Chaos in de Broglie - Bohm quantum mechanics

Christos Efthymiopoulos

Research Center for Astronomy and Applied Mathematics

Academy of Athens

Collaboration with: G. Contopoulos, N. Delis, and C Kalapotharakos

J.Phys.A39, 1819 (2006)J.Phys.A40, 12945 (2007)Cel. Mech. Dyn. Astron.102, 219 (2008)Nonlin. Ph. Compl. Sys.11, 107 (2008)Phys. Rev. E79, 036203 (2009)Annals of Physics 327, 438 (2012)Int. J. Bifurcations and Chaos (2012) J. Phys. A.: Math. Theor. (2012).

Bohm’s (de Broglie) theory

Demonstration that a non-local realistic theory is possible

Non-relativistic QM

Introduce polar co-ordinates

Quantum Hamilton-Jacobi equationContinuity equation

Quantum potential

Equations of motion(“pilot-wave”)

=| |2

Manifestedly non-local theory

Fully-consistentwith Bell-type correlationsBell (1987): “Speakable and un-speakable about QM”

Relativistic, spin and field theories

No “measurement problem”

Quantum relaxation - Quantum equilibrium

Serves as a “picture” of quantum mechanics

(solve Schrödinger’s equation by the quantum tajectories)

Quantum relaxation - quantum equilibrium

Quantum equilibrium hypothesis: if dBB(x,t=0) | (x,t=0)|2

then, under certain conditions, dBB(x,t) | (x,t)|2 as t

First result in this direction: Bohm - Vigier theory (1954)

Theorem (Bohm and Vigier 1954): one has dBB(x,t) | (x,t)|2 as tprovided that the following `mixing condition’is satisfied:

“a fluid element starting in an elementary element of volume dx’ , in a region where the fluid density is appreciable, has a nonzero probability of reaching any other element of volume dx in this region”

Typicality theorem (Dürr et al. 1992): in a large system, “most” Bohmian initial conditions imply extremely small fluctuationswith respect to the quantum equilibrium state.

Vallentini (1991): the “coarse-grained” particle distribution approaches | |2 on a finer and finer scale (works also for few degrees of freedom).

Valentini (1991)Valentini and Westman

(2005)

Quantum version of Boltzmann’s H-theorem

Timescale to relaxation for an -coarse graining:

Bohm’s method as a “propagator” of quantum dynamics

Consider particle-like “tracers” of the quantum flow

positions of particles at the time t:

=R2

Integrate the Newtonian equations for one time step

positions of particles at the time t+dt:

R(t+dt)= (t+dt)1/2

momenta of particles at the time t+dt:

S(x1,x2,...,xN, t+dt)

A two-fold approach to the importance of chaos in dBB theory:

1) Foundational aspects:dynamical relaxation to quantum equilibrium(Valentini)

2) Practical aspects:accuracy of Lagrangian (hydrodynamical) schemes for solving Schrödinger’s equation(Wyatt, Oriols, Towler)

New predictions?

I. Times of arrival

II. Non-equilibrium quantum physics (early cosmology)

Co-existence of ordered and chaotic trajectories

2D harmonic oscillator (Parmenter and Valentine 1995)

Superposition of three eigenstates

One moving nodal point

Domains of analyticity (devoid of nodal points)

a=b=1

c= 7 / 10 2/2=c

Theoretical limits on nodal point domains

Series expansions for regular orbits

(valid within the domain of analyticity)

where xn, yn are of order n in the superposition amplitudes a,b

Proposition: construction is consistent (no secular terms appear)

Convergence: guaranteed if c=rational (conjectured if c=irrational)

NumericalSeries

representation

Origin of chaos near moving nodal points

Early works: Frisk 1997, Konkel and Makowski 1998, Wu and Sprung 1999, Makowski et al. 2000, Falsaperla and Fonte 2003,

Wisniaski and Pujals 2005, Wisniaski et al. 2006

Goals

1) Study the quantum flow structure in a frame of reference moving with the local vortex speed

2) Unravel the mechanism of generation of chaos

3) Quantitative estimates on Lyapunov exponentsin terms of:

a) the local parametersof a vortex, and b) the number of vortices in a given state and time

Vortex speed:Vx=dx0/dt, Vy=dy0/dt

Introduce local coordinates: u=x(t)-x0(t), v=y(t)-y0(t)

Expand (x,y,t) around (x0,y0) and calculate dBB equations of motion

Determine quantum flow structure in an adiabatic approximation

Nodal point

X point

Nodal point: limit of a spiral (point attractor or point repellor)

where is the averaged distance from the nodal point as a function of the polar angle

Nodal point

X point

)(φR

R( )

<f3> is in general, non-zero, only if the vortex velocity is non-zero(moving vortices generate chaos)

<f3> changes sign quasi-periodically in time (Hopf bifurcation: nodal point turns from attractor to repellor)

Nodal point

X point

R( )

Frequencies: 1=1, 2= 2/2

Example of a Hopf bifurcation

Channelof inward flow(typically verynarrow )

Nodal point:attractor

Nodal point:repellor

Limit cycle

Limit cycle reachesouter separatrix

No inward flow

Nodal point:repellor

Nodal point:repellor

`Avoidance’ rule

As a rule, quantum trajectories in systems with moving vortices avoid approaching very close to nodal points

Most of the time, the nodal point:

i) is a repellor, or

ii) is an attractor protected by a limit cycle, or

iii) is accompanied by a very narrow channel of inward flow

Flow is regular very close to a nodal point

However,

quantum trajectories are chaotically scattered by X-points!

