Collective spontaneous emission from a system of two atoms with multiphoton transitions in a cavity

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Collective spontaneous emission from a system of two atoms with multiphoton transitions in a cavity

A.S. Shumovsky, Fam Le Kien, E.I. Aliskenderov

To cite this version:

A.S. Shumovsky, Fam Le Kien, E.I. Aliskenderov. Collective spontaneous emission from a system of two atoms with multiphoton transitions in a cavity. Journal de Physique, 1987, 48 (11), pp.1933-1937.

�10.1051/jphys:0198700480110193300�. �jpa-00210635�

Collective spontaneous emission from a system ^{of two atoms}

with multiphoton transitions in a cavity

A. S. Shumovsky, ^{Fam Le} Kien and E. I. Aliskenderov

Laboratory ^of Theoretical Physics, Joint Institute for Nuclear Research, ^{Head Post} Office, ^{P.O. Box} 79, Moscow, U.S.S.R.

(Requ le 23 avril 1987, accept6 ^{le 23} juin 1987)

Résumé.

On détermine les caractéristiques ^de l’émission spontanée ^d’un système ^{formé de deux atomes à}

deux niveaux avec transitions multiphotoniques excités dans une cavité résonnante sans perte. ^{On trouve} que le comportement collectif de la multiplicité ^de photons affecte considérablement la dynamique ^{et la} statistique

de photons.

Abstract.

Characteristics of the collective spontaneous ^{emission of a} system consisting ^{of two two-level} multiphoton-transition ^atoms excited in a lossless resonant cavity ^are calculated. The photon-multiplicity

collective behaviour is found to affect the photon dynamics ^and statistics considerably.

Classification

Physics ^Abstracts

32.80

42.50

Recent experimental observations [1-4] of spon- taneous emission of Rydberg ^atoms in cavities make

. possible ^the testing ^of simple exactly soluble models of interaction of a single-mode radiation field with ^a few atoms (for ^{a review see} [5]). ^Exact solutions for the collective spontaneous emission from ^{an assem-}

bly ^{of N atoms,} placed ^{into a lossless resonant} cavity

and excited to a state with only ^{one atom in the}

excited state, have been derived [6]. The influence of cavity damping ^{has been} rigorously examined for the case of symmetrical ^{Dicke one-atom initial}

excitation [7]. ^The exactly soluble model of two one-

photon-transition ^{two-level atoms in a lossless} cavity

has been discussed [8-12, 31]. ^{It has} been shown that

interesting quantum effects such ^as super-radiance,

radiation trapping, vacuum-field Rabi oscillations, collapse and revival of Rabi oscillations ^are also

possible. ^The emphasis, ^{however, has been on the}

time behaviour of the ^mean photon ^{number and}

atomic inversion. The spectrum and various charac- teristics of vacuum-field Rabi oscillations in ^one- and

two-photon-transition ^atoms have been examined

[31]. ^In this paper ^{we treat} collective spontaneous emission from a system ^{of two two-level atoms with} multiphoton transitions. The case when both atoms are excited is considered ^on the basis of an exact solution and a detailed analysis.

The Hamiltonian for two two-level atoms interact-

ing ^{with a resonant} single-mode radiation field in a

lossless resonant cavity ^via m-photon transitions in the rotating ^wave approximation ^{reads as :}

where Rj ^and Rf ^{are the} population inversion and transition operators ^of the j-th atom, a’ and a are

the photon creation and annihilation operators ^for the resonant field mode, w0 and «) ⁼ lù 0/ m ^{are the} frequencies of the atomic transition and the mode, g

is the atom-field coupling constant, ^{and m is the}

photon multiple of atomic transitions. We have

neglected the variation of the field over the distance between the two atoms. It should be mentioned here that the development of the laser has led to the study

and exploitation ^of highly ^nonlinear phenomena

such as multiphoton transitions and that the sequ-

ences of from 1- to as many ^as 28-photon transitions have recently been observed [13].

