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Bunching and antibunching of de-excited atoms leaving a micromaser
Optics Communications 85 ( 1991 ) 508-519 North-Holland
OPTICS COM MUN ICATIONS
Full length article
Bunching and antibunching of de-excited atoms leaving a micromaser H. Paul a n d Th. R i c h t e r ('entral Institute oJ Optics and Spectroscopy, Rudower Chaussee 6. O- 1199 Berlin, Germany Received 26 April 1991
The joint probability of detecting two successive atoms both in the lower state at the exit o f a micromaser is calculated. We find that this joint probability, in dependence on the atomic flux Nez, takes values greater (bunching) or smaller (antibunching) than that for uncorrelated events. The regions of Ne, where antibunching (bunching) occurs, correlate fairly well with the regions where sub(super)-poissonian statistics for the number fluctuations of atoms leaving the maser in the lower state has been predicted and recently observed. This coincidence is shown to result from the fact that both the bunching/antibunching phenomenon and the number fluctuations are governed by the photon statistics of the maser field in a similar way.
!. I n t r o d u c t i o n
The single-atom micromaser is a system of fundamental interest in quantum optics. It allows to study the interaction o f an atom with only a few photons and has recently been investigated in detail both experimentally [ 1-3] and theoretically [ 4 - I 1 ]. The m icromaser consists of a high-Q cavity into which highly excited Rydberg atoms are injected, one after the other, so that a single-mode microwave field is built up inside the cavity, provided the mean time interval between atoms is shorter than the cavity decay time. The micromaser field shows a number of non-classical properties that arc averaged out, however, in usual lasers and masers [4,5 ]. In particular, the photon statistics of thc micromaser field can be sub-poissonian [4,5 ]. In the experiments the energy state of the atoms after they have passed the cavity, is monitored by field ionization measurements, from which some information about the cavity field can be obtained. In particular, an analytic formula has been derived [12] which connects the steady-state photon statistics of a micromaser with the number fluctuations of de-excited atoms leaving the micromaser. Utilizing this formula, nonclassical photon statistics has recently been observed [3 ] via sub-poissonian atomic statistics in excellent agreement with the theoretical predictions [ 12 ]. In the derivation of the formula in question it proved sufficient to take into account the change of the probability P to detect an atom in the lower state merely from one measurement interval to the following, thus considering P as a function of the (fluctuating) number of excited atoms N that enter the cavity during a measurement interval. On the other hand, when the atomic analogs o f photon bunching and antibunching, i.e. correlations between the excitation states of successive atoms are studied, as will be done in the present paper, the basic effect is the change of the probability P for the second atom due to a change of the field resulting from its interaction with the first atom. So our procedure will be quite different from that adopted in ref. [ 12]. We will start from the well-known steady-state solution for the micromaser field [4] and calculate, via state reduction, the joint probability of finding two successive atoms in the lower state at the exit of the cavity. We will speak of bunching or antibunching when this joint probability is greater or smaller than that for uncorrelated events, i.e. the square of the probability for the corresponding single atom event. It will be shown that the bunching/antibunching phenomenon is directly connected with the steady-state 508
0030-4018/91/$ 03.50 © 1991 Elsevier Science Publishers B.V. All rights reserved.
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photon statistics. Moreover, it turns out that in certain operating regimes the bunching effect becomes enhanced with growing spacing between the two successive atoms. In this paper, an analytical formula for the joint probability will be derived. In doing so. we must explicitly take into account the damping of the cavity field. This gives us an opportunity to reconsider briefly the micromaser theory and to rederive the steady-state photon distribution in a slightly different way, compared to ref. [4].
2. Micromaser theory reconsidered We consider a situation in which a monoenergetic beam o f two-level Rydberg atoms prepared in the excited state is injected into a high-Q single-mode maser cavity. The injection rate will be so low that at most one atom is present in the cavity at a time. Moreover, we assume that the atom-field interaction time t,,, (the same for all atoms) is much shorter than the cavity damping time To= l/),, so that the damping of the cavity field can be ignored while an atom is inside the resonator. Then the interaction of the two-level atom with a resonant single mode of the cavity field is described by the well-known Jaynes-Cummings model [ 13 ]. Suppose that at time t, the ith excited atom enters the cavity containing a field that is diagonal with respect to the Fock states I n ) . Then the combined density operator for the atom in the upper state [e) and the field at time t, reads p (t,) --- 5"~ Pn ( t, ) [ e, n ) ( e, n I, where Pn gives the probability to find n photons in the field. After the interaction time tin, we have
p(t, +t,.,)-- Z p.(t,)
{cos2(x/n4-1~tm,) [e, n ) (e, nl +sin2(x/~-+
l-Qlint)
Ig, n + 1 ) (g. n + 1[
¢1
+ i sin ( x / - n + l ~ t , , , ) c o s ( x / / ~ l~2ti,,) [Ig, n + 1 ) (e, n l - le, n ) (g, n + 11 ]} ,
(1)
where [g) denotes the lower atomic state and g2 is the atom-field coupling constant. Forming the trace over the atomic states shows that the reduced density operator for the field remains diagonal during the interaction and that its matrix elements are given by
p,,(t, +tin,)
= ( 1 - / ~ . + , )p.(t,) + p. p . _ , ( t , ) .
(2)
where /~,, = sin 2 (x/ng2ti.,) •
(3)
In the interval tp between t, + tin, and t,+~ when the next atom enters the cavity, the field density matrix evolves according to the standard master equation for a damped harmonic oscillator. Again the diagonality of the field is preserved during its decay, and the equation of motion for the diagonal elements reads [14] .b,, = y ( n + 1 )p,,+ ~-7np..
(4)
where for the sake of simplicity we have assumed that the temperature of the cavity walls is low enough so that thermal photons can be neglected. The well-known solution to eq. (4) is given by
p.(t,+t,.,+tv)=
'~ B,,..+k(tp)p.+~(t,+t,.,).
(5)
k=0
where B ..... ~(lP)=(n+nk)exp(-nyt.)
[l--exp(--Ttp)] k,
(6)
as can be easily verified by insertion into eq. (4). Hence combining eqs. (2) and (5). the photon distribution at time t,+~=t,+t,.,+t w just before the next atom enters the cavity, is given by 509
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p,,(t,+,)= Z B ...... ,(t,,)[(l--fl,,+k+,)p,,.k(t,)+fl~+kp,,+k_,(t,)].
(7)
-: I)
where we have to put p _ , = 0 . Generally the atoms are not injected with equal spacings. Rather the injection times are distributed according to a Poisson process so that the probability distribution for the time intervals q, between successivc atoms is given by an exponential distribution P(tp)= ( l / t ~ ) exp(-to~i-p), where {p is the mean spacing between two atoms. Successive iterations of eq. (7) and averaging over the random time intervals t~ between the atoms, indicated by a horizontal bar, lead to the return map ~. /~,..... k[(l--fln+~+,)P,,+k(t,)+fl,+kP,,+~-,(t,)].
p,,(t,+~)=
(8)
k=O
which relates the average photon distribution p~(t,+,) after the passage of i atoms through the resonator to the average distribution/~(t,) after the passage o f i - 1 atoms. The average quantities/~,.,,÷k defined as .x,
(9)
fi . . . . ~ = j dto P(tp)B,,.,,+~(to) o
characterize the photon statistics which results, on the average, from the damping according to eq. (5) p,,(t,+,)=
Z fft,,.~+kp,,~(t,+t,n,).
(10)
k=O
Upon insertion of the binomial expansion of [ 1 - exp ( - 7I) ] ~ into (6) the integral (9) is easily evaluated to yield
where .~,'~,= "1"~./{o = l/yi-o is the mean number of atoms traversing thc cavity during thc decay time T~. of the maser field. The Appendix A shows that the quantities /~,,.,,+k fulfill the recurrence relation
n+k+ 1 /~,,.,,+~+ 1 - N,., + n + k +
B ..... ~ 1
(12)
"
and hence can also be written in the form k
n+m
g ...... k = B .... , , = 1 7 , / ~ + n + m
(k>~l) .
