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Title: Playing Billiards with Microwaves Quantum Manifestations of Classical Chaos


1
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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2

Classical Billiard
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3

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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4

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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5

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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6

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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7

The Quantum Billiard and its Simulation
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8

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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9

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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14

Nearest Neighbor Spacings Distribution
GOE and GUE ? ?Level Repulsion? Poissonian
Random Numbers ? ?Level Clustering?
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15

Nearest Neighbor Spacings Distribution
stadium billiard
nuclear data ensemble
  • Universal (generic) behaviour of the two systems

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16

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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18

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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20
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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21
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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23
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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27

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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28
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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30
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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31

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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33
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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35

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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36

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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37

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
SFB 634 C4 Quantum Chaos
39

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
40

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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42

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
SFB 634 C4 Quantum Chaos
43

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

SFB 634 C4 Quantum Chaos
44

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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45

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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47

Decay of S-matrix correlations
  • overlapping resonances
  • for G/Dgt1
  • Ericson fluctuations

isolated resonances for G/Dltlt1
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48

Spectra and Autocorrelation Function
  • Regime of isolated resonances
  • ?/D small
  • Resonances eigenvalues
  • Overlapping resonances
  • ?/D 1
  • Fluctuations ?coh

Correlation function
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49

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

SFB 634 C4 Quantum Chaos
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Spectra of S-Matrix Elements
Example 8-9 GHz
S12 ?
S
S11 ?
S22 ?
Frequency (GHz)
SFB 634 C4 Quantum Chaos
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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

SFB 634 C4 Quantum Chaos
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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


SFB 634 C4 Quantum Chaos
55

Comparison Experiment - VWZ
Time domain
Frequency domain
Example 8-9 GHz
? S12 ?
? S11 ?
? S22 ?
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56

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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57

Billiard with Broken Time Reversal Symmetry
Sab Sba
  • Clear violation of reciprocity in the regime of
    G/D gtgt 1

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58

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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61

S-Matrix Fluctuations and RMT
  • Pure GOE ? VWZ description 1984
  • Pure GUE ? V
    description 2007
  • Partial TRSB ? no analytical model
  • RMT ?

GOE
GUE
SFB 634 C4 Quantum Chaos
62

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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64

Wo das Chaos auf die Ordnung trifft, gewinnt
meist das Chaos, weil es besser organisiert
ist. Friedrich Nietzsche (1844-1900)
SFB 634 C4 Quantum Chaos
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