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Quantum Chaos
Dominique Delande
Laboratoire Kastler-Brossel
Université Pierre et Marie Curie and Ecole Normale Supérieure
Paris (European Union)
What is chaos?
What is quantum chaos? Is it useful?
Is quantum chaos relevant for the physics of cold atoms?
Are cold atoms useful for understanding quantum chaos?
Quantum Chaos – Les Houches – September 2004
Outline
First lecture:
What is classical chaos?
Quantum dynamics vs. classical dynamics
A simple example: the hydrogen atom in a magnetic field
Time scales – Energy scales Good systems for studying quantum chaos
Quantum chaos and cold atoms
Second lecture:
Random Matrix Theory
Semiclassical approximation(s)
Periodic orbit theory
Third lecture:
Chaos assisted tunneling
Transport properties; dynamical localization
Coupling to the environment; decoherence for chaotic systems
Loschmidt echo
Quantum Chaos – Les Houches – September 2004
Classical chaos
Temporal evolution of an initially small localized region of phase space:
Stretches in some directions (exponential sensitivity on initial conditions)
Must shrink in other directions to preserve phase space volume
If phase space is finite, exponential stretching cannot last for ever  folding
Very often, chaotic systems are ergodic and mixing (a typical trajectory uniformly fills the whole phase space).
In low­dimensional systems, the transition from regularity to chaos is smooth (when a parameter is varied). Mixed regular­classical dynamics is rather complicated.
Quantum Chaos – Les Houches – September 2004
The Wigner distribution
Try to formulate Quantum Mechanics in phase space
Let us define the Wigner distribution (for a pure state; extension for a general density matrix follows straightforwardly):
It is a phase space quasi­probability density; it is real but can be negative.
W occupies at least a volume (2pℏ)d, i.e. one Planck cell in phase space.
Evolution equation is easily obtained from Schrödinger equation:
where
Makes it possible an explicit expansion in powers of ℏ.
Lowest order: classical Liouville equation.
Quantum Chaos – Les Houches – September 2004
The Wigner distribution (continued)
p2
Assume . Then
H=
V q
2m
For polynomial potential up to degree 2, the Wigner distribution evolves exactly like the classical phase space density.
For chaotic systems, the classical phase space distributions develop structures at smaller and smaller scales as time increases  ∂3W/∂p3 becomes larger and larger. Even for tiny Planck constant, the corrections will eventually overcome the lowest­order contribution.
The break­time at which it occurs diverges as ℏ goes to zero.
The two limits t→∞ and ℏ→0 do not commute. Quantum (semiclassical) chaos is essentially a study of the asymptotic properties of Quantum Mechanis at small ℏ.
Quantum Chaos – Les Houches – September 2004
Some basic questions in quantum chaology
What are the appropriate quantum observables to detect the regular or chaotic classical behaviour of the system?
More precisely, how does the regular or chaotic classical behaviour translate in the energy levels and eigenstates of the (bound) system? For an open system, in the decay rates, in the S­matrix, in the transport properties? (lectures 1 and 2)
What kind of semiclassical approximations can be used? (lecture 2)
What is the long­time behaviour of a quantum system? (lecture 3)
For a macroscopic system, how is the classical behaviour recovered? (lecture 3)
Quantum Chaos – Les Houches – September 2004
Constant energy contours
we study the Lz=0 subspace
Quantum Chaos – Les Houches – September 2004
Poincaré surface of section for the hydrogen atom in a magnetic field
“Regular” trajectory
“Chaotic” trajectory
Quantum Chaos – Les Houches – September 2004
Poincaré surfaces of section for the hydrogen
atom in a magnetic field
e=­0.4
e=­0.5
e=­1
e=­0.3
e=­0.2
e=­0.1
Quantum Chaos – Les Houches – September 2004
Summary of “experimental” observations
When the classical dynamics is regular (i.e. not sensitive on initial conditions), the trajectory in configuration space looks ordered, with strong position­
momentum correlations.
When the classical dynamics is chaotic (sensitive on initial conditions), the trajectory in configuration space looks completely disordered, apparently erratic, without position­momentum correlations.