(13)
where
B,,.,, =N,.~/(N¢, + n ) .
(14)
3. Recovew of the steady-state photon statistics Driven by the excited atoms injected into the cavity, the maser field eventually evolves toward a quasi-steady state characterized by a mean photon distribution p, which is a solution to eq. (8) with Pn(t,+ ~ ) = p , ( t , ) = p ,
p,=
~ B,.,,+~[(1--/3,,+,÷,)p,,+~+[3,,+~P,+k_,]. k=O
510
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At first sight, it seems hopeless to solve this infinite system of equations. Nevertheless, it is possible to rewrite cq. (15) in such a form that immediately shows up the steady-state solution. Indeed, using the property (12) and rearranging the terms in the sum we can rewrite eq. (15), after some algebra, in the form (see Appendix B)
_ _
0=
~
1
N~, + n
nn,n+k
[ntO.,-N~B.tO._,]+ k~_,,N.~T~--k+ 1 [(n+k+l)p,,+k+,--N~x[3,,.k+,to.,+kl.
(16)
Clearly, the right-hand side of eq. (16) is zero if for successive occupation numbers to, the relation
np, =N,.xfl,,to~_,
( 17 )
holds, so that we arrive at the following expression for the photon probability distribution to, ( n = 0 , 1, 2, ...)
_ ~.,-.! N~', f i & tO, = Po •
(n>~l),
(18)
k=l
where p o = P o ( N ~ ) depends on N~ and is determined by the normalization condition Y~'Loto, = 1. The analytical form ( 18 ) of the steady-state photon statistics has been first derived in ref. [4] in a very elegant manner. (Actually, the treatment in ref. [4] is more general since it includes the case of arbitrary cavity temperature. ) The authors noticed that the inverse of the map (10) has a very simple form. They found the inverse transformation immediately by averaging the evolution operator exp(Lt o) over G, where L is the operator in the standard master equation of a damped harmonic oscillator. We reproduce their result (for zero cavity temperature), starting from eq. (10) which we write in the form
tO,,(t,+,)-
N~p~(t,+t,.t)+ Nex + rt
"y. B ...... k + , P . * k + t ( t , + t , n t ) .
(19)
k=O
Next we make use of the relation
B...+~+,-
n+l N,.x+nB.+,..+,+k
(20)
which can be readily deduced from ( ! 3), and obtain
tO,,(t,+,)-
~v~n",+--N~" p , , ( t , + t , . , ) +
n+l oo N~-~-_(_nk~=oB,+c~+ , +~p,+~+ , ( t , + G , )
.
(21)
Replacing the sum on the right-hand side by ton+l(t,+! ) according to eq. (10), we thus find that the inverse of the map (10) has indeed the simple form of a two-term recursion relation
p,,(t,+ti.,)=p,,(t,+t)+
NLt[np.(t,+,)-(n+l)to,,+,(t,.~)],
(22)
in agreement with ref. [4]. With the help of this inverse transformation the steady-state condition ( 15 ) can be rewritten in the simple form
to,, + N L ' [ n,O. - ( n + I )to,,+ ,] = ( 1 - fl,, + , )tO,, + fl,,tO,, _ ,
(23)
from which the steady-state photon statistics (18) is easily obtained [4]• We note in passing that eq. (17) has a simple physical interpretation, it states a balance between the transitions that take place between neighbouring photon states [ n - 1 ) and ] n ) only. The "flow" of probability from I n - l ) to I n ) due to atoms delivering their excitation energy to the cavity field is fl,15,,_l/i-p (see the discussion following eq. ( 2 4 ) ) . On the other hand, the "flow" of probability from [ n ) to [ n - I ) due to the 511
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damping of the cavity field s given by N,.~ = I/;'t~, yiclds, in fact. e q (17).
7nP,,, according to eq. (4). Equating both flows and noticing that
4. Correlations between the e~:citation states of successive atoms We now turn to the calcuktion of the joint probability 14"(2)(g, g, to) of finding, under steady-state conditions, two successive atoms in the lower level just after they have left the cavity. According to eq. ( 1 ) the probability to detect an atom in the lower state at the exit o f the cavity is given by W'l>(g)=
~
fl,,+Ip,,=NL' ~n),
(24)
where ( n ) is the mean photon number in the maser field. The second equality follows from eq. (17) and has a simple interpretation: On the average l/{p atoms traverse the cavity per second of which each has the probability W ~ ( g ) o f adding one photon to the field. The steady state is reached when the gain W ~ ( g ) / i - p compensates the cavity losses 7(t'). This requirement, combined with the definition of N~, N ~ = 1/7{o, immediately gives relation (24). Similarly, the probability H' ~~J (c) of finding an initially excited atom in the upper state is H'~"(e)=
Z
(I-fl,,+l)~O,,=l-(n)/N,.~.
(25)
The normalized photon distribution after the measurement becomes ,,+l~t,.... ) = [1/H'll ~(g) 1/3,,, ,/L,. p•g,
(26)
p~g'(t,ot) = 0
or
(27)
p},"'(t,.t)= [ 1 / W " ' ( e ) l ( 1 - f l , , + , ) ~ O , , .
depending upon whether the atom is found in the lower or the upper level. In the interval between two successive atoms the photon star:sties changes according to eq. (5) due to the damping o f thc maser field. As a result, after the time to+t,, , ~hen the next atom enters the cavity, the photon distribution is given by
P~3~(tp +trot)
=
~
B ...... ~ (to)~y,~3+~ (t,,,~) .
(28)
/,.=0
provided the preceding atom was found in the lower state. Thus the conditional probability P t 2~(g, g, to) that given the first atom was detected in the lower state the next atom arriving a time t o later will also be found in the lower state becomes
p(2)(g, gtp)
=
~ ,-,,6'+ ~ e"(s)(to +t,,,),,
.
(29)
¢1=0
We find the joint probability t4'¢e~(g, g, to ) from bining eqs. (29), (28) and (26) wc obtain
I "¢'-~(g, g, to) by multiplication with H ' ¢ ~ ( g ) so that com(30)
tt~O
k=O
In the limit to=0, when simultaneously the first atom leaves and the second atom enters the cavity we have
B,,,,,=~ .... (Kronecker symbol ). Hencc thc exprcssion (30) goes over into 512
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W~2~(g,g, to=O)= ~ fl.*2fl.+,P.-
10clobcr 1q91
(31)
n=O
On the other hand, when the time interval between both atoms is large compared with the cavity damping time (Tto>> 1 ) only Bok is different from zero (Bok= 1 ), and the expression (30) reduces to WI2'(g, g.
to = ~ ) =
W~lJ(g),8, .
(32)
Clearly, in this case the second atom finds the cavity field in the vacuum state, and after passage of the atom through the cavity the probability to detect it in the lower level is given by flu i.e. independent of the measurement on the foregoing atom. As a result, the joint probability of finding both atoms in the lower level is simply the product (32) of the probabilities for the single events. Extending the concept of bunching and antibunching from photon statistics to the case of de-excited atoms at the exit of the micromaser cavity, we compare the joint probability 14,'~2~(g, g, to ) with that for uncorrelated events (de-excitation of two non-successive atoms with a very large spacing), i.e. the square of the probability to find an atom in the lower state, [ W~ ~(g)]2. We are thus led to consider the normalized joint probability w~2~(g, g, to)= W~2~(g, g,
to)/[ W~l)(g)]2,
(33)
and we will speak of antibunching or bunching when w~2~(g, g, to) is smaller or greater than unity. In the case of antibunching (bunching) it is less (more) likely to find the second atom in the lower state, when the first atom has already been found in the lower state, compared to the detection probability (24) corresponding to steady-state conditions. Clearly, the reason for this effect is that the observed de-excitation of the first atom changes the field, making its photon statistics different from that for the steady state, and this, in turn, gives rise to a change of the probability for the second atom to make a transition to the lower state. In the literal sense, the terms bunching and antibunching refer to zero "delay time" t o. However, we find it convenient and instructive to use those terms also in case of arbitrary delay times. From the experimental point of view it is important to keep in mind that, in contrast to photon statistical measurements, we are dealing with correlations between successive atoms only and not between any two atoms with a prescribed spacing to. Only for t o < - .=,, ~ N - - / ~ + , P , , , 0N~
(34)
which by virtue of eq. (17) can be written as 0W~l~(g) 0N..=o
-
z, p
+2ao÷,po-,~,'~ .=o
o.,po.