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0198700480110193300

1934

Let at the initial time t ⁼ 0 both the atoms be

prepared in the excited state and the radiation field be in the vacuum state. We expect that, ^{in this} situation, self-Rabi oscillation phenomena arise, ^as in the single-atom one-photon-transition spontane-

ous emission : the field radiated by ^{the atoms is}

stored in the cavity ^a long ^{time, and is} eventually

reabsorbed by ^{the atoms.} The situation has been studied for the case of many ^atoms with one-photon-

transitions [14-16, 5], ^and recently, ^{for the case of} multiphoton transitions [17-20].

On introducing ^the eigenstates of the free part HA ⁺ HF ^{of the}

the wave function of the total system in the interac- tion picture ^is quickly ^{found to be :}

where

and

From these equations ^{we find the mean number of} photons in the field

It is seen that (N (t )) undergoes cosine oscillations at two frequencies 2 f2 R and ⁴ n R. The presence of two types of vacuum-initial-field Rabi oscillations in the temporal behaviour of the mean photon ^number

is a feature of collective spontaneous emission from

a fully ^{excited two-atom system} [8].

Using equation (6) ^we obtain the maximum value of (N(t»

Note that this value is smaller than 2 m. The latter is the number of photons potentially ^emitted by ^two

separate ^excited single-atom systems. Thus, ^the system ^{of two} initially ^{excited atoms} considered here cannot give ^{out all} stored energy ^to radiation. This

implies ^the occurrence of the radiation trapping

effect [6] ^{which is} possible ^{in the} system ^{due to} interference between emission processes of the ^two atoms. The degree ^of trapping ^can be defined by ^the

factor :

For the cases of single- ^and two-photon-transitions,

i. e. when m ⁼ 1 and m ⁼ 2, ^we have R = 1/9 and R ⁼ 25/49, respectively. ^{It is} easily shown that the

larger ^{is the} photon-multiple m ^{of atomic} transitions,

the larger is the factor R, ^and consequently, ^the higher ^{is the} degree ^of trapping.

The times at which the photon ^number (N (t))

reaches the maximum value (7) ^{are :}

where tR is the Rabi half-period ^defined by :

We also ^see from equation (6) ^that (N (t )) ^is equal

to zero at the times to N ^{= 2} K tR ( K ⁼ 0, 1, 2, ...).

So, ^the time tR is the characteristic duration of the collective spontaneous emission process in the sys- tem. The collective features become apparent ^{if we} compare the above results with the single-atom ^case,

where the wave function, ^mean photon number, radiation trapping factor and the characteristic time of emission are given by [18, 19, 5] :

respectively. ^From equations (10) ^and (lld) ^{we find}

that the characteristic time tR ^{of the two-atom case is}

shorter than the characteristic time t R (1) ^{of the} single-

atom case,

For the cases of single- ^and two-photon-transitions (m = 1, 2) ^{we have} tRltA!) ⁼ -V2/3, B/2/7, respect- ively. ^{It is seen from} equation (12) ^{that the} larger ^is

the photon-multiple m ^of transitions, the smaller is

the ratio tRltA!), ^and consequently, ^the higher ^{is the}

degree of atomic co-operation in the collective process.

We now consider the radiation rate I (t ) ^calculated

from equation (6),

It is easily established that I (t ) reaches the maximum value

at times

where the notation

and the characteristic time

Fig. ^1.

Evolution of the radiation rate I (t ) ^{in the case of} two-photon transitions (m ⁼ 2). ^{The time unit} is l/g. ^The

full line corresponds ^{to emission} (I > 0), the dotted line to

reabsorption (I 0). ^The asymmetry of the emission

curve relative to the peak ^{time is seen.}

have been introduced. Further, I (t ) ^is equal ^{to zero}

at the times tOJ = KtR’ K ⁼ 0, 1, 2,

Finally, ^at

times

the rate I(t) takes the negative minimum value

I miD ^{= -} I max’ This is in agreement with the reab-

sorption process.

The relation (17) between the characteristic time tD ^for peaks and the characteristic time tR for nulls of the radiation ^rate I(t) ^{shows that} t D - t Rand

tD > tR/2. The latter inequality indicates that the

shape ^{of the curve} describing the evolution of

I (t ) in the time interval 0 , t _ t R ^is asymmetrical

relative to the peak time tD (see Fig. 1). ^{This is a}

feature of the collective behaviour. In the case of ^a

single ^{atom in a lossless} cavity [21, 22, 5], ^the

radiation-rate oscillations described by ^{the sine-}

function are symmetrical ^{relative to the} peaks.

