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Photon distribution drift in multiphoton absorption-emission processes due to one-photon perturbations
a,._-
__ kB
ELSE3’IER
28 July 1997
PHYSICS
Physics Letters A 232 (1997)
LETTERS
A
175-182
Photon distribution drift in multiphoton absorption-emission processes due to one-photon perturbations E.C. Caparelli ‘, V.V. Dodonov 2-3, S.S. Mizrahi 4 Departamento de Fisica, CCT, Universidade Federal de Sdo Carlos. Via Washington Luiz km 235, 13565-905, Sio Carlos - SP, Brazil Received 6 March 1997; revised manuscript received 20 May 1997; accepted for publication 21 May 1997
Communicated by P.R. Holland
Abstract We obtain asymptotical solutions to the master equation for diagonal density matrix elements of the one-mode quantized field in the case when strong k-photon absorption and emission are disturbed by weak processes of lower orders. These
solutions demonstrate a slow drift of k quasiequilibrium distribution functions to the unique final stationary distribution. @ 1997 Elsevier Science B.V. PACS: 03.65.B~; 42.50.D~ Keywords: Master equation; Multiphoton processes
1. Introduction Under certain conditions, the evolution of a quantum system interacting with an environment can be described by means of the master equation in the socalled “Lindblad form” (n = 1) ,.
$+i[A,bl (1) k
It preserves hermiticity, normalization, and positive definiteness of the statistical operator /i, for any set of linear operators & [ l-31. If the system under study ’ E-mail: [email protected]. * On leave from: Lebedev Physics Institute and Moscow Institute of Physics and Technology, Russian Federation. 3 E-mail: [email protected]. 4 E-mail: [email protected]. 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO375-9601(97)00398-S
is an electromagnetic field mode (or an equivalent harmonic oscillator), then expressing the Hamiltonian fi and operators & in terms of the photon annihilation and creation operators, B and Zt, [G,Ct] = 1, one obtains a phenomenological equation describing an interaction between the field mode and an atomic system, the details of interaction being reduced to the effective coupling coefficients in the expressions for the operators &. In particular, operators A:“’ = [Dk(a)] 1/22k and A:’ = [ Dr’] ‘I*( at) k are used usually to describe k-photon absorption and emission by some atomic reservoir [ 41, with the positive constants D(a'e) being proportional to the probabilities of the k corresponding processes. It is clear that such a simple scheme cannot describe all the details of the interaction between an electromagnetic field and atoms, especially for cascade processes involving intermediate atomic levels, which only as an approximation can be replaced by a single multiphoton transition. In gen-
176
E.C. Caparelli et al. /Physics Letters A 232 (1997) 175-182
eral, a microscopic approach is necessary, which takes into account explicitly the atomic degrees of freedom, especially for calculating off-diagonal elements of the density matrix [5,6]. However, the evolution of the diagonal elements turns out to be not so sensitive to the details of the interaction, and in many cases the master equation with effective multiphoton absorption operators proportional to 2k leads to the same results as the microscopic approach [ 5,6], provided that the emission operators are modified in accordance with the (generalized) Scully-Lamb model [ 7-91, A:“’ = @-)]lP(at)k
accounting for the effect of saturated emission. In this case, assuming that the Hamiltonian is diagonal in the Fock basis, fi = A(A), we obtain from Eq. ( 1) a closed set of equations for the occupation probabilities Pn = (Q[n)
(n = 0, 1,. .I,
nj+)P,+k - pp,) (+I
_
0:“’
nk
n:-'pn-k
Pn
1 + ykn:-)
1 + Ykn:+)
’
(2)
where we define nk(+) =
(n + k)!/n!,
nj-’
= n!/(n
-k)!.
Although various special cases of Eq. (2) were studied (mainly in connection with the problem of multiphoton laser) by many authors during the last three decades (see, e.g. reviews [ 8,9] ), not many explicit analytical solutions were found until now. Exact timedependent solutions are known in the following special cases: k = 1, yt = 0 (unsaturated one-photon emission and absorption) [ 10,l l] ; k = 1, 0:“’ = 0 (one-photon emission with account of saturation but without absorption) [ 121; k = 2,Di”’ = 79 = 0 (twophoton unsaturated emission) [ 13,141; k = 2, 0:“’ = 0 (two-photon emission with account of saturation) [ 121; k = 2, 0:” = 0 (two-photon absorption) [ 14181; k = 1,2, D,(e) = 0:“’ = 0 (competition between one- and two-photon absorption) [ 19,201; k 3 2 0;“’ = 0 (k-photon absorption) [ 21,221; k > 2, i(a) _- yk = 0 (k-photon unsaturated emission) [ 221. k
Exact stationary solutions in the case k = 1 (stationary regime of the Scully-Lamb laser model) were found in Refs. [ 7,231. The detailed balance between k-photon emission and absorption was considered in Ref. [ 241 (see also Ref. [ 171 for k = 2 only, and Ref. [ 251 for yk = 0). The stationary regime of a two-photon laser with one-photon losses (k = 1,2, Dee) = 0:“’ = 0, y2 # 0) was studied in Ref. I [26], and the generalizations to the case of arbitrary k 3 1 (provided that the only non-zero coefficients were OF”‘, Dr’, and yk ) were given in Ref. [ 271. A stationary photon distribution in the case of onephoton absorption and emission accompanied by twophoton absorption was discussed in Refs. [ 28,291. Different schemes of finding approximate stationary solutions of Eq. (2) were given, e.g. in Ref. [ 301 (two-photon emission and single-photon absorption), Ref. [ 3 1 ] (m-photon emission and absorption or mphoton emission and single-photon absorption), Ref. [ 321 (m-photon emission and absorption plus additional k-photon absorption, including the cases m > k andm< k). In the present article we give asymptotical solutions to Eq. (2) in the case when the coupling constants D$,“‘, describing the processes with the maximal number of instantaneously emitted/absorbed photons, are much greater than all other coefficients with k < M, in the limit cases of unsaturated and completely saturated emission. New solutions have simple analytical forms, and they show explicitly, how an M-fold degeneracy inherent to pure M-photon interactions is eliminated due to weak processes of lower orders. In Section 2, we give a general scheme of finding asymptotical solutions, whereas Sections 3 and 4 are devoted to the analysis of the most interesting special cases, M = 2 and M = 3, including a comparison with the results of numerical calculations. Section 5 contains conclusions.
