Download Michal Lawniczak

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
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
Quantum graphs are excellent
examples of quantum chaotic
systems.
The idea of quantum graphs
was introduced by Linus
Pauling.
Fig. 1
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:
d2
2



x

k
i , j x   0,
i, j
2
dx
where
(1)
  2m  1
The telegraph equation for a microwave network’s bond
d2
 2
U ij x   2 U ij x   0
2
dx
c
If
ij x   U ij x ,
k2 
 2
c
2
the equations (1) and (2) are equivalent.
(2)
(3)
SMA cable cut-off
frequency:
r1
r2
c
C 
 33 GHz
 r1  r2  
Below this frequency
in a network may
propagate only a
Fig. 3
wave in a single TEM
mode
The level spacing distribution
Fig. 4a
where:
G - mean resonance width
D - mean level spacing
2G

D
Fig. 4b
(4)
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
S 1
K i
S 1
(5)
Scattering matrix S can be parametrized as:
i
S  Re
where, R is the reflection coefficient and  is the phase.
(6)
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)
P(v)
P(u)
– the
– –the
the
distribution
distribution
distribution
ofof
the
ofthe
the
reflection
imaginary
real part
coefficient
of
part
K of K
Fig.
Fig.9a
7a
8a
Fig.
Fig.8b
7b
9b
u
Re
vS  
RIm
eKK
i
Savin D. V., Sommers H.- J., Fyodorov Y.V., JETP Lett. 82, 544 (2005)
S 1
K i
S 1
The cross-correlation function
Fig. 12
Fig. 11
The cross–correlation function
The cross-correlation function c12 study the relation between S12 and
S21:
c12 v  
*
v  Dv 
S12 v  Dv S 21
S12 v  Dv 
2
Dv
Dv
S 21 v  Dv 
(10)
2
Dv
• 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)
The autocorelation function of level velocities
 i
 i
c( x) 
(x)
( x  x)
x
x
Fig. 14
(11)
The autocorrelation function
Fig. 15
Ei
i 
D
Fig. 16
X
x

X in
  x  
C 0dX , C0   i 
 x 
2
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