X-point: existence is generic

Nodal point

X point

X-point: local eigenvalue analysis

1 2<0

p=2 (analytic estimate), p=1.5 (numerically)

Unstable manifolds U, UU

Stable manifolds S, SS

Nodal point

X point

Rx

Size of the vortex is inversely proportional to its speed: RX~1/V

Compare systems with one, two, or three vortices

= 00 + a 10 + b 11one nodal point, exists for all times

= 00 + a 20 + b 11two nodal points, exist in particulartime intervals

= 00 + a 30 + b 11one or three nodal points

Correlation of Lyapunov exponents with the size of vortices

Most chaotic scattering events satisfy dmin<O(RX)

Quantum relaxation

First probe case dBB(0)=| (0)|2

for numerical accuracy

900 (30X30) initial conditions

Define `smoothed density’:

noise level

Probe, now, case dBB(0) | (0)|2

Relaxation timescale correlateswith the `time of stabilization’ of the trajectories (positive) Lyapunovcharacteristic number

Quantum Nekhoroshev time

T15( )

Tcross( )

6.0

*0 exp~

ρρ

TTrelax

Exponentially long in 1/

result analogous to the classical Nekhoroshevtheorem (1977)

Relaxation in non-completely chaotic regime?

III. charged particle diffraction

and arrival time measurements

Wavepacket description of electron scattering

ingoing electronwavepacket

D: transverse quantum coherence length

llll: transverse quantum coherence length

outgoing (radial) electron wavepacketin direction 2

at t=2l0/v0

outgoing (radial) electron wavepacketin direction 1

at t=2l0/v0

crystal: source of a radial wavepacket propagating from the center outwards

=crystal number densityd=crystal thickness

Experimental requirements

LASER-inducedfield emission

“Start” event determined with a psec accuracy

Highly coherent electron

nanotip

laser pulse with high time resolution

photon detector

E

Experimental requirements

(axisymmetric diffraction pattern)

crystal: thin metal foil or polycrystalline film

Experimental requirements

Single electron detectors

with psecresponse time

Wavefunction modelling

Potential Eigenfunctions (in Born approximation)

sum over atomic nodes in the crystal

Final model

radial (Gaussian) wavepacket

Gaussian wavepacket

sum of phasors (produces diffraction pattern)

Outgoing wavefunction: fitting model when l>>D

Fraunhofer factor(depending on the reciprocal lattice vector g)

at distancesr<k 0D2a/done has:

Seff~( D)1/2d

at large distances

)2/(sin4

),;(122

0 θ

θ

k

rgS

reff≈

Gaussianwavepacketpropagatingoutwards

Outgoing wavefunction: fitting model when l<<D

Radial extent of the outgoing wave

If l>>D lIf l<<D D

Quantum current structure, separator and quantum vortices

central beam axis

D

separator: locus where | in|=| out|

Quantum trajectories:horizontal up to the point where separator is encounteredRadial outward afterwards

O(D)1

Ingoing term:exponential fall

Outgoing term:power-law fall

22 /~ DRin e−ψ

rout /1~ψ

`hard’ deflectionsdue to the approach to nodal point -X-point complexes

2

Quantum vortices: structure of quantum currents

Size of quantum vortices

Expand the wavefunction around a nodal point

Second order perturbation theory

close toBragg angles

domain of diffusescattering

Size of vortices

Bohmian trajectories(test numerically preservation of the continuity of the quantum flow)

Separator evolution and quantum vortices

Analytic approach to thelocus of initial conditions leading toscattering in a particular angle

The problem of time in quantum mechanics

(reviewed in Muga and Leavens 2000, and Muga et al. 2002,2009)

1) Times are experimentally observed (e.g. TOF spectroscopy for heavy ions)

2) Time, however, is not an operator-valued observable(Pauli’s theorem).Furthermore, measurements in QM are considered to occur oninstants of time

3) Several approaches, still ambiguous (and controversial)

- Kijowski (1972) arrival times distribution (based on Aharonov-Bohm operators). For asymptotically free wave packets, it turns to be analogous to the classical flux approach

- Sum-over-historiesapproach (Yamada and Takagi 1992, based on works by Hartle, Gel-Mann, Griffiths and Omnes)

- POVMs deduced from histories (Anastopoulos and Savvidou 2006)

- Bohmian approach (Leavens): consistent and simple

Arrival time distributions

t

l>>D

P(t)

t

t~max(l,D)/v0 (of order ~1psec for cold field electrons)

t

D>>l