According ^to equation (llb), the radiation rate

1(1 >(t), its maximum value I maX ^{and the} characteristic time t bI ^{for the} single-atom ^{case are} given by

From equations (17), (18c) ^and (10) ^we find that the ratio tD/tbl) is smaller than unity and decreases

rapidly ^{when the} photon multiple m increases. In

particular, ^{for the cases of one-, two-,} three- and

ten-photon-transitions ^{one obtains} tD/tbl):::::: ^0.934,

0.648, 0.380 and 0.0004, respectively. Using equations (14), (16), (5) ^and (18b), ^{one can further}

show that 2 lmax/ImaX 4 ^{for the case m =1,} Imax/Imax 2 when m > 2, ^and Imax/I max ^{1 when}

m > 4. Moreover, numerical calculations for the

cases m =1, 2, 3, ^{4 and 10} give the values

Imax/I (m’a)x ^{2.241, 1.968, 1.284,} 0.727 and 0.014, respectively. Thus, the collective behaviour of the system considered here is strictly distinguished ^from

the super-fluorescent behaviour of an assembly ^of

excited atoms in a low-Q cavity [23-26, 5] ^{where the}

maximum radiation rate is proportional ^to the square of the atom number. The above results clearly ^show

the influence of the trapping effect which leads to limited superradiance ^{in the case m = 1 and to}

subradiance in the cases m > 2.

We now study ^the photon statistics in collective

spontaneous ^emission using the solution given by

equations (3) ^and (4). For this aim we calculate the

normalized correlation function defined as :

1936

Here N(t)> ^{is the mean photon number} given by equation (6), ^and (N2(t) ^{is the mean} square value of the photon number found from equations (3) ^and (4) ^{to be :}

With the definition (19), ^the photon antibunching (sub-Poissonian statistics) ^{condition can be written} simply ^as g2(t) ¹ [27, 28]. For t =F 2 KtR ; ^{K =} 0, 1, 2, ^{..., we can} easily ^{show that}

and

This means that the statistics of the photons ^emitted spontaneously ^and collectively by ^the system ^{of two}

atoms is, ^{in the case of} one-photon-transitions, ^sub- Poissonian, and in the other multiphoton-transition

cases, super-Poissonian, for all times except ^{for the}

moments at which the field is in the vacuum state.

The maximum amount of photon antibunching being possible only ^{in the case m = 1,} is characterized by

the factor gm2n ⁼ 9/16, ^see equation (21). ^The

situation is different for the system consisting ^{of a} single ^atom excited. In this case the correlation function g(2)(t) is found from equations (lla) ^and (19) ^{to read}

From equation (23) ^we find that the photon ^anti- bunching ^{effect can occur} for any photon multiple ^m

in the single-atom ^{case, and the} corresponding

maximum amount of photon antibunching ^{is deter-}

mined by ^{the factor} gm2n ⁼ 1 - 11m. ^{Thus, the}

effect of either cooperativity ^or photon multiplicity

is generally ^to reduce the possibility ^and magnitude

of photon antibunching. This reduction is somewhat similar to the reduction of antibunching and squeez-

ing ^{in resonance} fluorescence [19, 30].

In conclusion, ^{we have} presented ^a rigorous ^and

detailed treatment for collective spontaneous ^emis- sion from a system ^{of two two-level atoms with} multiphoton transitions in a lossless resonant cavity.

The characteristics of the photon dynamics ^and

statistics have been calculated. It has been shown that interesting quantum effects such as self-induced Rabi oscillations, ^radiation trapping, limited super- radiance (in ^{the case m =} 1), subradiance (in ^the

case m , 2) ^and photon antibunching (in ^{the case}

m ⁼ 1) ^are possible. The considerable influence of the cooperativity ^and photon multiplicity ^{on the}

behaviour of the emission has been found to reduce the characteristic times and the possibility ^and magnitude ^of photon antibuching. ^An asymmetry ^of the emission curve relative to the peak ^{time as a}

feature of the collective behaviour has been noted.

The effect of cavity damping ^{on the} dynamics ^and

statistics of photons ^{in the} system will be discussed in

a subsequent ^paper.

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