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Trapping conditions for a three-level atom interacting with cavity fields
Optics Communications 226 (2003) 285–296 www.elsevier.com/locate/optcom
Trapping conditions for a three-level atom interacting with cavity fields N.A. Enaki b
a,*
, V.I. Ciornea a, D.L. Lin
b
a Institute of Applied Physics, Academy of Sciences of Moldova, Moldova, Moldavia Department of Physics and Astronomy, University at Buffalo, State University of New York, Buffalo, NY 14260, USA
Received 30 January 2003; received in revised form 27 May 2003; accepted 30 June 2003
Abstract The interaction between a single-mode cavity field and a string of three-level atoms of cascade type with equally spaced energy levels is investigated. The matrix elements of dipole transitions between adjacent levels are different. Trapping conditions for the flying time in the cavity as well as the explicit state of the field are found. The results are shown to be qualitatively different from the case of two-level systems. The properties of cavity fields are examined. In various limits, the state exhibits vacuum nutation as well as sub-Poissonian statistics and squeezing properties. Ó 2003 Published by Elsevier B.V.
1. Introduction The creation of one-atom micromasers has prompted experimental studies of a single atom interacting with the resonant mode of cavity fields in recent years [1,2]. Much interest in quantum optics has been generated by such devices, especially, in connection with the trapping states in the micromaser. It has been shown that under certain circumstances, a simple quantum harmonic oscillator driven by a quantum current evolves to unique pure states even if it starts as a mixed state [3]. It has also been shown that in various limits, these states exhibit non-classical properties such as sub-Poissonian statistics [4], or more interestingly resemble a macroscopic superposition. Moreover, it is found from the analysis of coherent trapping states in a lossless two-photon micromaser that the field evolves to a pure state, which may be a superposition of even or odd photon number states [5]. These states may exhibit perfect second-order squeezing behavior as the upper limit when the trapping increases. The general properties of a three level atom interacting with cavity fields have been discussed in great detail in the past [6]. Consider a three-level atom of cascade type with the ground state, first and second excited states denoted by jgi, jii and jei, respectively. In the case the levels are equally spaced, the trapping condition in a cavity is already found when the dipole transition matrix elements between adjacent levels *
Corresponding author. Fax: +373-2-739907. E-mail address: [email protected] (N.A. Enaki).
0030-4018/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.optcom.2003.06.003
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are assumed to be the same [7]. The initial state ajei þ bjii þ cjgi differs from the state ajei bjii þ cjgi pffiffifinal ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by a phase change. trapping conditions for the flying time s are given by 2 sk 2N u þ 3 ¼ pp for even pffiffiffi The pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi integers p and 2sk 2Nd 1 ¼ qp for even integers q. Here, Nd and Nu are the lower and upper values of photon number states in the Fock space. It is shown that such a three-level atom is equivalent to a pair of indistinguishable two-level atoms [7]. This may seem to be obvious at a first glance. However, a more careful study reveals that it is not true in general even if the levels are equally spaced. When the matrix elements for different transitions are equal, the three-level results can never pass to the two-level results in any limit. On the other hand in the [8] it was shown that the second distinguishable atom, which enters the cavity before the first atom leaves, destroys the trapping effect. The aim of this paper is to find some connection between the trapping conditions for a three-level equidistant system and that of the two-level one. In doing this, one considers different dipole moment matrix elements of dipole transitions between adjacent levels. As a matter of fact, the passing to two-level atoms is impossible for fixed Nd and Nu , as we shall discuss below. In other words, passing to a pure cotangent state of the two-level atom [3,9,10] is practically unrealizable due to the fact that the recursion relation for a threelevel system with equal level spacing is quite different from that for a two-level system. For instance, when the dipole transition between the states jgi and jii tends to zero, the recursion relation and lower trapping conditions do not pass into that for a two-level system. In a similar fashion, it can be shown that the upper trapping condition for a three-level atom is not valid when the transition between jii and jei tends to zero. Perhaps, this is because the second atom, which enters the cavity before the first atom leaves, destroys the trapping effect [8]. Since we assume different transition matrix elements, the problem becomes more general and is expected to provide deeper physical insights in the understanding of trapping multilevel systems. In the present work, we obtain the trapping-state solution for a three-level atom in the cavity and investigate the properties of the field in the trapping state. The surprising result is that this state exhibits not only such properties as sub-Poissonian statistics and squeezing, but also vacuum nutation which, to our knowledge, has thus far been observed only in the case of two-level atom [3] and has not been reported in previous studies of the trapping effect in a micromaser [11].
2. Three-level atom with arbitrary matrix elements We consider, as usual, a micromaser in which a very high-Q cavity is pumped by a string of atoms at a rate so low that there is at most one atom at a time in the cavity [12,13]. The incident beam consists of threelevel atoms of cascade type with equally spaced levels but different dipole transition matrix elements. Suppose that the pumping is regular so that the time interval sp between the arrival of two successive atoms remains unchanged [14]. Assume further that the cavity is lossless. The system of the electron interacting with a cavity field is described by the Hamiltonian hx0 ðjeihej jgihgjÞ þ hk1 ðay jiihej þ ajeihijÞ þ hk2 ðay jgihij þ ajiihgjÞ; H ¼ hxay a þ
ð1Þ
where hx0 stands for the atomic level spacing, and the energy is measured from the middle level of the atom. The operator ay ðaÞ creates (annihilates) a photon of frequency x in the cavity, and hk1 and hk2 are the electron-field coupling energies. It is clear that k1 and k2 are proportional to the dipole transition matrix elements d32 and d21 , respectively. The state-vector of the system at time t is given by jWðtÞi ¼
Nu X n¼Nd
sn jniðajei þ bjii þ cjgiÞ;
ð2Þ
PNu sn jni. in which we have expanded the one-mode cavity field on the finite number of the Fock states n¼N d When the atom enters the cavity at time t0 , the state evolves according to the Schr€odinger equation