2. General scheme Complete information about the distribution {p”} is contained in the generating function (GF) F( Z, t) = cz* pn ( t ) zn. Its derivatives yield the probabilities p,, and the factorial l)...(n-m+l)p,:
moments
N,
G Cz,,
n( n -
EC. Caparelli et al. /Physics Letters A 232 (1997) 175-182
P”‘,I
1 #‘F G
,
Nm=g
z=0
.
(3)
z=l
If the coefficients at variables pj (t) in the right-hand side of the infinite system of ordinary differential equations (2) are polynomiufs of n, then one can replace this system by a single partial differential equation for F( 2, t). If ok = 0 (unsaturated emission), then dF at=
M (1 -&$[(D:“‘-D$‘)F], c k=,
(4)
M being the maximal number of instantaneously emitted or absorbed photons. Putting formally ok --f oo, D:“/yk -_) EC’ = const, we arrive at the case of completely saturated emission. Physically, it corresponds to large effective photon numbers, when one can neglect 1 in comparison with ykn:*” in the denominator of the Scully-Lamb factor for the most significant transitions. In this case we have aF
M
at=c
(1
_
zk> Dja)sdkF - DF)F
.
(5)
k.,
If Djave) z 0 for all (during the M-photon TM = (D$‘) -‘) goes in the case of Eq. (4)
k < M, then the GF rapidly relaxation time of the order of to a stationary solution, which reads
R s 0:)/D;‘. The state occupation excited states, pjfkM=AjRk,
probabilities
C$’ Aj(t>t’
t> =
1 -RzM
E - m&D:“3”)/D~’
+ef(z,t), < 1.
(9)
The assumption of a slow evolution means that the time derivatives of functions Aj (t) and f (z, t) are proportional to E. Consequently, putting function (9) into lZq. (4), we can neglect the term &f N E* in comparison with the other terms of the order of E. Thus we arrive at
Cz’
Aj(t)Zj
l-
RzM
M-’ (1 - zk) = ck=l
nothing but the of the first M - 1
+
-z”)$,(D;‘-
O(E2).
D:‘z”>fl (10)
(7)
Their concrete conditions,
are determined
from the initial
-R)epj+~k(O)v k=O
j=O,l,...,
F(z,
(6)
M-l,
k=O,l,...
A,=(1
t < TM these processes do not influence the evolution of the occupation probabilities. However, at t > TM they give rise to an interaction between all the levels (whereas pure M-photon processes mix only the levels whose numbers differ by gM, 4 being an integer) _This results in a slow (with the characteristic time much greater than TM) variation of the relative weights A, in the distribution (7), so that these weights become slowly varying functions of time tending to some final stationary distribution, which is determined by the relative strengths of coupling constants D:a3e), not by the initial conditions. Having in mind this qualitative picture, we look for the generating function in the form
+&(l
j=O,l,...,
1-n
M-
1.
(8)
Now let us admit the existence of other processes with small coupling constants DfSe) < OF’, k = 1,2,. . 1. During initial time interval 0 <
Now we take into account that the generating function has no singularities at Iz 1 = 1 (for instance, F( 1) 3 1, whereas the combinations F(1) i F ( -1) are proportional to the sums of probabilities over even or odd states, respectively). Consequently, the last term in the right-hand side of (10) disappears for M different values, z4 = exp( 2vriq/M), q 1, . . . , M - 1, of complex auxiliary variable z, and we obtain M linear differential equations for M unknown functions Aj(t) (q=O,l,...,M-l),
178
E.C. Caparelli et ul./Ptysic.~
.i=O
Letters A 232 (1997) 175-182
L=l
CFI!,-’Aj(t)zj
and then assumed D:‘) = 1 (in other words, we mea-
I-RzM
sure time in the units of (D:“‘)-‘). GF in the form
(11) These equations determine both an asymptotical behaviour of GF at t >> TM and its final stationary form. The equation for q = 0, c Aj = 0, is the consequence of the normalization condition C A,(r) E 1 - R (see Eqs. (7) and (8)). Therefore actually we have a set of M - 1 inhomogeneous linear differential equations of the first order, so that each function Aj( t) is a linear combination of a stationary solution and M - 1 exponential functions with coefficients determined by the initial conditions (8). After calculating the coefficients Aj ( t) , one can use Eq. ( 10) to find an expression for the correction f(z, t) in quadratures. However, in most of the cases this correction is not of immediate interest. A similar scheme can be applied to Eq. (5). The only difference is the form of the asymptotical solution,
= C
Aj(t)PjPi(z),
(12)
j=O
each function pj (z) being a superposition of Mindependent combinations of exponential functions exp(rz,) (q=O, I,...,M- 1,r = (5~)/D~‘)ll”), distinguished by the condition limz,o (Pi( Z) /zj = 1.
