InvestigationInvestigation of Wigner reaction of Wigner reaction matrix, cross- and velocity matrix, cross- and velocity correlators for microwave correlators for microwave

networks 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

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

Quantum graphs

Fig. 1

Quantum graphs are excellent

examples of quantum chaotic

systems.

The idea of quantum graphs

was introduced by Linus

Pauling.

Microwave networks

Fig. 2

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.

,0,2

,2

2

xkxdx

djiji

12 m

02

2

2

2

xUc

xUdx

dijij

2

22 ,

ckxUx ijij

SMA cable cut-off

frequency:

Below this frequency

in a network may

propagate only a

wave in a single TEM

mode

Fig. 3

r1

r2

GHz

rr

cC 33

21

The level spacing distribution

Fig. 4a Fig. 4b

- mean resonance width

- mean level spacing

(4)

where:

2

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)

1

1

S

SiK

ieRS

The experimental setup

Fig. 5

Hemmady S., Zheng X., Ott E., Antonsen T. M., Anlage S. M., Phys. Rev. Lett. 94, 014102 (2005)

Fig. 6

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

ieRS Ku Re Kv Im1

1

S

SiK

The cross-correlation function

Fig. 11

Fig. 12

The cross–correlation function

The cross-correlation function c12 study the relation between S12 and S21:

(10)

• for systems with time reversal symmetry (GOE) c12=1,

• for systems with broken time reversal symmetry (GUE) c12<1.

Fig. 13

Ławniczak M., Hul O., Bauch S. and Sirko L., Physica Scripta,T135, 014050 (2009)

vv

v

vvSvvS

vvSvvSvc

2

21

2

12

*2112

12

The autocorelation function of level velocities

(11)

Fig. 14

)()()( xxx

xx

xc ii

The autocorrelation function

Fig. 15 Fig. 16

2

0C ,0

x

xdXCx i

X

X in

i

i

E

Conclusions1. 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