(35)
Finally, using the relations (24) and (31) we arrive at the result
Wt2~(g, g, tp=O)= [ Wt.~(g) ]2+ OW~'~(g) 8N~
(36)
It follows from eq. (36) that the normalized joint probability Wt2)(g, g, to=0 ) takes values greater or smaller than 1 according to whether the derivative 0 I4' ~~~(g)/ONcx is positive or negative. Fig. 2 in which both w ~2~(g, 513
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OPTICS COMMtJNICATIONS
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20 30
50
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Fig. 1. Normalized joint probability w(2)(g, g, to=0) versus atomic flux N¢~for several values of g-2tm,.
0.6, .
0
.
-.
1
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.
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5
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.
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10
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20
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30
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50
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70
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Nex
Fig. 2. Normalized joint probability w c-")( g, g, to= 0 ) ( --- ) and single-atom probability 14(')( g ) ( - - - ) versus N,. for .Qt,,,= 1.54.
g, t o = 0 ) and W ( ' ) ( g ) are plotted as a function o f the atomic flux h,rx for -Qt,,,= 1.54 clearly displays this mutual relationship. The p a r a m e t e r -Qlint= 1.54 used in some o f our calculations is identical to that chosen in the experimental study o f the n u m b e r fluctuations o f the atoms leaving the maser cavity in the lower state [3 ]. A comparison with the experimental results shows that the regions o f h,~x where antibunching ( b u n c h i n g ) occurs correlate fairly well with the regions where s u b ( s u p e r ) - p o i s s o n i a n statistics was observed [3 ]. Next we investigate the d e p e n d e n c e o f the n o r m a l i z e d j o i n t probability w(2)(g, g, t o) on the length of the time interval tp between successive atoms. In fact, after a triggering event the next atom entering the cavity finds a field which d e p e n d s on the time tp elapsed since then due to the d a m p i n g o f the cavity mode. In fig. 3, w(2)(g, g, t o) is plotted as a function o f N~x for ,.Qlint= 1.54 and several values o f t o. Fig. 3 shows that, as the time interval t o between the a t o m s increases, so does the normalized j o i n t probability w(2)(g, g, to). thereby retaining its general structure as a function o f Nex for time intervals t o c o m p a r a b l e in magnitude to the mean spacing/-o between successive atoms. This means, in particular, that m a x i m u m bunching occurs not for zero delay time, t o = 0 , but for a finite value o f t w The antibunching effect, on the other hand, decreases with t o (see also fig. 4). It becomes obvious from fig. 3 that after a period o f d a m p i n g o f duration i-o the photon statistics has been reshuffled to such an extent that the antibunching effect occurring for small values o f tp vanishes. Ultimately, for large values o f t o, t o >> 1/7, the j o i n t probability 14' (2) (g, g, to) approaches the value (32). For the chosen p a r a m e t e r -Qlint= 1.54 in fig. 2 we have fl, ~ 1, so that w(2) (g, g, to = ov ) = 1/ W (') (g) > 1 for all values o f the atomic flux N,.~. It is interesting to note that instead of the m e n t i o n e d increase of w(2)(g, g, t o) with t o a decrease takes place for certain values o f the p a r a m e t e r s 12tin, and Nc~ (see fig. 4). In particular, after an initial period o f bunching an antibunching effect may show up. Two cxamples are shown in fig. 4 (curves b and c). However, we do not succeed in finding a case in which an initial antibunching effect increases with t o growing from zero. As already mentioned, the time intervals between successive a t o m s in the atomic beam are statistically distributed according to a Poisson process. In order to d e t e r m i n e the averaged j o i n t probability Wt-')(g, g) we have to average the weighted expression ( 3 0 ) over the t i m e interval t o with the result that the factors B , . n + k ( t o ) in (30) are replaced by their averages/~n .... , (eq. ( 1 3 ) ) 514
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i 81
-~
+
I
1.5-
~ --..,..b
1.0
. . . . . . . . .
s/
.....
""~
. . . .%. . . . . .%
...
/
I
,/ ,/ /
! 06~ 1
S 10 ZO iO
S-----~'7/
i
0.6i''/ o
/
i
i
~
Nex
Fig. 3. Normalizedjoint probabilityw<2~(g,g, In) versus A,'¢~for 1.54and several valuesoftp (in units of the mean spacing /- = 7~/~;~).
.,t'2lint =
Fig. 4. Normalizedjoint probability w(2)(g, g, t~) versus to for several values of the parameters £2t,,,and N¢~. (a) .Qt,.t= 1.54, N~= 16" (b) I2tm,=0.55,A,'~=4; (c) .f2t,m=0.8.N~=2.25; (d) .Ql,m= 1.54, Ne~=10.
w,2,(g.g)=
(37) n=O
k=O
This averaged joint probability normalized according to (33), is illustrated in fig. 5 as a function of the atomic flux Ncx for two different values of the parameter ~lint. Comparison with fig. 3 shows that averaging over all time intervals t~ between two atoms drastically reduces the pronounced antibunching effect of the deexcited atoms occurring for tp=0 for certain regions of Ncx. Finally, we would like to mention that the averaged joint probability of finding an atom in the upper level and the next atom a time h, later in the lower level can be easily evaluated too, the result being W ' 2 ' ( e , g ) = ~ fl.+, ~. /~...+k(l--fl.+k+,)/~.+k. n=O
(38)
k=O
One expects the joint probabilities W(2)(e, g) and W(2>(g, g) to fulfill the relation
W(2)(e, g ) +
W(2)(g, g ) = W (l)(g) .
(39)
Indeed, it follows from eqs. (37) and (38) that W~2'(e,g)+W'2'(g,g) =
~. fl,,+L ~ B,,.,,+k[(l--fl,,+k+,)fln+k+fln+kfln+k-,]
n=O
(40)
k=O
515
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I
i I I mox I .1.8/~ I
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Fig. 5. Normalized averaged joint probability w~2~(g,g) versus and 1.57 ( - - - ) .
N~ forI2t,.t= 1.54 ( - - )
and perlbrming here the summation over k according to ( 15 ) yields (39).
