Statistics of quantum resonances and fluctuations in chaotic - PowerPoint PPT Presentation

12:42:25 Statistics of quantum resonances and fluctuations in chaotic scattering Dmitry Savin Department of Mathematical Sciences, Brunel University, UK

12:42:25 Outline Two complementary viewpoints: from ‘inside’ from ‘outside’ S matrix ∼ outgoing wave local Green ′ s function ∼ field current incoming wave eigenmodes & eigenfunctions reflection & scattering phase Unified description: scattering theory + non-Hermitian RMT resonances = poles of S -matrix Main object: • Universalities in open chaotic systems • Mean resonance density, decay law & width fluctuations • Spectral correlations • Quasi-resonances Application: uniform vs non-uniform absorption D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 2/18

12:42:25 Open wave-chaotic systems • microwave cavities / billiards ( non-integrable shape ) • ultrasonics on elastodynamic billiards • light propagation in random media ( disorder / impurities ) • mesoscopic quantum dots ( interactions ) • compound nuclei Fluctuations in scattering observables delay time reflect statistics of resonance states. Aim is to study their statistical properties via distribution / correlation functions. energy D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 3/18

12:42:25 Resonance scattering V open system � resonances H S poles of the scattering matrix Scattering matrix = outgoing amplitude incoming amplitude : ( dim S = M : #channels ) 1 S res ( E ) = 1 − iV † with coupling amplitudes V c V , n E − H eff Separation of energy scales: potential vs resonance scattering Effective non-Hermitian Hamiltonian: ( dim H eff = N : #resonances ) 2 V V † , with H † = H � complex eigenvalues E n − i H eff = H − i 2 Γ n Mahaux, Weidenm¨ uller (1969); Livˇ sic (1973) Flux conservation (at zero absorption) = S matrix is unitary (at real E ): S res ( E ) = 1 − iK ( E ) with K ( E ) = 1 2 V † 1 1 + iK ( E ) , E − H V – reaction matrix D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 4/18

12:42:25 Closed chaotic cavities replace H with a random operator Statistical approach: Wigner, Dyson ( ∼ ’60); Bohigas, Giannoni, Schmidt (1984) H taken from appropriate ensemble of random matrices RMT � + symmetry constraints on H (e.g. H T = H for time-reversal systems) H † = H = H T H † = H H † = H = H R (GOE, β =1 ) (GUE, β =2 ) (GSE, β =4 ) Universality of spectral correlations: In the RMT limit N → ∞ , local fluctuations at the scale of mean level spacing ∆ are universal and described by those in Gaussian ensembles: dH ( · · · ) exp {− Nβ � 4 Tr H 2 } , � � ( · · · ) � = const dH = dH nm Examples: mean density (global, non-universal) and 2-point correlator (local, universal) n δ ( E − E n ) � = − 1 1 � 1 − ( E/ 2) 2 � ρ ( E ) � = � � π Im Tr � E − H � = ( N/π ) � enough considering E = E 1 + E 2 1 − ∆ 2 � ρ ( E 1 ) ρ ( E 2 ) � = Y 2 β ( ω ) with ω = ( E 2 − E 1 ) / ∆ = 0 2 D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 5/18

12:42:25 Open chaotic cavities 2 V V † requires statistical assumptions on coupling amplitudes H eff = H − i Fixed (‘f-case’) Random (‘r-case’) with ‘orthogonality’ condition gaussian, uncorrelated � N n =1 V a n V b n = 2 γ a δ ab � V a n V b m � = 2( γ a /N ) δ ab δ nm Verbaarschot, Weidenm¨ uller, Zirnbauer (1984) Sokolov, Zelevinsky (1988) Direct reaction absent: � S ab ( E ) � = δ ab 1 − γ a g ( E ) 1+ γ a g ( E ) , a = 1 , . . . , M Global E -dependence of g ( E ) not essential for local fluctuations at E = 0 Dependence of scattering observables via transmission coefficients: T a = 1 − |� S aa �| 2 = 4 γ eff (1+ γ eff ) 2 with γ eff = γ a g (0) Universality (model-independence): Lehmann, Saher, Sokolov, Sommers (1995) ⊲ ‘quantum’ case of finite M ( γ eff = γ a ) ⊲ ‘semiclassical’ case of M, N → ∞ with fixed m = M/N ≪ 1 ( γ eff ≈ γ a ) Qualitatively similar results for moderate m < 1 D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 6/18

12:42:25 Isolated resonances Porter-Thomas distribution appears at both γ ≪ 1 and γ ≫ 1 limits 2 ( V V † ) nm and treat V V † as a perturbation H eff = ε n δ nm − i Case γ ≪ 1 : → E n ≈ ε n (G β E) and Γ n ≈ ( V V † ) nn = � Mβ v 2 ֒ i i Distribution of widths P (Γ) is a χ 2 Mβ distribution � Mβ/ 2 − 1 � exp( − Mβ Γ Γ P (Γ) ∝ � Γ � ) with � Γ � = 2 γM/N � Γ � 2 ֒ → noting 4 γ ≈ T gives Weisskopf width Γ W = MT ∆ / 2 π ‘doorway’ representation in the eigenbasis of V V † Case γ ≫ 1 : Dynamical reorganization of resonance states: Sokolov, Zelevinsky (1989) ⊲ M collective states Γ coll ∼ (1 − 1 γ 2 )2 γ ≫ ∆ 1 M ⊲ N − M trapped states Γ n ∼ γ 2 2 γ N − M ≈ (2 /γ ) M/N ≪ ∆ ‘Overlapping’ is weaker than ‘interference’! Example: Absorption limit T → 0 and M → ∞ with fixed MT = 2 π Γ abs / ∆ D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 7/18

