PramS.ha- J. Phys., Vol. 27, Nos 1 & 2, July & August 1986, pp. 19-31.
(c) Printed in India.
An age-dependent model of cavity radiation and its detection
S K SRINIVASAN
Department of Mathematics, Indian Institute of Technology, Madras 600036, India Abstract. Cavity radiation is modelled as a population point process with age-dependent birth rate. A method of phases is introduced to analyze special types of age dependencies. The conditional life-time oftbe cavity photon is assumed to be in hypothetical phases, the life-span of the phases being distributed independently and exponentially. The analysis leads to an explicit differential equation for the generating function of the population size. The detection process is analyzed and an explicit expression for the correlation of the counts provided. By an appropriate choice of the parameters, the spectrum corresponding to Gaussian Lorentzian light is recovered.
Keywords. Age-dependent model; cavity radiation; cavity photon; Gaussian-Lorentzian light beam; detection process; population growth.
PACS No. 42.50
1. Introduction
Population point process models of cavity radiation are interesting by virtue of their viability and have acquired special importance (Townes 1984) in view of their intimate connection to light amplification and its detection. Shimoda et al (1957) had proposed a population process as a model for the evolution of cavity photons; a birth corresponds to a stimulated emission while an immigration, to a spontaneous emission. By an appropriate choice of the rates of birth, death and immigration (that are constants) Shimoda et al established many of the interesting properties of the amplifiers. It was realized even at that stage of development that the process of detection of radiation can be included in the model by introducing emigration at a constant rate. The model, in fact, provided a stimulus for the development of the rate equations and a semi-classical approach to the theory of masers and lasers in general (Haken 1970). In addition, attempts were made to arrive at a fully quantum-mechanical theory of amplifiers in general (Scully and Lamb 1969; Scully et al 1974). However in all these treatments, it is generally assumed that all interactions except that between the field and the detector are switched off during the process of detection. In the Shimoda-Takahasi-Townes model it is indeed possible to choose the rate constants in such a way as to include the field- detector interaction; nevertheless such a procedure will provide very little information relating to the detection process. The situation was improved considerably by Shepherd (1981) who provided an integrated approach by taking into account the varying nature of the field as well as the continual nature of the cavity-field and field-detector
The author felicitates Prof. D S Kothari on his eightieth birthday and dedicates this paper to him on this occasion,
19
interaction. The Shepherd model is close to the density matrix approach in as much it describes all the features relating to the transitions corresponding to the diagonal elements of the density matrix; moreover it has an advantage over the Scully-Lamb approach since the interaction between the cavity and field is not switched off during the process of detection. When the rate constants are chosen appropriately, the model does describe the photodetection process corresponding to the Gaussian-Lorentzian light beam.
The description of the evolution of cavity photons and the field-detector interaction has several interesting features. Apparently it is Markovian as it should be purely from the point of view of the statistical analysis of the problem. The modelling of the field- detector interaction as an emigration process with a constant rate per individual is appropriate in that it characterizes an ideal detector. However the existence of spectral profiles other than Lorentzian does point the need for non-Markov models of cavity evolution and detection. In the past non-Markov models were based on some kind of a Pauli master equation in which the transition rates are time-dependent (Bonifacio et al 1971; Huang et al 1981). The application of these models to processes like superfluores- cence has led to the inevitable conclusion that Markov description is perhaps the most appropriate one. Moreover the memory dependence as brought out by the time- dependent transition rates is rather unphysical. On the other hand age-dependent evolution is more appropriate since age can be interpreted as the extent of the time interval during which the photons are in interaction with the cavity. Moreover from a general point of view, phenomena like anti-bunching of photon statistics (Loudon 1980; Paul 1982) can arise only from non-Markov effects since inhibitory effects characterize memory dependence. Naturally non-Markov point processes have been studied quite extensively within the context of population evolution in the more recent past (Saleh et al 1983; Teich et al 1984; Srinivasan and Vasudevan 1985, 1986). The results of these investigations demonstrate that bunching is considerably reduced when non-Markov modelling is resorted to; however in no case anti-bunching was shown to be a natural consequence. A common feature of the non-Markov model is its ability to produce the spectrum of the detection process corresponding to several distinct widths.