N.A. Enaki et al. / Optics Communications 226 (2003) 285–296
ih
o jWðt0 Þi ¼ H jWðt0 Þi: ot0
287
ð3Þ
It is not difficult to observe that such a three-level system is equivalent to the two-level atomic system when either k1 ! 0 or k2 ! 0. We will study the trapping condition only for the resonance case x ¼ x0 . For the coupled system ‘‘atom + cavity’’ the state-vector at time t þ s is jWðt þ sÞi ¼ expðiH s= hÞjWðtÞi ¼ exp½iðx0 ðjeihej jgihgjÞ þ x0 ay aÞs ( " Nu X 2ðn þ 1Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Þ þ g2 ðn þ 1Þð1 þ g jni sn a 1 sin ðn þ 1Þð1 þ g2 Þ þ g2 2 n¼Nd rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ib sin k1 s n þ g2 ðn þ 1Þ jn 1i n þ g2 ðn þ 1Þ # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 nð1 þ g Þ 1 jn 2i jei sin 2cg 2 nð1 þ g2 Þ 1 " Nu pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X sn b cos k1 s n þ g2 ðn þ 1Þ jni þ n¼Nd
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1 ia sin k1 s ðn þ 1Þð1 þ g2 Þ þ g2 jn þ 1i ðn þ 1Þð1 þ g2 Þ þ g2 # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 icg sin k1 s nð1 þ g Þ 1 jn 1i jii nð1 þ g2 Þ 1 " Nu X n k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 nð1 þ g Þ 1 jni sin þ sn c 1 2g nð1 þ g2 Þ 1 2 n¼Nd pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn þ 1Þðn þ 2Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Þ þ g2 jn þ 2i ðn þ 1Þð1 þ g sin 2ag 2 ðn þ 1Þð1 þ g2 Þ þ g2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1 2 sin k1 s n þ g ðn þ 1Þ jn þ 1i jgi ; ibg n þ g2 ðn þ 1Þ
ð4Þ
where we have defined g ¼ k2 =k1 . To find the recursion formula, we first change the photon number states in every term in (Eq. (4)) to jni by shifting n accordingly in various terms, then replace sn by Sn ¼ sn Hðn Nd Þ HðNu nÞ, where HðxÞ is the step function. This enables us to include all nine term under a single sum over n from 0 to 1. More explicitly, Eq. (4) becomes jWðt þ sÞi ¼ expðiH s= hÞjWðtÞi ¼ exp i x0 ðjeihej jgihgjÞ þ x0 ay a s (" 1 X ffi 2ðn þ 1Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðn þ 1Þð1 þ g Þ þ g jni Sn a 1 sin ðn þ 1Þð1 þ g2 Þ þ g2 2 n¼0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1 sin k1 s ðn þ 1Þ þ g2 ðn þ 2Þ jni Snþ1 ib 2 ðn þ 1Þ þ g ðn þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ðn þ 2Þðn þ 1Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðn þ 2Þð1 þ g Þ 1 jni jei sin Snþ2 2cg 2 ðn þ 2Þð1 þ g2 Þ 1
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"
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n þ Sn b cos k1 s n þ g2 ðn þ 1Þ jni Sn1 ia sin k1 s nð1 þ g2 Þ þ g2 jni nð1 þ g2 Þ þ g2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1 sin k1 s ðn þ 1Þð1 þ g2 Þ 1 jni jii Snþ1 icg ðn þ 1Þð1 þ g2 Þ 1 " n k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2Þ 1 nð1 þ g sin þ Sn c 1 2g2 jni nð1 þ g2 Þ 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðn 1Þn k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Þ þ g2 jni ðn 1Þð1 þ g sin Sn2 2ag 2 ðn 1Þð1 þ g2 Þ þ g2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ) n Sn1 ibg ð5Þ sin k1 s ðn 1Þ þ g2 n jni jgi : ðn 1Þ þ g2 n The recursion formula for Sn follows if we require that the probability amplitude for the ground state remains unchanged when time changes. In other words, we impose the condition "
cSn 1 2g2
n k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2Þ 1 nð1 þ g sin jni nð1 þ g2 Þ 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 1Þn k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðn 1Þð1 þ g Þ þ g jni Sn2 2ag sin 2 ðn 1Þð1 þ g2 Þ þ g2 # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n sin k1 s ðn 1Þ þ g2 n jni jgi ¼ Sn jnicjgi: Sn1 ibg ðn 1Þ þ g2 n
ð6Þ
By noting the relations nð1 þ g2 Þ 1 ¼ ðn 1Þð1 þ g2 Þþ g2 ¼ n 1 þ g2 n, one can easily show that Eq. (6) reduces to rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nð1 þ g2 Þ 1 1 a n1 2 cot nð1 þ g Þ 1 Sn1 Sn ¼ ð7Þ Sn2 : gc 2 n gc n When this condition is satisfied, it is straightforward to show that the atom and cavity field become separable and Eq. (5) can be written as Nu X jWðt þ sÞi ¼ exp i x0 Sz þ x0 ay a s Sn jniðajei bjii þ cjgiÞ:
ð8Þ
n¼Nd
For the lower bound photon Nd , the corresponding coefficient Sn is determined from pffiffiffi number np¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (7) by multiplying it by n tanððk1 s=2Þ nð1 þ g2 Þ 1Þ and then setting SNd 1 ¼ SNd 2 ¼ 0. Thus, we have pffiffiffiffiffiffi Nd tan
k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nd ð1 þ g2 Þ 1 SNd ¼ 0; 2
ð9Þ
which implies for non-vanishing SNd that k1 s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nd ð1 þ g2 Þ 1 ¼ p0 p
ð10Þ
N.A. Enaki et al. / Optics Communications 226 (2003) 285–296
289
for even p0 , and Nd 6¼ 0. As follows from expression (6) and expression (9), lower bound photon number condition Nd ¼ 0 is satisfied for arbitrary flight time s and trapping condition depends only on the upper bound condition. In a similar fashion, we find for non-vanishing SNu corresponding to the upper bound of photon number Nu the condition pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s ðNu þ 1Þð1 þ g2 Þ þ g2 ¼ pp; ð11Þ for even p. The expressions (10) and (11) are the trapping conditions for the equidistant three-level atom with arbitrary dipole momentum transitions between the levels jei; jii p and jgi. From expression (9) follows ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that when Nu ¼ 0 one can obtain the vacuum trapping condition k1 s 1 þ 2g2 ¼ pp. It is interesting from physical point of view to found the transition of this conditions to two-level system. When k1 ¼ k2 one obtain the trapping condition for two-level system from obtained results. Let us consider the case when k1 ! 0 (g ! 1 evidently). In this case the recursion relation pass into relation for cotangent state of two-level atomic system b k2 s pffiffiffi n sn1 : sn ¼ i cot ð12Þ c 2 Wepobserve ffiffiffiffiffiffiffiffiffiffiffiffiffiffi that when n ¼ Nu þ 1 the upper trapping condition can be obtained for above relation k2 s Nu þ 1 ¼ pp for odd p. This condition violates the upper trapping condition for three-level atom when k1 ! 0 (see conditionp(11)). ffiffiffiffiffiffi Let us consider n ¼ Nd . From (12) it is follows the lower trapping condition for a two-level atom k2 s Nd ¼ qp for even q. This condition coincides with condition (10) for a three-level system, when k1 ! 0. Where is the reason that upper conditions are different for two-level and three-level systems? In order to answer this question let us represent all coefficients Sn through SNu , by using the recursion relation (7). Considering SNu 6¼ 0 we can express the SNu 1 through SNu by the following ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Þ 1 ffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN þ 1Þð1 þ g 1a Nu u cot ðNu þ 1Þð1 þ g2 Þ 1 SNu SN 1 SNu þ1 ¼ gc 2 Nu þ 1 g c Nu þ 1 u ¼ 0:
ð13Þ
From this equation one obtain that SNu 1
