PHYSICAL REVIEW A VOLUME 48, NUMBER 3 SEPTEMBER 1993

Effect of atomic decay in degenerate two-photon micromasers

E. S. Guerra, J. Pion, and N. ZaguryDepartatnento de Fisica, Pontificia Uni Uersidade Catolica do Rio de Janeiro, Caixa Postal 38071,

22452 Rio de Janeiro, Rio de Janeiro, Brazil(Received 1 March 1993)

We study the effect of the atomic decay and the pumping statistics in degenerate two-photon oscilla-tors. We obtain an expression for the gain that is valid for all values of the atomic lifetime. Numericalresults are presented for regular and Poissonian pumping statistics.

PACS number(s): 42.50.Dv, 42.52.+x

I. INTRODUCTION

Production of beams of Rydberg atoms [1] and the de-veloprnent of superconducting microwave cavities withhigh-Q quality factors have given an important contribu-tion to recent developments in quantum optics. Rydbergatoms have very large electric-dipole coupling to the radi-ation field, transition frequencies in the microwave re-gion, and long spontaneous-emission lifetimes in freespace. Thus, if a very-high-Q cavity is tuned in a low sin-gle mode to an atomic transition between Rydberg levels,the atom-cavity system may be described to a good ap-proximation by a two-level atom interacting with a singlemode of the electromagnetic field. The realization of thisvery fundamental system can be used to test basic resultsof quantum mechanics.

The injection of a beam of Rydberg atoms in supercon-ducting cavities resonant or quasiresonant with atomictransitions has rendered possible the realization [2,3] ofnew kinds of masers, the micromasers, that may operateat very low thresholds down to at most one atom at atime inside the cavity. The properties of the one-photonmicromasers have been predicted by Filipowicz,Javanainen, and Meystre [4], and the theory of the degen-erate two-photon rnicromaser was developed by Brune,Raimond, and Haroche [5] and Davidovich et al. [6].

Due to the quantum properties of the atoms and of theradiation, the field generated in microscopic maserspresent properties very different from those found inmacroscopic masers, like collapses and revivals [7] andsub-Poissonian or super-Poissonian statistics [4,5,8].Atoms that pass through the cavity may also be used asdetectors to manipulate the field inside the cavity [9],testing basic results of quantum mechanics. Sources ofAuctuations, such as thermal field, irregular distributionof atoms in the beam, wide atomic velocity distributions,radiation loss, and spontaneous atomic decay, however,contribute to wash out the specific quantum effects of theatom-field system. In the first theoretical studies of one-photon micromasers [4], two-photon degenerate micro-masers [5,6], and two-photon nondegenerated micro-masers [10],radiation loss, cavities in contact with a heatreservoir at a low temperature, and velocity distributionswere taken into account, but it was assumed that theatoms do not decay during their Aight in the cavity and

that the arrival of atoms in the cavity obeys a Poissonianstatistics. The effect of the pumping statistics in one- andtwo-photon micromasers was discussed recently [11—13],and it was shown that it may be more important than inmacroscopic lasers and masers [14]. The effect of bothspontaneous emission and pumping statistics for the one-photon micromaser was studied in Ref. [15] using amesoscopic approach to derive a master equation and anexpression for the gain valid for all values of the lifetime.

In this paper we analyze the role of spontaneous emis-sion and of the pumping statistics in degenerate two-photon micrornasers using a microscopic approach to de-scribe the field evolution. In Sec. II we describe our mod-el, obtain an expression for the gain valid for any value ofthe atomic decay constant, and discuss the limit of largephoton numbers. In Sec. III we apply the results of Sec.II to perform calculations for different values of the de-cay constant and for Poissonian and regular pumpingstatistics. In Sec. IV we present our conclusions.