A question which has not been satisfactorily answered is whether a non-Markov evolution can lead to a spectrum corresponding to thermal light and whether more general spectral profiles can be generated. We therefore consider it worthwhile to examine in depth non-Markov population point processes with special reference to cavity evolution and study the consequences of non-Markov interaction in general. It turns out as the results below will show that a spectrum with multimodes as well as a Gaussian-Lorentzian spectrum can be accommodated to at least second order within the framework of age-dependent population process.
Although the analysis of age-dependent population growth with special reference to age structure is rather intractable in its most general form (Harris 1963) the moments of the population size can be estimated for some specific age-dependent rate functions (Srinivasan and Rao 1968; Srinivasan 1974). Even so the analysis is rather cumbersome and we adopt a new method of phases to circumvent the difficulty. The layout of the paper is as follows. In §2 we formulate the phase approach to age dependent population growth and analyse the specific problem of the evolution of cavity photons. By an appropriate reformulation, we derive the equations satisfied by the appropriate generating functions. Section 3 deals with the moments and correlations of the cavity photons and general results relating to the equilibrium distribution of the cavity
Cavity radiation and detection 21 photons. The detection process is analyzed in § 4 and an explicit expression is derived for the second order correlation of the counting process. By an appropriate choice of the parameters, the spectrum corresponding to Gaussian-Lorentzian light is recovered.
The concluding section deals briefly with possible generalizations.
2. Age-dependent model of cavity evolution
Throughout we shall freely make use of the terminology of population growth. The photons constitute the members of the cavity population which is assumed to evolve according to the Kendall (1949, 1950) process of birth, death and immigration. The death (pure absorption) and immigration (spontaneous emission) rates are assumed to be constants equal respectively to/~ and v. The birth rate is assumed to be specified by the age dependent function 2(x) given by
, , I - ( 2 x r 7
2(x)- aexp(-
. )where a and 2 are two constants having the dimension of reciprocal of time and n is any positive integer. The major part of this paper will be devoted to the case n = 1; in the concluding section we shall outline the method of analysis for any arbitrary value of n.
The photo-detection process corresponds to emigration at a constant rate ~/. The form (1) is highly suggestive of a possible phase transformation by the cavity photons. For n = 1 the life-span of any photon conditional upon its survival {from death and detection) can be thought of as the sum of three phases. The first two phases have durations/'1 and T2 that are independent and exponentially distributed with the same parameter 2. In the first phase, the photons passively interact with the cavity producing no additional photons at all. In the second phase the photons are active and each photon produces an additional photon at a constant rate a. The third phase is assumed to have indefinite span during which no births are possible. This can be interpreted to mean that the photon is out of the main interaction region during the third phase.
Although the span of the third phase is indefinite, the photon will eventually be absorbed or detected. If we revert to the description in terms of age, a photon of age x is in
phase I with probability exp( - 2x), phase 2 with probability e x p ( - 2 x ) 2 x ,
phase 3 with the residual probability 1 - exp(-2x)(1 + 2x).
