Playing Billiards with Microwaves Quantum Manifestations of Classical Chaos

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Playing Billiards with Microwaves Quantum
Manifestations of Classical Chaos

2008

• Classical billiards and quantum billiards
• Random Matrix Theory (Wigner 1951 Dyson 1962)
• Spectral properties of billiards and mesoscopic
  systems
• Two examples
  - Spectra
  and wavefunctions of mushroom billiards- Friedel
  oscillations in microwave cavities
• S-Matrix fluctuations in the regime of
  overlapping resonances

Supported by DFG within SFB 634 S. Bittner, B.
Dietz, T. Friedrich, M. Miski-Oglu, P.
Oria-Iriarte, A. R., F. Schäfer A. Bäcker, H.L.
Harney, S. Tomsovic, H.A. Weidenmüller
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Classical Billiard
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Regular and Chaotic Dynamics
Regular
Bunimovich stadium (chaotic)

• Only energy is conserved

• Equations of motion are not integrable

• Equations of motion are integrable

• Predictable for a finite time only

• Predictable for infinite long times

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Tool Poincaré Sections of Phase Space

• Parametrization of billiard boundary L
• Momentum component along the boundary sin(f)

conjugatevariables
sin(f)
L
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Small Changes ? Large Actions

• Sensitivity of the solutions of a deterministic
  problem with respect to small changes in the
  initial conditions is called Deterministic Chaos.
• Beyond a fixed, for the system characteristic
  time becomes every prediction impossible. The
  system behaves in such a way as if not determined
  by physical laws but randomness.

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Our Main Interest

• How are these properties of classical systems
  transformed into corresponding quantum-mechanical
  systems? ? Quantum chaos?
• What might we learn from generic features of
  billiards and mesoscopic systems (hadrons,
  nuclei, atoms, molecules, metal clusters, quantum
  dots) ? 

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The Quantum Billiard and its Simulation
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Schrödinger ? Helmholtz
2D microwave cavity hz lt ?min/2
quantum billiard
Helmholtz equation and Schrödinger equation are
equivalent in 2D. The motion of the quantum
particle in its potential can be simulated by
electromagnetic waves inside a two-dimensional
microwave resonator.
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Superconducting Niobium Microwave Resonator
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Experimental Setup

• Superconducting cavities
• LHe (T 4.2 K)
• f 45 MHz ... 50 GHz
• 103...104 eigenfrequencies
• Q f/?f ? 106

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Stadium Billiard ? n 232Th
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Random Matrices ? Level Schemes
Random Matrix
Eigenvalues
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Nearest Neighbor Spacings Distribution
GOE and GUE ? ?Level Repulsion? Poissonian
Random Numbers ? ?Level Clustering?
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Nearest Neighbor Spacings Distribution
stadium billiard
nuclear data ensemble

• Universal (generic) behaviour of the two systems

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Universality in Mesoscopic Systems Quantum Chaos
in Hadrons

• Combined data from measured baryon and meson
  mass spectra up to 2.5 GeV (from PDG)
• Spectra can be organized into multiplets
  characterized by a set of definite quantum
  numbers isospin, spin, parity, strangeness,
  baryon number, ...

• Scale 10-16 m

P(s)
Pascalutsa (2003)
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Universality in Mesoscopic Systems Quantum Chaos
in Atoms

• 8 sets of atomic spectra of highly excited
  neutral and ionized rare earth atoms
  combined into a data ensemble
• States of same total angular momentum and parity

• Scale 10-10 m

Camarda Georgopulos (1983)
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Universality in Mesoscopic Systems Quantum Chaos
in Molecules

• Vibronic levels of NO2
• States of same quantum numbers

• Scale 10-9 m

Zimmermann et al. (1988)
S
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How is the Behavior of the Classical System
Transferred to the Quantum System?

• There is a one-to-one correspondence between
  billiards and mesoscopic systems.
• ? Bohigas conjecture
• The spectral properties of a generic chaotic
  system coincide with those of random matrices
  from the GOE.
• Next systems of mixed dynamics

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Bunimovich Mushroom Billiards Classical Dynamics
Variable width of stem
Semi-circle
Mushroom
Stadium
Regular
Mixed
Chaotic

• Candidate for new standard system generalized
  stadium

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Orbits in Mushroom Billiards

• Orbits with conservation of angular
  momentum in the hat ? regular dynamics
• Those orbits stay in the hat at all times

• Bunimovich orbits which enter into the
  stem at some time ? chaotic dynamics

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Phase Space Mushrooms
Poincaré section of a mushroom
chaotic
sinF
chaotic
regular
L

• Exact section (analytic, not simulated)
• Sharply divided areas in phase space
• No fractal properties

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Electromagnetic Cavities

• Lead plated copper
• Q 105-106
• 938 resonances found up to 22 GHz
• Spectrum complete so far (Weyls formula)

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Measured Spectra
Part of raw spectra from 10 antenna combinations

• Global fluctuations in the spacings small and
  large gaps (bunching)

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Fluctuation Properties
Nodes of beating

• Level number fluctuates oscillations and
  beating ? shell structures
• Do we see this behaviour in other mesoscopic
  systems?

