Reversing chaos

Boris Fine

Skolkovo Institute of Science and Technology

University of Heidelberg

2

Postdoctoral and Ph.D. positions are availablein the group of B.F.

What is quantum chaos?

No fully consistent answer. Fundamentally all quantum systems are nonchaotic at the level of the Schroedinger equation for many-particle wave functions. Yet, the averaged behavior of quantum systems is often strongly reminiscent of the behavior of classical chaotic systems.

Best empirical indication of quantum chaos is the Wigner-Dyson statistics of the spacings between quantum energy levels.

D. Poilblanc et al., EPL 22, 537 (1993)

P(Δ

)

Distribution of energy level spacings

Integrable Chaotic

ΔΔ

Δ

Problem: In many-particle systems Δ is exponentially small. The corresponding timescale is unphysically long.

Nuclear Magnetic Resonance (NMR)

H

μ

nucleus

HΩ

Ωh

Spin 1/2

Larmor precession

NMR free induction decay

nm

yn

ym

xn

xmmn

zn

zm

zmn SSSSJSSJ

H

3

222 cos 31 2

mn

mnmn

zmn r

gJJ

magnetic dipolar interaction:

t = 0 t ~ T2

T = ∞

Generic long-time behavior of nuclear spin decays:[B. F., Int. J. Mod. Phys. B 18, 1119 (2004)]

), ( cos ~ )( tetG tnnNJ ~ ,where

Markovian behavior on non-Markovian time scale is a manifestation of chaos.

Chaotic eigenmodes of the time evolution operator:Pollicott-Ruelle resonances [D. Ruelle, PRL 56, 405 (1986)]

Spins 1/2, experimentB. Meier et al., PRL 108, 177602 (2012).

Investigations of Lyapunov instabilities in classical spin systems A. de Wijn, B. Hess, and B. F., Phys. Rev. Lett. 109, 034101 (2012)

…. J. Phys. A 46, 254012 (2013)

Definition of Lyapunov exponents

NN

nm

zn

zmz

yn

ymy

xn

xmx SSJSSJSSJH

)0(X

)(tX

nnn hSS Equations of motion:

NN

m

mzz

myy

mxx

n

SJ

SJ

SJ

hLocal fields:

Phase space vectors:

Nz

Ny

Nx

z

y

x

z

y

x

S

S

S

S

S

S

S

S

S

2

2

2

1

1

1

X

nnnnn hShSS

NN

m

mzz

myy

mxx

n

SJ

SJ

SJ

h

Small deviations:

Nz

Ny

Nx

z

y

x

z

y

x

S

S

S

S

S

S

S

S

S

2

2

2

1

1

1

X

Largest Lyapunov exponent:

)0(

)(ln

1 lim

0)0(

tmax X

X

X

t

t

Lesson learned:

is an intensive quantity for spin lattices with short-range interactions

max

How to extract Lyapunov exponent from the observable behavior of a many-particle system?

Difficulties:

We cannot experimentally observe phase space coordinates of every particle

We cannot controllably prepare two close initial conditions in a many-particle phase space

Statistical behavior of many-particle systems masks Lyapunov instabilities

Relaxation path in many-body phase space

Mx

Solution: time-reversal of relaxation! (Loschmidt echo)

Mx

Sensitivity of Loschmidt echoes to small perturbations

Physica A 283, 166 (2000)

)0(X

)(tX

Time reversal of magnetization noise

B.F., T.A. Elsayed, C. M. Kropf, and A.S. de Wijn, Phys. Rev. E 89, 012923 (2014)

Observable exponential sensitivity of classical spin systems to small perturbations

B.F., T.A. Elsayed, C. M. Kropf, and A.S. de Wijn, Phys. Rev. E 89, 012923 (2014)

0H 0H-R

R

Each Ψν is obtained from Ψ- by complete flipping of

a small fraction of spins

5x5 lattice of spins 1/2

No exponential growth in the regime 1-F(τ) << 1

Absence of exponential sensitivity to small perturbationsin nonintegrable systems of spins 1/2

B.F., T.A. Elsayed, C. M. Kropf, and A.S. de Wijn, Phys. Rev. E 89, 012923 (2014)

Larger quantum spinsT.A. Elsayed, B. F., Phys. Scripta (2015)

Solid black: 6 spins 15/2

Blue: 6 classical spins

Dashed: 26 spins 1/2

Summary:

1. NMR experiments indicate that relaxation in quantum spin systems exhibits generic similarities with the relaxation in chaotic classical spin systems.

2. Loschmidt echoes in chaotic systems of classical spins exhibit exponential sensitivity to small perturbations controlled by twice the value of the largest Lyapunov exponent.

3. Loschmidt echoes in non-integrable systems of spins 1/2 exhibit power-law sensitivity to small perturbations.

- Implication for generic many-particle quantum systems: ergodicity ≠ exponential sensitivity to small perturbations

- Good news for the efforts to create quantum simulators!

Investigations of Lyapunov spectra in classical spin systems [A. de Wijn, B. Hess, and B. F., J. Phys. A 46, 254012 (2013)]

Lyapunov spectra for four different latticesSize dependence of the entire spectra (Heisenberg chains)

Size dependence of (Heisenberg Hamiltonian)Lattices considered max