b ¼ i cot a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ðNu þ 1Þ k21 þ k22 k21 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðNu þ 1Þðk21 þ k22 Þ k21 SNu : Nu k1
ð14Þ
One can observe here that when k1 ! 0, the coefficient SNu 1 tends to 1. This coefficient SNu 1 can achieve finite values which donÕt depend on the k1 only in the case when SNu is proportional to the k1 . But this is in conflict with our proposal that SNu 6¼ 0 for k1 ¼ 0. In the same way letÕs represent SNu 2 through SNu 1 when k1 ! 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Þ 1 i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ð1 þ g 1 a Nu 1 u cot Nu ð1 þ g2 Þ 1 SNu 1 SNu ¼ ð15Þ SNu 2 k1 ! 0: gc 2 Nu gc Nu In this case SNu 2 is
SNu 2
b ¼ i cot a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Nu k21 þ k22 k21 SNu 1 s 2 2 Nu k1 þ k2 k1 : 2 Nu 1 k1
ð16Þ
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In order to obtain the non-zero finite value for SNu 2 when k1 ! 0 we again require that in the right-hand side of Eq. (16), the coefficient will be proportional to k1 . In opposite case we obtain the divergent expression in (16). This requirement is impossible because we already defined SNu 1 to be a constant, which does not depend on k1 . In other words, we have only one possibility to put SNu ¼ SNu 1 ¼ 0. This possibility gives us the zero value of SNu 2 . By proceeding this procedure we obtain all zero coefficients, which is in contrast with normalized condition 2
2
2
jSNd j þ þ jSn j þ jSNu j ¼ 1: We conclude that trapping conditions for a three-level system are different and do not pass in trapping conditions for a two-level system when k1 ! 0. To obtain the analytical expression for the coefficient Sn , for arbitrary k1 and k2 , expressed via SNu , we observe that Sn can be represented through SNu in the following manner Sn ¼
NY u 1
ð17Þ
Qm SNu ;
m¼n
where the coefficient Qm can be expressed through continued fraction by the following: ( rffiffiffiffiffiffiffiffiffiffiffiffi) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Þ 1 b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 2Þð1 þ g c mþ2 ðm þ 2Þð1 þ g2 Þ 1 Qm ¼ i cot g a 2 mþ1 a mþ1 8 9 qffiffiffiffiffiffiffi > ,> c > > mþ3 < = g a mþ2 ; bðsÞ > > cðsÞ. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : ðNu þ1Þð1þg2 Þ1 ; . .ib cot k1 s ðN þ1Þð1þg2 Þ1 u a
2
ð18Þ
Nu
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 3Þð1 þ g2 Þ 1 2 ðm þ 3Þð1 þ g Þ 1 ; 2 mþ2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 4Þð1 þ g2 Þ 1 2 ðm þ 4Þð1 þ g Þ 1 : cðsÞ ¼ i cot a 2 mþ3
b bðsÞ ¼ i cot a
In the limit k2 ! 0, Qm reminds the corresponding coefficient of the tangent state of the two-level atom. But this is not so true, as we can take the limit k2 ! 0 andprepresent all coefficients Sn through SNd . We ffiffiffiffiffiffi ¼ ð2m þ 1Þp is violated by condition (10) observe that lower trapping condition for tangent states k s N 1 d pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s Nd 1 ¼ 2mp, where m is an integer. In this case the recursion relation (7) goes to recursion relation for tangent states Sn ¼ i
a k2 s pffiffiffi tan n Sn1 ; b 2
when k2 ¼ 0. Representing the first two coefficients SNd þ1 and SNd þ2 through SNd ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Þ 1 ffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN þ 1Þð1 þ g d cot ðNd þ 1Þð1 þ g2 Þ 1 SNd SNd þ1 ¼ gc 2 Nd þ 1 and
N.A. Enaki et al. / Optics Communications 226 (2003) 285–296
SNd þ2
i b cot ¼ gc
ffi k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðNd þ 2Þð1 þ g2 Þ 1 2 qffiffiffiffiffiffiffiffi 1 a g c
gi
b c
cot
291
ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðNd þ 2Þð1 þ g2 Þ 1 Nd þ 2
Nd þ1 Nd þ2
ffi SNd þ1 ; k s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Þ1 1 ðNd þ 1Þð1 þ g2 Þ 1 ðNd þ1Þð1þg 2 Nd þ1
one can conclude that when k2 tends to zero, the finite and different from zero SNd þ1 can be obtained only in the case when SNd / k2 . This fact is in conflict with the assumption that SNd must be non-zero, for k2 ¼ 0. In this case SNd þ1 from the first expression becomes finite and independent on k2 . From the second expression in order to obtain the finite SNd þ2 we again must suppose that SNd þ1 / k2 and this is in conflict with the conclusion of the first expression. In general, it is clear, that it is impossible to get the tangent states from our trapping conditions (10) and (11). From above considerations it is not difficult to write down the product SN ¼
N Y
ð19Þ
Pm SNd :
m¼Nd þ1
Here, the coefficient Pm can be expressed through continued fraction of the form ( rffiffiffiffiffiffiffiffiffiffiffiffi) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mð1 þ g2 Þ 1 1 a m1 2 cot mð1 þ g Þ 1 Pm ¼ gc 2 m gc m 8 9 q ffiffiffiffiffiffi ffi > ,> > > 1 a m2 < = g c m1 ; bðsÞ > > cðsÞ. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ðNd þ1Þð1þg2 Þ1 > : ; . . i b cot k1 s ðN þ1Þð1þg2 Þ1 d gc
2
ð20Þ
Nd þ1
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm 1Þð1 þ g2 Þ 1 ðm 1Þð1 þ g2 Þ 1 ; 2 m1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm 2Þð1 þ g2 Þ 1 cðsÞ ¼ cot ðm 2Þð1 þ g2 Þ 1 : gc 2 m2
i b bðsÞ ¼ cot gc
In the limit k1 ! 0, Pm tends to the corresponding coefficient of the cotangent state of the two-level atom. With the increases of m, Pm drastically differ from the same coefficient in cotangent state. We end this section with conclusion that trapping conditions for a three-level equidistant system with different dipole momentum transition elements, totally differ from that of a two-level system. These systems (two-level and three-level systems), in principle, have the same behavior only in a single case, when Nu ! 1. In this case upper trapping condition loses the sense.
3. Results and discussion We are now in a position to discuss the statistical properties of the cavity field in trapping states. First, we note that the coefficients obtained from Eqs. (7) and (19) are very different from their counter parts in the case of a two-level atom interacting with one-mode cavity fields [3]. The major reason is of course that in the present case all coefficients depend on a, b and c. Because these parameters represent the photon
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2
2
population probability, they satisfy the normalization condition jaj þ jbj þ jcj ¼ 1. Thus the electromagnetic field fluctuations depends on two independent parameters, say, a and b. As a consequence, the photon statistics in a cavity changes drastically when the two-level atom is replaced by a three-level atom. The mean photon number hni of the cotangent state is defined by hni ¼
Nu X
Sn Sn0 hnjay ajn0 i ¼
n;n0 ¼0
Nu X
2
jSn j n:
ð21Þ
n¼0
The fluctuation r in the mean excitations is defined by 2
r¼
hn2 i hni ; hni
Fig. 1. (a) Mean excitation hni of the cotangent state as a function of the populations of the second excited states jaj2 and jbj2 . (b) Fluctuations in the mean excitations r ¼ ðhn2 i hni2 Þ=hni. (c) The black region represents sub-Poisson statistics (r < 1) and white region represents super-Poisson statistics (r > 1) of cavity field, parameters used are Nu ¼ 20, p ¼ 10 and g ¼ 1:1.
