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Quantum statistics of a monomode-laser model
Physica 72 (1974) 578-596 0 North-Holland Publishing Co.
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
D. WALGRAEF* Universite’ Libre de Bruxelles, BruxelIes, Belgique
Received 25 September
1973
Synopsis A kinetic equation for the photon density matrix of a monomode laser model with exphclt heat baths is derived in the framework of the Prigogine-RBsibois theory of nonequdibrium statlstical mechanics. Approximate solutions are discussed and are found to agree with other theones. The phase transition behaviour and the creation of order which is due to the nonlinear structure of the macroscopic kinetic equations may be interpreted within the more recent theories of far-from-thermal-equilibrium open systems while further study is needed to fill completely the gap between microscopic and macroscopic description.
1. Znfroduction. Theoretical and experimental studies of the laser oscillator are still growingle3). The physical and the mathematical concepts appearing in its theoretical description make this subject fascinating. It is also a typical problem of nonequilibrium statistical mechanics as we are confronted with a system far from thermal equilibrium. Since the early works in this field, essentially three families of quantum-mechanical theories have been developed4-‘). These theories are almost equivalent but each of them is most appropriate to the study of particular aspects of the laser system, namely: the quantum-mechanical Langevin equations, for the correlation function9); the generalized Fokker-Planck equation for the linewidth problems5); the density-matrix equation, for the study of the photon statistics4). More recently, Lieb and Hepp derived exact results in the field*) which illuminated some aspects of the approximations made in all these theories. The merits of these methods have been discussed in great detaill) and we shall not go further in this subject here. Our aim is to present the derivation of a kinetic equation for the photon density matrix, starting from the Liouville-Von Neumann equation, in the framework of the Prigogine-Rtsiboisg) theory of nonequilibrium statistical mechanics in order to emphasize and to improve some typical aspects of the problem as a problem in far-from-equilibrium statistical mechanics. In * Charge de Recherches au Fonds National Belge de la Recherche Scientlfique. 578
QUANTUM
STATISTICS OF A MONOMODE-LASER
MODEL
579
fact, we hope to clarify some aspects of already developed theories, namely the one-particle character of the Scully-Lamb theory4) as well as the problem of the comparison of the laser transition with a second-order phase transitionlO). To do this we consider a monomode laser system described by a hamiltonian consisting of three parts: the Dicke hamiltonian’) which couples atoms and field; a coupling between the field and the cavity; a dissipative coupling of the atoms while the pumping effects appear in the initial density matrix. We develop on the basis of the iterative solution of the Liouville-von Neumann equation a graphical technique which allows us, in the spirit of the PrigogineResibois method, to classify the various contributions. After a partial resummation of the contributions due to the coupling with the reservoirs in the thermodynamic limit, we obtain a kinetic equation for the photon density matrix of which the diagonal part may be written as &eo (N, t) = j d-cc G (N, N’ 0
I t) ~0 (N', t -
t).
(1.1)
N'
It should be emphasized that the kernel is nonlinear in its parameters. The markovian limit of this equation is of Kolmogorov’s type and the “birth and death” approximation of it corresponds to the Scully-Lamb equation4). The solution is a functional of three parameters related to the coupling between atoms and field, the pumping processes and the dissipative losses (due to the coupling with the thermal reservoirs). In the Glauber-Sudarshanll) representation defined by
(6 a *, t) obeys a generalised Fokker-Planck tionary solution is given by
p
equation whose approximate
sta-
(1.3) This solution has the characteristics already discussed by Scully, Lamb and Arecchi and De Giorgio12), d and ,!?being functionals of the hamiltonian parameters. Namely, owing to the value of #?/A, the system goes from a black-body statistics to a Poisson one. The similarities between the laser transition and a second-order phase transition have been mainly discussed by De Giorgio, Scully, Haken and Graham1*3y10). This concept first appeared when neglecting the fluctuations in the equation for the mean values (i.e., putting (n“) N (n>‘). It is clear, however, that this approximation is not valid in the threshold region. The similarities are recovered for the function defined as Z = P(Z,E*)
= maxP(a,oL*),
(1.4)
580
D. WALGRAEF
in close analogy with the exact treatment of Lieb and Hepp*). In this picture, (Z, Z*) defines the macroscopic state of the system’. It is found that below the threshold value of ,9/O, this state, di = E * = 0, is stable and corresponds in fact to the thermal-equilibrium state. At threshold this state becomes unstable and a set of nonequilibrium states appears. These states are stable and of equal probability, as P ICX, a*) = ~&XI); moreover, each of them results from the phasesymmetry breaking. So one obtains a typical second-order phase-transition behaviour for a system being in a nonequilibrium stationary state far from thermal equilibrium. In the presence of an external driving field different stationary states appear but in this case only one of them is stable and may be continuously reached starting from the other ones. In section 2, we discuss the characteristics of the hamiltonian of the model. Section 3 is devoted to the derivation of the kinetic equation while in section 4 approximate solutions are proposed. The analogy with the second-order phase transition is discussed in section 5 while formal or technical problems have been postponed to the appendices. 2. The model. The hamiltonian of the monomode laser model we investigate may be written as the sum of the Dicke hamiltonian*) and an interaction hamiltonian which depicts the coupling with the heat baths, while the pumping effects on the atoms appear in the initial density matrix; so, in the rotating wave approximation
H = ?iJ2:
S: + fioJa+a + 1 i (&+a + ,!?;a+) i=l
t=1
+
C Ao~,a,+a, + C g, (a,+a + axa+) + C S&S,, x
x
(2.1)
i#J
where & hco,a~a, is the hamiltonian of the cavity modes, xX g, (aza + axa+) the interaction of the laser mode with the cavity modes and Ct+i S,A,Jj the interaction between the “spins”. Our aim is to study the time evolution of the density matrix r of the system and, more precisely, of the laser-radiation density matrix ,o(t) = Tr’ r(r),
(2.2)
where the trace is taken on all the other variables of the problem. As r(t) may formally be written as the iterative solution of the Liouville-von Neumann equa* Thus In P (CX,oc*) may be thought of as some kind of thermodynamic
potenttal.
QUANTUM
STATISTICS OF A MONOMODE-LASER
tion, we have, in the (v, N) representation13) in the interaction picture
MODEL
581
A,(N) = (N + 3~1A IN - jv) and
p, (N, t) = C 5 Tr’ (I/%)” j dt” (VI [y(r)]” Iv’>7,~
II’n=O
(N (9,
(2.3)
where the v, N variables are those of the complete system while Y, N belong to the laser mode only. (VI
$(T) Iv’)
=
f”&
(N, r) 7” -
qv’~v_,,(N, t)f,
(2.4)
(2.5) [Q)
= e’~SL~~~Ie
-
‘=q,
The initial density matrix is taken as F(O) = (1/2$11(1
+ ssf)ne& x
(2.6)
q may vary from -2 to +2 (r = -2 corresponds to all the atoms in the lower state while q = +2 corresponds to all the atoms in the upper state). In order to obtain laser action we must have at least 11> 0 and the pumping is evidently maximum for 7 = 2. Furthermore, the field density matrices e, and Q are equilibrium ones and will be supposed to be zero-photon density matrices. We are now able to derive a kinetic equation for p, (N, t) taking first the thermodynamic limit on the heat bath, then on the proper laser system itself. 3. Kinetic equations. Owing to the diagonal character of the initial density matrix and the structure of the hamiltonian the only nonvanishing part of Q will be its diagonal part e. (N, t). So we have to study
(3.1) We develop, in appendix A, a graph technique in order to represent the various contributions of this equation. The first orders are given as examples in fig. 1.