II. THE MICROSCOPICAND SEMICLASSICAL APPROACHES

Consider a fiux of three-level atoms (la), b), andlc ) ) being injected into a cavity tuned to a frequency co.la ) and c) are the upper and lower lasing levels, andlb ) is a relay intermediate state with energy fico& nearlyhalfway between the energies %co, and %co, of the lasinglevels. 61 ~ b ~ ~ ~b ~ and ~2= ~ b

=cob —co, —co are the detunings chosen in such way as toenhance the two-photon transition probability while atthe same time keeping negligible the resonant one-photoncascade a~b —+c. We will call A, b and Qb, the one-photon Rabi frequencies associated to the transitionsa~b and b~c respectively. The Hamiltonian that de-scribes the atom-field system, in the rotating-wave ap-proximation, is given by

H=HO+ V,where

H, =a~&'&+a~. la &&al+g~, lb&&bl+&~, lc&&cl

(2)

describes the free Hamiltonian and

1050-2947/93/48(3)/2261(9)/$06. 00 48 2261 1993 The American Physical Society

2262 E. S. GUERRA, J. PLON, AND N. ZAGURY 48

&=&&.b(A la &(b I+ A'Ib &(a I)

+an„(A lb&(el+A'Ic&(bl) . (3)

The operator A (A ) in Eqs. (2) and (3) is the annihila-tion (creation) operator for the electromagnetic fieldmode of frequency cu quasiresonant with the one-photontransitions a ~b and b —+c. In the interaction picturethe Hamiltonian becomes

V I=A'ge ' A Ia &(b +Age ' A Ib &(c +H.c. , (4)

where for simplicity we took Q,b=Ob, =g, which is ap-

proximately true in many cases of interestI 3,5,6].

We now take into consideration the spontaneous emis-sion from each level by using the Weisskopf-Wigner ap-proximation I16]. Consider that the level

Ij & (j=a, b, c)decays to another level Ij'& (j'=a', b', c'). The state of

I

the atom-field system at the time t; of arrival of the ithatom is

ItIt(t;) &= g I c., N, [pj(t; )I a&+cb, N, [Q[(t; )lb &

+c N [oj(t, )lc&]ln& .

C, N [oj (j=a,b, c) is the probability amplitude for theatom to be found in the j level without having emittedany photons by spontaneous decay and the laser fieldphoton number to have the value N. C ' N I & I

(j '=a', b', c') is the probability amplitude for the atom tobe found in the j ' level, having emitted one photon byspontaneous decay in the r mode and the laser field pho-ton number to have the value X. The equations ofmotion for the atom-field system in the interaction pic-ture are given by I16]

Va ib, )rCa, N, [oj(t;+7 )= C,N, [ jp(t;+&) —ige ' &N+1Cb, N+1, [oj( '+&)

+b —iA)w i h)~Cb, N+1, [pj(t;+7 )= Cb, N+1, [ p(jt; +&)—ige +N+1Ca, N, [oj(t, +r) ige —' &N+2Cc, N+2, [oj(t, +2-), (6)

Ve —i b, 2rC N+2 [pj(t;+r)= — C, N+2 [pj(t, +r) —ige ' &N+2Cb N+1 [pj(t;+r)

where the dot represents the derivative d /d ~ and y„yb, and y, are the spontaneous decay rates from levels a, b, and c,respectively.

The atom stays in the cavity during an interval of time r=t;„,. We will solve Eq. (6) in the case where the detuningsare chosen as 6,= —62 =6, which implies 2' =co, —co, . We also take equal decay constants from the lasing levels,

y, =y, =y. In this case the solutions of the above equations at the time t; + t;„, are given by

~ —,N ~+,N

( + )— 1' int C ( )

' '[ j t(

—, N int +,N int

)—yt 2

a N [Qj tt tint e 'a N [pj ~ —,N ~+,N

Nl (NC, N, [oj(t )+gN+1C N+'2, [oj(t)]'1+ 1—~+,N~ —,N

r t. r t-(r e +Nint r , e —,N int)—,N +,N

~b';nt b»+1 [oj "+ N'tnt "—N'tntCi N+1 [oj(t +t t) =e '"' ' ' (r Ne

+' '"' —r+ Ne ' '"')I —,N "+,N

[(NCa, N, [oj(ti)+kN+1Cc, N+2, [oj(ti)l —r Nt t—r+ Nt

I +,N

( + )— r int C ( )

