arXiv subjects

J. G. Hirsch

Publications and source records attributed to J. G. Hirsch.

At least 19 recordsLinked to original sources

arXiv2021

Quantum localization measures in phase space

Measuring the degree of localization of quantum states in phase space is essential for the description of the dynamics and equilibration of quantum systems, but this topic is far from being understood. There is no unique way to measure localization, and individual measures can reflect different aspects of the same quantum state. Here, we present a general scheme to define localization in measure spaces, which is based on what we call Rényi occupations, from which any measure of localization can be derived. We apply this scheme to the four-dimensional unbounded phase space of the interacting spin-boson Dicke model. In particular, we make a detailed comparison of two localization measures based on the Husimi function in the regime where the model is chaotic, namely one that projects the Husimi function over the finite phase space of the spin and another that uses the Husimi function defined over classical energy shells. We elucidate the origin of their differences, showing that in unbounded spaces the definition of maximal delocalization requires a bounded reference subspace, with different selections leading to contextual answers.

quant-ph↗

arXiv2020

Quantum Phase Transition and Berry Phase in an Extended Dicke Model

We investigate quantum phase transitions, quantum criticality, and Berry phase for the ground state of an ensemble of non-interacting two-level atoms embedded in a non-linear optical medium, coupled to a single-mode quantized electromagnetic field. The optical medium is pumped externally through a classical electric field, so that there is a degenerate parametric amplification effect, which strongly modifies the field dynamics without affecting the atomic sector. Through a semiclassical description the different phases of this extended Dicke model are described. The quantum phase transition is characterized with the expectation values of some observables of the system as well as the Berry phase and its first derivative, where such quantities serve as order parameters. It is remarkable that the model allows the control of the quantum criticality through a suitable choice of the parameters of the non-linear optical medium, which could make possible the use of a low intensity laser to access the superradiant region experimentally.

quant-ph↗

arXiv2019

Dynamical signatures of quantum chaos and relaxation timescales in a spin-boson system

Quantum systems whose classical counterparts are chaotic typically have highly correlated eigenvalues and level statistics that coincide with those from ensembles of full random matrices. A dynamical manifestation of these correlations comes in the form of the so-called correlation hole, which is a dip below the saturation point of the survival probability's time evolution. In this work, we study the correlation hole in the spin-boson (Dicke) model, which presents a chaotic regime and can be realized in experiments with ultracold atoms and ion traps. We derive an analytical expression that describes the entire evolution of the survival probability and allows us to determine the timescales of its relaxation to equilibrium. This expression shows remarkable agreement with our numerical results. While the initial decay and the time to reach the minimum of the correlation hole depend on the initial state, the dynamics beyond the hole up to equilibration is universal. We find that the relaxation time of the survival probability for the Dicke model increases linearly with system size.

cond-mat.stat-mech↗

arXiv2017

Dynamics of Coherent States in Regular and Chaotic Regimes of the Non-integrable Dicke Model

The quantum dynamics of initial coherent states is studied in the Dicke model and correlated with the dynamics, regular or chaotic, of their classical limit. Analytical expressions for the survival probability, i.e. the probability of finding the system in its initial state at time $t$, are provided in the regular regions of the model. The results for regular regimes are compared with those of the chaotic ones. It is found that initial coherent states in regular regions have a much longer equilibration time than those located in chaotic regions. The properties of the distributions for the initial coherent states in the Hamiltonian eigenbasis are also studied. It is found that for regular states the components with no negligible contribution are organized in sequences of energy levels distributed according to Gaussian functions. In the case of chaotic coherent states, the energy components do not have a simple structure and the number of participating energy levels is larger than in the regular cases.

