* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Download Chaotic Scattering of Microwaves in Billiards: Induced Time
Wave–particle duality wikipedia , lookup
Quantum chromodynamics wikipedia , lookup
Relativistic quantum mechanics wikipedia , lookup
Delayed choice quantum eraser wikipedia , lookup
Renormalization wikipedia , lookup
Basil Hiley wikipedia , lookup
Topological quantum field theory wikipedia , lookup
Bohr–Einstein debates wikipedia , lookup
Quantum decoherence wikipedia , lookup
Particle in a box wikipedia , lookup
Measurement in quantum mechanics wikipedia , lookup
Probability amplitude wikipedia , lookup
Bell test experiments wikipedia , lookup
Copenhagen interpretation wikipedia , lookup
Density matrix wikipedia , lookup
Quantum electrodynamics wikipedia , lookup
Quantum field theory wikipedia , lookup
Quantum dot wikipedia , lookup
Path integral formulation wikipedia , lookup
Renormalization group wikipedia , lookup
Coherent states wikipedia , lookup
Quantum entanglement wikipedia , lookup
Bell's theorem wikipedia , lookup
Scalar field theory wikipedia , lookup
Hydrogen atom wikipedia , lookup
Quantum fiction wikipedia , lookup
Many-worlds interpretation wikipedia , lookup
Orchestrated objective reduction wikipedia , lookup
Quantum computing wikipedia , lookup
Quantum teleportation wikipedia , lookup
EPR paradox wikipedia , lookup
Symmetry in quantum mechanics wikipedia , lookup
Interpretations of quantum mechanics wikipedia , lookup
Quantum machine learning wikipedia , lookup
Quantum key distribution wikipedia , lookup
History of quantum field theory wikipedia , lookup
Quantum group wikipedia , lookup
Canonical quantization wikipedia , lookup
Quantum state wikipedia , lookup
SCUOLA INTERNAZIONALE Varenna sul lago DI di FISICA como - 1965 “fermi" Dr. h.c. Oriol Bohigas TU Darmstadt 2001 Chaotic Scattering in Microwave Billiards Orsay 2008 • BGS conjecture, quantum billiards and microwave resonators • Chaotic microwave resonators as a model for the compound nucleus • Fluctuation properties of the S-matrix for weakly overlapping resonances (ΓD) in a T-invariant (GOE) and a Tnoninvariant (GUE) system • Test of model predictions based on RMT for GOE and GUE Supported by DFG within SFB 634 B. Dietz, T. Friedrich, M. Miski-Oglu, A. R., F. Schäfer H.L. Harney, J.J.M. Verbaarschot, H.A. Weidenmüller SFB 634 – C4: Quantum Chaos Conjecture of Bohigas, Giannoni + Schmit (1984) • For chaotic systems, the spectral fluctuation properties of eigenvalues coincide with the predictions of random-matrix theory (RMT) for matrices of the same symmetry class. • Numerous tests of various spectral properties (NNSD, Σ2, Δ3,...) and wave functions in closed systems exist • Our aim: to test this conjecture in scattering systems, i.e. in open chaotic microwave billiards in the regime of weakly overlapping resonances SFB 634 – C4: Quantum Chaos The Quantum Billiard and its Simulation Shape of the billiard implies chaotic dynamics Δp Δx 2 SFB 634 – C4: Quantum Chaos Schrödinger Helmholtz quantum billiard 2D microwave cavity: hz < min/2 k 0 k 2 Ez 0 2mE k 2 2f k c 2 Helmholtz equation and Schrödinger equation are equivalent in 2D. The motion of the quantum particle in its potential can be simulated by electromagnetic waves inside a two-dimensional microwave resonator. SFB 634 – C4: Quantum Chaos Microwave Resonator as a Model for the Compound Nucleus rf power rf power in out A+a Compound Nucleus • Microwave power is emitted into the resonator by antenna and the output signal is received by antenna Open scattering system • The antennas act as single scattering channels • Absorption into the walls is modelled by additive