Post on 04-Jan-2016
Are scattering properties Are scattering properties of graphs uniquely connected of graphs uniquely connected
to their shapes?to their shapes?
Leszek Sirko, Oleh HulLeszek Sirko, Oleh Hul
Michał Ławniczak, Szymon BauchMichał Ławniczak, Szymon BauchInstitute of Physics
Polish Academy of Sciences, Warszawa, Poland
Adam Sawicki, Marek KuśAdam Sawicki, Marek KuśCenter for Theoretical Physics, Polish Academy of Sciences,Center for Theoretical Physics, Polish Academy of Sciences,
Warszawa, Poland
Trento, 26 July, 2012Trento, 26 July, 2012
EUROPEAN UNION
Can one hear the shape of a drumCan one hear the shape of a drum??
Is the spectrum of the Laplace operator unique on the planar domain with Dirichlet boundary conditions?
Is it possible to construct differently shaped drums which have the same eigenfrequency spectrum (isospectral drums)?
M. Kac, Can one hear the shape of a drum?, Am. Math. Mon. (1966)
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OOnene can can’’tt hear the shape of a hear the shape of a ddrumrum C. Gordon, D. Webb, S. Wolpert, One can't hear the shape of a drum, Bull.
Am. Math. Soc. (1992)
C. Gordon, D. Webb, S. Wolpert, Isospectral plane domains and surfacesvia Riemannian orbifolds, Invent. Math. (1992)
T. Sunada, Riemannian coverings and isospectral manifolds, Ann. Math. (1985)
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Isospectral drumsIsospectral drums
S.J. Chapman, Drums that sound the same, Am. Math. Mon. (1995)
Pairs of isospectral domains could be constructed by concatenating an elementray „building block” in two different prescribed ways to form two domains. A building block is joined to another by reflecting along the common boundary.
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C. Gordon and D. Webb
TransplantationTransplantation
A
B
C
D
E
F G
A-B-G
A-D-F
B-E+F
D-E+G
-A-C-E
-B+C-D
C-F-G
For a pair of isospectral domains eigenfunctions corresponding to the same eigenvalue are related to each other by a transplantation
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OOnene cannot cannot hear the shape of a hear the shape of a ddrumrum
Authors used thin microwave cavities shaped in the form of two different domains known to be isospectral.
They checked experimentally that two billiards have the same spectrum and confirmed that two non-isometric transformations connect isospectral eigenfunction pairs.
S. Sridhar and A. Kudrolli, Experiments on not hearing the shape of drums, Phys. Rev. Lett. (1994)
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Can one hear the shape of a drumCan one hear the shape of a drum??Isospectral drums could be distinguished by measuring their scattering poles
Y. Okada, et al., “Can one hear the shape of a drum?”: revisited, J. Phys. A: Math. Gen. (2005)
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Quantum graphs and microwave Quantum graphs and microwave networksnetworks
What are quantum graphs?
Scattering from quantum graphs
Microwave networks
Isospectral quantum graphs
Scattering from isospectral graphs
Experimental realization of isoscattering graphs
Experimental and numerical results
Discussion
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Quantum graphsQuantum graphs
Quantum graphs were introduced to describe diamagnetic anisotropy in organic molecules:
Quantum graphs are excellent paradigms of quantum chaos:
In recent years quantum graphs have attracted much attention due to their applicability as physical models, and their interesting mathematical properties
T. Kottos and U. Smilansky, Phys. Rev. Lett. (1997)
L. Pauling, J. Chem. Phys. (1936)
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A graph consists of A graph consists of nn verticesvertices (nodes) (nodes) connected by connected by BB bonds (bonds (edgesedges))
On each bond of On each bond of aa graph graph the the one-dimensional one-dimensional Schrödinger equation is definedSchrödinger equation is defined
Topology is defined by nTopology is defined by nn connectivityn connectivity matrix matrix
The length The length matrixmatrix LLi,ji,j
Vertex scattering matrix Vertex scattering matrix ϭϭdefines boundary conditionsdefines boundary conditions
Quantum graphs, dQuantum graphs, definitionefinition
,
1, and are connected
0, otherwisei j
i jC
22
, ,2( ) ( )i j i j
dx k x
dx
, ' , '
2ij j j j
iv
, ' , 'ij j j j
Neumann b. c. Dirichlet b. c.