Thus the probability that a photon of age x at time t conditional upon its survival up to t produces a photon of age 0 in the time interval (t, t + A) is
exp(-2x)2xaA+o(A)
so that the rate is given by (1) when n = 1. It is to be specially noted that the third residual phase is necessary to ensure that the birth rate is precisely of the form (1). Thus the age-dependent population process is completely equivalent to the process described above in which each photon has a conditional life-span consisting of three phases. The
study of cavity evolution through phases is straightforward in as much as the population rate parameters are constants in each phase. If there is no immigration current (process) then the process is of the branching type in the sense that the population processes (of emigration as well as of the size of the cavity population in different phases) generated by different members are independent. Similarly the population processes generated by different members immigrating into the system are also independent. Thus it is sufficient to consider the populations generated by a single individual in each of the phases in the absence of immigration. The Poisson nature of the immigration process will in turn enable us to arrive at the appropriate connecting relations between the various probabilities. As we have remarked earlier, there are several probabilities of interest and it may be worthwhile to make a comprehensive analysis. However we will strictly confine ourselves to the main task ahead namely the characterization of the photo-count distribution with special reference to the correlation of photo-counts. Let X(t), Y(t) and Z(t) represent the size of the cavity population at time t in phases 1, 2 and 3 respectively. We introduce the following generating function characterizing the cavity population:
0i(Zl, Z2, Z3,
t)~- E[zX(t)Z~(t)zZ(t) Ix(O)
= 2 - - i , Y ( 0 ) = i - - | , v = 0],i = 1, 2 (2)
0(zl, z2, z3, t) = E[zX(°z ~OzZ(O I X(O) = 0 = Y(O), v ~ 0]. (3) We next obtain a relation connecting 0 and 01. We note that the conditioning on the right-side of (3) implies that the population process is generated by immigrants; noting that the time to the arrival of the first immigrant is exponentially distributed with parameter v and that the immigrant will generate a population independent of further immigrants, we obtain the following equation:
g(zl, z2, z3, t)=exp(-vt)+vfl
exp(-vu)g(zx, z2, z3, t - u )x 01 (zl, z2, z3, t - u) du. (4)
Solving the integral equation, we obtain
g(zl, z2, z3, t) = exp-v f l
[ 1 - 0 1 ( z l , z2, z3, u)]du. (5) The equilibrium distribution of the cavity is obtained by taking the limit as t goes to infinity.We next invoke the branching nature of the process to obtain an equation for gl and 02. The constancy of the population rates imply Markov nature and hence by analyzing the various possibilities in the infinitesimal interval (0, A), we obtain the following backward equation from the Chapman-Kolmogorov relation (Bartlett 1966)
g,(zl, z 2, z3, t) = {1 - ( 2 + p + q ) A } g x ( z l , z2, z3, t - A )
+2Ag2(zl, z2, z3, t - A ) + (/~ + r/)A + 0(A). (6) The above relation is obtained by arguing that the photon that is in phase I initially can,
Cavity radiation and detection 23 in the interval (0, A), (i) move to phase 2 with probability 2A + o(A); (ii) be absorbed or detected with probability (/~ + r/)A + o(A); and can continue to be in phase 1 with residual probability 1 - (2 + # + q)A + o(A). Proceeding to the limit as A --} 0, we obtain
001 (Zl, z2, z3, t)
~t = - (2 + / l + q)gl (zl, z2, z3, t)
+ J.g2(zl, 22, z3, t ) + # + t]. (7) In an exactly similar way, we obtain
002(Z1, Z2, Z3, t)
Ot = - - ( 2 + # - { - q W a ) f f 2 ( Z l , Z2, Z3, t) + ag2(zl, z2, z3, t)gt (Zl, z2, z3, t)
+293(zl, z2, z3, t)+la+rl, (8)
where 93 is the immigration-free generating function of the population size in different phases conditional upon a photon in phase 3 being present at t = 0. The product term ag29x arises because of the assumption that a photon in phase 2 can create a photon of age 0 (and hence in phase 1) at a constant rate a. The branching nature of the process leads to the product g2gl.
In view of our assumption that the photons in phase 3 have zero birth rates, the functions ga is independent of zl and z 2 and can be explicitly evaluated:
g3(Zl, z2, z3, t) = 1 + (z 3 - l)exp[ - (/l + r/)t]. (9) It is rather difficult to solve 91 and 92 explicitly from (7) and (8). However the moments of X (t) and Y(t) can be calculated.
