Quantum Chaos: An Overview

Sankhasubhra Nag sankhasubhra [email protected]

Department of Physics Sarojini Naidu College for Women 30, Jessore Road, Kolkata 700028

Quantum Chaos: – p. 1/15 A Popular Misconception!! Quantum Chaos is not chaotic quantum mechanics.

Quantum Chaos: – p. 2/15 A Popular Misconception!!

It deals with quantum mechanics of the systems which are chaotic in behaviour in classical regime.

Quantum Chaos: – p. 2/15 A Popular Misconception!!

It deals with quantum mechanics of the systems which are chaotic in behaviour in classical regime.

But what is chaos and chaotic behaviour?

Quantum Chaos: – p. 2/15 Chaos Some bound motion extremely sensitive to initial conditions is called chaotic. Chaos Some bound motion extremely sensitive to initial conditions is called chaotic. l(t) i.e. l(t)= l(0) exp(λt)

l(0)

Quantum Chaos: – p. 3/15 Chaos Some bound motion extremely sensitive to initial conditions is called chaotic. l(t) i.e. l(t)= l(0) exp(λt)

l(0) λ is called the Lyapunov exponent, which is a measure of chaos

Quantum Chaos: – p. 3/15 Chaos Some bound motion extremely sensitive to initial conditions is called chaotic. l(t) i.e. l(t)= l(0) exp(λt)

l(0) Though chaos is a general concept valid for any type of dynamical system, for the present context we restrict ourselves within the motion of particles under certain force field i.e. the Hamiltonian systems.

Quantum Chaos: – p. 3/15 Visual signature of Chaos If some cross section of the phase space is taken

for chaotic systems where the trajectories can diverge from each other in a bound region only by taking

v v v v v

sharp twists and turns, the section will show a scattered sea of points.

Quantum Chaos: – p. 4/15 Visual signature of Chaos If some cross section of the phase space is taken

In contrast regular kind of features are manifested by the integrable systems i.e. when N independent constants of motion in involution with each other exist for a system with N degrees of freedom or in other words there exist N relations

Fm(q,p)= Cm, and also,

{Fm,Fn}PB =0 for all m,n =1,...,N

Quantum Chaos: – p. 4/15 Visual signature of Chaos If some cross section of the phase space is taken

Now all linear systems can be separated out into a number of (autonomous) systems with single degree of freedom each. Automatically for each of them a single constant of motion exists i.e. the Hamiltonian itself; or in other words All linear systems are regular i.e. non-chaotic.

Quantum Chaos: – p. 4/15 Visual signature of Chaos If some cross section of the phase space is taken

In contrast regular kind of features are manifested by the integrable systems where the trajectories are constrained to remain on a set of surfaces (tori) and periodic or quasiperiodic types of motion are the only

//  

possible ones; giving rise to some regular curves on the section plot

Quantum Chaos: – p. 4/15 Visual signature of Chaos If some cross section of the phase space is taken

(a) (b) 1 1

0.8 0.8

0.6 0.6 1 1 p p 0.4 0.4

0.2 0.2

0 0 0 0.5 1 0 0.5 1 q q 1 1

Quantum Chaos: – p. 4/15 Visual signature of Chaos If some cross section of the phase space is taken

6 6 5 5 K=0.5 K=1.5 4 4 p p 3 3 2 2 1 1 0 0 0 2 4 6 0 2 4 6 q q

6 6 5 5 K=3.5 K=6.5 4 4 p p 3 3 2 2 1 1 0 0 0 2 4 6 0 2 4 6 q q Thus just by observing the cross section plot one can infer about the nature of the system.

Quantum Chaos: – p. 4/15 Quantization Scheme

H −→ Hˆ = Hˆ (ˆq, pˆ) Choose some suitable basis states to write the explicit form of Hˆ

Quantum Chaos: – p. 5/15 Quantization Scheme

H −→ Hˆ = Hˆ (ˆq, pˆ) Choose some suitable basis states to write the explicit form of Hˆ Calculate the eigenvalues and eigenvectors of Hˆ to find the energy spectra and stationary states which are associated with the static features.

Quantum Chaos: – p. 5/15 Quantization Scheme

H −→ Hˆ = Hˆ (ˆq, pˆ) Choose some suitable basis states to write the explicit form of Hˆ Calculate the eigenvalues and eigenvectors of Hˆ to find the energy spectra and stationary states which are associated with the static features. Evolve some initial state ψ(t = 0) according to the Schrödinger equation to find the dynamical features.

Quantum Chaos: – p. 5/15 Quantization Scheme

H −→ Hˆ = Hˆ (ˆq, pˆ) But Schrödinger equation, ∂ Hψˆ = i~ ψ, ∂t is a perfectly linear one.

Quantum Chaos: – p. 6/15 Quantization Scheme

H −→ Hˆ = Hˆ (ˆq, pˆ) But Schrödinger equation, ∂ Hψˆ = i~ ψ, ∂t is a perfectly linear one. Hence there is no scope of CHAOS in classical sense. So one should expect that the distinction between regular and chaotic systems should vanish, once they get quantised .

Quantum Chaos: – p. 6/15 Quantization Scheme

H −→ Hˆ = Hˆ (ˆq, pˆ) But Schrödinger equation, ∂ Hψˆ = i~ ψ, ∂t is a perfectly linear one.

But interestingly enough, in practice those persist both in static features and dynamical features of the corresponding quantum mechanical systems.

