DISTRIBUTION OF SCATTERING MATRIX ELEMENTS IN QUANTUM CHAOTIC SCATTERING

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1 DISTRIBUTION OF SCATTERING MATRIX ELEMENTS IN QUANTUM CHAOTIC SCATTERING Santosh Kumar Shiv Nadar University Dadri, India Joint work with A. Nock*, H.-J. Sommers, T. Guhr Universität Duisburg-Essen, Duisburg. *Queen Mary University of London, London Distribution of S-matrix element in Fourier Space B. Dietz, M. Miski-Oglu, A. Richter, F. Schäfer Institut für Kernphysik, Technische Universität Darmstadt IX Brunel-Bielefeld Workshop on Random Matrix Theory Bielefeld 2013

2 OUTLINE Introduction Brief sketch of the derivation Results and comparison with simulations and experiments Conclusion

3 SCATTERING Deviation of a wave or a particle from its trajectory because of some localized non-uniformity in the medium. Scattering of light waves by clouds Scattering of radio waves Electron scattering inside a quantum dot Scattering of laser beam Rutherford-Geiger-Marsden experiment A candidate scattering event at LHC leading to the discovery of Higgs Boson Image Sources: ttp_research_clip_image010.gif,

4 SCATTERING: GENERIC SET-UP States before the scattering event Interaction region (Scattering event) States after the scattering event Channels of reaction (Characterized by total energy E and other quantum numbers) Scattering matrix (S-matrix) connects states existing asymptotically before and after the scattering event (Relates incoming and outgoing waves)

5 SCATTERING MATRIX Relates incoming and outgoing waves 2 channel example: H 1(x) =A 1 (k)e ikx + B 1 (k)e ikx 2(x) =A 2 (k)e ikx + B 2 (k)e ikx apple B1 (k) A 2 (k) = apple S11 S 12 S 21 S 22 apple A1 (k) B 2 (k) S- matrix is unitary owing to the flux conservation, SS = S S = 1

6 HAMILTONIAN FORMULATION H = X lm + X l,c lih lm hm + X Z de c, EiEhc, E c Z li de W lc hc, E +herm. conj. N( 1) bound states : hl mi = lm M channels : ha, E 1 b, E 2 i = ab (E 1 E 2 ) GaAs based quantum dot (Source: e d f H c a Schematic view of the general scattering problem: Different channels of reaction (labeled a,b,...) are connected via a compact interaction region described by a Hamiltonian H. C. Mahaux and H. A. Weidenmüller, Shell Model Approach to Nuclear Reactions (North Holland, Amsterdam, 1969) b

7 SCATTERING MATRIX S ab (E) = ab i2 W ag(e)w b The resolvent G(E) is given by: G(E) = E1 N H + i MX c=1 W c W c! 1 The N-component coupling vectors are assumed to satisfy the orthogonality W c W d = c cd ; c, d =1,,M C. Mahaux and H. A. Weidenmüller, Shell Model Approach to Nuclear Reactions (North Holland, Amsterdam, 1969)

8 STATISTICAL DESCRIPTION Scattering process is quite often of chaotic nature. Complicated dependence on Parameters of incoming waves (e.g., energy) The scattering region (e.g., the form or strength of the scattering potential.) Scattering description of S-matrix is needed!

9 MEXICO APPROACH S-matrix, itself, is treated as a stochastic quantity and is described by the Poisson kernel. (Based on the assumption of minimal information content) P (S) / det 1 M ShS i ( M+2 ) M: Dimension of the S-matrix (Number of channels) <S>: Average S-matrix β: Symmetry class Dependence on the parameters not obvious! P. Mello, P. Pereyra and T. H. Seligman, Ann. Phys. (NY) 161, 254 (1985) H. U. Baranger and P. Mello, Phys. Rev. Lett. 73,142 (1994); Europh. Lett. 33, 465 (1996) P. Mello and H. Baranger, Physica A 220, 15 (1995)