» [ j t(

—, N int + N int

)/2 N+1 bN+1 0(i)

e, N+2, IOI t; t;n) =e e, N+2, IOI t; e ' e~+,N

N+1[(NCa, N, [0)(ti )+gN+1Cc, N+2 [Oj(tl )]+ 7

I +,N~ —,N

where gN =ig &N + 1 and

N P+ N

r t. r t.(r e +,N int r e —,N int)—,N +,N

f+ N

with

(5, bi+)+tItQ(5, bi+6, ) 4ANb, —2

g (2N+3) 'Ya Xband 5 b 2

Assuming that the atoms get into the cavity in the excited state Cb N+1 [pj(1' )=C N+2 [pj(t )=0]., the solution of Eq.

EFFECT OF ATOMIC DECAY IN DEGENERATE TWO-PHOTON. . . 2263

(7) simplifies to

—yt, „,/2.N [0[(;+ .t)= '"' .N [0[(kN

r+, N"—,N I"—,N "+,Nr t r t.

(r e +,N int r e —,N int)—,N +,N

—rbt;„, /2"—N' t "+N' t

Cb, N+l, [0[(t +ti t) 4N(~+,N

(10)

a, N, [0[ ti aNKN+11 int C ( )p

c, N+2, [0[ i int+,N I'+, N

r t. r t.(r e +,N int p, e —,N int)—,N +,N

In order to have the two-photon transition at frequency co=co,&/2, keeping at the same time the one-photon cascadea ~b ~c negligible, we consider the situation where flN/b, && 1. Also, we may take (QN/b)e " '"' &&1, since we areinterested in situations where yt;„t is not too large (otherwise the populations of the lasing atomic levels will be stronglydepleted). Using those approximations Eq. (10) may be written as

T

—y t» (N +1) inNt, „,Ca N [O[(tt tn, )=Ca N [O[(tt )e '"' + (e " '"' —1)

Cb, N+1, [0[( tt +tint ) ='0 t

+ ) C ( )rt;„,—» '1/(N + 1)(N +2)

(&Nt;„, 1)c,N+2, IOI i int a, N, IOI i (2N+ 3)

that is, we obtain the result for infinite lifetimes [6] times the usual exponential decay factor.To obtain the reduced density matrix of the field in the number representation after the passage of one atom, we

proceed as in Ref. [15] and get

int int

PN M(ti+ itlt ) Pa, NQM( i +tirtt )+1 a PaNaM( i ++)dr+PbN, bM( i + int )+ vb PbNbM( i

+p,N, M( ti + ttnt ) +yc f pcN, cM(t, + r)dr (12)

where p;N ~ =C;NC.~. We will be interested only in thediagonal elements of pNM. Substituting (11) in (12) andsolving the integrals, we have

micromaser:

4(N + 1)(N +2)N+1 (2N+3)2

(16)

PN N(t; +t;„)=[MP(t; )]N N

(1 PN+2)PN, N(t, )

+P p„,„,(t, )—=rwhere M is the gain operator and

(N+1)(N+2) (2N+3) /(2Nr)(2N+3) 1+ (2N+ 3)

(2N )

X 1 —e '"' cos(QN t;„t )

2N+(2N +3)

(13)

(14)

In the other limit, we assume that the interaction time ofthe atoms with the field inside the cavity is very large,and we obtain the expression of the gain for the two-photon laser:

2(N+1)(N+2)4N +(2N+3)

(17)

Now we include field dissipation. The usual loss con-tribution to the density operator is given by [17]

(p)i„,=Lp= (NT+1)(2ApA —A Ap —pA A )1

cay

+ NT(2A pA —AA p —pAA ),1

2t pgy

(18)

where we have defined

N =yA/2g

We consider two limits of this expression. In the firstwe assume that the atoms do not decay while they travelthrough the cavity. In this case we obtain the usual ex-pression [5,6] for the gain of the degenerate two-photon

where NT is the mean photon number for a cavity inequilibrium at temperature T. Both the finite tempera-ture and finite lifetimes will introduce fluctuations in thesystem. As we are interested in studying the influence offinite lifetimes, we set NT=O. We also assume that thefield losses may be treated independently of the interac-tion of the atoms with the field during the time the atom

