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Kinetic equations and light-matter interaction
Henin,
Physica
F.
39
599-649
1968
KINETIC EQUATIONS AND LIGHT-MATTER INTERACTION by F. HENIN Facultk des Sciences, Universit6 Libre de Bruxelles, Belgique
synopsis The evolution quasiparticles
of strongly
by a distribution
function
totic
bare particle
view
of
both
coupled systems
in the Boltzmann
which is obtained
distribution
their
is much clarified These
representation. function.
dynamical
(kinetic
by the introduction
quasiparticles
by a new transformation
They
are entities
equation)
and
which
problem
of interaction
for quasiparticles, the Pauli photon
The meaning ties
and
First, and
(Born
Weisskopf
levels, only
of these concepts
electrons.
is derived
describes
of
(entropy)
The kinetic
to the
equation
up to order 24. Whereas
one-photon
processes,
states, from the points of view of equilibrium
Rayleigh
Schrijdinger
model
pictures.
the contributions
is used to exhibit Then,
we restrict
and discuss mainly
formalism
perturbation
is then used to discuss the problem
into account
general
model
the T-matrix
with
equation
quasiparticles
and bound
in perturbed
approximation)
of the perturbed
state, taking a rather
radiation
the points
two-
are now included.
comparison
discussed. The kinetic excited
i.e. electron
equation
processes
between
from the asympfrom
thermodynamical
properties, behave like weakly coupled objects. The aim of the present work is to discuss the application
of
are described
is sketched;
expansion,
of spontaneous
ourselves
the sequential a more detailed
decay of an
of one- and two-photon
the main differences decay.
to
comparison
between
a generalized
A brief
properis briefly
processes. the bare Wigner-
comparison
with
is left out for another
paper.
1. Irttroa~ction. Since a few years, there exist systematic methods to derive kinetic equations, see refs. l-3. Recently, much attention has been devoted to the understanding of the asymptotic evolution of strongly or at least not too weakly coupled systems. Since field theoretical problems share one of the most important features of the systems usually considered in statistical mechanics, i.e. the large number of degrees of freedom, one may well wonder whether the methods used in the latter field could not be useful in the former one. A preliminary discussion of this problem has been presented by Prigogined); see also refs. 5-7. The present work will consider the application of these methods to the -
599 -
600
F. HENIN
discussion of the interaction field. This is one of the simplest
between problems
a bound electron
and the radiation
where the statistical
method may be
discussed in detail and compared with the conventional approach based on the T-matrix formalism, see, for instance, ref. 8. Another simple example is the Lee model which will be discussed by C. George elsewhereg). We also want to mention that Prigogine has already presented a brief report on these problemsh). We refer the reader to this paper for a discussion of the physical ideas on which our method is based. There exist at present compact, but rather formal methods to derive kinetic equations to describe the asymptotic behaviour of systems with a large number of degrees of freedom la) ii). However, the earlier methodsl-s), although much heavier from the formal point of view, exhibit perhaps better the physical ideas involved. The first important step is the recognition of the important dynamical role of correlations (as expressed by the off diagonal elements of the density matrix). One of the most important characteristics of systems discussed in statistical mechanics is their ability, for a large class of initial conditions, to reach, in the absence of external constraints, a state of thermodynamic equilibrium in which the memory of the initial correlations is lost. This feature is an important guide in the analysis of the formal solution of the Von Neumann-Liouville equation for the density matrix in the quantum case or for the distribution function in the classical case. The aim is to separate that part of the evolution for which the initial correlations are irrelevant from that which is essentially determined by those initial correlations. This leads to the following set of exact equations (1.3) : t
aPoP) =
at
s
(1.1)
dTG(t- 7) PO(T) + %v(O), 4,
0
PY(4 = P:(t) + P:‘(t),
ap: (4
at
-I- iLop:
=
(1.2)
t
s
dTGv(t -
T)p:(T) -I- %(p#),
t),
(1.3)
T) PO(T).
(1.4
0
p,“(t) = 2 v’*O
s” d&?w(t 0
T) p;,(T)
+ i dTgp(t 0
In these equations, PO(t) represents the set of diagonal elements of the density matrix while pp(t) refers to correlations (off-diagonal elements). G, 9 and V are operators acting on the occupation numbers. The dash on the summation in eq. (1.4) indicates that only those Y’ such that pv’ corresponds to a lower degree of correlations than pv must be taken into account. Let us first consider the diagonal elements. Eq. (1.1) expresses the fact
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
601
that the evolution of these elements is due (first term) to scattering processes which have a finite duration (hence the non markovian character of this contribution). However, besides these effects, we have a contribution expressing the destruction of the initial correlations. The set of equations (1.2)-( 1.4) d escribes the evolution of the correlations. These are split into two parts. The first one, pi depends upon the existence of initial correlations. The term iI.,& corresponds to free propagation. The two terms in the right hand side have an interpretation similar to those in eq. (1.1). The second part p: expresses the fact that fresh correlations may be created from lower order ones (this term depends on the presence of initial correlations) but also from the vacuum of correlations. This latter term exists even if there are no initial correlations. The forgetting of the initial correlations is now very easily expressed, through the assumptions that for long times. c%,p:
4-0
U-5)
(more precisely, the contribution of the p: part of the correlations to average values of observables goes to zero). Of course, such an assumption requires the existence of well separated time scales; indeed, the mechanism by which initial correlations are forgotten must be efficient on a time scale much shorter than the time required to bring the system to the steady state through collision processes. The existence of such time scales is easily understood in systems like moderately dense gases interacting through short range forces, with initial correlations of molecular range. The short time scale is then the duration of the collision while the long one is the relaxation time.The situationhas not been thoroughly investigated yet for the problem of interaction between light and matter but one may hope that these assumptions will be valid for situations not too far from thermal equilibrium. Also, in the absence of an initial finite electromagnetic energy density, the important processes are spontaneous emission processes. The long time scales are then given by the life times of the excited states while the short time scales are given by the inverse of the level spacings. However, if we irradiate the atom with a finite electromagnetic energy density, absorption and induced emission processes play an important role, as well as the type of initial correlations between the field and the particle. The validity of the assumption (1.5) cannot be ascertained generally. Also, situations where there are initial correlations among atomic levels might have to be investigated more carefully. Indeed, correlations between atom and field or between photons themselves always appear in integrals over the continous photon spectrum, in the computation of average values of observables. On the contrary, correlations between atomic levels appear in discrete summations. Arguments based on the spreading of wave packets then do no apply. For all these reasons, the connection between (1.5) and the initial conditions does deserve more study
602
F. HENIN
However, we shall assume here that eq. (1.5) is valid and discuss the behaviour of the system in that case. This should not give rise to difficulties in the applications we shall study in this paper: equilibrium properties or spontaneous decay. In subsequent publications, we hope to discuss more general situations in which the destruction fragment is expected to play an important role. When eq. (1.5) is valid, we have a set of equations (kinetic equations) much simpler than the exact set (I.])-(1.4) : aPoP) __ = at
s
dTG(t -
T) ,OO(T) = --iQa/+~~(t),
(1.6)
0
P4) = d(4J p:‘(t) = rd&(t
(1.7) - T) PO(T)= C,po(t),
0
(1.8)
where I$ is the limit for z -+ 0 of the Laplace transform $(z) of G(t). The operator D takes into account the finite duration of the collision processes and can be expressed in terms of #( z) and its derivatives for z + 0. The operator C describes the creation of correlations from the vacuum of correlations and is given by the limit for z + 0 of well defined off-diagonal elements of the resolvent operator. These equations must be solved with a redefined initial condition (post initial condition) 12)13), which corresponds to the extrapolation to t = 0 of the asymptotic behaviour :
PO(O) = A[po(O) + c &4w Y
(1.9)
The operator A is a functional of $J(z) and its derivatives for z -+ 0. The operator D, describes destruction of correlations. In the weak coupling limit, collisions may be considered as instantaneous and the evolution is described by the Pauli equation: -
a,0 (4
at
= -iA2&p0(t),
(1.10)
where $2 is the lowest order contribution to #. However, when the interaction is stronger and one wants to take into account higher order processes, the situation is much more complex. A clear intuitive picture does no longer correspond to (1.6) 4) 5). Much progress towards the understanding of these equations has been achieved recently through the introduction of a new transformation theory5) 6) 7)is) 14). The main idea is to introduce a dressing operator: x = XIX”
(1.11)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
603
where x’ is a hermitian operator such that A = x’x’
(1.12)
where the operator A is the operator which enters in (1.9). In eq. (1.11) x” is an arbitrary unitary operator. Then, if one defines the dressed distribution function : a(0
= X-+0(4
(1.13)
it satisfies a kinetic equation:
-a/7,(t)= -if&x(t) at
(1.14)
with % = 0Vx.
(1.15)
The operator -iyz is an hermitian operator14), a property which -iQ# didn’t have, and the evolution equation is already much more similar to that for weakly coupled system. Average values of an observable 0 can be written: (1.16) CO>= z Q% 0’) where the summation is over all possible values of occupation numbers and where 0; is a redefined operator: a; = X+(00 + z CJOV) Y
(1.17)
where 00 and 0, are the diagonal and off-diagonal elements of the operator 0. For instance, the energy operator is now: H, = x+(Ho + E C!K,, Y
(1.18)
which can be shownis) to be identical to: H, = x--lHo.
(1.19)
The analogy between (1.16) and the expression of an average value in weakly coupled systems (or more generally systems for which a random phase approximation is valid) should be noticed. An important property of weakly coupled systems is that their entropy is given by: Sh0 = --k~l$0(WN (1.20) In PO(@% One may ask whether there is a x operator such that the entropy of the system including correlations may be written in a similar purely combinatorial form: (1.21) S = --R~$WN ln MW
604
F. HENIN
If this is possible, the equilibrium distribution function of the system must be of the form: r?‘p = exp(-H,/kT)~~~~exp(-H,/kT),
(1.22)
which gives us a very simple criterion for the choice of the x” operator: --iqZ exp(--H,/kT)
= 0.
(1.23)
Several problems have been investigated up to order P. The answer has always been positive, with a unique x” operator. In fact, one obtains: x” = 1 + O(P).
(1.24)
With this choice of x”, we obtain what we have called the Boltzmann representationa). For this representation, we drop the index x and denote the energy operator by HR. In another paperl5) we shall discuss the 14 problem in the special case of the Wigner-Weisskopf two level model. Here, however, we shall restrict ourselves to the 12 problem for the distribution function, i.e. the 14 problem for the kinetic equation. As we shall show in this paper, in the Boltzmann representation, the quasiparticle distribution function obeys the equation:
aF -= at
-i@
(1.25)
with V=
12912+
14934+
(1.26)
qw.
Further analysisl6) of the scattering operator ~4 shows that it can be split up into two parts: one which describes processes which could not occur in a single step in the Born approximation (for instance, two photon processes in the interaction between light and atoms, if the A2 term is neglected in the hamiltonian); the other part describes the same processes as the Born approximation and is easily understood as giving rise to frequency and vertex renormalization. Thus, we may write:
aP -= at
--i@,
(1.27)
where ij?= v = 3r2[vz+ PAv2) +
14(94 -
Ap2) = Pi& +
A4$4.
(1.28
The cross section for lower order (one photon) processes is directly related to the transition probability in the operator ~2. This requires that this transition probability be positive definite, which is indeed verified. Two photon processes, in general, can occur through two different channels. The first one, which is the only one retained in the Born approximation, consists in a transition occurring in two steps with passage through a real intermediate
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
605
state during a time of the order of the lifetime of this intermediate state. But there exist also quasi instantaneous two-photon processes with a passage through a virtual intermediate state. The duration of the process is then of the order of the collision time, i.e. much shorter than the lifetime of the excited state. This channel is the only one which appears in the operator ~$4; to the two successive one photon processes corresponds a term @i in the solution of the kinetic equation. Therefore, the transition probability in e4 is only an excess transition probability, which need not be positive definite. Only the total cross section must be positive. When the process is at all possible in the Born approximation, the first type of effect will always be dominant and the positive definite character is trivially ensured. When the process is not possible in the Born approximation (as for instance in the case of resonance scattering of incident light with a frequency different from the level spacings), it can be shown that the corresponding transition probability in $?4is positive definite. (see section 6 and ref. 15). In the next paragraph, we specify the model and write down the hamiltonian and Von Neumann-Liouville equation. We also indicate briefly how the diagrammatic representation can be set up. The derivation of the kinetic equation, up to order 14, is quite analogous to the derivation of the kinetic equation for interacting phonons in solids. This problem has been discussed in great details in two other papers, refs. 16, 17*). There are, however, some features which are not common to both problems and which are mentioned in section 3. In section 4, we derive the weak coupling (Pauli) equation. The formal handling of the fourth order contributions is briefly recalled in section 5. The two photon scattering operator is discussed in section 6, while the analytic expressions for the dressing operator as well as for energy and vertex renormalizations are given in an appendix. The explicit computation of these quantities involves rather cumbersome algebraic manipulations and will be given in a more technical paper ls) . Then, we examine somewhat more closely the meaning of the quasiparticle distribution. We show that, for a quasiparticle in the ground state and no real photons present, the density matrix factorizes in a product of wave functions. This wave function is that obtained from ordinary perturbation theory for the physical ground state. The situation is not so simple when the particle is in an excited, unstable state; the corresponding density matrix does no longer factorize. The physical unstable states do not belong to Hilbert space; states and energies are no longer eigenfunctions and eigenvalues of the hamiltonian. These states appear in a very natural way in the equilibrium distribution (section 8); we have a Planck distribution for radiation and a Boltzmann distribution for matter. *) Further on, we shall refer to these papers as I and II.
606
F. HENIN
Finally, we use the kinetic equation to discuss the problem of spontaneous emission. First (section 9), we take a model which exhibits all the complicated features of the problem and specially shows the importance of the choice of the initial condition and the difference between the bare and quasiparticle pictures. Then, in section 10, through drastic assumptions on the Fourier coefficients of the potential, we very much simplify the model. This allows us to discuss the order of magnitude of the two photon contributions to the decay and to show the drawbacks of too intzGitivepictures, based on spectroscopic models rather than on a study of the dynamics of correlations. The main advantage of the formalism used here is that we work directly with the density matrix, i.e. observables and can discuss unambigously the time ordering of events. The price we have to pay is a large increase in the number of diagrams. We present some comments on the comparison with the T-matrix or level shift method which has been applied recently by Goldhaber and Watsonis) to the problem of sequential decay. Very interesting differences appear. However, this comparison to be developed fully would have required too much space and is reported to a further publication. 2. Hamiltonian
and Von Neumann-Liouville
eqzlation.
We consider a single bound electron with non degenerate unequally spaced”) unperturbed levels 1,~) with energy wp and discuss its interaction with radiation in the non relativistic limit. The unperturbed hamiltonian Ho is the sum of the unperturbed atom hamiltonian Ht and of the free transverse field hamiltonian H ‘0. If we introduce occupation numbers n, for the atomic levels and %k for the photons polarization index), we have :
(k denotes both wave vector and
(2.1)
(2.4
As we have only one electron we have the conditions:
and as all levels but one are necessarily
ncl = 0, 1;
x?zp= P
empty,
(2.3)
1.
For the photons, we have the usual condition that ?&kis zero or a positive integer. The only physically meaningful unperturbed states are thus necessarily of the form: ~bd{nk~) --*) The condition cumbersome
of unequally
notations.
=
8N,,1
n %I~‘,o P’=+C
spaced levels could be relaxed
(2.4
l@k)>.
easily
but this would lead to
rather
-
KINETIC EQUATIONS
AND
~TGHT-I~ATTZR INTERACTION
60’1
If we restrict ourselves to first order in the coupling constant e, the interaction hamiltonian is given by V = -e(p. A)/m. This hamiltonian induces transitions from a state 1,~) to a state Iv) through emission or absorption of one photon k. In terms of creation and annihilation operators a+i, a-1(& = f 1)
(2.6) we have: v = c,
x
lrvk
c=fl
v~&z;u,“u,”
(2.7)
with VZ”k” = [vii;)]*.
(2.8)
Detailed expressions of the coefficients VeIykwill not be needed here and can be found elsewhere (see Hei tler 20) for instance). The only properties we need are that these coefficients are of order L-1 and that, because the electron is bound, they do not contain any condition of momentum conservation; any excess momentum can be taken up by the nucleus. In order to write the Von Neumann-Liouville equation, we define the v -N representation in the usual waysi) : v=n-n
N = (n + n’)/2,
=
= A,-,#
,
(2.9)
A,(N).
(2.10)
The conditions on the occupation numbers are such that the only physically meaningful elements pty)({N}) correspond to : either N,=O, with
1 andvp=O
xNp=l, P
orN,=$andv,=
zvfi=O, B
fl, (2.11)
Nk = 0, a positive integer, or a positive half integer; and Vk = zero or an integer. The Von Neumann-Liouville
equation
(2.12) is then: (2.13)
with <@I ILo1 +‘j)
= @Y,Y’}(~ VP’%~+ x P
k
vkWk)r
(2.14)
608
F. HENIN
sNII+YPIP,~l+c~12~N”--Y”,2,~1+E~,2 x
The unperturbed operator LO is diagonal while 6L describes one photon processes inducing transitions ,u - Y in the atom. The only p+,({N}) which we must consider must of course be such that the values of N and v fulfil eqs. (2.11) and (2.12). In particular, we see that all correlation states j{v}> must be such that either all Q’S are zero or two of them are different from zero, one being equal to + 1, the other to - 1. There is no restriction on the Vk’s except that they must be integers. Therefore, if we associate straight lines with the atomic levels and wavy lines with photons, the only relevant correlation states are those given in fig. 1.
Fig. 1. Possible correlation states for light-matter interaction.
These are the only allowed intermediate states in any diagrammatic representation of the formal solution of the Von Neumann-Liouville equation. From eqs. (2.14) and (2.1~3, we also see that the only elementary vertices are those given in fig. 2. involving two atom lines and one photon line.
Fig. 2. Elementary vertices for light-matter interaction (in each case, the photon line may have both directions).
Let us stress the fact that the diagrams of fig. 2 describe modifications in the cowehtiolz stateof the system. For instance, reading the first diagram
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
609
from the right to the left, we understand that we perform a transition from the vacuum of correlation (diagonal elements of the density matrix) to a state where there is a correlation between the atom and one photon (described by off diagonal elements of the density matrix). These diagrams are thus entirely different from the Feynman graphs, used in discussions of the evolution of the wave function, which represent transitions between different eigenstates of the unperturbed hamiltonian. 3. Analogy with anharmonic
solids.
The kinetic equation, up to order il4, has been discussed previously for the case of anharmonic solidsrs). When the cubic anharmonic term only is retained, the phonon hamiltonian has a strong similarity with the hamiltonian considered here (unperturbed hamiltonian: quadratic in the creation and annihilation operators; perturbed hamiltonian: cubic in these operators). One may thus be tempted to use the results derived previously in a straightforward way. There are, however, some differences between both problems. In the phonon case, all three lines at a vertex play a similar role. This is not true here: we have quite restrictive conditions on the atom lines but not on the photon lines. For instance, the two diagrams (a, b) in fig. 3 are topologically equivalent when we deal with phonons; here, however, (b) is allowed but (a) is not, see fig. 1. Another difference is that we have no condition of momentum conservation at the vertices while we have such a condition in the anharmonic solid problem. This enables us to construct some diagrams which would be forbidden in the phonon case such as, for instance, diagram (c), fig. 3.
ee* (a)
(b)
(cl
Fig. 3. Examples of differences between phonons and the present problem.
It is quite obvious that these differences will only play a role at order P and higher. Indeed, at order ils, the only diagram in both cases is given in fig. 4.
e P
Q
Fig. 4. Second order diagram.
However, these differences play a minor role. A detailed derivation of all expressions used here will be given elsewhere 18). The general structure of the results will always be that derived in the phonon case.
610
F. HENIN
More important is the fact that the unperturbed electron spectrum, even in the limit of a large box, is discrete. The consequences of this are twofold. 1) Let us consider, for instance, the second order diagram. In the phonon case, forgetting the operational character of &$zj, we have an expression of the form:
&h(Z) = kFkMf(kh’, h”)(z +
wok+ mk'
- (-k*)-'-
(3.1)
Using as variable: 0 = Ok + tik’ in the limit of an infinite
volume,
F,,(z)
0.,k”
(3.2)
we have:
=_$w F(co)(z+
co-l.
(3.3)
This is a Cauchy integral with an infinite cut along the real axis. The singularities of #(z) and its analytic continuation lie in the lower half plane and are determined by those of F( co) in the lower half plane. They define the collision time scale. In the present problem, however, instead of eq. (3. l), we have (see fig. 4) :
F&z)
= x f,&)(z
+ ‘=,u-
‘& f
Wk)-l.
(3.5)
k
In the limit of an infinite volume, the photon spectrum becomes continuous in the interval (0, co) and we obtain an expression of the form (we consider only the minus sign in (3.5); the case with the plus sign may be discussed in a similar way) : F&Z)
= rdw
F/w(u))@ + wp -
wy -
co)-1.