3. Moments of the cavity population We introduce the moments
A~(t), B~J(t), A'(t), B'i(t)
og _
A~,(t) = c ~ z l z , = z , = z~ = 1
( i , j = l , 2 , 3 ; k = 1,2) by
~0
Ai(t) = ~zi z, = z, = z3 = 1' (10)
A ' ( t ) = v f l A~(u)du,
B ° ( t ) = v f l B~J(u)du+ A'(t)AJ(t) •
(12)
(13)
02gk zl ' BiJ(t)= 02g . (11)
We first connect the moments of X(t) and Y(t) with the corresponding conditional moments by differentiating both sides of (5) appropriately:
We next differentiate (7) and (8) to obtain dA](t)
- - dt + (2 + U + q)A~ (t) -- 2A~(t), i = 1, 2, 3, (14) dA~(t)
d - - - ~ + ( 2 + # + q ) A ~ ( t ) - - a A ] ( t ) , i = 1,2, (15) dA](t)
d----~- + (2 + II + q)A ~(t) = aA~(t) + ;texp [ - ~ + q)t]. (16) We note that the initial conditions are given by
A ~ ( 0 ) = A 2 ( 0 ) = 1, A a 2 ( 0 ) = A 2 ( 0 ) = 0 = A ~ ( 0 ) = A 2 3(0). (17) Solving the system of linear differential equations (14) through (16) we obtain
Al(t)
= A~(t) = ½[p(t)+ q(t)],(18)
A~(t)= a-{a) ½ * \ ; t J [p(t) - q(t)], (19)
[ 2 \ I
Al(t )
=~la l\l
[p(t)-q(t)], (20)A~ (t) = ~ {exp[- ~ + q)t] - A~ (t) -
AI (t)},
(21)A~(t)
=A~(t)+ A~(t),
(22)where p(t) and q(t) are given by
p(t) = e x p [ - ( 2 + / * + q - V / ~ ) t ] , (23)
q(t) = e x p [ - ( 2 + U + q + w/r~)t]. (24)
Again differentiating (7) and (8) successively, we obtain for i, j = 1, 2, 3 dBi~(t)
B2 it),
dt ~_(2+p+q)Bi~(t)= ij (25)
dB~(t)
- - + dt
(2
+/~ + q)By (t) = aB~ (t) + a[ A ~ (t)A ~ (t) + A ~ it)A i2 (t)]. (26) The system of equations (25) and (26) can be solved to arrive at an explicit expression for B[ j (t). However we don't need them; we shall obtain the Laplace transform(LT) of B~(t) from which the relevant information regarding the detection process can be extracted. Denoting by B~*(s) the LT of B~J(t), we obtain for i,j = 1, 2, 3B'~*(s) = 2a L ij (s)/D(s), (27)
B~* Is) = (; + ~ + .~) 8~ Is)/;, (28)
Cavity radiation and detection 25 where
D(s) = (s + 2 + .1 +/,)2 _ 2a, (29)
22 L 11, (s) = 2a L z 2, (s) = 2a/D (s/2), (30)
4 L 21, (s) = 4 L 12, (s) = (s + 22 + 2/* + 2.1)/D (s/2), (31) L3'*(s) = LlS*(s ) = ~ 2 {Al*(s+/*+*1)+A~*(s+/*+*1)
1 r l 1 . / ~ 1
- ~ * . t~)-~L22*(s)-L12*(s)}, (32)
{AI*(s+ +.1)+ A~*(s+/* +.1)
L 23. (s) = L 32. (s) = ~ /*
- L 22. (s) - L 12. (s) }, (33)
• ( 2 ) 2 { 1 _ 2 A ~ , ~ + * 1 + s ) _ ( 2 _ ~ ) A ~ , ~ + ~ I + s ) L 33. (s) = ~ s + 21, + 2.1
/:2 + a'~ 22,
+ A~*(s+/*+*1+ 2 + ~/2-aa)]+ l, (!)i[A~*(s+/*+*1+ 2-~/~-a)
- A~* (s +/* ÷ .1 + 2 + ~-a)] }. (34)
The moments and cross correlations of the equilibrium distribution can be obtained from (12) and (13) by proceeding to the limit as t-+ oo. Thus we have
A * ( ~ ) = v(2 +/*+*1)/D(O), A2(oo)= vA/O(O) A 3 (~) = v22/[ ~ + *1)D(0)],
B x* (oc.) = vAa2/2[D(O)] 2 + [A ~ ( ~ ) ] 2 , (35) B22(oo) = v22a/2[D(O)] 2 + [ A 2 ( ~ ) ] 2,
B 21 (or) = B12(~) = vA(:t +/* + q)a/2[D(O)] 2 + A 1 (~)A2(ov).