2264 E. S. GUERRA, J. PLON, AND N. ZAGURY 48

passes through the cavity (a condition sufficient for thisto happen has been discussed in Ref. [11]). Therefore, thechange in the reduced density operator for the field dueto the passage of the ith atom is given by

Lht, .

p(t, +, )=e 'Mp(t; ), (19)

where b, t; =t;+, t; an—d L is given by Eq. (18).We study two extreme cases of the pumping statistics:

Poissonian and regular. In the Poissonian case, the dis-tribution of time intervals b, t; is given by r exp( rb, t;—).Taking the statistic average of Eq. (19) over this distribu-tion, we have for the change in the mean-field operator

p( t, +, ) = (1 L /r —) 'M p( t; ) . (20)

We are interested only in the diagonal elements of thedensity matrix and, from now on, we use p»=~~.From (13) and (20) we get the photon population at thetime of arrival, I;;+ &, of the i + 1 atom in terms of the pho-ton populations at time t;:

25

&n )„=P„, (27)

and for the regular case we have

e'+1 8 1 N(29)

Now we assume that the population distribution ischaracterized by a narrow peak at N=N= (N ). We ex-pand the gain coefficients Ijiv+2 in the neighborhood ofN+2 to first order in X—N and retain only terms up to(N N) in—Eqs. (26) —(29). In this approximation theprobabilities of finding the atom in the lower level are

where n =N5 =N /(2N, „).We may also obtain from Eqs. (21) and (22) expressions

for the variance X=(N ) —(N) . In the Poissonianpumping case we have

(28)

N(ti+1) (1 pN+2)&iv(ti )+pN&N —2(ti )

1+ [(N+1)sr~+, (t;+, ) N~iv(t—;+,)],P =&n)

26

25

(30a)

(30b)

(21)

where

—6AXMp(t)(A )ie (22)

(23)

The diagonal elements of Eq. (22) are

where N,„=~t„, is the number of excited atoms thatenters the cavity during t„,.

For the regular case ht; =t„= const. Equation (18)may be integrated [11],and we have

(1——2s)~(+„)=g ."

( )'j=0

and the normalized variances of the field, a =X/(N),are given by

3

2(1 —P'— )

1r=f iv+2

(31)

(32)

where

i)13iv+ z~N+2 N=X

(33)

(34)

As we shall show below 1 —P~+z is positive for stationary

stable solutions. From Eqs. (31) and (32) we have

(24)

P(t+t;„, ) = +A+i'iriv(t)N

(25)

in the steady state, and taking into account Eq. (13), wehave for the Poissonian case

(n), =P,and for the regular case

(26)

where I iv is given by Eq. [13].As it was shown in Ref. [12], it is possible to obtain a

relation between the total probability P that an atom bedetected in the lower-energy state just after its passagethrough the cavity and the mean photon number. Start-ing from

Similar results have already been obtained in the case ofno atomic decay using a mesoscopic approach [11]. Thelargest noise reduction is —,

' and occurs when (n )„=l.For lasers operating high above threshold (N ))Nr ) wehave P~ = —,', and therefore in this case

(n )„=(n ) =P, =P = —,',(35)

o-„=1, o =1.5 .

The solutions of Eqs. (30a) and (30b) in the semiclassi-cal limit may be obtained graphically. In Fig. 1 we plotthe right-hand side (gain) and left-hand side (loss) of Eq.(30a) for several values of N& and for a fixed value of thereduced interaction time P;„,=N,„yt;„,/N =3m. In Fig.1(a) we show the micromaser limit N =0 (y=0), and in

Fig. 1(b) we plot the loss and gain curves for Nz =3, 5,7.5, and 15. The stable solutions are indicated by circles

48 EFFECT OF ATOMIC DECAY IN DEGENERATE TWO-PHOTON ~ ~ ~ 2265

N,„Np ( 1 + [ 1 ( 2N /Ne ) ] j (36)

in the case of Poissonian pumping and

1 1N„= (1+[1—[2N~(e 1)] ]' )— (37)

in the case of regular pumpin . We h'

g. e ave a nontrivial

1.0

O. BV3V3C3 0.6

0.4

in the f urgures. As we increase the value of 1V

e c aracteristic multistable solutions (circles) of the mi-cromaser to a single nontrivial (NWO) blsta e solution at