(3.6)
0
Such an expression has a semi infinite cut from (oy - o,J to +co. As a consequence, besides the singularities we meet in the phonon case, we now have a branch point at z = cc),, - wg, which introduces new time scales and gives rise to non exponential contributions to the evolution of the distribution function. For times t of the order of the relaxation times of the system or for t -+ 00, they play no role. (This problem will be discussed more thoroughly in another paper 15); it is also often discussed in the literature in connection with the problem of spontaneous decay of an unstable state, see for instance refs. 8 and 22. Such kind of effects are not taken into account in the kinetic equation). 2) The operator &(z), where 1z > 2, may have poles on the real axis. This is the case for instance, if we consider the diagram given in fig. 5. There, the
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
611
P’ * Fig. 5. Example
propagator
of contribution
to ye
with a pole on the real axis.
is :
kk’ Z +
Wp’
-
1
1
1 22
Ov +
COk
.Z +
Wp -
WV
2 +
a,”
-
WV
+
ok’
(3.7)
The first and the third factor may be incorporated in expressions of the type (3.6) and no new problem arises. But this is not so for the second factor. Because of this factor, the corresponding contribution to &(z’) has a pole on the real axis at z = oy - C+. However, the corresponding contributions to the formal solution of the Liouville equation will be of order A4. Therefore, as long as we do not discuss the kinetic equation at orders higher than A4, i.e. the solution at order highers than 12, we have not to take such terms into account *) . Besides this important difference, the presence of terms like (3.7) introduces also slight modifications in the formulae derived for the phonon case. Indeed, if we apply without care these formulae, we shall write the limit for z -+ 0 of (3.7), (see ref. 16), 1 4
( W’ -
‘% +
1
X9
ql - Qh>
Yi
--Xi 8(W,$ ok
CUB”-
{C
COy+ ok’ >
kk’
1
Bu
W/L’
-
WV
+
Wk
XB I(
)
--xi 6(0& -
I
x
up + w&’ 1
WY+ Ok’) , 1
(3.8)
that we have assumed that the
01 + Ok)
1 cc)p”-
~0”)
-7Li 6(Wp -
whereas the correct expression is (remember levels are nondegenerate) 1s) : I;
x 1
--xi B(wLc-
1
XB
COY + Ok)
>
1 1
OJp -
---xi 6(w@” -
X WY
my+ Wk’) . 1
(3.9)
We notice that we can easily go from the “incorrect” expression obtained by straightforward application of the phonon formula to the correct one where proper account is taken of the discrete character of the atom spectrum *) When order of magnitudes are discussed, we must keep in mind that, in the derivation of kinetic equations, we take the limit As: finite.
612
F. HENIN
if we write that, for OJ~+ w,,: (3. IO)
V% - coy)--f
0.
After these preliminaries, let us now apply the formulae of I to our specific problem. Before we proceed, it might, however, be convenient to recall the meaning of the notations used in that paper and illustrate them for the diagrams of fig. 6.
e I
e
R
P
r
P’ I
@
V
I
(b)
(a)
Fig. 6. Diagrams illustrating the notations.
First of all, vertices are numbered 1, 2 . . . starting from the left. If a vertex happens to be the image of another one, say i, previously met, it is denoted by i. For instance, in diagram (a), the four vertices would be labelled (123 4) while in diagram (b), we have (12 2 1).Lines entering a vertex from the right are called ingoing lines while lines leaving it to the left are called outgoing lines. At each vertex, we associate a set ~2of three numbers, one for each line forming the vertex. For outgoing lines, these numbers are equal to - 1 if the arrow is towards the left, + 1 if the arrow is towards the right; if we intercharzge the words outgoing and &corn&, we m%stalso interchange left and right. For instance, for the third vertex in the diagrams of fig. 6 we have: (a) :
&fi =
(b)
&fi= 1, &/J’= -1,
1, &fl’ = -1,
E/p =
1;
&fi= -1.
The notations E~COZ or @/aNi) are abbreviations for a summation of the quantities ECO or .@/aN) for the three lines of the corresponding vertex i. For instance, for the third vertex of diagram a: ~3~3 = wp - CO&$ + Wk’while for diagram b: ~30s = cu, - o+* - Ok. Also we have: <&i&s. . . G> =
+
81 +
-52+
... +
~r-~
~2) . . .
IV(yN * + ~1 + ~2+ . . . + 8,. = N>,
(3.11)
i.e. a < > involving Y factors correspond to a well defined diagonal element of V*. Finally, all expressions in I contain summations & ..,. E81,+.8,.
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
613
These mean summations over all possible types of diagrams and, for each type, summations over all line indices and over all possible directions of the arrows on the lines. 4. Weak cozGpling afifiroximation. In the weak coupling approximation, the bare and quasi particle distribution functions are identical: po(N, t) = P(N, t).
(4.1)
They satisfy the kinetic equation:
am, t)
= -ii12&j(N,
at
(4.4
t),
where the hermitian operator --iv2 is given by (see I, eq. 5.4) : -ig1s = -2x
E E +r~r) 1
El
[
1-
exp (-.l
&)I.
(4.3)
From the diagram of fig. 4, we may easily construct the four diagrams of fig. 7.
&a@@ I (a) Fig. 7. S~!zd
orde:“,iagra.ms’:’
However, as ,Uand Yplay the same role, (a-d) and (b-c) are equivalent and we only have to consider a and c for instance. Now, c is the image of a and therefore we only have to write (4.3) for a, using the rules of section 3 and then add the same contribution with f 1 --f r 1. Thus, we have: -iv2 x
=
-~TC x Z x S(0, pv k: s=*l
IVI NP
-
E, Nv
+
x [l-exP{-+&
-
e, Nk
-&
my +
wk) x
&>
-
E,
Nv + E, Nk + E IV] N> x
-&)j].
(4.4
With (2.5)-(2.8), we obtain: -iq2
= -2~
E 2 9~ k
E lVpjyk12 6(wfl - WY- 08) X a=z!cl ’
‘Np, (1+8)/Z
‘N.,
(l-92
Nfi+_
I+&
x )
X[I--exp{-+&-&-&-)l].
(4.5)
614
F. HENIN
This operator describes transitions p - v with emission or absorption of one photon k. This process conserves the energy. The transition probability is the usual expression in the Born approximation. If we insert (4.5)into (4.2),perform the summation over E and let the displacement operators act on p”(N), we obtain the weak coupling kinetic equation :
x F({N}>4 - p”({N)‘> Nfi - 1, Nv + 1, Nk + 1,t)l + 6N,,,o%&W({N},
t) - i%N}', Nu +
1,Nv -
1,N/c -
1,t)]},
(4.6)
with y&4 = 2xS(ou -
% -
WC) IVfllrkls.
(4.7)
Eq. (4.6)is the familiar Pauli equation; terms proportional to p”((N}, t) are the familiar loss terms while the other terms are gain terms due to absorption of a photon and transition from a lower state v to an higher state p or emission of a photon and a transition from a state p to a less excited state v. 5. Kinetic
eqtiation
at order 14.
For the bare particle distribution aP0 __ = at
The post initial condition equation is l4) :
function,
-iA2$2p0
which
-
iA4($4
operator
+
Y%#z)
PO.
(5-l)
has to be used in the solution
pop) = (1 + Wi){po(O) Applying the dressing distribution function :
the kinetic equation isis) is) :
x-
+ c ap@)).
of this (5.2)
the quasiparticle
(5.3) The post initial condition
is here:
(5.4)
The quasiparticle
distribution
function obeys the kinetic equation : (5.5)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
615
where : -
iv2 = -i*s,
-
iv4
(5.6) After close inspection, (see ref. 18),
=
-@4-
+
[&,
+21+.
one finds out that the operator -iv4
= -i(Avs)
w -i(Avs)v
-
-iv4
$4
can be written, (5.7)
where (h~s)~ and (A174v represent frequency and vertex renormalizations of the operator ~2 while $j4 describes higher order scattering processes (i.e. processes involving two photons). With -i$s
= --ipz -
ils(Avs)w
-
iP(Aqs)v
(5.8)
i.e.
(5.9)
x(Nk++)[l-exp{-~(&-&-&)]]3 where
&qYk = 2xqb, 6,
tiv -
equation
(5.10)
= ofi + i2sAWi4,,
l~~,vklz + .IVplvklZ= the kinetic
.I,
WC)
(5.11)
i12A(2)I~~~4 2 9
(5.12)
-
(5.13)
becomes: ap -=
at
-iJ2Qsp
iPfj@.
The operator -i@s describes real one-photon processes; the electron makes a transition from the renormalized state with energy 6, to the renormalized state with energy G,,. The cross section involves the renormalized quantity LF. In the appendix, we give the detailed renormalizations and dressing operator.
expressions
of the second order
6. Fourth order scattering operator. In I, all fourth order diagrams have been split into three classes; in classes A and B, we have diagrams which can be reduced to a superposition of two second order diagrams and correspond respectively to the ordering (122i) and (2121).
616
F. HENIN
All other diagrams (fully connected diagrams) belong to the third class C. (For instance, in fig. 6, (a) is of class C while (b) is of class A). Their contributions to the scattering operator @4 have been shown to be (see I, 6. 5. and 1.6.6. *) :
-i(&
+a
(& >
=x&291
QlW
a x ( ?&y-x
X
[
l--q
+
E2O2)
x
a aE202 >
<[E2, 811-
a
a r
[62,
-~1~-62~
El]-> x
, 11
(6.1)
(6.2)
<[Es, &4]- [Ez, El]->
In each case, we have a delta function and a displacement operator which involve the particles participating to two successive elementary transitions. This operator therefore describes processes where two Photons play a role. In agreement with intuition, one finds out very easily that these processes are of three types: 1) double emission or absorption and a transition ,u c-) v in the atom; 4 Raman scattering i.e. emission (absorption) of one photon and absorption (emission) of another photon with a different energy and a transition ,u +-+vin the atom; 3) resonance scattering i.e. emission (absorption) of one photon and absorption (emission) of another photon with the same energy; the atom is left in its initial state. One indeed obtains:
I+& Nn,+yj--
- exp --E {
(
a
a
aN,,-aN,,-aNk,-aNk,
a
1
>[
-
a
II
+
*) There is no contribution of class C of the type (I. 6. 16). Indeed, such contributions involve products of three delta functions, one for each elementary step. Now, for all diagrams in this problem, it can be shown that such contributions, when one takes into account 8(a) 6(b) = 6(a + b) 6(b) always contain a delta function which does not involve photons, hence vanishes. A more detailed discussion is given elsewhere.
KINETIC
+
EQUATIONS
c z
PI ~a+;~1
x
Ix
klkr
Nk,+---
AND
x
LIGHT-MATTER
6Nup,(l-E)/2 x hdP*k, 4vs,,(1+ev2
e=+l
I+&
l--E
Nk,+T
2
617
INTERACTION
>[
I--
+ II I-& +2 x ( I+& >( (
>(
a - exp -_E ___-i
klkn
x
[I
8=&l
a
( 8%
ckllk,
-~
2%
Nk,
+
2
a
8%
Nk,
a
’
a&,
+
-y-
X
)
-e+(&
-
(6.3)
&)}]*
In agreement with our discussion of section 1, the transition probabilities C, 5, 5 which appear in the rhs represent contributions from simultaneozls two photon processes and are thus excess transition probabilities. The quantity OflllDpklke is the excess transition probability for simultaneous emission or absorption of two photons:
(6.6) and EPlk,lh,klis the excess transition probability for Raman scattering: k&k,
=
E$:,k,
+
(6.7)
l:kn,fiak,*
with
h,lrk,12
lb,l~k,12
+
(6.8)
618
F. BENIN
(6.9) where z’ means that ~1must also be excluded in terms where the arguments of the principal parts in the product would be equal, if one takes the delta function into account (this happens for instance in the product of the first terms in each ( }). Finally, ck, Ik2 is the excess transition probability for resonance scattering : ck,lk,
=
ckdka A+B
+
c&k,
(6.10)
with ri$,” =
--xa(Wk,
x
+
&vp,,l
-
wk,)
%$*,1)
x
(6.11)
and
where, again, x’ means that one must take ,us + ~1 in all terms where the arguments of the principal parts would be equal if ,us = ~1. As mentioned in section 1, these excess transition probabilities are not necessarily positive definite. We must require the positive definiteness only for processes which cannot occur in two steps in the lowest order approxi-
619
KINETIC EQUATIONS AND LIGHT-MATTER INTERACTION
mation, i.e. which cannot be described by I#$. This is the case if we irradiate the atom in a level ,Uwith a photon il such that cue is different from all level spacings and consider the probability [,,,b) of resonant scattering (i.e. a transition A --f k, with u)~ = O.Q, and no real transition in the atom). Then all energy denominators are nonvanishing ; if we use *)
0
1 B----t--, x
1
a
ax
0 1 B-+_-,
we easily verify that,
for that case, we may write:
hlk(iU)
c;kb)
x
=
c
I v*lr 1
S;(zBb)
&Jv;rk
+
=
x8bA
1 Oy -
W,.,-
Wk
1
x
9
-
-
uk)
(6.13)
X
x
~P,“k%
-
2
1 ofi -
u)y -
01
>
0.
(6,14)
Another important feature of the above results is linked with the contribution of the fully connected diagrams (class C). If we compare any of these contributions with the corresponding terms from classes A and B, we notice that the main difference is essentially that, for fully connected diagrams, we have a product of four different V factors while for classes A and B, we have a product of two factors IV12. The transition probability for one-photon processes is proportional to the square modulus of the corresponding Fourier coefficient of the potential. If we draw “spectroscopic pictures” of the kind given in fig. 8 to represent the effects of light matter interaction, we are tempted to consider twophoton processes as some kid of firoduct of one-fihoton processes. This is certainly true in the Born approximation where two-photon processes appear
I
1
a) one-photon emission or absorption
I
i
b) two-photon emission or absorption
L
-
7[I
F
c) Raman scattering
I
-
tl
d) Resonance scattering.
Fig. 8. Spectroscopic pictures of one- and two-photon processes *) Rememberthat all asymptotic expressions must be used to compute average values of observables, i.e. will involve integrals over the photon spectrum. Our choice of WA is such that x = 0 is not in the integration domain.
620
F. HENIN
as a sequence of two, well separated in time, one-photon processes, with energy conservation at each step. When higher order processes are kept, simultaneous processes play a role. Such processes require only a global energy conservation between the initial and final states; the intermediate state appears only virtually. If we think in these terms we might easily guess that the corresponding transition probability will be analogous to that of a product of two onephoton processes (i.e. proportional to (r/j2 IV(s) bzct zeritIzlIze firodtict of delta functions which a#pear in the Born approximation replaced by a si@e delta ficnction times some other propagator, takirtg into account the finite width of the intermediate state. Contributions of classes A and B clearly correspond to such a guess. Those of fully connected diagrams do not and are thus far less intuitive when we try to use such kind of arguments. They would be easily overlooked in the usual description whereas, when we discuss the evolution of the system in terms of the dynamics of correlations, all contributions appear on the same footing. We shall come back to this point in section 10, where we shall, on a simple model, discuss spontaneous decay of an excited state, taking into account two-photon processes. 7. Link
with stationary
state perturbation
theory. grozcnd state with no real
A pure case where the atom is in the perturbed photon present is given by:
(7.1)
!%A% 4 = %VO,lI~O%V.,Ol-J ~h%,O’
Such a distribution is a stationary solution of the kinetic equation. Now, for a stable state, it is well-known that stationary state perturbation theory in the Schrodinger formalism may be used without any difficulties. Therefore, it is interesting to make the connection between the two formalisms for this situation. Let us first go back to the bare particle distribution function p. The diagonal matrix elements are easily obtained by inversion of (5.3). We obtain :
(7.2) Using (A.3) and the notation
{Sk}
lpi
b,
{nk)>
=
(see also eq. (2.9))
pO(b,
{flkh
t) =
we
obtain as only nonvanishing
:
=
hvp,l
n %G,o v=W
y
h5,nr
PO(Pl~
Ipl
lo>
=
po(loa
t)
=
(7.3)
diagonal matrix elements: 2
1
t)*
1 -
A2 2
0.Q -
LDy-
o.)k >
1VO[vkl’s
(7.4)
KINETIC
EQUATIONS
lk JpI 44, lk>
To obtain
=
LIGHT-MATTER
00 -
we notice that,
621
INTERACTION
1
lit, t) = P
pop,,
these results,
AND
2
Wc)p - Wk >
within
I~OlP7c12.
the integration
(7.5)
domain
for
Wk we have: WO <
0s
+
Wk
(7.6)
and use eq. (6.13). In order to obtain the complete bare particle distribution function, we still have to find out the off diagonal matrix elements. They can be obtained through relation (1.8). The detailed calculation will be given elsewhereis). One finds as only nonvanishing off-diagonal matrix elements:
IpI
I,,
lk>
=
lk
IpI
= Pie,-I,,,-ik
If’1 l,,
lk,
lk’>
=
IO>*
=
lk,
lk’
IpI
IO>*
-
0,
-
cc)k
VOILck,
1 W,, -
1 WO -
00
W/J-
Lc)k-
X
Wk’
1
wv -
VvlUk
Wk’
*= = 2s
VOlvk’
+
wo -
Pl,,-l&&)0,
00 -
WV -
(a&)
1 0,
(7.7)
=
= Pio,--lG,--lk,-ir, @)OI (#fit &)k, &)k’) = I2 XZ
1
((+)a, (+)fi> @)k) = A
wk
v”l8k’
VOlvk
,
=
1
z
vk
(7.8)
Wo -
Wy -
(7.9)
Ok
It is easily verified that those contributions are the only nonvanishing matrix elements of the following density matrix:
lG> 41
p =
(7.10)
with
+
’ 2 w,, -
ifi
-
VO[lck
IMP, 1 k> +
Wk
1 +P
+
+
c pvkk’ ‘00 -
Wo/1-
1 wk
-
Wk’
{
WO -
Wv -
ok’
VOlvk’
Il,jfik
+
1 00
-
Wv -
wk
VOlvk
“2,‘?o it ’ w I.4 00
WO
vv/;Ifik’
-
Ilcc,
iv-
wk
Ikr
lk*>
VOlvk
+
v&k
] 1,).
(7.11)
622
F. HENIN
It is of course a straightforward calculation to verify that /To> is identical with the perturbed ground state wave function obtained by ordinary perturbation calculus in the Schrijdinger formalism. Similarly one can verify that A(s)00 is also identical with the perturbed ground state energy obtained in the Schrodinger formalism. In the perturbed state representation, the density matrix, as expressed by eq. (7. IO), corresponds to a #ure case (one diagonal element equal to one, all other elements equal to zero). The canonical form (7. IO) can be obtained from the density matrix in the bare particle representation by means of a canonical transformation Q which will be given elsewhere. The situation is rather different if we consider a pure case where the atom, at a given time to, is in the p-th perturbed excited state (7.12)
p”(W), to) = KVJ rI %V”,OrI &*,O. k v+P First of all, this is no longer a stationary Of course, we can again, through the corresponding bare particle distribution be that, instead of a simple factor (~0 we shall have expressions of the form B(wu Similarly, in the off-diagonal we shall have {CP(ou -
oy -
c+ -
solution of the kinetic equation. same steps as above, find out the function. The main differences will ou - ok)-2 in the diagonal terms,
Cl&1 a/awk.
elements, instead of factors (wa c0k)-r -
xi8(0,
-
may-
W, -
wk)-1
0.~)).
As a result, we no longer have a factorization property analogous to eq. (7.10). Diagonalization of the density matrix shows that we are no longer dealing with a pure case in the bare particle representation. Dissi$ative processes are entirely responsible for the departwe from the pwe case. If we consider a pure case like eq. (7.1) or eq. [7.12), we expect to find that the average occupation number for the perturbed ground state (for the perturbed p-th excited state) is one while the average occupation number for the other states as well as the average occupation number for photons is zero. This is indeed the result we obtain if we define the average numberof quasiparticles in the following way: = Z N&N)> 0’)
t),
(7.13)
where (RP> is thus the average occupation number for the ,u-th perturbed state; is the average occupation number for real photons. We notice that eq. (7.13) is not identical with the average number of bare particles. Indeed, this number is given by:
KINETIC
The physical
EQUATIONS
meaning
instance the distribution
ANb
1NTERACTION
623
is quite clear if we consider
for
LIGHT-MATTER
of this difference
(7.1). There, we find:
=
which is a situation where the atom is in the perturbed no real photon present. On the other hand, we find:
=
1-
w(,
-
WV
-
Ok
A2 X k
CO,-) -
0~
-
Ok
>
=
A2 x
P
00 -
(7.16)
1v01”k12,
ivOlfik12,
(7.17)
il”Olak/2j
(7.18)
2
1
>
2
1 =
ground state with
2
1
i1s c vk
(7.15)
1, <&> = 0, = 0,
cc)fi-
cc)k>
which reflects the fact that the perturbed ground state is a superposition of the unperturbed ground state and of the excited states with virtual photons. 8. Equilibrium
distribtition.
One verifies easily that any function I, where HR is given by (1.19j (see also appendix), is an eigenfunction with eigenvalue zero of the scattering operator. Indeed, for the second order part, for instance, we take into account (see A.7 and A.8):
[l-exp{-(& -& - &)]]fWRj = = a
- ~(HR -
E(& -
c& -
wk)).
(8.1)
Then using the relation: VxjU(a)
-0
f
x)1 = 0
(8.2)
we easily obtain: -i&f(HR)
= 0.
The proof for the fourth order contribution
(8.3)
involves exactly the same steps
An Z-theorem can be established in the same sense as in the phonon case. As a special case, the canonical equilibrium distribution is given by: ,‘& = exP(--BHR)/C If we expand
Peq =
W)
exP(--HR),
(/5 = l/H).
(8.4)
(8.4) in a power series of r22,up to qrder 12, we have: exP(-_gHoj
E exp(-@Ha) WI
Z AH exp(--BHaj 1-
a2BAH + A28 ‘N;~lexp~_BHo~
> I
(8.5)
624
F.
