How to treat negative diffusion problems: the anharmonic oscillator in the Q-representation

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

2. 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) .

3. The problem: How can I obtain stochastic equations reproducing exact moments of a pseudo-Fokker-Planck equation with negative diffusion?

4. 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

5. Linear systems (quadratic Hamiltonian) • Fokker-Planck equation in W-repr. LANGEVIN EQUATIONS • Non-linear systems • -- P singular,D<0 • -- Q D<0NO Fokker-Planck equation! • -- W d3,negative with b * a Drummond,Gardiner, J.Phys.A,13,2353(80) -- 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.

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

7. 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) ord(aIm-a0Im) in either quadratures (0<Q<1)!! 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

8. System: ANHARMONIC OSCILLATOR (undamped) Why? Non-linear exact solvable model! (Milburn, PRA 33 674 (‘86 ), cl//qu ) Hamiltonian in the interaction picture N = â†â constant of motion exact solution of N.L. Heisemberg equations In phase space (forQ): Coherent initial state

9. Exact solution n=2mt n=p n=p/2 n=p/2 p n=p q n=3p/2 n=2p n=3p/2 n=2p • Quantum recurrence, interference • Non-gaussian squeezing • Dissipative case classical ‘whorl’ structure restored (PRL,56,2237(86))

10. 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 <a+(t)a2(t)>xx= <a* (t)a2 (t)>Q • 1 Combining a) and b) : =d2a(Q(,*) a* (t)a2 (t)) c)

11. 2 Solution of a) using c) Taylor expansion • 3 Calculation of moment(s): • - stochastic average -------> over realization of noise ,+ • <...> using moments of ,+ • - initial condition average -------> over (0) • <...>a(0) =da(0)da+(0) (... Q(,+,0))

12. Importanceof order of operations (<...>,<...>a(0), ?) < (t) >=ei3t (0)n|(0)|2n(1-ei2t)n / n! • IF: • 1 <...>a(0) • 2 SUM n • IF: • 1 SUM n • 2 <...>a(0) n--> exp(|(0)|2(1-ei2t)), then d2a(0) exp(|(0)|2(1-ei2t)-|a(0)-a0|2) undefined for times such that cos(2mt)<0

13. Simulations showed the divergence of trajectories

14. 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