10 HEIDELBERG APPROACH Introduces stochasticity on the level of the Hamiltonian describing the scattering center. Random Matrix Universality: Universal and generic features can be extracted by modeling the Hamiltonian describing the scattering center using appropriate ensemble of Random matrices P(H) / exp N: Dimension of the matrices H v: Energy scale β: Symmetry class β=1 Time reversal invariant spinless systems (Real-Symmetric H) β=2 Time reversal noninvariant systems (Hermitian H) D. Agassi, H. A. Weidenmüller and Z. Mantzouranis, Phys. Rep. 22, 145 (1975) N 4v 2 trh2

11 KNOWN RESULTS Two-point correlation function (β=1) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer Phys. Rep. 129, 367 (1985) Up to fourth moment (β=1) E. D. Davis and D. Boose Phys. Lett. B 211, 379 (1988); Z. Phys. A 332, 427 (1989) Two-point correlation function (β=2) D. V. Savin, Y. V. Fyodorov, H.-J. Sommers Acta Phys. Pol. A 109, (2006) Two-point correlation function (β=2, Arbitrary U N invariant distribution for H) S. Mandt and M. R. Zirnbauer, J. Phys. A: Math. Theor. 43, (2010)

12 KNOWN RESULTS Two-point correlation function (β=1) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer Phys. Rep. 129, 367 (1985) Up to fourth moment (β=1) E. D. Davis and D. Boose Phys. Lett. B 211, 379 (1988); Z. Phys. A 332, 427 (1989) Two-point correlation function (β=2) D. V. Savin, Y. V. Fyodorov, H.-J. Sommers Acta Phys. Pol. A 109, (2006) Two-point correlation function (β=2, Arbitrary U N invariant distribution for H) S. Mandt and M. R. Zirnbauer, J. Phys. A: Math. Theor. 43, (2010) Distribution of diagonal elements (β=1, 2): Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers J. Phys. A: Math. Gen 38, (2005)

13 KNOWN RESULTS Two-point correlation function (β=1) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer Phys. Rep. 129, 367 (1985) Up to fourth moment (β=1) E. D. Davis and D. Boose Phys. Lett. B 211, 379 (1988); Z. Phys. A 332, 427 (1989) Two-point correlation function (β=2) D. V. Savin, Y. V. Fyodorov, H.-J. Sommers Acta Phys. Pol. A 109, (2006) Two-point correlation function (β=2, Arbitrary U N invariant distribution for H) S. Mandt and M. R. Zirnbauer, J. Phys. A: Math. Theor. 43, (2010) Distribution of diagonal elements (β=1, 2): Y. V. Fyodorov, D. V. Savin, and H.-J. Sommers J. Phys. A: Math. Gen 38, (2005) The problem of finding distribution of off-diagonal S-matrix elements remained unsolved!

14 SOME OBSERVATIONS Already in 1975 numerical simulations revealed that The distributions, in general, exhibit Non - Gaussian behavior (expected because of Unitarity constraint). The real and imaginary parts show different deviations from the Gaussian behavior for β=1. Similar conclusions were arrived at from the data obtained from experiments on microwave resonators. J.W.Tepel, Z.Physik A 273, 59 (1975). J.Richert, M. H. Simbel, and H. A. Weidenmüller,Z.Physik A 273, 195 (1975). B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, and H. A. Weidenmüller, Phys. Rev. E 81, (2010).

15 SOME OBSERVATIONS Already in 1975 numerical simulations revealed that The distributions, in general, exhibit Non - Gaussian behavior (expected because of Unitarity constraint). The real and imaginary parts show different deviations from the Gaussian behavior for β=1. Similar conclusions were arrived at from the data obtained from experiments on microwave resonators. J.W.Tepel, Z.Physik A 273, 59 (1975). J.Richert, M. H. Simbel, and H. A. Weidenmüller,Z.Physik A 273, 195 (1975). B. Dietz, T. Friedrich, H. L. Harney, M. Miski-Oglu, A. Richter, F. Schäfer, and H. A. Weidenmüller, Phys. Rev. E 81, (2010).