HENIN
where
AH = HR - Ho. From this expression, at equilibrium :
we easily obtain
(8.6)
the mean number
= (1 -
exp(-j?cok))-l
of real photons
+ 0(L-3).
k
(8.7)
We obtain a Planck distribution for radiation. The average occupation number for the ,u-th state is given by: <%>,,
= {exp(++)/I:
exp(+W)
x (1 1
x + n2g Z A(2b(
d2BA(2)o,(<%>,,)
x
exp(--B41~= "
x [e4--B4/2j
= exp(-@Eq)/x
Y
exp(-@jzq),
(8.8)
where ‘;)E’J is given by (A.6) with Nk replaced by
emission.
We now want to discuss the decay of one excited state by spontaneous emission. Usually, one assumes that the atom is initially in the ,u-th unperturbed excited states) 19) 2s) and there are no photon present .e. one takes : PO@% t = ‘) = and compute
the probability
sN,.,l
rl: v*P
%‘,,I_l
n k
(9.1)
‘%%,!I~
to reach some final unperturbec
state.
However, when radiative corrections are taken into account we should think that the correct procedure is to discuss transitions between perturbed states. As mentioned by Heitler, ref. 20, p. 163: ((a bound electron is like a free electron, accompanied by its virtual field”. There are no reasons while we should work with dressed free electrons and with bare bound particles! Of course, if one adopts this point of view, one needs a definition of the perturbed states. This presents no difficulty in our formalism; the initial condition which describes a particle in its ,u-th perturbed state, in the absence of photons is simply:
p”(@h‘1 =
'Nu,l
n:
v*B
sNv,O
n.‘Nr,O’ k
(9.2)
Both conditions (9.1) and (9.2) will be considered below. As we shall see, in case (9.2) all line broadening is due to simultaneous twophoton processes. In case (9.1) we also have broadening due to emission
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
625
processes in the “dressing” period which follows the switching on of the interaction, i.e. before the asymptotic regime is reached. All this shows us that any satisfactory treatment of processes like spontaneous emission should take into account the excitation mechanism. This feature has already been stressed some years ago by H ei tler se)24) and LCvys5), although it is sometimes regarded as unimportant provided the life time of the excited state is long enoughs). In order to avoid too lengthy expressions, we restrict ourselves to a few level model (0, 1, 2, 3) and assume that the atom is initially in the second unperturbed state. With the notation (8.3), we thus have: all other
pe(ls, t = 0) = 1,
pa = 0.
(9.3)
The post initial condition for the quasiparticle distribution function is now (see 5.3) : 1
P(ls, 0) = 1 + L2 zg
W2 -
vk
p”(lv, lk, 0) = -P
0.l~ -
Ok
>
1
B
02
-
(9.4
Wy -
all other
“.lk
po = 0
In order to exhibit the dependence of the decay process on the initial conditions, rather than eq. (9.4), we shall take: _ p”(lZ,o)
=
1 +~2~/%k~
w2_;v_w,
& >
( j$lv,
1 k, 0) =
--‘/bk
9
1 ‘02 -
a
wv -
ok
aWk
IV21vk12,
(9.5)
with all &k’s of the same order of magnitude. All &k’s equal to one correspond to (9.1) while all /&k’s equal to zero correspond to the post initial condition (9.2) (in the bare particle representation, we have then both diagonal and off-diagonal elements different from zero). As all photons play the same role in this problem, we must have at any time t: P(lv, t) = O(l), p”(lv, lk, p(b’,
lkl,
t) =
@(L-3),
. ..> lk,, t) =
(9.6) o(L-3r).
These conditions insure that the atom reduced distribution function is independent of the volume of the box. If we take (9.6) into account, as well as the fact that (v&lvk12 is of order
626
F. HENIN
(i.e. Ttiirlvk of order L-3,crplrkk', &klrk', &k' of order L-6) and neglect terms of order L-3 in the equation for p”(ly, t), L-s@+l) in the equation for p”(l,, Ikl, . . . . Ik;, t), we obtain the much simpler set of equations: L-3
(9.7)
ap”(l@, lk, lk’, t)
at
= -12r$(l,,
lk, Ik’, t) +
+ A2 x {j%yltikp(lv> lk’j t) + $&k’ p”(h lk> t)} + + A4 ;
-+‘vlckk’
+
G+k’k}
(9.9)
p”(lv> t),
Y
@(l/h lk, lk’, lk”, t) at
= -~2r$(lu,
+
A2c
{j%lbk p”(l”,
+
?vI,In
p”(b,
1 k,
1 k’,
lk> lk’, lk@, t) +
1 k”‘, t, +
jhk’
1k’r t)} + A4 c {(‘hlakk’ Y
~(1%
lk,
lk”,
t) +
+ &+k’k) p( 1%‘~ 1k”,
t) +
+ (%(bkk” + cvlak”k) p”(Iv, 1k’, t) + + (ovl,uk’k” + @vlbk’k’) p( 1%~1k, t)},
(9.10)
etc., with (9.11) vk
(9.12) (9.13)
Only spontaneous emission processes are kept in the evolution equations. This is a consequence of the fact that, in the absence of initial finite electromagnetic energy density, absorption and induced emission processes are of order L-3 compared to spontaneous emission. The first term in the rhs of each equation describes processes whereby the atom leaves the state ,u and emits one (7) or two (u) photons. The other terms describe the fact that a state with at least one (two) photons may be reached from another state with emission of one (two) photons. The set of equations (9.7) etc. is easily solved. For our four level model, with the initial condition (9.5), we obtain as only nonvanishing matrix
KINETIC
EQUATIONS
AND
LIGHT-MATTER
627
INTERACTION
elements (we neglect A4): p(l2, t) = exp(-A2F2t)
p(13,
1 k, t) =
x
A2 exp(-12r3t)
I
b3k (02
-
03
(9.15)
1v213k/2,
mk)2
-
I 1 +
A2
2 /f&k’ 9 k’
c v=O,1,3
--A2expk-2ht) P(lo9lk,t) x
I
1 +
==&-
(1
o2
-
_
WV -
_
_
bl
Ok’
ok-
&
Iv21vk’12
>
p,
1
(9.16)
/v211k12,
>
exp(-n2r2t)}p210k
I2 vcg , , 3 F
--‘fiOkg
o2
,8lk 9
Lo2 -
bk’
B
(
_-
;, -
ok
p”(12,lk, lk!, t) = “2+lr-m
X
w2 _
m; _
ok ,
&
iv210k/2,
&
Iv2,vk’,2
-
>
(9.17)
>
3 {exp(-A2P$)
- exp(-A2r2t)}
bkjh2k
w’,
X
2
x
+
i
(m2 _
_ __ ok)2
iv213ki2
id. with permutation of k and k’ , 1 3 {exp(-A2f
p(ll, lk, lkt, t) = iz2 ,‘,-
+
(9.18)
St) - exp(- ATIt)} x
1
x
+
I
ihd311k’
-(02_
w’, _
okj2
iv213k12
+
id. with permutation of k and k’ + 1 ,
{exp(--2rlt)
-
exp(--2r2t)){~211kk’
f
qlk’k>,
(9.19)
628
F. HENIN
exp(--ilTzt)
P(lo, Ifi.,Iv, q =
exp(-AsPIt) +
{ (F2 --l)rz
x
{Y211k~llo~n+ I%lk'~llOk}
rl-r2)fl
1 + {
a
1
XB W2
p(11, lk,
lk’,
>
aOk
-wor-wk
Ik”rt)
=
+
1 x
IX /f&k" x i12 z v=o,1,3 k"
(v2\vkj2
+
id.
with permutation of k and k’ , (9.20)
exp(-A2rst)
A2
r:r,
(r3 --Pz)(F3
exp(-A2F2t)
-F1)
+
+
(r2--3)(p241)
exp(-A2Plt) +
(&--3)(&-~2)
1 (02
-
W3 -
cOk”)2
/V213k”(2
+ id. with all permutations of k, k’, k” , p(lOt Ikr lk’,
lklt
+
(9.21)
t) =
1 (02
+
-
cc)3 -
Wk’)2
1v213k”/2
+
id. with all permutations of k, k’, k” +
+12
X {~llOkC%,Wk”-i- ?211&(TlJOk’E” + ti permutations Of k, k’, k”}, (9.22)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
629
INTERACTION
p(10,lkr lk’, lk”, lk”, t) = exp( - A2Fst)
exp(- A2f st)
-
i12
(r3
-
rZ)(r3
-
rl)
r3
+
(r2
-
rl)(rz
-
r3)
r2
+
’
(9.23)
+ all permutations of k, k’, K”, k” , I
In these expressions, we have taken into account the fact that 8(6, -
-
%lfik
a
1
B OV
flvlfikk’
-
rnfi
-
6(ov
> - hk
Ok
-
-
O/J -
for
Ok) --f 0
8, -
1 +
ok
-
-
(coy -
wk’)
+
W/J -
for ’ <
mk)2
for
0
(9.24)
v < /&
”
(9.25) (9.26)
V < fA.
To obtain these results, we assume that the perturbation is sufficiently small to avoid crossing of levels (i.e. for instance ~2 > ~3). Such an assumption is of course unavoidable in a theory based on perturbation expansion. Moreover, in eq. (9.25), we take into account the fact that all matrix elements will always be used to compute average quantities, hence will involve a summation over K and we shall be able to perform integrations by parts. (This has already been taken into account in the evaluation of the asymptotic propagators.) In the long-time limit, t > r,i,
r,i,
r,i,
(9.27)
the decay is completed and we obtain (we only write nonvanishing matrix elements) : 1 p(lO,
lk,
8 +
a)
=
-
p210k
x
r2
a
1
x
1 + A2 I
x
v=o,1,3
&t?vk’~ k
cc)2-
WY-
-
cOk’> ihks
Iv21vkk’12
-
(9.28)
630
F.
p(lO,
lk> lk’,t
-+=‘)
=
HENIN
@Zllk~llOk’
-‘-
+
~2(lk’$%lOk}
x
r1r2
+
-
+
A2 __
(c210kk’
+
a2lOk’k)
-
~2~~,3~{~~k~~,0k~~(.2_~~_.k)~~~2~~k~2+
id. with permutation
of k, k’ ,
1 = A2v=l x a ryr3
p”(la, lk, lk’, lk”,t -too)
(9.29)
&‘ldW~:‘~~k”
x
1 -
x
(m2
-
W3
-
IV213k”i2
+
A2 F~-
+
id. with all permutations
@I), lk, lk’, lk”, I&‘“, t +W)
+
Wk”)2
(j%IOk’JZllk’k”
+
?2llk~llOk’k”)
+
of k, k’, k”,
(9.30)
1
= a2
Y”1~0k~211k’~3~2kw
x
rlrZr3
Iv213k"i2 + + id. with all permutations
of k, k’, k”, k”‘.
(9.31)
The atom is in the ground state. The maximum number of photons is four. At first sight, this latter result is not intuitive. However, this is easily explained if we look at the initial condition. Then, we notice that we have some probability that the atom is in the third perturbed excited state with one photon. Decay of this state, in the Born approximation, happens through emission of one, two, three photons. We notice that if we take as our initial condition that the atom is in the second perturbed excited state (i.e. all &k’s equal to zero), the maximum number of photons is three. In fact, the only contributions which remain are proportional to j+~Ok (straight decay to the ground state by emission of one photon), 7$ikj7ilOk* (cascade decay to the ground state with emission of one photon at each step). These are the only contributions in the Born approximation. At order 12, we also have terms proportional to Usiakk’ (straight decay to the ground state with simultaneous emission of two photons) and jj21ikUilak’k” or j?llOkUslik’k”
KINETIC
EQUATIONS
AND
LIGHT-MATTER
631
INTERACTION
(cascade decay to the ground state with emission of one photon at one step and of two simultaneous photons at the other step). From the results for the quasiparticle distribution function, we may easily, by inversion of (5.3), find out the nonvanishing diagonal matrix elements in the bare particle representation. For long times, we obtain: I
1k,
pO(lO,
t -+
00)
=
-
$hlOk
x
r2
-
1
A2.=F2,3 z (cDo -
-
WV
-
Ok
>
1
--A29 02
-
00
-
‘Vo~vk”2 -
wk>)2
d$
tv2lOkt2
lk,
lk’,
f +
m)
=
1
~2--?210k’ r2
(a0
-
id. with permutations of R, k’
+
19.32)
BOkr
1 pO(lv,
lJ721Vk’12 -
&
wk’ >
WV -
mk12
ivOlvk12
(v = 1, 2, 3),
(9.33) 1
f’O(lO,
lk,
Ik’>t-tCQ)
=
-A2
r, v=1,3
j&
+
j%,Ok/%k
-._
9 04
V
-
WV
-
X Wk’
>
a
(l/r21vk’12 + id. with permutations of k, k’ + aOkc
x +
&
x
bhky”llOk
I
+
f2llk’$hlOk)
x x /%k" @ t’=O,1,3 k”
1
1 + a2
-
A2v=i52,3
+
A2 &
i?
b2IOkk’
(WO -
+
m;
-
x
c02 -
C%v -
wk”
&
lk,
X
lk’,
lk”,t+W)
1vOlvk”(2
+
-
Wke)2
(9.34)
02(Ok’k),
1
1 pO(lv,
l~!Zlvk")2
>
=a2--__ FIT2
Y”2llkhW
(0.h) -
id. with all permutations of k, k’, k”
coy -
Ok”)2
X
(v = 1) 2, 3),
(9.35)
632
F. HENIN
PO(l0,lk, lk’, 1k”,t+co)
1
= 12 x
__
v=l,2
~~lOk”~3(Vk’/93k
rpr3
x
1 x
(02
+ a2-
-
w3 -
* r1r2
IV213kj2
Ok)2
(02Jlkk’jhlOk”
+
+
?72llkUllOk’k”)
+
+ id. with all permutations of k, k’, k”,
po(l0, It, lk’, lk”, lk”, t -+m) = a2
x
P312k”
(w2
-
r,;2r3
w:
-
~3k”jhlOk~2)lk’
Ok”)2
(9.34)
x
Iv213k”/2 +
+ id. with all permutations of k, k’, k”, k”‘,
(9.37)
We notice that, even for t +CQ, we have components where the atom is in the excited state. This corresponds to the physical fact that the perturbed ground state is a mixture of the unperturbed ground state and excited states. Besides these diagonal nonvanishing matrix elements, we also have off diagonal elements describing correlations between the bare atomic levels and the electromagnetic field. Let us now compute the average number of real photons as defined by (7.13) emitted during the asymptotic evolution. In other words, we compute the difference between the number of photons at t --co and the number of photons already present in the post initial conditionp.r.. These are emitted during the short time interval following the switching on of the interaction at t = 0. We obtain:
,,
x +
-
I
P.I.
1 + a2
3L2&
=
-&-
c xk’
lr=o,1,3
{?2lOk
+
?211k)
x
1 &k’ i?
5 p312k’ 5 B3k”
(w2
02
_
-
cr),~ -
w;
_
wk’
Ok,,)2
&
I~21rckd2
> tv213k"j2
+ 3
+
KINETIC
r2r3
A2 ‘c /%k’ 9 k
AND
LIGHT-MATTER
633
INTERACTION
1
z 7211k’ k’k”
+ P---- * -
EQUATIONS
7312k”
5
p3k”
1 )
( 02-ml-Wk’
(co2 -
&
’
03
-
WkW)2
Iv211k’j2
+
Iv213k112
12&-k:_
-
u2,lk’k”
+
1 +
a2 &
5
;83k”
7311k’
-
(w2 -
o3 _
okn)2
‘v213k”12
1
+
1 +
a2 &
{73,Ok
+
A2 &
z
+
A2 &
+
p311k +
(02lOkk’
5?2,lk”
-
<2Gfkh’.I.
b2lOk’k
f$
In the weak coupling,,
+
p312k} F p3k’
+
(UllOkk
g2Ilkk’
+
(W2 -
+
t03 -
@2llk’k)
Wkp)2
1v213k’12
+
+
(9.38)
~l[Ok’k).
case, this expression reduces to: =
,,
=
+-
=
(72,Ok +
t-+oo
Y211k) +
=
-!-
YlY2
)‘l,Ok
5
7’21lk’.
(9.39)
We have three sharp lines at the frequencies 02 - cc)a, ws - 01 and o1 - ~0. They correspond to the three spectroscopic lines one expects in such a situation. When we take higher order effects into account, the situation is somewhat more complicated. Let us first consider the initial situation where the atom is in the second perturbed excited state (all pvk’s equal to zero). Then to the sharp contributions of the weak coupling case, displaced by the second order level shifts, we have to add a continuous background due to the simultaneous two-photon transitions; broadening of the lines is essentially due to this. This kind of effect will be discussed more thoroughly in the next section on a simplified model. If we start from the unperturbed excited state (all &‘s equal to one), besides these effects, we have to add some contributions due to transitions from higher excited states (here the third excited level), corresponding to the fact that the second unperturbed level is a mixture of all levels. Also, the intensity of the various lines is modified. This picture, where all line broadening is obtained through two-photon processes might appear in contradiction with other results. For instance, in a two-level model, with a single matrix element different from zero, all u contributions vanish and the line remains sharp. However, we notice that if we compute the number of photons at t +oo without subtracting the contribution of photons emitted before the asymptotic regime is reached,
634
F. HENIN
we have to add to (9.4) :P.I.
= -ii2
2
,&‘ky
v=o,1,3
~-;-_y;
&
(
k
>
(9.40)
1vWd2.
This introduces a continuous background, mostly important around the lines involving the transitions 2-1, 2-O. This effect depends on the choice of the initial condition. It is a similar effect which gives rise to the line shape in the two-level model when one starts from the unperturbed excited state. This will be further discussed in another paper-is). All this emphasizes the importance of the initial condition, i.e. of the excitation mechanism in the description. From that point of view, a rather clear problem is that of resonance scattering. This will be discussed in another paper is), using the Wigner-Weisskopf model. Then simultaneous absorption and emission will be possible (the c operator in (6.3)) and will allow scattering of nonresonant light. 10. S$ontaneozls emission - simplified model. We shall simplify furthermore the model used in the previous section. We assume : V211k>
VllOk
=b 0,
i all other vfljVk = 0.
(10.1)
(k arbitrary)
In other words, the atom may decay only step by step. We also forget about possible contributions from higher excited states. Moreover, we put equal to zero all matrix elements which describe only virtual transitions (v#lvk = 0 if ,u < v). With this restriction, the unperturbed ground state is identical to the perturbed ground state as soon as we neglect absorption processes (see eq. 7.3), This model is a generalization of the two-level Wigner-Weisskopf modelse). Another way to describe it is to say that the only possible elementary Feynman
graphs are the graphs of fig. 9.
Fig. 9. Feynman
As a consequence
graphs for the particular
of these restrictions
problem
of section
on the potential
10.
we have, see eqs.
(5.10) and (6.4), PalOk= 0,
Y”311k =
0,
f312k
=
0,
7210k
=
0,
02llkk’
=
0,
OllOkk’
i(a210kkr
+
~210k’k)>
=
0.
(10.2)
Hence, rl= r2 r3
pl=
xj%IOk, k
= =
72 0.
+
A202
=
x y211k + k
2' x kk’
(10.3)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
635
~__
We easily verify that, in this case: 1
ts =ws+PCP
/Vzllr=
-
Wl
-
Wk
-
Lc)k)
1
91 =oi+1acg f_zO =
02
k
C01 -
k
00
-
lJ7211k12*
(10.4)
IT"llOkl2,
(10.5) (10.6)
WO,
Iv2llki 2{1+~2~~(~_-Lljl_Wk,)&
iv211k'i2+
+g (
1
-2i12 z B
Co2 -
k’
01
-
1
B-
Wk’
)(
v21lkv;,lk
'Vl(Ok'v;,0k,
aO+Wk'--ml
(10.7)
In all expressions (10.4) to (10.8), the dependence on the photon state has been neglected; indeed, when there is only a finite number of photons, this is of order L--3. Also, we have (see 6.4) : 0210kk’ =
XB
=
o$-kf,
+
2X8(WJ -
O&,kk’
Wa -
Wk -
1
cc)2--~~--~k
I(
>(
1
+y (
=
-
Wk’) x
a
aOk
-
>
i1"211ki2Ivl10kd2 + )
1
B
Us-Ol-Ok
&
( Ol+Ok'-W2
The post initial condition
>
‘V211kV;,lk’
h1Ok’V;,ijk
becomes:
a p(l2,o) =
1 +
~2~/%k~
r;zk
v211k(2,
-
1 =
(10.10)
) aOk
( p"(ll,lk,o)
(10.9)
* I
+lkg
& W2-Wl-Cok
/T"211k12.
(10.11)
)
Let us now write down the solution of the kinetic equation for the quasiparticle distribution function. The exponential decay of the perturbed excited state is given by P(ls,t)
=
exp(-A2r2t)
1 1 + A2 IZ&ky k
co2-w1-ok
& >
~~211k~2
.
