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arXiv:1503.02027 · Annalen der Physik

Quantum efficiencies in finite disordered networks connected by many‐body interactions

Adrian Ortega, Manan Vyas, Luis Benet

Published Jun 30, 2015• 17 citations
arXivHTMLPDF

Abstract

The quantum efficiency in the transfer of an initial excitation in disordered finite networks, modeled by the k‐body embedded Gaussian ensembles of random matrices, is studied for bosons and fermions. The influence of the presence or absence of time‐reversal symmetry and centrosymmetry/centrohermiticity are addressed. For bosons and fermions, the best efficiencies of the realizations of the ensemble are dramatically enhanced when centrosymmetry (centrohermiticity) is imposed. For few bosons distributed in two single‐particle levels this permits perfect state transfer for almost all realizations when one‐particle interactions are considered. For fermionic systems the enhancement is found to be maximal for cases when all but one single particle levels are occupied. image

DOI
10.1002/andp.201500140

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The quantum efficiency in the transfer of an initial excitation in disordered finite networks, modeled by the k‐body embedded Gaussian ensembles of random matrices, is studied for bosons and fermions. The influence of the presence or absence of time‐reversal symmetry and centrosymmetry/centrohermiticity are addressed. For bosons and fermions, the best efficiencies of the realizations of the ensemble are dramatically enhanced when centrosymmetry (centrohermiticity) is imposed. For few bosons distributed in two single‐particle levels this permits perfect state transfer for almost all realizations when one‐particle interactions are considered. For fermionic systems the enhancement is found to be maximal for cases when all but one single particle levels are occupied. image
The coherent transport of n fermions in disordered networks of l single-particle states connected by k-body interactions is studied. These networks are modeled by embedded Gaussian random matrix ensemble (EGE). The conductance bandwidth and the ensemble-averaged total current attain their maximal values if the system is highly filled n∼l-1 and k∼n/2. For the cases k=1 and k=n the bandwidth is minimal. We show that for all parameters the transport is enhanced significantly whenever centrosymmetric embedded Gaussian ensemble (csEGE) are considered. In this case the transmission shows numerous resonances of perfect transport. Analyzing the transmission by spectral decomposition, we find that centrosymmetry induces strong correlations and enhances the extrema of the distributions. This suppresses destructive interference effects in the system and thus causes backscattering-free transmission resonances that enhance the overall transport. The distribution of the total current for the csEGE has a very large dominating peak for n=l-1, close to the highest observed currents.
The robustness of quantum transport under various perturbations is analyzed in disordered interacting many-body systems, which are constructed from the embedded Gaussian random matrix ensembles (EGEs). The transport efficiency can be enhanced drastically, if centrosymmetry (csEGE) is imposed. When the csEGE is perturbed with an ordinary EGE, the transport efficiency in the optimal cases is reduced significantly, while in the suboptimal cases the changes are less pronounced. Qualitatively the same behavior is observed, when parity and centrosymmetry are broken by block perturbations. Analyzing the influence of the environment coupling, optimal transport is observed at a certain coupling strength, while too weak and too strong coupling reduce the transport. Taking into account the effects of decoherence, in the EGE the transport efficiency approaches its maximum at a finite nonzero decoherence strength (environment-assisted transport). In the csEGE the efficiency decays monotonically with the decoherence but is always larger than in the EGE.
Eigenstate structure in many-body bosonic systems: Analysis using random matrices and qqq-Hermite polynomials
We analyze the structure of eigenstates in many-body bosonic systems by modeling the Hamiltonian of these complex systems using Bosonic Embedded Gaussian Orthogonal Ensembles (BEGOE) defined by a mean-field plus kkk-body random interactions. The quantities employed are the number of principal components (NPC), the localization length (lHl_HlH​) and the entropy production S(t)S(t)S(t). The numerical results are compared with the analytical formulas obtained using random matrices which are based on bivariate qqq-Hermite polynomials for local density of states Fk(E∣q)F_k(E|q)Fk​(E∣q) and the bivariate qqq-Hermite polynomial form for bivariate eigenvalue density ρbiv:q(E,Ek)ρ_{biv:q}(E,E_k)ρbiv:q​(E,Ek​) that are valid in the strong interaction domain. We also compare transport efficiency in many-body bosonic systems using BEGOE in absence and presence of centrosymmetry. It is seen that the centrosymmetry enhances quantum efficiency.
Embedded random matrix ensembles generated by random interactions (of low body rank and usually two-body) in the presence of a one-body mean field, introduced in nuclear structure physics, are now established to be indispensable in describing statistical properties of a large number of isolated finite quantum many-particle systems. Lie algebra symmetries of the interactions, as identified from nuclear shell model and the interacting boson model, led to the introduction of a variety of embedded ensembles (EEs). These ensembles with a mean field and chaos generating two-body interaction generate in three different stages, delocalization of wave functions in the Fock space of the mean-field basis states. The last stage corresponds to what one may call thermalization and complex nuclei, as seen from many shell model calculations, lie in this region. Besides briefly describing them, their recent applications to nuclear structure are presented and they are (i) nuclear level densities with interactions; (ii) orbit occupancies; (iii) neutrinoless double beta decay nuclear transition matrix elements as transition strengths. In addition, their applications are also presented briefly that go beyond nuclear structure and they are (i) fidelity, decoherence, entanglement and thermalization in isolated finite quantum systems with interactions; (ii) quantum transport in disordered networks connected by many-body interactions with centrosymmetry; (iii) semicircle to Gaussian transition in eigenvalue densities with [Formula: see text]-body random interactions and its relation to the Sachdev–Ye–Kitaev (SYK) model for majorana fermions.
We study the fidelity decay of the k‐body embedded ensembles of random matrices for bosons distributed over two single‐particle states. Fidelity is defined in terms of a reference Hamiltonian, which is a purely diagonal matrix consisting of a fixed one‐body term and includes the diagonal of the perturbing k‐body embedded ensemble matrix, and the perturbed Hamiltonian which includes the residual off‐diagonal elements of the k‐body interaction. This choice mimics the typical mean‐field basis used in many calculations. We study separately the cases k = 2 and 3. We compute the ensemble‐averaged fidelity decay as well as the fidelity of typical members with respect to an initial random state. Average fidelity displays a revival at the Heisenberg time, t = tH = 1, and a freeze in the fidelity decay, during which periodic revivals of period tH are observed. We obtain the relevant scaling properties with respect to the number of bosons and the strength of the perturbation. For certain members of the ensemble, we find that the period of the revivals during the freeze of fidelity occurs at fractional times of tH. These fractional periodic revivals are related to the dominance of specific k‐body terms in the perturbation.
Embedded ensembles or random matrix ensembles generated by k-body interactions acting in many-particle spaces are now well established to be paradigmatic models for many-body chaos and thermalization in isolated finite quantum (fermion or boson) systems. In this article, briefly discussed are (i) various embedded ensembles with Lie algebraic symmetries for fermion and boson systems and their extensions (for Majorana fermions, with point group symmetries etc.); (ii) results generated by these ensembles for various aspects of chaos, thermalization and statistical relaxation, including the role of q-hermite polynomials in k-body ensembles; and (iii) analyses of numerical and experimental data for level fluctuations for trapped boson systems and results for statistical relaxation and decoherence in these systems with close relations to results from embedded ensembles.