1. Investigation of Wigner reaction matrix, cross- and velocity correlators for microwave networks M. Ławniczak, S. Bauch, O. Hul, A. Borkowska and L. Sirko Institute of Physics, Polish Academy of Sciences, Aleja Lotników 32/46, 02-668 Warszawa

2. Plan of the talk: 1.Quantum graphs and microwave networks 2.Level spacing distributions 3.Statistics of Wigner reaction matrix K and reflection coefficient R 4.The cross-correlation function of scattering matrix 5.The autocorrelation function of level velocities 6.Conclusions

3. Quantum graphs Fig. 1 Quantum graphs are excellent examples of quantum chaotic systems. The idea of quantum graphs was introduced by Linus Pauling.

4. Microwave networks Fig. 2

5. For each graph’s bond connecting the vertices i and j the wave function is a solution of the one-dimentional Schrödinger equation: where (1) The telegraph equation for a microwave network’s bond (2) If (3) the equations (1) and (2) are equivalent.

6. SMA cable cut-off frequency: Below this frequency in a network may propagate only a wave in a single TEM mode Fig. 3 r1r1 r2r2

7. The level spacing distribution Fig. 4aFig. 4b - mean resonance width - mean level spacing (4) where:  

8. Wigner reaction matrix in the presence of absorption In the case of a single-channel antenna experiment the Wigner reaction matrix K and the measured matrix S are related by (5) Scattering matrix S can be parametrized as: where, R is the reflection coefficient and is the phase. (6)

9. The experimental setup Fig. 5 Hemmady S., Zheng X., Ott E., Antonsen T. M., Anlage S. M., Phys. Rev. Lett. 94, 014102 (2005)

10. Fig. 6

11. P(R) – the distribution of the reflection coefficientP(u) – the distribution of the real part of KP(v) – the distribution of the imaginary part of K Fig. 7aFig. 9aFig. 8a Savin D. V., Sommers H.- J., Fyodorov Y.V., JETP Lett. 82, 544 (2005) Fig. 7bFig. 8bFig. 9b

12. Fig. 10

13. The cross-correlation function Fig. 11 Fig. 12

14. The cross–correlation function The cross-correlation function c 12 study the relation between S 12 and S 21 : (10) for systems with time reversal symmetry (GOE) c 12 =1, for systems with broken time reversal symmetry (GUE) c 12 <1. Fig. 13 Ławniczak M., Hul O., Bauch S. and Sirko L., Physica Scripta,T135, 014050 (2009)

15. The autocorelation function of level velocities (11) Fig. 14

16. The autocorrelation function Fig. 15Fig. 16

17. Conclusions 1.We show that quantum graphs can be simulated experimentally by microwave networks Microwave networks without microwave circulators simulate quantum graphs with time reversal symmetry Microwave networks with microwave circulators simulate quantum graphs with broken time reversal symmetry 2.We measured and calculated numerically the distribution of the reflection coefficient P(R) and the distributions of Wigner reaction matrix P(v) and P(u) for hexagonal graphs The experimental results are in good agreement with the theoretical predictions 3.We show that the cross-correlation function can be used for identifying a system’s symmetry class but is not satisfactory in the presence of the strong absorption 4.We show that the autocorrelation function has universal shape of a distribution for systems with time reversal symmetry