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293
where the mean square is given by hn2 i ¼
Nu X
2
jSn j n2 :
n¼0
When r < 1, the photon statistics is sub-Poisonian. The cavity field behavior is investigated by calculating 2 2 hni and r as a function of the photon population probability jaj and jbj . When k2 k1 or the transition matrix element dig die . It seen that for Nu ¼ 20 the mean photon number decrease from maximum value to zero with a sharp transition in the domain 0:5b2 < a2 < 0:7b2 . The fluctuations r takes the maximum values in this domain too. The cavity field obeys the sub-Poisson statistics in the domain a2 > 0:6b2 . The most interesting results are obtained when the dipole transition matrix elements are roughly of the same size. We consider the case for g ¼ 1:1, and present the results for Nu ¼ 20 in Fig. 1. In both Figs. 1(a)
ð2Þ
ð2Þ
Fig. 2. Second-order squeezings D1 (a) and D2 (c) of the cotangent state as a function of the populations of the second excited states jaj2 and jbj2 . The shaded regions in (b) and (c) represent, respectively, the negative values of the first and second quadratures of amplitude-squared squeezing. Here, Nu ¼ 20, p ¼ 10 and g ¼ 1:1.
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and (b), a more oscillatory behavior of the mean photon number hni and r are observed, especially in the region a2 ; b2 < 0:6. In this region, the sub- (black) and super-Poisonian (white) statistics show interesting oscillatory behavior as illustrated in Fig. 1(c). These results indicate that the anomalous oscillatory behavior of the fluctuation takes place when the two dipole moments are of the same size. In what follows, we study the second-order squeezing and amplitude-squared squeezing. The two slowly varying Hermitian quadrature components of the field amplitude are defined by d1 ¼ ðA þ Aþ Þ=2;
d2 ¼ ðA Aþ Þ=2i;
ð22Þ N
N
where A ¼ a expðix0 tÞ. Whenever the condition hðDdi Þ i < ðN 1Þ!!=2 is satisfied, the corresponding state is called the Nth-order squeezing state [15]. The degree of squeezing is measured by the parameter
Fig. 3. Amplitude-squared squeezings Q1 (a) and Q2 (c) of the cotangent state as a function of the populations of the second excited states jaj2 and jbj2 . The shaded regions in picture (b) and (d) represent, respectively, the negative values of the first and second quadratures of the amplitude-squared squeezing. Here, Nu ¼ 20, p ¼ 10 and g ¼ 1:1.
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295
N
ðN Þ
Di
¼
2N hðDdi Þ i 1; ðN 1Þ!!
i ¼ 1; 2:
ð23Þ ðN Þ
According to the definition in Eq. (22), it is easily seen that squeezing appears when 1 < Di In a similar fashion, we define two operators Y1 ¼ ðA2 þ Ay2 Þ=2;
Y2 ¼ ðA2 Ay2 Þ=2i:
< 0. ð24Þ
The condition for the amplitude-squared squeezing is [16] 2 hðDYi Þ i < hN^ þ 1=2i; i ¼ 1; 2: ð25Þ y ^ where N ¼ a a represents the photon number operator. The amplitude-squared squeezing is measured by the operators 2
Qi ¼
hðDYi Þ i hN^ þ 1=2i ; hN^ þ 1=2i
i ¼ 1; 2:
ð26Þ
Thus, the field is in an amplitude-squared squeezed state when 1 < Qi < 0. The smaller the Qi , the stronger the amplitude-squared squeezing is. In the numerical calculation, we assume real probability amplitudes a, b, c and g ¼ 1:1. The secondð2Þ order squeezing Di is calculated for Nu ¼ 20 and results are presented in Fig. 2. It is observed in Figs. 2(a) ð2Þ ð2Þ and (c) that both D1 and D2 shoe oscillations in the region b2 < 0:6. The shaded area in Figs. 2(b) and (d) ð2Þ ð2Þ shows the squeezing found in this calculation. It is seen that D2 has a stronger squeezing than D1 . The results of numerical simulation of Qi are presented in Fig. 3, and the amplitude-squared squeezed states are shaded in Figs. 3(b) and (d). It is also observed that both Q1 and Q2 show oscillatory behavior in the region b2 < 0:6, and that Q2 has a stronger squeezing than Q1 . 4. Conclusion We have developed a model to describe the three-level atom in a microcavity which exhibits the trapping effect. We have obtained the optical trapping conditions for the three-level atom passing through a cavity. Such trapping effect can be realized for higher excited Rydberg states of atoms, for which the transition energies between the transitions jn; si ! jn 1; pi and jn 1; pi ! jn 1; si are approximately equal. The dependence of the detuning xei xig on the excitation number for Rydberg atoms has been described in [4]. The detuning tends to zero around n 40. It should be noted that the cavity electromagnetic field state obtained from the trapping conditions of the three-level atom has interesting statistical pecularities that are not observed in the trapping effect of the two-level atom. We have studied the quantum statistical properties of the such cavity field, which provide the trapping conditions for the atom. We have investigated the mean photon number, the fluctuations, the second-order squeezing and amplitude-squared squeezing of the cavity field in a cotangent state. It is shown that the behavior of such a field characteristics totally differ from the case of two-level trapping. The oscillatory behavior of the fluctuations as function of parameters a; b and c is a specific behavior in the three level optical trapping effect. Although the model with a lossless cavity may seem to be academic, actually it may not be too far from the reality as experiments on micromasers have been performed with extremely high Q values as Q 1011 [1]. Experimental verification of the results obtained here requires very low temperatures such that the number of thermal photons is much less than 1. This can be achieved [1,12]. It is already pointed out that the existence of trapping states does not seem to be very sensitive to small velocity fluctuation in the beam [17]. On the other hand, one would expect these states to be robust to cavity damping [18]. In any case, we believe that the experimental difficulties can all be overcome by the present day technology.
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References [1] [2] [3] [4] [5] [6]
[7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18]
G. Rempe, H. Walther, N. Klein, Phys. Rev. Lett. 58 (1987) 353. M. Brune, J.M. Raimond, P. Goy, L. Davidovich, S. Haroche, Phys. Rev. Lett. 59 (1987) 1899. J.J. Slosser, P. Meystre, S. Braunstein, Phys. Rev. Lett. 63 (1989) 934. R. Short, L. Mandel, Phys. Rev. Lett. 51 (1983) 384; G. Rempe, F. Schmidt-Kaler, H. Walther, Phys. Rev. Lett. 64 (1990) 2783. M. Orszag, R. Ramirez, J.C. Retamal, L. Roa, Phys. Rev. A 45 (1992) 6717. X. Li, D.L. Lin, C.D. Gong, Phys. Rev. A 36 (1987) 5209; Z.D. Liu, X. Li, D.L. Lin, Phys. Rev. A 36 (1987) 5220; D.L. Lin, X. Li, T.N. Peng, Phys. Rev. A 39 (1989) 1933; X. Li, D.L. Lin, T.F. George, Phys. Rev. A 40 (1989) 228. N.A. Enaki, V.I. Koroli, J. Opt. Soc. Am. A 16 (1999) 2973. E. Wehner, R. Seno, N. Sterpi, B. Englert, H. Walther, Opt. Commun. 110 (1994) 655. F. Li, D.L. Lin, J. Phys. B: At. Mol. Opt. Phys. 30 (1997) 3719. M. Hillery, J. Skvarcek, J. Mod. Opt. 45 (1998) 1717. P. Filipowicz, J. Javanainen, P. Meystre, J. Opt. Soc. Am. B 3 (1986) 906. D. Meschede, H. Walther, G. Muller, Phys. Rev. Lett. 54 (1985) 551. P. Filipowicz, J. Javanainen, P. Meystre, Phys. Rev. A 34 (1986) 3077. R.R. Puri, C.K. Law, J.H. Eberly, Phys. Rev. A 50 (1994) 4212. C.K. Hong, L. Mandel, Phys. Rev. Lett. 54 (1985) 323; Phys. Rev. A 32 (1985) 974. M. Hillery, Opt. Commun. 62 (1987) 135; Phys. Rev. A 36 (1987) 3796. M. Orszag, J.C. Retamal, C. Saavedra, Phys. Rev. A 45 (1992) 2118. J.J. Slosser, P. Meystre, Phys. Rev. A 41 (1990) 3867.