(10.12)
636
F. HENIN
Also p(11,1k,t)=
2 {exp(-12rst)
r
-
exp( - n2rrt)} 721lk x
1 1 x
1 + n2z
/9lk'P
k’
1
;k; (
~2-wl-Ok
lV2,1kf12)-
> 1
-
exp(-_‘rlt)
12filk
9
/v2/1k/2
2;
e&J-~r---wk
(
(10.13)
>
gives us the probability that the atom spends some time in the intermediate state after emission of one photon at the frequency (62 - ~1) (first term) plus the exponential decay of that part which is already present in the post initial condition (see eq. 10.11). The probability to reach the ground state after emission of two photons is given by: exp( - J2rst) p(lO, lk,
x
Ik’,
t)
=
i
(r2
-
rl)
r2
+
(PI
-
r2)
r1
+
r:r,
x
{~2jlkj%lOk' + ~2llk'Y"llOk) x
1
1 +;22C/&“g W2 -
k”
+ id. with permutation +
exp(-_2rrt)
A2 &
(1
-
WI-
Ok”
&
11/‘211k”)2
+
>
of k, K’ +
eXp(--‘r2t)
}{U2IOkk
+
(10.14)
~ZIOk’k}.
For times t much longer than the lifetimes
of the two excited states, we have :
p(lO> lk, lk’,
Y"2llk'Y"llOk) x
t-too)
=
&
{?2llk~llOk’
+
1 & m2-m()1-t%kN
+ id. with permutation
+
Aa&
(~2jOkk’
+
of k, k’
u2lOk’k).
Iv211kd2 -
>
+ (10.15)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
637
INTERACTION
Let us now discuss somewhat more thoroughly the role of two-photon processes. First of all, we examine their contribution to the lifetime. For this purpose, we use the dipolar approximation and introduce a cutoff at high frequencies (much higher than the level spacing) : V 21lk:
=
W2ll(e)
__ Sk
q(K
-
(10.16)
4,
where Wslr contains the polarization and wave vector direction dependence but is independent of the wavelength k and where q(x) is the Heaviside step function. One easily verifies that one has: 4’”
= 4 X (&$z kk 00 =
xz
AA’s
co
dwfi, 6(w2 - w. - wk - ok’) x
dmk
0
X
+ $$k&) =
s
0
JIW2ded12
IWl10(er)12
\
-__
B
W2-01-Ok
-
- -
a awkl > a
+
X
aWk
(10.17)
x WkWk&J- Wk)q(Q - Ok’),
where the sums over 1, 1 indicate summations over all polarizations and wave vector directions. Now, we have: 00 co
ss ss do
dw’ S(ws - w. - w - w’) B
0 0 x 7&Q - 0) &2 - w’) = 00 m =
dw
0
1
dw’F(ws-woo-w-w’)9
ws-01-w
0
M-Q - 4 Il(Q - 0’) x
x (0’ - w) - ww’[S(Q - 0) ?#2 - 0’) - ?j(Q - w) S(l2 - co’)]} = OQ
=
1 ws-wr-w
1 ( dwB
>
q(w2
-
00
-
w)
x
0 x {(wz - 00 - 2w) 7#2 - w) V#2 - 0’) - w(w2 - WI)- 0) x X P(Q - 4 Il(Q + w - 0s + 00) - ?&I2- 0) x x 6(Q + w -
w2 +
wo)]}.
(10.18)
638
F. HENIN
If we take into account
the fact that Q>
w2 -
(10.19)
wo,
this becomes : wa--wa
s(
1
~ (ws ws - co1 - 0 )
dwB
0
wa - 2w) =
si
=
dm 2 + (2w1-
w,, -
1
w2) z?
02 - co1 - 0J >i
0
,
(10.20)
and we obtain: G?+B = 2~ C IW211 (eA)12 lW1l0 (e,t,)j2 x
LA’
X
i
2(co2 -
u)o) + (2~x1 -
00 -
cog) In (, 1; :
(10.21)
Ii)}.
In a similar way, we obtain:
e = ikS b$Okk, + 4fOk’k) = =
-2~
C W211(e~) Wi,, (en,)w;,,(eA) Wl10 (el>)x 1.X'
ss 03
Co
X
cr)s-W1-Wk
0
0
1
1
dwk dwk’ 8(U&Mi.&)-C0k--u)k~) 9
ws--w1-~k’
X
x mkWk’?‘/(G- Wk)Tj(.n - wk’) =
J+?&,(w)wT,,(e)
= -2~ C W2ll(e,t)
x
Wlfo(w)
Both terms have the same order of magnitz&e. The main difference is the angular dependence. This will of course depend on the choice of the three levels. The importance of the contribution of fully connected diagrams appears in a similar way if we are interested in the line shape. For our present model (9.40) becomes :+,
-
,.,.
=
1 1 +
+j&~l,Okjj2 12
~2~/hk'~
+ i.'$
~2--~1-cc)k'
z (a2lOkk' +
>
('2lOk'k).
(10.23)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
639
INTERACTION
The first two terms give us sharp contributions at the frequencies ~2 - i3r and ~1 - ~0. Let us now consider with some more details the two-photon contributions to the line shape. With the same model as above, we have:
CO2-Wl-wk 1
~
XB
-
__
COkWk’Tj(G
-
Wk) ?j'(Q-
ok') =
02-01-0.)k
2x1s = 7+2
-
Ul -
lW211(el)12 IWlio(eA,)12 X
Wk) F
1 XB (
x
w2 - ml-
+
Wk
x
IWlj0(ed129
W2ded12
(10.24)
(m2--Coo--mk),
while J2r,
1
5 @lOkV + GlOk’k) = ou
= -,-F
x
X9
2x1s
40(w)
?,o(e3
J%0(w)
> -
+
WO -
W2llh)
w:,,(e)
1
00
Ql(ed
-
(
~2-cOl-Wk’
JfG,,(d
~s-Wr--Ok
w;,,(w)
__
WG1(en’)
wljo(e,)}
Wk’“l;l(g-
wk
Wk) q(G
w&(eAr)
w;,,(w)
w;,O(eA)
Wlfo(e,)}
=
02
-
Wk,)
=
x
>
of both contributions -
(10.25)
Wk(W‘Z--O-Wk).
cc)l-me--w]c
‘Ok
x
x
1
Let us now consider the importance
x
)
Wk) ~(W2Il(eA)
1 X9
~k’){W2Il(eR)
ok -
B
( U)2--l-Wk
v(“2
W2 -
+
1
2xas = 7 X
dwk’a(
s
01
f
CL
for (10.26)
F. HENIN
640
with cc small, but different from zero. If we forget the angular dependence, we find that the dominant contribution of &+n is: &+B
-
(10.27)
+ (201 - IBe- ws),
while the dominant contribution to crCis: & _ L o!
(w2 -
-
W)bl
wo)
2wi - we - 0s
(10.28)
*
The ratio of both contributions to the broadening of the (~2 - &) line depends on the level spacings and the matrix elements of (e1.x). A similar result holds for the broadening of the (61 - 60) line. When we vary the frequency, one or the other of the contributions will become dominant. On the extreme boundary of the line, at ok = 0s - LOO the C contribution vanishes while at cok = (02 - 0,-J/2 the (A + B) contribution vanishes. Let us now go back to the long time quasiparticle distribution function as given by (10.15) and consider the case of the post initial condition (9.2) ; in other words, we assume that the atom is initially in the physical second excited state and take all /l’s equal to zero. Then, using (10.9)) we may write : = - 27c S(G, -Go-Wok--Ok’)
p(10, lk, lk’, t So)
f
xv1
1
9 (
(
-
)(
W2-01-Ok
1
+ izT,B
x
rlr2
a
- & - -a$
>( aOk +
+
>
a
hIOk’V;,Ok
1~2llkl2 l~llOk~l2
aWk
__
0.&---ml-Wk’
~2rl(v211kv;jlks
-
l~2IlVl2 I~lIOA2
-
1
c-c.)
9
(~2-~l-~~~(W2-W:-~k~}.
(10.29) One can verify that this is the limit for z + 0 of the following expression (here again, we take account of the fact that all expressions will be used to evaluate average values of observable% i.e., will involve integrations over k, k’) :
2xi p(lO,
lk,
Ik’,
t +=‘)
=
rlr2
8(&3
-
GO--Wk-Ok’)
1
1
Xh
+ 5-O
,?+&-&--k
x
-1
l~211k12 IVl10k't2 +
z-&+f%l+~k
KINETIC
EQUATIONS
AND
LIGHT-MATTER
1
1 +
+
z+82-c51--tok
[
ii1Tl
-_
x
[
I~Zllkl2
ii12f
--
2
lV2llk’l2
z-f52+8l+ok’
+
IVllOk’l2
(Z --2+
w+
1
-
y
x
[ z+W2-O,l--cc)k
Wk)2
1 x
l~llOk’l2
1
1
(z + w2 - 01-
IV211k’l2
-
-
1 [
I~llOkl2
1
1
(Z+~z--l--k)2
x
Ok<)2+‘(Z-co2+Wl+Wk~)2
x
-
(~2llk~~,lk’~llOk’~~,~k
1
+
C.C.)
o2
w; _
2col
x
+
Z-mZ+wl+wk
11
1
+
z+CO2--~Cc)l--~k’
+
1
-
1
+
I_
1
2
641
INTERACTION
z--w2+ol+wk’
(10.30)
’
which by addition of some terms of order A4and higher can be written: 2xi
p( 10,
1 k, 1 k’, t -+m)
=
8(92
-
50
1
x lim Z-+0 i[
--
r1r2
Z +
&2 -
Ok +
&l -
i1TI
-
ok
+-----
ia2rl/2
Z -
-
Wk’)
x
1 02
+
81+
1
2
Cok +
1 -
Z+W2-Wl-Wk+ij22r1/2
Z-0J2+Ol+Ok+i~2rl/2
iA2rl/2
1 X
+
1 +
Z+wz--l
-
Z -
x
[~2llkV~,lk’Vr,OkVllOk’
-
Ok’
+
iA2rl/2
1 02
+
01
+
Wk’ +
ii22r1/2
+
c.c.1
-
1
cc)2 +
.
1 COO-
2~01
X
(10.31)
642
F. HENIN
Performing
the limit z + 0, we obtain:
P(lO> lk, l/c’> t -=) =_--
= --
2x12 r2
~2llk~liOk’ b2
-
G1 -
ok
+
-7&2 -
bl
-
Wk’ +
t
2
~2llk’~llOk +-
iLsri/2
iA2ri/2
8(82
-
Go -
u)k -
Lc)k’) =
(10.32) This expression indicates the link with the usual Green function technique and Feynman graphs. Indeed, the T matrix element for the double emission in this model is given by the superposition of the two Feynman graphs of fig. 10.
(a)
(b)
Fig. 10. Feynman graphs for double emission.
In the Born approximation, v2llk
we have:
VllOk'
(a): _______
W2--w1-wk
The expression in (10.32) takes into account the fact that we start from the second perturbed level (hence ~2 and not 04. It also takes into account renormalization of vertices. The renormalization of the energy of the intermediate level and the line width i1sri/2 correspond to the introduction of the level shift operator Rr(z). In the usual technique*), this procedure avoids the divergence of the denominators in the Born approximation. Also, one usually does the approximation (10.33)
RI E ;12Ai - ii1sr1/2.
(Notice, however, that the r’s in eq. (10.32) involve two photon corrections). This comparison shows us that one must be careful in the T-matrix approach. Indeed, as we have seen the dominant contributions come from either Ok = ij, and wk’ s &j2 -
Cui
and
mc)f’= bi -
&a
or
W,$= 61 -
dl. Now, when one considers the Feynman
tia
graphs alone,
KINETIC
EQUATIONS
AND
LIGHT-MATTER
643
INTERACTION
one is tempted in the first case to keep only diagram (a) and in the second case (b) and add the cross sections For both channels, one then obtains:
I~2ilk12
.~___ x
(62
-
81-
l~l10k~12 Ok)2
+
24r34
+
id. with k, k’ interchanged
.
(10.33)
This corresponds to the guess one might make if one uses the spectroscopic picture discussed in section 9. Such an expression is analogous to that derived by Goldhaber and Watsonls) (see eq. (3.10) of that paper; the main difference seems to be the replacement of the delta function of eq. (10.33) by a broadened function). From the derivation of eq. (10.32), it is quite clear that such an approximation amounts to the neglect of the contribution of fully connected diagrams (UC). However, a closer examination shows us that the square terms contain both the Born approximatizm and, as corrections to it, the twophoton contribution uA+~ . If one assumes that the coupling constant is small (what one is always forced to do, at one stage or another, for instance, in the evaluation of ri), the above discussion shows us that as soon as we want to keep corrections to the Born approximation, we have to keep also the fully connected diagrams i.e. the cross terms in eq. (10.32). In this sense eq. (10.33) is strictly valid only in the Born approximation. Another interesting feature of eq. (10.32) is that this expression has been obtained for the quasiparticle distribution function at t -+ 00, starting with the initial condition (9.2). In other words, this expression gives us the cross section for the transition from the $hysical second excited state to the physica final state (perturbed ground state and two real photons). If we start from the unperturbed excited state, we have to add the terms proportional to @rk and pik, (with ,%l~ = &r, = 1 in the right hand side of (10.15). From the post initial condition (9.3), taking into account eq. (10.1) we easily see that those terms describe the decay, in the Born approximation, of the post initial component in the first perturbed excited state. (Because of our restrictions on the matrix elements of the potential, the first excited state is the only one which is mixed with the second excited state in the definition of perturbed states). Acknowledgements. I am specially grateful to Prof. I. Prigogine for many helpful and interesting discussions. I also appreciated very much his help for the preparation of this manuscript. This work is part of a larger program developed at Brussels. Collaboration with C. George, P. Resibois, F. Mayne and M. de Haan has been very fruitful.
644
F. HENIN
APPENDIX
-
1. Dressing
DRESSING
OPERATOR,
ENERGY
AND
VERTEX
operator.
The operator #i differs from the operator -i#s pagator : -i&
RENORMALIZATIONS
-1 -+*i : lim z-0 (Z$_Wfl--Or--#’
-i
: lim
only through the pro-
z-to z+~g-oY--u)k
(A-1)
When one takes into account the fact that the propagators appear in integrals over WR,one easily shows that this amounts to: 1 --x8(ofl-
coy -
Wk) +B
(
COB--c&--c#k
a ---> ) acOk
(A4
provided this operator acts on a function which vanishes at the boundaries of the integration domain. This is the case if we include in I/;livka suitable cutoff function at high energies. Therefore, we have:
*i=2z: pv
z
k a=%1
B
wfi -
l
oy -
Wk
>&
lvi4d2 ‘&,~l+s~,~ x (A.3)
x,,,,-.,,,~k+~)[l-eXP{-~(~-~-~)}].
We notice that this operator displaces the variables N only by 0 or + 1 or - 1. Therefore, as the only relevant values of the N to be considered in pa(N) are integers (these are diagonal elements in the ordinary representation), this will be true also for p”(N). A given state Nfi = 1, {Nk: = %k) in the quasiparticle representation is connected with the same bare particle state but also with bare particle state where the atom is in any other state and where there is one more or one less (if possible) photon. 2. Energy renormalization. The operator (Ar~7.a)~ is given in the phonon case by the operator IL74(see I. A.3.4) *). The only diagrams which contribute are those of class A.
x [l -exp(-El&)]. l)
This operator is also given in (I. 6.7) but with a misprint (2 instead of - 1).
(A.4)
KINETIC
When this
EQUATIONS
AND LIGHT-MATTER
INTERACTION
645
is written explicitly for all the relevant diagrams one finds:
(Ay2)m = -2x
‘c
x
uvk a=+1 x
6’(qt -
l~m12
oy -
CO~)[A%~ -
hv#,(l+E)/2 h,
A(2)co,] x
(l-.3)/2
(A-5)
X[l--exp(-e(&-&-Y&J}], with 1 lVplvk12 (Nk Ofi -
WV -
+
1) +
Lc)k
+g
oc _
iv+,,>
ll"r,ak12Nk}.
(A.6)
If we compare this with (4.5), we easily see that (A.5) indeed introduces frequency renormalization in the second order scattering operator. There is another way to obtain (A.6). Indeed, it has been shownrs) that the renormalized energies are also given by:
where the renormalized energy operator is given by (1.19). (Notice that are defined for integer values of the N’s only). HR({N}) as well as HO(W)) In the present case, we easily obtain, using (A.3) : HR({N})
=
ZNcc% P
+
~NN~WC k
-
1 -PXIZ
x9 pv k e=fl
Lop -
x %7u,~1-4,2 Nk + +)
0,s -
Ok >
+$
-
l~fllvki2%7,,,(1+s)/2 x
&
WV-
Wk).
Now: m
Co
Jdxg(-&) 0
$(~)(a
- x) = - j~$-$+(~)
(A.9)
0
when f(x) vanishes at the boundaries of the integration domain. Also, taking into account eq. (2.1 l), we may write for integer values of N,:
646
F. HENIN
Therefore, H&N>)
= X N,w, P
Upon derivation
+ C Nkok + k
with respect to NW, we obtain: w1 = wfi + FA(s)oG
with A(a),, given by (A.6). Upon derivation with respect
(A. 12)
to Nk, we obtain:
&,lc = Cok= 8(L-3). Photon frequency renormalization is negligible in the volume. The renormalized energy levels depend on the state of field. As a special case, in vacuum (all Nk’s equal to zero) known second order correction to the energy of the ,u-th approximation : 1VfiIvkj’
-
(A.13) limit
of an infinite
the electromagnetic we recover the welllevel. In the dipolar
k-l
(A. 14)
we have a linearly divergent contribution (electron self-mass) and a logarithmically divergent contribution, identical to the usual expression for the Lamb shift in the non relativistic limit 26). At finite temperature, the perturbed states energies depend on the photon numbers. The average level spacing at equilibrium will thus be a function of temperature. Also, in the presence of incident light, there is an induced level shift, proportional to the intensity of the light beam. This shift has been observed on the Zeeman splitting of the ground state level of mercury by CohenTannoudjis7). 3. Vertex renormalization. In this case, we have contributions diagrams A and B (see (I.A.4.7)): (A&$+B
from all diagrams.
We obtain,
for
= 2x C ;r: S(E~CO~) 9 12 Elm
x
[I -exp(-El&)],
(A.15)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
647
which may be written as:
(Nk+I)[l x 8Nd1+E,12 %v.,(l-G/2
-exl’(-E(&-&i-&)}], (A. 16)
with
1
+p
u)p - Wfi’ - Ok’ > 1
+9 (
cc)p’ -
Way-
cc)k’
>
1
i-9 (
WV- “,p’ - ok’ >
&
I’v,I,‘k’12
(Nk,
&
)vfi’lvk’12
Nk,
&
I’Vvl@‘k’l’
(Nk’
+
1)
t
+
+
I)}
IV,uIvk12.
(A.17)
The diagrams of class C lead us to (see (I.A.5.14)) :
x
<[~2,[~3,e41-]-~1>[I
-=p(-w&)].
which may be written in a form similar to (A.16) with #‘I
vfl,vk12
=
C-4.18)
(A. 19) The renormalized potential jr/lU,vk12
=
Ivfilvk12 +
i22A’2’ ~+jql&vk12
+
~2A~)Iblvk12
(A.20)
is the quantity which will, up to the order considered, determine the transition probability for one-photon processes, i.e. for processes where one photon k is emitted or absorbed and where a transition between the true levels B,, C%occurs in the atom. Received 6-3-68 REFERENCES
1) Prigogine,
I., Non Equilibrium Statistical Mechanics, Interscience Publ. (New-York, 1962). R., Statistical Mechanics of Charged Particles, Interscience Publ. (New-York, 1963). P., in Physics of many Particle Systems: Methods and Problems, ed. by E. Meeron, 3) Resibois, Gordon and Breach (New-York, 1966). I., Report at the XIVtk Solvay Conference, Brussels, 1967, Interscience Publ. 4) Prigogine, (New-York, in press). I. and Henin, F., Proceedings of the IUPAP InternationalConferenceonStatistical 5) Prigogine, Mechanics and Thermodynamics, Copenhagen 1966, ed. T. Bak, Benjamin (New-York, 1967). I., in Fast Reactions and Primary Processes in Chemical Kinetics, Nobel Sympo6) Prigogine, sium 5, ed. S. Claesson, Interscience Publ. (New-York, 1967). I., Henin, F. and George, C., Proc. Nat. Acad. Sci. U.S.A. 59 (1968) 7. 7) Prigogine, M. and Watson, K., Collision Theory, John Wiley and Sons (New -York, 1964). 8) Goldberger, 9) George, C., to be published. R., Physica 36 (1967) 433. 18) Balescu, (1964) 1109. 11) Zwansig, R., Physica 12) George, C., Physica 37 (1967) 182. 13) George, C. Physica 30 (1964) 1513. 14) George, C., Bull. Acad. Roy. Belg. Cl. Sci. 53 (1967) 623. 15) Henin, F. and De Haan, M., Physica 40 (1968) (to be published). I., George, C. and Mayne, F., Physica 32 (1966) 1828. 16) Henin, F., Prigogine, I., Henin, F. and George, C., Physica 32 (1966) 1873. 17) Prigogine, 18)dHenin, F., Bull. Acad. Roy. Belg. Cl. Sci., to be published. 19) Goldhaber, A. and Watson, K., Phys. Rev.160 (1967) 1151.
2) Balescu,
KINETIC
EQUATIONS
AND LIGHT-MATTER
INTERACTION
649
20) Heitler, W., The Quantum Theory of Radiation (3rd ed.), Clarendon Press (Oxford, 1954). 21) Resibois, P., Physica 27 (1961) 541. 22) Hahler, G., Z. Phys. 152 (1958) 546. Henin, F. and De Haan, M., Bull. Acad. Roy. Belg. Cl. Sci. (to be published). Arnous, E. and Heitler, W., Proc. Roy. Sot. A220 (1953) 290. Levy, M., Nuovo Cimento 14 (1959) 612. Schweber, S., Bethe, H. and De Hoffman, F., Mesons and Fields, vol. I, Row Peterson and Company (Evanston, Illinois, 1956). C., Ann. Physique 7 (1962) 423. 27) Cohen-Tannoudji,
23) 24) 25) 26)
Physica
F.