16 GOAL Distribution of off-diagonal S-matrix elements Z P s (x s )= d[h]p(h) (x s } s (S ab )); s =1, 2 Real part of S ab : } 1 (S ab )= S ab + S ab 2 = i (W agw b W b G W a ) Imaginary part of S ab : } 2 (S ab )= S ab S ab 2i = (W agw b + W b G W a )

17 CHARACTERISTIC FUNCTIONS R s (k) = Z d[h]p(h)exp( ik} s (S ab )); s =1, 2 The characteristic function also serves as the moment generating function The distributions can be obtained as the Fourier transform of R s (k): P s (x s )= 1 2 Z 1 1 dkr s (k)exp(ikx s )

18 CHARACTERISTIC FUNCTIONS Z R s (k) = d[h]p(h)exp( ik} s (S ab )); s =1, 2 Introduce a 2N-dimensional vector W and a 2N 2N dimensional matrix A s W = apple Wa W b A s = R s (k) = Z apple 0 ( i) s G i s G 0 d[h]p(h)exp( ik W A s W ) P(H) / exp N 4v 2 trh2 G = E1 N H + i MX W c W c c=1! 1 H appears in the denominator of G: Ensemble averaging nontrivial!

19 SUPERMATHEMATICS Anticommuting (Grassmann or Fermionic) variables: Any function of the anticommuting variables is a finite polynomial, e.g., 1 2 = =0 exp(α)=1+α Complex conjugate Conventions: ( ) = ( ) = ( ) =( ) = = F. A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987) K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, 1997)

20 SUPERMATHEMATICS Integrals (Berezin Integrals): Z d =0, Z Z d d exp(iq )= q 2 i In contrast, for the ordinary complex variables Z Z Z d = 1 p 2 dz dz exp(iq z z)= 2 i q Z Z Z Z Superintegral: dz dz d d exp(iq(z z + )) = 1 F. A. Berezin, Introduction to Superanalysis (Reidel, Dordrecht, 1987) K. B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, 1997)

21 SUPERMATHEMATICS Supervectors: = apple z =[z ] Supermatrices: = apple a µ b T = apple a T T µ T b T = apple a µ b Supertrace: Superdeterminant: Definitions str =tra trb sdet = det(a µb 1 ) det b = det a det(b a 1 µ)

22 SUPERMATHEMATICS Multivariate Gaussian Integrals: Using vectors with commuting entries: Z! i i d[z]exp 4 k z A 1 s z exp 2 (W z + z W ) =det 1 As 1 exp( ik W A i8 2 s W ) k Using vectors with anticommuting entries: Z i d[ ]exp 4 k A 1 s =det! As 1 i8 2 k z = apple za z b = apple a b A 1 s = apple 0 ( i) s (G 1 ) i s G 1 0 W = apple Wa W b

23 SUPERMATHEMATICS Combining the above integral results we obtain Z Z d[z] i i d[ ]exp 4 k (z A 1 s z + A 1 s ) exp 2 (W z + z W ) =exp( ik W A s W )

24 SUPERMATHEMATICS Combining the above integral results we obtain Z Z d[z] i i d[ ]exp 4 k (z A 1 s z + A 1 s ) exp 2 (W z + z W ) =exp( ik W A s W ) The exponential on RHS is exactly the factor in our expression for R s (k) R s (k) = Z d[h]p(h)exp( ik W A s W )

25 Z R s (k) = W = 2 W a 6W b d[ ]exp i 2 (W = z a z b a b A 1 G 1 = E1 N Z i W) d[h]p(h)exp 4 k s = 2 0 ( i) s (G 1 ) i s G H + i MX W c W c c=1 0 A 1 s 0 ( i) s (G 1 ) i s G

26 Z R s (k) = W = 2 W a 6W b d[ ]exp i 2 (W = z a z b a b A 1 G 1 = E1 N Z i W) d[h]p(h)exp 4 k s = 2 0 ( i) s (G 1 ) i s G H + i MX c=1 W c W c 0 A 1 s 0 ( i) s (G 1 ) i s G 1 0 H is now linear in the exponent containing the supervectors 3 7 5

