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Transitions de phase quantiques dans des modèles de Spin collectif : applications au calcul adiabatique

Pedro Ribeiro

Published Jan 1, 2008• 0 citations

Abstract

Abstract Part I: Quantum Collective Spin Systems We use a coherent spin state formalism in order to study collective spin models, which have many applications in physics. In particular, the Lipkin-Meshkov-Glick (LMG) has been analyzed in the thermodynamic limit. The method developed during this work can, in principle, be used for more general Hamiltonians written as a function of the su(2) algebra generators. We have derived exact expressions for the integrated density of states for this model. The nature of the density of states singularities has been detailed. The first order finite size corrections as well as observables expectation values were also computed. The standard Bohr-Sommerfeld quantization approach, adapted to the spin case, is no longer valid near the spectral singularities. In order to treat these cases, we have developed a new approach that permits to describe the spectrum in the neighbourhood of the critical points. Part II: Adiabatic Quantum Computation We have proposed a simple model which highlights the relations between quantum phase transitions and adiabatic (quantum) computation. This model puts in evidence the importance of the choice of the initial Hamiltonian and of the adiabatic path in the parameter space; it should be helpful as a toy model for more realistic cases. We have also studied the state populations dynamics when a quantum phase transition point is crossed, for the LMG model studied in the first part. Numerical simulations show that the dynamics of the populations is very sensitive to the presence of exceptional points in the spectrum, which a simplified model for the quantum evolution already suggests.

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Abstract Part I: Quantum Collective Spin Systems We use a coherent spin state formalism in order to study collective spin models, which have many applications in physics. In particular, the Lipkin-Meshkov-Glick (LMG) has been analyzed in the thermodynamic limit. The method developed during this work can, in principle, be used for more general Hamiltonians written as a function of the su(2) algebra generators. We have derived exact expressions for the integrated density of states for this model. The nature of the density of states singularities has been detailed. The first order finite size corrections as well as observables expectation values were also computed. The standard Bohr-Sommerfeld quantization approach, adapted to the spin case, is no longer valid near the spectral singularities. In order to treat these cases, we have developed a new approach that permits to describe the spectrum in the neighbourhood of the critical points. Part II: Adiabatic Quantum Computation We have proposed a simple model which highlights the relations between quantum phase transitions and adiabatic (quantum) computation. This model puts in evidence the importance of the choice of the initial Hamiltonian and of the adiabatic path in the parameter space; it should be helpful as a toy model for more realistic cases. We have also studied the state populations dynamics when a quantum phase transition point is crossed, for the LMG model studied in the first part. Numerical simulations show that the dynamics of the populations is very sensitive to the presence of exceptional points in the spectrum, which a simplified model for the quantum evolution already suggests.

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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.

Driven-dissipative collective spin systems can host several long-lived metastable states; rare switching between them governs slow relaxation and first-order dissipative phase transitions associated with changes in the dominant stationary state. In the companion Letter [arXiv:2604.15091], we introduced a quantum instanton framework based on exact quasiprobability dynamics to describe this switching and illustrated it with a minimal model possessing a simplifying symmetry. Here we develop its numerical implementation, combining collocation and shooting methods, and apply it to two microscopically motivated systems: the Lipkin-Meshkov-Glick model with purely collective dissipation and the cooperative resonance fluorescence with collective and local dissipation. By determining the leading exponential scaling of switching rates between metastable attractors, we predict the stationary magnetization, the first-order dissipative phase-transition point, and the exponential scaling of slow relaxation rates. These predictions agree with quantum master-equation calculations and an exact analytical result for the second model without dephasing. We also examine how the choice of quasiprobability representation affects numerical convergence and find that the alternative semiclassical Wigner approach, based on Gaussian truncation, has varying accuracy across models and parameter regimes. These results substantiate the framework's applicability to experimentally relevant collective spin systems with both collective and local dissipation.

It is of high interest, in the context of adiabatic quantum computation, to better understand the complex dynamics of a quantum system subject to a time-dependent Hamiltonian, when driven across a quantum phase transition. We present here such a study in the Lipkin-Meshkov-Glick (LMG) model with one variable parameter. We first display numerical results on the dynamical evolution across the LMG quantum phase transition, which clearly shows a pronounced effect of the spectral avoided level crossings. We then derive a phenomenological (classical) transition model, which already shows some closeness to the numerical results. Finally, we show how a simplified quantum transition model can be built which strongly improve the classical approach, and shed light on the physical processes involved in the whole LMG quantum evolution. From our results, we argue that the commonly used description in term of Landau-Zener transitions is not appropriate for our model.

Quantum and classical pairwise correlations in two typical collective spin systems (i.e., the Dicke model and the Lipkin-Meshkov-Glick model) are discussed. These correlations in the thermodynamical limit are obtained analytically and in a finite-size system are calculated numerically. Large-size scaling behavior for the quantum discord itself is observed, which has never been reported in another critical system. A logarithmic diverging behavior for the first derivative of the quantum discord is also found in both models, which might be universal in the second-order quantum phase transition. It is suggested that the pronounced maximum or minimum of first derivative of quantum discord signifies the critical point. Comparisons between the quantum discord and the scaled concurrence are performed. It is shown that the quantum discord is very small in one phase and robust in the other phase, while the scaled concurrence shows maximum at the critical point and decays rapidly when away from the the critical point.