5. Discussion
In the present paper we have studied, under steady-state conditions, correlations between the excitation states of successive atoms at the exit of the micromaser cavity. In particular, we have derived a closed-form expression for the joint probability W<2~(g, g, Ip) o f detecting two successive atoms with spacing t, in the lower state after they passed the cavity. Our calculations indicated that bunching as well as antibunching - in the sense that 14"t2)(g, g, Ip) exceeds or falls below the joint probability for uncorrelated events, i.e. the square of the corresponding single-atom probability - occurs, dependent on the relevant physical parameters. Comparing our results with those recently obtained in an experimental [3] and a theoretical [ 12] study of the fluctuations of the number of de-excited atoms detected in a fixed measurement interval, we found a precise correspondence between antibunching and sub-poissonian statistics on the one hand, and between bunching and super-poissonian statistics on the other hand. O f course, this does not come as a surprise. Actually, this correlation is a direct consequence o f the fact that both the bunching/antibunching properties and the characteristics of the number statistics are determined by the photon statistics of the micromaser field. Indeed, the authors in ref. [ 12] succeeded in deriving, in some approximation, a formula that establishes a rather simple relationship between the normalized variance of the number o f photons n being present in the maser cavity Qr = ( < n 2 ) - < n > 2 - ( n ) ) / ( t l )
,
(41)
and the corresponding quantity for the atoms Qa = ( ( m 2 ) - ( m > 2 - ( m > ) / ( r r l )
516
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(where m is the number of atoms detected in the lower state during the measurement interval ), namely Qa = w ( ' ~ ( g ) Q,.(2+ Q,.),
(43)
where 14' t ' ~(g) is the probability for an atom to make a transition to the lower state under steady-state conditions. Essential is the proportionality of the right-hand side in eq. (43) with Qf from which it immediately follows that Q, and Qr have the same sign so that the photons and the de-excited atoms show both either subpoissonian (Qr, Q a < 0 ) or super-poissonian (Qr, Q a > 0 ) statistics. On the other hand, it proves very easy to derive a relationship between antibunching/bunching properties of the de-excited atoms and the photon statistical properties of the maser field. For the special case to= 0 (the second atom enters the cavity when the first leaves it) we found the joint probability of detecting both atoms in the lower state to be given by (31 ). Making use of the relationship (17) we can rewrite cq. (31 ) as 14I(21(g, g, tp = 0 ) =
~, n (g~~ n - 1 ) Pn -- (n 2 ) r~- (n)
,[~--0
(44)
Observing, in addition, eq. (24) we arrive, without any approximation, at the strikingly simple result w~2~(g, g, to=O)= ( ( n : > - (n> )/ (n)2= 1+Qr/ ( n> ,
(45)
which indicates that the photon statistical properties of the maser field (sub- or super-poissonian statistics corresponding to Of< 0 or Of> 0) are faithfully reflected by the antibunching or bunching properties of the deexcited atoms (w~-'~(g, g, t o = 0 ) < 1 or > 1). Moreover, the special feature of the atomic correlation phenomenon we have found in our study, namely the increase or decrease of the normalized joint probability w(2~(g, g, tp) with to growing from zero, is also easily explained analytically. Starting from eq. (30) and differentiating it with respect to t o at tp=0, we get 0 W(Z)(g' g' /P) ,~-o ~" ~-~ (n+l)flT,+~p.. ~ 0t-~ _ = - Y n=ofl~+'nfl"lS"-~ +7.=o
(46)
Here, it has been noticed that only the derivatives
0 B...(tn) ,0=o = --n7 0t~
(47)
and
Ot--~pB...+,(tp)
Ip=0
=(n+l)7
(48)
are different from zero. Making still use ofeq. (17), we can write eq. (46) in the form
7 ~ (fl,-fl,+~)n2P, ~t-~ W~Z~(g'g' t p ) , __ ° - N,x tl=0
(49)
from which it becomes obvious that the slope of W(2)(g, g, tp) at tp=0 may take positive as well as negative values, dependent on the parameters f2t,,, and N~. So it becomes possible to choose operating regimes of the micromaser in which the normalized joint probability w~2~(g, g, to) increases or decreases, as a function of the time tp, in the vicinity of to=0 (see fig. 4).
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Appendix A.
l'roof o f eq. (12). We start from eq. ( I 1 )
U~ •
J=\
n
/ t / : =\o
(AI)
N¢~+n+l
and show that eq. (12) holds. First we decompose the series into two parts
(. k).+k+ll- £ ( k + l ) ) , ....
+ ./~:~-5-L,=o\
*+'=
/
(-I
N~, No.+n+/
+(_1),+,
N..
N~.,+n+k+l
] "
(A2)
Making use of the identity (A3) we find
/~,,,,+,+~= •
?:k)
(n+k+l)
- N~,+n+k+l
~ (-1) / : ~
(') k +"l - l L Nr~ x + n,+ l
'
N~,+n+k+l
]} (A4)
N~.,+n+l
/=o
and using ( A l ) we arrive at n+k+l . . . . k+I-- N ¢ x + n + k +
(A5)
l ~ ..... k.
A p p e n d i x B.
t'roof o f eq. (16). Here we derive eq. (16), starting from eq. ( 15 ). Using the decompositions
(Bl) k=O
k=O
k=O
a ....~p.+~:o+~_,=ao.opo:o_, + ~ a .... ~ + : o + ~ + , : o + ~ ,
(B2)
k=O
we rewrite eq. (15) in the form
: ~ = a o . ° : ~ + a j . : ° _ , + ~ ~ .... ~÷,:°+k+, + E. k=O
k=O
~ao.°+~+,-aoo+k)/~°+k+,:o+~.
(B3)
Substituting the expression (12) for B . . . . k+ ~ into eq. (B3), we readily obtain
B,.,+k 0 = (B,,.,,- 1 ).0, +B,,,,p,p,,_, + k~'o.= Ncx + n + k + 518
1 [ (n+k+
1 )lS,+k+, --N~x/3,+k+ ,P.+k] •
(B4)
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Finally, r e p l a c i n g / ~ . . , by e x p r e s s i o n ( 1 4 ) we i m m e d i a t e l y a r r i v e at eq. ( 1 6 ) .
Note added in proof. R e c e n t l y , a n t i c o r r e l a t i o n s o f a t o m s l e a v i n g the m a s e r cavity in the l o w e r state h a v e been o b s e r v e d [ 15], in r a t h e r fair a g r e e m e n t with o u r t h e o r e t i c a l p r e d i c t i o n s for t p = 0 (fig. 1 ). Actually, a c o m parison with o u r results m a k e s sense o n l y for v e r y short t i m e spacings tp since, d u e to the low d e t e c t i o n efficiency o f a b o u t 10%, it is o n l y in this case that m o s t l y pairs o f successive a t o m s that were exclusively s t u d i e d in the present paper, are o b s e r v e d .
References [ I ] D. Meschede, H. Wahher and G. Miiller, Phys. Rev. Lett. 54 (1984) 551. [2] G. Rempe, H. Walther and G. Miiller, Phys. Rev. Left. 58 (1987) 353. [3 ] G. Rempe, F. Schmidt-Kaler and H. Walther, Phys. Rev. Lett. 64 (1990) 2783. [4] P. Filipowicz, J. Javanainen and P. Meystre, Phys. Rev. A 36 (1986) 4547. [ 5 ] L.A. Lugiato, M.O. Scully and H. Walther, Phys. Rev. A 36 ( 1987 ) 740. [6] F. Filipowicz, J. Javanainen and P. Meystre, J. Opt. Soc. Am. B 3 (1986) 906. [ 7 ] J. Krause, M.O. Scully and H. Walther, Phys. Rev. A 36 ( 1987 ) 4547. [ 8 ] P. Meystre, G. Rempe and H. Walther, Optics Lett. 13 ( 1988 ) 1078. [ 9 ] P. Meystre, Optics Lett. 7 ( 1987 ) 365. [ 10] P. Meystre and E.M. Wright, Phys. Rev. A 37 (1988) 2524. [ 11 ] H. Paul, J. Mod. Optics 36 (1989) 515. [ 12 ] G. Rempe and H. Walther, Phys. Rev. A 42 ( 1990 ) 1650. [ 13 ] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 ( 1963 ) 89. [ 14] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973 ). [ 15 ] G. Rempe and H. Wahher, Proc. Intern. Conf. on Quantum Optics, Jan. 1991, Hyderabad.