Expressions for the cross-correlations of the type B ~3 do not take a simple form;
nevertheless they can be evaluated by the combined use of (13) and (27) through (34).
We obtain an interesting result if we neglect the cavity photons in phase 3. Let W(t) represent the total number of cavity photons in phases 1 and 2. Then from (35) we have
E [ W (~:~)] = v(22 + p + *1)/D(0), (36)
2a(32 + 2/, + 2*t + a) "~
E [ W ( w . J { W ( c , : , ) - I } ] = ( E [ W ( ~ ) ] ) 2 1 + ~ v ~ j . + - + ~ / ) ~ j . (37) The quantity wilhin the braces on the right side of (37) is a measure of bunching of the
cavity photon population. If we choose 2 very large bunching can be considerably reduced by choosing a small. If we set a = 2, it reduces to 1 + ;t2/[v(22 +/t + q)]. The persistence o f bunching is to be expected from a model of population growth. The motivation to set a = 2 will become clear in the section to follow; however this makes the model a little bit perplexing; for if we choose v = 22/22 +/~ + ~/, we can recover the result corresponding to Bose-Einstein distribution. An interesting feature is that v must be chosen smaller (than say 2) in order to achieve a bunching factor 2.
4. Detection process
We are generally interested in N (to, t) the number o f photons detected in a destructive scheme of counting over the interval (to, t o + t ) (Shepherd 1981; Jakeman and Shepherd 1984). In the model under consideration, the process is stationary and hence the distributional characteristics of N(to, t) are independent of t o. The physically important case corresponds to the situation when the cavity is maintained in a state of equilibrium say at the time point to. Now there are two ways of dealing with the detection process. The first is to deal directly with the probability distribution of the number of photo-counts over the interval (to, t o + T) (Mandel 1958, 1959); the second method consists in dealing with the point process generated by the epochs of detection (Kelly and Kleiner 1964; Srinivasan and Vasudevan 1967). The probability distribution of N(to, t) or rather N(t) is best studied by introducing a more comprehensive generating function G~ (zl, z2, z3, z, t) where
Gi(ZI,
Z2, Z3, Z, t) =E[zXl (t)z g2(t)zZ(t)z s(t) l x (0)---
2 - i,Y ( 0 ) = i - l , v = 0 ] ( i = 1 , 2 ) , (38) G(zl, z2, z s, z, t) = E[zX(Ozr(OzZ(t)zNtt)l X(O) = Y(O) = O, v vL 0]. (39) N o w the photo count distribution corresponding to equilibrium condition can be obtained from the generating function Gr(z, t) defined by
Ge(z, t) = E[zS~O]cavity population in equilibrium
at the time origin]. (40)
The equilibrium distribution of the cavity population can be obtained by setting z = 1 and taking the limit as t ~ ~ in (39). If we use the notation
Gi(z , t) = Gi(1, 1, 1, z, t),
(41)
p~(zl, z2, z3) = G(zl, z2, z3, 1, ~). (42) We can use the independence o f population evolution process to obtain (see for example Srinivasan and Vasudevan 1985)
GE(Z , t ) = G(z, t)pE[Gl(z, t), G2(z , t), G3(z, t)]. (43) We can proceed as in §2 to obtain differential equations satisfied by the generating
Cavity radiation and detection 27 functions. Since the structure of the differential equations is going to be the same, explicit solution is again a difficult task. We shall not proceed in this direction any further.