Quantum Chaos: – p. 6/15 Static Features Level Repulsion The energy levels of a regular system cluster together so that nearest neighbour level spacing distribution becomes f(s) ∼ exp(−s) which is maximum for zero spacing. For chaotic Hamiltonian, the distribution f(s) ∼ sν exp(−s2) it is zero. It seems energy levels repels each other.

Quantum Chaos: – p. 7/15 Static Features Level Repulsion The energy levels of a regular system cluster together so that nearest neighbour level spacing distribution becomes f(s) ∼ exp(−s) which is maximum for zero spacing.

Quantum Chaos: – p. 7/15 Static Features Features of Eigenvectors The elements of the eigenvectors are also distributed in some charateristics distribution

1 d 2 dx2 P (x)= exp − . 2π 2     for chaotic Hamiltonians.

Quantum Chaos: – p. 8/15 Static Features Features of Eigenvectors The elements of the eigenvectors are also distributed in some charateristics distribution

1 d 2 dx2 P (x)= exp − . 2π 2     for chaotic Hamiltonians. There are certain other static aspects like scarring or residual parameters for eigenvectors are associated with the quantum chaos

Quantum Chaos: – p. 8/15 Dynamical Features: Example One of the simplest Hamiltonian systems is pendulum, where p2 H = + K cos q 2I which is a regular system.

Quantum Chaos: – p. 9/15 Dynamical Features: Example One of the simplest Hamiltonian systems is pendulum, where p2 H = + K cos q δ(t − nτ) I 2 n X Kicked Rotor Its chaotic counterpart is found by multiplying periodic δ kicks with the potential part.

Quantum Chaos: – p. 9/15 Quantization of Such Systems The Hamiltonian may be quantized by taking the operator form,

pˆ2 Hˆ = + K cosq ˆ δ(t − nτ) I 2 n X The quantization is done in basis states φm, such that

pφˆ m = m~φm

Quantum Chaos: – p. 10/15 Quantization of Such Systems The Hamiltonian may be quantized by taking the operator form,

pˆ2 Hˆ = + K cosq ˆ δ(t − nτ) I 2 n X The time evolution of a state vector ψ is given by

ψ(t) = Uˆ(0,t)ψ(0) ıpˆ2τ ıK cosq ˆ = exp − exp − ψ(0) 2I~ ~    

Quantum Chaos: – p. 10/15 Dynamical Localisation Quantum-Classical correspondence violated ???

Here the energy is scaled by a factor k2 ∝ K2 and t˜= n.[Ref. F.M. Izrailev, Physics Reports,196, Nos. 5 & 6 (1990) 299 -U˚ 392.]

Quantum Chaos: – p. 11/15 Dynamical Localisation Quantum-Classical correspondence violated ???

100

80

60 E 40 classical η=1.000 quantum η=1.000 20 classical η=0.001 quantum η=0.001

0 0 10 20 30 40 n Here the system is coupled to an external heat bath with coupling strength η.

Quantum Chaos: – p. 11/15 RDM fluctuation If two systems A and B are chosen with density matrices ρA and ρB respectively, the density matrix of the composite system S may be written as

ρS = ρA ⊗ ρB,

Quantum Chaos: – p. 12/15 RDM fluctuation But always one can calculate reduced density matrix for the system A (or for system B) by taking partial trace of ρS over the space of the system B (or the system A). To study the time evolution one may choose the representative value for the wave-function as

2 SL =1 − Tr(ρR); which is also a measure of entanglement between A and B when the composite system is in pure state.

Quantum Chaos: – p. 12/15 RDM fluctuation

ρ(t=T) Unitary Evolution

ρ(t=0)

Reduction Reduction

Non−Unitary ρ (t=0) Evolution ρ (t=T) R R

Quantum Chaos: – p. 12/15 RDM fluctuation

(a) (b) 1 1

0.8 0.8

0.6 0.6 L L S S 0.4 1 0.4 1

0.2 0.5 0.2 0.5 0 0 0 10 20 0 10 20 0 0 0 50 100 0 50 100 time time

Quantum Chaos: – p. 12/15 Feature of Hamiltonian Matrix Correlation among Hamiltonian matrix elements are measured by

∗ AH (m)= Hl+m,k+mHl,k, l,k X

(a) (b) 1 1 1

0.8 0.5 0.8

0.6 0 0.6 0 5 10 A l =63.39 A H c H 0.4 0.4 l =6.13 c

0.2 0.2

0 0 0 50 100 0 50 100 m m

Quantum Chaos: – p. 13/15 Feature of Hamiltonian Matrix Bohigas, Giannoni and Schmit conjectured that the spectra of sufficiently chaotic Hamiltonian may be explained with Random Matrix spectra

2 Tr ρR = (ρR)m,n (ρR)n,m m,n
X   ab a′b′ = φm,nφn,m a,b,a′,b′ m,n ! X X ′ ′ Ea − Eb Ea − Eb × exp −i( ~ + ~ )t .  

Quantum Chaos: – p. 13/15 References E. Ott.,Chaos in dynamical systems 1997. H. Stockmann.,Quantum chaos An Introduction 2009. M.L. Mehta, Random Matrices 2004

Quantum Chaos: – p. 14/15 Conclusions Quantum Measurement

3 & Decoherence

Quantum Quantum Chaos - Information Theory

s Mesoscopic Systems

Quantum Chaos: – p. 15/15 Conclusions

OK, it’s over at last!

Quantum Chaos: – p. 15/15