Nonlinear Quantum Dissipative Systems

Nonlinear Quantum Dissipative Systems


Abstract In the preceding chapter we derived linearised solutions to the quantum fluctuations occurring in some nonlinear systems in optical cavities. In these solu- tions the quantum noise has been treated as a small perturbation to the solutions of the corresponding nonlinear classical problem. It is not possible, in general, to find exact solutions to the nonlinear quantum equations which arise in nonlinear optical interactions. It has, however, been possible to find solutions to some specific sys- tems. These solutions provide a test of the region of validity of the linearised solu- tions especially in the region of an instability. Furthermore they allow us to consider the situation where the quantum noise is large and may no longer be treated as a perturbation. In this case, manifestly quantum mechanical states may be produced in a nonlinear dissipative system.
We shall give solutions to the nonlinear quantum equations for two of the prob-
lems considered in Chap. 8, namely, the parametric oscillator and dispersive optical bistability.



9.1 Optical Parametric Oscillator: Complex P Function


We shall first solve for the steady state of the parametric oscillator using the com- plex P function. Then, we show, using the positive P function, that the steady state subharmonic field is in a superposition state. We go on to calculate the tunnelling time between the two states in the superposition.
We consider the degenerate parametric oscillator described in Chap. 8, following the treatment of Drummond et al. [1]. The Hamiltonian is






where

3
H = ? Hi (9.1)
i=0


H0 = k?a†a1 + 2k?a†a2 , (9.2)

1 2


177


? †2 2 †
H1 = ik 2 (a1 a2 ? a1a2) , (9.3)
H2 = ik(?2 a†e?2i?t ? ??a2e2i?t ) , (9.4)
2 2

H3 = a1?† + a2?† + h.c . (9.5)
1 2

where a1 and a2 are the boson operators for two cavity modes of frequency ? and 2?, respectively. ? is the coupling constant for the nonlinear coupling between the modes. The cavity is driven externally by a coherent driving field with frequency 2? and amplitude ?2 . ?1, ?2 are the bath operators describing the cavity damping of the two modes.
We recall from Chap. 8 that there are two stable steady state solutions depending on whether the driving field amplitude is above or below the threshold amplitude 2 = ?1?2/? . In particular, the steady states for the low frequency mode ?1 are

?0 c
1 = 0, ?2 < ?2 ,

. 2
?0

1/2
c c

1 = ±

(?2 ? ?2 )

, ?2 ? ?2 . (9.6)


The master equation for the density operator of the two modes is

? 1 † † †
? t ? = ik [H0 + H1 + H2, ? ]+ ?1(2a1? a1 ? a1a1? ? ? a1a1)
+ ?2(2a2? a† ? a†a2? ? ? a†a2) (9.7)
2 2 2

where the irreversible part of the master equation follows from (6.44) for a zero- temperature bath. ?1, ?2 are the cavity damping rates.
This equation may be converted to a c-number Fokker–Planck equation using the
generalized P representation discussed in Chap. 6. Using the operator-algebra rules described in Chap. 6, we arrive at the Fokker–Planck equation

? . ?
P(?)=



(? ?


?
? ?? ? )+



(? ?


? ? ? ? )

? t ? ?1 1 1
+ ? .

1 2 ? ?1 1 1 1 2
? . ? . ? .


? ?2

?2?2 ? ?2 + 2

+
? ?2

?2?2 ? ?? + 2

1 . ? 2
+

? 2
(? ?2)+

..
(??2)


P(?) (9.8)

2 ? ?2

?? 2


where (?)= [?1, ?1, ?2, ?2].
An attempt to find the steady state solution of this equation by means of a poten-
tial solution fails since the potential conditions (6.73) are not satisfied.
We proceed by adiabatically eliminating the high-frequency mode. This may be accomplished best in the Langevin equations equivalent to (9.8).

9.1 Optical Parametric Oscillator: Complex P Function 179

? .?1. = .??1?2 ? ?1?1 +

??? ? [? (t)].

? t ?1

? ?1?2 ? ?1?1 + ,??2[?? 1(t)]

? .?2. = .?2 ? 2 ?1 ? ?2?2 .

? t ?2

? 2
? ? 2
2 2 2


? ?2?2


(9.9)


where ?1(t), ?? 1(t) are delta correlated stochastic forces with zero mean
(?1(t)) = (?? 1(t)) = (?1(t)?1(t?)) = (?? 1(t)?? 1(t?)) = 0 , (9.10)
(?1(t)?? 1(t)) = ? (t ? t?) . (9.11)
Under the conditions ?2 ± ?1 we can adiabatically eliminate ?2 and ?2 which gives the resultant Langevin equation for ?1 and ?1

?. ?

? 2 .1/2 ? (t)?

. ? .?2 ? ?

2. ?1 ? ?1?1 .