39
599-649
1968
KINETIC EQUATIONS AND LIGHT-MATTER INTERACTION by F. HENIN Facultk des Sciences, Universit6 Libre de Bruxelles, Belgique
synopsis The evolution quasiparticles
of strongly
by a distribution
function
totic
bare particle
view
of
both
coupled systems
in the Boltzmann
which is obtained
distribution
their
is much clarified These
representation. function.
dynamical
(kinetic
by the introduction
quasiparticles
by a new transformation
They
are entities
equation)
and
which
problem
of interaction
for quasiparticles, the Pauli photon
The meaning ties
and
First, and
(Born
Weisskopf
levels, only
of these concepts
electrons.
is derived
describes
of
(entropy)
The kinetic
to the
equation
up to order 24. Whereas
one-photon
processes,
states, from the points of view of equilibrium
Rayleigh
Schrijdinger
model
pictures.
the contributions
is used to exhibit Then,
we restrict
and discuss mainly
formalism
perturbation
is then used to discuss the problem
into account
general
model
the T-matrix
with
equation
quasiparticles
and bound
in perturbed
approximation)
of the perturbed
state, taking a rather
radiation
the points
two-
are now included.
comparison
discussed. The kinetic excited
i.e. electron
equation
processes
between
from the asympfrom
thermodynamical
properties, behave like weakly coupled objects. The aim of the present work is to discuss the application
of
are described
is sketched;
expansion,
of spontaneous
ourselves
the sequential a more detailed
decay of an
of one- and two-photon
the main differences decay.
to
comparison
between
a generalized
A brief
properis briefly
processes. the bare Wigner-
comparison
with
is left out for another
paper.
1. Irttroa~ction. Since a few years, there exist systematic methods to derive kinetic equations, see refs. l-3. Recently, much attention has been devoted to the understanding of the asymptotic evolution of strongly or at least not too weakly coupled systems. Since field theoretical problems share one of the most important features of the systems usually considered in statistical mechanics, i.e. the large number of degrees of freedom, one may well wonder whether the methods used in the latter field could not be useful in the former one. A preliminary discussion of this problem has been presented by Prigogined); see also refs. 5-7. The present work will consider the application of these methods to the -
599 -
600
F. HENIN
discussion of the interaction field. This is one of the simplest
between problems
a bound electron
and the radiation
where the statistical
method may be
discussed in detail and compared with the conventional approach based on the T-matrix formalism, see, for instance, ref. 8. Another simple example is the Lee model which will be discussed by C. George elsewhereg). We also want to mention that Prigogine has already presented a brief report on these problemsh). We refer the reader to this paper for a discussion of the physical ideas on which our method is based. There exist at present compact, but rather formal methods to derive kinetic equations to describe the asymptotic behaviour of systems with a large number of degrees of freedom la) ii). However, the earlier methodsl-s), although much heavier from the formal point of view, exhibit perhaps better the physical ideas involved. The first important step is the recognition of the important dynamical role of correlations (as expressed by the off diagonal elements of the density matrix). One of the most important characteristics of systems discussed in statistical mechanics is their ability, for a large class of initial conditions, to reach, in the absence of external constraints, a state of thermodynamic equilibrium in which the memory of the initial correlations is lost. This feature is an important guide in the analysis of the formal solution of the Von Neumann-Liouville equation for the density matrix in the quantum case or for the distribution function in the classical case. The aim is to separate that part of the evolution for which the initial correlations are irrelevant from that which is essentially determined by those initial correlations. This leads to the following set of exact equations (1.3) : t
aPoP) =
at
s
(1.1)
dTG(t- 7) PO(T) + %v(O), 4,
0
PY(4 = P:(t) + P:‘(t),
ap: (4
at
-I- iLop:
=
(1.2)
t
s
dTGv(t -
T)p:(T) -I- %(p#),
t),
(1.3)
T) PO(T).
(1.4
0
p,“(t) = 2 v’*O
s” d&?w(t 0
T) p;,(T)
+ i dTgp(t 0
In these equations, PO(t) represents the set of diagonal elements of the density matrix while pp(t) refers to correlations (off-diagonal elements). G, 9 and V are operators acting on the occupation numbers. The dash on the summation in eq. (1.4) indicates that only those Y’ such that pv’ corresponds to a lower degree of correlations than pv must be taken into account. Let us first consider the diagonal elements. Eq. (1.1) expresses the fact
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
601
that the evolution of these elements is due (first term) to scattering processes which have a finite duration (hence the non markovian character of this contribution). However, besides these effects, we have a contribution expressing the destruction of the initial correlations. The set of equations (1.2)-( 1.4) d escribes the evolution of the correlations. These are split into two parts. The first one, pi depends upon the existence of initial correlations. The term iI.,& corresponds to free propagation. The two terms in the right hand side have an interpretation similar to those in eq. (1.1). The second part p: expresses the fact that fresh correlations may be created from lower order ones (this term depends on the presence of initial correlations) but also from the vacuum of correlations. This latter term exists even if there are no initial correlations. The forgetting of the initial correlations is now very easily expressed, through the assumptions that for long times. c%,p:
4-0
U-5)
(more precisely, the contribution of the p: part of the correlations to average values of observables goes to zero). Of course, such an assumption requires the existence of well separated time scales; indeed, the mechanism by which initial correlations are forgotten must be efficient on a time scale much shorter than the time required to bring the system to the steady state through collision processes. The existence of such time scales is easily understood in systems like moderately dense gases interacting through short range forces, with initial correlations of molecular range. The short time scale is then the duration of the collision while the long one is the relaxation time.The situationhas not been thoroughly investigated yet for the problem of interaction between light and matter but one may hope that these assumptions will be valid for situations not too far from thermal equilibrium. Also, in the absence of an initial finite electromagnetic energy density, the important processes are spontaneous emission processes. The long time scales are then given by the life times of the excited states while the short time scales are given by the inverse of the level spacings. However, if we irradiate the atom with a finite electromagnetic energy density, absorption and induced emission processes play an important role, as well as the type of initial correlations between the field and the particle. The validity of the assumption (1.5) cannot be ascertained generally. Also, situations where there are initial correlations among atomic levels might have to be investigated more carefully. Indeed, correlations between atom and field or between photons themselves always appear in integrals over the continous photon spectrum, in the computation of average values of observables. On the contrary, correlations between atomic levels appear in discrete summations. Arguments based on the spreading of wave packets then do no apply. For all these reasons, the connection between (1.5) and the initial conditions does deserve more study
602
F. HENIN
However, we shall assume here that eq. (1.5) is valid and discuss the behaviour of the system in that case. This should not give rise to difficulties in the applications we shall study in this paper: equilibrium properties or spontaneous decay. In subsequent publications, we hope to discuss more general situations in which the destruction fragment is expected to play an important role. When eq. (1.5) is valid, we have a set of equations (kinetic equations) much simpler than the exact set (I.])-(1.4) : aPoP) __ = at
s
dTG(t -
T) ,OO(T) = --iQa/+~~(t),
(1.6)
0
P4) = d(4J p:‘(t) = rd&(t
(1.7) - T) PO(T)= C,po(t),
0
(1.8)
where I$ is the limit for z -+ 0 of the Laplace transform $(z) of G(t). The operator D takes into account the finite duration of the collision processes and can be expressed in terms of #( z) and its derivatives for z + 0. The operator C describes the creation of correlations from the vacuum of correlations and is given by the limit for z + 0 of well defined off-diagonal elements of the resolvent operator. These equations must be solved with a redefined initial condition (post initial condition) 12)13), which corresponds to the extrapolation to t = 0 of the asymptotic behaviour :
PO(O) = A[po(O) + c &4w Y
(1.9)
The operator A is a functional of $J(z) and its derivatives for z -+ 0. The operator D, describes destruction of correlations. In the weak coupling limit, collisions may be considered as instantaneous and the evolution is described by the Pauli equation: -
a,0 (4
at
= -iA2&p0(t),
(1.10)
where $2 is the lowest order contribution to #. However, when the interaction is stronger and one wants to take into account higher order processes, the situation is much more complex. A clear intuitive picture does no longer correspond to (1.6) 4) 5). Much progress towards the understanding of these equations has been achieved recently through the introduction of a new transformation theory5) 6) 7)is) 14). The main idea is to introduce a dressing operator: x = XIX”
(1.11)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
603
where x’ is a hermitian operator such that A = x’x’
(1.12)
where the operator A is the operator which enters in (1.9). In eq. (1.11) x” is an arbitrary unitary operator. Then, if one defines the dressed distribution function : a(0
= X-+0(4
(1.13)
it satisfies a kinetic equation:
-a/7,(t)= -if&x(t) at
(1.14)
with % = 0Vx.
(1.15)
The operator -iyz is an hermitian operator14), a property which -iQ# didn’t have, and the evolution equation is already much more similar to that for weakly coupled system. Average values of an observable 0 can be written: (1.16) CO>= z Q% 0’) where the summation is over all possible values of occupation numbers and where 0; is a redefined operator: a; = X+(00 + z CJOV) Y
(1.17)
where 00 and 0, are the diagonal and off-diagonal elements of the operator 0. For instance, the energy operator is now: H, = x+(Ho + E C!K,, Y
(1.18)
which can be shownis) to be identical to: H, = x--lHo.
(1.19)
The analogy between (1.16) and the expression of an average value in weakly coupled systems (or more generally systems for which a random phase approximation is valid) should be noticed. An important property of weakly coupled systems is that their entropy is given by: Sh0 = --k~l$0(WN (1.20) In PO(@% One may ask whether there is a x operator such that the entropy of the system including correlations may be written in a similar purely combinatorial form: (1.21) S = --R~$WN ln MW
604
F. HENIN
If this is possible, the equilibrium distribution function of the system must be of the form: r?‘p = exp(-H,/kT)~~~~exp(-H,/kT),
(1.22)
which gives us a very simple criterion for the choice of the x” operator: --iqZ exp(--H,/kT)
= 0.
(1.23)
Several problems have been investigated up to order P. The answer has always been positive, with a unique x” operator. In fact, one obtains: x” = 1 + O(P).
(1.24)
With this choice of x”, we obtain what we have called the Boltzmann representationa). For this representation, we drop the index x and denote the energy operator by HR. In another paperl5) we shall discuss the 14 problem in the special case of the Wigner-Weisskopf two level model. Here, however, we shall restrict ourselves to the 12 problem for the distribution function, i.e. the 14 problem for the kinetic equation. As we shall show in this paper, in the Boltzmann representation, the quasiparticle distribution function obeys the equation:
aF -= at
-i@
(1.25)
with V=
12912+
14934+
(1.26)
qw.
Further analysisl6) of the scattering operator ~4 shows that it can be split up into two parts: one which describes processes which could not occur in a single step in the Born approximation (for instance, two photon processes in the interaction between light and atoms, if the A2 term is neglected in the hamiltonian); the other part describes the same processes as the Born approximation and is easily understood as giving rise to frequency and vertex renormalization. Thus, we may write:
aP -= at
--i@,
(1.27)
where ij?= v = 3r2[vz+ PAv2) +
14(94 -
Ap2) = Pi& +
A4$4.
(1.28
The cross section for lower order (one photon) processes is directly related to the transition probability in the operator ~2. This requires that this transition probability be positive definite, which is indeed verified. Two photon processes, in general, can occur through two different channels. The first one, which is the only one retained in the Born approximation, consists in a transition occurring in two steps with passage through a real intermediate
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
605
state during a time of the order of the lifetime of this intermediate state. But there exist also quasi instantaneous two-photon processes with a passage through a virtual intermediate state. The duration of the process is then of the order of the collision time, i.e. much shorter than the lifetime of the excited state. This channel is the only one which appears in the operator ~$4; to the two successive one photon processes corresponds a term @i in the solution of the kinetic equation. Therefore, the transition probability in e4 is only an excess transition probability, which need not be positive definite. Only the total cross section must be positive. When the process is at all possible in the Born approximation, the first type of effect will always be dominant and the positive definite character is trivially ensured. When the process is not possible in the Born approximation (as for instance in the case of resonance scattering of incident light with a frequency different from the level spacings), it can be shown that the corresponding transition probability in $?4is positive definite. (see section 6 and ref. 15). In the next paragraph, we specify the model and write down the hamiltonian and Von Neumann-Liouville equation. We also indicate briefly how the diagrammatic representation can be set up. The derivation of the kinetic equation, up to order 14, is quite analogous to the derivation of the kinetic equation for interacting phonons in solids. This problem has been discussed in great details in two other papers, refs. 16, 17*). There are, however, some features which are not common to both problems and which are mentioned in section 3. In section 4, we derive the weak coupling (Pauli) equation. The formal handling of the fourth order contributions is briefly recalled in section 5. The two photon scattering operator is discussed in section 6, while the analytic expressions for the dressing operator as well as for energy and vertex renormalizations are given in an appendix. The explicit computation of these quantities involves rather cumbersome algebraic manipulations and will be given in a more technical paper ls) . Then, we examine somewhat more closely the meaning of the quasiparticle distribution. We show that, for a quasiparticle in the ground state and no real photons present, the density matrix factorizes in a product of wave functions. This wave function is that obtained from ordinary perturbation theory for the physical ground state. The situation is not so simple when the particle is in an excited, unstable state; the corresponding density matrix does no longer factorize. The physical unstable states do not belong to Hilbert space; states and energies are no longer eigenfunctions and eigenvalues of the hamiltonian. These states appear in a very natural way in the equilibrium distribution (section 8); we have a Planck distribution for radiation and a Boltzmann distribution for matter. *) Further on, we shall refer to these papers as I and II.
606
F. HENIN
Finally, we use the kinetic equation to discuss the problem of spontaneous emission. First (section 9), we take a model which exhibits all the complicated features of the problem and specially shows the importance of the choice of the initial condition and the difference between the bare and quasiparticle pictures. Then, in section 10, through drastic assumptions on the Fourier coefficients of the potential, we very much simplify the model. This allows us to discuss the order of magnitude of the two photon contributions to the decay and to show the drawbacks of too intzGitivepictures, based on spectroscopic models rather than on a study of the dynamics of correlations. The main advantage of the formalism used here is that we work directly with the density matrix, i.e. observables and can discuss unambigously the time ordering of events. The price we have to pay is a large increase in the number of diagrams. We present some comments on the comparison with the T-matrix or level shift method which has been applied recently by Goldhaber and Watsonis) to the problem of sequential decay. Very interesting differences appear. However, this comparison to be developed fully would have required too much space and is reported to a further publication. 2. Hamiltonian
and Von Neumann-Liouville
eqzlation.
We consider a single bound electron with non degenerate unequally spaced”) unperturbed levels 1,~) with energy wp and discuss its interaction with radiation in the non relativistic limit. The unperturbed hamiltonian Ho is the sum of the unperturbed atom hamiltonian Ht and of the free transverse field hamiltonian H ‘0. If we introduce occupation numbers n, for the atomic levels and %k for the photons polarization index), we have :
(k denotes both wave vector and
(2.1)
(2.4
As we have only one electron we have the conditions:
and as all levels but one are necessarily
ncl = 0, 1;
x?zp= P
empty,
(2.3)
1.
For the photons, we have the usual condition that ?&kis zero or a positive integer. The only physically meaningful unperturbed states are thus necessarily of the form: ~bd{nk~) --*) The condition cumbersome
of unequally
notations.
=
8N,,1
n %I~‘,o P’=+C
spaced levels could be relaxed
(2.4
l@k)>.
easily
but this would lead to
rather
-
KINETIC EQUATIONS
AND
~TGHT-I~ATTZR INTERACTION
60’1
If we restrict ourselves to first order in the coupling constant e, the interaction hamiltonian is given by V = -e(p. A)/m. This hamiltonian induces transitions from a state 1,~) to a state Iv) through emission or absorption of one photon k. In terms of creation and annihilation operators a+i, a-1(& = f 1)
(2.6) we have: v = c,
x
lrvk
c=fl
v~&z;u,“u,”
(2.7)
with VZ”k” = [vii;)]*.
(2.8)
Detailed expressions of the coefficients VeIykwill not be needed here and can be found elsewhere (see Hei tler 20) for instance). The only properties we need are that these coefficients are of order L-1 and that, because the electron is bound, they do not contain any condition of momentum conservation; any excess momentum can be taken up by the nucleus. In order to write the Von Neumann-Liouville equation, we define the v -N representation in the usual waysi) : v=n-n
N = (n + n’)/2,
=
,
(2.9)
A,(N).
(2.10)
The conditions on the occupation numbers are such that the only physically meaningful elements pty)({N}) correspond to : either N,=O, with
1 andvp=O
xNp=l, P
orN,=$andv,=
zvfi=O, B
fl, (2.11)
Nk = 0, a positive integer, or a positive half integer; and Vk = zero or an integer. The Von Neumann-Liouville
equation
(2.12) is then: (2.13)
with <@I ILo1 +‘j)
= @Y,Y’}(~ VP’%~+ x P
k
vkWk)r
(2.14)
608
F. HENIN
sNII+YPIP,~l+c~12~N”--Y”,2,~1+E~,2 x
The unperturbed operator LO is diagonal while 6L describes one photon processes inducing transitions ,u - Y in the atom. The only p+,({N}) which we must consider must of course be such that the values of N and v fulfil eqs. (2.11) and (2.12). In particular, we see that all correlation states j{v}> must be such that either all Q’S are zero or two of them are different from zero, one being equal to + 1, the other to - 1. There is no restriction on the Vk’s except that they must be integers. Therefore, if we associate straight lines with the atomic levels and wavy lines with photons, the only relevant correlation states are those given in fig. 1.
Fig. 1. Possible correlation states for light-matter interaction.
These are the only allowed intermediate states in any diagrammatic representation of the formal solution of the Von Neumann-Liouville equation. From eqs. (2.14) and (2.1~3, we also see that the only elementary vertices are those given in fig. 2. involving two atom lines and one photon line.
Fig. 2. Elementary vertices for light-matter interaction (in each case, the photon line may have both directions).
Let us stress the fact that the diagrams of fig. 2 describe modifications in the cowehtiolz stateof the system. For instance, reading the first diagram
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
609
from the right to the left, we understand that we perform a transition from the vacuum of correlation (diagonal elements of the density matrix) to a state where there is a correlation between the atom and one photon (described by off diagonal elements of the density matrix). These diagrams are thus entirely different from the Feynman graphs, used in discussions of the evolution of the wave function, which represent transitions between different eigenstates of the unperturbed hamiltonian. 3. Analogy with anharmonic
solids.
The kinetic equation, up to order il4, has been discussed previously for the case of anharmonic solidsrs). When the cubic anharmonic term only is retained, the phonon hamiltonian has a strong similarity with the hamiltonian considered here (unperturbed hamiltonian: quadratic in the creation and annihilation operators; perturbed hamiltonian: cubic in these operators). One may thus be tempted to use the results derived previously in a straightforward way. There are, however, some differences between both problems. In the phonon case, all three lines at a vertex play a similar role. This is not true here: we have quite restrictive conditions on the atom lines but not on the photon lines. For instance, the two diagrams (a, b) in fig. 3 are topologically equivalent when we deal with phonons; here, however, (b) is allowed but (a) is not, see fig. 1. Another difference is that we have no condition of momentum conservation at the vertices while we have such a condition in the anharmonic solid problem. This enables us to construct some diagrams which would be forbidden in the phonon case such as, for instance, diagram (c), fig. 3.
ee* (a)
(b)
(cl
Fig. 3. Examples of differences between phonons and the present problem.
It is quite obvious that these differences will only play a role at order P and higher. Indeed, at order ils, the only diagram in both cases is given in fig. 4.
e P
Q
Fig. 4. Second order diagram.
However, these differences play a minor role. A detailed derivation of all expressions used here will be given elsewhere 18). The general structure of the results will always be that derived in the phonon case.
610
F. HENIN
More important is the fact that the unperturbed electron spectrum, even in the limit of a large box, is discrete. The consequences of this are twofold. 1) Let us consider, for instance, the second order diagram. In the phonon case, forgetting the operational character of &$zj, we have an expression of the form:
&h(Z) = kFkMf(kh’, h”)(z +
wok+ mk'
- (-k*)-'-
(3.1)
Using as variable: 0 = Ok + tik’ in the limit of an infinite
volume,
F,,(z)
0.,k”
(3.2)
we have:
=_$w F(co)(z+
co-l.
(3.3)
This is a Cauchy integral with an infinite cut along the real axis. The singularities of #(z) and its analytic continuation lie in the lower half plane and are determined by those of F( co) in the lower half plane. They define the collision time scale. In the present problem, however, instead of eq. (3. l), we have (see fig. 4) :
F&z)
= x f,&)(z
+ ‘=,u-
‘& f
Wk)-l.
(3.5)
k
In the limit of an infinite volume, the photon spectrum becomes continuous in the interval (0, co) and we obtain an expression of the form (we consider only the minus sign in (3.5); the case with the plus sign may be discussed in a similar way) : F&Z)
= rdw
F/w(u))@ + wp -
wy -
co)-1.