3. Strong two-photon
processes
Consider Eq. (4) with M = 2. It describes strong two-photon unsaturated emission and absorption accompanied by weak one-photon processes,
(13) To simplify the expressions, dimensionless coefficients
we have introduced
new
A(r) + Btt)z 1 -pz=
for the
-tvf(z,t),
we arrive at two linear differential equations for slowly varying functions A(t) c Ao(t) and B(t) = Al(r), whose solutions read A(t)
= [A(O) - Am]e-“x’
B(t)
= [B(O) - Boo]eWYX’ + B,,
+ A,,
( 16) (17)
where the initial values A (0) and B (0) are given by Eq. (8), and 1+3p+35+p5 l-p
x=
A
=
00
B
M-l F(z,~)
F(z,t) =
Looking
cc
(1 -P)(l
+P+25)
1+3p+3!5+p5 = Cl- P)(5+2P+P5) 1+3p+3t+p5
(18)
’
’ ’
(19)
The final state turns out a mixture of even and odd thermal states, considered for the first time (in the even case) in Ref. [ 331. The final probabilities satisfy the inequalities p;“’ > pl(O”) > pirn’ > . . . for 0 < p < 1 and 0 < 5 < co. In the special case p = 0 we have B, = t/( 1 + 35), in accordance with the exact solution found in Ref. [ 201. In the opposite case. 1 - p < 1, the final distribution does not depend on n$,AmzBB,x ( 1 - p) /2, however, the parameter .$ influences the relaxation time: x = 4( 1 +[) / ( 1 - p) Figs. 1 and 2 show an excellent agreement between the asymptotical solution given above and the numerical results obtained by solving the system (2). A simple solution can be found also in the case of a completely saturated two-photon emission. Assuming that the one-photon emission is described by a sum of two terms, corresponding to the unsaturated and completely saturated processes, with the coefficients 0:” and 6;“’ respectively, we obtain (in the same notation as above)
EC. Caparelli
et al.
0
/Physics Letters A 232 (I 997) 175-182
b
-,
-_-,,-
-.,.-
01
n
Fig. 1. Evolution of the photon distribution function in the case of strong unsaturated two-photon processes, found by numerically solving the system (2) truncated at n = 22, for different values of the slow time 7 = YC; Y = 0.01, p = [ = 0.5, D:“) = 1. The initial condition was ~1, (0) = 1. The asymptotical time-dependent regime (16)-( 19) is established at 7 z 0.01. Empty circles correspond to odd values of n, and crossed circles correspond to even ones.
= A(t) cosh(rz)
+ vf(z,
+ B(t)
x = 2(5 + CT) + 2r( 1 + 5) coth(2r), rcoshr
A,=
B,
=
rcosh(2r) rsinhr
s=
(5_tU)/(5f
1).
(23)
with time-dependent coefficients of the form ( 16), (17). Here g = 4,/T, and F(a,b;c;x) means the Gauss hypergeometric function. The coefficients x, A, and B, are expressed in this case in
’
r cosh( 2r) + S sinh( 2r) ’
0.5
Fig. 2. The probabilities po and p1 versus the slow time 7 = vt, found by numerically solving the system (2) truncated at n = 22. in the case of strong unsaturated two-photon processes with the parameters v = 0.01, p = 6 = 0.5, Dfl = 1, and the initial condition p,,(O) = 1. The functions A(T) and B(r) are given by the asymptotical solutions ( 16)-( 19). The curve p;(r) corresponds to pure two-photon processes. The asymptotical time-dependent regime is established at 7 z 0.02.
1
+ Ssinhr + Ssinh(2r) + Scoshr
~__’ 0.4
(22)
(21)
we get for A ( t) and B ( t) the same expressions and (17), but now
,-
lffiz
sinh(rz)
t) 1
03
( 16)
(20)
where a prime means the differentiation over z, fl E fi~“‘/D~“‘, r2 s bie)/D:a). Looking for the GF in the form F(z,t)
02
Note that the GF cosh( rz ) and sinh( rz ) describe the same photon distributions that in the even and odd coherent states were introduced in Ref. [ 341. If r --f 0, then p1 s rB, + S/( 1 + 2s). The final distribution becomes insensitive to the values of 8 and v at r > 1, going to the Poisson distribution with F( z, 00) = exp [ r( z - 1) 1, but the relaxation constant x essentially depends on 5 and g. When both, unsaturated and completely saturated, two-photon emission processes are present, the GF F( z, t) can be written as a linear combination of the even and odd parts of the function
$V(l-z)[(1-zS)F’-(6+fl)F1 +(l-z2)(F”-r’F).
179
F($-g, -
;+g;2;
;(l-fiz))
(24)
180
E.C. Caparelli
terms of function (24) z = 1 and z = -1.
and its derivatives
4. Strong three-photon
processes
et al. /Physics
Letters A 232 (I 997) 175-182
at points
o’25r---7-m
In the case of strong three-photon processes accompanied by weak single-photon ones, the asymptotics of the GF in the unsaturated regime reads
F(z,t)
=
Ao(r) + Adz
+
&(t>z*
l-Rz3
I
0.15
_-
I~_.
_
1
0
+vf(z,t)* (25) t
where R E Die)/D $“‘, and now Y - D,‘“‘/D$“‘. Normalizing the time scale by the three-photon absorption coefficient (i.e. putting formally 0:“’ = l), we can write the solution to Eq. ( 11) in the form (j = 0, 1,2)
i
0.05 I
0.00 t 0
A,(t) I
= A(m) +epa’ I
+ sinh(/?t)
cosh(Pt)AAj )
(UjAAt + UjAA2)
9
Fig. 3. The I strong unsatl tally solving
>
where AAj E Aj (0) - A:“‘, are defined as follows,
R = 0.5, [ =
while other coefficients
LY= 2(131R)(1+2R+2[+R&),
p=
q= [l -2OR-8R2
+2&8+
-[*(8+20R-
R*)J’/*,
uo=-l+4R-45+Rt,
” 2(1 -R)’
llR+8R2)
vo=-3+2[+R[,
UI = 1+2R-3R(,
v1 =4+2R-2(-4R&,
~2=-6R+45+2Rl,
+2R)(2+
R) +3&(1+2R)
wi,
Aim) = !$3R(2+R)+5(1+2R)(2+R) +35*(1 A’“’ 2
+2R)], 1 + 2R) + 3R[(2
= 9[3R(
+&*(I
G=2+ +c*(ll
+2R)(2+ 14R+
es form
The final monotono for 0 < R parameter in Fig. 3. depend 01 coefficien forms for
+ R)
R)],
llR*+&5+
+ 14R+2R2).
