How to treat negative diffusion problems:

the anharmonic oscillator in the Q-representation

http://www.imedea.uib.es

IMEDEAR. Zambrini

S. Barnett

• Aim

To study the possibilities of quantum non-linear EXACT treatements in phase space.

It exists a method to study quantum optics systems by mean of classical stochastic differential equations. BUT this method can be useless for NON-LINEAR systems.

• Idea

Use a technique proposed by Yuen-Tombesi to assign stochastic equations associated with (quantum) Fokker-Planck equation with negative diffusion, for a positive and regular representation (Q) .

• Problem

How can I obtain stochastic

equations reproducing exact

moments of a pseudo-Fokker-

Planck equation with negative

diffusion?

The problem:

)(ˆ 2 Pd

PHASE SPACE PICTURE OF QUANTUM OPTICS (uncertainty principle!)

Liouville equation

,Hi

t

time evolution of a time evolution of a

quasi-probability quasi-probability distributiondistribution

? ),( sWt

ˆ1

)(Q

*),()())ˆ,ˆ(ˆ ˆ( 2 OWdAAOtr sFrom operators to classical functions

(open systems: Master equation)

-- P+ -

- positive not ! (ex. Gardiner!!!)

- trajectories in unphysical regions: moments are meaningful!

But problems in simulation of Langevin equations! Diverging

trajectories!

For many non-linear Q.O. problems there isn’t an exact solution.

? ),( sWt• Linear systems (quadratic Hamiltonian)

Fokker-Planck equation in W-repr. LANGEVIN LANGEVIN EQUATIONSEQUATIONS

• Non-linear systems

-- P singular,D<0

-- Q D<0 NO Fokker-Planck equation!

-- W ,negative

,*

*ˆ 22 Pdd Drummond,Gardiner,

J.Phys.A,13,2353(80)with*

Yuen-Tombesi recipe

Langevin equations with negative diffusion coefficients. A new approach to quantum optics.Opt.Comm.59,155(1986)

•1 Take the Q representation equation with negative diffusion

pseudo-FPE

independently of sign of D

•2 Map pseudo-FPE onto (Ito-)Langevin equation

where and

From applying Ito’s formula, we obtain , i.e. correct

averages!!

But now dx is complex because z(t)= i W(t) (W Wigner process)

PxDx

PxAxt

Q)()(

2

2

dxxMtxQxM )(),()(

)()()()(),(

2

2xM

xxDxM

xxA

t

txM

If Q,M,D are smooth and obey proper boundary conditions at infinity

zxGtxAx d)(d)(d DG 2 ttztz )( ,0)( 2

Why the Q? Positive + smooth !

xtxtxtx

txD )(

2

0

)()(),( limGenerally ,

but for the Q the conditioning is not defined!!

QRe,Imcan never be a sharp (Re-Re) or (Im-Im)

in either quadratures (0<Q<1)!!

((1) Tombesi, “Parametric oscillator in squeezed bath”, Phys.Lett.A 132,241(1988))

Applications of this method:

squeezing in linear problems(1) , with Q gaussian.

Our aim: check the validity of this method for non-linear problems

System: ANHARMONIC OSCILLATOR (undamped)

Why? Non-linear exact solvable model! (Milburn, PRA 33 674 (‘86 ), cl//qu )

2† ˆˆ aaH Hamiltonian in the interaction picture

)exp()0*,,(

.23

20

22

22

Q

ccQiQiQt

N = â†â constant of motion exact solution of N.L. Heisemberg equations )0(ˆ)(ˆ 2/1ˆˆ2 aeta aati

* DD

In phase space

(forQ):

Coherent initial state

2

0

0

220

2

!

*

])exp[()*,,(

ni

n

ne

nS

StQ

Exact solution

• Quantum recurrence, interference

• Non-gaussian squeezing

Dissipative case classical ‘whorl’ structure

restored (PRL,56,2237(86))

/2 /2

/2

t

q

ppiq

22

/2

Anharmonic Oscillator + Tombesi recipe

)()()(1)()(2

)()()(1)()(2

tttttidt

d

tttttidt

d

0)'()( )'(2)'()( )'(2)'()( ttttittttitt

Langevin equations

(Strathonovich)

* ( apart the initial time:(0)=(0) and +(0)=*(0))

! Correct averages equations if <+(t)2(t)>= <* (t)2 (t)>Q

• 1 Combining a) and b) :

a)

b)

......

dt

d

t

ttdttt0

* )'()'('exp)0()0()()(c)

= d2(Q(,*) * (t)2

(t))

t ttdtt

edti0

)''()''(''2'

0')0(2 t dttti eeet 0 ')'(2 )0())0(,;(

• 2 Solution of a) using c)

• 3 Calculation of moment(s):

- stochastic average -------> over realization of noise ,+

<...> using moments of ,+

- initial condition average -------> over (0)

<...> d(0)d+(0) (... Q(,+,0))

Taylor expansion

Importance of order of operations (<...> ,

<...>

< t > ei3t n |n1-ei2t)n / n!IF:

•1 SUM n

•2 <...>

IF:

•1 <...>

•2 SUM n

0

1

)0(,

220

)(

tieti ee

t n --> exp(|1-ei2t)),

then d2() exp(|1-ei2t)|(0)-0|2)

undefined for times such that cos(2t)<0

Simulations showed

the divergence of

trajectories

Summary

• we have presented a way to obtain Langevin equations from pseudo Fokker-Planck equations

• we have seen that WITH SOME CARE we can obtain the correct moments

• for highly non-linear (undamped?) systems stochastic trajectories can diverge