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arXiv:2002.02491 · Physica Scripta

Quantum phase diagrams of matter-field Hamiltonians I: Fidelity, Bures distance, and entanglement

Sergio Cordero, Eduardo Nahmad-Achar, Ramón López–Peña, Octavio Castaños

Published Dec 23, 2020• 13 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1402-4896/abd653

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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.
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.
Abstract The quantum phase diagram for a finite three-level system in the Λ configuration, interacting with a two-mode electromagnetic field in a cavity, is determined by means of information measures such as fidelity, fidelity susceptibility and entanglement, applied to the reduced density matrix of the matter sector of the system. The quantum phases are explained by emphasizing the spontaneous symmetry breaking along the separatrix. Additionally, a description of the reduced density matrix of one atom in terms of a simplex allows a geometric representation of the entanglement and purity properties of the system. These concepts are calculated for both, the symmetry-adapted variational coherent states and the numerical diagonalisation of the Hamiltonian, and compared. The differences in purity and entanglement obtained in both calculations can be explained and visualised by means of this simplex representation.
Quantum phase transition in Ξ\XiΞ-configuration Tavis-Cummings model driven by an electromagnetic field for a finite number of atoms
The quantum phase transitions of a three-level Tavis-Cummings model in the \Xi Ξ -configuration driven by a single-mode electromagnetic field in a QED cavity are studied for n_a=1,\,2 n a = 1 , 2 , and 4 4 atoms. Although conventional observables like photon number and energy level populations suffice to locate quantum phase transitions, they often lack sensitivity to finite-size effects. Here, we show that the expectation values of the quadratic Casimir invariants associated with the atomic subsystems offer a more robust measure, exhibiting sharp signatures of criticality that are validated by fidelity calculations. Additionally, the von Neumann entropy is employed to map the entanglement landscape between matter and the quantized field. A key finding is that the external driving field acts as a control parameter that shrinks the normal phase region; as the driving intensity increases, the separable ground state vanishes, leading to the dominance of field-matter entanglement across the parameter space.
The reduced density fidelity is a measure of distance between two reduced density matrix, which can be used to characterize quantum phase transitions in quantum many-body systems. In this paper, we use the multi-scale entanglement reorganization ansatz (MERA) algorithm to simulate the spin 1 quantum Blume-Capel model and determine its ground-state phase diagram through calculating the reduced density fidelity. The qualitative relevant information contained in one site reduced density matrix is different from that contained two-site reduced density matrix, which can be detected by using the reduced density fidelity. In addition, we also characterize quantum phase transitions in quantum many-body systems by using the local parameters and energy gaps.
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.
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.