Are scattering properties of graphs uniquely connected to their shapes?

1
Are scattering properties of graphs uniquely
connected to their shapes?

• Leszek Sirko, Oleh Hul
• Michal Lawniczak, Szymon Bauch
• Institute of Physics
• Polish Academy of Sciences, Warszawa, Poland
• Adam Sawicki, Marek Kus
• Center for Theoretical Physics, Polish Academy of
  Sciences,
• Warszawa, Poland

EUROPEAN UNION
Trento, 26 July, 2012
2
Can one hear the shape of a drum?
M. Kac, Can one hear the shape of a drum?, Am.
Math. Mon. (1966)

• 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)?

Trento, 26 July, 2012
3
One cant hear the shape of a drum
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 surfaces via Riemannian orbifolds,
Invent. Math. (1992)
T. Sunada, Riemannian coverings and isospectral
manifolds, Ann. Math. (1985)
Trento, 26 July, 2012
4
Isospectral drums
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.

C. Gordon and D. Webb
S.J. Chapman, Drums that sound the same, Am.
Math. Mon. (1995)
Trento, 26 July, 2012
5
Transplantation

• For a pair of isospectral domains eigenfunctions
  corresponding to the same eigenvalue are related
  to each other by a transplantation

Trento, 26 July, 2012
6
One cannot hear the shape of a drum
S. Sridhar and A. Kudrolli, Experiments on not
hearing the shape of drums, Phys. Rev. Lett.
(1994)

• 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.

Trento, 26 July, 2012
7
Can 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)
Trento, 26 July, 2012
8
Quantum graphs and microwave networks

• 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

Trento, 26 July, 2012
9
Quantum 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

L. Pauling, J. Chem. Phys. (1936)
T. Kottos and U. Smilansky, Phys. Rev. Lett.
(1997)
Trento, 26 July, 2012
10
Quantum graphs, definition

• A graph consists of n vertices (nodes) connected
  by B bonds (edges)
• On each bond of a graph the one-dimensional
  Schrödinger equation is defined
• Topology is defined by n x n connectivity matrix
• The length matrix Li,j
• Vertex scattering matrix ? defines boundary
  conditions

Neumann b. c. Dirichlet
b. c.
Trento, 26 July, 2012
11
Spectrum and wavefunctions
Spectral properties of graphs can be written in
terms of 2Bx2B bond scattering matrix U(k)
Trento, 26 July, 2012
12
Scattering from graphs
 
Trento, 26 July, 2012
13
Microwave networks
O. Hul et al., Phys. Rev. E (2004)
Quantum graphs can be simulated by microwave
networks
Microwave network (graph) consists of coaxial
cables connected by joints
Trento, 26 July, 2012
14
Hexagonal microwave network
Trento, 26 July, 2012
15
Equations for microwave networks

• Continuity equation for charge and current
• Potential difference

Trento, 26 July, 2012
16
Equivalence of equations
Microwave networks
Quantum graphs
Current conservation
Neumann b. c.
Equations that describe microwave networks with
R0 are formally equivalent to these for quantum
graphs with Neumann boundary conditions
Trento, 26 July, 2012
17
Can one hear the shape of a graph?
B. Gutkin and U. Smilansky, Can one hear the
shape of a graph?, J. Phys. A Math. Gen. (2001)

• 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

Trento, 26 July, 2012
18
Isospectral quantum graphs
R. 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
Trento, 26 July, 2012
19
Isoscattering quantum graphs
R. Band, A. Sawicki, U. Smilansky, Scattering
from isospectral quantum graphs, J. Phys. A (2010)

• Authors presented examples of isoscattering
  graphs
• Scattering matrices of those graphs are connected
  by transplantation relation

Trento, 26 July, 2012
20
Isoscattering graphs, definition
Two graphs are isoscattering if their scattering
phases coincide
Trento, 26 July, 2012
21
Experimental set-up
Trento, 26 July, 2012
22
Isoscattering microwave networks
Network I
Network II
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.
Trento, 26 July, 2012
23
Measurement of the scattering matrix
Trento, 26 July, 2012
24
The scattering phase
Two microwave networks are isoscattering if for
all values of ?
Trento, 26 July, 2012
25
Importance of the scattering amplitude
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
Trento, 26 July, 2012
26
Scattering amplitudes and phases

• O. Hul, M. Lawniczak, S. Bauch,
• Sawicki, M. Kus, and L. Sirko,
• accepted to Phys. Rev. Lett. 2012

Isoscattering networks
Networks with modified boundary conditions
Trento, 26 July, 2012
27
Transplantation relation
Trento, 26 July, 2012
28
Summary

• Are scattering properties of graphs uniquely
  connected to their shapes? in general NO!
• The concept of isoscattering graphs is not only
  theoretical idea but could be also realized
  experimentally
• Scattering amplitudes and phases obtained from
  the experiment are the same within the
  experimental errors
• Using transplantation relation it is possible to
  reconstruct the scattering matrix of each network
  using the scattering matrix of the other one

EUROPEAN UNION
Trento, 26 July, 2012