l~ehruary 1974. STATIONARY MOMENTUM SPACE SOLUTION OF THE FOKKER-PLANCK. EQUATION. FOR A SIMPLE MODEL OF A LASER OSCILLATOR.
Volume
10. nutnber
2
STATIONARY
OPTl(‘S
(‘OMMUNICATlONS
l~ehruary
MOMENTUM SPACE SOLUTION OF THE FOKKER-PLANCK FOR A SIMPLE MODEL OF A LASER OSCILLATOR EXHIBITING SPATIAL DISPERSION
Kcccivcd
5 November
1974
EQUATION
1Y73
Starting from a multimode hamiltonian for a system of radiating oscillators coupled with atomic rcservoirr, the secular master equation for the radiation-density operator i, calculated in the interaction picture after elimination of the atomic variables. Using the differential operator rcprcscntation for coherent states this cquatlon is transcribed into a multimode t okkcr Planck equation. The stationary solution m momentum space is gvcn for the threshold region. Fourier transformation to configuration space results in a quabl-free energy formula for a laser oscillator exhibiting spatial dispersion.
1. Introduction It is well known iour. However,
that the semiclassical
in order
to understand
theory
of radiation
the coherence
was very successful
and the fluctuations
in its description
of laser radiation
of laser behav-
a fully quantun-
mechanical theory is needed. A nice formalism to describe these statistical features is provided by the Fokker Planck equation. One can, in principle, arrive at a Fokker-Planck equation in two ways. The first method is to determine the quantum-noise operator equation for the laser oscillator (see for example ref. [ 1 J). From this Langevin-type dures
equation
its stochastically
[Z. 31. In the second
refs. [ 1,4]).
Transcription
method
equivalent
one calculates
of this socalled
master
Fokker--Planck equation
yields a Fokker -Planck equation for the quasi-probability In this paper we use the second method. The multimode presenting operator
the pumped is calculated
medium following
(the A-reservoir’) a treatment
equation
the density-operator
can be derived
equation
into the diagonal
coherent-state
distribution of the fields. radiation field is coupled
and the cavity losses (the C-reservoir).
by Scully and Lamb
[ 1.4.71.
by standard
of the system
The atomic
proce-
(see for example
P-representation
with atomic The motion variables
reservoirs,
[5,6] rc-
of the density
are eliminated
by
tracing over the reservoirs. The resulting radiation density operator equation is then transcribed into the coherentstate formalism using the differential-operator representation [6]. Thus a multimode Fokker---Planck equation in the diffusion approximation is obtained. Its stationary solution is given for the threshold region. Finally this momentum-space solution is transformed into configuration space for an infinite one-dimensional medium. The result can be compared with the formula given by Graham and Haken [8 1, obtained via Langevin equations and a configuration-space solution of the equivalent Fokker-Planck equation. Special care is taken in this paper to keep the formulae as simple as possible in order to provide good insight in considering boundary problems and coupled the theory. The final result will be used in a forthcoming publication, oscillators.
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A- RESERVOIR ATOMIC
SYSTEMS
RADIATION OSCILLATOR SYSTEM
C-RESERVOIR
Fig. 1. Laser model, where the radiation reservoir, representing the cavity losses.
2. The density-operator
oscillators
are coupled
to the A-reservoir,
corresponding
atoms, and the C.
equation
The laser model we wish to use is depicted in fig. 1. The total hamiltonian Jc=;ic,+~KJsR+Jc~) where r = radiation,
to the king
may be written as (1)
R = reservoirs, I = interaction:
t&3,4) The index k refers to the radiation system, 1-1refers to the atoms. It is assumed that each mode k is coupled to its own subreservoir. These subreservoirs are taken to be statistically independent [9]. Furthermore the rotatingwave approximation has been used in eq. (4). We transform from the SchrGdinger picture to the interaction picture in the usual manner with the unitary operator U = exp [-(i/h)
(X, t K,)t]
.
(5)
For the A-reservoir, corresponding to the lasing atoms, we take ckP = fiw,, cd0 being the atomic transition frequency. For the C-reservoir, representing the broad-band cavity losses, we take EkC(= AU,. We first consider the action on the field by one atom. The equation of motion for the total density operator of the system in the interaction picture reads:
(6) The interaction
i$=
ck
gk
*btkak
hamiltonian e:Kp
[-
i(ok-
is now: ek/fi)t]
+ h.c.