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
583
We may now perform a partial resummation of the perturbation series by defining the operator associated with a heavy-photon line in the weak-coupling limit as shown in fig. 3. F(N,r&).
_ _ _ -= 1
_ -
l f
Fig. 3. Renormalization
-2:
1
_ -+‘-2 ,- *
K. -
-,-
-
L,__+....
of the photon line by the coupling with the cavity.
Owing to the thermodynamical limit and according to the time dependence of these graphs, F(t) obeys the following kinetic equation &I; (IV, t) = kdr (2/A*) 1 gx”cos (~0%- 0) (t - 7) x x [N+~-(N+~)3(N+3)3y*]I:(N,t).
(3.4)
As the heat bath has a very short memory, this equation may be used in its markovian approximation &P(N, t) = -y [N + 3 - (iv + +>*(N + 4)+ $1 F(N, t) and F (IV, t) may formally be written as exp [-A(N)
(3.5)
t] with
A(N) = y [N + 3 - (N + 3)” (N + 4)+ ?*I.
to) D-w-..
IGI_---a (b)
Fig, 4. Graphical representation
of the terms including spin-photon interaction.
and explicit photon-cavity
The remaining graphs now have to be constructed on two types: the first one may be represented as in fig. 4a; the partial trace on the heat-bath variables may also be taken and the contribution of this graph is proportional to 2 c gx”cos (0, - m) (f1 - t2) x x [N’G (N - 3) IV’ -(IV+
l)+G(N+
*)(N+
l)*$],
and will be represented as 4b; the second ones are represented in fig. 5.
(3.6)
584
D. WALGRAEF
(a)
Fig. 5. Graphical
(4
(b)
representation
of the terms mcluding only spin-photon renormalized picture.
mteractlons
in the
So we are left with graphs containing only spin and renormalized laser-mode lines. As the number of spins tends to infinity the only contributing graphs are those where all the spins are different (this limit is similar to the Weiss hmit taken in the pure-spin problems). So the partial trace on the spin operators may be taken and the operators q * “c?~‘,O are to be replaced by 1 and dMM1, Oq*‘I by Q* where Q* = f (1 + 7). The perturbation series for Q,,(N, t) is now
+ jd& ydt, rdt, 0
0
Jhf4 Go (tI - tJ Go (t3 - t,+)Q,,(IV, 0) + ... , 0
0
(3.7) where G, is represented in fig. 6.
__&gg_,_ Gp(t):
=
_..$&E+ii_ +__&?jgg+r_
Fig. 6. Graphical representation
of the kernel of the kinetic equation.
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
585
So Q,,(N, t) obeys the following kinetic equation
where the Laplace transform of the kernel, derived in appendix B is formally defined as a continued fraction
yo(4 =
co+ A,(z) (1 -
b - wp(41-‘Ao(z>>-‘~
where A,(z) = -2;1*
(z + Y) (z + y)” + (J-2- co)*
x {(N - Me-
- (N2 - tP*)*e+T-*
- [(N + l)* - +$]Q-$
+ (N + 1 +
!A
e+>
(3.8)
and c,=
-r(N-3E”)+Yr(N+1)*--~*1~~*,
while (z - 0)-l is a formal notation for I$, O”/z”+l. It should be noted that the main effect of the introduction of the spin-spin coupling would only change during time the relative strength of Q+ versus Q-. This coupling, if too intense, is of course able to destroy the pumping effects. In the future we shall consider it to be sufficiently small as to always have Q+ $ Q-. We are now able to derive approximate solutions for this equation. This will be done in the next section. 4. Approximate solutions. The Laplace transform of eq. (3.8) may be written as
The complexity as well as the structure of the kernel allow no inversion or diagonalization of the evolution operator z - ye(z), or any explicit calculation of its singularities. We may only say that, for $2 - o sufficiently weak (i.e., near resonance), this equation admits a markovian limit and that the solution tends to a stationary one independent of the initial state15). We are thus led to try obtaining approximate solutions. We shall only be concerned with analytical approximations in order to facilitate our discussion bearing in mind that in this case only qualitative agreement with the experimental results may be obtained. When quantitative comparison is needed, numerical calculations have to be performed.
D. WALGRAEF
586
For weak coupling and weak nonlinearity approximated by the Scully-Lamb one
-
a(N+l) 1 + q(N+
1)
this equation may reasonably be
+ -- CXN 7 -2 1 + qN
1
@o(N, t).
(4.2)
This equation has been extensively studied16*17). Two limiting cases are of particular interest. 1) For intense dissipation Q,(N,t)=
[--yN+y(N+
~)$-cx(N+
l)+olNy-‘]eo(N,t).
(4.3)
In this case the evolution is governed by a series of exponentials, the leading one being exp [-(7 - a) t] and the stationary distribution is a black-body one: eo (N, st) = (1 - N/Y)WY.
(4.4)
2) For weak dissipation and appreciable nonlinearity we have %eo(N,t)=
l)~2-B+B~-‘leoW~~).
]--yN+y(N+
(4.5)
In this case, the evolution of the characteristic function ‘j& eiNseo(N, t) is exp (-B/v) (1 - e
-“I
~1
_
(1
_
eis>
e-~‘]
(ref. 15), the leading term being eeyr and the stationary distribution is e. (N 3 st) = e-@” (/?/r>“/N .1.
(4.6)
In general the evolution of the various components of the density matrix is very difficult to evaluatel’). It is in fact easier to obtain directly the evolution of mean values which are to good approximation solutions of coupled nonlinear differential equations. However, if we defined a characteristic function @(s, z) = EN eiNs$o(N, z) we may write owing to (4.1) P (s, z) = f(s, z 1e(O)) + ;,d.s’ G (s, s’, z) &s’, z), -m where G (s, s’, z) =
c eiNS
NN’
cNN’@)
z-
eNNtZ)
e-
IN’S’
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
587
and fh
4 e 10) = C eiNS z FF
(z)
(4.7)
.
NN
A rough approximation to GNN,which preserves its general behaviour in N and N - N’ leads to the following approximation in the markovian limit Go (s, z) P (S, z) + GI (s, z) aSP (8, z) = +I-(&z] @10)) where G,, (s, 0) behaves like Y e-iS -e%?o is a/(1 + 40) > ( Y
l-
and G1 (s, 0) like 1
(
y
emiS
41 + %I
-
(1 -
q1 e’3
>
,
(4.8)
where 01= 2Py/[yz + (Q - w)‘] N 2L2/y near resonance, q. and q1 being nonlinearity coefficients [q. - (a/~)“/(1 + qly/a)]. Hence,
p(s,z)
= z-‘exp
_
(-(d$-Gz)
’ ds’ f(“, ’ s 0
I @to))
G,(i)
exp (jdo$$$
(4.9
The characteristic function F (s, t) admits obviously a stationary limit exp (j”odo I- Go @,O)/GI (~,@I) independent of the initial condition. Unfortunately, the problem of the explicit time evolution in the threshold region remains open. The stationary function results mainly from the superposition of a black-body distribution function and a fully coherent one: l-6
F (s, st) N , _ 6 eIs exp [B (e” -
01,
(4.10)
with 6 = a/y (1 + qo) and fi = Lyqo/y.Hence e. (N, st) =“io(l
- 6) eeb (/P/n!) ~3~~“.