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Bunching and antibunching of de-excited atoms leaving a micromaser H. Paul a n d Th. R i c h t e r ('entral Institute oJ Optics and Spectroscopy, Rudower Chaussee 6. O- 1199 Berlin, Germany Received 26 April 1991
The joint probability of detecting two successive atoms both in the lower state at the exit o f a micromaser is calculated. We find that this joint probability, in dependence on the atomic flux Nez, takes values greater (bunching) or smaller (antibunching) than that for uncorrelated events. The regions of Ne, where antibunching (bunching) occurs, correlate fairly well with the regions where sub(super)-poissonian statistics for the number fluctuations of atoms leaving the maser in the lower state has been predicted and recently observed. This coincidence is shown to result from the fact that both the bunching/antibunching phenomenon and the number fluctuations are governed by the photon statistics of the maser field in a similar way.
!. I n t r o d u c t i o n
The single-atom micromaser is a system of fundamental interest in quantum optics. It allows to study the interaction o f an atom with only a few photons and has recently been investigated in detail both experimentally [ 1-3] and theoretically [ 4 - I 1 ]. The m icromaser consists of a high-Q cavity into which highly excited Rydberg atoms are injected, one after the other, so that a single-mode microwave field is built up inside the cavity, provided the mean time interval between atoms is shorter than the cavity decay time. The micromaser field shows a number of non-classical properties that arc averaged out, however, in usual lasers and masers [4,5 ]. In particular, the photon statistics of thc micromaser field can be sub-poissonian [4,5 ]. In the experiments the energy state of the atoms after they have passed the cavity, is monitored by field ionization measurements, from which some information about the cavity field can be obtained. In particular, an analytic formula has been derived [12] which connects the steady-state photon statistics of a micromaser with the number fluctuations of de-excited atoms leaving the micromaser. Utilizing this formula, nonclassical photon statistics has recently been observed [3 ] via sub-poissonian atomic statistics in excellent agreement with the theoretical predictions [ 12 ]. In the derivation of the formula in question it proved sufficient to take into account the change of the probability P to detect an atom in the lower state merely from one measurement interval to the following, thus considering P as a function of the (fluctuating) number of excited atoms N that enter the cavity during a measurement interval. On the other hand, when the atomic analogs o f photon bunching and antibunching, i.e. correlations between the excitation states of successive atoms are studied, as will be done in the present paper, the basic effect is the change of the probability P for the second atom due to a change of the field resulting from its interaction with the first atom. So our procedure will be quite different from that adopted in ref. [ 12]. We will start from the well-known steady-state solution for the micromaser field [4] and calculate, via state reduction, the joint probability of finding two successive atoms in the lower state at the exit of the cavity. We will speak of bunching or antibunching when this joint probability is greater or smaller than that for uncorrelated events, i.e. the square of the probability for the corresponding single atom event. It will be shown that the bunching/antibunching phenomenon is directly connected with the steady-state 508
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photon statistics. Moreover, it turns out that in certain operating regimes the bunching effect becomes enhanced with growing spacing between the two successive atoms. In this paper, an analytical formula for the joint probability will be derived. In doing so. we must explicitly take into account the damping of the cavity field. This gives us an opportunity to reconsider briefly the micromaser theory and to rederive the steady-state photon distribution in a slightly different way, compared to ref. [4].
2. Micromaser theory reconsidered We consider a situation in which a monoenergetic beam o f two-level Rydberg atoms prepared in the excited state is injected into a high-Q single-mode maser cavity. The injection rate will be so low that at most one atom is present in the cavity at a time. Moreover, we assume that the atom-field interaction time t,,, (the same for all atoms) is much shorter than the cavity damping time To= l/),, so that the damping of the cavity field can be ignored while an atom is inside the resonator. Then the interaction of the two-level atom with a resonant single mode of the cavity field is described by the well-known Jaynes-Cummings model [ 13 ]. Suppose that at time t, the ith excited atom enters the cavity containing a field that is diagonal with respect to the Fock states I n ) . Then the combined density operator for the atom in the upper state [e) and the field at time t, reads p (t,) --- 5"~ Pn ( t, ) [ e, n ) ( e, n I, where Pn gives the probability to find n photons in the field. After the interaction time tin, we have
p(t, +t,.,)-- Z p.(t,)
{cos2(x/n4-1~tm,) [e, n ) (e, nl +sin2(x/~-+
l-Qlint)
Ig, n + 1 ) (g. n + 1[
¢1
+ i sin ( x / - n + l ~ t , , , ) c o s ( x / / ~ l~2ti,,) [Ig, n + 1 ) (e, n l - le, n ) (g, n + 11 ]} ,
(1)
where [g) denotes the lower atomic state and g2 is the atom-field coupling constant. Forming the trace over the atomic states shows that the reduced density operator for the field remains diagonal during the interaction and that its matrix elements are given by
p,,(t, +tin,)
= ( 1 - / ~ . + , )p.(t,) + p. p . _ , ( t , ) .
(2)
where /~,, = sin 2 (x/ng2ti.,) •
(3)
In the interval tp between t, + tin, and t,+~ when the next atom enters the cavity, the field density matrix evolves according to the standard master equation for a damped harmonic oscillator. Again the diagonality of the field is preserved during its decay, and the equation of motion for the diagonal elements reads [14] .b,, = y ( n + 1 )p,,+ ~-7np..
(4)
where for the sake of simplicity we have assumed that the temperature of the cavity walls is low enough so that thermal photons can be neglected. The well-known solution to eq. (4) is given by
p.(t,+t,.,+tv)=
'~ B,,..+k(tp)p.+~(t,+t,.,).
(5)
k=0
where B ..... ~(lP)=(n+nk)exp(-nyt.)
[l--exp(--Ttp)] k,
(6)
as can be easily verified by insertion into eq. (4). Hence combining eqs. (2) and (5). the photon distribution at time t,+~=t,+t,.,+t w just before the next atom enters the cavity, is given by 509
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p,,(t,+,)= Z B ...... ,(t,,)[(l--fl,,+k+,)p,,.k(t,)+fl~+kp,,+k_,(t,)].
(7)
-: I)
where we have to put p _ , = 0 . Generally the atoms are not injected with equal spacings. Rather the injection times are distributed according to a Poisson process so that the probability distribution for the time intervals q, between successivc atoms is given by an exponential distribution P(tp)= ( l / t ~ ) exp(-to~i-p), where {p is the mean spacing between two atoms. Successive iterations of eq. (7) and averaging over the random time intervals t~ between the atoms, indicated by a horizontal bar, lead to the return map ~. /~,..... k[(l--fln+~+,)P,,+k(t,)+fl,+kP,,+~-,(t,)].
p,,(t,+~)=
(8)
k=O
which relates the average photon distribution p~(t,+,) after the passage of i atoms through the resonator to the average distribution/~(t,) after the passage o f i - 1 atoms. The average quantities/~,.,,÷k defined as .x,
(9)
fi . . . . ~ = j dto P(tp)B,,.,,+~(to) o
characterize the photon statistics which results, on the average, from the damping according to eq. (5) p,,(t,+,)=
Z fft,,.~+kp,,~(t,+t,n,).
(10)
k=O
Upon insertion of the binomial expansion of [ 1 - exp ( - 7I) ] ~ into (6) the integral (9) is easily evaluated to yield
where .~,'~,= "1"~./{o = l/yi-o is the mean number of atoms traversing thc cavity during thc decay time T~. of the maser field. The Appendix A shows that the quantities /~,,.,,+k fulfill the recurrence relation
n+k+ 1 /~,,.,,+~+ 1 - N,., + n + k +
B ..... ~ 1
(12)
"
and hence can also be written in the form k
n+m
g ...... k = B .... , , = 1 7 , / ~ + n + m
(k>~l) .
(13)
where
B,,.,, =N,.~/(N¢, + n ) .
