Inhomogeneous losses and complexness of wave functions in chaotic cavities

Savin, Dmitry V.; Legrand, Olivier; Mortessagne, Fabrice

doi:10.1209/epl/i2006-10358-3

Cited by 33 publications

(63 citation statements)

References 27 publications

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“…The resonance widths Γ n > 0 bring fundamentally new features to the open system in comparison with its closed counterpart. In particular, width fluctuations govern the decay law [8,9] and give rise to nonorthogonal modes [10,11] leading to enhanced sensitivity to perturbations [12]. Various aspects of lifetime and width statistics are a subject of intensive research, both theoretically and experimentally, with recent studies motivated by practical applications including optical microresonators [13], superconductor superlattices [14], many-body fermionic systems [15], microwave billiards [16,17], and dissipative quantum maps [18,19], as well as by long-standing interest in superradiance-like "resonance trapping" phenomena [4,15,[20][21][22].…”

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confidence: 99%

Resonance width distribution in RMT: Weak-coupling regime beyond Porter-Thomas

Fyodorov¹,

Savin²

2015

EPL

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-We employ the random matrix theory (RMT) framework to revisit the distribution of resonance widths in quantum chaotic systems weakly coupled to the continuum via a finite number M of open channels. In contrast to the standard first-order perturbation theory treatment we do not a priory assume the resonance widths being small compared to the mean level spacing. We show that to the leading order in weak coupling the perturbative χ 2 M distribution of the resonance widths (in particular, the Porter-Thomas distribution at M = 1) should be corrected by a factor related to a certain average of the ratio of square roots of the characteristic polynomial ('spectral determinant') of the underlying RMT Hamiltonian. A simple single-channel expression is obtained that properly approximates the width distribution also at large resonance overlap, where the Porter-Thomas result is no longer applicable.Introduction.-Scattering of both classical and quantum waves in systems with chaotic intrinsic dynamics is characterised by universal statistical properties [1][2][3]. Those can be understood by studying the properties of quasi-stationary states in an open system formed at the intermediate stage of the scattering process [4][5][6][7]. Such states are associated with the resonances which correspond to the complex poles E n = E n −

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Resonance width distribution in RMT: Weak-coupling regime beyond Porter-Thomas

Fyodorov¹,

Savin²

2015

EPL

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“…Their nonorthogonality is crucial in many applications; it influences nuclear cross sections [5], features in decay laws of quantum chaotic systems [6], and yields excess quantum noise in open laser resonators [7]. For systems invariant under time reversal, like open microwave cavities studied below, the nonorthogonality is due to the complex wave functions, yielding the so-called phase rigidity [8][9][10] and mode complexness [11,12]. Nonorthogonal mode patterns also appear in reverberant dissipative bodies [13], elastic plates [14], optical microstructures [15] and lossy random media [16].…”

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“…We also took into account the absorption width Γ (w) n due to the finite conductivity of the metallic walls, but neglected its variations, since Γ (w) n as a function of the parameter induces much smaller changes than those due to the coupled antennas. Note that we do not assume that Γ (w) n is the same for all resonances [11,32,33].…”

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Experimental Width Shift Distribution: A Test of Nonorthogonality for Local and Global Perturbations

et al. 2014

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The change of resonance widths in an open system under a perturbation of its interior has been recently introduced by Fyodorov and Savin [Phys. Rev. Lett. 108, 184101 (2012)] as a sensitive indicator of the nonorthogonality of resonance states. We experimentally study universal statistics of this quantity in weakly open two-dimensional microwave cavities and reverberation chambers realizing scalar and electromagnetic vector fields, respectively. We consider global as well as local perturbations, and also extend the theory to treat the latter case. The influence of the perturbation type on the width shift distribution is more pronounced for many-channel systems. We compare the theory to experimental results for one and two attached antennas and to numerical simulations with higher channel numbers, observing a good agreement in all cases. The most general feature of open quantum or wave systems is the set of complex resonances. They manifest themselves in scattering through sharp energy variations of the observables and correspond to the complex poles of the S matrix. Theoretically, the latter are given by the eigenvalues E n = E n − i 2 Γ n of the effective non-Hermitian Hamiltonian H eff of the open system [1-4]. The antiHermitian part of H eff originates from coupling between the internal (bound) and continuum states, giving rise to finite resonance widths Γ n > 0. The other key feature is that the eigenfunctions of H eff are nonorthogonal [2,4]. Their nonorthogonality is crucial in many applications; it influences nuclear cross sections [5], features in decay laws of quantum chaotic systems [6], and yields excess quantum noise in open laser resonators [7]. For systems invariant under time reversal, like open microwave cavities studied below, the nonorthogonality is due to the complex wave functions, yielding the so-called phase rigidity [8][9][10] and mode complexness [11,12] Recently, such nonorthogonality was identified as the root cause for enhanced sensitivity to perturbations in open systems [17], see also Ref. [18]. Consider the parametric motion of complex resonances under internal perturbations. This can be modeled by a Hermitian term V added to H eff , so H eff = H eff + V . The complex energy shift δE n = E n − E n of the nth resonance is then given by perturbation theory for non-Hermitian operators [17,19], yielding in the leading order δE n = L n |V |R n , where L n | and |R n are the left and right eigenfunctions of H eff corresponding to E n . They form a biorthogonal system; in particular, L n |R m = δ nm but U nm ≡ L n |L m = δ nm in general. U is known in nuclear physics as the Bell-Steinberger nonorthogonality matrix [2,5], see also [20]. Crucially, a nonzero width shift δΓ n = −2Im δE n is induced solely by the off-diagonal elements of U [17]where V nm = R n |V |R m = V * mn . It vanishes only if the resonance states were orthogonal (all U m =n = 0).Note that the nonorthogonality measures studied in Refs. [7][8][9][10][11][12] are related to the diagonal elements U nn . Those and the width s...