In phase space, a regular trajectory seems to fill a two­dimensional surface (a torus), very regularly.
In phase space, a chaotic trajectory seems to fill a subspace with non­zero volume.
Poincaré surfaces of section are very useful to discriminate between regular and chaotic behaviour.
At low scaled energy, the dynamics looks fully regular. Chaos requires (at least) two forces with comparable strength and different symmetries.
Above e=­0.5, some chaotic regions appear.
Between e=­0.5 and e=­0.13, regular and chaotic regions (depending on the initial conditions) coexist peacefully.
Above e=­0.13, the classical dynamics looks fully chaotic.
Quantum Chaos – Les Houches – September 2004
Time scales – Energy scales
●
The Ehrenfest time TEhr. It is the time for a minimum wavepacket to spread in the full phase space.
TEhr
Quantum Chaos – Les Houches – September 2004
Time scales: the Ehrenfest time (continued)
Examples for the Ehrenfest time:
Wavepacket in configuration space for the stadium billiard (see viewgraph)
Regular circular billiard
Chaotic stadium billiard
Wigner function in phase space for a non­linear oscillator (Zurek et al, Rev. Mod. Phys., 75, 715 (2003)):
2
2
p
x
H=
− cos  x−l sin t a
2m
2
with m=1,=0.36, l=3, a=0.01
Quantum Chaos – Les Houches – September 2004
Quantum
Wigner
distribution
Classical
Liouville density
Quantum Chaos – Les Houches – September 2004
Time scales: the Heisenberg time
Quantum autocorrelation function for
a chaotic system (Hydrogen atom +
magnetic field)
Fourier transform at short time
The peaks associated with individual energy levels are not resolved!
Fourier transform at long time (longer
than the Heisenberg time)
The peaks are resolved!
Quantum Chaos – Les Houches – September 2004
Time scales – Energy scales (summary)
N.B.: In mutidimensional systems, the Heisenberg time is much longer than the
Ehrenfest time.
Quantum Chaos – Les Houches – September 2004
Good systems for studying quantum chaos
What is required:
Classically chaotic dynamics; at least two strongly coupled degrees of freedom (2d time­independent or 1d time­dependent system).
Controlled preparation of the initial state.
Controlled analysis of the final state.
Well controlled Hamiltonian with tunable parameters.
Tunable effective Planck's constant.
Interaction time sufficiently long (classical chaos is an asymptotic property for long times).
System well isolated from its environment.
Toys for theorists:
Billiards.
Coupled harmonic oscillators.
Internal dynamics of atomic nuclei.
Electronic dynamics in atoms:
Atoms in external field.
Three­body system (helium atom). Quantum Chaos – Les Houches – September 2004
Good systems for studying quantum chaos
Electronic dynamics in molecules or clusters.
Nuclear motion in excited molecules.
Electronic dynamics in (clean) solid­state samples (mesoscopy).
External dynamics (i.e. motion of the center of masses) of cold atoms in external fields.
Other wave equations:
Microwave billiards;
Acoustic waves;
...
Quantum Chaos – Les Houches – September 2004
Quantum chaos and cold atoms
Forget internal structure of the atoms (excited electronic states), so that the atom can be considered as a single particle. Fine and hyperfine structures may add some complications.
Control of the dynamics with laser fields, magnetic fields, gravitational field. If far detuned lasers are used, the interaction is simply a time and space dependent optical potential.
Orders of magnitude:
Velocity: cm/s
Temperature: mK De Broglie wavelength: mm
Time: ms­ms
Frequency: kHz­MHz
Energy: neV
Very favorable!
Quantum Chaos – Les Houches – September 2004
Quantum chaos and cold atoms
Advantages:
Time scales;
Wafefunction can be measured;
Transport properties can be measured;
The basic ingredients are well known and under control.
Disadvantgaes of cold atoms:
Selective preparation of the initial state is not obvious.
Gravity.
Typical spatial dimensions not very favorable and not very tunable.
Relatively small number of atoms in an experiment (signal/noise problem).