17R+5R2)
a
gl+R
q=3(1-
uo/q =-& w/q= f, uo/q= -;, o/q = $9
again
.he influence of the :ibution can be seen listribution does not M (1 - p)/3. The ssume the simplest
1-R’
v2=-1-2R+3Rt,
A;(“) = ’ - R ~[(1 +35*(2+
function in the case of (the result of numeriII = 22). for Y = 0.01, :ircles )
u*/q=
$
v*/q = -$.
Fig. 4 shows the dependence of the first seven probabilities and the final weights Al”’ = A;“‘/( I - R), j=0,1,2,ontheparameterRat~=l.Forl-R< 1, all the three weights are close to l/3 independently on 5. For certain values of R or 6, the coefficient q can be pure imaginary. Then one should replace the hyperbolic functions in Eq. (26) by their trigonometrical counterparts. Nonetheless, no significant oscillations of functions Aj( t) are observed due to the inequality CY> IpI, although the evolution of these functions
E.C. Caparelli et al. /Physics Letters A 232 (1997) 175-182
181
0.6
... --
P”lV
P,@)
p,ir,
\
(1 ‘\
P.
0.3
:
‘\\ ,,/
‘b’e~s ‘.\
l’ 0.2
:--::-:
.-..
‘\ ‘l.
,,” i’
\
.....
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
00
1.0
R Fig. 4. The stationary probabilities pi’“‘, j = 0.1,. ,6 (solid cmves, from top to bottom), and the stationary weight functions A!“), j = 0.1.2 (dashed curves, from top to bottom), versus thk parameter R in the case of strong unsaturated three-photon processes with 5 = 1.
(here r3 3
pa(z)
fi:“‘/D$“‘) ,
= $[erZ +2e-‘z12cos($JSrz)],
m(z) =
k{erz
- e-“/*[cos(
rp2Cz)
iv?rz>
- &sin(i&z)]},
= ${erz
-ee-‘z~2[cos(~J?;rz)
+ &sin(+&z)]}.
0.1
0.2
0.3
0.4
i !
0.5
--.--.__._._
0.6
0.7
i
0.6
0.9
1.0
7
Fig. 5. The probabilities po, PI, and p2 versus the slow time r = it in the case of strong unsaturated three-photon processes, obtained by numerically solving the system (2) truncated at n = 22, with the initial conditions PO(O) = 0.2, pi (0) = 0.2, pz(O) = 0.6; for v = 0.1, R = 0.25, 5 = 0, Df) = 1. The asymptotical solutions (26) practically coincide with the numerical results for r > 0.05. 5.
may be not so monotone as in the two-photon case: see Fig. 5. An account of weak two-photon processes does not change the time dependence of slowly varying weights Aj ( t) (26)) but the explicit expressions of the coefficients Q, p, etc.., become too cumbersome to give them here. The same is true for completely saturated three-photon emission, when the asymptotical GF is given by Eq. ( 12) with the following functions pj (z )
0.0
i
A_
v
0.1
I
Conclusion
We have found analytical asymptotical solutions of the master equation for the diagonal elements of the density matrix, when M-photon processes dominate over the processes with k < M photons. These approximate solutions are in excellent agreement with the results of exact numerical calculations. Relatively simple explicit expressions for all the coefficients, determining the form of solutions, are given for M = 2 and M = 3. In a generic case, one has to solve a system of M - 1 linear inhomogeneous differential equations of the first order with constant coefficients. It is worth to note that the proposed approach cannot be used in the opposite case, when k-photon processes are disturbed by weak processes of the order M > k. Exact stationary solutions in the case M = 2, k = 1, given in Refs. [ 28,291, show that even very weak higher-order processes change the picture drastically.
182
EC. Caparelli et al./Plysics
Acknowledgement This research was supported by FAPESP (Brazil), projects 199513843-9 and 96/05437-O. SSM thanks CNPq, Brasil, for partial financial support.
References A.A. Belavin, B. Ya. Zel’dovich, A.M. Perelomov and VS. Popov, Zhum. Eksp. Teor. Fiz. 56 ( 1969) 264 [Sov. Phys. JETP 29 (1969) 1451. 121 G. Lindblad, Commun. Math. Phys. 48 ( 1976) 119. [31 E.B. Davies, Quantum Theory of Open Systems (Academic, London, 1976). I41 Y.R. Shen. Phys. Rev. 155 (1967) 921. 151 A.W. Boon and S. Swain, Quant. Opt. 1 (1989) 27. [61 A.H. Toor and M.S. Zubairy, Phys. Rev. A 45 ( 1992) 4951. 171 M.O. Scully and W.E. Lamb, Phys. Rev. 159 (1967) 208. 181 H. Paul, Rev. Mod. Phys. 54 ( 1982) 1061. 191 J. Perina. Quantum Statistics of Linear and Non-linear Optical Phenomena (Kluwer, Dordrecht, 1991). [lOI B. Ya. Zel’dovich, A.M. Perelomov and V.S. Popov, Zhum. Eksp. Tear. Fiz. 55 (1968) 589 [ Sov.Phys. JETP 28 (1969) 3081. 1111 A. Schell and R. Bar&at, J. Phys. A 6 (1973) 826.