.?2 ? 2 ?1. 1

? .?1. = ?2

2 ?1

+ ? ?2

? . (9.12)

? t ?1

? .??

? 2. ?1 ? ?1?1

?. ? .?

? 2..1/2 ??

(t)?

?2 2 ? 2 ?1

?2 2 ? 2 ?1 1

The Fokker–Planck equation corresponding to these equations is

? . ? .
P(? , ? )= ? ?

? .?

? ?2. ? .

? t 1 1

? ?1

1 1 ? ?2

2 ? 2 1 1

? .
+ ? ?

? . ?
??

2. ? .

? ?1

1 1 ? ?2

2 ? 2 1 1

1 . ? 2 ? .

? 2.

? ? .

? 2...
2 ?

P(?1?1) .

+ 2 ? ?2 ?

?2 ? 2 ?1

?
?? 2 ? 2 1

1 2 1 2

(9.13)

We set ? P(?1, ?1)= 0 and attempt to find a potential solution as given by (6.72). It is found as

? 2?2 .

?2 .
?

F1 = ?2 ??1 ?

?1
2??2 ?

2?2 1
? 2?2

? 2?2 .?1 ? ? . ?

F2 = ?2 ??1 ?

2???

2?2 1
2

2 ? ? 2?1

It follows that the potential conditions
? F1 = ? F2

(9.16)

are satisfied.

? ?1

? ?1

The potential is obtained on integrating (9.14 and 9.15)



P(?)= N exp where

.
2?1?1 +


c = ,2??2 ,



? ?1)



?¯1 = ?1 ?






? 2
.
2?2

.


(9.17)


It is clear that this function diverges for the usual integration domain of the complex plane with ?1 = ??. The observable moments may, however, be obtained by use of
the complex P representation. The calculations are described in Appendix 9.A.
The semi-classical solution for the intensity exhibits a threshold behaviour at
?2 = ?c = ?1?2/? . This is compared in Fig. 9.1 with the mean intensity I = (?1?1)
calculated from the solution (9.17), as shown in the Appendix 9.A. For comparison,
the mean intensity when thermal fluctuations are dominant (Exercise 9.4) is also plotted. The mean intensity with thermal fluctuations displays the rounding of the transition familiar from classical fluctuation theory. The quantum calculation shows a feature never observed in a classical system where the mean intensity actually drops below the semi-classical intensity. This deviation from the semi-classical be- haviour is most significant for small threshold photon numbers. As the parameter ?1?2/? 2 is increased the quantum mean approaches the semi-classical value.













Fig. 9.1 A plot of the mean intensity for the degenerate parametric oscillator versus the scaled driving field ? . (a) The case of zero thermal fluc- tuations. The dashed curve represents the semi-classical intensity, the solid curve is the exact quantum result. In both cases ? 2 = 2?c /? = 5.0. Note that above threshold the exact quantum result is less than the semi-classical prediction. (b) The case of dominant thermal fluctuations. The mean ther- mal photon number is 10.0 and ? 2 = 2?c/? = 100.0


Fig. 9.2 The log variance of the squeezed (solid) and
unsqueezed (dashed) quadra- ture in a degenerate para- metric amplifier versus the scaled driving field with
? 2 = 2?c /? = 5.0





The variance of fluctuations in the quadratures X1 = a1 + a† and X2 = (a1 ? a†)/i
1 1
is given by
?X 2 = [((?1 + ?1)2)? ((?1 + ?1))2]+ 1 , (9.18)
?X 2 = ?[((?1 ? ?1)2)? ((?1 ? ?1))2]+ 1 . (9.19)

The variance in the quadratures is plotted in Fig. 9.2a versus the scaled driving field ? . The variance in the phase quadrature X2 reaches a minimum at threshold. This minimum approaches 1 as the threshold intensity is increased [10]. The value of one
half in the variance of the intracavity field corresponds to zero fluctuations found at the resonance frequency in the external field. The fluctuations in the amplitude quadrature X1 increase dramatically as the threshold is approached. However, unlike the calculation where the pump is treated classically the fluctuations do not diverge. This is because (9.17) is an exact solution to the nonlinear interaction including pump depletion. As the threshold value increases and therefore the number of pump photons required to reach threshold increases, the fluctuations become larger. In
the limit ?1?2/? 2 ? ? the fluctuations diverge, as this corresponds to the classical
pump (infinite energy). The variance in the amplitude quadrature above threshold
continues to increase as the distribution is then double-peaked at the two stable output amplitudes.
The above solution demonstrates the usefulness of the complex P representation. Although the solution obtained for the steady state distribution has no interpreta- tion in terms of a probability distribution, the moments calculated by integrating the distribution on a suitable manifold correspond to the physical moments. We have demonstrated how exact moments of a quantized intracavity field undergoing a nonlinear interaction may be calculated. To calculate the moments of the external field however, we must resort to linearization techniques.