Spontaneous emission acts as a source of decoherence (and damping at small detuning).
Atom­atom interaction:
Acts as a source of decoherence.
A BEC could be used. The GP equation is non linear and its dynamics can be chaotic. It is however an approximate description of a full many­body dynamics. Be careful. Quantum Chaos – Les Houches – September 2004
A simple experiment on chaos with cold atoms
Build a “billiard” with walls made of light
gravity
Hole
Classical dynamics in the “gravitational billiard”:
Depending on the angle of the wedge, the motion alternates between regularity and chaos, with mixed intermediate regime.
Quantum Chaos – Les Houches – September 2004
Experimental result on the gravitational billiard
Prepare a cloud of cold Cs atoms in the billiard: temperature few mK, size 250 mm (large!);
Launch the atoms and wait 300 ms;
Measure the number of atoms still trapped in the billiard; the survival probability is larger when the dynamics is regular and practically zero when it is fully chaotic.
Typical atomic velocity (mainly due to gravity): few cm/s
Classical
simulation
De Broglie wavelength: less than 0.1 mm Action of a typical classical orbit: more than 1000ℏ
Period of a typical classical orbit: 10 to 100 ms
Heisenberg time: more than 10s
Experimental result
No hope to reach the quantum regime
Quantum Chaos – Les Houches – September 2004
A typical experiment on quantum chaos with cold atoms
Expose a cold atomic cloud to a time­dependent standing wave
Temporal modulation of the
standing wave
W. Hensinger et al, 2002
Quantum Chaos – Les Houches – September 2004
Statistical properties of energy levels
Outline
Level dynamics (qualitative)
Spectral fluctuations
Spectral fluctuations in a regular system
Spectral fluctuations in a chaotic system ­ Random Matrix Theory
Relevance of Random Matrix Theory for chaotic systems
Quantum Chaos – Les Houches – September 2004
Level dynamics (qualitative)
Plot the energy levels of a classically chaotic system versus a parameter; in our example, energy levels of the hydrogen atom vs. magnetic field.
* Energy levels apparently cross (actually
tiny avoided crossings)
* Eigenstates smoothly change with
parameter
* Apparently easy to label the various states
* Energy levels avoid each other (no real
crossing)
* Eigenstates change rapidly and apparently
erratically with parameter
* No simple label for eigenstates
Obviously very different...
Quantum Chaos – Les Houches – September 2004
Spectral fluctuations in the regular regime
P(s)=exp(­s)
Numerical experiment on the
hydrogen atom in a magnetic
field in the regular regime
(scaled energy e<-0.5)
Few thousands level spacings
Small spacings are most probable
 quasi-degeneracies
 no level repulsion
The same distribution is observed on many systems
Universal behaviour in the regular regime
Quantum Chaos – Les Houches – September 2004
Spectral fluctuations in the chaotic regime
Random Matrix Theory
The level spacing distribution is known in closed form for large N, but the expression is very complicated. For all practical purposes, it is equal to the Wigner distribution.
Comparison with numerically obtained spacing distributions.
Lack of small spacings
(Linear) level repulsion
No degeneracy
Numerical results for the hydrogen
atom in a magnetic field, at scaled
energy e>-0.13
The same distribution is observed on many systems.
Universal behaviour for classically chaotic systems
Quantum Chaos – Les Houches – September 2004
Experimental observation
Rydberg atom in a magnetic field
s
N  s=∫0 P  x dx
H. Held et al, Europhysics Lett.(1998)
Quantum Chaos – Les Houches – September 2004
Practical use of Random Matrix Theory
Spectroscopy of chaotic states of (for example) the hydrogen atom in a magnetic field
Chaotic
The excitation probability I is proportional to:
2
∣〈 Ground state∣Dipole operator ∣Excited state 〉∣
If the excited states are chaotic (i.e. the classical dynamics at this energy is chaotic), the matrix element will have fluctuations described by Random Matrix Theory. It is predicted to be Gaussian distributed.
