Reversing chaos Boris Fine Skolkovo Institute of Science and Technology University of Heidelberg.
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Transcript of Reversing chaos Boris Fine Skolkovo Institute of Science and Technology University of Heidelberg.
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.
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
)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!