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Spectrum and wavefunctionsSpectrum and wavefunctions
,
( ) , ,, ,
ikL ji jU k ej j i mi j j m
, , ,( )
in ikx ou t ikx
i j i j i jx a e a e
( ), ', ',
'
out i inj jj ji j
j
a a
det 0I U k ki
a U k a
Spectral properties of graphs can be written in terms of 2Bx2B bond scattering matrix U(k)
(1)
(3)
( 2 )
( 4 )
1,4L
1,2L
1,3L
2,5L
2,6L
1 2
5
6
4
3 ( 6 )
(5)
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Scattering from graphsScattering from graphs
1 1
in ikx ou t ikxc e c e
2 2
in ikx ou t ikxc e c e
(1)
(3)
( 2 )
( 4 )
1,4L 1,2
L
1,3L
2,5L
2,6L
1 2
5
6
4
3 ( 6 )
(5)
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Microwave networksMicrowave networks
Microwave network (graph) consists of coaxial cables connected by joints
O. Hul et al., Phys. Rev. E (2004)
Quantum graphs can be simulated by microwave networks
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Hexagonal microwave networkHexagonal microwave network
12
3
4
5
6
6 vertices
15 bonds
n
B
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Equations for mEquations for microwave networkicrowave networkssContinuity equation for charge and current:Continuity equation for charge and current:
Potential difference:Potential difference:
1r
2r
, ,( , ) ( , )i j i jdq x t dJ x t
dt dx
,,
( , )( , ) i j
i j
q x tV x t
C
, ,( , ) ( )i ti j i jq x t e q x
, ,( , ) ( )i ti j i jV x t e V x
0R
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Equivalence of equationsEquivalence of equations
2 2
, ,2 2( ) ( ) 0i j i j
dV x V x
dx c
2, 2
,2
( )( ) 0i j
i j
d xk x
dx
, ,( ) ( )i j i jx V x 2
22
kc
Current conservation:
Microwave networks Quantum graphs
Neumann b. c.
,, , , , 0
0i j
i j j i i j i jx L xj i j i
d dC V x C V x
dx dx
Equations that describe microwave networks with R=0 are formally equivalent to these for quantum graphs with Neumann
boundary conditions
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Can one hear the shape of a Can one hear the shape of a graph?graph?
One can hear the shape of the graph if the graph is simple and bonds lengths are non-commensurate
Authors showed an example of two isospectral graphs
B. Gutkin and U. Smilansky, Can one hear the shape of a graph?, J. Phys. A: Math. Gen. (2001)
b
aa
2a+2b
b
2a+ba+2b
a
b2a+3b
2a
ba
a+2b
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Isospectral quantum Isospectral quantum graphgraphssR. Band, O. Parzanchevski, G. Ben-Shach, The isospectral fruits of representation theory: quantum graphs and drums, J. Phys. A (2009)
Authors presented new method of construction of isospectral graphs and drums
b
c
2a
c b
D
N
N
D
N2b
2c
aD
a
ND
N
D
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Isoscattering quantum Isoscattering quantum graphgraphss
Authors presented examples of isoscattering graphs
Scattering matrices of those graphs are connected by transplantation relation
R. Band, A. Sawicki, U. Smilansky, Scattering from isospectral quantum graphs, J. Phys. A (2010)
b
c
2a
cb
D
N
N
D
N2b
2c
aD
a
1( ) ( ) , for II IS k T S k T k
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IsoscatteringIsoscattering graphs, definitiongraphs, definition
Two graphs are isoscattering if their scattering phases coincide
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)) ((Im log det ( ) Im log det ( )IIIS S
det ( ) iS A e
11 12
21 22
S S
S S
S
Experimental set-up
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Isoscattering microwave networksIsoscattering microwave networks
4 321a a
2c
2b
2a
c
c
b
b6 4
35
1
2
Two isoscattering microwave networks were constructed using microwave cables. Dirichlet boundary conditions were prepared by soldering of the internal and external leads. In the case of Neumann boundary conditions, vertices 1 and 2, internal and external leads of the cables were soldered together, respectively.
Network I Network II
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Measurement of the scattering Measurement of the scattering matrixmatrix
2a
c
c
b
b6 4
35
1
2
4 321a a
2c
2b
S
11 12
21 22
S S
S S
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The scatteringThe scattering phase phase
)) ((Im log det ( ) Im log det ( )IIIS S
Two microwave networks are isoscattering if for all values of ν:
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Importance of the scattering Importance of the scattering aamplitudemplitude
In the case of lossless quantum graphs the scattering matrix is unitary. For that reason only the scattering phase is of interest.
However, in the experiment we always have losses. In such a situation not only scattering phase, but the amplitude as well gives the insight into resonant structure of the system
det ( ) iS A e
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( () )det ( ) det ( )II ISS
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Scattering amplitudes and phases
Isoscattering networks
Networks with modifiedboundary conditions
O. Hul, M. Ławniczak, S. Bauch, A.Sawicki, M. Kuś, and L. Sirko,accepted to Phys. Rev. Lett. 2012
Transplantation relationTransplantation relation
32b
2c
a4
a1 2
b
c1
2a4
23
5 c 6
b
1( ( ))II ITS S T
1 1
1 1T
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SummarySummary
Are scatteringAre scattering propertiesproperties of graphs uniquely connected of graphs uniquely connected to their shapes? – to their shapes? – in general in general NONO!!
The concept of isoscattering The concept of isoscattering graphs graphs is not only theoretical is not only theoretical idea but could be also realized experimentallyidea but could be also realized experimentally
Scattering amplitudes and phases obtained from Scattering amplitudes and phases obtained from the the experiment are the same withinexperiment are the same within the the experimental errors experimental errors
Using transplantation relation it is possible to reconstruct Using transplantation relation it is possible to reconstruct the scattering matrix of each network using the scattering the scattering matrix of each network using the scattering matrix of the other onematrix of the other one
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