The second line of approach is fruitful particularly in view of the results that are already available in §§2 and 3. The point process o f detection can be characterized in terms of the sequence of the coincidence functions of Kelly and Kleiner (1964)and these are in fact known as product densities (Ramakrishnan (1950)) in the literature on point processes and are defined by
ft (t) = lim Pr {N(t + A ) - N(t) = 1 ]cavity in equilibrium
~ o initially}/&
(44)
j~(tt, tz) lim Pr {N(h + A 1 ) - N(q) = 1,
A,,A2~0
N(t2 + Az) - N(t2) = 1 {cavity in equilibrium.
initially}/A, A2
(45)
Higher order product densities are defined in a similar way. It is clear from the construction of our model, that the process is stationary and hence we have
f~ (t) = a constant,
f2(q, t2) = a function of It 2 - t ~ [ = hsty(] t 2 - t 1 I).
(46) (47) Our primary object is to identify the constant on the right side of (46) and obtain the function hsty(. ) explicitly.
We proceed, as in §2, to obtain these functions by appropriate conditioning at the origin and then revert back to equilibrium condition. Accordingly we define
h](t)
= l i mPr{N(t+A)-N(t)
= 1 I X ( O ) = 2 - i , A--*0Y(O) = i-- I, v = 0}/A (i = 1, 2), (48) h~(t) = lim P r { N ( t + A ) - N ( t ) = 1 IX(0) = 0 = Y(0), v # 0}/A. (49)
A--,0
We note that a photon in phase 3 is subject to absorption or detection and hence it can be dealt with directly. To start with we shall assume that photons in all the phases are subject to detection.
Next we note that hi (.) and f~ (.) are connected by
f~ (.) = lim h x (t). (50)
The Poisson nature o f the immigration process implies
f
thl(t ) = v e x p ( - v z ) [ h l ( t - z ) + h [ ( t - z ) ] d z .
0
(51)
The above relation is obtained by arguing that the event corresponding to the detection at the epoch t may be generated by first immigration photon or a subsequent one.
Solving the above relation we obtain
hi(t) = v hl (~)d~. (52)
0
Using the definition of h~ (.) we directly obtain
hi (t) = n[A~ (t) + a~(t) + a~(t)], (53)
where A ~(.) are the conditional first moments introduced in § 3 and explicitly given by (18), (19) and (20). Using those results, we obtain
[ oh ] (54)
A ( ' ) = # - ~ l + ( 2 + / ~ + t / ) 2 _ a 2 '
where the first term on the fight side corresponds to the contribution from the third phase.
In order to obtain an expression for hsty(t ) (t > 0) we note that for the two detections separated by t, the contribution can arise from the same population tree or different trees. Using combinatorial arguments, (Srinivasan 1974) we have
hsty(t) = fl (.)h, (t) + r/~,, B°h{(t), (55)
i,j
where B ° are the equilibrium second moments of the cavity population and the summation over i,j runs from 1 to 3. The function h 2 (.) is obtained on the same lines as before:
h i (t) = r/[A2' (t) + A2(t) + A~(t)]. (56)
On the other hand the function h~ (.) is obtained directly by observing the initial photon in phase 3 must survive absorption and detection right upto t.
hi(t ) = ,7 exp[ - 0t + r/)t]. (57)
The constant B ° is obtained by taking the limit as t ---, oo of B~J(t) defined by (23).
A measure ~ of bunching of the point events of the detection process can be defined by
= hsty(O)/hsty(OO). (58)
Observing
we have
ht(0 ) = 0, h~(oo) = 0, h~(0) = r/, (59)
3
8 = ~ B'~/[f,(.)] z. (60)
i , j = l
Using the relation (13) we can conclude that ~ > 1.
Next we consider the situation when the population in phase 3 is lost due to flight away from the interaction region. We further set 2 = a; then (55) can be cast in the form.