27 Z R s (k) = W = 2 W a 6W b d[ ]exp i 2 (W = z a z b a b A 1 G 1 = E1 N Z i W) d[h]p(h)exp 4 k s = 2 0 ( i) s (G 1 ) i s G H + i MX c=1 W c W c 0 A 1 s 0 ( i) s (G 1 ) i s G 1 0 H is now linear in the exponent containing the supervectors A s -1 is not block diagonal! 3 7 5

28 Ψ and Ψ can be treated as independent complex quantities and therefore admit independent transformations.!! apple ± = apple 0 ±( i) s 1 N i s 1 N 0 Jacobian: (-1) N 2-2 N for β= 1 and (-1) N for β= 2 The choice of Ξ ± ensures proper convergence requirements for the supermatrix introduced later

29 β = 2 (HERMITIAN H) R s (k) =( 1) N Z d[ ]exp i 2 (U s + Z i W) d[h]p(h)exp 4 k A 1 A 1 = diag[ (G 1 ),G 1, (G 1 ), G 1 ]

30 β = 2 (HERMITIAN H) R s (k) =( 1) N Z d[ ]exp i 2 (U s + Z i W) d[h]p(h)exp 4 k A 1 A 1 = diag[ (G 1 ),G 1, (G 1 ), G 1 ] = 2 3 z a 6z b a b

31 β = 2 (HERMITIAN H) R s (k) =( 1) N Z d[ ]exp i 2 (U s + Z i W) d[h]p(h)exp 4 k A 1 A 1 = diag[ (G 1 ),G 1, (G 1 ), G 1 ] H-part in exponent involving the supervectors: z ahz a z b Hz b + ah a + b H b =tr(hd) where D = z a z a z b z b a a b b

32 β= 1 (REAL SYMMETRIC H) R s (k) =( 1) N Z d[ ]exp i V s Z i d[h]p(h)exp 4 k A 1 A 1 = diag( (G 1 ),G 1, (G 1 ), G 1 ) 1 2

33 R s (k) =( β= 1 (REAL SYMMETRIC H) 1) N Z d[ ]exp i V s Z i d[h]p(h)exp 4 k A 1 = diag( (G 1 ),G 1, (G 1 ), G 1 ) 1 2 = A 1 2 x a 3 y a x b y b a 6 a 7 4 b 5 b

34 β= 1 (REAL SYMMETRIC H) R s (k) =( 1) N Z d[ ]exp i V s Z i d[h]p(h)exp 4 k A 1 A 1 = diag( (G 1 ),G 1, (G 1 ), G 1 ) 1 2 H-part in exponent involving the supervectors: x T a Hx a + y T a Hy a x T b Hx b y T b Hy b + ah a T a H a + b H b T b H b =tr(hd) where D = x a x T a + y a y T a x b x T b y b y T b a a + a T a b b + b T b

35 ENSEMBLE AVERAGING Z d[h]p(h)exp i 4 k trhd =exp =exp 1 4r trd2 1 4r str (K1/2 BK 1/2 ) 2 where r = 4 2 k 2 N B mn = v 2 NX ( m) j ( j=1 n) j ; m, n =1, 2,..,8/ K = diag(1, 1, 1, 1) 1 2/

36 HUBBARD-STRATONOVICH TRANSFORMATION exp 1 4r str (K1/2 BK 1/2 ) 2 = Z d[ ]exp r str 2 + i str K 1/2 BK 1/2 = Z d[ ]exp r str 2 + i K 1/2 ( 1 N )K 1/2 σ is an 8/β-dimensional supermatrix having same structure as B, and K = K 1 2/

37 β=1 R s (k) =( 1) N Z Z d[ ]exp( r str 2 ) d[ ]exp i K 1/2 K 1/2 + i V s β=2 R s (k) =( 1) N Z Z d[ ]exp( r str 2 ) d[ ]exp h i K 1/2 K 1/2 + i 2 (U s + i W) = E 4 k 1 8/ 1 N + i 4k L M X c=1 W c W c L = diag(1, 1, 1, 1) 1 2/

38 β=1 R s (k) =( 1) N Z Z d[ ]exp( r str 2 ) d[ ]exp i K 1/2 K 1/2 + i V s β=2 R s (k) =( 1) N Z Z d[ ]exp( r str 2 ) d[ ]exp h i K 1/2 K 1/2 + i 2 (U s + i W) = E 4 k 1 8/ 1 N + i 4k L M X c=1 W c W c L = diag(1, 1, 1, 1) 1 2/ Integral over the supervector can now be performed

