Statistics of Resonance Width Shifts as a Signature of Eigenfunction Nonorthogonality

Abstract

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 ...

Information Geometry of Complex Hamiltonians and Exceptional Points

“…The reason for introducing the additional states {|χ n } is because the eigenstates {|φ n } ofK are in general not orthogonal, and hence conventional projection techniques so commonly used in many calculations of quantum mechanics, in particular, in perturbation theory, are not effective when dealing with the eigenstates of a complex Hamiltonian [43][44][45][46][47][48]. With the introduction of the states {|χ n }, however, we have the relations:…”

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

“…[46]. Basically, that proposal coincides with the form of the "naive" (not-normalized) expression (12) in which the correct factor Φ(y) was however replaced by a heuristic expression sinh y y β/2 . Although such a form at β = 1 shares similar large-y asymptotic with Φ(y), it is manifestly different from the latter, rendering the proposal [46], in the strict sense, invalid.…”

“…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].…”

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

“…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].…”

“…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.…”