Ill
Letters A 232 (1997) 175-182 I 121 A. BandiIJa and H. Voigt, Opt. Comm. 43 ( 1982) 277. P Lambropoulos, Phys. Rev. 156 ( 1967) 286. K.J. McNeil and D.F. Walls. J. Phys. A 7 ( 1974) 6 17. G.S. Agarwal, Phys. Rev. A 1 ( 1970) 1445. N. Tomau and A. Bach. Opt. Comm. 11 ( 1974) 46. 1171 H.D. Simaan and R. Loudon, J. Phys. A 8 ( 1975) 539. I 181 A. Bandilla and H.-H. Ritze, Ann. Physik (Leipzig) 33 ( 1976) 207. [ 191 A. Bandilla, Opt. Comm. 23 ( 1977) 299. [20] V.V. Dodonovand S.S. Mizrahi. J. Phys. A 30 (1997) 2915. [21] H. Voigt. A. Bandilla and H.-H. Ritze, Zs. Phys. B 36 (1980) 295. (221 M.S. Zubairy and J.J. Yeh. Phys. Rev. A 21 (1980) 1624. 1231 R. Landauer, J. Appl. Phys. 33 ( 1962) 2209. [24] K.J. McNeil and DF Walls, J. Phys. A 8 (1975) 104. [ 25 ] V.V. Dodonov and S.S. Mizrahi, Physica A 214 ( 199.5) 619. 1261 M.S. Zubairy, Phys. L&t. A 80 ( 1980) 225. [27] A. Bandilla and H. Voigt, Zs. Phys. B 58 (198.5) 165. I28 ] A. Bandilla and H.-H. Ritze, Opt. Comm. 19 ( 1976) 169. 1291 G.P Hildred, Optica Acta 27 ( 1980) 1621. 1301 R. GGrtz and D.F. Walls, Z. Phys. 25 (1976) 423. [31] P.S. Gupta and B.K. Mohanty, Optica Acta 28 (1981) 521. [ 321 U. Herzog. Optica Acta 30 (1983) 639. [ 331 K.E. Cahill and R.J. Glauber Phys. Rev. 177 ( 1969) 1882. [34] V.V. Dodonov, LA. Malkin and V.I. Man’ko, Physica 72 ( 1974) 597.
[ 13I 114 I [ 151 [ 161
__ kB
ELSE3’IER
28 July 1997
PHYSICS
Physics Letters A 232 (1997)
LETTERS
A
175-182
Photon distribution drift in multiphoton absorption-emission processes due to one-photon perturbations E.C. Caparelli ‘, V.V. Dodonov 2-3, S.S. Mizrahi 4 Departamento de Fisica, CCT, Universidade Federal de Sdo Carlos. Via Washington Luiz km 235, 13565-905, Sio Carlos - SP, Brazil Received 6 March 1997; revised manuscript received 20 May 1997; accepted for publication 21 May 1997
Communicated by P.R. Holland
Abstract We obtain asymptotical solutions to the master equation for diagonal density matrix elements of the one-mode quantized field in the case when strong k-photon absorption and emission are disturbed by weak processes of lower orders. These
solutions demonstrate a slow drift of k quasiequilibrium distribution functions to the unique final stationary distribution. @ 1997 Elsevier Science B.V. PACS: 03.65.B~; 42.50.D~ Keywords: Master equation; Multiphoton processes
1. Introduction Under certain conditions, the evolution of a quantum system interacting with an environment can be described by means of the master equation in the socalled “Lindblad form” (n = 1) ,.
$+i[A,bl (1) k
It preserves hermiticity, normalization, and positive definiteness of the statistical operator /i, for any set of linear operators & [ l-31. If the system under study ’ E-mail: [email protected]. * On leave from: Lebedev Physics Institute and Moscow Institute of Physics and Technology, Russian Federation. 3 E-mail: [email protected]. 4 E-mail: [email protected]. 0375-9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved. PIISO375-9601(97)00398-S
is an electromagnetic field mode (or an equivalent harmonic oscillator), then expressing the Hamiltonian fi and operators & in terms of the photon annihilation and creation operators, B and Zt, [G,Ct] = 1, one obtains a phenomenological equation describing an interaction between the field mode and an atomic system, the details of interaction being reduced to the effective coupling coefficients in the expressions for the operators &. In particular, operators A:“’ = [Dk(a)] 1/22k and A:’ = [ Dr’] ‘I*( at) k are used usually to describe k-photon absorption and emission by some atomic reservoir [ 41, with the positive constants D(a'e) being proportional to the probabilities of the k corresponding processes. It is clear that such a simple scheme cannot describe all the details of the interaction between an electromagnetic field and atoms, especially for cascade processes involving intermediate atomic levels, which only as an approximation can be replaced by a single multiphoton transition. In gen-
176
E.C. Caparelli et al. /Physics Letters A 232 (1997) 175-182
eral, a microscopic approach is necessary, which takes into account explicitly the atomic degrees of freedom, especially for calculating off-diagonal elements of the density matrix [5,6]. However, the evolution of the diagonal elements turns out to be not so sensitive to the details of the interaction, and in many cases the master equation with effective multiphoton absorption operators proportional to 2k leads to the same results as the microscopic approach [ 5,6], provided that the emission operators are modified in accordance with the (generalized) Scully-Lamb model [ 7-91, A:“’ = @-)]lP(at)k
accounting for the effect of saturated emission. In this case, assuming that the Hamiltonian is diagonal in the Fock basis, fi = A(A), we obtain from Eq. ( 1) a closed set of equations for the occupation probabilities Pn = (Q[n)
(n = 0, 1,. .I,
nj+)P,+k - pp,) (+I
_
0:“’
nk
n:-'pn-k
Pn
1 + ykn:-)
1 + Ykn:+)
’
(2)
where we define nk(+) =
(n + k)!/n!,
nj-’
= n!/(n
-k)!.