(7)
The general solution of (6) may be written as:
p((+~)=~(Ot~(-i,~)“~Td~~jl(lf2~~~~-1di,[~~(tl). t t
[zI(t2)... t
ends. where t is the time at which the interaction with the atom starts and t t r is the time at which the interaction So the change in p due to one atom is 6p = ~(t t 7) - p(t). In the Markov approximation, where the relaxation time 7 of the atoms is small compared to the characteristic times of the radiation field (Q/w), we may simply add the contributions of all atoms. Thus we calculate the socalled course-grained or secular motion of the density operator i, = w6p, where w is the number of atoms ‘injected’ into the laser per second (e.g. w 1 in their upper state).
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The motion of the radiation field alone is obtained by tracing over the reservoirs. i.e. pI = trap. accounting for the anti-commutation of the O,, and taking p = prpK at the start of the interaction when the radiation and rescrvou systems arc still independent. One finds. using (7), up to the fourth ICI-~ in (8) and expanding time exponcntials to fourth order in the detuning Auk = wxwo, that: p, = $2) + $4,
(‘)I
where
Bhh.’ =~(7k/h)2(7k,/fi)*1Rk1* Ig+J, The expansion
in the dctuning
.
is motivated
the saturation
toe fficien t
by the fact that in the vicinity
configuration is not too far from a plane wave with wavenumber dispersion has been neglected.
3. The Fokker-Planck
of threshold
the most probable
wt,/~. [8]. 111the saturation
cocfficlcn
field
t U,,,, the
equation
The master equation (9) can be written operator can be expanded as:
Pr=.
( 14)
i dl.l,P((~h})I{oYk})(!“h.}l.
in the diagonal
coherent-state
representation
15.61, where
the densit)
(15)
dp, = II, d?c~. d2afi = d Ke ak * d In1 ak, and where ax_ is the eigenvalue belonging to the eigenstate 1ak ) of the annihilation operator ux-, After substitution of the representation (15) into cqs. (9)- (I 1) and operation on an arbitrary normally ordered functional %({~i.,a~j) we take the trace over the field variables. Noting that where
and that
161
tr/(Y)(~lI(a+.a)=(crl7(a’.a)la)=F(a*,cyta/acu*) for any arbitrary
functional
( 17)
7. we obtain:
where N(a*, a) is the associated classical functional of the operator % (a’ ,a), and some contributions in the saturation term which are very small in the vicinity of threshold have been neglected. We may integrate the first terms at the right-hand side once and the last term twice. The surface term vanishes and one finds,N being arbitrary, that 116
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a2 [(Reill()P] aBk 30;
+-----
I:ebruarp
\ + C.C. I
1974
(19)
In the single-mode cast eq. (19) describes the well-known rotating wave Van der Pol oscillator 13, 101. It may bc noted in passing that this oscillator is useful in the description of frequency locking (automatic spnchronization) phenomena [ 111. We hope to return to this problem in the near future. The single-mode solution of the FokkerPlanck equation (10) is obtained by a simple integration procedure (see for example ref. [3 I). The stationary part reads : P(a*, 0) =
‘4 -c
exp
r)
-7
la12 +
[ where 7 is a normalization
4. Stationary
solution
l&l4
1)
(20)
factor (see also refs. [12, 131)
of the Fokker-Planck
We look for a solution
equation
of eq. (19) in the form
(21)
P = r7 exp [+({crkII~] Introducing
(22)
into (19) with aP/ilt = 0.one obtains:. 0
=
c
(23)
k Since + is supposed to be a real functional, gk = -iCY/,(lmAk) where a&/&,
+&
we may write:
,
is real. Inserting (24) in (23) one verifies that (23) reduces to:
It is consistent with our quasi-plane-wave approximation (QPWA) to assume now that ck provides only a small correction to the single or independent mode solution (20). It is noted that & = 0 is a special solution of eq. (2.5). Following our assumptions we take each term in (25) equal to zero, make use of (22) for &D/acrk and neglect terms of second order in &. The resulting differential equation is easily integrated: gk = G exp
[-@‘“)(Iak>)l ,
(2h)
where G is an integration constant and @(‘I follows from (22) for & = 0. However, this solution is in contradiction with the requirement ,$k/a,, being real, except for G = 0. Thus we see that @ = @co). Since it is within the approximations to take A, = Re A, as independent of the wave number in the saturation terms, one finds: (27)
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5. Projection
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February
I974
onto one spatial dimension
The solution (27) can bc transformed to configuration space using the Fourier transforms of the radiation creation and annihilation operators 0; and ak. and the associated numbers CY~,crYk(see for example ref. 1131). Since our interest lies in the laser we specify to a one-dimensional medium IX] : cY,=I,
II2
.Idzcu(z) e- I’,- 1
(38)
where (fi/l~~w~~)“* (u(z) represents the positive frequency part of the vector potential. Insert (28) and its conjugate into (27), and take the limit for an unbounded medium. W’e then put in the QPWA: (y(z~ = (u(z) e-ikOz representing relation
(29)
a running wave with wave number we/c. where G(z) is a slowly varying complex amplitude.