quant-ph↗

arXiv2017

Regularity and chaos in cavity QED

The interaction of a quantized electromagnetic field in a cavity with a set of two-level atoms inside can be described with algebraic Hamiltonians of increasing complexity, from the Rabi to the Dicke models. Their algebraic character allows, through the use of coherente states, a semiclassical description in phase space, where the non-integrable Dicke model has regions associated with regular and chaotic motion. The appearance of classical chaos can be quantified calculating the largest Lyapunov exponent in the whole available phase space for a given energy. In the quantum regime, employing efficient diagonalization techniques, we are able to perform a detailed quantitative study of the regular and chaotic regions, where the quantum Participation Ratio (PR) of coherent states on the eigenenergy basis plays a role equivalent to the Lyapunov exponent. It is noted that, in the thermodynamic limit, dividing the Participation Ratio by the number of atoms leads to a positive value in chaotic regions, while it tends to zero in the regular ones.

quant-ph↗

arXiv2016

Adiabatic invariants for the regular region of the Dicke model

Adiabatic invariants are introduced and shown to provide an approximate second integral of motion for the non-integrable Dicke model, in the energy region where the system exhibits a regular dynamics. This low-energy region is always present and has been described both in a semiclassical and a full quantum analysis. Its Peres lattices exhibit that many observables vary smoothly with energy, along lines which beg for a formal description. It is shown how the adiabatic invariants provide a rationale to their presence in many cases. They are built employing the Born-Oppenheimer approximation, valid when a fast system is coupled to a much slower one. As the Dicke model has a one bosonic and one fermionic degree of freedom, two versions of the approximation are used, depending on which one is the faster. In both cases a noticeably accord with exact numerical results is obtained. The employment of the adiabatic invariants provides a simple and clear theoretical framework to study the physical phenomenology associated to this energy regime, far beyond the energies where the quadratic approximation can be employed.

quant-ph↗

arXiv2016

Classical chaos in atom-field systems

The relation between the onset of chaos and critical phenomena, like Quantum Phase Transitions (QPT) and Excited-State Quantum Phase transitions (ESQPT), is analyzed for atom-field systems. While it has been speculated that the onset of hard chaos is associated with ESQPT based in the resonant case, the off-resonant cases show clearly that both phenomena, ESQPT and chaos, respond to different mechanisms. The results are supported in a detailed numerical study of the dynamics of the semiclassical Hamiltonian of the Dicke model. The appearance of chaos is quantified calculating the largest Lyapunov exponent for a wide sample of initial conditions in the whole available phase space for a given energy. The percentage of the available phase space with chaotic trajectories is evaluated as a function of energy and coupling between the qubit and bosonic part, allowing to obtain maps in the space of coupling and energy, where ergodic properties are observed in the model. Different sets of Hamiltonian parameters are considered, including resonant and off-resonant cases.

nlin.CD↗

arXiv2015

Delocalization and quantum chaos in atom-field systems

Employing efficient diagonalization techniques, we perform a detailed quantitative study of the regular and chaotic regions in phase space in the simplest non-integrable atom-field system, the Dicke model. A close correlation between the classical Lyapunov exponents and the quantum Participation Ratio of coherent states on the eigenenergy basis is exhibited for different points in the phase space. It is also shown that the Participation Ratio scales linearly with the number of atoms in chaotic regions, and with its square root in the regular ones.

quant-ph↗

arXiv2014

Comparative quantum and semi-classical analysis of Atom-Field Systems I: density of states and excited-state quantum phase transitions

We study the non-integrable Dicke model, and its integrable approximation, the Tavis-Cummings model, as functions of both the coupling constant and the excitation energy. Excited-state quantum phase transitions (ESQPT) are found analyzing the density of states in the semi-classical limit and comparing it with numerical results for the quantum case in large Hilbert spaces, taking advantage of efficient methods recently developed. Two different ESQPTs are identified in both models, which are signaled as singularities in the semi-classical density of states, one {\em static} ESQPT occurs for any coupling, whereas a dynamic ESQPT is observed only in the superradiant phase. The role of the unstable fixed points of the Hamiltonian semi-classical flux in the occurrence of the ESQPTs is discussed and determined. Numerical evidence is provided that shows that the semi-classical result describes very well the tendency of the quantum energy spectrum for any coupling in both models. Therefore the semi-classical density of states can be used to study the statistical properties of the fluctuation in the spectra, a study that is presented in a companion paper.

