Z. Physik B 37, 339-341 (1980) Zeitschrift

Physik B © by Springer-Verlag 1980

RPA for the Linewidth of the van der Pol Oscillator

Klaus Ziegler and Heinz Horner

Institut fiir Theoretische Physik, Universit~it Heidelberg, Heidelberg, Federal Republic of Germany

Received November 28, 1979

The linewidth of the n component van der Pol Oxcillator is calculated in random phase approximation. For n = 2 this model describes a single mode laser. This approximation is asymptotically correct for weak and strong pumping and also in the large n limit for all pumping. The result is in good agreement with numerical results for n =2 in the whole range including the threshold. The corresponding vertex renormalized approximation on the other hand yields poor agreement and the wrong asymptotic value in the limit of strong pumping. The reason for this failure is discussed.

Classical stochastic systems described by nonlinear equations of motion, Langevin or Fokker-Planck equations, are subject of growing theoretical interest. A prototype of such a system is the noisy van der Pol oscillator used as a model for a single mode laser [1, 2]. Because of its relative simplicity it is particu- larly well suited as a test object for various approxi- mation schemes and different methods have indeed been used. Among those are numerical solutions of the Fokker-Planck equation by Risken [2], mode coupling and continued fraction expansion by Gross- mann [31 and renormalized perturbation theory by Deker and Haake [4] which has recently been extend- ed to fifth order [5]. As function of the Pumping strength one can distinguish three regimes, pumping below the threshold which formally resembles the high temperature limit of a system undergoing a phase transition, pumping near the laser threshold in some analogy to the critical point and pumping above threshold corresponding to the low tempera- ture limit. Both approximation schemes, mode cou- pling [3] and renormalized perturbation theory [4, 5] are asymptotically correct for low pumping strength. This holds also for the unrenormalized perturbation expansion. In the strong pumping (low temperature) limit only the mode coupling scheme is asymptoti- cally correct whereas renormalized perturbation theory gives finite but incorrect results. The latter has still to be considered as a substantial improvement over the unrenormalized expansion which diverges in

this limit in each order. In the threshold region which is the most critical one, the renormalized expansion in fifth order yields results which are closer to Risken's [2] numerical data than those of the cou- pling or the continued fraction calculation [3]. In our present study we investigate a partial sum- mation of the unrenormalized perturbation theory of the random phase type. Extending the original two component model to n components this approxima- tion is not only asymptotically correct in the low and high pumping limit but also for any pumping includ- ing the threshold region in the limit n--, Go up to order 1In. Despite its simplicity the agreement with Risken's result [2] is of the same quality or even better than the above schemes [3-5]. The model is described by a Langevin equation

1 2 6~= - L a q ~ - 2 n L U q ) ~ ~ ~o~ + ~ ~ (1)

where ~0~ are the components of a real n-dimensional vector. The fluctuating forces (~ are Gauss-correlated with zero mean

(~(t) ~ (t')) = 2L6~ 6(t - t'). (2)

For n = 2 and b=qol+iqo 2 one recovers the usual model for the single mode laser. Low pumping corre- sponds to a -* + oo, strong pumping to a ~ - oo. In the context of perturbation theory it is convenient to use a path integral formulation [6] which is

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340 K. Ziegler and H. Horner: RPA for the Linewidth of the van der Pol Oscillator

, %--<

a b c

Fig. 1 a-e. Elements of the diagrams: a correlation propagator C=# (t), b response propagator R~¢(t), e interaction vertex + Lu/n

equivalent to the Langevin equation. Correlation or response functions are represented as averages over paths with a statistical weight exp J((0, ~). The action integral for the present case is

[2 u 2

In this formulation response variables (~=(t) have been introduced in addition to the original variable q~=(t). Defining the path integral we assume prepoint discret- ization [7]. Correlation and response function are defined as

Q#(t, t')=6~# C(t ± t ') = 5 ~ {(p; i(}} ~G(t) cpe(t') exp a

R~e(t , t') = 6~# R(t -- t') =5 @{(P; i(}} (p~(t)qS~(t') exp a (4)

where R(t)=0 for t < 0 because of causality. The perturbation theory is constructed by expanding (4) in powers of the coupling u. It is conveniently represented by diagrams constructed from the two propagators C and R, Fig. l a and b, and the vertex, Fig. 1 c. The correlation and response function are related by a fluctuation-dissipation theorem [4]

d C(t) =L {R( - t) - R (t)}. (5)

As shown by Deker and Haake [4] the original equations of motion of the propagators [6] can be written with the help of the fluctuation-dissipation theorem for t > 0 in the form

d C(t)=-faC(t) 1 Sdz~c(t_,) C(z) (6) dt - L o

where

1 = oo! dtR(t)=lc(t=O)" (7)

This means f2 is determined by the static expectation value C( t=0) 2 =<~G>. ~ ( t ) is the selfenergy of the correlation function given by the sum of all proper (one particle irreducible) diagrams with two external response terminals.