Although various special cases of Eq. (2) were studied (mainly in connection with the problem of multiphoton laser) by many authors during the last three decades (see, e.g. reviews [ 8,9] ), not many explicit analytical solutions were found until now. Exact timedependent solutions are known in the following special cases: k = 1, yt = 0 (unsaturated one-photon emission and absorption) [ 10,l l] ; k = 1, 0:“’ = 0 (one-photon emission with account of saturation but without absorption) [ 121; k = 2,Di”’ = 79 = 0 (twophoton unsaturated emission) [ 13,141; k = 2, 0:“’ = 0 (two-photon emission with account of saturation) [ 121; k = 2, 0:” = 0 (two-photon absorption) [ 14181; k = 1,2, D,(e) = 0:“’ = 0 (competition between one- and two-photon absorption) [ 19,201; k 3 2 0;“’ = 0 (k-photon absorption) [ 21,221; k > 2, i(a) _- yk = 0 (k-photon unsaturated emission) [ 221. k
Exact stationary solutions in the case k = 1 (stationary regime of the Scully-Lamb laser model) were found in Refs. [ 7,231. The detailed balance between k-photon emission and absorption was considered in Ref. [ 241 (see also Ref. [ 171 for k = 2 only, and Ref. [ 251 for yk = 0). The stationary regime of a two-photon laser with one-photon losses (k = 1,2, Dee) = 0:“’ = 0, y2 # 0) was studied in Ref. I [26], and the generalizations to the case of arbitrary k 3 1 (provided that the only non-zero coefficients were OF”‘, Dr’, and yk ) were given in Ref. [ 271. A stationary photon distribution in the case of onephoton absorption and emission accompanied by twophoton absorption was discussed in Refs. [ 28,291. Different schemes of finding approximate stationary solutions of Eq. (2) were given, e.g. in Ref. [ 301 (two-photon emission and single-photon absorption), Ref. [ 3 1 ] (m-photon emission and absorption or mphoton emission and single-photon absorption), Ref. [ 321 (m-photon emission and absorption plus additional k-photon absorption, including the cases m > k andm< k). In the present article we give asymptotical solutions to Eq. (2) in the case when the coupling constants D$,“‘, describing the processes with the maximal number of instantaneously emitted/absorbed photons, are much greater than all other coefficients with k < M, in the limit cases of unsaturated and completely saturated emission. New solutions have simple analytical forms, and they show explicitly, how an M-fold degeneracy inherent to pure M-photon interactions is eliminated due to weak processes of lower orders. In Section 2, we give a general scheme of finding asymptotical solutions, whereas Sections 3 and 4 are devoted to the analysis of the most interesting special cases, M = 2 and M = 3, including a comparison with the results of numerical calculations. Section 5 contains conclusions.
2. General scheme Complete information about the distribution {p”} is contained in the generating function (GF) F( Z, t) = cz* pn ( t ) zn. Its derivatives yield the probabilities p,, and the factorial l)...(n-m+l)p,:
moments
N,
G Cz,,
n( n -
EC. Caparelli et al. /Physics Letters A 232 (1997) 175-182
P”‘,I
1 #‘F G
,
Nm=g
z=0
.
(3)
z=l
If the coefficients at variables pj (t) in the right-hand side of the infinite system of ordinary differential equations (2) are polynomiufs of n, then one can replace this system by a single partial differential equation for F( 2, t). If ok = 0 (unsaturated emission), then dF at=
M (1 -&$[(D:“‘-D$‘)F], c k=,
(4)
M being the maximal number of instantaneously emitted or absorbed photons. Putting formally ok --f oo, D:“/yk -_) EC’ = const, we arrive at the case of completely saturated emission. Physically, it corresponds to large effective photon numbers, when one can neglect 1 in comparison with ykn:*” in the denominator of the Scully-Lamb factor for the most significant transitions. In this case we have aF
M
at=c
(1
_
zk> Dja)sdkF - DF)F
.
(5)
k.,
If Djave) z 0 for all (during the M-photon TM = (D$‘) -‘) goes in the case of Eq. (4)
k < M, then the GF rapidly relaxation time of the order of to a stationary solution, which reads
R s 0:)/D;‘. The state occupation excited states, pjfkM=AjRk,
probabilities
C$’ Aj(t>t’
t> =
1 -RzM
E - m&D:“3”)/D~’
+ef(z,t), < 1.
(9)
The assumption of a slow evolution means that the time derivatives of functions Aj (t) and f (z, t) are proportional to E. Consequently, putting function (9) into lZq. (4), we can neglect the term &f N E* in comparison with the other terms of the order of E. Thus we arrive at
Cz’
Aj(t)Zj
l-
RzM
M-’ (1 - zk) = ck=l
nothing but the of the first M - 1
+
-z”)$,(D;‘-
O(E2).
D:‘z”>fl (10)
(7)
Their concrete conditions,
are determined
from the initial
-R)epj+~k(O)v k=O
j=O,l,...,
F(z,
(6)
M-l,
k=O,l,...
A,=(1
t < TM these processes do not influence the evolution of the occupation probabilities. However, at t > TM they give rise to an interaction between all the levels (whereas pure M-photon processes mix only the levels whose numbers differ by gM, 4 being an integer) _This results in a slow (with the characteristic time much greater than TM) variation of the relative weights A, in the distribution (7), so that these weights become slowly varying functions of time tending to some final stationary distribution, which is determined by the relative strengths of coupling constants D:a3e), not by the initial conditions. Having in mind this qualitative picture, we look for the generating function in the form
+&(l
j=O,l,...,
1-n
M-
1.