channels SFB 634 – C4: Quantum Chaos B+b Scattering Matrix Description • Scattering matrix for both scattering processes Ŝ(E) = - 2i ŴT (E - Ĥ + i ŴŴT)-1 Ŵ Compound-nucleus reactions nuclear Hamiltonian coupling of quasi-bound states to channel states Microwave billiard Ĥ resonator Hamiltonian Ŵ coupling of resonator states to antenna states and to the walls complex S-matrix elements • Experiment: • RMT description: replace Ĥ by a GOE T-inv matrix for systems GUE T-noninv SFB 634 – C4: Quantum Chaos Excitation Spectra atomic nucleus microwave cavity overlapping resonances for G/D>1 Ericson fluctuations isolated resonances for G/D<<1 ρ ~ exp(E1/2) ρ~f • Universal description of spectra and fluctuations: Verbaarschot, Weidenmüller + Zirnbauer (1984) SFB 634 – C4: Quantum Chaos Spectra and Correlation of S-Matrix Elements • Regime of isolated resonances • Overlapping resonances • Г/D small • Г/D ~ 1 • Resonances: eigenvalues • Fluctuations: Гcoh Correlation function: C ( ) S ( f ) S ( f ) S ( f ) S ( f ) SFB 634 – C4: Quantum Chaos Ericson’s Prediction for Γ > D • Ericson fluctuations (1960): 2 Gcoh C ( ) 2 Gcoh 2 2 • Correlation function is Lorentzian • Measured 1964 for overlapping compound nuclear resonances P. v. Brentano et al., Phys. Lett. 9, 48 (1964) • Now observed in lots of different systems: molecules, quantum dots, laser cavities… • Applicable for Г/D >> 1 and for many open channels only SFB 634 – C4: Quantum Chaos Fluctuations in a Fully Chaotic Cavity with T-Invariance • Tilted stadium (Primack + Smilansky, 1994) • Height of cavity 15 mm • Becomes 3D at 10.1 GHz • GOE behaviour checked • Measure full complex S-matrix for two antennas: S11, S22, S12 SFB 634 – C4: Quantum Chaos Spectra of S-Matrix Elements Example: 8-9 GHz S11 |S| S12 S22 Frequency (GHz) SFB 634 – C4: Quantum Chaos Distributions of S-Matrix Elements • Ericson regime: Re{S} and Im{S} should be Gaussian and phases uniformly distributed • Clear deviations for Γ/D 1 but there exists no model for the distribution of S SFB 634 – C4: Quantum Chaos Road to Analysis Of the Measured Fluctuations • Problem: adjacent points in C() are correlated ~ • Solution: FT of C() uncorrelated Fourier coefficients C(t) Ericson (1965) • Development: Non Gaussian fit and test procedure SFB 634 – C4: Quantum Chaos Fourier Transform vs. Autocorrelation Function Time domain Frequency domain Example 8-9 GHz S12 S11 S22 SFB 634 – C4: Quantum Chaos Exact RMT Result for GOE systems • Verbaarschot, Weidenmüller and Zirnbauer (VWZ) 1984 for arbitrary Г/D : • VWZ-integral: C = C(Ti, D ; ) Transmission coefficients Average level distance • Rigorous test of VWZ: isolated resonances, i.e. Г << D • First test of VWZ in the intermediate regime, i.e. Г/D 1, with high statistical significance only achievable with microwave billiards • Note: nuclear cross section fluctuation experiments yield only |S|2 SFB 634 – C4: Quantum Chaos Corollary: Hauser-Feshbach Formula • For Γ>>D: Sab ( f ) Sab ( f ) iG TaTb (1 ab ) iG Tc c • Distribution of S-matrix elements yields W S fl 2 11 S fl 2 22 1/ 2 fl 2 12 /S 2 • Over the whole measured frequency range 1 < f < 10 GHz we find 3.5 > W > 2 in accordance with VWZ SFB 634 – C4: Quantum Chaos What Happens in the Region of 3D Modes? ~ • VWZ curve in C(t) progresses through the cloud of points but it passes too high GOF test rejects VWZ • This behaviour is clearly visible in C() GOE H • Behaviour can be modelled through Hˆ 1 0 SFB 634 – C4: Quantum Chaos 0 GOE H2 Distribution of Fourier Coefficients • Distributions are Gaussian with the same variances • Remember: Measured S-matrix elements were non-Gaussian • This still remains to be understood SFB 634 – C4: Quantum Chaos Induced Time-Reversal Symmetry Breaking (TRSB) in Billiards F • T-symmetry breaking caused by a magnetized ferrite • • a b • Coupling of microwaves to the ferrite depends on the direction a Sab b a Sba • Principle of detailed balance: • Principle of reciprocity: SFB 634 – C4: Quantum Chaos b Search for Time-Reversal Symmetry Breaking in Nuclei SFB 634 – C4: Quantum Chaos TRSB in the Region of Overlapping Resonances (ΓD) 1 2 F • Antenna 1 and 2 in a 2D tilted stadium billiard • Magnetized ferrite F in the stadium • Place an additional Fe - scatterer into the stadium and move it up to 12 different positions in order to improve the statistical significance of the data sample distinction between GOE and GUE behaviour becomes possible SFB 634 – C4: Quantum Chaos Violation of Reciprocity S12 S12 • Clear violation of reciprocity in the regime of Γ/D 1 SFB 634 – C4: Quantum Chaos Quantification of Reciprocity Violation • The violation of reciprocity reflects degree of TRSB • Definition of a contrast function Sab Sba | Sab | | Sba | • Quantification of reciprocity violation via Δ SFB 634 – C4: Quantum Chaos Magnitude and Phase of Δ Fluctuate B 200 mT SFB 634 – C4: Quantum Chaos B 0 mT: no TRSB S-Matrix Fluctuations and RMT • Pure GOE VWZ 1984 • Pure GUE FSS (Fyodorov, Savin + Sommers) 2005 V (Verbaarschot) 2007 • Partial TRSB • RMT analytical model under development (based on Pluhař, Weidenmüller, Zuk + Wegner, 1995) Hˆ Hˆ s i Hˆ a 0 GOE 1 GUE • Full T symmetry breaking sets in experimentally already for λ α/D 1 SFB 634 – C4: Quantum Chaos Crosscorrelation between S12 and S*21 at = 0 { 1 for GOE 0 for GUE • Data: TRSB is incomplete mixed GOE / GUE system *)= • C(S12, S21 SFB 634 – C4: Quantum Chaos Test of VWZ and FSS / V Models VWZ VWZ FSS/V VWZ VWZ • Autocorrelation functions of S-matrix fluctuations can be described by VWZ for weak TRSB and by FSS / V for strong TRSB SFB 634 – C4: Quantum Chaos First Approach towards the TRSB Matrix Element based on RMT maximal observed T-symmetry breaking • RMT Hˆ Hˆ s i Hˆ a 0 GOE 1 GUE • Full T-breaking already sets in for α D SFB 634 – C4: Quantum Chaos α (MHz) Determination of the rms value of T-breaking matrix element SFB 634 – C4: Quantum Chaos Summary • Investigated a chaotic T-invariant microwave resonator (i.e. a GOE system) in the regime of weakly overlapping resonances (Γ D) • Distributions of S-matrix elements are not Gaussian • However, distribution of the 2400 uncorrelated Fourier coefficients of the scattering matrix is Gaussian • Data are limited by rather small FRD errors, not by noise • Data were used to test VWZ theory of chaotic scattering and the predicted non-exponential decay in time of resonator modes and the frequency dependence of the elastic enhancement factor are confirmed • The most stringend test of the theory yet uses this large number of data points and a goodness-of-fit test SFB 634 – C4: Quantum Chaos Summary ctd. • Investigated furthermore a chaotic T-noninvariant microwave resonator (i.e. a GUE system) in the regime of weakly overlapping resonances • Principle of reciprocity is strongly violated (Sab ≠ Sab) • Data show, however, that TRSB is incomplete mixed GOE / GUE system • Data were subjected to tests of VWZ theory (GOE) and FFS / V theory (GUE) of chaotic scattering • S-matrix fluctuations are described in spectral regions of weak TRSB by VWZ and for strong TRSB by FSS / V • Analytical model for partial TRSB is under development • First approach using RMT shows that full TRSB sets already in when the symmetry breaking matrix element is of the order of the mean level spacing of the overlapping resonances SFB 634 – C4: Quantum Chaos