(14)
3. Recovew of the steady-state photon statistics Driven by the excited atoms injected into the cavity, the maser field eventually evolves toward a quasi-steady state characterized by a mean photon distribution p, which is a solution to eq. (8) with Pn(t,+ ~ ) = p , ( t , ) = p ,
p,=
~ B,.,,+~[(1--/3,,+,÷,)p,,+~+[3,,+~P,+k_,]. k=O
510
(15)
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At first sight, it seems hopeless to solve this infinite system of equations. Nevertheless, it is possible to rewrite cq. (15) in such a form that immediately shows up the steady-state solution. Indeed, using the property (12) and rearranging the terms in the sum we can rewrite eq. (15), after some algebra, in the form (see Appendix B)
_ _
0=
~
1
N~, + n
nn,n+k
[ntO.,-N~B.tO._,]+ k~_,,N.~T~--k+ 1 [(n+k+l)p,,+k+,--N~x[3,,.k+,to.,+kl.
(16)
Clearly, the right-hand side of eq. (16) is zero if for successive occupation numbers to, the relation
np, =N,.xfl,,to~_,
( 17 )
holds, so that we arrive at the following expression for the photon probability distribution to, ( n = 0 , 1, 2, ...)
_ ~.,-.! N~', f i & tO, = Po •
(n>~l),
(18)
k=l
where p o = P o ( N ~ ) depends on N~ and is determined by the normalization condition Y~'Loto, = 1. The analytical form ( 18 ) of the steady-state photon statistics has been first derived in ref. [4] in a very elegant manner. (Actually, the treatment in ref. [4] is more general since it includes the case of arbitrary cavity temperature. ) The authors noticed that the inverse of the map (10) has a very simple form. They found the inverse transformation immediately by averaging the evolution operator exp(Lt o) over G, where L is the operator in the standard master equation of a damped harmonic oscillator. We reproduce their result (for zero cavity temperature), starting from eq. (10) which we write in the form
tO,,(t,+,)-
N~p~(t,+t,.t)+ Nex + rt
"y. B ...... k + , P . * k + t ( t , + t , n t ) .
(19)
k=O
Next we make use of the relation
B...+~+,-
n+l N,.x+nB.+,..+,+k
(20)
which can be readily deduced from ( ! 3), and obtain
tO,,(t,+,)-
~v~n",+--N~" p , , ( t , + t , . , ) +
n+l oo N~-~-_(_nk~=oB,+c~+ , +~p,+~+ , ( t , + G , )
.
(21)
Replacing the sum on the right-hand side by ton+l(t,+! ) according to eq. (10), we thus find that the inverse of the map (10) has indeed the simple form of a two-term recursion relation
p,,(t,+ti.,)=p,,(t,+t)+
NLt[np.(t,+,)-(n+l)to,,+,(t,.~)],
(22)
in agreement with ref. [4]. With the help of this inverse transformation the steady-state condition ( 15 ) can be rewritten in the simple form
to,, + N L ' [ n,O. - ( n + I )to,,+ ,] = ( 1 - fl,, + , )tO,, + fl,,tO,, _ ,
(23)
from which the steady-state photon statistics (18) is easily obtained [4]• We note in passing that eq. (17) has a simple physical interpretation, it states a balance between the transitions that take place between neighbouring photon states [ n - 1 ) and ] n ) only. The "flow" of probability from I n - l ) to I n ) due to atoms delivering their excitation energy to the cavity field is fl,15,,_l/i-p (see the discussion following eq. ( 2 4 ) ) . On the other hand, the "flow" of probability from [ n ) to [ n - I ) due to the 511
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damping of the cavity field s given by N,.~ = I/;'t~, yiclds, in fact. e q (17).
7nP,,, according to eq. (4). Equating both flows and noticing that
4. Correlations between the e~:citation states of successive atoms We now turn to the calcuktion of the joint probability 14"(2)(g, g, to) of finding, under steady-state conditions, two successive atoms in the lower level just after they have left the cavity. According to eq. ( 1 ) the probability to detect an atom in the lower state at the exit o f the cavity is given by W'l>(g)=
~
fl,,+Ip,,=NL' ~n),
(24)
where ( n ) is the mean photon number in the maser field. The second equality follows from eq. (17) and has a simple interpretation: On the average l/{p atoms traverse the cavity per second of which each has the probability W ~ ( g ) o f adding one photon to the field. The steady state is reached when the gain W ~ ( g ) / i - p compensates the cavity losses 7(t'). This requirement, combined with the definition of N~, N ~ = 1/7{o, immediately gives relation (24). Similarly, the probability H' ~~J (c) of finding an initially excited atom in the upper state is H'~"(e)=
Z
(I-fl,,+l)~O,,=l-(n)/N,.~.
(25)
The normalized photon distribution after the measurement becomes ,,+l~t,.... ) = [1/H'll ~(g) 1/3,,, ,/L,. p•g,
(26)
p~g'(t,ot) = 0
or
(27)
p},"'(t,.t)= [ 1 / W " ' ( e ) l ( 1 - f l , , + , ) ~ O , , .
depending upon whether the atom is found in the lower or the upper level. In the interval between two successive atoms the photon star:sties changes according to eq. (5) due to the damping o f thc maser field. As a result, after the time to+t,, , ~hen the next atom enters the cavity, the photon distribution is given by
P~3~(tp +trot)
=
~
B ...... ~ (to)~y,~3+~ (t,,,~) .
(28)
/,.=0
provided the preceding atom was found in the lower state. Thus the conditional probability P t 2~(g, g, to) that given the first atom was detected in the lower state the next atom arriving a time t o later will also be found in the lower state becomes
p(2)(g, gtp)
=
~ ,-,,6'+ ~ e"(s)(to +t,,,),,
.
(29)
¢1=0
We find the joint probability t4'¢e~(g, g, to ) from bining eqs. (29), (28) and (26) wc obtain
I "¢'-~(g, g, to) by multiplication with H ' ¢ ~ ( g ) so that com(30)
tt~O
k=O
In the limit to=0, when simultaneously the first atom leaves and the second atom enters the cavity we have
B,,,,,=~ .... (Kronecker symbol ). Hencc thc exprcssion (30) goes over into 512
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W~2~(g,g, to=O)= ~ fl.*2fl.+,P.-
10clobcr 1q91
(31)
n=O
On the other hand, when the time interval between both atoms is large compared with the cavity damping time (Tto>> 1 ) only Bok is different from zero (Bok= 1 ), and the expression (30) reduces to WI2'(g, g.
to = ~ ) =
W~lJ(g),8, .
(32)
Clearly, in this case the second atom finds the cavity field in the vacuum state, and after passage of the atom through the cavity the probability to detect it in the lower level is given by flu i.e. independent of the measurement on the foregoing atom. As a result, the joint probability of finding both atoms in the lower level is simply the product (32) of the probabilities for the single events. Extending the concept of bunching and antibunching from photon statistics to the case of de-excited atoms at the exit of the micromaser cavity, we compare the joint probability 14,'~2~(g, g, to ) with that for uncorrelated events (de-excitation of two non-successive atoms with a very large spacing), i.e. the square of the probability to find an atom in the lower state, [ W~ ~(g)]2. We are thus led to consider the normalized joint probability w~2~(g, g, to)= W~2~(g, g,
to)/[ W~l)(g)]2,
(33)
and we will speak of antibunching or bunching when w~2~(g, g, to) is smaller or greater than unity. In the case of antibunching (bunching) it is less (more) likely to find the second atom in the lower state, when the first atom has already been found in the lower state, compared to the detection probability (24) corresponding to steady-state conditions. Clearly, the reason for this effect is that the observed de-excitation of the first atom changes the field, making its photon statistics different from that for the steady state, and this, in turn, gives rise to a change of the probability for the second atom to make a transition to the lower state. In the literal sense, the terms bunching and antibunching refer to zero "delay time" t o. However, we find it convenient and instructive to use those terms also in case of arbitrary delay times. From the experimental point of view it is important to keep in mind that, in contrast to photon statistical measurements, we are dealing with correlations between successive atoms only and not between any two atoms with a prescribed spacing to. Only for t o < - .=,, ~ N - - / ~ + , P , , , 0N~
(34)
which by virtue of eq. (17) can be written as 0W~l~(g) 0N..=o
-
z, p
o.,po.