39 REPRESENTATION IN SUPERMATRIX SPACE Z R s (k) = d[ ]exp r str 2 2 str ln i 4 F s = F s = E 4 k 1 8/ 1 N + i 4k L M X c=1 W c W c ( V T s L 1/2 1 L 1/2 V s, =1 U sl 1/2 1 L 1/2 W, =2 L = L 1 N, L = diag(1, 1, 1, 1) 1 2/ β=1 32 independent integration variables β=2 16 independent integration variables

40 REPRESENTATION IN SUPERMATRIX SPACE Z R s (k) = d[ ]exp r str 2 2 str ln i 4 F s = F s = E 4 k 1 8/ 1 N + i 4k L M X c=1 W c W c ( V T s L 1/2 1 L 1/2 V s, =1 U sl 1/2 1 L 1/2 W, =2 L = L 1 N, L = diag(1, 1, 1, 1) 1 2/ β=1 32 independent integration variables β=2 16 independent integration variables Drastic reduction in the number of integration variables!

41 REPRESENTATION IN SUPERMATRIX SPACE Z R s (k) = d[ ]exp r str 2 2 str ln i 4 F s = F s = E 4 k 1 8/ 1 N + i 4k L M X c=1 W c W c ( V T s L 1/2 1 L 1/2 V s, =1 U sl 1/2 1 L 1/2 W, =2 L = L 1 N, L = diag(1, 1, 1, 1) 1 2/ β=1 32 independent integration variables β=2 16 independent integration variables Drastic reduction in the number of integration variables! Form similar to that of generating function for correlations

42 REPRESENTATION IN SUPERMATRIX SPACE Z R s (k) = d[ ]exp r str 2 2 str ln i 4 F s = F s = E 4 k 1 8/ 1 N + i 4k L M X c=1 W c W c ( V T s L 1/2 1 L 1/2 V s, =1 U sl 1/2 1 L 1/2 W, =2 L = L 1 N, L = diag(1, 1, 1, 1) 1 2/ β=1 32 independent integration variables β=2 16 independent integration variables Drastic reduction in the number of integration variables! Form similar to that of generating function for correlations (Verbaarschot, Weidenmüller, Zirnbauer) apart from the Fs part

43 L = SADDLE POINT ANALYSIS We are interested in N >> M limit. We fix M and let N R s (k) = Z d[ ]exp( L L) L = N 4 2 k 2 str 2 + N str ln 2 MX str ln c=1 v 2 1 8/ + i c 4 k E 4 k 1 8/ E 4 k 1 8/ 1L 0 = 1 8 k E ± ip 4v 2 E 2 + i 4 F s F s is a linear combination of matrix elements of multiplied with γ c, where c=a, b. 8 2 k 2 E 1 Saddle point equation: v k 1 8/ =0 Scalar solution:

44 MANIFOLD OF SOLUTIONS G = 1 8 k E1 p 8/ 4v 2 E 2 Q Q = it 1 LT; strq = 0; Q 2 = 1 8/ L = diag(1, 1, 1, 1) 1 2/ The dominant part of the free energy is invariant under the application of T β=1: T belongs to Lie supergroup UOSP(2,2/4) Q belongs to the coset superspace UOSP(2,2/4)/(UOSP(2/2) UOSP(2/2)) β=2: T belongs to Lie supergroup U(1,1/2) Q belongs to the coset superspace U(1,1/2)/(U(1/1) U(1/1)) K. B. Efetov, Adv. Phys. 32, 53 (1983) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985) Y. V. Fyodorov, and H.-J. Sommers, J. Math. Phys. 38,1918 (1997)

45 SEPARATING GOLDSTONE AND MASSIVE MODES = G + Expand up to the second power in δσ. The integrals involving Goldstone and Massive modes factorize. Symbolically: Z ( )= Z ( G ) Z ( ) The part involving Massive modes are Gaussian integrals and yields unity. L. Schäfer and F. Wegner, Z. Phys. B 38, 113 (1980) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985)