Prediction for the statistical distribution of excitation probability:
I
N  I =∫0 P  I  dI
Porter-Thomas distribution
Excellent agreement with numerical and experimental observations
Quantum Chaos – Les Houches – September 2004
excited
states
Ground state
Semiclassical Approximations
Outline
WKB approximation
EBK approximation
Semiclassical propagator
Semiclassical Green's function
Periodic Orbit Theory – Gutzwiller Trace Formula
Use of the trace formula
Quantum Chaos – Les Houches – September 2004
Semiclassical approximation
Some useful(?) but complicated formula
WKB (Wentzel, Kramers, Brillouin) approximate solution of the time­
independent Schrödinger equation for a one­dimensional system:
where i s the classical momentum at energy E.
Semi­classical Van Vleck propagator for a time­dependent multi­dimensional system: where i s the classical reduced action along the classical trajectory and n the Morse index.
Quantum Chaos – Les Houches – September 2004
Semiclassical approximation
Some useful(?) but complicated formula
Semiclassical Green's function (multi­dimensional time­independent system):
with is the classical action and n the Maslov index along the classical trajectory. Gutzwiller trace formula:
where the sum in over all primitive periodic orbits and their repetitions at energy E, with:
– Action Sk, period Tk
–
Maslov index nk
–
Stability matrix Mk
is the mean density of states (Weyl rule)
Quantum Chaos – Les Houches – September 2004
A simple application of Periodic Orbit Theory
Photo­ionization cross­section of the hydrogen atom in a magnetic field at positive scaled energy e=0.2
Semiclassics uses circa 1000 periodic orbits
and reproduces most apparently random
spectral fluctuations
Quantum Chaos – Les Houches – September 2004
Energy levels of the Helium atom
Semiclassical Calculations Smart
Naive
Energies in a.u. (=2 Rydberg)
D. Wintgen et al. (1992)
Quantum Chaos – Les Houches – September 2004
References
L. Landau and Lifshitz, Classical Mechanics, Ed. Mir
P. Cvitanovic et al, Chaos classical and quantum, http://www.nbi.dk/ChaosBook (very useful, especially for semiclassical approximations)
H.­J. Stöckmann, Quantum Chaos: An Introduction, Cambridge University Press (1999)
F. Haake, Quantum Signatures of Chaos, 2nd edition, Springer­Verlag (2001)
M. Brack and R.K. Bhaduri, Semiclassical Physics, Addison­Wesley (1997)
M. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer­Verlag (1991)
D. Delande, Quantum Chaos in Atomic Physics, Les Houches Summer School 1999, session LXXII Coherent Atomic Matter Waves, Springer (2001)
W. Zurek, Rev. Mod. Phys. 75, 715 (2003) (on decoherence)
Quantum Chaos – Les Houches – September 2004
Kicked rotor
Eigenstates of the evolution operator
Linear
scale
2
2
∣ p∣
∣ p∣
Logarithmic
scale
Momentum p
Momentum p
A typical (Floquet) eigenstate of the chaotic quantum kicked rotor
showing exponential localization in momentum space
2

∣ p− p 0∣
∣ p∣ ~exp −
l

Quantum Chaos – Les Houches – September 2004
Experimental observation of dynamical
localization with cold atoms
Temporal modulation of the
standing wave
∞
I t =I 0 ∑n=−∞ t−nT 
The atoms are initially prepared in a thermal distribution of momentum (Gaussian) before the modulated optical potential is applied. Then, let the system evolve with the modulated potential during few tens of periods.
The standing wave is modulated at a frequency of the order of 10­100 kHz (Cs atoms) or 100­200 kHz (Na atoms)  effective ℏ of the order of 0.2 to 1.
Switch off abruptly the optical potential and analyze the momentum distribution by a time of flight or a velocity selective Raman technique.
Quantum Chaos – Les Houches – September 2004
Experimental observation of dynamical
localization with cold atoms
Initial momentum distribution (Gaussian)
Final momentum distribution
(exponentially localized)
Time
(number of kicks)
Momentum (in units of
2 recoil momenta)
M. Raizen et al (1995)
Quantum Chaos – Les Houches – September 2004
Experimental observation of dynamical
localization with cold atoms
Energy
Classical chaotic diffusion
Localization time of the order of 10 kicks
N.B.: Nothing special happens
at the Ehrenfest time
M. Raizen et al (1995)
(Number of kicks)
Quantum Chaos – Les Houches – September 2004
Destruction of dynamical localization by
breaking periodicity
●
●
First method: change kick strength at each kick
–
The evolution is completely Hamiltonian, but the evolution operator over one kick varies.
