Chaos and Irreversibility:
An introduction to the Loschmidt echo
Diego A. Wisniacki
UBA
Overview
● Introduction ● Loschmidt echo● Loschmidt echo and chaos● Regimes of Loschmidt echo ● Decoherence and Loschmidt echo● Experiments● Final Remarks
Colaboradores-ReferenciasColaborators
●Horacio Pastawski (UNC)
●Fernando Cuccietti (Los Alamos)
●Eduardo Vergini (TANDAR, Buenos Aires)
●Doron Cohen (BGU)
●Florentino Borondo (UAM, Madrid)
●Rosa Benito (UPM, Madrid)
Colaboradores-ReferenciasIntroduction
What is chaos in classical mechanics?
Colaboradores-ReferenciasIntroduction
Colaboradores-ReferenciasIntroduction
x , p x 1, p2and
Colaboradores-ReferenciasIntroduction
Colaboradores-ReferenciasIntroduction
Colaboradores-ReferenciasIntroduction
Sensitivity to initial conditions
How it can be measure?
Colaboradores-ReferenciasIntroduction
How it can be measure?
Liapunov ExponentsLiapunov Exponents
x ,w= limt∞ d 0 0
1
tln
d x0
, t
d x0,0
∃1≥
2≥
3....≥
M
Colaboradores-ReferenciasIntroduction
Lets make the same program in quantum mechanics:
∣ ⟩ ∣ ⟩and
d , =∣⟨∣ ⟩∣2
So d t =d 0
d t =∣⟨ t ∣ t ⟩∣2=∣⟨∣U t U∣ ⟩∣2=∣⟨ 0 ∣ 0 ⟩∣2
Colaboradores-ReferenciasLoschmidt Echo
In 1984 A. Peres proposed:
M t =∣⟨∣ U t U∣ ⟩∣2
U=exp−i H0t
U=exp[−i H0 V t ] Perturbed evolution
Josef Loschmidt (1821-1895)
Colaboradores-ReferenciasLoschmidt Echo
∣ ⟩ ∣⟨∣ U t U∣ ⟩∣2
U
USensitivity to perturbations
Colaboradores-ReferenciasLoschmidt Echo
∣ ⟩
U
U
Irreversibility
∣ t ⟩U t∣ t ⟩
Colaboradores-ReferenciasLoschmidt EchoPeres, 1984 PRA
Coupled rotator model: H=LzM
zL
xM
x
Colaboradores-ReferenciasLoschmidt Echo and Chaos
Jalabert-Pastawski PRL 2001
● Initial state: localized state r , t=0=1
2d /4
exp [ i p0.r−r0
−1
2 2r−r 0
2]
r , t =∫ d r ' K r , r ' , t r , t=0
K r ,r ' , t =∑ sr ,r ' ,t
1
2 i ℏ
d / 2
Cs
1/ 2 exp [1ℏ
Ssr ,r ' , t −
i
2
s]
● Semiclassical aproximation for propagator K
● Analytical semiclassical study of the LE
Cs=del
−∂2 Ssr ,r ' , t
∂ ri∂ r '
j
r '
rt
Colaboradores-ReferenciasLoschmidt Echo and Chaos
Jalabert-Pastawski PRL 2001
● Perturbation: static disordered potential
M t =Mdt M
ndt
V r =∑
222
exp [−1
22r− R
2]
● The Loschmidt echo has two contributions:
M t ≈Mdt ≈ Aexp [− t ]
is the Lyapunov exponent of the unperturbed Hamiltonian!!!!
● For strong perturbation:
● The LE results
M t ≈ 2
ℏ2
d
∫ d r∫ d r '∑s ,s 'C
sC
s 'exp [
iℏ S
s− S
s']exp [
− 2
ℏ2[ ps
− p02 p s '
− p02]]
≫≫F
Colaboradores-ReferenciasLoschmidt Echo and Chaos
Jacquod et al 2003
What is the behavior of LE if H0 is integrable?
M t =Mdt M
ndt ● The Loschmidt echo has two contributions:
M t ≈Mdt ≈ At−3d /2
Power law decay
● For strong perturbation:
● Semiclassical aproximation for K idem Jalabert-Pastawski
Colaboradores-ReferenciasLoschmidt Echo and ChaosJacquod et al 2003
Increase of
the perturbation
● Numerical check: kicked top H 0=/2 SyK / 2 S S
z
2∑ t−n
perturbation= Sx∑ t−n−
Colaboradores-ReferenciasLoschmidt Echo and ChaosJacquod et al 2003
Colaboradores-ReferenciasRegimes of the LE
M t =∣⟨∣exp[ i H 0V t ]exp −i H 0 t ∣ ⟩∣2
The LE depends on
●The perturbation ● The initial state ● The time t
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE with perturbation
Jacquod Silvestrov Beenakker PRE 2001
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE with perturbation
If the perturbation matrix element is much smaller than
M t ≈∣∑ ∣b m∣
2exp −i t V
m m/ ℏ ∣
2
H =H0V
M t ≈exp −2E
2 t 2/ℏ2 Gaussian decay
Variance of level velocities
Perturbation theory
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE with perturbation
H =H0VIf V
ij
LDOS
∣ni ⟩
Relates old and new eigenstates
Ei
E
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE with perturbation
H =H0VIf V
ij
LDOS ∣ni ⟩
Pi , j =∣⟨n
i0 ∣n
j ⟩∣2
Relates old and new eigenstates
Width of LDOS
M t ≈exp − t FGR decay
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE with perturbation
If
M t ≈exp − t
Liapunov Regime !!!!!!!!!