%,,,/< = % - - < + > - - - ( ~ - <

÷ > - - - ( 1 1 > - < i > - < +-"

Fig. 2. Selfenergy X c in order 1/n

The leading order in a 1/n expansion is the partial sum of diagrams shown in Fig. 2. It is equivalent to the usual random phase approximation. In addition we assume that the time dependence of the cor- relation function is sufficiently well represented by an exponential [3-5]

C(t)~ C(t =0) exp ( - c~Olt]) (8)

where the parameter c~ is determined selfconsistently such that

C(t O) J dt C(t)- (9) o ~Q

From the fluctuation-dissipation theorem (5) we find with (7) and (8)

R(t) = c~ exp ( - 0~Qt) (10)

for t > 0 and R( t )=0 for t__<0. Realizing that the partial sum of bubbles in the selfenergy (Fig. 2) is just a geometric series and using the above expressions (7-10) one finds after some manipulations the result

L2u 2 C 2 ( t = 0 ) zc( t ) = n{2+uCZ(t=O)} e x p ( - el2 {3 + u C2(t = 0)} Itl). (11)

Inserting this expression into the equation of motion the linewidth factor e becomes with (9)

c~= l_ [ n i l 4 2 3 -1 uC2(t=O)}{l+ucZ(t=O)}] (12)

In the limit of strong pumping ucg(t=O)---~oo and one finds the asymptotically correct result ~ 1 - 1 / n .

For n = 2 [3, 4]

2a c ( t=0 ) -

u

+ ~ e x p ( - a Z / u ) / ~ dxexp(-x2/2). (13)

K. Ziegler and H. Homer: RPA for the Linewidth of the van der Pol Oscillator 341

2 ~

~.8 ~ \ b

0.8 t- 2 a/,/ff

Fig. 3. Linewidth factor as function of pumping strength for n = 2. a) present calculation, unrenormalized; b) present calculation, re- normalized; c) Risken [2]; d) continued fractions [3]; e) mode coupling [3]; f) renormalized perturbation theory in fifth order [5]

With this expression the linewidth factor is plotted as function of the pumping strength in Fig. 3. Also shown is the numerical result of Risken [-2], the mode coupling and continued fraction result [-3] and the renormalized perturbation theory in fifth order [5]. The agreement with the numerical result is quite good over the whole range of the pumping parameter and comparable or even better than the other ap- proaches mentioned. It is of course tempting to try the same partial summation based on the vertex renormalized per- turbation theory of Deker and Haake [-4]. Their renormalized coupling u R is defined as the zero fre- quency limit of the four point vertex function with one response and three correlation terminals. This function can again be expressed by static correlation functions via a fluctuation-dissipation theorem [-4].

1 (~02 @2) c

uR- n + 2 C4(t=0)

n (n+2)uCZ(t=O)-2naC(t=O)-2n - n + 2 u C 4 ( t = 0) (14)

The second expression is due to an identity among the static expectation values derived in [3]. The computation of the selfenergy (11) and the linewidth factor (12) differs from the unrenormalized calcu- lation by the replacement of u by u R and by counter terms which have to be included in the sum of bubbles. The resulting expressions are identical to (11) and (12) if u is replaced by

u R nR = 1-½u R C2(t =0)" (15)

This yields the wrong asymptotic value n

~ 1 (16) (n+2) (n+3)

in the strong pumping limit, and for n =2 the curve shown in Fig. 3. This rather poor result surprises only at first sight. The present model might be interpreted as the dif- fusion of a particle in a spherically symmetric well (for ~<0). Accordingly there are two characteristic frequencies, a slow one equal e~2 due to the motion in the well and a fast one of the order LuC(t=0)>>f2 (in the limit a ~ - oo) due to perpendicular motions. This latter is also the characteristic frequency of the bubble chain in Fig. 2. If one therefore tries to im- prove the unrenormalized theory one should use a coupling constant renormalized at this large fre- quency, not at zero frequency as done in (14). On the other hand such a high frequency renormalized cou- pling is identical to the unrenormalized interaction in the limit of infinite frequency. The model described by the Langevin equation (1) can be visualized as a zero dimensional version of the timedependent n-component Ginzburg-Landau mod- el. The Goldstone modes which show up in this model below its critical temperature are the d-dimen- sional analog to the slow fluctuations in the van der Pol oscillator above the threshold. The method used in this case [8] is indeed a summation of bubble diagrams also, and the renormalization of the vertex, which is necessary because of ultraviolet divergencies, has been performed at a temperature above the criti- cal temperature where no Goldstone modes exist.

References

1. Haken, H.: Handbuch der Physik, Vol. XXV/2c. Berlin, Heidel- berg, New York: Springer 1970

2. Risken, H.: Forstschr. Phys. 16, 161 (1968); Z. Physik 191, 302 (1966)

3. Grossmann, S.: Phys. Rev. A17, 1123 (1978) 4. Deker, U., Haake, F.: Phys. Rev. A12, 1629 (1975) 5. King, H., Deker, U., Haake, F.: Z. Physik B36, 205 (1979) 6. Janssen, H.K., Z. Physik B23, 377 (1976)

Bausch, R., Janssen, H.K., Wagner, H.: Z. Physik B24, 113 (1976) 7. Haken, H.: Z. Physik B24, 321 (1976)

Leschke, H., Schmutz, M.: Z. Physik B27, 85 (1977) Langouche, F., Roekaerts, D., Tirapegui, E.: Physika 95A, 252 (1979)

8. Sch~ifer, L., Horner, H.: Z. Physik B29, 251 (1978) Schgfer, L.: Z. Physik B31,289 (1978)

Klaus Ziegler Heinz Horner Institut fiir Theoretische Physik Universit~it Heidelberg Philosophenweg 19 D-6900 Heidelberg 1 Federal Republic of Germany