, and 7.5 in Fig. 1(b)] and finall to thbelow the

a y o e regione threshold for oscillation (N )7.5). Notice

that for stable solutions the slo f h 1

Fi. 1

e s ope o t e loss curve (1 inig. 1) should be greater than the 1 f h

curve, and therefore 1—e s ope or the gain—

Pz+2 is positive. Similar resultshold for the solutions of Eq. (30b). In h'

of the losn t is case the slope

age values o X fo e oss ine is greater than one and th fere ore tne aver-age va ues o X for the regular pumping are smaller thanthose obtained in the Poissonia Whsimple analytical solution of Eqs. (30a) and (30b) may befound. For 1V)&1 the nontrivial t bl 1s a e so ution is given

N, „/2 for Poissonian pumping

1

2(e —1)for regular pumping . (3&)

At threshold the equals sign holds and X hvalue as' .

o s an as the same

r

III. NUMERICAL RESULTS

In this section wewe present numerical results for theprobability of finding the atom th 1m in e ower lasing levelafter it crosses the cavity P whi h

'd

able, w ic is a directly measur-

or e normalizeda e quantity for micromasers, and for thevariance of the field a in the stationa t t . Wry s a e. e analyze

e behavior of degenerated two- hotonvary the lifetirn

-p o on osci lators as wevary e i etirne of the decaying levels in the case where

InFi s. 2—7weg . — present results as a function of the re-duced interaction time;„t=X t N .s ow the e6'ect of a finite lifetime in th b h

e statistics, we calculate ourN =yb. /2 =0.0 . org = . , 0.1, and 1.0. The results for 1V =0had already been obtained in Ref. [11].

r

gion close to the semiclassicalFirst we discuss the ret reshold for micromasers. In F' 2 d 3suits for P and o.. The lines for N =0

igs. an we show re-

~ ~ ~

nes or =0 and 0.1 are versimilar, while the ones for N =1.0r = . are displaced in Por a small amount.

int

that, in this r1 ount. This is a consequence of th f

is region, the variable yt;„, =N P /N ise o e act

small even for N =1.0int r int ex

, and from the expression for the

0.2

0.0 I i I i I

0.0 0.2 0.4 0.6 O. B 1.0

0.6

V20.4

C3 0.5—

0.00.0 0.5 1.0

0.8

(n)FICz. 1. Loss and ain c

~ =3, 5, 7.5 and 15. The circles indicate stable solutions

FICr. 2. Stationary probability to measure the low tower atomic„( gular pumping statistics) and P (P

'oissonian statis-

n the re a res o as a function ofn the region of the semiclassical thr h ld

dN =10(d h dl' )

a „, (b P, P~. N~=0 (solid line), N =0.1 (d

2266 E. S. GUERRA, J. PLON, AND N. ZAGURY 48

(b) (b)

b

1.5

20

0.4 0,6 0.8 0 3I

4

FIG. 3. Stationary normalized photon-number variance forcr„regular pumping statistics) and o.

~ (Poissonian statistics) int e region of the semiclassical threshold as a functio f +~ ( )unc ion 0 glut. (a)

(b) ~ I/ wp &y:0 (solid line), &~ =0. 1 (dotted line), andN~ = 1.0 (dashed line).

gain, Eq. (14), we expect results very close to the infinite-lifetime limit.

As we see in Fig. 2, the total probability that an atomis detected in the lower-energy state for regular pumpingP„starts to build up before the total probability that an

FIG. 5. Stationary probability to measure the lower atomicstate P„(regular pumping statistics) and P„(Poissonian statis-tics) as a function of P;„,. (a) P„, (b) P„/P . N =1.0.

atom is detected in the lower-energy state for Poissonianpumping P . This means that the oscillation threshold islowered as the pumping is regularized. This had been no-ticed already, for infinite lifetimes, in Ref. [11]. These re-sults are valid for the steady state only, and one should becareful in comparing them with experiments, since thetime to reach the steady state may be very large in this

(b)

1.5

(b)

0.5

0 0

010 P, O 30

0 ~/~

40 50 60

FIG. 4. Stationary probability to measure the lower atomicstate P„(regular pumping statistics) and P~ (Poissonian statis-tics) as a function of P;„,. (a) P„, (b) P„/P . N~ =0 (solid line)and N~ =0.1 (dotted line).