(3.6)
0
Such an expression has a semi infinite cut from (oy - o,J to +co. As a consequence, besides the singularities we meet in the phonon case, we now have a branch point at z = cc),, - wg, which introduces new time scales and gives rise to non exponential contributions to the evolution of the distribution function. For times t of the order of the relaxation times of the system or for t -+ 00, they play no role. (This problem will be discussed more thoroughly in another paper 15); it is also often discussed in the literature in connection with the problem of spontaneous decay of an unstable state, see for instance refs. 8 and 22. Such kind of effects are not taken into account in the kinetic equation). 2) The operator &(z), where 1z > 2, may have poles on the real axis. This is the case for instance, if we consider the diagram given in fig. 5. There, the
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
611
P’ * Fig. 5. Example
propagator
of contribution
to ye
with a pole on the real axis.
is :
kk’ Z +
Wp’
-
1
1
1 22
Ov +
COk
.Z +
Wp -
WV
2 +
a,”
-
WV
+
ok’
(3.7)
The first and the third factor may be incorporated in expressions of the type (3.6) and no new problem arises. But this is not so for the second factor. Because of this factor, the corresponding contribution to &(z’) has a pole on the real axis at z = oy - C+. However, the corresponding contributions to the formal solution of the Liouville equation will be of order A4. Therefore, as long as we do not discuss the kinetic equation at orders higher than A4, i.e. the solution at order highers than 12, we have not to take such terms into account *) . Besides this important difference, the presence of terms like (3.7) introduces also slight modifications in the formulae derived for the phonon case. Indeed, if we apply without care these formulae, we shall write the limit for z -+ 0 of (3.7), (see ref. 16), 1 4
( W’ -
‘% +
1
X9
ql - Qh>
Yi
--Xi 8(W,$ ok
CUB”-
{C
COy+ ok’ >
kk’
1
Bu
W/L’
-
WV
+
Wk
XB I(
)
--xi 6(0& -
I
x
up + w&’ 1
WY+ Ok’) , 1
(3.8)
that we have assumed that the
01 + Ok)
1 cc)p”-
~0”)
-7Li 6(Wp -
whereas the correct expression is (remember levels are nondegenerate) 1s) : I;
x 1
--xi B(wLc-
1
XB
COY + Ok)
>
1 1
OJp -
---xi 6(w@” -
X WY
my+ Wk’) . 1
(3.9)
We notice that we can easily go from the “incorrect” expression obtained by straightforward application of the phonon formula to the correct one where proper account is taken of the discrete character of the atom spectrum *) When order of magnitudes are discussed, we must keep in mind that, in the derivation of kinetic equations, we take the limit As: finite.
612
F. HENIN
if we write that, for OJ~+ w,,: (3. IO)
V% - coy)--f
0.
After these preliminaries, let us now apply the formulae of I to our specific problem. Before we proceed, it might, however, be convenient to recall the meaning of the notations used in that paper and illustrate them for the diagrams of fig. 6.
e I
e
R
P
r
P’ I
@
V
I
(b)
(a)
Fig. 6. Diagrams illustrating the notations.
First of all, vertices are numbered 1, 2 . . . starting from the left. If a vertex happens to be the image of another one, say i, previously met, it is denoted by i. For instance, in diagram (a), the four vertices would be labelled (123 4) while in diagram (b), we have (12 2 1).Lines entering a vertex from the right are called ingoing lines while lines leaving it to the left are called outgoing lines. At each vertex, we associate a set ~2of three numbers, one for each line forming the vertex. For outgoing lines, these numbers are equal to - 1 if the arrow is towards the left, + 1 if the arrow is towards the right; if we intercharzge the words outgoing and &corn&, we m%stalso interchange left and right. For instance, for the third vertex in the diagrams of fig. 6 we have: (a) :
&fi =
(b)
&fi= 1, &/J’= -1,
1, &fl’ = -1,
E/p =
1;
&fi= -1.
The notations E~COZ or @/aNi) are abbreviations for a summation of the quantities ECO or .@/aN) for the three lines of the corresponding vertex i. For instance, for the third vertex of diagram a: ~3~3 = wp - CO&$ + Wk’while for diagram b: ~30s = cu, - o+* - Ok. Also we have: <&i&s. . . G> =
+
81 +
-52+
... +
~r-~
~2) . . .
IV(yN * + ~1 + ~2+ . . . + 8,. = N>,
(3.11)
i.e. a < > involving Y factors correspond to a well defined diagonal element of V*. Finally, all expressions in I contain summations & ..,. E81,+.8,.
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
613
These mean summations over all possible types of diagrams and, for each type, summations over all line indices and over all possible directions of the arrows on the lines. 4. Weak cozGpling afifiroximation. In the weak coupling approximation, the bare and quasi particle distribution functions are identical: po(N, t) = P(N, t).
(4.1)
They satisfy the kinetic equation:
am, t)
= -ii12&j(N,
at
(4.4
t),
where the hermitian operator --iv2 is given by (see I, eq. 5.4) : -ig1s = -2x
E E +r~r)
El
[
1-
exp (-.l
&)I.
(4.3)
From the diagram of fig. 4, we may easily construct the four diagrams of fig. 7.
&a@@ I (a) Fig. 7. S~!zd
orde:“,iagra.ms’:’
However, as ,Uand Yplay the same role, (a-d) and (b-c) are equivalent and we only have to consider a and c for instance. Now, c is the image of a and therefore we only have to write (4.3) for a, using the rules of section 3 and then add the same contribution with f 1 --f r 1. Thus, we have: -iv2 x
=
-~TC x Z x S(0, pv k: s=*l
IVI NP
-
E, Nv
+
x [l-exP{-+&
-
e, Nk
-&
my +
wk) x
&>
-
E,
Nv + E, Nk + E IV] N> x
-&)j].
(4.4
With (2.5)-(2.8), we obtain: -iq2
= -2~
E 2 9~ k
E lVpjyk12 6(wfl - WY- 08) X a=z!cl ’
‘Np, (1+8)/Z
‘N.,
(l-92
Nfi+_
I+&
x )
X[I--exp{-+&-&-&-)l].
(4.5)
614
F. HENIN
This operator describes transitions p - v with emission or absorption of one photon k. This process conserves the energy. The transition probability is the usual expression in the Born approximation. If we insert (4.5)into (4.2),perform the summation over E and let the displacement operators act on p”(N), we obtain the weak coupling kinetic equation :
x F({N}>4 - p”({N)‘> Nfi - 1, Nv + 1, Nk + 1,t)l + 6N,,,o%&W({N},
t) - i%N}', Nu +
1,Nv -
1,N/c -
1,t)]},
(4.6)
with y&4 = 2xS(ou -
% -
WC) IVfllrkls.
(4.7)
Eq. (4.6)is the familiar Pauli equation; terms proportional to p”((N}, t) are the familiar loss terms while the other terms are gain terms due to absorption of a photon and transition from a lower state v to an higher state p or emission of a photon and a transition from a state p to a less excited state v. 5. Kinetic
eqtiation
at order 14.
For the bare particle distribution aP0 __ = at
The post initial condition equation is l4) :
function,
-iA2$2p0
which
-
iA4($4
operator
+
Y%#z)
PO.
(5-l)
has to be used in the solution
pop) = (1 + Wi){po(O) Applying the dressing distribution function :
the kinetic equation isis) is) :
x-
+ c ap@)).
of this (5.2)
the quasiparticle
(5.3) The post initial condition
is here:
(5.4)
The quasiparticle
distribution
function obeys the kinetic equation : (5.5)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
615
where : -
iv2 = -i*s,
-
iv4
(5.6) After close inspection, (see ref. 18),
=
-@4-
+
[&,
+21+.
one finds out that the operator -iv4
= -i(Avs)
w -i(Avs)v
-
-iv4
$4
can be written, (5.7)
where (h~s)~ and (A174v represent frequency and vertex renormalizations of the operator ~2 while $j4 describes higher order scattering processes (i.e. processes involving two photons). With -i$s
= --ipz -
ils(Avs)w
-
iP(Aqs)v
(5.8)
i.e.
(5.9)
x(Nk++)[l-exp{-~(&-&-&)]]3 where
&qYk = 2xqb, 6,
tiv -
equation
(5.10)
= ofi + i2sAWi4,,
l~~,vklz + .IVplvklZ= the kinetic
.I,
WC)
(5.11)
i12A(2)I~~~4 2 9
(5.12)
-
(5.13)
becomes: ap -=
at
-iJ2Qsp
iPfj@.
The operator -i@s describes real one-photon processes; the electron makes a transition from the renormalized state with energy 6, to the renormalized state with energy G,,. The cross section involves the renormalized quantity LF. In the appendix, we give the detailed renormalizations and dressing operator.
expressions
of the second order
6. Fourth order scattering operator. In I, all fourth order diagrams have been split into three classes; in classes A and B, we have diagrams which can be reduced to a superposition of two second order diagrams and correspond respectively to the ordering (122i) and (2121).
616
F. HENIN
All other diagrams (fully connected diagrams) belong to the third class C. (For instance, in fig. 6, (a) is of class C while (b) is of class A). Their contributions to the scattering operator @4 have been shown to be (see I, 6. 5. and 1.6.6. *) :
-i(&
+a
(& >
=x&291
QlW
a x ( ?&y-x
X
[
l--q
+
E2O2)
x
a aE202 >
<[E2, 811-
a
a r
[62,
-~1~-62~
El]-> x
, 11
(6.1)
(6.2)
<[Es, &4]- [Ez, El]->
In each case, we have a delta function and a displacement operator which involve the particles participating to two successive elementary transitions. This operator therefore describes processes where two Photons play a role. In agreement with intuition, one finds out very easily that these processes are of three types: 1) double emission or absorption and a transition ,u c-) v in the atom; 4 Raman scattering i.e. emission (absorption) of one photon and absorption (emission) of another photon with a different energy and a transition ,u +-+vin the atom; 3) resonance scattering i.e. emission (absorption) of one photon and absorption (emission) of another photon with the same energy; the atom is left in its initial state. One indeed obtains:
I+& Nn,+yj--
- exp --E {
(
a
a
aN,,-aN,,-aNk,-aNk,
a
1
>[
-
a
II
+
*) There is no contribution of class C of the type (I. 6. 16). Indeed, such contributions involve products of three delta functions, one for each elementary step. Now, for all diagrams in this problem, it can be shown that such contributions, when one takes into account 8(a) 6(b) = 6(a + b) 6(b) always contain a delta function which does not involve photons, hence vanishes. A more detailed discussion is given elsewhere.
KINETIC
+
EQUATIONS
c z
PI ~a+;~1
x
Ix
klkr
Nk,+---
AND
x
LIGHT-MATTER
6Nup,(l-E)/2 x hdP*k, 4vs,,(1+ev2
e=+l
I+&
l--E
Nk,+T
2
617
INTERACTION
>[
I--
+ II I-& +2 x ( I+& >( (
>(
a - exp -_E ___-i
klkn
x
[I
8=&l
a
( 8%
ckllk,
-~
2%
Nk,
+
2
a
8%
Nk,
a
’
a&,
+
-y-
X
)
-e+(&
-
(6.3)
&)}]*
In agreement with our discussion of section 1, the transition probabilities C, 5, 5 which appear in the rhs represent contributions from simultaneozls two photon processes and are thus excess transition probabilities. The quantity OflllDpklke is the excess transition probability for simultaneous emission or absorption of two photons:
(6.6) and EPlk,lh,klis the excess transition probability for Raman scattering: k&k,
=
E$:,k,
+
(6.7)
l:kn,fiak,*
with
h,lrk,12
lb,l~k,12
+
(6.8)
618
F. BENIN
(6.9) where z’ means that ~1must also be excluded in terms where the arguments of the principal parts in the product would be equal, if one takes the delta function into account (this happens for instance in the product of the first terms in each ( }). Finally, ck, Ik2 is the excess transition probability for resonance scattering : ck,lk,
=
ckdka A+B
+
c&k,
(6.10)
with ri$,” =
--xa(Wk,
x
+
&vp,,l
-
wk,)
%$*,1)
x
(6.11)
and
where, again, x’ means that one must take ,us + ~1 in all terms where the arguments of the principal parts would be equal if ,us = ~1. As mentioned in section 1, these excess transition probabilities are not necessarily positive definite. We must require the positive definiteness only for processes which cannot occur in two steps in the lowest order approxi-
619
KINETIC EQUATIONS AND LIGHT-MATTER INTERACTION
mation, i.e. which cannot be described by I#$. This is the case if we irradiate the atom in a level ,Uwith a photon il such that cue is different from all level spacings and consider the probability [,,,b) of resonant scattering (i.e. a transition A --f k, with u)~ = O.Q, and no real transition in the atom). Then all energy denominators are nonvanishing ; if we use *)
0
1 B----t--, x
1
a
ax
0 1 B-+_-,
we easily verify that,
for that case, we may write:
hlk(iU)
c;kb)
x
=
c
I v*lr 1
S;(zBb)
&Jv;rk
+
=
x8bA
1 Oy -
W,.,-
Wk
1
x
9
-
-
uk)
(6.13)
X
x
~P,“k%
-
2
1 ofi -
u)y -
01
>
0.
(6,14)
Another important feature of the above results is linked with the contribution of the fully connected diagrams (class C). If we compare any of these contributions with the corresponding terms from classes A and B, we notice that the main difference is essentially that, for fully connected diagrams, we have a product of four different V factors while for classes A and B, we have a product of two factors IV12. The transition probability for one-photon processes is proportional to the square modulus of the corresponding Fourier coefficient of the potential. If we draw “spectroscopic pictures” of the kind given in fig. 8 to represent the effects of light matter interaction, we are tempted to consider twophoton processes as some kid of firoduct of one-fihoton processes. This is certainly true in the Born approximation where two-photon processes appear
I
1
a) one-photon emission or absorption
I
i
b) two-photon emission or absorption
L
-
7[I
F
c) Raman scattering
I
-
tl
d) Resonance scattering.
Fig. 8. Spectroscopic pictures of one- and two-photon processes *) Rememberthat all asymptotic expressions must be used to compute average values of observables, i.e. will involve integrals over the photon spectrum. Our choice of WA is such that x = 0 is not in the integration domain.
620
F. HENIN
as a sequence of two, well separated in time, one-photon processes, with energy conservation at each step. When higher order processes are kept, simultaneous processes play a role. Such processes require only a global energy conservation between the initial and final states; the intermediate state appears only virtually. If we think in these terms we might easily guess that the corresponding transition probability will be analogous to that of a product of two onephoton processes (i.e. proportional to (r/j2 IV(s) bzct zeritIzlIze firodtict of delta functions which a#pear in the Born approximation replaced by a si@e delta ficnction times some other propagator, takirtg into account the finite width of the intermediate state. Contributions of classes A and B clearly correspond to such a guess. Those of fully connected diagrams do not and are thus far less intuitive when we try to use such kind of arguments. They would be easily overlooked in the usual description whereas, when we discuss the evolution of the system in terms of the dynamics of correlations, all contributions appear on the same footing. We shall come back to this point in section 10, where we shall, on a simple model, discuss spontaneous decay of an excited state, taking into account two-photon processes. 7. Link
with stationary
state perturbation
theory. grozcnd state with no real
A pure case where the atom is in the perturbed photon present is given by:
(7.1)
!%A% 4 = %VO,lI~O%V.,Ol-J ~h%,O’
Such a distribution is a stationary solution of the kinetic equation. Now, for a stable state, it is well-known that stationary state perturbation theory in the Schrodinger formalism may be used without any difficulties. Therefore, it is interesting to make the connection between the two formalisms for this situation. Let us first go back to the bare particle distribution function p. The diagonal matrix elements are easily obtained by inversion of (5.3). We obtain :
(7.2) Using (A.3) and the notation
{Sk}
lpi
b,
{nk)>
=
(see also eq. (2.9))
pO(b,
{flkh
t) =
we
obtain as only nonvanishing
:
=
hvp,l
n %G,o v=W
y
h5,nr
PO(Pl~
Ipl
lo>
=
po(loa
t)
=
(7.3)
diagonal matrix elements: 2
1
t)*
1 -
A2 2
0.Q -
LDy-
o.)k >
1VO[vkl’s
(7.4)
KINETIC
EQUATIONS
lk JpI 44, lk>
To obtain
=
LIGHT-MATTER
00 -
we notice that,
621
INTERACTION
1
lit, t) = P
pop,,
these results,
AND
2
Wc)p - Wk >
within
I~OlP7c12.
the integration
(7.5)
domain
for
Wk we have: WO <
0s
+
Wk
(7.6)
and use eq. (6.13). In order to obtain the complete bare particle distribution function, we still have to find out the off diagonal matrix elements. They can be obtained through relation (1.8). The detailed calculation will be given elsewhereis). One finds as only nonvanishing off-diagonal matrix elements:
IpI
I,,
lk>
=
lk
IpI
= Pie,-I,,,-ik
If’1 l,,
lk,
lk’>
=
IO>*
=
lk,
lk’
IpI
IO>*
-
0,
-
cc)k
VOILck,
1 W,, -
1 WO -
00
W/J-
Lc)k-
X
Wk’
1
wv -
VvlUk
Wk’
VOlvk’
+
wo -
Pl,,-l&&)0,
00 -
WV -
(a&)
1 0,
(7.7)
=
= Pio,--lG,--lk,-ir, @)OI (#fit &)k, &)k’) = I2 XZ
1
((+)a, (+)fi> @)k) = A
wk
v”l8k’
VOlvk
,
=
1
z
vk
(7.8)
Wo -
Wy -
(7.9)
Ok
It is easily verified that those contributions are the only nonvanishing matrix elements of the following density matrix:
lG> 41
p =
(7.10)
with
+
’ 2 w,, -
ifi
-
VO[lck
IMP, 1 k> +
Wk
1 +P
+
+
c pvkk’ ‘00 -
Wo/1-
1 wk
-
Wk’
{
WO -
Wv -
ok’
VOlvk’
Il,jfik
+
1 00
-
Wv -
wk
VOlvk
“2,‘?o it ’ w I.4 00
WO
vv/;Ifik’
-
Ilcc,
iv-
wk
Ikr
lk*>
VOlvk
+
v&k
] 1,).
(7.11)
622
F. HENIN
It is of course a straightforward calculation to verify that /To> is identical with the perturbed ground state wave function obtained by ordinary perturbation calculus in the Schrijdinger formalism. Similarly one can verify that A(s)00 is also identical with the perturbed ground state energy obtained in the Schrodinger formalism. In the perturbed state representation, the density matrix, as expressed by eq. (7. IO), corresponds to a #ure case (one diagonal element equal to one, all other elements equal to zero). The canonical form (7. IO) can be obtained from the density matrix in the bare particle representation by means of a canonical transformation Q which will be given elsewhere. The situation is rather different if we consider a pure case where the atom, at a given time to, is in the p-th perturbed excited state (7.12)
p”(W), to) = KVJ rI %V”,OrI &*,O. k v+P First of all, this is no longer a stationary Of course, we can again, through the corresponding bare particle distribution be that, instead of a simple factor (~0 we shall have expressions of the form B(wu Similarly, in the off-diagonal we shall have {CP(ou -
oy -
c+ -
solution of the kinetic equation. same steps as above, find out the function. The main differences will ou - ok)-2 in the diagonal terms,
Cl&1 a/awk.
elements, instead of factors (wa c0k)-r -
xi8(0,
-
may-
W, -
wk)-1
0.~)).
As a result, we no longer have a factorization property analogous to eq. (7.10). Diagonalization of the density matrix shows that we are no longer dealing with a pure case in the bare particle representation. Dissi$ative processes are entirely responsible for the departwe from the pwe case. If we consider a pure case like eq. (7.1) or eq. [7.12), we expect to find that the average occupation number for the perturbed ground state (for the perturbed p-th excited state) is one while the average occupation number for the other states as well as the average occupation number for photons is zero. This is indeed the result we obtain if we define the average number
t),
(7.13)
where (RP> is thus the average occupation number for the ,u-th perturbed state;
KINETIC
The physical
EQUATIONS
meaning
instance the distribution
ANb
1NTERACTION
623
is quite clear if we consider
for
LIGHT-MATTER
of this difference
(7.1). There, we find:
which is a situation where the atom is in the perturbed no real photon present. On the other hand, we find:
=
1-
w(,
-
WV
-
Ok
A2 X k
CO,-) -
0~
-
Ok
>
=
A2 x
P
00 -
(7.16)
1v01”k12,
ivOlfik12,
(7.17)
il”Olak/2j
(7.18)
2
1
>
2
1 =
ground state with
2
1
i1s c vk
(7.15)
1, <&> = 0,
cc)fi-
cc)k>
which reflects the fact that the perturbed ground state is a superposition of the unperturbed ground state and of the excited states with virtual photons. 8. Equilibrium
distribtition.
One verifies easily that any function I, where HR is given by (1.19j (see also appendix), is an eigenfunction with eigenvalue zero of the scattering operator. Indeed, for the second order part, for instance, we take into account (see A.7 and A.8):
[l-exp{-(& -& - &)]]fWRj = = a
- ~(HR -
E(& -
c& -
wk)).
(8.1)
Then using the relation: VxjU(a)
-0
f
x)1 = 0
(8.2)
we easily obtain: -i&f(HR)
= 0.
The proof for the fourth order contribution
(8.3)
involves exactly the same steps
An Z-theorem can be established in the same sense as in the phonon case. As a special case, the canonical equilibrium distribution is given by: ,‘& = exP(--BHR)/C If we expand
Peq =
W)
exP(--HR),
(/5 = l/H).
(8.4)
(8.4) in a power series of r22,up to qrder 12, we have: exP(-_gHoj
E exp(-@Ha) WI
Z AH exp(--BHaj 1-
a2BAH + A28 ‘N;~lexp~_BHo~
> I
(8.5)
624
F.