(8)
Now let us admit the existence of other processes with small coupling constants DfSe) < OF’, k = 1,2,. . 1. During initial time interval 0 <
Now we take into account that the generating function has no singularities at Iz 1 = 1 (for instance, F( 1) 3 1, whereas the combinations F(1) i F ( -1) are proportional to the sums of probabilities over even or odd states, respectively). Consequently, the last term in the right-hand side of (10) disappears for M different values, z4 = exp( 2vriq/M), q 1, . . . , M - 1, of complex auxiliary variable z, and we obtain M linear differential equations for M unknown functions Aj(t) (q=O,l,...,M-l),
178
E.C. Caparelli et ul./Ptysic.~
.i=O
Letters A 232 (1997) 175-182
L=l
CFI!,-’Aj(t)zj
and then assumed D:‘) = 1 (in other words, we mea-
I-RzM
sure time in the units of (D:“‘)-‘). GF in the form
(11) These equations determine both an asymptotical behaviour of GF at t >> TM and its final stationary form. The equation for q = 0, c Aj = 0, is the consequence of the normalization condition C A,(r) E 1 - R (see Eqs. (7) and (8)). Therefore actually we have a set of M - 1 inhomogeneous linear differential equations of the first order, so that each function Aj( t) is a linear combination of a stationary solution and M - 1 exponential functions with coefficients determined by the initial conditions (8). After calculating the coefficients Aj ( t) , one can use Eq. ( 10) to find an expression for the correction f(z, t) in quadratures. However, in most of the cases this correction is not of immediate interest. A similar scheme can be applied to Eq. (5). The only difference is the form of the asymptotical solution,
= C
Aj(t)PjPi(z),
(12)
j=O
each function pj (z) being a superposition of Mindependent combinations of exponential functions exp(rz,) (q=O, I,...,M- 1,r = (5~)/D~‘)ll”), distinguished by the condition limz,o (Pi( Z) /zj = 1.
3. Strong two-photon
processes
Consider Eq. (4) with M = 2. It describes strong two-photon unsaturated emission and absorption accompanied by weak one-photon processes,
(13) To simplify the expressions, dimensionless coefficients
we have introduced
new
A(r) + Btt)z 1 -pz=
for the
-tvf(z,t),
we arrive at two linear differential equations for slowly varying functions A(t) c Ao(t) and B(t) = Al(r), whose solutions read A(t)
= [A(O) - Am]e-“x’
B(t)
= [B(O) - Boo]eWYX’ + B,,
+ A,,
( 16) (17)
where the initial values A (0) and B (0) are given by Eq. (8), and 1+3p+35+p5 l-p
x=
A
=
00
B
M-l F(z,~)
F(z,t) =
Looking
cc
(1 -P)(l
+P+25)
1+3p+3!5+p5 = Cl- P)(5+2P+P5) 1+3p+3t+p5
(18)
’
’ ’
(19)
The final state turns out a mixture of even and odd thermal states, considered for the first time (in the even case) in Ref. [ 331. The final probabilities satisfy the inequalities p;“’ > pl(O”) > pirn’ > . . . for 0 < p < 1 and 0 < 5 < co. In the special case p = 0 we have B, = t/( 1 + 35), in accordance with the exact solution found in Ref. [ 201. In the opposite case. 1 - p < 1, the final distribution does not depend on n$,AmzBB,x ( 1 - p) /2, however, the parameter .$ influences the relaxation time: x = 4( 1 +[) / ( 1 - p) Figs. 1 and 2 show an excellent agreement between the asymptotical solution given above and the numerical results obtained by solving the system (2). A simple solution can be found also in the case of a completely saturated two-photon emission. Assuming that the one-photon emission is described by a sum of two terms, corresponding to the unsaturated and completely saturated processes, with the coefficients 0:” and 6;“’ respectively, we obtain (in the same notation as above)
EC. Caparelli
et al.
0
/Physics Letters A 232 (I 997) 175-182
b
-,
-_-,,-
-.,.-
01
n
Fig. 1. Evolution of the photon distribution function in the case of strong unsaturated two-photon processes, found by numerically solving the system (2) truncated at n = 22, for different values of the slow time 7 = YC; Y = 0.01, p = [ = 0.5, D:“) = 1. The initial condition was ~1, (0) = 1. The asymptotical time-dependent regime (16)-( 19) is established at 7 z 0.01. Empty circles correspond to odd values of n, and crossed circles correspond to even ones.
= A(t) cosh(rz)
+ vf(z,
+ B(t)
x = 2(5 + CT) + 2r( 1 + 5) coth(2r), rcoshr
A,=
B,
=
rcosh(2r) rsinhr
s=
(5_tU)/(5f
1).
(23)
with time-dependent coefficients of the form ( 16), (17). Here g = 4,/T, and F(a,b;c;x) means the Gauss hypergeometric function. The coefficients x, A, and B, are expressed in this case in
’
r cosh( 2r) + S sinh( 2r) ’
0.5
Fig. 2. The probabilities po and p1 versus the slow time 7 = vt, found by numerically solving the system (2) truncated at n = 22. in the case of strong unsaturated two-photon processes with the parameters v = 0.01, p = 6 = 0.5, Dfl = 1, and the initial condition p,,(O) = 1. The functions A(T) and B(r) are given by the asymptotical solutions ( 16)-( 19). The curve p;(r) corresponds to pure two-photon processes. The asymptotical time-dependent regime is established at 7 z 0.02.
1
+ Ssinhr + Ssinh(2r) + Scoshr
~__’ 0.4
(22)
(21)
we get for A ( t) and B ( t) the same expressions and (17), but now
,-
lffiz
sinh(rz)
t) 1
03
( 16)
(20)
where a prime means the differentiation over z, fl E fi~“‘/D~“‘, r2 s bie)/D:a). Looking for the GF in the form F(z,t)
02
Note that the GF cosh( rz ) and sinh( rz ) describe the same photon distributions that in the even and odd coherent states were introduced in Ref. [ 341. If r --f 0, then p1 s rB, + S/( 1 + 2s). The final distribution becomes insensitive to the values of 8 and v at r > 1, going to the Poisson distribution with F( z, 00) = exp [ r( z - 1) 1, but the relaxation constant x essentially depends on 5 and g. When both, unsaturated and completely saturated, two-photon emission processes are present, the GF F( z, t) can be written as a linear combination of the even and odd parts of the function
$V(l-z)[(1-zS)F’-(6+fl)F1 +(l-z2)(F”-r’F).