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“…[27] and references therein. Note that the spatial properties related to the associated bi-orthogonal eigenvectors are known to a much lesser extent [18,[28][29][30].…”

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“…In contrast to the level curvature distribution, the broadness of the width shift distribution (13) can be additionally controlled and is proportional to √ M ∼ var(κ). Physically, this gives the variance of widths the role of a universal parameter that controls the degree of nonorthogonality in weakly open chaotic systems [29,30]. In the limit of many weakly open channels M ≫ 1 (but still M ≪ N ) the widths cease to fluctuate, so distribution (8) becomes very narrow and peaked around its mean value κ = M .…”

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Statistics of Resonance Width Shifts as a Signature of Eigenfunction Nonorthogonality

2012

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We consider an open (scattering) quantum system under the action of a perturbation of its closed counterpart. It is demonstrated that the resulting shift of resonance widths is a sensitive indicator of the nonorthogonality of resonance wavefunctions, being zero only if those were orthogonal. Focusing further on chaotic systems, we employ random matrix theory to introduce a new type of parametric statistics in open systems, and derive the distribution of the resonance width shifts in the regime of weak coupling to the continuum. PACS numbers: 05.45.Mt, 03.65.Nk, 05.60.Gg The classical question of how energy levels of a quantum system get shifted under the action of a perturbation kept attracting renewed attention during the last two decades, mostly due to the established universality of such a parametric motion for systems with chaotic dynamics or intrinsic disorder [1,2]. In particular, the distributions and correlation functions of parametric derivatives of energy levels ("level velocities") [1][2][3] and their second derivatives ("level curvatures") [4] were found explicitly using the methods of random matrix theory (RMT) [5], and also verified, e.g., in microwave billiard experiments [6]. The other reason for such an interest is the recent development of the fidelity concept as the measure of the susceptibility of internal dynamics to perturbations [7].Experimentally, the energy levels are mostly accessible by means of a scattering setup [8]. From such a viewpoint, parametric dependencies of scattering characteristics, like phase shifts and time delays [9], conductances [10] and S matrix elements [11] were under intensive study. As to the parental discrete energy levels, they are converted into the resonances with finite lifetimes, since the original closed system becomes open (unstable). Such resonances manifest themselves in the energy-dependent S matrix as its poles in the complex energy plane, and can be analytically described as the complex eigenvalues of an effective non-Hermitian Hamiltonian [12][13][14]. Notably, the corresponding eigenfunctions are not orthogonal in the conventional sense but rather form a biorthogonal system. Their nonorthogonality plays an important role in many applications, e.g., describing interference in neutral kaon systems [15], influencing branching ratios of nuclear cross sections [16], and yielding excess noise in open laser resonators [17,18]. It also features in decay laws of quantum chaotic systems [19] and in dissipative quantum chaotic maps [20].In such a context the question of parametric motion of resonances and associated resonance states in open systems arises very naturally, but to the best of our knowledge has never been properly addressed. Our goal here is to begin filling in that gap by considering universal statistics of the shifts in the resonance widths under a generic perturbation in chaotic systems. In particular, we will demonstrate that such shifts are a clear manifestation of eigenstate nonorthogonality, thus providing a promising way to probe this ...

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Inhomogeneous losses and complexness of wave functions in chaotic cavities

Cited by 33 publications

References 27 publications

Resonance width distribution in RMT: Weak-coupling regime beyond Porter-Thomas

Resonance width distribution in RMT: Weak-coupling regime beyond Porter-Thomas

Experimental Width Shift Distribution: A Test of Nonorthogonality for Local and Global Perturbations

Statistics of Resonance Width Shifts as a Signature of Eigenfunction Nonorthogonality

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