–
Scrambles the phases and kills destructive interference effects  restauration of chaotic diffusion.
Experimental observation using noise on the kick strength
Classical
diffusion
Increasing
noise level
M. Raizen et al (1998)
Number of kicks
Quantum Chaos – Les Houches – September 2004
Destruction of dynamical localization by
breaking periodicity (continued)
●
Second method: change time interval between kicks (no randomness at all):
Standing
wave
amplitude
Time
Quasi-periodic
Hamiltonian
If r is rational: the system is periodic  Dynamical localization at long time.
If r is irrational: the system is quasiperiodic  no dynamical localization(?).
Rational r=p/q; for large q, the period is very long and the localization time will be extremely long, i.e. not observable.
The classical diffusion constant is not sensitive to the rational or irrational character of r.
Quantum Chaos – Les Houches – September 2004
Experimental observation of quasi-periodic kicks on cold atoms
2
∣ p∣
Logarithmic
scale
Initial distribution (Gaussian)
Quasi-periodic kicks (r=1.083)
Dynamical localization
Periodic kicks r=1
When there is dynamical localization, the number of atoms with zero velocity increase  peak.
The peaks appear at the simple rational r.
The longer the experiment, the more peaks are visible.
J. Ringot et al (1999)
Quantum Chaos – Les Houches – September 2004
How fast does a quantum chaotic system
recognize a rational number?
Take r=1+e. How long will it take for the system to recognize that r is not equal to 1? Naive answer: 1/e kicks (Fourier limit). Wrong! Much less...
Experimental observation: “rational” peaks narrow ~1/(Number of kicks)2.
This is due to the long­range phase coherence induced in the wavefunction by the chaotic dynamics. Energy
r=1.002
r=0.998
r=1.001
r=0.999
Fourier limit
r=1.000
Experimental sub-Fourier resonance line
Quantum Chaos – Les Houches – September 2004
Evolution of the Wigner distribution in the
presence of decoherence
Hamiltonian evolution
Damping
toward p=0
Decoherence
where g and D are constants specific of the model, which can be explicitely calculated knowing the properties of the reservoir and its coupling with the system.
For small coupling, the damping is negligible and the decoherence term
prevents the Wigner distribution from becoming “too narrow”.
Quantum Chaos – Les Houches – September 2004
Decoherence of a quantum chaotic system
2
p
4
2
H=
 A x −B x Cx cos  t 
2m
m=1 A=0.5 B=10 C=10 w=6.07
Classical phase space density
after 8 driving periods
Quantum Wigner distribution
Quantum Wigner distribution in the
presence of decoherence
W. Zurek (2003)
Quantum Chaos – Les Houches – September 2004
Experimental observation of decoherence on kicked atoms
Add some spontaneous emission events. One event is enough to kill the phase coherence of the atomic wavefunction (with negligible energy transfer) and destroy dynamical localization.
Energy
One spontaneous emission
every 20 kicks
One spontaneous emission
every 130 kicks
Ammann et al (1998)
Quantum Chaos – Les Houches – September 2004
Time scales – Energy scales
Quantum Chaos – Les Houches – September 2004
Some basic questions in quantum chaology
What are the appropriate quantum observables to detect the regular or chaotic classical behaviour of the system?
More precisely, how does the regular or chaotic classical behaviour translate in the energy levels and eigenstates of the (bound) system? For an open system, in the decay rates, in the S­matrix, in the transport properties? (lectures 1 and 2)
What kind of semiclassical approximations can be used? What is the long­time behaviour of a quantum system? (lecture 3)
For a macroscopic system, how is the classical behaviour recovered? (lecture 3)
Quantum Chaos – Les Houches – September 2004