00

I i I

10 20 30 40 50 600 t/~

FIG. 6. Stati'onary normalized photon-number variance forcr„(regular pumping statistics) and o.

~ (Poissonian statistics) as afunction of P;„,. (a) o „(b) o „/o~. N~ =0 (solid line) and

X~ =0. 1 (dotted line).

48 EFFECT OF ATOMIC DECAY IN DEGENERATE TWO-PHOTON. . . 2267

b

b

0.1. Only in certain regions of P;„, is the singularbehavior that appears in the infinite-lifetime case(N =0) present when N =0.1. This is particularly truein the region of the semiclassical threshold, as we havediscussed above, and around P;„,=30m for N~ =0.0. ForP;„,=20~ and 40m, the. sharp decrease in P„and P~ (Fig.4) for N =0, associated with trapping states [18] atN=O, disappears when N~=0. 1, while the other one atP;„,=60~ still remains, but the minimum is no longerequal to zero when Nz =0.1. There is a sharp rise in P atP;„,= 30m. At this point Eq. (14) reduces to

20—(N+2)(N+1) 1+e(2N+3) +4N

(39)

io—

FIG. 7. Stationary normalized photon-number variance foro.„(regular pumping statistics) and o.

~ (Poissonian statistics) as afunction of P;„,. (a) rr„, (h) o „/cr~. Nr = 1.0.

region [12]. The steady-state ratio P„/P may reach verylarge values, like =4 ( =3), for P;„,=0.53~ (0. 57m ) when

N~ =0 or 0.1 (1.0).In Fig. 3 we plot the results for o.„and the ratio

o „/o. . We notice that in this region very large values ofo.„and o. occur due to the existence of more than onepeak in the photon distribution (one of them associated tothe stable solution at N=O in the semiclassical limit).Also, noise reduction occurs for P;„,~0. 5m. and reachesextremely small values, like o.„/o „=0.08 (0.12) atP;„,=0.63m. (0.68m. ) for N =0 or 0.1 (1.0).

%'e now discuss the behavior of P and o. outside the re-gion of the semiclassical threshold. Some features of thebehavior of P and the scaled variables o. for infinite life-times still show up when N~=0. 1 (Figs. 4 and 6). ForNz =1.0, P and o. present features similar to the ones forthe two-photon micromaser in the case of an incomingbeam with a large dispersion in velocities (Figs. 5 and 7).This is due to the fact that both kinds of atomic Auctua-tions induce fluctuations in the gain coefficient, whichtends to wash out the X dependence of the effective two-photon Rabi frequency A~. From Eq. (16) we see im-mediately that when Nr =0 (no spontaneous decay fromthe lasing levels), the gain coefficients P~+z are periodicfunctions of P;„, with period 4vrN, „and are symmetricwith respect to 2~N„. These symmetries are shown ex-plicitly in Figs. 4 and 6 for the curves with X =0, whichshow a period of 60~ and are symmetric with respect tothe point 30~. Some remnants of these symmetries arestill present for Nz =0.1, but they disappear completelywhen Nz = 1.0 (Figs. 5 and 7).

In Fig. 4 we show the steady-state results for P for bothregular and Poissonian pumping and for iV =0.0 and

Therefore, for N =0, Pz+2=1, that is, the atoms leavethe cavity with a high probability of being in the lowerstate (an "opaque cavity"). When Nr =0.1, however,Pz+z --0.77, as predicted by Eq. (39), and the atoms arenot able to deliver all their initial energies to the cavity.

In Fig. 5 we show the steady-state results for P„andthe ratio P„/P at N ='1.0, a situation that approaches alaser model. The results are qualitatively similar to thoseobtained in Refs. [11] and [12] for an injection of atomswith large dispersion in velocities. As we expected fromSec. II, we have P„=P = —,', except for small values of P;„,( & 2.0~, in this case).