quant-ph↗

arXiv2014

Comparative quantum and semi-classical analysis of Atom-Field Systems II: Chaos and regularity

The non-integrable Dicke model and its integrable approximation, the Tavis-Cummings (TC) model, are studied as functions of both the coupling constant and the excitation energy. The present contribution extends the analysis presented in the previous paper by focusing on the statistical properties of the quantum fluctuations in the energy spectrum and their relation with the excited state quantum phase transitions (ESQPT). These properties are compared with the dynamics observed in the semi-classical versions of the models. The presence of chaos for different energies and coupling constants is exhibited, employing Poincaré sections and Peres lattices in the classical and quantum versions, respectively. A clear correspondence between the classical and quantum result is found for systems containing between $\mathcal{N} = 80$ to $200$ atoms. A measure of the Wigner character of the energy spectrum for different couplings and energy intervals is also presented employing the statistical Anderson-Darling test. It is found that in the Dicke Model, for any coupling, a low energy regime with regular states is always present. The richness of the onset of chaos is discussed both for finite quantum systems and for the semi-classical limit, which is exact when the number of atoms in the system tends to infinite.

quant-ph↗

arXiv2013

Neutrinoless $ββ$ decay transition matrix elements within mechanisms involving light Majorana neutrinos, classical Majorons and sterile neutrinos

In the PHFB model, uncertainties in the nuclear transition matrix elements for the neutrinoless double-$β$ decay of $\ ^{94,96}$Zr, $^{98,100}$Mo, $^{104}$Ru, $^{110}$Pd, $^{128,130}$Te and $^{150}$Nd isotopes within mechanisms involving light Majorana neutrinos, classical Majorons and sterile neutrinos are statistically estimated by considering sets of sixteen (twenty-four) matrix elements calculated with four different parametrization of the pairing plus multipolar type of effective two-body interaction, two sets of form factors and two (three) different parameterizations of Jastrow type of short range correlations. In the mechanisms involving the light Majorana neutrinos and classical Majorons, the maximum uncertainty is about 15% and in the scenario of sterile neutrinos, it varies in between approximately 4 (9)%--20 (36)% without(with) Jastrow short range correlations with Miller-Spencer parametrization, depending on the considered mass of the sterile neutrinos.

nucl-th↗

arXiv2013

Fidelity, susceptibility and critical exponents in the Dicke model

We calculate numerically the fidelity and its susceptibility for the ground state of the Dicke model. A minimum in the fidelity identifies the critical value of the interaction where a quantum phase crossover, the precursor of a phase transition for finite number of atoms N, takes place. The evolution of these observables is studied as a function of N, and their critical exponents evaluated. Using the critical exponents the universal curve for the specific susceptibility is recovered. An estimate to the precision to which the ground state wave function is numerically calculated is given, and found to have its lowest value, for a fixed truncation, in a vicinity of the critical coupling.

quant-ph↗

arXiv2012

Uncertainties in nuclear transition matrix elements for $β^{+}β^{+}$ and $\varepsilon β^{+}$ modes of neutrinoless positron double-$β$ decay within PHFB model

Uncertainties in the nuclear transition matrix elements $M^{(0ν)}$ and $M^{(0N)}$ of the double-positron emission $(β^{+}β^{+})_{0ν}$ and electron-positron conversion $(\varepsilon β^{+})_{0ν}$ modes due to the exchange of light and heavy Majorana neutrinos, respectively, are calculated for $^{96}$Ru, $^{102}$Pd, $^{106}$Cd, $^{124}$Xe, $^{130}$Ba and $^{156}$Dy isotopes by employing the PHFB model with four different parameterization of the pairing plus multipolar two-body interactions and three different parameterizations of the Jastrow short range correlations. In all cases but for $^{130}$Ba, the uncertainties are smaller than 14% for light Majorana neutrino exchange and 35% for the exchange of a heavy Majorana neutrino.