HENIN
where
AH = HR - Ho. From this expression, at equilibrium :
we easily obtain
(8.6)
the mean number
= (1 -
exp(-j?cok))-l
of real photons
+ 0(L-3).
k
(8.7)
We obtain a Planck distribution for radiation. The average occupation number for the ,u-th state is given by: <%>,,
= {exp(++)/I:
exp(+W)
x (1 1
x + n2g Z A(2b(
d2BA(2)o,(<%>,,)
x
exp(--B41~= "
x [e4--B4/2j
= exp(-@Eq)/x
Y
exp(-@jzq),
(8.8)
where ‘;)E’J is given by (A.6) with Nk replaced by
emission.
We now want to discuss the decay of one excited state by spontaneous emission. Usually, one assumes that the atom is initially in the ,u-th unperturbed excited states) 19) 2s) and there are no photon present .e. one takes : PO@% t = ‘) = and compute
the probability
sN,.,l
rl: v*P
%‘,,I_l
n k
(9.1)
‘%%,!I~
to reach some final unperturbec
state.
However, when radiative corrections are taken into account we should think that the correct procedure is to discuss transitions between perturbed states. As mentioned by Heitler, ref. 20, p. 163: ((a bound electron is like a free electron, accompanied by its virtual field”. There are no reasons while we should work with dressed free electrons and with bare bound particles! Of course, if one adopts this point of view, one needs a definition of the perturbed states. This presents no difficulty in our formalism; the initial condition which describes a particle in its ,u-th perturbed state, in the absence of photons is simply:
p”(@h‘1 =
'Nu,l
n:
v*B
sNv,O
n.‘Nr,O’ k
(9.2)
Both conditions (9.1) and (9.2) will be considered below. As we shall see, in case (9.2) all line broadening is due to simultaneous twophoton processes. In case (9.1) we also have broadening due to emission
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
625
processes in the “dressing” period which follows the switching on of the interaction, i.e. before the asymptotic regime is reached. All this shows us that any satisfactory treatment of processes like spontaneous emission should take into account the excitation mechanism. This feature has already been stressed some years ago by H ei tler se)24) and LCvys5), although it is sometimes regarded as unimportant provided the life time of the excited state is long enoughs). In order to avoid too lengthy expressions, we restrict ourselves to a few level model (0, 1, 2, 3) and assume that the atom is initially in the second unperturbed state. With the notation (8.3), we thus have: all other
pe(ls, t = 0) = 1,
pa = 0.
(9.3)
The post initial condition for the quasiparticle distribution function is now (see 5.3) : 1
P(ls, 0) = 1 + L2 zg
W2 -
vk
p”(lv, lk, 0) = -P
0.l~ -
Ok
>
1
B
02
-
(9.4
Wy -
all other
“.lk
po = 0
In order to exhibit the dependence of the decay process on the initial conditions, rather than eq. (9.4), we shall take: _ p”(lZ,o)
=
1 +~2~/%k~
w2_;v_w,
& >
( j$lv,
1 k, 0) =
--‘/bk
9
1 ‘02 -
a
wv -
ok
aWk
IV21vk12,
(9.5)
with all &k’s of the same order of magnitude. All &k’s equal to one correspond to (9.1) while all /&k’s equal to zero correspond to the post initial condition (9.2) (in the bare particle representation, we have then both diagonal and off-diagonal elements different from zero). As all photons play the same role in this problem, we must have at any time t: P(lv, t) = O(l), p”(lv, lk, p(b’,
lkl,
t) =
@(L-3),
. ..> lk,, t) =
(9.6) o(L-3r).
These conditions insure that the atom reduced distribution function is independent of the volume of the box. If we take (9.6) into account, as well as the fact that (v&lvk12 is of order
626
F. HENIN
(i.e. Ttiirlvk of order L-3,crplrkk', &klrk', &k' of order L-6) and neglect terms of order L-3 in the equation for p”(ly, t), L-s@+l) in the equation for p”(l,, Ikl, . . . . Ik;, t), we obtain the much simpler set of equations: L-3
(9.7)
ap”(l@, lk, lk’, t)
at
= -12r$(l,,
lk, Ik’, t) +
+ A2 x {j%yltikp(lv> lk’j t) + $&k’ p”(h lk> t)} + + A4 ;
-+‘vlckk’
+
G+k’k}
(9.9)
p”(lv> t),
Y
@(l/h lk, lk’, lk”, t) at
= -~2r$(lu,
+
A2c
{j%lbk p”(l”,
+
?vI,In
p”(b,
1 k,
1 k’,
lk> lk’, lk@, t) +
1 k”‘, t, +
jhk’
1k’r t)} + A4 c {(‘hlakk’ Y
~(1%
lk,
lk”,
t) +
+ &+k’k) p( 1%‘~ 1k”,
t) +
+ (%(bkk” + cvlak”k) p”(Iv, 1k’, t) + + (ovl,uk’k” + @vlbk’k’) p( 1%~1k, t)},
(9.10)
etc., with (9.11) vk
(9.12) (9.13)
Only spontaneous emission processes are kept in the evolution equations. This is a consequence of the fact that, in the absence of initial finite electromagnetic energy density, absorption and induced emission processes are of order L-3 compared to spontaneous emission. The first term in the rhs of each equation describes processes whereby the atom leaves the state ,u and emits one (7) or two (u) photons. The other terms describe the fact that a state with at least one (two) photons may be reached from another state with emission of one (two) photons. The set of equations (9.7) etc. is easily solved. For our four level model, with the initial condition (9.5), we obtain as only nonvanishing matrix
KINETIC
EQUATIONS
AND
LIGHT-MATTER
627
INTERACTION
elements (we neglect A4): p(l2, t) = exp(-A2F2t)
p(13,
1 k, t) =
x
A2 exp(-12r3t)
I
b3k (02
-
03
(9.15)
1v213k/2,
mk)2
-
I 1 +
A2
2 /f&k’ 9 k’
c v=O,1,3
--A2expk-2ht) P(lo9lk,t) x
I
1 +
==&-
(1
o2
-
_
WV -
_
_
bl
Ok’
ok-
&
Iv21vk’12
>
p,
1
(9.16)
/v211k12,
>
exp(-n2r2t)}p210k
I2 vcg , , 3 F
--‘fiOkg
o2
,8lk 9
Lo2 -
bk’
B
(
_-
;, -
ok
p”(12,lk, lk!, t) = “2+lr-m
X
w2 _
m; _
ok ,
&
iv210k/2,
&
Iv2,vk’,2
-
>
(9.17)
>
3 {exp(-A2P$)
- exp(-A2r2t)}
bkjh2k
w’,
X
2
x
+
i
(m2 _
_ __ ok)2
iv213ki2
id. with permutation of k and k’ , 1 3 {exp(-A2f
p(ll, lk, lkt, t) = iz2 ,‘,-
+
(9.18)
St) - exp(- ATIt)} x
1
x
+
I
ihd311k’
-(02_
w’, _
okj2
iv213k12
+
id. with permutation of k and k’ + 1 ,
{exp(--2rlt)
-
exp(--2r2t)){~211kk’
f
qlk’k>,
(9.19)
628
F. HENIN
exp(--ilTzt)
P(lo, Ifi.,Iv, q =
exp(-AsPIt) +
{ (F2 --l)rz
x
{Y211k~llo~n+ I%lk'~llOk}
rl-r2)fl
1 + {
a
1
XB W2
p(11, lk,
lk’,
>
aOk
-wor-wk
Ik”rt)
=
+
1 x
IX /f&k" x i12 z v=o,1,3 k"
(v2\vkj2
+
id.
with permutation of k and k’ , (9.20)
exp(-A2rst)
A2
r:r,
(r3 --Pz)(F3
exp(-A2F2t)
-F1)
+
+
(r2--3)(p241)
exp(-A2Plt) +
(&--3)(&-~2)
1 (02
-
W3 -
cOk”)2
/V213k”(2
+ id. with all permutations of k, k’, k” , p(lOt Ikr lk’,
lklt
+
(9.21)
t) =
1 (02
+
-
cc)3 -
Wk’)2
1v213k”/2
+
id. with all permutations of k, k’, k” +
+12
X {~llOkC%,Wk”-i- ?211&(TlJOk’E” + ti permutations Of k, k’, k”}, (9.22)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
629
INTERACTION
p(10,lkr lk’, lk”, lk”, t) = exp( - A2Fst)
exp(- A2f st)
-
i12
(r3
-
rZ)(r3
-
rl)
r3
+
(r2
-
rl)(rz
-
r3)
r2
+
’
(9.23)
+ all permutations of k, k’, K”, k” , I
In these expressions, we have taken into account the fact that 8(6, -
-
%lfik
a
1
B OV
flvlfikk’
-
rnfi
-
6(ov
> - hk
Ok
-
-
O/J -
for
Ok) --f 0
8, -
1 +
ok
-
-
(coy -
wk’)
+
W/J -
for ’ <
mk)2
for
0
(9.24)
v < /&
”
(9.25) (9.26)
V < fA.
To obtain these results, we assume that the perturbation is sufficiently small to avoid crossing of levels (i.e. for instance ~2 > ~3). Such an assumption is of course unavoidable in a theory based on perturbation expansion. Moreover, in eq. (9.25), we take into account the fact that all matrix elements will always be used to compute average quantities, hence will involve a summation over K and we shall be able to perform integrations by parts. (This has already been taken into account in the evaluation of the asymptotic propagators.) In the long-time limit, t > r,i,
r,i,
r,i,
(9.27)
the decay is completed and we obtain (we only write nonvanishing matrix elements) : 1 p(lO,
lk,
8 +
a)
=
-
p210k
x
r2
a
1
x
1 + A2 I
x
v=o,1,3
&t?vk’~ k
cc)2-
WY-
-
cOk’> ihks
Iv21vkk’12
-
(9.28)
630
F.
p(lO,
lk> lk’,t
-+=‘)
=
HENIN
@Zllk~llOk’
-‘-
+
~2(lk’$%lOk}
x
r1r2
+
-
+
A2 __
(c210kk’
+
a2lOk’k)
-
~2~~,3~{~~k~~,0k~~(.2_~~_.k)~~~2~~k~2+
id. with permutation
of k, k’ ,
1 = A2v=l x a ryr3
p”(la, lk, lk’, lk”,t -too)
(9.29)
&‘ldW~:‘~~k”
x
1 -
x
(m2
-
W3
-
IV213k”i2
+
A2 F~-
+
id. with all permutations
@I), lk, lk’, lk”, I&‘“, t +W)
+
Wk”)2
(j%IOk’JZllk’k”
+
?2llk~llOk’k”)
+
of k, k’, k”,
(9.30)
1
= a2
Y”1~0k~211k’~3~2kw
x
rlrZr3
Iv213k"i2 + + id. with all permutations
of k, k’, k”, k”‘.
(9.31)
The atom is in the ground state. The maximum number of photons is four. At first sight, this latter result is not intuitive. However, this is easily explained if we look at the initial condition. Then, we notice that we have some probability that the atom is in the third perturbed excited state with one photon. Decay of this state, in the Born approximation, happens through emission of one, two, three photons. We notice that if we take as our initial condition that the atom is in the second perturbed excited state (i.e. all &k’s equal to zero), the maximum number of photons is three. In fact, the only contributions which remain are proportional to j+~Ok (straight decay to the ground state by emission of one photon), 7$ikj7ilOk* (cascade decay to the ground state with emission of one photon at each step). These are the only contributions in the Born approximation. At order 12, we also have terms proportional to Usiakk’ (straight decay to the ground state with simultaneous emission of two photons) and jj21ikUilak’k” or j?llOkUslik’k”
KINETIC
EQUATIONS
AND
LIGHT-MATTER
631
INTERACTION
(cascade decay to the ground state with emission of one photon at one step and of two simultaneous photons at the other step). From the results for the quasiparticle distribution function, we may easily, by inversion of (5.3), find out the nonvanishing diagonal matrix elements in the bare particle representation. For long times, we obtain: I
1k,
pO(lO,
t -+
00)
=
-
$hlOk
x
r2
-
1
A2.=F2,3 z (cDo -
-
WV
-
Ok
>
1
--A29 02
-
00
-
‘Vo~vk”2 -
wk>)2
d$
tv2lOkt2
lk,
lk’,
f +
m)
=
1
~2--?210k’ r2
(a0
-
id. with permutations of R, k’
+
19.32)
BOkr
1 pO(lv,
lJ721Vk’12 -
&
wk’ >
WV -
mk12
ivOlvk12
(v = 1, 2, 3),
(9.33) 1
f’O(lO,
lk,
Ik’>t-tCQ)
=
-A2
r, v=1,3
j&
+
j%,Ok/%k
-._
9 04
V
-
WV
-
X Wk’
>
a
(l/r21vk’12 + id. with permutations of k, k’ + aOkc
x +
&
x
bhky”llOk
I
+
f2llk’$hlOk)
x x /%k" @ t’=O,1,3 k”
1
1 + a2
-
A2v=i52,3
+
A2 &
i?
b2IOkk’
(WO -
+
m;
-
x
c02 -
C%v -
wk”
&
lk,
X
lk’,
lk”,t+W)
1vOlvk”(2
+
-
Wke)2
(9.34)
02(Ok’k),
1
1 pO(lv,
l~!Zlvk")2
>
=a2--__ FIT2
Y”2llkhW
(0.h) -
id. with all permutations of k, k’, k”
coy -
Ok”)2
X
(v = 1) 2, 3),
(9.35)
632
F. HENIN
PO(l0,lk, lk’, 1k”,t+co)
1
= 12 x
__
v=l,2
~~lOk”~3(Vk’/93k
rpr3
x
1 x
(02
+ a2-
-
w3 -
* r1r2
IV213kj2
Ok)2
(02Jlkk’jhlOk”
+
+
?72llkUllOk’k”)
+
+ id. with all permutations of k, k’, k”,
po(l0, It, lk’, lk”, lk”, t -+m) = a2
x
P312k”
(w2
-
r,;2r3
w:
-
~3k”jhlOk~2)lk’
Ok”)2
(9.34)
x
Iv213k”/2 +
+ id. with all permutations of k, k’, k”, k”‘,
(9.37)
We notice that, even for t +CQ, we have components where the atom is in the excited state. This corresponds to the physical fact that the perturbed ground state is a mixture of the unperturbed ground state and excited states. Besides these diagonal nonvanishing matrix elements, we also have off diagonal elements describing correlations between the bare atomic levels and the electromagnetic field. Let us now compute the average number of real photons as defined by (7.13) emitted during the asymptotic evolution. In other words, we compute the difference between the number of photons at t --co and the number of photons already present in the post initial condition
x +
-
I
1 + a2
3L2&
=
-&-
c xk’
lr=o,1,3
{?2lOk
+
?211k)
x
1 &k’ i?
5 p312k’ 5 B3k”
(w2
02
_
-
cr),~ -
w;
_
wk’
Ok,,)2
&
I~21rckd2
> tv213k"j2
+ 3
+
KINETIC
r2r3
A2 ‘c /%k’ 9 k
AND
LIGHT-MATTER
633
INTERACTION
1
z 7211k’ k’k”
+ P---- * -
EQUATIONS
7312k”
5
p3k”
1 )
( 02-ml-Wk’
(co2 -
&
’
03
-
WkW)2
Iv211k’j2
+
Iv213k112
12&-k:_
-
u2,lk’k”
+
1 +
a2 &
5
;83k”
7311k’
-
(w2 -
o3 _
okn)2
‘v213k”12
1
+
1 +
a2 &
{73,Ok
+
A2 &
z
+
A2 &
+
p311k +
(02lOkk’
5?2,lk”
-
<2Gfkh’.I.
b2lOk’k
f$
In the weak coupling
+
p312k} F p3k’
+
(UllOkk
g2Ilkk’
+
(W2 -
+
t03 -
@2llk’k)
Wkp)2
1v213k’12
+
+
(9.38)
~l[Ok’k).
case, this expression reduces to: =
=
+-
=
(72,Ok +
Y211k) +
=
-!-
YlY2
)‘l,Ok
5
7’21lk’.
(9.39)
We have three sharp lines at the frequencies 02 - cc)a, ws - 01 and o1 - ~0. They correspond to the three spectroscopic lines one expects in such a situation. When we take higher order effects into account, the situation is somewhat more complicated. Let us first consider the initial situation where the atom is in the second perturbed excited state (all pvk’s equal to zero). Then to the sharp contributions of the weak coupling case, displaced by the second order level shifts, we have to add a continuous background due to the simultaneous two-photon transitions; broadening of the lines is essentially due to this. This kind of effect will be discussed more thoroughly in the next section on a simplified model. If we start from the unperturbed excited state (all &‘s equal to one), besides these effects, we have to add some contributions due to transitions from higher excited states (here the third excited level), corresponding to the fact that the second unperturbed level is a mixture of all levels. Also, the intensity of the various lines is modified. This picture, where all line broadening is obtained through two-photon processes might appear in contradiction with other results. For instance, in a two-level model, with a single matrix element different from zero, all u contributions vanish and the line remains sharp. However, we notice that if we compute the number of photons at t +oo without subtracting the contribution of photons emitted before the asymptotic regime is reached,
634
F. HENIN
we have to add to (9.4) :
= -ii2
2
,&‘ky
v=o,1,3
~-;-_y;
&
(
k
>
(9.40)
1vWd2.
This introduces a continuous background, mostly important around the lines involving the transitions 2-1, 2-O. This effect depends on the choice of the initial condition. It is a similar effect which gives rise to the line shape in the two-level model when one starts from the unperturbed excited state. This will be further discussed in another paper-is). All this emphasizes the importance of the initial condition, i.e. of the excitation mechanism in the description. From that point of view, a rather clear problem is that of resonance scattering. This will be discussed in another paper is), using the Wigner-Weisskopf model. Then simultaneous absorption and emission will be possible (the c operator in (6.3)) and will allow scattering of nonresonant light. 10. S$ontaneozls emission - simplified model. We shall simplify furthermore the model used in the previous section. We assume : V211k>
VllOk
=b 0,
i all other vfljVk = 0.
(10.1)
(k arbitrary)
In other words, the atom may decay only step by step. We also forget about possible contributions from higher excited states. Moreover, we put equal to zero all matrix elements which describe only virtual transitions (v#lvk = 0 if ,u < v). With this restriction, the unperturbed ground state is identical to the perturbed ground state as soon as we neglect absorption processes (see eq. 7.3), This model is a generalization of the two-level Wigner-Weisskopf modelse). Another way to describe it is to say that the only possible elementary Feynman
graphs are the graphs of fig. 9.
Fig. 9. Feynman
As a consequence
graphs for the particular
of these restrictions
problem
of section
on the potential
10.
we have, see eqs.
(5.10) and (6.4), PalOk= 0,
Y”311k =
0,
f312k
=
0,
7210k
=
0,
02llkk’
=
0,
OllOkk’
i(a210kkr
+
~210k’k)>
=
0.
(10.2)
Hence, rl= r2 r3
pl=
xj%IOk, k
= =
72 0.
+
A202
=
x y211k + k
2' x kk’
(10.3)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
635
~__
We easily verify that, in this case: 1
ts =ws+PCP
/Vzllr=
-
Wl
-
Wk
-
Lc)k)
1
91 =oi+1acg f_zO =
02
k
C01 -
k
00
-
lJ7211k12*
(10.4)
IT"llOkl2,
(10.5) (10.6)
WO,
Iv2llki 2{1+~2~~(~_-Lljl_Wk,)&
iv211k'i2+
+g (
1
-2i12 z B
Co2 -
k’
01
-
1
B-
Wk’
)(
v21lkv;,lk
'Vl(Ok'v;,0k,
aO+Wk'--ml
(10.7)
In all expressions (10.4) to (10.8), the dependence on the photon state has been neglected; indeed, when there is only a finite number of photons, this is of order L--3. Also, we have (see 6.4) : 0210kk’ =
XB
=
o$-kf,
+
2X8(WJ -
O&,kk’
Wa -
Wk -
1
cc)2--~~--~k
I(
>(
1
+y (
=
-
Wk’) x
a
aOk
-
>
i1"211ki2Ivl10kd2 + )
1
B
Us-Ol-Ok
&
( Ol+Ok'-W2
The post initial condition
>
‘V211kV;,lk’
h1Ok’V;,ijk
becomes:
a p(l2,o) =
1 +
~2~/%k~
r;zk
v211k(2,
-
1 =
(10.10)
) aOk
( p"(ll,lk,o)
(10.9)
* I
+lkg
& W2-Wl-Cok
/T"211k12.
(10.11)
)
Let us now write down the solution of the kinetic equation for the quasiparticle distribution function. The exponential decay of the perturbed excited state is given by P(ls,t)
=
exp(-A2r2t)
1 1 + A2 IZ&ky k
co2-w1-ok
& >
~~211k~2
.
(10.12)
636
F. HENIN
Also p(11,1k,t)=
2 {exp(-12rst)
r
-
exp( - n2rrt)} 721lk x
1 1 x
1 + n2z
/9lk'P
k’
1
;k; (
~2-wl-Ok
lV2,1kf12)-
> 1
-
exp(-_‘rlt)
12filk
9
/v2/1k/2
2;
e&J-~r---wk
(
(10.13)
>
gives us the probability that the atom spends some time in the intermediate state after emission of one photon at the frequency (62 - ~1) (first term) plus the exponential decay of that part which is already present in the post initial condition (see eq. 10.11). The probability to reach the ground state after emission of two photons is given by: exp( - J2rst) p(lO, lk,
x
Ik’,
t)
=
i
(r2
-
rl)
r2
+
(PI
-
r2)
r1
+
r:r,
x
{~2jlkj%lOk' + ~2llk'Y"llOk) x
1
1 +;22C/&“g W2 -
k”
+ id. with permutation +
exp(-_2rrt)
A2 &
(1
-
WI-
Ok”
&
11/‘211k”)2
+
>
of k, K’ +
eXp(--‘r2t)
}{U2IOkk
+
(10.14)
~ZIOk’k}.