12:42:25 Mean resonance density Idea: electrostatic analogy Sommers, Crisanty, Somplinsky, Stein (1988) ֒ → average Green’s function as a 2D field Sokolov, Zelevinsky (1988) g ( z ) = 1 1 N � Tr � = ℜ g ( x, y ) + i ℑ g ( x, y ) z − H eff • Maxwell eqs = Cauchy-Riemann for ρ ( x, y ) ≡ 0 • ‘charge’ density: ρ ( E, Γ) = − 1 4 π ( ∂ 2 x + ∂ 2 y )Φ( x, y ) | x = E,y = − Γ / 2 ‘Electrostatic’ potential Φ( x, y ) = � ln Det[( z − H eff ) † ( z − H eff ) + δ 2 ] � ֒ → relation to a 2-point correlator problem non-perturbative perturbative ‘weak’ non-Hermiticity ‘strong’ non-Hermiticity SUSY calculation mean-field approach Z = � det[ ... ] det[ ... ] � � ln( . . . ) � = ln � ( . . . ) � saddle-point manifold appears no ‘soft’ mode D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 8/18

12:42:25 Strongly overlapping resonances Formation of the gap Γ g in the spectrum Haake et al. (1992) Nonzero density ρ ( x, y ) = ρ r , f ( y ) (universal at m ≪ 1 ): Lehmann, Saher, Sokolov, Sommers (1995) Redistribution of states at γ ∼ 1 γ cr 1 = 1 − 1 2 m 1 / 3 , m ≪ 1 γ cr 2 = 1 + 3 2 m 1 / 3 , m ≪ 1 density inside upper cloud 1 m ρ ( y ) = y 2 4 π • Γ g = Γ corr correlation length of fluctuations in scattering ( � = Γ W !) Γ 2 T ( γ eff ) | 2 = ⊲ S -matrix correlator = | i Γ( ǫ ) T ( ǫ ) at ǫ ≪ 1 corr ǫ 2 +Γ corr ǫ + i Γ( ǫ ) Γ 2 corr − ǫ 2 ⊲ time-delay correlator = Lehmann, Savin, Sokolov, Sommers (1995) ( ǫ 2 +Γ 2 corr ) 2 D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 9/18

12:42:25 Width distribution Exact GUE result valid at any T a , a = 1 , . . . , M Fyodorov, Sommers (1997) Equivalent channels, g = 2 /T − 1 ≥ 1 : P ( y ) = ( − 1) M Γ( M ) y M − 1 d M � e − gy sinh y � , y = π Γ / ∆ dy M y Limiting cases of isolated and many strongly overlapping resonances: then y ∼ T ≪ 1 so sinh y ≈ 1 � χ 2 • T ≪ 1 : 2 M (Porter-Thomas) y P (Γ) = M/ (2 y 2 ) only for 1 MT • M ≫ 1 : 2 MT < y < 2(1 − T ) cloud ր with upper bound → ∞ at T = 1 • Moldauer-Simonius relation as a consequence of y − 2 tail � Γ � = − ∆ � a ln(1 − T a ) 2 π GOE result is also known Sommers, Fyodorov, Titov (1999) D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 10/18

12:42:25 Decay law ... is directly related to fluctuations of the widths! Gap in spectrum shows up as classical (exponential) decay When (and how) does quantum (power law) decay appear? The ‘norm-leakage’ decay function: Savin, Sokolov (1997) N � Tr e i H † P ( t ) = � ψ ( t ) | ψ ( t ) � = 1 eff t e − i H eff t � P closed ( t ) ≡ 1 � time-dependence is due to the openness only Consider the eigenbasis of H eff H eff | n � = E n | n � and � ˜ n |H eff = E n � ˜ n | n | � = | n � † � ˜ n | m � = δ nm but � ˜ (bi-orthogonal) ֒ → U nm = � n | m � non-orthogonality matrix Bell, Steinberger (1959) Express P ( t ) in terms of resonances: N � � U 2 N � � ′ U 2 P ( t ) = 1 nn e − Γ n t � + 1 nm e i ( E n − E m ) t e − (Γ n +Γ m ) t/ 2 � D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 11/18

12:42:25 Hierarchy of time scales Qualitative: diagonal approximation N � � e − Γ n t � = � ∞ P d ( t ) = 1 0 d Γ e − Γ t P (Γ) (exact at t → ∞ ) � T/ (1 − T ) (1+ ξ ) 2 exp[ − M ln(1 + 1+ ξ dξ = 1 M Γ W t )] T 0 P (Γ) P ( t ) Semiclassical regime of M ≫ 1 strongly overlapping resonances: formation of exponential the gap decay κ = MT ≫ 1 √ Mt cl = � κ t H Sub-gap resonances slow down decay at t q = T t cl = √ κT Exact: SUSY calculation suggests D V Savin: Statistics of quantum resonances and fluctuations in chaotic scattering 12/18