46 R s (k) = Z G = 1 8 k E1 8/ NONLINEAR SIGMA MODEL dµ( G )e if s/4 β=1: Eight commuting variables Eight anticommuting variables β=2: Four commuting variables, Four anticommuting variables MY c=1 sdet 2 1 8/ + i c 4 k 1 E L p 4v 2 E 2 Q E = G E 4 k 1 8/ Parametrization of Q: K. B. Efetov, Adv. Phys. 32, 53 (1983) J. J. M. Verbaarschot, H. A. Weidenmüller and M. R. Zirnbauer, Phys. Rep. 129, 367 (1985) Y. V. Fyodorov, and H.-J. Sommers, J. Math. Phys. 38,1918 (1997)

47 RESULTS (β=2) Identical results for real (s=1) and imaginary (s=2) parts R s (k) =1 Z 1 1 Characteristic Function d 1 Z 1 1 k 2 q d 2 4( 1 2 ) 2 F U( 1, 2) t 1 at 1 b + t 2 at 2 b J 0 k t 1 at 1 b f(x) =x (x)+ F U = MY c=1 Z 1 1 P s (x s f(x s 2 s d 1 Z 1 1 Distribution, d 2 F U ( 1, 2) 4 ( 1 2 ) 2 t 1 at 1 b + t2 at 2 b t 1 at 1 b x 2 1/2 (t1 at 1 b x 2 ) q v 2 + c 2 g c + g c = 2 2 p j 1 c 4v 2 E 2 g c + 1 t j c = (g = 2 c + j ),j=1, 2 1 T c

48 R s (k) = Z 1 1 Z 1 Z 1 Z 2 d 0 d 1 d 2 d J ( 0, 1, 2)F O ( 0, 1, 2) 1 RESULTS (β=1) Different results for real (s=1) and imaginary (s=2) parts Characteristic Function 1 0 4X n=1 apple (s) n kn P s (x s )= (x s )+ (s) s f (s) 2 s f (s) 3 s f (s) 4 s F O = J = MY c=1 g c + 0 (g c + 1 ) 1/2 (g c + 2 ) 1/2 (1 2 0 ) 1 2 2( 2 1 1) 1/2 ( 2 2 1) 1/2 ( 1 0 ) 2 ( 2 0 ) 2

49 EXPERIMENTS WITH MICROWAVE RESONATORS Equivalence in mathematical structure of the time-independent Schrödinger and Hemholtz equations (two-dimensions) (r 2 + k 2 ) =0 (r 2 + k 2 )E z =0 The shape of microwave cavity is such that the dynamics of the corresponding classical billiard is chaotic Not only moduli, but both real and imaginary parts of the S- matrix elements can be measured

50 COMPARISON WITH EXPERIMENTAL DATA (β=1) Characteristic functions for the real and imaginary parts of S 12 for the frequency range GHz Characteristic functions for the real and imaginary parts of S 12 for the frequency range GHz

51 COMPARISON WITH EXPERIMENTAL DATA (β=1) Distributions for the real and imaginary parts of S 12 for the frequency range GHz Distributions for the real and imaginary parts of S 12 for the frequency range GHz

52 COMPARISON WITH NUMERICAL SIMULATIONS (β=1)

53 COMPARISON WITH NUMERICAL SIMULATIONS (β=2)

54 CONCLUSION We solved a long-standing problem of finding the exact results (in the N limit) for distributions of off-diagonal S- matrix elements. We accomplished this task using a novel route to the nonlinear sigma model based on the characteristic function. We validated our results with experimental data obtained with chaotic microwave billiards, and thus presented a new confirmation of the random matrix universality conjecture. S. Kumar, A. Nock, H.-J. Sommers, T. Guhr, B. Dietz, M. Miski-Oglu, A. Richter, and F. Schäfer, Phys. Rev. Lett. 111, (2013) A. Nock, S. Kumar, H.-J. Sommers, T. Guhr, Ann. Phys. (In press); Preprint: arxiv:

55 Thank You!