(4.11)
588
D. WALGRAEF
In the Glauber-Sudarshan
representation’l)
defined by
P (U, U*) = j dl j dl* Tr (efitA(a+-ur)l efitA*@-+)]e),
(4.12)
we obtain (4.13) or in analytical approximation P (24,u*, St) N 24 -I& (Z!E$!_)
exp (-!t$!d),
(4.14)
where A = S/(1 - 6). The structure of the kinetic eq. (4.8) leads to the following equation
for
P (u, u*, t): d,P (u, u*, t) = 3 {a” [uy + d-2 (24,u*, a,, &*)I +
a,*[u*y+ a9+
(u, u*,a,,a”*)]} P (u, u*, t).
(4.15)
When expanding the operator 52 in powers of d, and a,, up to order two we obtain the usual Fokker-Planck equation which may be written in this case
a,p(2.4,u*, t) = + (auu +
aud4*) r(t241)P
(u, u*, t) + 01-&
p (us u*, t),
(4.16)
11 u* with, in the approximation
scheme used here,
It is obvious that for /I/A + 1, P (u, u *, st) has the black-body behaviour exp (- lu12A - ‘) while for /?/A % 1 the behaviour is that of a coherent field lima,o 6-l exp (-([ul -p*)“/S). Furthermore, the maximum of P (u, u*, st) is different from zero when p/d > 1 and this can be thought of as the emergence of a macroscopic state. So this function seems useful in discussing the problems related to the threshold and the stability of the stationary state. 5. Phase-transition behaviour. The complexity of the microscopic equations leads rapidly to untractable expressions for the density matrix or the mean values of the field observables in the threshold region. However, the various approxima-
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
589
tions made and the detailed experimental studies12) show that the macroscopic behaviour is governed only by a few variables whose evolution is markovian and slow compared to the other ones. The most interesting feature is the possible emergence of a macroscopic variable for sufficiently long times. This may be thought of as the apparition of a dissipative structure in agreement with the Prigogine-Glansdorff theoryl*) as extensively discussed by Grahamlg). Let us only show here how these concepts may appear through the theory developed in the preceding sections. If the macroscopic state of the system is defined by the values U and 1* of u and U* that maximize the pseudodistribution function P (u, u*) [P (ii, zTi*)= max P (u, u*)], we see that for p/A < 1, ii = ii* = 0, in agreement with the nearly black-body behaviour of the system. In this case the macroscopic behaviour is always the thermal-equilibrium one. For /l/A > 1, this state becomes unstable as it turns out in /3/A = 1 to a minimum and a set of new maxima appear that may be written as ii = 161eip, with r(liil) = 0. This is typical of the apparition of a dissipative structure: driven by the fluctuation, as we shall see below, the system passes through an instability and reaches a macroscopicstable state (fig. 7) and this state may not be obtained by a simple extrapolation ofthe equilibrium properties. lUl#
0
.
____________________------
(a)
h
Fig. 7. The macroscopic field variable of the system versus the strength of the driving force (arbitrary scale). The plain line represents the stable states, the dashed one the unstable state and the dotted one the mean value of jul (a) is the instability or bifurcation point while the origin corresponds to the thermodynamic equilibrium.
In this picture, the fluctuations of C are such that a, IWI = -
MOI ~(lfi(Ol)
or the macroscopic state of the system maximizes the pseudodistribution P(u,u*,t)
=24-l
(5.1) function
590
D. WALGRAEF
The presence of the fluctuations is a consequence of the restriction of the description of the system to the variables u and U* only. In the presence of an external resonant driving field at the frequency of the laser mode, P (u, u*, st) becomes P (u, u*, st) = 2A- 110(T)
exp (
FluI cos 8 - (lul’ + p) ), d
(5.2)
where F is proportional to the intensity of the field and 13is the angle between the direction of polarization of the external field and of the laser mode. The time evolution of the fluctuations of the state of the system is governed by the following equations 8,
lu(t)l = -
[u(t)]
r (lu(t)l) + d --IF
cos
e(t), (5.3)
liT(
a,o (t) = -A -‘F sin e(t).
We obtain in this case two classes of stationary states. The first ones characterized by 8 = f2nx are stable, the second ones characterized by s = ) (2n + 1) pi are unstable. The corresponding intensity ii may be graphically determined as shown in fig. 8.
2nll e
Fig. 8. Graphical determination of the stationary state of the system with external drlvmg field. The states (a) are stable; the states (b) are unstable for the phase variable, while the states (c) are completely unstable (arbitrary scale).
For weak 1~1(i.e., for B/,4 near 1 in the threshold region) the time-evolution equations may be interpreted as deriving from a gradient dynamics with potential
QUANTUM
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OF A MONOMODE-LASER
MODEL
591
where A = (a/d) (1 - /3/A), B = I”B~/~A~, C = -F/A. Without driving field (C = 0), the transition may be interpreted as a Riemann-Hugoniot catastrophe2’): for A > 0 we have only one possible state for the system ([iii = 0) while for A = 0 we have a bifurcation point and for A < 0 we have a set of possible states in competition, corresponding to conflict points in close analogy with second-order phase transition, the symmetry breaking being the phase-symmetry breaking. With an external driving field the situation is somewhat different. The stationary states are defined by Aliil
+ B(ii13 + Ccos 6 = 0
with
8 = +nx.
These states are stable in t9 for n even and unstable for n odd. For & (A/B)3 + 4 (C/B)2 > 0 we have only one possible state which is completely stable while for (l/27) (A/B)3 + 4 (C/B)’ < 0 we have one completely stable state and two unstable states corresponding to a maximum and a saddle point of the potential. For (l/27) (A/B)3 + -$(C/B)2 = 0 the two unstable states coincide. For each situation it is always possible to reach continuously the stable state so in this case the second-order phase-transition behaviour disappears. 6. Conclusion. In the first part of this paper, we derived a kinetic equation for the reduced field density matrix of a monomode quantum laser with explicit forms for the heat baths in the framework of the general theory of nonequilibrium statistical mechanics developed by the Brussels group. We recover the principal features of already developed theories and in particular the Scully-Lamb theory, as weak nonlinear and weak-coupling approximation. Unfortunately the complexity and the formal character of most of the expression appearing in this microscopic approach allows no explicit solution. However, several, and even rough, approximations can be made in order to obtain more tractable expressions which lead to a fair agreement with the experimental data. On the other hand, the introduction of such approximations allows a macroscopic description of the system in the framework of the modern theories of far-from-thermal-equilibrium systems, Furthermore, the possibility for a microscopic variable to become a macroscopic one for sufficiently long times elicits analogies with the more recent theories of phase transitions where the space variables play the role of the time in this problem21*22). So the laser model described here seems very appropriate to elicit further study of the link between microscopic and macroscopic description of the behaviour of far-from-equilibrium open systems. Acknowledgments. acknowledged.
Fruitful discussions with Dr. P. Mandel are gratefully
592
D. WALGRAEF APPENDIX
A
The matrix elements (p( l(t) I/L’) may be represented by the various elementary vertices appearing in fig. A.1. Their time-independent operator contributions, owing to the fact that (pi ~4 I/L’) = q@‘A,_,, (N) q-” - q-“‘AA,-,s (IV) $’ are :
Cd) <-li,,4
%a+lO,,p -
1)
= 6&O [$‘(N + +p)+q-l- p(N (e>
aa,+ IP +
-
$f.J+
1,~ - 1)
= [(N + 1 + i/4) (iv + +j&J”q-lXq - W - 3/J) (N - 3P +
(a)
(b)
~M3Ti+r1,
(4 Fig. A.l. The elementary vertices.
lp#J,
QUANTUM
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OF A MONOMODE-LASER APPENDIX
MODEL
593
B
We have to evaluate the Laplace transform of the kernel described in fig. 6: y,,(z) = jz dt e-“G, (t). Consider first the so-called connected graphs as defined in fig. B.l. _
1 c
= ..;...+ f D__i_.