Trapping conditions for a three-level atom interacting with cavity fields N.A. Enaki b
a,*
, V.I. Ciornea a, D.L. Lin
b
a Institute of Applied Physics, Academy of Sciences of Moldova, Moldova, Moldavia Department of Physics and Astronomy, University at Buffalo, State University of New York, Buffalo, NY 14260, USA
Received 30 January 2003; received in revised form 27 May 2003; accepted 30 June 2003
Abstract The interaction between a single-mode cavity field and a string of three-level atoms of cascade type with equally spaced energy levels is investigated. The matrix elements of dipole transitions between adjacent levels are different. Trapping conditions for the flying time in the cavity as well as the explicit state of the field are found. The results are shown to be qualitatively different from the case of two-level systems. The properties of cavity fields are examined. In various limits, the state exhibits vacuum nutation as well as sub-Poissonian statistics and squeezing properties. Ó 2003 Published by Elsevier B.V.
1. Introduction The creation of one-atom micromasers has prompted experimental studies of a single atom interacting with the resonant mode of cavity fields in recent years [1,2]. Much interest in quantum optics has been generated by such devices, especially, in connection with the trapping states in the micromaser. It has been shown that under certain circumstances, a simple quantum harmonic oscillator driven by a quantum current evolves to unique pure states even if it starts as a mixed state [3]. It has also been shown that in various limits, these states exhibit non-classical properties such as sub-Poissonian statistics [4], or more interestingly resemble a macroscopic superposition. Moreover, it is found from the analysis of coherent trapping states in a lossless two-photon micromaser that the field evolves to a pure state, which may be a superposition of even or odd photon number states [5]. These states may exhibit perfect second-order squeezing behavior as the upper limit when the trapping increases. The general properties of a three level atom interacting with cavity fields have been discussed in great detail in the past [6]. Consider a three-level atom of cascade type with the ground state, first and second excited states denoted by jgi, jii and jei, respectively. In the case the levels are equally spaced, the trapping condition in a cavity is already found when the dipole transition matrix elements between adjacent levels *
Corresponding author. Fax: +373-2-739907. E-mail address: [email protected] (N.A. Enaki).
0030-4018/$ - see front matter Ó 2003 Published by Elsevier B.V. doi:10.1016/j.optcom.2003.06.003
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are assumed to be the same [7]. The initial state ajei þ bjii þ cjgi differs from the state ajei bjii þ cjgi pffiffifinal ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi by a phase change. trapping conditions for the flying time s are given by 2 sk 2N u þ 3 ¼ pp for even pffiffiffi The pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi integers p and 2sk 2Nd 1 ¼ qp for even integers q. Here, Nd and Nu are the lower and upper values of photon number states in the Fock space. It is shown that such a three-level atom is equivalent to a pair of indistinguishable two-level atoms [7]. This may seem to be obvious at a first glance. However, a more careful study reveals that it is not true in general even if the levels are equally spaced. When the matrix elements for different transitions are equal, the three-level results can never pass to the two-level results in any limit. On the other hand in the [8] it was shown that the second distinguishable atom, which enters the cavity before the first atom leaves, destroys the trapping effect. The aim of this paper is to find some connection between the trapping conditions for a three-level equidistant system and that of the two-level one. In doing this, one considers different dipole moment matrix elements of dipole transitions between adjacent levels. As a matter of fact, the passing to two-level atoms is impossible for fixed Nd and Nu , as we shall discuss below. In other words, passing to a pure cotangent state of the two-level atom [3,9,10] is practically unrealizable due to the fact that the recursion relation for a threelevel system with equal level spacing is quite different from that for a two-level system. For instance, when the dipole transition between the states jgi and jii tends to zero, the recursion relation and lower trapping conditions do not pass into that for a two-level system. In a similar fashion, it can be shown that the upper trapping condition for a three-level atom is not valid when the transition between jii and jei tends to zero. Perhaps, this is because the second atom, which enters the cavity before the first atom leaves, destroys the trapping effect [8]. Since we assume different transition matrix elements, the problem becomes more general and is expected to provide deeper physical insights in the understanding of trapping multilevel systems. In the present work, we obtain the trapping-state solution for a three-level atom in the cavity and investigate the properties of the field in the trapping state. The surprising result is that this state exhibits not only such properties as sub-Poissonian statistics and squeezing, but also vacuum nutation which, to our knowledge, has thus far been observed only in the case of two-level atom [3] and has not been reported in previous studies of the trapping effect in a micromaser [11].
