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arXiv:2009.13663 · Physica Scripta

Quantum phase diagrams of matter-field Hamiltonians II: Wigner function analysis

R López-Peña, S Cordero, E Nahmad-Achar, O Castaños

Published Dec 23, 2020• 17 citations
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Abstract

Abstract Non-classical states are of practical interest in quantum computing and quantum metrology. These states can be detected through their Wigner function negativity in some regions. We show that the surfaces of minimum fidelity or maximum Bures distance constitute a signature of quantum phase transitions. Additionally the behaviour of the Wigner function associated to the field modes carry the information of both, the entanglement properties between matter and field sectors, and the regions of the parameter space where the quantum phase transitions take place. A finer classification for the continuous phase transitions is obtained through the computation of the surface of maximum Bures distance.

DOI
10.1088/1402-4896/abd654

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Abstract Non-classical states are of practical interest in quantum computing and quantum metrology. These states can be detected through their Wigner function negativity in some regions. We show that the surfaces of minimum fidelity or maximum Bures distance constitute a signature of quantum phase transitions. Additionally the behaviour of the Wigner function associated to the field modes carry the information of both, the entanglement properties between matter and field sectors, and the regions of the parameter space where the quantum phase transitions take place. A finer classification for the continuous phase transitions is obtained through the computation of the surface of maximum Bures distance.
A general procedure is established to calculate the quantum phase diagrams for finite matter-field Hamiltonian models. The minimum energy surface associated to the different symmetries of the model is calculated as a function of the matter-field coupling strengths. By means of the ground state wave functions, one looks for minimal fidelity or maximal Bures distance surfaces in terms of the parameters, and from them the critical regions of those surfaces characterize the finite quantum phase transitions. Following this procedure for N a = 1 and N a = 4 particles, the quantum phase diagrams are calculated for the generalised Tavis-Cummings and Dicke models of 3-level systems interacting dipolarly with 2 modes of electromagnetic field. For N a = 1, the reduced density matrix of the matter allows us to determine the phase regions in a 2-simplex (associated to a general three dimensional density matrix), on the different 3-level atomic configurations, together with a measurement of the quantum correlations between the matter and field sectors. As the occupation probabilities can be measured experimentally, the existence of a quantum phase diagram for a finite system can be established.
Quantum phase transitions are currently studied in several fields of physics, as examples one finds: in nuclear physics the shape phase transitions within the Interacting Boson Model; in quantum optics two level atoms interacting with a one mode electromagnetic radiation in a cavity; and in condensed matter in the analysis of the behavior of spin systems. In this contribution we employ the fidelity and its susceptibility, two concepts widely used in quantum information theory, to determine the quantum phase transitions of the Lipkin-Meshkov-Glick (LMG) model together with its scaling properties. By means of these concepts, we propose a quantum method to determine the crossings and anti-crossings present in a model Hamiltonian as a function of the control parameters of the model. A review of the separatrix of the LMG model is done to compare the results obtained by means of quantum information concepts with those related with the singular behavior of the energy surface of the model, which is the expectation value of the Hamiltonian with respect to spin coherent states.
We establish, within the second quantization method, the general dipole-dipole Hamiltonian interaction of a system of nnn-level atoms. The variational energy surface of the nnn-level atoms interacting with $\ensuremath{\ell}$-mode fields and under the Van Der Waals forces is calculated with respect to the tensorial product of matter and electromagnetic field coherent states. This is used to determine the quantum phase diagram associated with the ground state of the system and quantify the effect of the dipole-dipole Hamiltonian interaction. By considering real induced electric dipole moments, we find the quantum phase transitions for 2- and 3-level atomic systems interacting with one and two modes of the electromagnetic field, respectively. The corresponding order of the transitions is established by means of Ehrenfest classification; for undetermined cases, we propose two procedures: one based on the difference of the expectation value of the Casimir operators of the 2-level subsystems, and another based on maximizing the Bures distance between neighbor variational solutions.