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?
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
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 low dimensional systems, the transition from regularity to chaos is smooth (when a parameter is varied). Mixed regular classical dynamics is rather complicated.
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 quasi probability 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.
The Wigner distribution (continued) p2 Assume H = V q. Then 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 lowest order contribution. The break time 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.
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 S matrix, in the transport properties? (lectures 1 and 2) What kind of semiclassical approximations can be used? (lecture 2) What is the long time behaviour of a quantum system? (lecture 3) For a macroscopic system, how is the classical behaviour recovered? (lecture 3)
Constant energy contours we study the Lz=0 subspace
Poincaré surface of section for the hydrogen atom in a magnetic field Regular trajectory Chaotic trajectory
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
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 position momentum correlations. In phase space, a regular trajectory seems to fill a two dimensional surface (a torus), very regularly. In phase space, a chaotic trajectory seems to fill a subspace with non zero 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.
Time scales Energy scales The Ehrenfest time TEhr. It is the time for a minimum wavepacket to spread in the full phase space. TEhr
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 non linear 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 Wigner distribution Classical Liouville density
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!
Time scales Energy scales (summary) N.B.: In mutidimensional systems, the Heisenberg time is much longer than the Ehrenfest time.
Good systems for studying quantum chaos What is required: Classically chaotic dynamics; at least two strongly coupled degrees of freedom (2d time independent or 1d time dependent 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. Three body system (helium atom).
Good systems for studying quantum chaos Electronic dynamics in molecules or clusters. Nuclear motion in excited molecules. Electronic dynamics in (clean) solid state 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 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: ms ms Frequency: khz MHz Energy: nev Very favorable!
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). Atom atom 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 many body dynamics. Be careful.
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.
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
A typical experiment on quantum chaos with cold atoms Expose a cold atomic cloud to a time dependent standing wave Temporal modulation of the standing wave W. Hensinger et al, 2002
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
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...
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
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
Experimental observation Rydberg atom in a magnetic field s N s = 0 P x dx H. Held et al, Europhysics Lett.(1998)
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 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
Semiclassical approximation Some useful(?) but complicated formula WKB (Wentzel, Kramers, Brillouin) approximate solution of the time independent Schrödinger equation for a one dimensional system: where i s the classical momentum at energy E. Semi classical Van Vleck propagator for a time dependent multi dimensional system: where i s the classical reduced action along the classical trajectory and n the Morse index.
Semiclassical approximation Some useful(?) but complicated formula Semiclassical Green's function (multi dimensional time independent 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)
A simple application of Periodic Orbit Theory Photo ionization cross section 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
Energy levels of the Helium atom Semiclassical Calculations Smart Naive Energies in a.u. (=2 Rydberg) D. Wintgen et al. (1992)
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, Springer Verlag (2001) M. Brack and R.K. Bhaduri, Semiclassical Physics, Addison Wesley (1997) M. Gutzwiller, Chaos in Classical and Quantum Mechanics, Springer Verlag (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)
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
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 10 100 khz (Cs atoms) or 100 200 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.
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)
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)
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
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.
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)
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 long range 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
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.
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)
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)
Time scales Energy scales
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 S matrix, in the transport properties? (lectures 1 and 2) What kind of semiclassical approximations can be used? What is the long time behaviour of a quantum system? (lecture 3) For a macroscopic system, how is the classical behaviour recovered? (lecture 3)