Liapunov exponent
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE in the stadium billiard
l=r
a=1
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE in the stadium billiard
Exp(- t)
Exp(- t)
Non-universal
exp(-t)
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE in the Lorentz gas
M(t)=0.09
Colaboradores-ReferenciasRegimes of the LE
Regimes of the LE in the Lorentz gas
Colaboradores-ReferenciasRegimes of the LE
Dependence of the LE with the initial state
Wisniacki-Cohen 2002
Is universal the Lyapunov regime?
∣ ⟩=∣n i ⟩ Initial state: eigenstate
M t =∣⟨∣ U t U∣ ⟩∣2=∣⟨ni∣ U t∣n
i⟩∣2=Sp
i t
But Spit =∣FT LDOS∣2 Then Physics of the LE = LDOS??
Colaboradores-ReferenciasRegimes of the LE
Dependence of the LE with the initial state
Wisniacki-Cohen 2002
H =H0V New V_ij=random(-1)*V_ij
No lyapunov regime!!!!
Colaboradores-ReferenciasRegimes of the LE
Dependence of the LE with the initial stateWisniacki-Cohen 2002
No lyapunov regime!!!!
Colaboradores-ReferenciasRegimes of the LE
Short time decay of the LE
Wisniacki 2003
Why? Experimental relevant regime??
M t ≈exp [− 2 t2 ]H =H
0V
Perturbed Hamiltonian
= V
V=⟨∣V 2∣ ⟩−⟨∣V∣ ⟩2
We show = V= Width of LDOS
depends on and V
Colaboradores-ReferenciasRegimes of the LE
Short time decay of the LE
Wisniacki 2003
Colaboradores-ReferenciasRegimes of the LE
Short time decay of the LE
Initial state: eigenfunction of Ho
Colaboradores-ReferenciasRegimes of the LE
Short time decay of the LE
Initial state: gaussian wave packet Initial state: evolved gaussian wave packet
Colaboradores-ReferenciasDecoherence and the LE
Zurek-Paz (1994)
Environment
ChaoticSystem
S t =Tr [slog
s]≈ t
Lyapunov exponent independent of the coupling with the
environment
As Loschmidt echo but with non-unitary evolution
Perturbation independent regime
S
t
Decoherence -> lost of quantum coherence -> quantum-classical transition
Colaboradores-ReferenciasDecoherence and the LECucchietti et al (2003)
M t =∫ D P ∣⟨ ∣U t t U t ∣ ⟩∣2
t =∫ D P U t ∣ ⟩ ⟨ ∣U t t
M t =Tr [t 0t ]
t =1
i ℏ[H
0, t ]−D [V x ,[V x , t ] ]
Unitary evo. Non Unitary evo.
M t =Tr [t t ]≈Tr [ t t ]≈a exp [− t ]b exp −D kp
2 t They showed
Direct connection between decoherence and the LE
Density matrix evolved by unperturbed U
Lyapunov regime FGR
Colaboradores-ReferenciasExperiments
● MNR polarization echo Physica A 00 Pastawski
● Microwave cavity PRL 05 Stockmann
● NMR Information processor PRL 05 Laflamme
Colaboradores-ReferenciasExperiments
●Single crystal of ferrocene●Many-body system●Gaussian decay●Perturbation independent
regime
MNR polarization echo Physica A 00 Pastawski
Colaboradores-ReferenciasExperiments
● Electromagnetic cavity: equivalence of
Helmholtz and Schrodinger eq.
● Measure the stationary scattering matrix element● RMT theoretical result
438 mm
200 mm
M t =exp [−8 2 2t 2t
2−∫0
t
∫0
b
2 'd ' d ]
Microwave cavity PRL 05 Stockmann
=0.01
=0.6
=0.8
Colaboradores-ReferenciasExperiments
● Measure of the LE in an Nuclear Magnetic Resonance experiment. ● Idea: Characterization of Complex Quantum Dynamics with a
Scalable NMR Information Processor ---> 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
NMR Information processor PRL 05 Laflamme et al
Colaboradores-ReferenciasExperiments
Regular U
Chaotic U
NMR Information processor PRL 05 Laflamme et al
different perturbations
FGR decay
FGR decay different perturbations
Colaboradores-ReferenciasFinal Remarks
● Is the LE a good measure of 'quantum chaos'?
● Regimes of the LE ---> 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,...