Cavity radiation and detection 29 hsty(t ) = h I (oo)h I (t) + r/(B 11 + BZl)h~ (t) + r/(B 22 + B 12)h2(t), (61) where
h~ ( ~ ) = vr/(22 +/z + ~l)/D(O),
h ] (t) = h2 (t) = t/exp [ - ~ + r/)t]. (62)
Substituting the values for B 'j from (35), we finally arrive at the following simple formula:
vr/2 f 2 % x p [ - ~ + t/)t]
hsty(t) = ~ ~ ,[ v 4 2 - T ~ F 7.1.-- ~ j . (63) Finally if we make the choice
4 2
v = ( 6 4 )
2 2 + / z + r / we arrive at
hsty(t ) = (h(oo))2t/2 { 1 + e x p [ - (# + r/)t] }, (65) a formula corresponding to thermal light with a Lorentzian spectrum. This is an interesting result. The second order properties of the detection process arising from an age-dependent (and hence non-Markov) population point process coincides with those of the detection process arising from a population point process with constant rates.
The choice 2 = a makes the process even in the sense that the rate of production equals the rate of loss to the third phase. However the choice of v is rather mute and does not admit of a direct interpretation. Further research is necessary to settle the question whether the resulting detection process can be made identical with the photodetection process of thermal light.
Finally we retain the choice 2 = a but take into account the photons in phase 3. In this case we have
A](t)
= e x p [ - ~ + tot] [ e x p ( - 220 + 22t - 1], (66) and consequently h~ (t) and hi(t) are given byh~ (t) = r/exp[ - (F + t/)t] [ e x p ( - 220 + 22t + 3], (67) 4
~/exp[ (z + ~/)t]
h2(t) = (5 - 2 e x p t - 2 2 0 + 220. (68)
4 Thus ht (t) takes the form
t / ( 1 3(#+ r/)+ 22 e x p [ - (# + 0 + 22)t]
hi(t) = ~
22 + 2 i , + , 1 + 22 [(# + t/) (1 + 22 0 + 22] e x p [ - (# + r/)t]
(69)
We can substitute (67) through (69) in (55) to arrive at the final formula for hsty (t). It is to be specially noted that the spectrum corresponding t o hsty(t ) contains new features characteristic ofnon-Markov evolution. In particular the spectrum shows the existence of terms corresponding to Poisson profiles. Such spectra were generated earlier in the context of Gaussian light (Srinivasan and Sukavanam 1971).
5. Summary and conclusion
To sum up we have modelled the evolution of the cavity photons as a population process with constant death and immigration rates; the detection process is taken to be an emigration process with a constant rate so that it corresponds to detection by an ideal detector. Stimulated emission is modelled as an age-dependent birth process, the dependency showing a peak rate at a particular age and a decline thereafter. The particular setting in which the idea of phases is invoked is essentially a device in analysis.
The model is able to reproduce the Gaussian-Lorentzian spectrum at least to second order. The model is also a bit versatile; the second-order stationary correlation of the counts contain terms o f the form t exp( - Ft) which may lead to spectral profiles with multimodes. We have not examined this aspect in great detail. May be with differential rates of absorption in the different phases, we may be able to restore the thermal nature.
Further research is perhaps necessary to be able to make precise statements on this aspect.
The extension of the phase approach to general age dependency of the form (1) is straightforward. The conditional life span of any cavity photon can be thought of as the sum of :vans arising from (n + 2) phases. The first n + 1 phases have durations that are independent and exponentially distributed with the common parameter 2, the (n + 2)th phase being defined as the residue. Emissions or births are assumed to be possible only in the (n + l)th phase at a constant rate a. We can proceed exactly as in §2 and obtain the equations satisfied by the generating function of the population sizes in various phases. We have checked and found that most of the analysis goes through. We have also verified that the first moments can be explicitly obtained as weighted negative exponentials under 2 +/~ + r / - ~w where w is the nth root of unity and 0t" = 2"- la. This is rather encouraging; with some more effort we can obtain explicitly the structure of h~ (t) corresponding to the detection process. Further generalizations are possible by ta~ing the phases to be distributed differently; this will o f course pave the way for interpreting more general spectral profiles. However the analysis is by no means simple even if we confine ourselves to small values of n.
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