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Supershell Structures in Metal Clusters ?

• Consider the fluctuating part of the binding
  energy of valence electrons in a Na
  cluster

• Scale 10-7-10-6 m

Bjørnholm et al. (1990) Nishioka, Hansen
Mottelson (1990)
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Periodic Orbit Theory (POT)
Motivation description of quantum spectra in
terms of classical closed orbits (Gutzwillers
trace formula)
spectrum
spectral density
length spectrum
Peaks at the lengths x of POs
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Length Spectrum
Mushroom
?(x)

Quarter Circle

• Part of the peaks are common to both the
  mushroom and the quarter circle ?
  regular orbits can be distinguished from the
  chaotic ones
• Isolated pair of peaks at 0.7m causes the
  beating
• Metal clusters

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Classification of Eigenstates
Subspectrum Regular modes ? Poisson Subspectrum
Chaotic modes ? GOE

• Spectrum

154 regular states
784 chaotic states

• Semiclassical limit reached for low quantum
  numbers
• Ratios 154/938 and 784/938 correspond to the
  respective parts of the phase space
• ?Percivals conjecture holds

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Cavities for Field Intensity Measurements

• XXL billiard ? get a quality factor Q104
  even at room temperature
• Scaled copies of superconducting cavities

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Maier-Slater Theorem
Dielectric perturbation body e.g. magnetic rubber
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Measured Wave Functions
(2,40) mode
N135
N136
N408
N176

• Perturbation body method ? measure E2?2
• Modes are regular, chaotic or (though rare)
  mixed
• Regular modes correspond to modes of a
  circular billiard

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Where in the mushroom is how much Chaos?

• Regularity is restricted to the hat ? Less
  chaos in the hat!

• Orbits with angular momentum smaller than
  Lcritr are chaotic
• pChaos is constant in the stem,
  decreasing in the hat
• Should be detectable in wave functions

?
r
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Averaged Intensity Distribution
Consider
of 239 measured chaotic modes
?

• For ? small, ? large and at the knee ? quantum
  effects, i.e. deviations of order of the
  shortest ?
• Behavior of averaged intensity distribution
  attributed to the separability of the
  classical phase space

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Friedel Oscillations

• The electronic wave function about an impurity
  atom in an alloy oscillates ? Friedel
  oscillations
• Oscillations also occur along spatial potential
  steps
• Electronic wave function on metallic surfaces
  are strongly influenced by defect atoms
  ? oscillating behavior due to diffraction
  
• Oscillations might be studied through scanning
  tunneling microscopy

J. Friedel, Nuovo Cim. Suppl. 2, 287 (1951)
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Scanning Tunneling Microscopy

• Experimental approach scanning tunneling
  microscopy

G.A. Fiete and E.J. Heller, Rev. Mod. Phys. 75,
933 (2003)

• For cold surfaces, the tunnel current
• Method to probe asymptotic properties of the
  wave functions

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Some Breathtaking Pictures
http//www.almaden.ibm.com/vis/stm/corral.html
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Modelling Friedel Oscillations

• Theoretic description Friedel
  1951 rectangular box Bäcker, Ullmo,
  Tomsovic 2008 random wave model
• Friedel oscillations are due to the perimeter
  correction of the Weyl
  formula and are
  thus independend of the dynamics of the system
• Friedel oscillations are a local phenomenon and
  are connected to
  the shortest POs near the surface

Dirichlet b.c. - Neumann b.c.x
distance to boundary
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Berrys Random Plane Wave Model (RPWM) for
Chaotic Systems

• Wave function of a chaotic system can be
  modeled by a random
• superposition of plane waves

• and are uniformly distributed in 0,2
  
• But the intensity of chaotic wave
  functions decreases towards the
  boundary ? are restricted in
  mixed systems

T
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Friedel Oscillations in the Mushroom Billiard

• Superposition of 239 intensity distributions of
  chaotic modes

x
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Chaotic Modes in the Mushroom Billiard and RPWM
Restricted RPWM
1
1
RPWM
I(x)
I(x)
circular boundary
straight boundary
0
0
0
0.1
0
0.1
x (m)
x (m)

• Strong enhancement of oscillations along the
  circular boundary due to restricted
  accessible directions of interfering waves

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Friedel Oscillations in the Barrier Billiard

• Billiard is pseudointegrable (contains corners ?
  ?/ n) and the barrier induces a particular
  structure of the wave functions (Bogomolny and
  Schmit, 2004)
• Superposition of 129 (non scarring) intensity
  distributions
• Dirichlet and Neumann boundary conditions

x
x
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Friedel Oscillations in the Barrier Billiard
2
1
I(x)
I(x)
1
Dirichlet
Neumann
0
0
0
0.1
0
0.1
x (m)
x (m)

• Enhancement observed in the Neumann case
  according to the model

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Application Ground State Energy Fluctuations in
Quantum Dots
Bäcker, Tomsovic, Ullmo (2008)

• Residual two electron Coulomb interaction in
  1st order perturbation theory

with
local density of particles
billiard area

• Our measurements

• Ground state energy fluctuations ? variance of
  Si
• Experiment on the way

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Microwave Resonator as a Model for the Compound
Nucleus

• Microwave power is emitted into the resonator by
  antenna ?
• and the output signal is received by antenna ?
  