For times t much longer than the lifetimes
of the two excited states, we have :
p(lO> lk, lk’,
Y"2llk'Y"llOk) x
t-too)
=
&
{?2llk~llOk’
+
1 & m2-m()1-t%kN
+ id. with permutation
+
Aa&
(~2jOkk’
+
of k, k’
u2lOk’k).
Iv211kd2 -
>
+ (10.15)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
637
INTERACTION
Let us now discuss somewhat more thoroughly the role of two-photon processes. First of all, we examine their contribution to the lifetime. For this purpose, we use the dipolar approximation and introduce a cutoff at high frequencies (much higher than the level spacing) : V 21lk:
=
W2ll(e)
__ Sk
q(K
-
(10.16)
4,
where Wslr contains the polarization and wave vector direction dependence but is independent of the wavelength k and where q(x) is the Heaviside step function. One easily verifies that one has: 4’”
= 4 X (&$z kk 00 =
xz
AA’s
co
dwfi, 6(w2 - w. - wk - ok’) x
dmk
0
X
+ $$k&) =
s
0
JIW2ded12
IWl10(er)12
\
-__
B
W2-01-Ok
-
- -
a awkl > a
+
X
aWk
(10.17)
x WkWk&J- Wk)q(Q - Ok’),
where the sums over 1, 1 indicate summations over all polarizations and wave vector directions. Now, we have: 00 co
ss ss do
dw’ S(ws - w. - w - w’) B
0 0 x 7&Q - 0) &2 - w’) = 00 m =
dw
0
1
dw’F(ws-woo-w-w’)9
ws-01-w
0
M-Q - 4 Il(Q - 0’) x
x (0’ - w) - ww’[S(Q - 0) ?#2 - 0’) - ?j(Q - w) S(l2 - co’)]} = OQ
=
1 ws-wr-w
1 ( dwB
>
q(w2
-
00
-
w)
x
0 x {(wz - 00 - 2w) 7#2 - w) V#2 - 0’) - w(w2 - WI)- 0) x X P(Q - 4 Il(Q + w - 0s + 00) - ?&I2- 0) x x 6(Q + w -
w2 +
wo)]}.
(10.18)
638
F. HENIN
If we take into account
the fact that Q>
w2 -
(10.19)
wo,
this becomes : wa--wa
s(
1
~ (ws ws - co1 - 0 )
dwB
0
wa - 2w) =
si
=
dm 2 + (2w1-
w,, -
1
w2) z?
02 - co1 - 0J >i
0
,
(10.20)
and we obtain: G?+B = 2~ C IW211 (eA)12 lW1l0 (e,t,)j2 x
LA’
X
i
2(co2 -
u)o) + (2~x1 -
00 -
cog) In (, 1; :
(10.21)
Ii)}.
In a similar way, we obtain:
e = ikS b$Okk, + 4fOk’k) = =
-2~
C W211(e~) Wi,, (en,)w;,,(eA) Wl10 (el>)x 1.X'
ss 03
Co
X
cr)s-W1-Wk
0
0
1
1
dwk dwk’ 8(U&Mi.&)-C0k--u)k~) 9
ws--w1-~k’
X
x mkWk’?‘/(G- Wk)Tj(.n - wk’) =
J+?&,(w)wT,,(e)
= -2~ C W2ll(e,t)
x
Wlfo(w)
Both terms have the same order of magnitz&e. The main difference is the angular dependence. This will of course depend on the choice of the three levels. The importance of the contribution of fully connected diagrams appears in a similar way if we are interested in the line shape. For our present model (9.40) becomes :
-
=
1 1 +
+j&~l,Okjj2 12
~2~/hk'~
+ i.'$
~2--~1-cc)k'
z (a2lOkk' +
>
('2lOk'k).
(10.23)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
639
INTERACTION
The first two terms give us sharp contributions at the frequencies ~2 - i3r and ~1 - ~0. Let us now consider with some more details the two-photon contributions to the line shape. With the same model as above, we have:
CO2-Wl-wk 1
~
XB
-
__
COkWk’Tj(G
-
Wk) ?j'(Q-
ok') =
02-01-0.)k
2x1s = 7+2
-
Ul -
lW211(el)12 IWlio(eA,)12 X
Wk) F
1 XB (
x
w2 - ml-
+
Wk
x
IWlj0(ed129
W2ded12
(10.24)
(m2--Coo--mk),
while J2r,
1
5 @lOkV + GlOk’k) = ou
= -,-F
x
X9
2x1s
40(w)
?,o(e3
J%0(w)
> -
+
WO -
W2llh)
w:,,(e)
1
00
Ql(ed
-
(
~2-cOl-Wk’
JfG,,(d
~s-Wr--Ok
w;,,(w)
__
WG1(en’)
wljo(e,)}
Wk’“l;l(g-
wk
Wk) q(G
w&(eAr)
w;,,(w)
w;,O(eA)
Wlfo(e,)}
=
02
-
Wk,)
=
x
>
of both contributions -
(10.25)
Wk(W‘Z--O-Wk).
cc)l-me--w]c
‘Ok
x
x
1
Let us now consider the importance
x
)
Wk) ~(W2Il(eA)
1 X9
~k’){W2Il(eR)
ok -
B
( U)2--l-Wk
v(“2
W2 -
+
1
2xas = 7 X
dwk’a(
s
01
f
CL
for (10.26)
F. HENIN
640
with cc small, but different from zero. If we forget the angular dependence, we find that the dominant contribution of &+n is: &+B
-
(10.27)
+ (201 - IBe- ws),
while the dominant contribution to crCis: & _ L o!
(w2 -
-
W)bl
wo)
2wi - we - 0s
(10.28)
*
The ratio of both contributions to the broadening of the (~2 - &) line depends on the level spacings and the matrix elements of (e1.x). A similar result holds for the broadening of the (61 - 60) line. When we vary the frequency, one or the other of the contributions will become dominant. On the extreme boundary of the line, at ok = 0s - LOO the C contribution vanishes while at cok = (02 - 0,-J/2 the (A + B) contribution vanishes. Let us now go back to the long time quasiparticle distribution function as given by (10.15) and consider the case of the post initial condition (9.2) ; in other words, we assume that the atom is initially in the physical second excited state and take all /l’s equal to zero. Then, using (10.9)) we may write : = - 27c S(G, -Go-Wok--Ok’)
p(10, lk, lk’, t So)
f
xv1
1
9 (
(
-
)(
W2-01-Ok
1
+ izT,B
x
rlr2
a
- & - -a$
>( aOk +
+
>
a
hIOk’V;,Ok
1~2llkl2 l~llOk~l2
aWk
__
0.&---ml-Wk’
~2rl(v211kv;jlks
-
l~2IlVl2 I~lIOA2
-
1
c-c.)
9
(~2-~l-~~~(W2-W:-~k~}.
(10.29) One can verify that this is the limit for z + 0 of the following expression (here again, we take account of the fact that all expressions will be used to evaluate average values of observable% i.e., will involve integrations over k, k’) :
2xi p(lO,
lk,
Ik’,
t +=‘)
=
rlr2
8(&3
-
GO--Wk-Ok’)
1
1
Xh
+ 5-O
,?+&-&--k
x
-1
l~211k12 IVl10k't2 +
z-&+f%l+~k
KINETIC
EQUATIONS
AND
LIGHT-MATTER
1
1 +
+
z+82-c51--tok
[
ii1Tl
-_
x
[
I~Zllkl2
ii12f
--
2
lV2llk’l2
z-f52+8l+ok’
+
IVllOk’l2
(Z --2+
w+
1
-
y
x
[ z+W2-O,l--cc)k
Wk)2
1 x
l~llOk’l2
1
1
(z + w2 - 01-
IV211k’l2
-
-
1 [
I~llOkl2
1
1
(Z+~z--l--k)2
x
Ok<)2+‘(Z-co2+Wl+Wk~)2
x
-
(~2llk~~,lk’~llOk’~~,~k
1
+
C.C.)
o2
w; _
2col
x
+
Z-mZ+wl+wk
11
1
+
z+CO2--~Cc)l--~k’
+
1
-
1
+
I_
1
2
641
INTERACTION
z--w2+ol+wk’
(10.30)
’
which by addition of some terms of order A4and higher can be written: 2xi
p( 10,
1 k, 1 k’, t -+m)
=
8(92
-
50
1
x lim Z-+0 i[
--
r1r2
Z +
&2 -
Ok +
&l -
i1TI
-
ok
+-----
ia2rl/2
Z -
-
Wk’)
x
1 02
+
81+
1
2
Cok +
1 -
Z+W2-Wl-Wk+ij22r1/2
Z-0J2+Ol+Ok+i~2rl/2
iA2rl/2
1 X
+
1 +
Z+wz--l
-
Z -
x
[~2llkV~,lk’Vr,OkVllOk’
-
Ok’
+
iA2rl/2
1 02
+
01
+
Wk’ +
ii22r1/2
+
c.c.1
-
1
cc)2 +
.
1 COO-
2~01
X
(10.31)
642
F. HENIN
Performing
the limit z + 0, we obtain:
P(lO> lk, l/c’> t -=) =_--
= --
2x12 r2
~2llk~liOk’ b2
-
G1 -
ok
+
-7&2 -
bl
-
Wk’ +
t
2
~2llk’~llOk +-
iLsri/2
iA2ri/2
8(82
-
Go -
u)k -
Lc)k’) =
(10.32) This expression indicates the link with the usual Green function technique and Feynman graphs. Indeed, the T matrix element for the double emission in this model is given by the superposition of the two Feynman graphs of fig. 10.
(a)
(b)
Fig. 10. Feynman graphs for double emission.
In the Born approximation, v2llk
we have:
VllOk'
(a): _______
W2--w1-wk
The expression in (10.32) takes into account the fact that we start from the second perturbed level (hence ~2 and not 04. It also takes into account renormalization of vertices. The renormalization of the energy of the intermediate level and the line width i1sri/2 correspond to the introduction of the level shift operator Rr(z). In the usual technique*), this procedure avoids the divergence of the denominators in the Born approximation. Also, one usually does the approximation (10.33)
RI E ;12Ai - ii1sr1/2.
(Notice, however, that the r’s in eq. (10.32) involve two photon corrections). This comparison shows us that one must be careful in the T-matrix approach. Indeed, as we have seen the dominant contributions come from either Ok = ij, and wk’ s &j2 -
Cui
and
mc)f’= bi -
&a
or
W,$= 61 -
dl. Now, when one considers the Feynman
tia
graphs alone,
KINETIC
EQUATIONS
AND
LIGHT-MATTER
643
INTERACTION
one is tempted in the first case to keep only diagram (a) and in the second case (b) and add the cross sections For both channels, one then obtains:
I~2ilk12
.~___ x
(62
-
81-
l~l10k~12 Ok)2
+
24r34
+
id. with k, k’ interchanged
.
(10.33)
This corresponds to the guess one might make if one uses the spectroscopic picture discussed in section 9. Such an expression is analogous to that derived by Goldhaber and Watsonls) (see eq. (3.10) of that paper; the main difference seems to be the replacement of the delta function of eq. (10.33) by a broadened function). From the derivation of eq. (10.32), it is quite clear that such an approximation amounts to the neglect of the contribution of fully connected diagrams (UC). However, a closer examination shows us that the square terms contain both the Born approximatizm and, as corrections to it, the twophoton contribution uA+~ . If one assumes that the coupling constant is small (what one is always forced to do, at one stage or another, for instance, in the evaluation of ri), the above discussion shows us that as soon as we want to keep corrections to the Born approximation, we have to keep also the fully connected diagrams i.e. the cross terms in eq. (10.32). In this sense eq. (10.33) is strictly valid only in the Born approximation. Another interesting feature of eq. (10.32) is that this expression has been obtained for the quasiparticle distribution function at t -+ 00, starting with the initial condition (9.2). In other words, this expression gives us the cross section for the transition from the $hysical second excited state to the physica final state (perturbed ground state and two real photons). If we start from the unperturbed excited state, we have to add the terms proportional to @rk and pik, (with ,%l~ = &r, = 1 in the right hand side of (10.15). From the post initial condition (9.3), taking into account eq. (10.1) we easily see that those terms describe the decay, in the Born approximation, of the post initial component in the first perturbed excited state. (Because of our restrictions on the matrix elements of the potential, the first excited state is the only one which is mixed with the second excited state in the definition of perturbed states). Acknowledgements. I am specially grateful to Prof. I. Prigogine for many helpful and interesting discussions. I also appreciated very much his help for the preparation of this manuscript. This work is part of a larger program developed at Brussels. Collaboration with C. George, P. Resibois, F. Mayne and M. de Haan has been very fruitful.
644
F. HENIN
APPENDIX
-
1. Dressing
DRESSING
OPERATOR,
ENERGY
AND
VERTEX
operator.
The operator #i differs from the operator -i#s pagator : -i&
RENORMALIZATIONS
-1 -+*i : lim z-0 (Z$_Wfl--Or--#’
-i
: lim
only through the pro-
z-to z+~g-oY--u)k
(A-1)
When one takes into account the fact that the propagators appear in integrals over WR,one easily shows that this amounts to: 1 --x8(ofl-
coy -
Wk) +B
(
COB--c&--c#k
a ---> ) acOk
(A4
provided this operator acts on a function which vanishes at the boundaries of the integration domain. This is the case if we include in I/;livka suitable cutoff function at high energies. Therefore, we have:
*i=2z: pv
z
k a=%1
B
wfi -
l
oy -
Wk
>&
lvi4d2 ‘&,~l+s~,~ x (A.3)
x,,,,-.,,,~k+~)[l-eXP{-~(~-~-~)}].
We notice that this operator displaces the variables N only by 0 or + 1 or - 1. Therefore, as the only relevant values of the N to be considered in pa(N) are integers (these are diagonal elements in the ordinary representation), this will be true also for p”(N). A given state Nfi = 1, {Nk: = %k) in the quasiparticle representation is connected with the same bare particle state but also with bare particle state where the atom is in any other state and where there is one more or one less (if possible) photon. 2. Energy renormalization. The operator (Ar~7.a)~ is given in the phonon case by the operator IL74(see I. A.3.4) *). The only diagrams which contribute are those of class A.
x [l -exp(-El&)]. l)
This operator is also given in (I. 6.7) but with a misprint (2 instead of - 1).
(A.4)
KINETIC
When this
EQUATIONS
AND LIGHT-MATTER
INTERACTION
645
is written explicitly for all the relevant diagrams one finds:
(Ay2)m = -2x
‘c
x
uvk a=+1 x
6’(qt -
l~m12
oy -
CO~)[A%~ -
hv#,(l+E)/2 h,
A(2)co,] x
(l-.3)/2
(A-5)
X[l--exp(-e(&-&-Y&J}], with 1 lVplvk12 (Nk Ofi -
WV -
+
1) +
Lc)k
+g
oc _
iv+,,>
ll"r,ak12Nk}.
(A.6)
If we compare this with (4.5), we easily see that (A.5) indeed introduces frequency renormalization in the second order scattering operator. There is another way to obtain (A.6). Indeed, it has been shownrs) that the renormalized energies are also given by:
where the renormalized energy operator is given by (1.19). (Notice that are defined for integer values of the N’s only). HR({N}) as well as HO(W)) In the present case, we easily obtain, using (A.3) : HR({N})
=
ZNcc% P
+
~NN~WC k
-
1 -PXIZ
x9 pv k e=fl
Lop -
x %7u,~1-4,2 Nk + +)
0,s -
Ok >
+$
-
l~fllvki2%7,,,(1+s)/2 x
&
WV-
Wk).
Now: m
Co
Jdxg(-&) 0
$(~)(a
- x) = - j~$-$+(~)
(A.9)
0
when f(x) vanishes at the boundaries of the integration domain. Also, taking into account eq. (2.1 l), we may write for integer values of N,:
646
F. HENIN
Therefore, H&N>)
= X N,w, P
Upon derivation
+ C Nkok + k
with respect to NW, we obtain: w1 = wfi + FA(s)oG
with A(a),, given by (A.6). Upon derivation with respect
(A. 12)
to Nk, we obtain:
&,lc = Cok= 8(L-3). Photon frequency renormalization is negligible in the volume. The renormalized energy levels depend on the state of field. As a special case, in vacuum (all Nk’s equal to zero) known second order correction to the energy of the ,u-th approximation : 1VfiIvkj’
-
(A.13) limit
of an infinite
the electromagnetic we recover the welllevel. In the dipolar
k-l
(A. 14)
we have a linearly divergent contribution (electron self-mass) and a logarithmically divergent contribution, identical to the usual expression for the Lamb shift in the non relativistic limit 26). At finite temperature, the perturbed states energies depend on the photon numbers. The average level spacing at equilibrium will thus be a function of temperature. Also, in the presence of incident light, there is an induced level shift, proportional to the intensity of the light beam. This shift has been observed on the Zeeman splitting of the ground state level of mercury by CohenTannoudjis7). 3. Vertex renormalization. In this case, we have contributions diagrams A and B (see (I.A.4.7)): (A&$+B
from all diagrams.
We obtain,
for
= 2x C ;r: S(E~CO~) 9 12 Elm
x
[I -exp(-El&)],
(A.15)
KINETIC
EQUATIONS
AND
LIGHT-MATTER
INTERACTION
647
which may be written as:
(Nk+I)[l x 8Nd1+E,12 %v.,(l-G/2
-exl’(-E(&-&i-&)}], (A. 16)
with
1
+p
u)p - Wfi’ - Ok’ > 1
+9 (
cc)p’ -
Way-
cc)k’
>
1
i-9 (
WV- “,p’ - ok’ >
&
I’v,I,‘k’12
(Nk,
&
)vfi’lvk’12
Nk,
&
I’Vvl@‘k’l’
(Nk’
+
1)
t
+
+
I)}
IV,uIvk12.
(A.17)
The diagrams of class C lead us to (see (I.A.5.14)) :
x
<[~2,[~3,e41-]-~1>[I
-=p(-w&)].
which may be written in a form similar to (A.16) with #‘I
vfl,vk12
=
C-4.18)
(A. 19) The renormalized potential jr/lU,vk12
=
Ivfilvk12 +
i22A’2’ ~+jql&vk12
+
~2A~)Iblvk12
(A.20)
is the quantity which will, up to the order considered, determine the transition probability for one-photon processes, i.e. for processes where one photon k is emitted or absorbed and where a transition between the true levels B,, C%occurs in the atom. Received 6-3-68 REFERENCES
1) Prigogine,
I., Non Equilibrium Statistical Mechanics, Interscience Publ. (New-York, 1962). R., Statistical Mechanics of Charged Particles, Interscience Publ. (New-York, 1963). P., in Physics of many Particle Systems: Methods and Problems, ed. by E. Meeron, 3) Resibois, Gordon and Breach (New-York, 1966). I., Report at the XIVtk Solvay Conference, Brussels, 1967, Interscience Publ. 4) Prigogine, (New-York, in press). I. and Henin, F., Proceedings of the IUPAP InternationalConferenceonStatistical 5) Prigogine, Mechanics and Thermodynamics, Copenhagen 1966, ed. T. Bak, Benjamin (New-York, 1967). I., in Fast Reactions and Primary Processes in Chemical Kinetics, Nobel Sympo6) Prigogine, sium 5, ed. S. Claesson, Interscience Publ. (New-York, 1967). I., Henin, F. and George, C., Proc. Nat. Acad. Sci. U.S.A. 59 (1968) 7. 7) Prigogine, M. and Watson, K., Collision Theory, John Wiley and Sons (New -York, 1964). 8) Goldberger, 9) George, C., to be published. R., Physica 36 (1967) 433. 18) Balescu, (1964) 1109. 11) Zwansig, R., Physica 12) George, C., Physica 37 (1967) 182. 13) George, C. Physica 30 (1964) 1513. 14) George, C., Bull. Acad. Roy. Belg. Cl. Sci. 53 (1967) 623. 15) Henin, F. and De Haan, M., Physica 40 (1968) (to be published). I., George, C. and Mayne, F., Physica 32 (1966) 1828. 16) Henin, F., Prigogine, I., Henin, F. and George, C., Physica 32 (1966) 1873. 17) Prigogine, 18)dHenin, F., Bull. Acad. Roy. Belg. Cl. Sci., to be published. 19) Goldhaber, A. and Watson, K., Phys. Rev.160 (1967) 1151.
2) Balescu,
KINETIC
EQUATIONS
AND LIGHT-MATTER
INTERACTION
649
20) Heitler, W., The Quantum Theory of Radiation (3rd ed.), Clarendon Press (Oxford, 1954). 21) Resibois, P., Physica 27 (1961) 541. 22) Hahler, G., Z. Phys. 152 (1958) 546. Henin, F. and De Haan, M., Bull. Acad. Roy. Belg. Cl. Sci. (to be published). Arnous, E. and Heitler, W., Proc. Roy. Sot. A220 (1953) 290. Levy, M., Nuovo Cimento 14 (1959) 612. Schweber, S., Bethe, H. and De Hoffman, F., Mesons and Fields, vol. I, Row Peterson and Company (Evanston, Illinois, 1956). C., Ann. Physique 7 (1962) 423. 27) Cohen-Tannoudji,
23) 24) 25) 26)