on the modified Dicke model.…”

Large-Nscaling behavior of the ground-state energy, fidelity, and the order parameter in the Dicke model

“…By a modified Holstein-Primakoff approach, Vidal and Dusuel has predicted theoretically the nontrivial scaling exponent for several quantities in the Dicke model [8]. To our knowledge, the finite-size studies in the Dicke model were previously limited to numerical diagonalization in Bosonic Fock state [3,4,5] in small size system N ≤ 35, the adiabatic approximation [6]. Recently, by using extended bosonic coherent states, a numerically exact technique to solve the Dicke model for large system size was developed by the present authors [11].…”

The Critical Point Entanglement and Chaos in the Dicke Model

“…However, the level statistics only becomes meaningful when a sufficient number of excited states in the model is known. On the other hand, for finite N cases, the system is non-integrable in general [16], with the finite-size effect shown to be crucial in understanding the properties of entanglement [5,7,8,[17][18][19]. As has been shown in [20] recently, the only exception is the model with the j = 1/2 case, which is not only exactly solvable, but also integrable.…”