• ? Open scattering system
• The antennas act as single scattering channels
• Absorption into the walls is modelled by
  additive channels

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Scattering Matrix Description

• Scattering matrix for both scattering processes

S(E) ? - 2pi WT (E? - H ip WWT)-1 W
Compound-nucleus reactions
Microwave billiard
nuclear Hamiltonian coupling of
quasi-bound states to channel states
resonator Hamiltonian coupling of resonator
states to antenna states and to the walls
? H ?
? W ?

• RMT description replace H by a GOE (GUE) matrix

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Decay of S-matrix correlations

• overlapping resonances
• for G/Dgt1
• Ericson fluctuations

isolated resonances for G/Dltlt1
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Spectra and Autocorrelation Function

• Regime of isolated resonances
• ?/D small
• Resonances eigenvalues

• Overlapping resonances
• ?/D 1
• Fluctuations ?coh

Correlation function
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Ericsons Prediction

• Ericson fluctuations (1960)

• Correlation function is Lorentzian
• Measured 1964 for overlapping
  compound nuclear resonances

P. v. Brentano et al., Phys. Lett. 9, 48 (1964)

• Now observed in lots of different systems
  molecules, quantum dots, laser cavities
• Applicable for ?/D gtgt 1 and for many open
  channels only

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Exact Random Matrix Theory Result

• For GOE systems and arbitrary ?/D
  Verbaarschot, Weidenmüller and Zirnbauer (VWZ)
  1984
• VWZ-integral

C C(Ti, D ?)
Transmission coefficients
Average level distance

• Rigorous test of VWZ isolated resonances, i.e.
  ? ltlt D
• Our goal test VWZ in the intermediate regime,
  i.e. ?/D 1

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Experimental Realisation in a Fully Chaotic Cavity

• Tilted stadium (Primack Smilansky, 1994)

• Height of cavity 15 mm
• Becomes 3D at 10.1 GHz

• GOE behaviour checked
• Measure full complex S-matrix for two antennas
  S11, S22, S12

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Spectra of S-Matrix Elements
Example 8-9 GHz
S12 ?
S
S11 ?
S22 ?
Frequency (GHz)
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Distributions of S-Matrix Elements

• Ericson regime ReS and ImS should be
  Gaussian and phases uniformly
  distributed
• Clear deviations but no model for general case

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Road to Analysis

• Problem adjacent points in C(?) are correlated
• Solution FT of C(?) ? uncorrelated Fourier
  coefficients C(t)
  Ericson (1965)
• Development Non Gaussian fit and test procedure

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Comparison Experiment - VWZ
Time domain
Frequency domain
Example 8-9 GHz
? S12 ?
? S11 ?
? S22 ?
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Induced Time-Reversal Symmetry Breaking (TRSB) in
Billiards

• T-symmetry breaking caused by a magnetized
  ferrite
• Coupling of microwaves to the ferrite depends on
  the direction a b

Sab
a
b
Sba

• Principle of detailed balance

• Principle of reciprocity

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Billiard with Broken Time Reversal Symmetry
Sab Sba

• Clear violation of reciprocity in the regime of
  G/D gtgt 1

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Quantification of Reciprocity Violation

• The violation of reciprocity reflects degree of
  TRSB
• Definition of a contrast function
• Quantification of reciprocity violation via ?

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Magnitude and Phase of ? Fluctuate
? B ? 200 mT
B ? 0 mTno TRSB
?
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Crosscorrelation between S12 and S21 at ? 0

1 for GOE0 for GUE

• C(S12, S21)
• Data TRSB is incomplete

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S-Matrix Fluctuations and RMT

• Pure GOE ? VWZ description 1984
• Pure GUE ? V
  description 2007
• Partial TRSB ? no analytical model
• RMT ?

GOE
GUE
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Test of VWZ and V Models
VWZ
VWZ
VWZ
VWZ
V
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Summary

• Quantum chaos manifests itself in the universal
  (generic) behavior of spectral properties
  and wave functions.
• Analogue experiments with microwave
  resonators provide interesting models for
  mesoscopic systems.

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Wo das Chaos auf die Ordnung trifft, gewinnt
meist das Chaos, weil es besser organisiert
ist. Friedrich Nietzsche (1844-1900)
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