2. Three-level atom with arbitrary matrix elements We consider, as usual, a micromaser in which a very high-Q cavity is pumped by a string of atoms at a rate so low that there is at most one atom at a time in the cavity [12,13]. The incident beam consists of threelevel atoms of cascade type with equally spaced levels but different dipole transition matrix elements. Suppose that the pumping is regular so that the time interval sp between the arrival of two successive atoms remains unchanged [14]. Assume further that the cavity is lossless. The system of the electron interacting with a cavity field is described by the Hamiltonian hx0 ðjeihej jgihgjÞ þ hk1 ðay jiihej þ ajeihijÞ þ hk2 ðay jgihij þ ajiihgjÞ; H ¼ hxay a þ
ð1Þ
where hx0 stands for the atomic level spacing, and the energy is measured from the middle level of the atom. The operator ay ðaÞ creates (annihilates) a photon of frequency x in the cavity, and hk1 and hk2 are the electron-field coupling energies. It is clear that k1 and k2 are proportional to the dipole transition matrix elements d32 and d21 , respectively. The state-vector of the system at time t is given by jWðtÞi ¼
Nu X n¼Nd
sn jniðajei þ bjii þ cjgiÞ;
ð2Þ
PNu sn jni. in which we have expanded the one-mode cavity field on the finite number of the Fock states n¼N d When the atom enters the cavity at time t0 , the state evolves according to the Schr€odinger equation
N.A. Enaki et al. / Optics Communications 226 (2003) 285–296
ih
o jWðt0 Þi ¼ H jWðt0 Þi: ot0
287
ð3Þ
It is not difficult to observe that such a three-level system is equivalent to the two-level atomic system when either k1 ! 0 or k2 ! 0. We will study the trapping condition only for the resonance case x ¼ x0 . For the coupled system ‘‘atom + cavity’’ the state-vector at time t þ s is jWðt þ sÞi ¼ expðiH s= hÞjWðtÞi ¼ exp½iðx0 ðjeihej jgihgjÞ þ x0 ay aÞs ( " Nu X 2ðn þ 1Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Þ þ g2 ðn þ 1Þð1 þ g jni sn a 1 sin ðn þ 1Þð1 þ g2 Þ þ g2 2 n¼Nd rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ib sin k1 s n þ g2 ðn þ 1Þ jn 1i n þ g2 ðn þ 1Þ # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nðn 1Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 nð1 þ g Þ 1 jn 2i jei sin 2cg 2 nð1 þ g2 Þ 1 " Nu pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X sn b cos k1 s n þ g2 ðn þ 1Þ jni þ n¼Nd
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1 ia sin k1 s ðn þ 1Þð1 þ g2 Þ þ g2 jn þ 1i ðn þ 1Þð1 þ g2 Þ þ g2 # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n 2 icg sin k1 s nð1 þ g Þ 1 jn 1i jii nð1 þ g2 Þ 1 " Nu X n k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 nð1 þ g Þ 1 jni sin þ sn c 1 2g nð1 þ g2 Þ 1 2 n¼Nd pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn þ 1Þðn þ 2Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Þ þ g2 jn þ 2i ðn þ 1Þð1 þ g sin 2ag 2 ðn þ 1Þð1 þ g2 Þ þ g2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ) pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1 2 sin k1 s n þ g ðn þ 1Þ jn þ 1i jgi ; ibg n þ g2 ðn þ 1Þ
ð4Þ
where we have defined g ¼ k2 =k1 . To find the recursion formula, we first change the photon number states in every term in (Eq. (4)) to jni by shifting n accordingly in various terms, then replace sn by Sn ¼ sn Hðn Nd Þ HðNu nÞ, where HðxÞ is the step function. This enables us to include all nine term under a single sum over n from 0 to 1. More explicitly, Eq. (4) becomes jWðt þ sÞi ¼ expðiH s= hÞjWðtÞi ¼ exp i x0 ðjeihej jgihgjÞ þ x0 ay a s (" 1 X ffi 2ðn þ 1Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðn þ 1Þð1 þ g Þ þ g jni Sn a 1 sin ðn þ 1Þð1 þ g2 Þ þ g2 2 n¼0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1 sin k1 s ðn þ 1Þ þ g2 ðn þ 2Þ jni Snþ1 ib 2 ðn þ 1Þ þ g ðn þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ðn þ 2Þðn þ 1Þ k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ðn þ 2Þð1 þ g Þ 1 jni jei sin Snþ2 2cg 2 ðn þ 2Þð1 þ g2 Þ 1
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"
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n þ Sn b cos k1 s n þ g2 ðn þ 1Þ jni Sn1 ia sin k1 s nð1 þ g2 Þ þ g2 jni nð1 þ g2 Þ þ g2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nþ1 sin k1 s ðn þ 1Þð1 þ g2 Þ 1 jni jii Snþ1 icg ðn þ 1Þð1 þ g2 Þ 1 " n k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2Þ 1 nð1 þ g sin þ Sn c 1 2g2 jni nð1 þ g2 Þ 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ðn 1Þn k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 Þ þ g2 jni ðn 1Þð1 þ g sin Sn2 2ag 2 ðn 1Þð1 þ g2 Þ þ g2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi # ) n Sn1 ibg ð5Þ sin k1 s ðn 1Þ þ g2 n jni jgi : ðn 1Þ þ g2 n The recursion formula for Sn follows if we require that the probability amplitude for the ground state remains unchanged when time changes. In other words, we impose the condition "
cSn 1 2g2
n k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2Þ 1 nð1 þ g sin jni nð1 þ g2 Þ 1 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðn 1Þn k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðn 1Þð1 þ g Þ þ g jni Sn2 2ag sin 2 ðn 1Þð1 þ g2 Þ þ g2 # rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n sin k1 s ðn 1Þ þ g2 n jni jgi ¼ Sn jnicjgi: Sn1 ibg ðn 1Þ þ g2 n
ð6Þ
By noting the relations nð1 þ g2 Þ 1 ¼ ðn 1Þð1 þ g2 Þþ g2 ¼ n 1 þ g2 n, one can easily show that Eq. (6) reduces to rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nð1 þ g2 Þ 1 1 a n1 2 cot nð1 þ g Þ 1 Sn1 Sn ¼ ð7Þ Sn2 : gc 2 n gc n When this condition is satisfied, it is straightforward to show that the atom and cavity field become separable and Eq. (5) can be written as Nu X jWðt þ sÞi ¼ exp i x0 Sz þ x0 ay a s Sn jniðajei bjii þ cjgiÞ:
ð8Þ
n¼Nd
For the lower bound photon Nd , the corresponding coefficient Sn is determined from pffiffiffi number np¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Eq. (7) by multiplying it by n tanððk1 s=2Þ nð1 þ g2 Þ 1Þ and then setting SNd 1 ¼ SNd 2 ¼ 0. Thus, we have pffiffiffiffiffiffi Nd tan
k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nd ð1 þ g2 Þ 1 SNd ¼ 0; 2
ð9Þ
which implies for non-vanishing SNd that k1 s
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Nd ð1 þ g2 Þ 1 ¼ p0 p
ð10Þ
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289
