Random Matrix Theory of Resonances: an Overview

Yan V. Fyodorov∗ ∗ King’s College London, Department of Mathematics, London WC2R 2LS, United Kingdom e-mail: [email protected]

Abstract—Scattering of electromagnetic waves in billiard-like channels can be taken either as fixed orthogonal vectors systems has become a standard experimental tool of studying PN satisfying i=1 W aiWbi = γaδab, γa > 0 ∀a = 1,...,M or properties associated with Quantum Chaos. Random Matrix alternatively the entries are chosen to be independent random Theory (RMT) describing statistics of eigenfrequencies and −1 associated eigenfunctions remains one of the pillars of theoretical Gaussian [18] : W aiWbj = N γaδabδij, i, j = understanding of quantum chaotic systems. In a scattering 1,...,N, with angular brackets standing for the ensemble av- system coupling to continuum via antennae converts real eigen- eraging. Equivalently, entries Sab(E) of the scattering matrix frequencies into poles of the scattering matrix in the complex can be rewritten as [5] frequency plane and the associated eigenfunctions into decaying X  1  resonance states. Understanding statistics of these poles, as well as S (E) = δ − 2i W ∗ W , (2) associated non-orthogonal resonance eigenfunctions within RMT ab ab ai jb E − Heff ij approach is still possible, though much more challenging task. ij with an effective non-Hermitian Hamiltonian I.INTRODUCTIONANDFORMALISM H = H − iΓ, Γ = WW † ≥ 0 (3) Scattering of electromagnetic waves in billiard-like mi- eff Γn crowave resonators or, in optical range, dielectric microcav- whose N complex eigenvalues zn = En−i 2 provide poles of ities, has become a standard experimental tool of studying the scattering matrix in the complex energy plane, commonly properties associated with Quantum Chaos, see e.g. reviews referred to as the resonances. The goal is to describe the [1]–[4]. Random Matrix Theory (RMT) describing statistics of statistics of positions of these poles in the complex plane, eigenfrequencies (referred to as ”energy levels” in the quantum as well as statistics of the associated residues related to non- mechanical context, the terminology to be used in the rest of orthogonal eigenvectors of non-Hermitian Hamiltonian Heff , the review) and associated eigenfunctions is one of the pillars using the random matrix statistics for H as an input. of theoretical understanding of quantum chaotic systems. II.STATISTICS OF S-MATRIXPOLESINSOMELIMITING Scattering set up drastically changes the system properties by CASES. coupling to the continuum via antennae, thereby converting discrete energy levels into decaying resonance states. Such A. Weak-coupling regime and Porter-Thomas distribution states can be associated with poles of the scattering matrix When W → 0 the anti-Hermitian part −iW W † can be in the complex energy plane. Understanding statistics of such treated as a perturbation of the Hermitian part H. The latter poles, as well as associated residues related to non-orthogonal matrix is characterized by real eigenvalues En and orthonor- eigenfunctions within RMT approach is still possible, but in mal eigenvectors |ni which are random vectors uniformly full generality remains a challenging task for theoreticians. spanning the N−dimensional unit sphere. In this regime one This presentation aims at giving a short overview of present expects the resonances to be largely non-overlapping, that is knowledge in this field. Γn  ∆, where ∆ is the mean spacing between neighbouring The starting point is an RMT model introduced in [5] real eigenvalues for H. To the first order and described in detail in [6], and more recently in [7], [8]. Γn † The model deals with the unitary M × M energy-dependent En → zn = En − i , with Γn = 2 n|WW |n 2 scattering matrix S(E) for quantum-chaotic system with M implying for M equivalent channels with coupling constants open channels 2 γa = γ  1 the χ distribution of scaled resonance widths 1 − iK 1 S(E) = , where K = W † W (1) yn = πΓn/∆  1 [9]:

arXiv:1606.03124v2 [cond-mat.dis-nn] 8 May 2021 1 + iK E − H Mβ Mβ/2   2 −1 (β) (β/2) y − βy N ×N N  1 4γ with , random Gaussian (real symmetric GOE, PM (y) = e (4) β = 1, Hermitian GUE, β = 2, or real quaternion GSE, β = 4) γΓ(Mβ/2) γ matrix H used to model spectral properties of the Hamiltonian This expression known as the Porter-Thomas distribution of closed system of quantum-chaotic nature. The columns of favourably agrees with a lot of experimental data in billiards, N ×M matrix W of coupling amplitudes to M open scattering from [10] to recent experiments [11] in a stadium billiard embedded into a two dimensional photonic crystal realized and a cloud of N − 1 long lived ones (analogue of Dicke on a silicon-on-insulator substrate. Note however recently superradiance), see recent optical experiments in [19]. reported deviations in high-precision neutron scattering [12], Similar, but somewhat simpler expression can be derived for see also discussion below and references in [13]. β = 2. In that case the limiting distribution of the resonance widths Γ can be derived non-perturbatively for any fixed B. Limiting case of very many open channels n number of channels M  N → ∞ [20]. To that end one On the other hand when couplings γ > 0 are fixed and the 1  1  defines ”renormalized coupling strengths” ga = γc + number M of open channels is very large and comparable with 2 γc for all channels a = 1,...,M. Then the probability density the number N  1 of internal states we expect that typically of the scaled resonance widths yn = πΓn/∆ is given for M resonances overlap strongly: Γn  ∆. The mean density of † equivalent channels with g1 = ... = gM ≡ g by [20] complex eigenvalues of Heff = H − iΓ, Γ = WW for M M    many channels 0 < m = M/N < 1 was found analytically (β) (−1) d sinh y P (y) = yM−1 e−yg (6) in [14], [15]. Generically, the density of resonances has the M (M − 1)! dyM y form of a cloud separated by a gap from the real axis. For For weak coupling γ  1 we have g ∼ γ−1  1 hence larger couplings γ second cloud emerges, as depicted in Fig.1 typically y ∼ g−1  1 and we are back to the Porter- taken from [15]. The gap has the physical meaning of a Thomas distribution (4). In contrast, for the perfect coupling (β) 2 case g = 1 the power-law tail emerges PM (y) ∝ 1/y so that some resonances may overlap strongly. This favourably agrees with the numerics for quantum chaotic graphs [21], see Fig.2. In fact not only the mean density of resonances, but also all

Fig. 1. The density of resonances in RMT model with many open channels, from [15]. Fig. 2. Top: 5000 resonances for a single realization of β = 2 chaotic scattering in quantum graphs ( taken from T. Kottos and U. Smilansky correlation length in energy-dependent scattering observables, [21]). Bottom: Resonance widths distribution as compared with the see [15] and also matches semiclassical considerations [16] RMT analytical prediction (6). and is observed experimentally in microwave billiards [17]. higher correlation functions can be found explicitly [22] as III.NON-PERTURBATIVE RESULTS FOR STATISTICS OF resonances form for β = 2 a determinantal process in the S-MATRIX POLES complex plane. For β = 1 only the mean density of resonances The first systematic non-perturbative investigation of reso- in the complex plane is known explicitly for 1 ≤ M < ∞ [23] nances within RMT was undertaken by Sokolov & Zelevin- and is given by a rather complicated expression. E.g. in the sky [18] for the special case M = 1. They provided an simplest case M = 1 the probability density of the scaled explicit expression for the joint probability density of positions resonance widths yn is given by † z1, . . . , zN of all N eigenvalues of Heff = H − iW W in 2 Z 1 (β=1) 1 d 2 2λy the complex energy plane assuming random Gaussian coupling PM=1 (y) = 2 (1 − λ )e (g − λ)F(λ, y) dλ (7) γ 4π dy −1 amplitudes hWiWji = N with fixed γ > 0. For β = 1 they found where ∞ −yp N Z 1 − Imzk 2 e (β=1) Y e γ Y |zi − zj| F(λ, y) = dp1 (8) P (z , . . . , z ) ∝ √ 2p 2 M=1 1 N g (λ − p1) (p1 − 1)(p1 − g) γImzk |zi − zj| k=1 i

