Sin t

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How to treat negative diffusion problems the
anharmonic oscillator in the Q-representation
S. Barnett
IMEDEA
R. Zambrini
http//www.imedea.uib.es
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• Problem

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.

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

• 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) .

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The problem
How can I obtain stochastic equations reproducing
exact moments of a pseudo-Fokker-Planck equation
with negative diffusion?
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PHASE SPACE PICTURE OF QUANTUM OPTICS
(uncertainty principle!)
From operators to classical functions
Liouville equation
(open systems Master equation)
time evolution of a quasi-probability
distribution
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• Linear systems (quadratic Hamiltonian)
• Fokker-Planck equation in W-repr.
  LANGEVIN EQUATIONS
• Non-linear systems
• -- P singular,Dlt0
• -- Q Dlt0 NO Fokker-Planck equation!
• -- W d3 ,negative

Drummond,Gardiner, J.Phys.A,13,2353(80)
with b ? a
-- 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 isnt an exact solution.
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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 Itos formula, we obtain
  , i.e. correct averages!!
• But now dx is complex because z(t) i W(t) (W
  Wigner process)

If Q,M,D are smooth and obey proper boundary
conditions at infinity
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Why the Q? Positive smooth !
Generally , but for the Q the
conditioning is not defined!! Q(aRe,aIm) can
never be a sharp d(aRe-a0Re) or d(aIm-a0Im) in
either quadratures (0ltQlt1)!!
Applications of this method squeezing in linear
problems(1) , with Q gaussian.
((1) Tombesi, Parametric oscillator in squeezed
bath, Phys.Lett.A 132,241(1988))
Our aim check the validity of this method for
non-linear problems
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System ANHARMONIC OSCILLATOR (undamped)
Why? Non-linear exact solvable model!
(Milburn, PRA 33 674 (86 ), cl//qu )
Hamiltonian in the interaction picture
N ââ constant of motion exact solution
of N.L. Heisemberg equations
In phase space (forQ)
Coherent initial state
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Exact solution
n2mt
np/2
np
np/2
p
np
q
n2p
n3p/2
n3p/2
n2p

• Quantum recurrence, interference
• Non-gaussian squeezing
• Dissipative case classical whorl
  structure restored (PRL,56,2237(86))

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Anharmonic Oscillator Tombesi recipe
Langevin equations (Strathonovich)
a)
b)
( apart the initial time a(0)a(0) and
a(0)a(0))

• ! Correct averages equations if lta(t)a2(t)gtxx
  lta (t)a2 (t)gtQ
• 1 Combining a) and b)

? d2a(Q(?,?) a (t)a2 (t))
c)
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• 2 Solution of a) using c)

Taylor expansion

• 3 Calculation of moment(s)
• - stochastic average -------gt over
  realization of noise ?,?
• lt...gt? using moments of ?,?
• - initial condition average -------gt over
  ?(0)
• lt...gta(0) ? da(0)da(0) (... Q(?,?,0))

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Importance of order of operations (lt...gt? ,
lt...gta(0),? ?) lt ?(t) gt? ei3?t ?(0)?n
?(0)2n(1- ei2?t)n / n!

• IF
• 1 SUM ?n
• 2 lt...gta(0)

• IF
• 1 lt...gta(0)
• 2 SUM ?n

?n --gt exp(?(0)2(1- ei2?t)), then ? d2a(0)
exp(?(0)2(1- ei2?t)-a(0)-a02) undefined for
times such that cos(2mt)lt0
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Simulations showed the divergence of trajectories
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• 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