nucl-th↗

arXiv2012

Phase transitions with finite atom number in the Dicke Model

Two-level atoms interacting with a one mode cavity field at zero temperature have order parameters which reflect the presence of a quantum phase transition at a critical value of the atom-cavity coupling strength. Two popular examples are the number of photons inside the cavity and the number of excited atoms. Coherent states provide a mean field description, which becomes exact in the thermodynamic limit. Employing symmetry adapted (SA) SU(2) coherent states (SACS) the critical behavior can be described for a finite number of atoms. A variation after projection treatment, involving a numerical minimization of the SA energy surface, associates the finite number phase transition with a discontinuity in the order parameters, which originates from a competition between two local minima in the SA energy surface.

quant-ph↗

arXiv2012

Shell model description of Ge isotopes

A shell model study of the low energy region of the spectra in Ge isotopes for $38\leq N\leq 50$ is presented, analyzing the excitation energies, quadrupole moments, $B(E2)$ values and occupation numbers. The theoretical results have been compared with the available experimental data. The shell model calculations have been performed employing three different effective interactions and valence spaces.We have used two effective shell model interactions, JUN45 and jj44b, for the valence space $f_{5/2} \, p \,g_{9/2}$ without truncation. To include the proton subshell $f_{7/2}$ in valence space we have employed the $fpg$ effective interaction due to Sorlin {\it et al.}, with $^{48}$Ca as a core and a truncation in the number of excited particles.

nucl-th↗

arXiv2012

Uncertainties in nuclear transition matrix elements for neutrinoless $ββ$ decay II: the heavy Majorana neutrino mass mechanism

Employing four different parametrization of the pairing plus multipolar type of effective two-body interaction and three different parametrizations of Jastrow-type of short range correlations, the uncertainties in the nuclear transition matrix elements $M_{N}^{(0ν)}$ due to the exchange of heavy Majorana neutrino for the $0^{+}\rightarrow 0^{+}$ transition of neutrinoless double beta decay of $^{94}$Zr, $^{96}$Zr, $^{98}$Mo, $^{100}$Mo, $^{104}$Ru, $^{110}$Pd, $^{128,130}$Te and $^{150}$Nd isotopes in the PHFB model are estimated to be around 25%. Excluding the nuclear transition matrix elements calculated with Miller-Spenser parametrization of Jastrow short range correlations, the uncertainties are found to be 10%-15% smaller.

nucl-th↗

arXiv2011

On the Superradiant Phase in Field-Matter Interactions

We show that semi-classical states adapted to the symmetry of the Hamiltonian are an excellent approximation to the exact quantum solution of the ground and first excited states of the Dicke model. Their overlap to the exact quantum states is very close to 1 except in a close vicinity of the quantum phase transition. Furthermore, they have analytic forms in terms of the model parameters and allow us to calculate analytically the expectation values of field and matter observables. Some of these differ considerably from results obtained via the standard coherent states, and by means of Holstein-Primakoff series expansion of the Dicke Hamiltonian. Comparison with exact solutions obtained numerically support our results. In particular, it is shown that the expectation values of the number of photons and of the number of excited atoms have no singularities at the phase transition. We comment on why other authors have previously found otherwise.

quant-ph↗

arXiv2011

Uncertainties in nuclear transition matrix elements for neutrinoless $ββ$ decay within the PHFB model

The nuclear transition matrix elements $M^{(0ν)}$ for the neutrinoless double beta decay of $^{94,96}$Zr, $^{98,100}$Mo, $^{104}$Ru, $^{110}$Pd, $^{128,130}$Te and $^{150}$Nd isotopes in the case of $0^{+}\rightarrow 0^{+}$ transition are calculated using the PHFB wave functions, which are eigenvectors of four different parameterizations of a Hamiltonian with pairing plus multipolar effective two-body interaction. \QCOM{35}{In addition, the consideration of} Employing two (three) different parameterizations of Jastrow-type short range correlations, \QCOM{19}{provides us with} a set of eight (twelve) different nuclear transition matrix elements $M^{(0ν)}$ is built for each decay, whose averages in conjunction with their standard deviations provide an estimate of the model uncertainties.

nucl-th↗