for even p0 , and Nd 6¼ 0. As follows from expression (6) and expression (9), lower bound photon number condition Nd ¼ 0 is satisfied for arbitrary flight time s and trapping condition depends only on the upper bound condition. In a similar fashion, we find for non-vanishing SNu corresponding to the upper bound of photon number Nu the condition pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s ðNu þ 1Þð1 þ g2 Þ þ g2 ¼ pp; ð11Þ for even p. The expressions (10) and (11) are the trapping conditions for the equidistant three-level atom with arbitrary dipole momentum transitions between the levels jei; jii p and jgi. From expression (9) follows ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi that when Nu ¼ 0 one can obtain the vacuum trapping condition k1 s 1 þ 2g2 ¼ pp. It is interesting from physical point of view to found the transition of this conditions to two-level system. When k1 ¼ k2 one obtain the trapping condition for two-level system from obtained results. Let us consider the case when k1 ! 0 (g ! 1 evidently). In this case the recursion relation pass into relation for cotangent state of two-level atomic system b k2 s pffiffiffi n sn1 : sn ¼ i cot ð12Þ c 2 Wepobserve ffiffiffiffiffiffiffiffiffiffiffiffiffiffi that when n ¼ Nu þ 1 the upper trapping condition can be obtained for above relation k2 s Nu þ 1 ¼ pp for odd p. This condition violates the upper trapping condition for three-level atom when k1 ! 0 (see conditionp(11)). ffiffiffiffiffiffi Let us consider n ¼ Nd . From (12) it is follows the lower trapping condition for a two-level atom k2 s Nd ¼ qp for even q. This condition coincides with condition (10) for a three-level system, when k1 ! 0. Where is the reason that upper conditions are different for two-level and three-level systems? In order to answer this question let us represent all coefficients Sn through SNu , by using the recursion relation (7). Considering SNu 6¼ 0 we can express the SNu 1 through SNu by the following ffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Þ 1 ffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN þ 1Þð1 þ g 1a Nu u cot ðNu þ 1Þð1 þ g2 Þ 1 SNu SN 1 SNu þ1 ¼ gc 2 Nu þ 1 g c Nu þ 1 u ¼ 0:
ð13Þ
From this equation one obtain that SNu 1
b ¼ i cot a
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ðNu þ 1Þ k21 þ k22 k21 2
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðNu þ 1Þðk21 þ k22 Þ k21 SNu : Nu k1
ð14Þ
One can observe here that when k1 ! 0, the coefficient SNu 1 tends to 1. This coefficient SNu 1 can achieve finite values which donÕt depend on the k1 only in the case when SNu is proportional to the k1 . But this is in conflict with our proposal that SNu 6¼ 0 for k1 ¼ 0. In the same way letÕs represent SNu 2 through SNu 1 when k1 ! 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Þ 1 i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N ð1 þ g 1 a Nu 1 u cot Nu ð1 þ g2 Þ 1 SNu 1 SNu ¼ ð15Þ SNu 2 k1 ! 0: gc 2 Nu gc Nu In this case SNu 2 is
SNu 2
b ¼ i cot a
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Nu k21 þ k22 k21 SNu 1 s 2 2 Nu k1 þ k2 k1 : 2 Nu 1 k1
ð16Þ
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In order to obtain the non-zero finite value for SNu 2 when k1 ! 0 we again require that in the right-hand side of Eq. (16), the coefficient will be proportional to k1 . In opposite case we obtain the divergent expression in (16). This requirement is impossible because we already defined SNu 1 to be a constant, which does not depend on k1 . In other words, we have only one possibility to put SNu ¼ SNu 1 ¼ 0. This possibility gives us the zero value of SNu 2 . By proceeding this procedure we obtain all zero coefficients, which is in contrast with normalized condition 2
2
2
jSNd j þ þ jSn j þ jSNu j ¼ 1: We conclude that trapping conditions for a three-level system are different and do not pass in trapping conditions for a two-level system when k1 ! 0. To obtain the analytical expression for the coefficient Sn , for arbitrary k1 and k2 , expressed via SNu , we observe that Sn can be represented through SNu in the following manner Sn ¼
NY u 1
ð17Þ
Qm SNu ;
m¼n
where the coefficient Qm can be expressed through continued fraction by the following: ( rffiffiffiffiffiffiffiffiffiffiffiffi) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Þ 1 b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 2Þð1 þ g c mþ2 ðm þ 2Þð1 þ g2 Þ 1 Qm ¼ i cot g a 2 mþ1 a mþ1 8 9 qffiffiffiffiffiffiffi > ,> c > > mþ3 < = g a mþ2 ; bðsÞ > > cðsÞ. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > : ðNu þ1Þð1þg2 Þ1 ; . .ib cot k1 s ðN þ1Þð1þg2 Þ1 u a
2
ð18Þ
Nu
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 3Þð1 þ g2 Þ 1 2 ðm þ 3Þð1 þ g Þ 1 ; 2 mþ2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm þ 4Þð1 þ g2 Þ 1 2 ðm þ 4Þð1 þ g Þ 1 : cðsÞ ¼ i cot a 2 mþ3
b bðsÞ ¼ i cot a
In the limit k2 ! 0, Qm reminds the corresponding coefficient of the tangent state of the two-level atom. But this is not so true, as we can take the limit k2 ! 0 andprepresent all coefficients Sn through SNd . We ffiffiffiffiffiffi ¼ ð2m þ 1Þp is violated by condition (10) observe that lower trapping condition for tangent states k s N 1 d pffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s Nd 1 ¼ 2mp, where m is an integer. In this case the recursion relation (7) goes to recursion relation for tangent states Sn ¼ i
a k2 s pffiffiffi tan n Sn1 ; b 2
when k2 ¼ 0. Representing the first two coefficients SNd þ1 and SNd þ2 through SNd ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2Þ 1 ffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðN þ 1Þð1 þ g d cot ðNd þ 1Þð1 þ g2 Þ 1 SNd SNd þ1 ¼ gc 2 Nd þ 1 and
N.A. Enaki et al. / Optics Communications 226 (2003) 285–296
SNd þ2
i b cot ¼ gc
ffi k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðNd þ 2Þð1 þ g2 Þ 1 2 qffiffiffiffiffiffiffiffi 1 a g c
gi
b c
cot
291
ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðNd þ 2Þð1 þ g2 Þ 1 Nd þ 2
Nd þ1 Nd þ2
ffi SNd þ1 ; k s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 Þ1 1 ðNd þ 1Þð1 þ g2 Þ 1 ðNd þ1Þð1þg 2 Nd þ1
one can conclude that when k2 tends to zero, the finite and different from zero SNd þ1 can be obtained only in the case when SNd / k2 . This fact is in conflict with the assumption that SNd must be non-zero, for k2 ¼ 0. In this case SNd þ1 from the first expression becomes finite and independent on k2 . From the second expression in order to obtain the finite SNd þ2 we again must suppose that SNd þ1 / k2 and this is in conflict with the conclusion of the first expression. In general, it is clear, that it is impossible to get the tangent states from our trapping conditions (10) and (11). From above considerations it is not difficult to write down the product SN ¼
N Y
ð19Þ
Pm SNd :
m¼Nd þ1
Here, the coefficient Pm can be expressed through continued fraction of the form ( rffiffiffiffiffiffiffiffiffiffiffiffi) rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mð1 þ g2 Þ 1 1 a m1 2 cot mð1 þ g Þ 1 Pm ¼ gc 2 m gc m 8 9 q ffiffiffiffiffiffi ffi > ,> > > 1 a m2 < = g c m1 ; bðsÞ > > cðsÞ. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > ðNd þ1Þð1þg2 Þ1 > : ; . . i b cot k1 s ðN þ1Þð1þg2 Þ1 d gc
2
ð20Þ
Nd þ1
where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm 1Þð1 þ g2 Þ 1 ðm 1Þð1 þ g2 Þ 1 ; 2 m1 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i b k1 s pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðm 2Þð1 þ g2 Þ 1 cðsÞ ¼ cot ðm 2Þð1 þ g2 Þ 1 : gc 2 m2
i b bðsÞ ¼ cot gc
In the limit k1 ! 0, Pm tends to the corresponding coefficient of the cotangent state of the two-level atom. With the increases of m, Pm drastically differ from the same coefficient in cotangent state. We end this section with conclusion that trapping conditions for a three-level equidistant system with different dipole momentum transition elements, totally differ from that of a two-level system. These systems (two-level and three-level systems), in principle, have the same behavior only in a single case, when Nu ! 1. In this case upper trapping condition loses the sense.