The results for the normalized variance for the regularpumping and the ratio o., /o. at Xz =0 and N& =0. 1 areshown in Fig. 6. The main features observed in theinfinite-lifetime case (N =0) [11] are still present for

=0.1; in general, we have noise reduction as we regu-larize the beam, but noise arnplification may occur whenmore than one peak is important in the stationary photon

0.5—

0.4

0.3

0.2

0.1

10

FIG. 8. Stationary probability to measure the lower atomicstate. The solid line corresponds to regular pumping statisticsand the dashed line to Poissonian pumping statistics. P;„,=3m. .

2268 E. S. GUERRA, J. PLON, AND N. ZAGURY 48

1.2

0 9

0.8

0.7

0.6—

that P„=P and that it goes to zero as we increase Xand pass the semiclassical threshold for oscillation. o.,[Fig. 9(a)] and the ratio cr„/o [Fig. 9(b)] change fromtheir values in the micromaser case, N~ =0, to approxi-mately the semiclassical values of 1 and of —,', respectively,in the region 2~%A ~4, which corresponds to lasersoperating above threshold.

IV. CONCLUSIONS

10

FIG. 9. Stationary normalized photon-number variance as afunction of X~ for P;„,=3'. (a) cr„, (b) cr„/cr .

distribution. Besides the small values of a, /o. attainedin the semiclassical threshold region, noise reduction isalso important for an "opaque cavity" (P;„,=30m) andfor two symmetrical points around P;„,=28~ and 32trthat are associated with dominance of low-N peaksaround N =6 in the photon distribution [11]. Fortt;„,=30tr, as we would expect from the approximate re-sults of Eqs. (34) and (39), we obtain cr„/o.~ =0.33 (0.5)for N =0.0 (0.1).

In Fig. 7 we show the results for the normalized vari-ances at N =1.0. For $,„,~2' we have a noise reduc-tion of =—', , which is the value predicted for the laser.

Until now we have shown results for fix %„=15andfor three values of N as a function of P;„,. In Figs. 8 and9 we show results for fix P;„,= 3' and N,„=15 as a func-tion of Nz, which is proportional to y for fixed P;„, andN„, from N =0 to 10. This corresponds to change con-tinuously from a micromaser model (N =0) to a lasermodel operating below threshold (Nr )7.5) as we de-crease the lifetime of the atoms. We observe in Fig. 8

In this paper we have studied the field statistics oftwo-photon degenerate micromasers, taking into con-sideration the finite lifetime of the atoms. We show thatthe field statistics is very sensitive to the value ofyt;„, =Nrem;„, /N, „[see Eq. (14)]. In typical micro-maser experiments one uses Rydberg atoms. Theirspontaneous-decay constant in free space is [19] @=10s '/n for atomic transitions from a state with principalquantum number n and a low orbital angular momentumI = 1 and are y = 10 s '/n for circular Rydberg states(l=n —1). Typical interaction times for atoms withthermal velocities =300 ms crossing microwave cavi-ties are of the order of 10 s, therefore the product y t,„,varies as =10 /n for Rydberg atoms with low l and as=10"/n for circular atoms, which shows that atomicdecay effects may be important even when we are dealingwith micromasers.

We have considered the effect of finite lifetimes only inthe case where that atomic decay constants from the las-ing levels are equal. New important effects should appearwhen they are different. For example, we would expectnoise reduction when the rate of decay from the upperlasing level is smaller than the one from the lower lasinglevel, y, )y„since there is an inhibition of photon reab-sorption in this case.

ACKNOWLEDGMENTS

The authors wish to thank Marcia Fontenelle for herhelp in preparing the figures. One of the authors (N.Z.)wishes to thank Professor A. F. T. Piza and the Institutode Fisica da Universidade de Sao Paulo for the hospitali-ty extended to him, during which this work was complet-ed. N.Z. was supported in part by Fundagao de Amparoa Pesquisa do Estado de Sao Paulo and Conselho Nacion-al de Desenvolvimento Cientifico e Technologico.

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