Fig. B.l. The “connected”
graphs of the kernel of the kmetic equation for e. (N,r).
For the graph (b), we have to associate R(z)=P(z+i(Q-o))+F(z-i(Q--o)) with each propagator containing the same number of photon lines and spin lines and z-l with the others. The particular structure of these graphs, imposed by the thermodynamical limit (succession of sequences composed by a one-photon, onespin line propagator followed by a two-spins, zero- or one-photon line propagator), lead to the following contribution for a graph with 2n + 2 vertices:
A364rz-‘A0 Ml”, where A,(z) = -A2 (N + 1) (e+ - e-q’) R(z) - AZN(e- - e+TT2) R(z).
(B.1)
The second series may be expressed, in time variables, as
CCsx”expi+i(~, - 4(tl f
t2)l
x
x [N*G*
(N - _5, t, -
t2)
N* - (N + 1)’ G*
(N + 3, t, -
tz)
03.2) where Gf (N, r) = F* (IV, z) - 212”; dt, j’dt, F* (IV, t - td 0 x
COS@
-
m)((tl -
0 t2>((N
+
$)e-
-[(~+~)((N+~)~e_11~-(N+3)e+}G*(N,t,).
-
(IV2 -
+)'e+q‘-2
(B.3)
D. WALGRAEF
594
If we define K* (N, t) = N3G* (N - 3, t) N3, we have in the Laplace-transform formulation K* (N, d = N+F* (N, z) N* -
Z
212 Z2 +
x N+F’
(Q
-
ojy
(N, z) N” (1 - ~+q-~ - ~-7~) K* (N, z).
(B.4)
The total contribution of this graph is thus the sum of the convolutions of the Laplace transform of cX g,” exp [i (0, - o) t] with the K+ (N, z) terms and of 2% g,” exp [-i (w, - o) t] with the K- (N, z) terms. In the short-memory limit for the heat bath, this reduces to -yN+y(N+
1)q2.
(B.5)
So ~,&) for the connected graphs may be written as A,(z) f (l/z”) A”,(z) - yN + y (N + 1) q2.
VW
n=O
If, furthermore we take into account the remaining or “non-connected” graphs, the propagator l/z has to be replaced by the corresponding Crzo (I/z”+~) [y,(z)]” and the kernel may be expressed as
Yo(4 = A,(z) f
n=o
f
m=O
&
’ A”,(z)-
IY,,(W
YN +
Y
(N + 1)v2
(B.7)
>
The same evaluation for y,, leads to the introduction
to the operator
A,(z) = -f?‘R (z) {(N + 1 + 3,~) Q+ - [(N + 1)’ - &413e-q2 + (N - +,/A)Q- - (N2 - $,u’)+Q’~-‘1,
(B-8)
while the dissipation term is defined as C,,=
-~(N-~~)+y[(N+1)2-~~21q~2.
(B.9)
Moreover, as we work on time scales much longer than the heat-bath characteristic time, and as the resummation makes sense for N great, F (N, t) may be approximated by exp (-rt) and R(z) = 2(z + r)[(z
(B. 10)
+ y)’ + (0 - w)“].
It should be noted that, in the absence of a heat bath, A,(z) may be written as 212z - z2 + (Q - W)2
[N(Q- - e+q-‘)
+ (N + l)(e+
-
e-~‘11,
(B.11)
QUANTUM
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MODEL
595
which leads to ~~(0) = 0 typical of nondissipative systems, in complete agreement with the periodic time evolution of the Dicke model*). Note furthermore that in the one-particle problem, the kernel of the density-matrix kinetic equation is (N being the photon variable and A4 the spin variable) -
212z z2 + (Q - c0)2 - NT,-~~&+-),
EN~Af++.cJ+ (N + 1) &f_+,o
g/z,-
(N + 1) +BM_*, 0~-2”].
(B. 12)
It is easy to check that the nontrivial roots of the dispersion equation z - v(z) for the Q;(Z) and Q;(Z) components of the density matrix are z,(N) = &i [(Q - 0)” + 4A2(N + l)]‘,
(B.13)
as they shouldZ3).
REFERENCES 1) Haken, H., in Encyclopaedia of Physics, Vol. XXV/ZC, Springer Verlag (Berlm-HeidelbergNew York, 1970), and in Synergetics, Cooperative Phenomena in Multi-Component Systems, B.G. Teubner (Stuttgart, 1973). 2) Scully, M.O., in Quantum Optics, Intern. School of Physics “Enrico Fermi” Course XLII, R.Glauber, ed., Academic Press (New York, 1969). 3) Graham, R., in Springer Tracts on Modern Physics, Vol. 65 (Berlin-Heidelberg-New York, 1972). 4) Scully, M.O. and Lamb, Jr., W.E., Phys. Rev. 159 (1967) 208; 166 (1968) 246; 179 (1969) 368. 5) Haken, H. and Weidhch, R., in Quantum Optics, Intern. School of Physics “Enrico Fermi” Course XLII, R. Glauber, ed., Academic Press (New York, 1969). 6) Gordon, J.P., Phys. Rev. 161 (1967) 367. 7) Korenman, V., Ann. Physics 39 (1966) 72. 8) Dicke, R.H., Phys. Rev. 93 (1954) 99. Hepp, K. and Lieb, E.H., Ann. Physics 76 (1973) 360. Hepp, K. and Lieb, E.H., Helv. phys. Acta, to be published. 9) Prigogine, I., Non Equilibrium Statistical Mechanics, Wiley (New York, 1962). 10) De Giorgio, V. and Scully, M.O., Phys. Rev. A2 (1970) 1170. 11) Glauber, R. J., Phys. Rev. Letters 10 (1963) 84. Sudarshan, E.C.G., Phys. Rev. Letters 10 (1963) 277. Klauder, J.R., Phys. Rev. Letters 16 (1966) 534. 12) Arecchi, F.T. and De Giorgio, V., Phys. Rev. A3 (1971) 1108. 13) Restbois, P. and De Leerier, M., Phys. Rev. 152 (1966) 305. 14) Walgraef, D. and Borckmans, P., Phys. Rev. 187 (1969) 421, 430. 15) Feller, W., An Introduction to Probability Theory and its Applications, Vol. 1, J. Wiley, Inc. (New York, 1968). 16) Scully, M.O., in Laser Handbook, North-Holland Publ. Comp. (Amsterdam, 1972).
596
D. WALGRAEF
17) Mandel, P., Phystca 63 (1973) 553. 18) Glansdortf, P. and Prigogine, I., Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience (New York, 1971). 19) Graham, R., in Synergettcs, Cooperative Phenomena in Multi-Component Systems, B.G. Teubner (Stuttgart, 1973). 20) Thorn, R., in Stabthte Structurelle et Morphogenese, W. A.Benjamin Inc. (New York, 1972). 21) Wtlson, K.G , Phys. Rev. B4 (1971) 3174. 22) Wilson, K.G. and Fisher, M.E., Phys. Rev. Letters 28 (1972) 4. 23) Davidson, R. and Rae, J., Bull. Acad. Roy. Belgique Cl. Sci., LVII (1971) 3, 325.