(35)
Finally, using the relations (24) and (31) we arrive at the result
Wt2~(g, g, tp=O)= [ Wt.~(g) ]2+ OW~'~(g) 8N~
(36)
It follows from eq. (36) that the normalized joint probability Wt2)(g, g, to=0 ) takes values greater or smaller than 1 according to whether the derivative 0 I4' ~~~(g)/ONcx is positive or negative. Fig. 2 in which both w ~2~(g, 513
Volume .~5. number 5.6
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1.5 p~
,I
I
1 October 1991
' 1.0
I I
I 161 I
t
\ I I
=~
I
\
)
I
~.
1.o.
I I
\ \
l
\\
l
/^x
) I
_i~ ./__ \
I
5
0.5
/ \
/ \\
•
/
\k
"\xx
/
/
/"
\
/
x
//
,
/
. . . . . . . . . . . . . . . .
1
\
i
/
0.5, 0
,
:
/ ¢
/
1.0: -
1
i/
~\\
10
20 30
50
o
70
Nex
Fig. 1. Normalized joint probability w(2)(g, g, to=0) versus atomic flux N¢~for several values of g-2tm,.
0.6, .
0
.
-.
1
.
.
.
5
.
.
.
10
.
.
.
20
.
.
30
.
.
.
.
50
.
70
0.1
Nex
Fig. 2. Normalized joint probability w c-")( g, g, to= 0 ) ( --- ) and single-atom probability 14(')( g ) ( - - - ) versus N,. for .Qt,,,= 1.54.
g, t o = 0 ) and W ( ' ) ( g ) are plotted as a function o f the atomic flux h,rx for -Qt,,,= 1.54 clearly displays this mutual relationship. The p a r a m e t e r -Qlint= 1.54 used in some o f our calculations is identical to that chosen in the experimental study o f the n u m b e r fluctuations o f the atoms leaving the maser cavity in the lower state [3 ]. A comparison with the experimental results shows that the regions o f h,~x where antibunching ( b u n c h i n g ) occurs correlate fairly well with the regions where s u b ( s u p e r ) - p o i s s o n i a n statistics was observed [3 ]. Next we investigate the d e p e n d e n c e o f the n o r m a l i z e d j o i n t probability w(2)(g, g, t o) on the length of the time interval tp between successive atoms. In fact, after a triggering event the next atom entering the cavity finds a field which d e p e n d s on the time tp elapsed since then due to the d a m p i n g o f the cavity mode. In fig. 3, w(2)(g, g, t o) is plotted as a function o f N~x for ,.Qlint= 1.54 and several values o f t o. Fig. 3 shows that, as the time interval t o between the a t o m s increases, so does the normalized j o i n t probability w(2)(g, g, to). thereby retaining its general structure as a function o f Nex for time intervals t o c o m p a r a b l e in magnitude to the mean spacing/-o between successive atoms. This means, in particular, that m a x i m u m bunching occurs not for zero delay time, t o = 0 , but for a finite value o f t w The antibunching effect, on the other hand, decreases with t o (see also fig. 4). It becomes obvious from fig. 3 that after a period o f d a m p i n g o f duration i-o the photon statistics has been reshuffled to such an extent that the antibunching effect occurring for small values o f tp vanishes. Ultimately, for large values o f t o, t o >> 1/7, the j o i n t probability 14' (2) (g, g, to) approaches the value (32). For the chosen p a r a m e t e r -Qlint= 1.54 in fig. 2 we have fl, ~ 1, so that w(2) (g, g, to = ov ) = 1/ W (') (g) > 1 for all values o f the atomic flux N,.~. It is interesting to note that instead of the m e n t i o n e d increase of w(2)(g, g, t o) with t o a decrease takes place for certain values o f the p a r a m e t e r s 12tin, and Nc~ (see fig. 4). In particular, after an initial period o f bunching an antibunching effect may show up. Two cxamples are shown in fig. 4 (curves b and c). However, we do not succeed in finding a case in which an initial antibunching effect increases with t o growing from zero. As already mentioned, the time intervals between successive a t o m s in the atomic beam are statistically distributed according to a Poisson process. In order to d e t e r m i n e the averaged j o i n t probability Wt-')(g, g) we have to average the weighted expression ( 3 0 ) over the t i m e interval t o with the result that the factors B , . n + k ( t o ) in (30) are replaced by their averages/~n .... , (eq. ( 1 3 ) ) 514
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i 81
-~
+
I
1.5-
~ --..,..b
1.0
. . . . . . . . .
s/
.....
""~
. . . .%. . . . . .%
...
/
I
,/ ,/ /
! 06~ 1
S 10 ZO iO
S-----~'7/
i
0.6i''/ o
/
i
i
~
Nex
Fig. 3. Normalizedjoint probabilityw<2~(g,g, In) versus A,'¢~for 1.54and several valuesoftp (in units of the mean spacing /- = 7~/~;~).
.,t'2lint =
Fig. 4. Normalizedjoint probability w(2)(g, g, t~) versus to for several values of the parameters £2t,,,and N¢~. (a) .Qt,.t= 1.54, N~= 16" (b) I2tm,=0.55,A,'~=4; (c) .f2t,m=0.8.N~=2.25; (d) .Ql,m= 1.54, Ne~=10.
w,2,(g.g)=
(37) n=O
k=O
This averaged joint probability normalized according to (33), is illustrated in fig. 5 as a function of the atomic flux Ncx for two different values of the parameter ~lint. Comparison with fig. 3 shows that averaging over all time intervals t~ between two atoms drastically reduces the pronounced antibunching effect of the deexcited atoms occurring for tp=0 for certain regions of Ncx. Finally, we would like to mention that the averaged joint probability of finding an atom in the upper level and the next atom a time h, later in the lower level can be easily evaluated too, the result being W ' 2 ' ( e , g ) = ~ fl.+, ~. /~...+k(l--fl.+k+,)/~.+k. n=O
(38)
k=O
One expects the joint probabilities W(2)(e, g) and W(2>(g, g) to fulfill the relation
W(2)(e, g ) +
W(2)(g, g ) = W (l)(g) .
(39)
Indeed, it follows from eqs. (37) and (38) that W~2'(e,g)+W'2'(g,g) =
~. fl,,+L ~ B,,.,,+k[(l--fl,,+k+,)fln+k+fln+kfln+k-,]
n=O
(40)
k=O
515
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85. number 5.6
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I
i I I mox I .1.8/~ I
•
I I I I
1.3
/ 12!
I I I'
I 1.0 \ \..
/ x
0.91
o i
5
,I
! ! f • I
lO
j
1.o
20
60
80
Nex
Fig. 5. Normalized averaged joint probability w~2~(g,g) versus and 1.57 ( - - - ) .
N~ forI2t,.t= 1.54 ( - - )
and perlbrming here the summation over k according to ( 15 ) yields (39).