3. Results and discussion We are now in a position to discuss the statistical properties of the cavity field in trapping states. First, we note that the coefficients obtained from Eqs. (7) and (19) are very different from their counter parts in the case of a two-level atom interacting with one-mode cavity fields [3]. The major reason is of course that in the present case all coefficients depend on a, b and c. Because these parameters represent the photon
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2
2
population probability, they satisfy the normalization condition jaj þ jbj þ jcj ¼ 1. Thus the electromagnetic field fluctuations depends on two independent parameters, say, a and b. As a consequence, the photon statistics in a cavity changes drastically when the two-level atom is replaced by a three-level atom. The mean photon number hni of the cotangent state is defined by hni ¼
Nu X
Sn Sn0 hnjay ajn0 i ¼
n;n0 ¼0
Nu X
2
jSn j n:
ð21Þ
n¼0
The fluctuation r in the mean excitations is defined by 2
r¼
hn2 i hni ; hni
Fig. 1. (a) Mean excitation hni of the cotangent state as a function of the populations of the second excited states jaj2 and jbj2 . (b) Fluctuations in the mean excitations r ¼ ðhn2 i hni2 Þ=hni. (c) The black region represents sub-Poisson statistics (r < 1) and white region represents super-Poisson statistics (r > 1) of cavity field, parameters used are Nu ¼ 20, p ¼ 10 and g ¼ 1:1.
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where the mean square is given by hn2 i ¼
Nu X
2
jSn j n2 :
n¼0
When r < 1, the photon statistics is sub-Poisonian. The cavity field behavior is investigated by calculating 2 2 hni and r as a function of the photon population probability jaj and jbj . When k2 k1 or the transition matrix element dig die . It seen that for Nu ¼ 20 the mean photon number decrease from maximum value to zero with a sharp transition in the domain 0:5b2 < a2 < 0:7b2 . The fluctuations r takes the maximum values in this domain too. The cavity field obeys the sub-Poisson statistics in the domain a2 > 0:6b2 . The most interesting results are obtained when the dipole transition matrix elements are roughly of the same size. We consider the case for g ¼ 1:1, and present the results for Nu ¼ 20 in Fig. 1. In both Figs. 1(a)
ð2Þ
ð2Þ
Fig. 2. Second-order squeezings D1 (a) and D2 (c) of the cotangent state as a function of the populations of the second excited states jaj2 and jbj2 . The shaded regions in (b) and (c) represent, respectively, the negative values of the first and second quadratures of amplitude-squared squeezing. Here, Nu ¼ 20, p ¼ 10 and g ¼ 1:1.
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and (b), a more oscillatory behavior of the mean photon number hni and r are observed, especially in the region a2 ; b2 < 0:6. In this region, the sub- (black) and super-Poisonian (white) statistics show interesting oscillatory behavior as illustrated in Fig. 1(c). These results indicate that the anomalous oscillatory behavior of the fluctuation takes place when the two dipole moments are of the same size. In what follows, we study the second-order squeezing and amplitude-squared squeezing. The two slowly varying Hermitian quadrature components of the field amplitude are defined by d1 ¼ ðA þ Aþ Þ=2;
d2 ¼ ðA Aþ Þ=2i;
ð22Þ N
N
where A ¼ a expðix0 tÞ. Whenever the condition hðDdi Þ i < ðN 1Þ!!=2 is satisfied, the corresponding state is called the Nth-order squeezing state [15]. The degree of squeezing is measured by the parameter
Fig. 3. Amplitude-squared squeezings Q1 (a) and Q2 (c) of the cotangent state as a function of the populations of the second excited states jaj2 and jbj2 . The shaded regions in picture (b) and (d) represent, respectively, the negative values of the first and second quadratures of the amplitude-squared squeezing. Here, Nu ¼ 20, p ¼ 10 and g ¼ 1:1.
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N
ðN Þ
Di
¼
2N hðDdi Þ i 1; ðN 1Þ!!
i ¼ 1; 2:
ð23Þ ðN Þ
According to the definition in Eq. (22), it is easily seen that squeezing appears when 1 < Di In a similar fashion, we define two operators Y1 ¼ ðA2 þ Ay2 Þ=2;
Y2 ¼ ðA2 Ay2 Þ=2i:
< 0. ð24Þ
The condition for the amplitude-squared squeezing is [16] 2 hðDYi Þ i < hN^ þ 1=2i; i ¼ 1; 2: ð25Þ y ^ where N ¼ a a represents the photon number operator. The amplitude-squared squeezing is measured by the operators 2
Qi ¼
hðDYi Þ i hN^ þ 1=2i ; hN^ þ 1=2i
i ¼ 1; 2:
ð26Þ
Thus, the field is in an amplitude-squared squeezed state when 1 < Qi < 0. The smaller the Qi , the stronger the amplitude-squared squeezing is. In the numerical calculation, we assume real probability amplitudes a, b, c and g ¼ 1:1. The secondð2Þ order squeezing Di is calculated for Nu ¼ 20 and results are presented in Fig. 2. It is observed in Figs. 2(a) ð2Þ ð2Þ and (c) that both D1 and D2 shoe oscillations in the region b2 < 0:6. The shaded area in Figs. 2(b) and (d) ð2Þ ð2Þ shows the squeezing found in this calculation. It is seen that D2 has a stronger squeezing than D1 . The results of numerical simulation of Qi are presented in Fig. 3, and the amplitude-squared squeezed states are shaded in Figs. 3(b) and (d). It is also observed that both Q1 and Q2 show oscillatory behavior in the region b2 < 0:6, and that Q2 has a stronger squeezing than Q1 . 4. Conclusion We have developed a model to describe the three-level atom in a microcavity which exhibits the trapping effect. We have obtained the optical trapping conditions for the three-level atom passing through a cavity. Such trapping effect can be realized for higher excited Rydberg states of atoms, for which the transition energies between the transitions jn; si ! jn 1; pi and jn 1; pi ! jn 1; si are approximately equal. The dependence of the detuning xei xig on the excitation number for Rydberg atoms has been described in [4]. The detuning tends to zero around n 40. It should be noted that the cavity electromagnetic field state obtained from the trapping conditions of the three-level atom has interesting statistical pecularities that are not observed in the trapping effect of the two-level atom. We have studied the quantum statistical properties of the such cavity field, which provide the trapping conditions for the atom. We have investigated the mean photon number, the fluctuations, the second-order squeezing and amplitude-squared squeezing of the cavity field in a cotangent state. It is shown that the behavior of such a field characteristics totally differ from the case of two-level trapping. The oscillatory behavior of the fluctuations as function of parameters a; b and c is a specific behavior in the three level optical trapping effect. Although the model with a lossless cavity may seem to be academic, actually it may not be too far from the reality as experiments on micromasers have been performed with extremely high Q values as Q 1011 [1]. Experimental verification of the results obtained here requires very low temperatures such that the number of thermal photons is much less than 1. This can be achieved [1,12]. It is already pointed out that the existence of trapping states does not seem to be very sensitive to small velocity fluctuation in the beam [17]. On the other hand, one would expect these states to be robust to cavity damping [18]. In any case, we believe that the experimental difficulties can all be overcome by the present day technology.
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