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
D. WALGRAEF* Universite’ Libre de Bruxelles, BruxelIes, Belgique
Received 25 September
1973
Synopsis A kinetic equation for the photon density matrix of a monomode laser model with exphclt heat baths is derived in the framework of the Prigogine-RBsibois theory of nonequdibrium statlstical mechanics. Approximate solutions are discussed and are found to agree with other theones. The phase transition behaviour and the creation of order which is due to the nonlinear structure of the macroscopic kinetic equations may be interpreted within the more recent theories of far-from-thermal-equilibrium open systems while further study is needed to fill completely the gap between microscopic and macroscopic description.
1. Znfroduction. Theoretical and experimental studies of the laser oscillator are still growingle3). The physical and the mathematical concepts appearing in its theoretical description make this subject fascinating. It is also a typical problem of nonequilibrium statistical mechanics as we are confronted with a system far from thermal equilibrium. Since the early works in this field, essentially three families of quantum-mechanical theories have been developed4-‘). These theories are almost equivalent but each of them is most appropriate to the study of particular aspects of the laser system, namely: the quantum-mechanical Langevin equations, for the correlation function9); the generalized Fokker-Planck equation for the linewidth problems5); the density-matrix equation, for the study of the photon statistics4). More recently, Lieb and Hepp derived exact results in the field*) which illuminated some aspects of the approximations made in all these theories. The merits of these methods have been discussed in great detaill) and we shall not go further in this subject here. Our aim is to present the derivation of a kinetic equation for the photon density matrix, starting from the Liouville-Von Neumann equation, in the framework of the Prigogine-Rtsiboisg) theory of nonequilibrium statistical mechanics in order to emphasize and to improve some typical aspects of the problem as a problem in far-from-equilibrium statistical mechanics. In * Charge de Recherches au Fonds National Belge de la Recherche Scientlfique. 578
QUANTUM
STATISTICS OF A MONOMODE-LASER
MODEL
579
fact, we hope to clarify some aspects of already developed theories, namely the one-particle character of the Scully-Lamb theory4) as well as the problem of the comparison of the laser transition with a second-order phase transitionlO). To do this we consider a monomode laser system described by a hamiltonian consisting of three parts: the Dicke hamiltonian’) which couples atoms and field; a coupling between the field and the cavity; a dissipative coupling of the atoms while the pumping effects appear in the initial density matrix. We develop on the basis of the iterative solution of the Liouville-von Neumann equation a graphical technique which allows us, in the spirit of the PrigogineResibois method, to classify the various contributions. After a partial resummation of the contributions due to the coupling with the reservoirs in the thermodynamic limit, we obtain a kinetic equation for the photon density matrix of which the diagonal part may be written as &eo (N, t) = j d-cc G (N, N’ 0
I t) ~0 (N', t -
t).
(1.1)
N'
It should be emphasized that the kernel is nonlinear in its parameters. The markovian limit of this equation is of Kolmogorov’s type and the “birth and death” approximation of it corresponds to the Scully-Lamb equation4). The solution is a functional of three parameters related to the coupling between atoms and field, the pumping processes and the dissipative losses (due to the coupling with the thermal reservoirs). In the Glauber-Sudarshanll) representation defined by
(6 a *, t) obeys a generalised Fokker-Planck tionary solution is given by
p
equation whose approximate
sta-
(1.3) This solution has the characteristics already discussed by Scully, Lamb and Arecchi and De Giorgio12), d and ,!?being functionals of the hamiltonian parameters. Namely, owing to the value of #?/A, the system goes from a black-body statistics to a Poisson one. The similarities between the laser transition and a second-order phase transition have been mainly discussed by De Giorgio, Scully, Haken and Graham1*3y10). This concept first appeared when neglecting the fluctuations in the equation for the mean values (i.e., putting (n“) N (n>‘). It is clear, however, that this approximation is not valid in the threshold region. The similarities are recovered for the function defined as Z = P(Z,E*)
= maxP(a,oL*),
(1.4)
580
D. WALGRAEF
in close analogy with the exact treatment of Lieb and Hepp*). In this picture, (Z, Z*) defines the macroscopic state of the system’. It is found that below the threshold value of ,9/O, this state, di = E * = 0, is stable and corresponds in fact to the thermal-equilibrium state. At threshold this state becomes unstable and a set of nonequilibrium states appears. These states are stable and of equal probability, as P ICX, a*) = ~&XI); moreover, each of them results from the phasesymmetry breaking. So one obtains a typical second-order phase-transition behaviour for a system being in a nonequilibrium stationary state far from thermal equilibrium. In the presence of an external driving field different stationary states appear but in this case only one of them is stable and may be continuously reached starting from the other ones. In section 2, we discuss the characteristics of the hamiltonian of the model. Section 3 is devoted to the derivation of the kinetic equation while in section 4 approximate solutions are proposed. The analogy with the second-order phase transition is discussed in section 5 while formal or technical problems have been postponed to the appendices. 2. The model. The hamiltonian of the monomode laser model we investigate may be written as the sum of the Dicke hamiltonian*) and an interaction hamiltonian which depicts the coupling with the heat baths, while the pumping effects on the atoms appear in the initial density matrix; so, in the rotating wave approximation
H = ?iJ2:
S: + fioJa+a + 1 i (&+a + ,!?;a+) i=l
t=1
+
C Ao~,a,+a, + C g, (a,+a + axa+) + C S&S,, x
x
(2.1)
i#J
where & hco,a~a, is the hamiltonian of the cavity modes, xX g, (aza + axa+) the interaction of the laser mode with the cavity modes and Ct+i S,A,Jj the interaction between the “spins”. Our aim is to study the time evolution of the density matrix r of the system and, more precisely, of the laser-radiation density matrix ,o(t) = Tr’ r(r),
(2.2)
where the trace is taken on all the other variables of the problem. As r(t) may formally be written as the iterative solution of the Liouville-von Neumann equa* Thus In P (CX,oc*) may be thought of as some kind of thermodynamic
potenttal.
QUANTUM
STATISTICS OF A MONOMODE-LASER
tion, we have, in the (v, N) representation13) in the interaction picture
MODEL
581
A,(N) = (N + 3~1A IN - jv) and
p, (N, t) = C 5 Tr’ (I/%)” j dt” (VI [y(r)]” Iv’>7,~
II’n=O
(N (9,
(2.3)
where the v, N variables are those of the complete system while Y, N belong to the laser mode only. (VI
$(T) Iv’)
=
f”&
(N, r) 7” -
qv’~v_,,(N, t)f,
(2.4)
(2.5) [Q)
= e’~SL~~~Ie
-
‘=q,
The initial density matrix is taken as F(O) = (1/2$11(1
+ ssf)ne& x
(2.6)
q may vary from -2 to +2 (r = -2 corresponds to all the atoms in the lower state while q = +2 corresponds to all the atoms in the upper state). In order to obtain laser action we must have at least 11> 0 and the pumping is evidently maximum for 7 = 2. Furthermore, the field density matrices e, and Q are equilibrium ones and will be supposed to be zero-photon density matrices. We are now able to derive a kinetic equation for p, (N, t) taking first the thermodynamic limit on the heat bath, then on the proper laser system itself. 3. Kinetic equations. Owing to the diagonal character of the initial density matrix and the structure of the hamiltonian the only nonvanishing part of Q will be its diagonal part e. (N, t). So we have to study
(3.1) We develop, in appendix A, a graph technique in order to represent the various contributions of this equation. The first orders are given as examples in fig. 1.