179
F($-g, -
;+g;2;
;(l-fiz))
(24)
180
E.C. Caparelli
terms of function (24) z = 1 and z = -1.
and its derivatives
4. Strong three-photon
processes
et al. /Physics
Letters A 232 (I 997) 175-182
at points
o’25r---7-m
In the case of strong three-photon processes accompanied by weak single-photon ones, the asymptotics of the GF in the unsaturated regime reads
F(z,t)
=
Ao(r) + Adz
+
&(t>z*
l-Rz3
I
0.15
_-
I~_.
_
1
0
+vf(z,t)* (25) t
where R E Die)/D $“‘, and now Y - D,‘“‘/D$“‘. Normalizing the time scale by the three-photon absorption coefficient (i.e. putting formally 0:“’ = l), we can write the solution to Eq. ( 11) in the form (j = 0, 1,2)
i
0.05 I
0.00 t 0
A,(t) I
= A(m) +epa’ I
+ sinh(/?t)
cosh(Pt)AAj )
(UjAAt + UjAA2)
9
Fig. 3. The I strong unsatl tally solving
>
where AAj E Aj (0) - A:“‘, are defined as follows,
R = 0.5, [ =
while other coefficients
LY= 2(131R)(1+2R+2[+R&),
p=
q= [l -2OR-8R2
+2&8+
-[*(8+20R-
R*)J’/*,
uo=-l+4R-45+Rt,
” 2(1 -R)’
llR+8R2)
vo=-3+2[+R[,
UI = 1+2R-3R(,
v1 =4+2R-2(-4R&,
~2=-6R+45+2Rl,
+2R)(2+
R) +3&(1+2R)
wi,
Aim) = !$3R(2+R)+5(1+2R)(2+R) +35*(1 A’“’ 2
+2R)], 1 + 2R) + 3R[(2
= 9[3R(
+&*(I
G=2+ +c*(ll
+2R)(2+ 14R+
es form
The final monotono for 0 < R parameter in Fig. 3. depend 01 coefficien forms for
+ R)
R)],
llR*+&5+
+ 14R+2R2).
17R+5R2)
a
gl+R
q=3(1-
uo/q =-& w/q= f, uo/q= -;, o/q = $9
again
.he influence of the :ibution can be seen listribution does not M (1 - p)/3. The ssume the simplest
1-R’
v2=-1-2R+3Rt,
A;(“) = ’ - R ~[(1 +35*(2+
function in the case of (the result of numeriII = 22). for Y = 0.01, :ircles )
u*/q=
$
v*/q = -$.
Fig. 4 shows the dependence of the first seven probabilities and the final weights Al”’ = A;“‘/( I - R), j=0,1,2,ontheparameterRat~=l.Forl-R< 1, all the three weights are close to l/3 independently on 5. For certain values of R or 6, the coefficient q can be pure imaginary. Then one should replace the hyperbolic functions in Eq. (26) by their trigonometrical counterparts. Nonetheless, no significant oscillations of functions Aj( t) are observed due to the inequality CY> IpI, although the evolution of these functions
E.C. Caparelli et al. /Physics Letters A 232 (1997) 175-182
181
0.6
... --
P”lV
P,@)
p,ir,
\
(1 ‘\
P.
0.3
:
‘\\ ,,/
‘b’e~s ‘.\
l’ 0.2
:--::-:
.-..
‘\ ‘l.
,,” i’
\
.....
0.2
0.3
0.4
0.5
0.6
0.7
0.6
0.9
00
1.0
R Fig. 4. The stationary probabilities pi’“‘, j = 0.1,. ,6 (solid cmves, from top to bottom), and the stationary weight functions A!“), j = 0.1.2 (dashed curves, from top to bottom), versus thk parameter R in the case of strong unsaturated three-photon processes with 5 = 1.
(here r3 3
pa(z)
fi:“‘/D$“‘) ,
= $[erZ +2e-‘z12cos($JSrz)],
m(z) =
k{erz
- e-“/*[cos(
rp2Cz)
iv?rz>
- &sin(i&z)]},
= ${erz
-ee-‘z~2[cos(~J?;rz)
+ &sin(+&z)]}.
0.1
0.2
0.3
0.4
i !
0.5
--.--.__._._
0.6
0.7
i
0.6
0.9
1.0
7
Fig. 5. The probabilities po, PI, and p2 versus the slow time r = it in the case of strong unsaturated three-photon processes, obtained by numerically solving the system (2) truncated at n = 22, with the initial conditions PO(O) = 0.2, pi (0) = 0.2, pz(O) = 0.6; for v = 0.1, R = 0.25, 5 = 0, Df) = 1. The asymptotical solutions (26) practically coincide with the numerical results for r > 0.05. 5.
may be not so monotone as in the two-photon case: see Fig. 5. An account of weak two-photon processes does not change the time dependence of slowly varying weights Aj ( t) (26)) but the explicit expressions of the coefficients Q, p, etc.., become too cumbersome to give them here. The same is true for completely saturated three-photon emission, when the asymptotical GF is given by Eq. ( 12) with the following functions pj (z )
0.0
i
A_
v
0.1
I
Conclusion
We have found analytical asymptotical solutions of the master equation for the diagonal elements of the density matrix, when M-photon processes dominate over the processes with k < M photons. These approximate solutions are in excellent agreement with the results of exact numerical calculations. Relatively simple explicit expressions for all the coefficients, determining the form of solutions, are given for M = 2 and M = 3. In a generic case, one has to solve a system of M - 1 linear inhomogeneous differential equations of the first order with constant coefficients. It is worth to note that the proposed approach cannot be used in the opposite case, when k-photon processes are disturbed by weak processes of the order M > k. Exact stationary solutions in the case M = 2, k = 1, given in Refs. [ 28,291, show that even very weak higher-order processes change the picture drastically.
182
EC. Caparelli et al./Plysics
Acknowledgement This research was supported by FAPESP (Brazil), projects 199513843-9 and 96/05437-O. SSM thanks CNPq, Brasil, for partial financial support.
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[ 13I 114 I [ 151 [ 161