5. Discussion
In the present paper we have studied, under steady-state conditions, correlations between the excitation states of successive atoms at the exit of the micromaser cavity. In particular, we have derived a closed-form expression for the joint probability W<2~(g, g, Ip) o f detecting two successive atoms with spacing t, in the lower state after they passed the cavity. Our calculations indicated that bunching as well as antibunching - in the sense that 14"t2)(g, g, Ip) exceeds or falls below the joint probability for uncorrelated events, i.e. the square of the corresponding single-atom probability - occurs, dependent on the relevant physical parameters. Comparing our results with those recently obtained in an experimental [3] and a theoretical [ 12] study of the fluctuations of the number of de-excited atoms detected in a fixed measurement interval, we found a precise correspondence between antibunching and sub-poissonian statistics on the one hand, and between bunching and super-poissonian statistics on the other hand. O f course, this does not come as a surprise. Actually, this correlation is a direct consequence o f the fact that both the bunching/antibunching properties and the characteristics of the number statistics are determined by the photon statistics of the micromaser field. Indeed, the authors in ref. [ 12] succeeded in deriving, in some approximation, a formula that establishes a rather simple relationship between the normalized variance of the number o f photons n being present in the maser cavity Qr = ( < n 2 ) - < n > 2 - ( n ) ) / ( t l )
,
(41)
and the corresponding quantity for the atoms Qa = ( ( m 2 ) - ( m > 2 - ( m > ) / ( r r l )
516
(42)
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(where m is the number of atoms detected in the lower state during the measurement interval ), namely Qa = w ( ' ~ ( g ) Q,.(2+ Q,.),
(43)
where 14' t ' ~(g) is the probability for an atom to make a transition to the lower state under steady-state conditions. Essential is the proportionality of the right-hand side in eq. (43) with Qf from which it immediately follows that Q, and Qr have the same sign so that the photons and the de-excited atoms show both either subpoissonian (Qr, Q a < 0 ) or super-poissonian (Qr, Q a > 0 ) statistics. On the other hand, it proves very easy to derive a relationship between antibunching/bunching properties of the de-excited atoms and the photon statistical properties of the maser field. For the special case to= 0 (the second atom enters the cavity when the first leaves it) we found the joint probability of detecting both atoms in the lower state to be given by (31 ). Making use of the relationship (17) we can rewrite cq. (31 ) as 14I(21(g, g, tp = 0 ) =
~, n (g~~ n - 1 ) Pn -- (n 2 ) r~- (n)
,[~--0
(44)
Observing, in addition, eq. (24) we arrive, without any approximation, at the strikingly simple result w~2~(g, g, to=O)= ( ( n : > - (n> )/ (n)2= 1+Qr/ ( n> ,
(45)
which indicates that the photon statistical properties of the maser field (sub- or super-poissonian statistics corresponding to Of< 0 or Of> 0) are faithfully reflected by the antibunching or bunching properties of the deexcited atoms (w~-'~(g, g, t o = 0 ) < 1 or > 1). Moreover, the special feature of the atomic correlation phenomenon we have found in our study, namely the increase or decrease of the normalized joint probability w(2~(g, g, tp) with to growing from zero, is also easily explained analytically. Starting from eq. (30) and differentiating it with respect to t o at tp=0, we get 0 W(Z)(g' g' /P) ,~-o ~" ~-~ (n+l)flT,+~p.. ~ 0t-~ _ = - Y n=ofl~+'nfl"lS"-~ +7.=o
(46)
Here, it has been noticed that only the derivatives
0 B...(tn) ,0=o = --n7 0t~
(47)
and
Ot--~pB...+,(tp)
Ip=0
=(n+l)7
(48)
are different from zero. Making still use ofeq. (17), we can write eq. (46) in the form
7 ~ (fl,-fl,+~)n2P, ~t-~ W~Z~(g'g' t p ) , __ ° - N,x tl=0
(49)
from which it becomes obvious that the slope of W(2)(g, g, tp) at tp=0 may take positive as well as negative values, dependent on the parameters f2t,,, and N~. So it becomes possible to choose operating regimes of the micromaser in which the normalized joint probability w~2~(g, g, to) increases or decreases, as a function of the time tp, in the vicinity of to=0 (see fig. 4).
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Appendix A.
l'roof o f eq. (12). We start from eq. ( I 1 )
U~ •
J=\
n
/ t / : =\o
(AI)
N¢~+n+l
and show that eq. (12) holds. First we decompose the series into two parts
(. k).+k+ll- £ ( k + l ) ) , ....
+ ./~:~-5-L,=o\
*+'=
/
(-I
N~, No.+n+/
+(_1),+,
N..
N~.,+n+k+l
] "
(A2)
Making use of the identity (A3) we find
/~,,,,+,+~= •
?:k)
(n+k+l)
- N~,+n+k+l
~ (-1) / : ~
(') k +"l - l L Nr~ x + n,+ l
'
N~,+n+k+l
]} (A4)
N~.,+n+l
/=o
and using ( A l ) we arrive at n+k+l . . . . k+I-- N ¢ x + n + k +
(A5)
l ~ ..... k.
A p p e n d i x B.
t'roof o f eq. (16). Here we derive eq. (16), starting from eq. ( 15 ). Using the decompositions
(Bl) k=O
k=O
k=O
a ....~p.+~:o+~_,=ao.opo:o_, + ~ a .... ~ + : o + ~ + , : o + ~ ,
(B2)
k=O
we rewrite eq. (15) in the form
: ~ = a o . ° : ~ + a j . : ° _ , + ~ ~ .... ~÷,:°+k+, + E. k=O
k=O
~ao.°+~+,-aoo+k)/~°+k+,:o+~.
(B3)
Substituting the expression (12) for B . . . . k+ ~ into eq. (B3), we readily obtain
B,.,+k 0 = (B,,.,,- 1 ).0, +B,,,,p,p,,_, + k~'o.= Ncx + n + k + 518
1 [ (n+k+
1 )lS,+k+, --N~x/3,+k+ ,P.+k] •
(B4)
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Finally, r e p l a c i n g / ~ . . , by e x p r e s s i o n ( 1 4 ) we i m m e d i a t e l y a r r i v e at eq. ( 1 6 ) .
Note added in proof. R e c e n t l y , a n t i c o r r e l a t i o n s o f a t o m s l e a v i n g the m a s e r cavity in the l o w e r state h a v e been o b s e r v e d [ 15], in r a t h e r fair a g r e e m e n t with o u r t h e o r e t i c a l p r e d i c t i o n s for t p = 0 (fig. 1 ). Actually, a c o m parison with o u r results m a k e s sense o n l y for v e r y short t i m e spacings tp since, d u e to the low d e t e c t i o n efficiency o f a b o u t 10%, it is o n l y in this case that m o s t l y pairs o f successive a t o m s that were exclusively s t u d i e d in the present paper, are o b s e r v e d .
References [ I ] D. Meschede, H. Wahher and G. Miiller, Phys. Rev. Lett. 54 (1984) 551. [2] G. Rempe, H. Walther and G. Miiller, Phys. Rev. Left. 58 (1987) 353. [3 ] G. Rempe, F. Schmidt-Kaler and H. Walther, Phys. Rev. Lett. 64 (1990) 2783. [4] P. Filipowicz, J. Javanainen and P. Meystre, Phys. Rev. A 36 (1986) 4547. [ 5 ] L.A. Lugiato, M.O. Scully and H. Walther, Phys. Rev. A 36 ( 1987 ) 740. [6] F. Filipowicz, J. Javanainen and P. Meystre, J. Opt. Soc. Am. B 3 (1986) 906. [ 7 ] J. Krause, M.O. Scully and H. Walther, Phys. Rev. A 36 ( 1987 ) 4547. [ 8 ] P. Meystre, G. Rempe and H. Walther, Optics Lett. 13 ( 1988 ) 1078. [ 9 ] P. Meystre, Optics Lett. 7 ( 1987 ) 365. [ 10] P. Meystre and E.M. Wright, Phys. Rev. A 37 (1988) 2524. [ 11 ] H. Paul, J. Mod. Optics 36 (1989) 515. [ 12 ] G. Rempe and H. Walther, Phys. Rev. A 42 ( 1990 ) 1650. [ 13 ] E.T. Jaynes and F.W. Cummings, Proc. IEEE 51 ( 1963 ) 89. [ 14] W.H. Louisell, Quantum statistical properties of radiation (Wiley, New York, 1973 ). [ 15 ] G. Rempe and H. Wahher, Proc. Intern. Conf. on Quantum Optics, Jan. 1991, Hyderabad.
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