Chaos and Irreversibility:

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Chaos and Irreversibility An introduction to
the Loschmidt echo
Diego A. Wisniacki
UBA
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Overview

• Introduction
• Loschmidt echo
• Loschmidt echo and chaos
• Regimes of Loschmidt echo
• Decoherence and Loschmidt echo
• Experiments
• Final Remarks

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Colaboradores-Referencias
Colaborators

• Horacio Pastawski (UNC)
• Fernando Cuccietti (Los Alamos)
• Eduardo Vergini (TANDAR, Buenos Aires)
• Doron Cohen (BGU)
• Florentino Borondo (UAM, Madrid)
• Rosa Benito (UPM, Madrid)

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Colaboradores-Referencias
Introduction
What is chaos in classical mechanics?
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Colaboradores-Referencias
Introduction
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Colaboradores-Referencias
Introduction
and
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Colaboradores-Referencias
Introduction
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Colaboradores-Referencias
Introduction
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Colaboradores-Referencias
Introduction
Sensitivity to initial conditions
How it can be measure?
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Colaboradores-Referencias
Introduction
How it can be measure?
Liapunov Exponents
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Colaboradores-Referencias
Introduction
Lets make the same program in quantum mechanics
and
So
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Colaboradores-Referencias
Loschmidt Echo
In 1984 A. Peres proposed
Perturbed evolution
Josef Loschmidt (1821-1895)
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Colaboradores-Referencias
Loschmidt Echo
Sensitivity to perturbations
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Colaboradores-Referencias
Loschmidt Echo
Irreversibility
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Colaboradores-Referencias
Loschmidt Echo
Peres, 1984 PRA
Coupled rotator model
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Loschmidt Echo and Chaos
Colaboradores-Referencias
Jalabert-Pastawski PRL 2001

• Analytical semiclassical study of the LE

• Initial state localized state

• Semiclassical aproximation for propagator K

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Loschmidt Echo and Chaos
Colaboradores-Referencias
Jalabert-Pastawski PRL 2001

• Perturbation static disordered potential

• The LE results

• The Loschmidt echo has two contributions

• For strong perturbation

l is the Lyapunov exponent of the unperturbed
Hamiltonian!!!!
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Loschmidt Echo and Chaos
Colaboradores-Referencias
What is the behavior of LE if H0 is integrable?
Jacquod et al 2003

• Semiclassical aproximation for K idem
  Jalabert-Pastawski

• The Loschmidt echo has two contributions

• For strong perturbation

Power law decay
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Loschmidt Echo and Chaos
Colaboradores-Referencias
Jacquod et al 2003

• Numerical check kicked top

Increase of the perturbation
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Loschmidt Echo and Chaos
Colaboradores-Referencias
Jacquod et al 2003
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Colaboradores-Referencias
Regimes of the LE
The LE depends on

• The perturbation S
• The initial state Y
• The time t

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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE with perturbation S
Jacquod Silvestrov Beenakker PRE 2001
S
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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE with perturbation S
If the perturbation matrix
Perturbation theory
element is much smaller than D
Gaussian decay
Variance of level velocities
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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE with perturbation S

If
S
Relates old and new eigenstates
LDOS
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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE with perturbation S

If
LDOS
Relates old and new eigenstates
Width of LDOS
FGR decay
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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE with perturbation S

If
Liapunov exponent
Liapunov Regime !!!!!!!!!
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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE in the stadium billiard
S
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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE in the stadium billiard
g
exp(-g t)
l
G(S)/2
S
Non-universal
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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE in the Lorentz gas
M(t)0.09
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Colaboradores-Referencias
Regimes of the LE
Regimes of the LE in the Lorentz gas
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Colaboradores-Referencias
Regimes of the LE
Dependence of the LE with the initial state
Wisniacki-Cohen 2002
Is universal the Lyapunov regime?
Initial state eigenstate
But
Then Physics of the LE LDOS??
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Colaboradores-Referencias
Regimes of the LE
Dependence of the LE with the initial state
Wisniacki-Cohen 2002
New V_ijrandom(-1)V_ij
No lyapunov regime!!!!
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Colaboradores-Referencias
Regimes of the LE
Dependence of the LE with the initial state
Wisniacki-Cohen 2002
No lyapunov regime!!!!
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Colaboradores-Referencias
Regimes of the LE
Short time decay of the LE
Wisniacki 2003
Why?
Experimental relevant regime??
Perturbed Hamiltonian
Width of LDOS
We show
depends on Y and V
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Colaboradores-Referencias
Regimes of the LE
Short time decay of the LE
Wisniacki 2003
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Colaboradores-Referencias
Regimes of the LE
Short time decay of the LE
Initial state eigenfunction of Ho
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Colaboradores-Referencias
Regimes of the LE
Short time decay of the LE
Initial state gaussian wave packet
Initial state evolved gaussian wave packet
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Colaboradores-Referencias
Decoherence and the LE
Decoherence -gt lost of quantum coherence -gt
quantum-classical transition
Zurek-Paz (1994)
S
Environment
Chaotic
System
t
Lyapunov exponent
independent of the coupling with the
environment
Perturbation independent regime
As Loschmidt echo but with non-unitary evolution
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Colaboradores-Referencias
Decoherence and the LE
Direct connection between decoherence and the LE
Cucchietti et al (2003)
Density matrix evolved by unperturbed U
Unitary evo.
Non Unitary evo.
Lyapunov regime
FGR
They showed
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Colaboradores-Referencias
Experiments

• MNR polarization echo Physica A 00 Pastawski
• Microwave cavity PRL 05 Stockmann
• NMR Information processor PRL 05 Laflamme

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Colaboradores-Referencias
Experiments
MNR polarization echo Physica A 00 Pastawski

• Single crystal of ferrocene
• Many-body system
• Gaussian decay
• Perturbation independent
• regime

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Colaboradores-Referencias
Experiments
Microwave cavity PRL 05 Stockmann

• Electromagnetic cavity equivalence of
• Helmholtz and Schrodinger eq.
• Measure the stationary scattering matrix element
• RMT theoretical result

200 mm
438 mm
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Colaboradores-Referencias
Experiments
NMR Information processor PRL 05 Laflamme et al

• Measure of the LE in an Nuclear Magnetic
  Resonance experiment.
• Idea Characterization of Complex Quantum
  Dynamics with a
• Scalable NMR Information Processor ---gt
  understanding the
• performance and improvement of the device
• It is implementing in an scalable circuit in
  which the measure is
• done in one q-bit
• U unitary map, P perturbation
• U chaotic o regular
• 5 q-bits

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Colaboradores-Referencias
Experiments
NMR Information processor PRL 05 Laflamme et al
Regular U
different perturbations
FGR decay
Chaotic U
different perturbations
FGR decay
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Colaboradores-Referencias
Final Remarks

• Is the LE a good measure of 'quantum chaos'?
• Regimes of the LE ---gt complex behaviour
• Irreversibility and LE
• Experiments -nobody see the Lyapunov regime
• -microwave billiards and
  NMR processor
• FGR regime
• -PID in the many body
  system
• Other works LE in a many body system, LE
  freeze,...