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
583
We may now perform a partial resummation of the perturbation series by defining the operator associated with a heavy-photon line in the weak-coupling limit as shown in fig. 3. F(N,r&).
_ _ _ -= 1
_ -
l f
Fig. 3. Renormalization
-2:
1
_ -+‘-2 ,- *
K. -
-,-
-
L,__+....
of the photon line by the coupling with the cavity.
Owing to the thermodynamical limit and according to the time dependence of these graphs, F(t) obeys the following kinetic equation &I; (IV, t) = kdr (2/A*) 1 gx”cos (~0%- 0) (t - 7) x x [N+~-(N+~)3(N+3)3y*]I:(N,t).
(3.4)
As the heat bath has a very short memory, this equation may be used in its markovian approximation &P(N, t) = -y [N + 3 - (iv + +>*(N + 4)+ $1 F(N, t) and F (IV, t) may formally be written as exp [-A(N)
(3.5)
t] with
A(N) = y [N + 3 - (N + 3)” (N + 4)+ ?*I.
to) D-w-..
IGI_---a (b)
Fig, 4. Graphical representation
of the terms including spin-photon interaction.
and explicit photon-cavity
The remaining graphs now have to be constructed on two types: the first one may be represented as in fig. 4a; the partial trace on the heat-bath variables may also be taken and the contribution of this graph is proportional to 2 c gx”cos (0, - m) (f1 - t2) x x [N’G (N - 3) IV’ -(IV+
l)+G(N+
*)(N+
l)*$],
and will be represented as 4b; the second ones are represented in fig. 5.
(3.6)
584
D. WALGRAEF
(a)
Fig. 5. Graphical
(4
(b)
representation
of the terms mcluding only spin-photon renormalized picture.
mteractlons
in the
So we are left with graphs containing only spin and renormalized laser-mode lines. As the number of spins tends to infinity the only contributing graphs are those where all the spins are different (this limit is similar to the Weiss hmit taken in the pure-spin problems). So the partial trace on the spin operators may be taken and the operators q * “c?~‘,O are to be replaced by 1 and dMM1, Oq*‘I by Q* where Q* = f (1 + 7). The perturbation series for Q,,(N, t) is now
+ jd& ydt, rdt, 0
0
Jhf4 Go (tI - tJ Go (t3 - t,+)Q,,(IV, 0) + ... , 0
0
(3.7) where G, is represented in fig. 6.
__&gg_,_ Gp(t):
=
_..$&E+ii_ +__&?jgg+r_
Fig. 6. Graphical representation
of the kernel of the kinetic equation.
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
585
So Q,,(N, t) obeys the following kinetic equation
where the Laplace transform of the kernel, derived in appendix B is formally defined as a continued fraction
yo(4 =
co+ A,(z) (1 -
b - wp(41-‘Ao(z>>-‘~
where A,(z) = -2;1*
(z + Y) (z + y)” + (J-2- co)*
x {(N - Me-
- (N2 - tP*)*e+T-*
- [(N + l)* - +$]Q-$
+ (N + 1 +
!A
e+>
(3.8)
and c,=
-r(N-3E”)+Yr(N+1)*--~*1~~*,
while (z - 0)-l is a formal notation for I$, O”/z”+l. It should be noted that the main effect of the introduction of the spin-spin coupling would only change during time the relative strength of Q+ versus Q-. This coupling, if too intense, is of course able to destroy the pumping effects. In the future we shall consider it to be sufficiently small as to always have Q+ $ Q-. We are now able to derive approximate solutions for this equation. This will be done in the next section. 4. Approximate solutions. The Laplace transform of eq. (3.8) may be written as
The complexity as well as the structure of the kernel allow no inversion or diagonalization of the evolution operator z - ye(z), or any explicit calculation of its singularities. We may only say that, for $2 - o sufficiently weak (i.e., near resonance), this equation admits a markovian limit and that the solution tends to a stationary one independent of the initial state15). We are thus led to try obtaining approximate solutions. We shall only be concerned with analytical approximations in order to facilitate our discussion bearing in mind that in this case only qualitative agreement with the experimental results may be obtained. When quantitative comparison is needed, numerical calculations have to be performed.
D. WALGRAEF
586
For weak coupling and weak nonlinearity approximated by the Scully-Lamb one
-
a(N+l) 1 + q(N+
1)
this equation may reasonably be
+ -- CXN 7 -2 1 + qN
1
@o(N, t).
(4.2)
This equation has been extensively studied16*17). Two limiting cases are of particular interest. 1) For intense dissipation Q,(N,t)=
[--yN+y(N+
~)$-cx(N+
l)+olNy-‘]eo(N,t).
(4.3)
In this case the evolution is governed by a series of exponentials, the leading one being exp [-(7 - a) t] and the stationary distribution is a black-body one: eo (N, st) = (1 - N/Y)WY.
(4.4)
2) For weak dissipation and appreciable nonlinearity we have %eo(N,t)=
l)~2-B+B~-‘leoW~~).
]--yN+y(N+
(4.5)
In this case, the evolution of the characteristic function ‘j& eiNseo(N, t) is exp (-B/v) (1 - e
-“I
~1
_
(1
_
eis>
e-~‘]
(ref. 15), the leading term being eeyr and the stationary distribution is e. (N 3 st) = e-@” (/?/r>“/N .1.
(4.6)
In general the evolution of the various components of the density matrix is very difficult to evaluatel’). It is in fact easier to obtain directly the evolution of mean values which are to good approximation solutions of coupled nonlinear differential equations. However, if we defined a characteristic function @(s, z) = EN eiNs$o(N, z) we may write owing to (4.1) P (s, z) = f(s, z 1e(O)) + ;,d.s’ G (s, s’, z) &s’, z), -m where G (s, s’, z) =
c eiNS
NN’
cNN’@)
z-
eNNtZ)
e-
IN’S’
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
587
and fh
4 e 10) = C eiNS z FF
(z)
(4.7)
.
NN
A rough approximation to GNN,which preserves its general behaviour in N and N - N’ leads to the following approximation in the markovian limit Go (s, z) P (S, z) + GI (s, z) aSP (8, z) = +I-(&z] @10)) where G,, (s, 0) behaves like Y e-iS -e%?o is a/(1 + 40) > ( Y
l-
and G1 (s, 0) like 1
(
y
emiS
41 + %I
-
(1 -
q1 e’3
>
,
(4.8)
where 01= 2Py/[yz + (Q - w)‘] N 2L2/y near resonance, q. and q1 being nonlinearity coefficients [q. - (a/~)“/(1 + qly/a)]. Hence,
p(s,z)
= z-‘exp
_
(-(d$-Gz)
’ ds’ f(“, ’ s 0
I @to))
G,(i)
exp (jdo$$$
(4.9
The characteristic function F (s, t) admits obviously a stationary limit exp (j”odo I- Go @,O)/GI (~,@I) independent of the initial condition. Unfortunately, the problem of the explicit time evolution in the threshold region remains open. The stationary function results mainly from the superposition of a black-body distribution function and a fully coherent one: l-6
F (s, st) N , _ 6 eIs exp [B (e” -
01,
(4.10)
with 6 = a/y (1 + qo) and fi = Lyqo/y.Hence e. (N, st) =“io(l
- 6) eeb (/P/n!) ~3~~“.