Fig. 3. Resonance widths distribution for β = 1,M = 1 scattering Suppose we slightly perturb the scattering system: Heff → in a microwave resonator as compared with the RMT analytical pre- Heff + αV (e.g. by moving scatterers, or billiard walls) diction for perfect coupling, taken from [24]. The inset shows direct † RMT simulations. The characteristic tail 1/Γ2 is well-developed. with a Hermitian V = V and α → 0. The width of each resonance Γn changes typically linearly in the perturbation strength: δΓn ∝ α, and one can show that such ”shifts” are Recently reported deviations from Porter-Thomas distribu- non-vanishing only due to non-orthogonality of the resonance tion in neutron scattering by Koehler et al. [12] stimulated eigenffunctions [28]. In the regime of non-overlapping reso- many heuristic proposals to modify Porter-Thomas distribution nances one can further show that beyond the small widths region. Motivated by that we analyzed X hm|G|mi the exact formulae for β = 1 in order to extract a non- δΓn = iα , (11) En − Em perturbative asymptotics of P(y) valid for g  1 and any m6=n scaled widths y. For M = 1 and g ∼ (2γ)−1  1 we have where real eigenvalues En and orthonormal eigenvectors |ni found [13] characterize the ”closed” chaotic system and (β=1) d  p  P (y) = − Φ(y)erfc gy/2 (9) G = WW †|n hn|V + V |ni n|WW † M=1 dy ∞ 2 One can use RMT results for En and |ni to derive the where erfc(z) = √2 R e−t dt and the correction factor π z distribution of ”shifts” δΓn which shows a power-law tail [28] −(β+2) 1  y   sinh y  y   ∼ (δΓn) . For experimental verification see [33]. Φ(y) = K cosh y − + K sinh y 2 0 2 y 1 2 0 10 For small widths Φ(y  1) ≈ 1 and we are back to the

−1 Porter-Thomas distribution (4). 10 ) y Finally, due coupling to continuum statistics of the real ( global local 1

M local 2 P −2 parts Re zn is significantly modified in comparison with eigen- 10 frequencies En of the closed systems. Although no results

−3 derived from the first principles are yet available, an interesting 10 generalization of the Wigner surmise to open system was

−4 suggested in [25], and compared with experimental data. 10 −10 −5 0 y 5 10 IV. RESONANCE EIGENFUNCTIONNON-ORTHOGONALITY Fig. 4. Resonance widths ”shifts” distribution for β = 1,M = 1 The effective non-Hermitian Hamiltonian Heff = H − scattering in a microwave billiard resonator as compared with the † iW W has a set of ”right” |Rni and ”left” hLn | eigenvectors: RMT analytical prediction, for various types of perturbations (global vs. local). From Gros et al. [33] Heff |Rni = zn |Rni , hLn |Heff = zn hLn| (10) V. CLOSELY RELATED TOPICS. with bi-orthogonality properties: There are quite a few works on resonances in wave chaotic N X scattering related to the topic of this review but going beyond hLn|Rni = δmn, |Rni hLn| = 1 its direct remit. Further information can be found in the n=1 references below. Note however that hLn|Lmi= 6 δnm 6= hRn|Rmi showing that • Resonance theory for time-periodic (Floquet) or strobo- the eigenmodes of open systems are no longer orthogonal. The scopic systems, relations to truncations of unitary random corresponding non-orthogonality overlap matrix matrices: [34]–[37], [39]. • Semiclassical structure of resonances in chaotic systems O = hL |L i hR |R i mn m n m n and Fractal Weyl law, see [38] and further references in shows up in various physical observables of quantum chaotic the review [39]. For a recent experiment see [40]. systems, such as e.g. decay laws [26], ”Petterman factors” • Resonance properties in systems with diffusion of waves describing excess noise in open laser resonators [27], as well [41], [42], with wave scattering generated by point- as in sensitivity of the resonance widths to small perturbations like active scatterers [43], and in interacting many-body [28]. RMT can be most efficiently applied to statistics of Omn chaotic systems [44]. • Resonances in systems with Anderson localization [45], [22] Y. V. Fyodorov, B. A. 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