and neglecting as before dispersion in the saturation
Using the
term one obtains after some partial integration: (30)
Here A, B and C arc read from eqs. ( 12) -( 14) for k = k,, 1 i.e. resonant
tuning.
6. Discussion We have calculated the stationary solution of the Fokker-Planck equation in the threshold region for a onedimensional unbounded laser medium exhibiting spatial dispersion within the QPWA. The solution (30) fulfils the boundary condition P({~(z)}) + 0 for i;(z) / + 00 also above threshold where A > N, due to the fourth-order term in the field amplitude. The spatial derivative in (30) leads to phase diffusion. restricting the spatial coherence even well above threshold [ 8 1. The quasi-probability function (30) is positive definite [S] and therefore can be interpreted as a classical probability density. It is completely analogous to the probability density for the order parameter in thermal equilibrium in a second-order phase transition, for example in superconductivity 1141, ferromagnetism [ 151 or liquid crystals [ 16. 171. Thus, the functional @ can be regarded as a quasi-free energy for a system in non-thermal equilibrium [181. In a forthcoming publication we will discuss some interesting boundary problems and coupled systems.
consequences
of expression (30). concerning
References M.O. Scully and K.G. Whitney. Progress m optics, Vol. X, (ed. E. Wolf) (North-Holland. 1972) pp. 89 135. K.L. Stratonovich. Topic? in the theory of random noise, Vol. 1 (Gordon and Breach. 1963). M. Las and W.H. Louisell, Quantum noise IX, IEEE .I. Quantum Electron. 3, 2 (1967) pp. 47--58. M. Sargent III and M.O. Scully, Laser handbook (eds. F.7‘. Arecchi and E.O. Schulz-DuBois) (North-Holland, 1972) Vol. 1 pp. 45 114. [S] R.J. Clauber, Phys. Rev. 131 (1963) pp. 2766-88; Quantum optics and electronics, Les Houches teds. C. Dewitt et al.) (Gordon and Breach. 1965) pp. 63-185. [I ] 121 [3] (41
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[6] J.R. Klauder and E.C.G. Sudarshan, Fundamentals of quantum optics (W.A. Benjamin, 1968). [7] M.O. Scully andP1.E. Lamb, Jr., Phys. Rev. 159 (1967) 208-26; 166 (1968) 246649: 179 (1969) [8] R. Graham and H. Haken, Z. Phys. 237 (1970) 31-46. [9] R. Graham and H. Haken, Z. Phys. 213 (1968) 420-50. [lo] B. van der Pol, Proc. Inst. Radio Engrs. Vol. 22 (1934) 1051-86. [ll] H. Dekker,Phys. Lettcrs42A (1973)410-12. [ 121 H. Risken and H.D. Vollmer. Z. Phys. 201 (1967) 323; 204 (1967) 240. [ 13 ] H. Haken, Encyclopedia of physics. Vol. XXV/2c (ed. S. Fliigge) (Springer-Verlag, 1970). [14] L.D. Landau and V.L. Ginzburg, JETP 20 (1950) 1064. 1151 V. DeCiorgio and M.O. Scully, Phys. Rev. A2,4 (1970) 1170-77. [16] P.G. de Gennes, Solid State Commun. 10 (1972) 753. [17] L. Leger, Phys. Lctters44A (1973) 53556. [ 18 1 R. Graham, Sprin;zer tracts in modern physics, Vol. 66 (ed. G. Hohler) (Springer-Verlag. 1973).
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