(4.11)
588
D. WALGRAEF
In the Glauber-Sudarshan
representation’l)
defined by
P (U, U*) = j dl j dl* Tr (efitA(a+-ur)l efitA*@-+)]e),
(4.12)
we obtain (4.13) or in analytical approximation P (24,u*, St) N 24 -I& (Z!E$!_)
exp (-!t$!d),
(4.14)
where A = S/(1 - 6). The structure of the kinetic eq. (4.8) leads to the following equation
for
P (u, u*, t): d,P (u, u*, t) = 3 {a” [uy + d-2 (24,u*, a,, &*)I +
a,*[u*y+ a9+
(u, u*,a,,a”*)]} P (u, u*, t).
(4.15)
When expanding the operator 52 in powers of d, and a,, up to order two we obtain the usual Fokker-Planck equation which may be written in this case
a,p(2.4,u*, t) = + (auu +
aud4*) r(t241)P
(u, u*, t) + 01-&
p (us u*, t),
(4.16)
11 u* with, in the approximation
scheme used here,
It is obvious that for /I/A + 1, P (u, u *, st) has the black-body behaviour exp (- lu12A - ‘) while for /?/A % 1 the behaviour is that of a coherent field lima,o 6-l exp (-([ul -p*)“/S). Furthermore, the maximum of P (u, u*, st) is different from zero when p/d > 1 and this can be thought of as the emergence of a macroscopic state. So this function seems useful in discussing the problems related to the threshold and the stability of the stationary state. 5. Phase-transition behaviour. The complexity of the microscopic equations leads rapidly to untractable expressions for the density matrix or the mean values of the field observables in the threshold region. However, the various approxima-
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
589
tions made and the detailed experimental studies12) show that the macroscopic behaviour is governed only by a few variables whose evolution is markovian and slow compared to the other ones. The most interesting feature is the possible emergence of a macroscopic variable for sufficiently long times. This may be thought of as the apparition of a dissipative structure in agreement with the Prigogine-Glansdorff theoryl*) as extensively discussed by Grahamlg). Let us only show here how these concepts may appear through the theory developed in the preceding sections. If the macroscopic state of the system is defined by the values U and 1* of u and U* that maximize the pseudodistribution function P (u, u*) [P (ii, zTi*)= max P (u, u*)], we see that for p/A < 1, ii = ii* = 0, in agreement with the nearly black-body behaviour of the system. In this case the macroscopic behaviour is always the thermal-equilibrium one. For /l/A > 1, this state becomes unstable as it turns out in /3/A = 1 to a minimum and a set of new maxima appear that may be written as ii = 161eip, with r(liil) = 0. This is typical of the apparition of a dissipative structure: driven by the fluctuation, as we shall see below, the system passes through an instability and reaches a macroscopicstable state (fig. 7) and this state may not be obtained by a simple extrapolation ofthe equilibrium properties. lUl#
0
.
____________________------
(a)
h
Fig. 7. The macroscopic field variable of the system versus the strength of the driving force (arbitrary scale). The plain line represents the stable states, the dashed one the unstable state and the dotted one the mean value of jul (a) is the instability or bifurcation point while the origin corresponds to the thermodynamic equilibrium.
In this picture, the fluctuations of C are such that a, IWI = -
MOI ~(lfi(Ol)
or the macroscopic state of the system maximizes the pseudodistribution P(u,u*,t)
=24-l
(5.1) function
590
D. WALGRAEF
The presence of the fluctuations is a consequence of the restriction of the description of the system to the variables u and U* only. In the presence of an external resonant driving field at the frequency of the laser mode, P (u, u*, st) becomes P (u, u*, st) = 2A- 110(T)
exp (
FluI cos 8 - (lul’ + p) ), d
(5.2)
where F is proportional to the intensity of the field and 13is the angle between the direction of polarization of the external field and of the laser mode. The time evolution of the fluctuations of the state of the system is governed by the following equations 8,
lu(t)l = -
[u(t)]
r (lu(t)l) + d --IF
cos
e(t), (5.3)
liT(
a,o (t) = -A -‘F sin e(t).
We obtain in this case two classes of stationary states. The first ones characterized by 8 = f2nx are stable, the second ones characterized by s = ) (2n + 1) pi are unstable. The corresponding intensity ii may be graphically determined as shown in fig. 8.
2nll e
Fig. 8. Graphical determination of the stationary state of the system with external drlvmg field. The states (a) are stable; the states (b) are unstable for the phase variable, while the states (c) are completely unstable (arbitrary scale).
For weak 1~1(i.e., for B/,4 near 1 in the threshold region) the time-evolution equations may be interpreted as deriving from a gradient dynamics with potential
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
591
where A = (a/d) (1 - /3/A), B = I”B~/~A~, C = -F/A. Without driving field (C = 0), the transition may be interpreted as a Riemann-Hugoniot catastrophe2’): for A > 0 we have only one possible state for the system ([iii = 0) while for A = 0 we have a bifurcation point and for A < 0 we have a set of possible states in competition, corresponding to conflict points in close analogy with second-order phase transition, the symmetry breaking being the phase-symmetry breaking. With an external driving field the situation is somewhat different. The stationary states are defined by Aliil
+ B(ii13 + Ccos 6 = 0
with
8 = +nx.
These states are stable in t9 for n even and unstable for n odd. For & (A/B)3 + 4 (C/B)2 > 0 we have only one possible state which is completely stable while for (l/27) (A/B)3 + 4 (C/B)’ < 0 we have one completely stable state and two unstable states corresponding to a maximum and a saddle point of the potential. For (l/27) (A/B)3 + -$(C/B)2 = 0 the two unstable states coincide. For each situation it is always possible to reach continuously the stable state so in this case the second-order phase-transition behaviour disappears. 6. Conclusion. In the first part of this paper, we derived a kinetic equation for the reduced field density matrix of a monomode quantum laser with explicit forms for the heat baths in the framework of the general theory of nonequilibrium statistical mechanics developed by the Brussels group. We recover the principal features of already developed theories and in particular the Scully-Lamb theory, as weak nonlinear and weak-coupling approximation. Unfortunately the complexity and the formal character of most of the expression appearing in this microscopic approach allows no explicit solution. However, several, and even rough, approximations can be made in order to obtain more tractable expressions which lead to a fair agreement with the experimental data. On the other hand, the introduction of such approximations allows a macroscopic description of the system in the framework of the modern theories of far-from-thermal-equilibrium systems, Furthermore, the possibility for a microscopic variable to become a macroscopic one for sufficiently long times elicits analogies with the more recent theories of phase transitions where the space variables play the role of the time in this problem21*22). So the laser model described here seems very appropriate to elicit further study of the link between microscopic and macroscopic description of the behaviour of far-from-equilibrium open systems. Acknowledgments. acknowledged.
Fruitful discussions with Dr. P. Mandel are gratefully
592
D. WALGRAEF APPENDIX
A
The matrix elements (p( l(t) I/L’) may be represented by the various elementary vertices appearing in fig. A.1. Their time-independent operator contributions, owing to the fact that (pi ~4 I/L’) = q@‘A,_,, (N) q-” - q-“‘AA,-,s (IV) $’ are :
Cd) <-li,,4
%a+lO,,p -
1)
= 6&O [$‘(N + +p)+q-l- p(N (e>
aa,+ IP +
-
$f.J+
1,~ - 1)
= [(N + 1 + i/4) (iv + +j&J”q-lXq - W - 3/J) (N - 3P +
(a)
(b)
~M3Ti+r1,
(4 Fig. A.l. The elementary vertices.
lp#J,
QUANTUM
STATISTICS
OF A MONOMODE-LASER APPENDIX
MODEL
593
B
We have to evaluate the Laplace transform of the kernel described in fig. 6: y,,(z) = jz dt e-“G, (t). Consider first the so-called connected graphs as defined in fig. B.l. _
1 c
= ..;...+ f D__i_.
