Surprising Relations between Parametric Level Correlations and Fidelity Decay

Abstract

Relations among fidelity, cross-form-factor (i.e., parametric level correlations), and level velocity correlations are found both by deriving a Ward identity in a two-matrix model and by comparing exact results, using supersymmetry techniques, in the framework of random matrix theory. A power law decay near Heisenberg time, as a function of the relevant parameter, is shown to be at the root of revivals recently discovered for fidelity decay. For cross-form-factors the revivals are illustrated by a numerical study of a multiply kicked Ising spin chain. DOI: 10.1103/PhysRevLett.100.190404 PACS numbers: 03.65.Sq, 03.65.Yz, 05.30.Ch, 05.45.Mt Fidelity decay presently attracts considerable attention [1]. It measures the change of quantum dynamics of a state under a modification of the Hamiltonian. In quantum information, fidelity measures the deviation between a mathematical algorithm and its physical implementation. From a different point of view, important insight into the properties of the underlying systems is provided by the studies of correlations between spectra of random and/or chaotic Hamiltonians which differ by a parameterdependent perturbation [2]. Since statistical properties of fidelity decay in random or chaotic systems involve both spectra and eigenfunctions of the original and perturbed Hamiltonians, the existence of any connections between fidelity and purely spectral correlations is not a priori obvious.Random matrix theory (RMT) has been successful in describing quantum many-body systems and as a model for the spectral properties of single particle systems whose classical analogue is chaotic [3]. Within RMT fidelity was analyzed in linear response approximation [4] and both fidelity [5][6][7] and parametric correlations [8] were calculated exactly using the supersymmetry method. An unexpected fidelity revival at Heisenberg time was encountered [5] within RMT and confirmed in a dynamical coupled spin chain model [9].Earlier, differential relations between parametric spectral correlations and parametric density correlations were established [10,11]. By relating the latter to the fidelity amplitude via Fourier transform, we show in this Letter that the existence of these relations opens a crucial insight into the properties of fidelity decay. By analyzing the characteristic features of the parametric correlations in the time domain, the cross-form-factor, we discover a new, simple interpretation of the previously puzzling phenomenon of revival [5]. These relations follow directly from the basic definitions and symmetries of the underlying matrix models, being essentially Ward identities. We show that they are valid under very general assumptions. No explicit (e.g., supersymmetric) calculation is required; however, they rely on the universality of the parametric spectral correlations at the scale of mean level spacing. We thus explain the origin of various relations connecting spectral and wave-function correlations, and establish a unified framework for their analysis and generalizations. A relat...

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“…For strong perturbations the cross formfactor develops peaks at Heisenberg time and for case II at twice the Heisenberg time (not shown here). It has its cause in the algebraic decay of the cross form-factor at these specific times [8]. At all other times it decays exponentially.…”

Fidelity Decay in Chaotical and Random Systems

“…We do not state the rather complicated expressions here, but refer the reader to a compilation of the pertinent formulae in Ref. [20]. In …”

“…A parametric spectral form factor can be dened for parametric energy correlations by the Fourier transformation in the same way as for pure, non-parametric RMT ensembles. It is called cross-form factor in [20]. Formally it is dened as the ensemble average of…”

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Experimental Determination of the Autocorrelation Function of Level Velocities for Microwave Networks Simulating Quantum Graphs

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