Chaos and Irreversibility: An introduction to the Loschmidt echo

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?



x , p x 1, p2and




Sensitivity to initial conditions

How it can be measure?


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


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)


∣ ⟩ ∣⟨∣ U t U∣ ⟩∣2

U

USensitivity to perturbations


∣ ⟩

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


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


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


Regimes of the LE with perturbation

Jacquod Silvestrov Beenakker PRE 2001



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



H =H0VIf V

ij

LDOS

∣ni ⟩

Relates old and new eigenstates

Ei

E



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



If

M t ≈exp − t

Liapunov Regime !!!!!!!!!

Liapunov exponent


Regimes of the LE in the stadium billiard

l=r

a=1


Regimes of the LE in the stadium billiard

Exp(- t)

Exp(- t)

Non-universal

exp(-t)


Regimes of the LE in the Lorentz gas

M(t)=0.09


Regimes of the LE in the Lorentz gas


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??


Dependence of the LE with the initial state

Wisniacki-Cohen 2002

H =H0V New V_ij=random(-1)*V_ij

No lyapunov regime!!!!


Dependence of the LE with the initial stateWisniacki-Cohen 2002

No lyapunov regime!!!!


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



Wisniacki 2003



Initial state: eigenfunction of Ho



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


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

regime

MNR polarization echo Physica A 00 Pastawski


● 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


● 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


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,...

Documents

Chaos and Irreversibility: An introduction to the Loschmidt echo