Fig. B.l. The “connected”
graphs of the kernel of the kmetic equation for e. (N,r).
For the graph (b), we have to associate R(z)=P(z+i(Q-o))+F(z-i(Q--o)) with each propagator containing the same number of photon lines and spin lines and z-l with the others. The particular structure of these graphs, imposed by the thermodynamical limit (succession of sequences composed by a one-photon, onespin line propagator followed by a two-spins, zero- or one-photon line propagator), lead to the following contribution for a graph with 2n + 2 vertices:
A364rz-‘A0 Ml”, where A,(z) = -A2 (N + 1) (e+ - e-q’) R(z) - AZN(e- - e+TT2) R(z).
(B.1)
The second series may be expressed, in time variables, as
CCsx”expi+i(~, - 4(tl f
t2)l
x
x [N*G*
(N - _5, t, -
t2)
N* - (N + 1)’ G*
(N + 3, t, -
tz)
03.2) where Gf (N, r) = F* (IV, z) - 212”; dt, j’dt, F* (IV, t - td 0 x
COS@
-
m)((tl -
0 t2>((N
+
$)e-
-[(~+~)((N+~)~e_11~-(N+3)e+}G*(N,t,).
-
(IV2 -
+)'e+q‘-2
(B.3)
D. WALGRAEF
594
If we define K* (N, t) = N3G* (N - 3, t) N3, we have in the Laplace-transform formulation K* (N, d = N+F* (N, z) N* -
Z
212 Z2 +
x N+F’
(Q
-
ojy
(N, z) N” (1 - ~+q-~ - ~-7~) K* (N, z).
(B.4)
The total contribution of this graph is thus the sum of the convolutions of the Laplace transform of cX g,” exp [i (0, - o) t] with the K+ (N, z) terms and of 2% g,” exp [-i (w, - o) t] with the K- (N, z) terms. In the short-memory limit for the heat bath, this reduces to -yN+y(N+
1)q2.
(B.5)
So ~,&) for the connected graphs may be written as A,(z) f (l/z”) A”,(z) - yN + y (N + 1) q2.
VW
n=O
If, furthermore we take into account the remaining or “non-connected” graphs, the propagator l/z has to be replaced by the corresponding Crzo (I/z”+~) [y,(z)]” and the kernel may be expressed as
Yo(4 = A,(z) f
n=o
f
m=O
&
’ A”,(z)-
IY,,(W
YN +
Y
(N + 1)v2
(B.7)
>
The same evaluation for y,, leads to the introduction
to the operator
A,(z) = -f?‘R (z) {(N + 1 + 3,~) Q+ - [(N + 1)’ - &413e-q2 + (N - +,/A)Q- - (N2 - $,u’)+Q’~-‘1,
(B-8)
while the dissipation term is defined as C,,=
-~(N-~~)+y[(N+1)2-~~21q~2.
(B.9)
Moreover, as we work on time scales much longer than the heat-bath characteristic time, and as the resummation makes sense for N great, F (N, t) may be approximated by exp (-rt) and R(z) = 2(z + r)[(z
(B. 10)
+ y)’ + (0 - w)“].
It should be noted that, in the absence of a heat bath, A,(z) may be written as 212z - z2 + (Q - W)2
[N(Q- - e+q-‘)
+ (N + l)(e+
-
e-~‘11,
(B.11)
QUANTUM
STATISTICS
OF A MONOMODE-LASER
MODEL
595
which leads to ~~(0) = 0 typical of nondissipative systems, in complete agreement with the periodic time evolution of the Dicke model*). Note furthermore that in the one-particle problem, the kernel of the density-matrix kinetic equation is (N being the photon variable and A4 the spin variable) -
212z z2 + (Q - c0)2 - NT,-~~&+-),
EN~Af++.cJ+ (N + 1) &f_+,o
g/z,-
(N + 1) +BM_*, 0~-2”].
(B. 12)
It is easy to check that the nontrivial roots of the dispersion equation z - v(z) for the Q;(Z) and Q;(Z) components of the density matrix are z,(N) = &i [(Q - 0)” + 4A2(N + l)]‘,
(B.13)
as they shouldZ3).
REFERENCES 1) Haken, H., in Encyclopaedia of Physics, Vol. XXV/ZC, Springer Verlag (Berlm-HeidelbergNew York, 1970), and in Synergetics, Cooperative Phenomena in Multi-Component Systems, B.G. Teubner (Stuttgart, 1973). 2) Scully, M.O., in Quantum Optics, Intern. School of Physics “Enrico Fermi” Course XLII, R.Glauber, ed., Academic Press (New York, 1969). 3) Graham, R., in Springer Tracts on Modern Physics, Vol. 65 (Berlin-Heidelberg-New York, 1972). 4) Scully, M.O. and Lamb, Jr., W.E., Phys. Rev. 159 (1967) 208; 166 (1968) 246; 179 (1969) 368. 5) Haken, H. and Weidhch, R., in Quantum Optics, Intern. School of Physics “Enrico Fermi” Course XLII, R. Glauber, ed., Academic Press (New York, 1969). 6) Gordon, J.P., Phys. Rev. 161 (1967) 367. 7) Korenman, V., Ann. Physics 39 (1966) 72. 8) Dicke, R.H., Phys. Rev. 93 (1954) 99. Hepp, K. and Lieb, E.H., Ann. Physics 76 (1973) 360. Hepp, K. and Lieb, E.H., Helv. phys. Acta, to be published. 9) Prigogine, I., Non Equilibrium Statistical Mechanics, Wiley (New York, 1962). 10) De Giorgio, V. and Scully, M.O., Phys. Rev. A2 (1970) 1170. 11) Glauber, R. J., Phys. Rev. Letters 10 (1963) 84. Sudarshan, E.C.G., Phys. Rev. Letters 10 (1963) 277. Klauder, J.R., Phys. Rev. Letters 16 (1966) 534. 12) Arecchi, F.T. and De Giorgio, V., Phys. Rev. A3 (1971) 1108. 13) Restbois, P. and De Leerier, M., Phys. Rev. 152 (1966) 305. 14) Walgraef, D. and Borckmans, P., Phys. Rev. 187 (1969) 421, 430. 15) Feller, W., An Introduction to Probability Theory and its Applications, Vol. 1, J. Wiley, Inc. (New York, 1968). 16) Scully, M.O., in Laser Handbook, North-Holland Publ. Comp. (Amsterdam, 1972).
596
D. WALGRAEF
17) Mandel, P., Phystca 63 (1973) 553. 18) Glansdortf, P. and Prigogine, I., Thermodynamic Theory of Structure, Stability and Fluctuations, Wiley-Interscience (New York, 1971). 19) Graham, R., in Synergettcs, Cooperative Phenomena in Multi-Component Systems, B.G. Teubner (Stuttgart, 1973). 20) Thorn, R., in Stabthte Structurelle et Morphogenese, W. A.Benjamin Inc. (New York, 1972). 21) Wtlson, K.G , Phys. Rev. B4 (1971) 3174. 22) Wilson, K.G. and Fisher, M.E., Phys. Rev. Letters 28 (1972) 4. 23) Davidson, R. and Rae, J., Bull. Acad. Roy. Belgique Cl. Sci., LVII (1971) 3, 325.