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Velocity selection for ultracold atoms using mazer action in a bimodal cavity
Optics Communications 283 (2010) 54–59
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Velocity selection for ultracold atoms using mazer action in a bimodal cavity Afshan Irshad a, Sajid Qamar b, Shahid Qamar a,c,* a
Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan Centre for Quantum Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan c The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy b
a r t i c l e
i n f o
Article history: Received 22 April 2009 Received in revised form 21 August 2009 Accepted 10 September 2009
a b s t r a c t In this paper, we discuss the velocity selection of ultracold three-level atoms in K configuration using a mazer. Our model is the same as discussed by Arun et al. [R. Arun, G.S. Agarwal, M.O. Scully, H. Walther, Phys. Rev. A 62 (2000) 023809] for mazer action in a bimodal cavity. We show that the initial Maxwellian velocity distribution of ultracold atoms can be narrowed due to the presence of resonances in the transmission through dressed-state potential. When the atoms are initially prepared in one of the two lower atomic states then significantly better velocity selectivity is obtained due to the presence of dark states. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Certain applications in atom optics, like matter-wave interferometry [1] and using light as lenses for submicron neutral atom lithography [2] desire production of atomic beams and the control of their motion. These require long coherence length, i.e., a small velocity spread of the atomic beam. It is known that in most cases the velocity of the atoms follow a Maxwellian distribution with a velocity spread equivalent to the most probable velocity. It is also possible to get ultracold atoms by using laser cooling techniques [3]. From some earlier studies, it has been realized that micromaser systems are of considerable interest in the field of quantum optics [4]. This is due to the fact that a number of features of the radiation field could be predicted in the micromaser systems. For example, sub-Poissonian photon statistics [5], trapping states, number states [6], and others [7]. The idea of trapping state has been realized in experiments [8]. When the micromaser is injected with ultracold atoms such that their kinetic energy is smaller than the atom–field interaction energy then their motion has to be described quantum mechanically [9]. It has been shown by Scully et al. [10] that the quantization of the center of mass motion leads to a completely new kind of induced emission called mazer (microwave amplification via z-induced emission of radiation). In ordinary maser, stimulated emission prevails as the mechanism for amplification of radiation, but in case of ultracold atoms the physics of the induced emission process is associated with the quantization of the center of mass mo-
* Corresponding author. Address: Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan. Tel.: +92 51 2207381; fax: +92 51 2208070. E-mail address: [email protected] (S. Qamar). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.09.022
tion. The physical mechanism responsible for the induced emission is the abrupt change in the atom–field coupling strength, which an atom experiences upon passing through the cavity. The quantum theory of mazer has been established by Scully and co-workers [11–13] and a number of interesting features, like emission probability, photon statistics and transmission of the atoms through the mazer cavity has been studied therein. Later, the idea of mazer was used for velocity selection of ultracold atoms [14]. It was shown that the velocity distribution of the two-level atoms passing through a resonant single mode microwave cavity can be significantly narrowed. The velocity selection appears due to resonances in the transmission through dressedstate potential. Through this passive device, the coherence length of the atomic beam can be made macroscopic. In fact, it is an atom-optical analogue of a Fabry–Perot resonator. In another study the detuning effects on the velocity selection of two-level atoms has also been discussed [15]. The point to be mentioned here is that all these mazer studies are based on a single photon resonance process. The idea has been extended to two photon resonance case either for ultracold atoms with cascade three-level configuration using two micromaser cavities [16] or using V-type three-levelatom mazer with atomic coherence [17]. Former proposal exhibits that the velocity selectivity is better for ultracold three-level atoms using two-cavity system than for ultracold two-level atoms using one-cavity system. However, the latter scheme shows that the atomic coherence does not play a favorable role on the velocity selectivity. In both these schemes the two photon resonance corresponds to a single mode of the micromaser cavities. Recently, the operation of a two-mode mazer with particular reference to the interesting question of mode–mode correlation has been studied [18] using three-level K configuration where the atom–field system is a two photon resonance process. It has
A. Irshad et al. / Optics Communications 283 (2010) 54–59
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been observed that the degree of anticorrelation between the cavity modes increases when the micromaser is pumped by ultracold atoms instead of fast atoms. This exhibits much stronger sub-Poissonian photon statistics for each mode. Motivated from this work, we study velocity selection of the ultracold atoms using mazer action in a bimodal cavity with three-level atoms in the K configuration. Here, instead of considering the same velocity for the injected atoms, we consider an initial Maxwellian velocity distribution. Our results show that the initial velocity distribution of the atoms can be narrowed down to many orders in magnitudes after passing through the cavity. The dressed state analysis exhibits dark state when the atoms are initially prepared in one of the two lower energy levels [18]. Therefore, we also discuss the effect of dark states on the velocity selection in this system.
We use the atom–field dressed states j/0n1 þ1;n2 þ1 i, j/ n1 þ1;n2 þ1 i, detailed analysis is available in [18], which reduces the center of mass motion to a scattering problem. We consider that the atom with initial state jai and center of mass momentum h k enters into the mazer cavity containing n1 and n2 photons in mode 1 and mode 2, respectively, and is reflected or transmitted during its passage through the cavity. There are three possible choices for the atom during its motion through the mazer cavity, first; it does not change its initial state ja; n1 ; n2 i, second; it makes a transition to the lower level jb1 ; n1 þ 1; n2 i, third; it makes a transition to the lower level jb2 ; n1 ; n2 þ 1i. The corresponding probability amplitudes for the reflected or transmitted atom can be written as [18]
2. Model
Ra;n1 ;n2 ¼
ð4Þ
We consider a beam of ultracold, monoenergetic three-level atoms with K-type configuration (as shown in Fig. 1) which are injected in high-Q two-mode mazer cavity of length L [18]. It is assumed that the atomic flux is adjusted such that there is only one atom at a time inside the cavity. The transition between the two lower states jb1 i and jb2 i is dipole forbidden, however, the transition from upper to lower levels jb1 i and jb2 i is allowed. The modes 1 and 2 of the cavity are assumed to be resonant with the transition frequencies of the levels jai ! jb1 i and jai ! jb2 i. The Hamiltonian for the atom-cavity system including the effects of quantized center of mass (CM) motion is given by [18]
T a;n1 ;n2
ð5Þ
2 2 X X p2 H ¼ z þ hXa jaihaj þ hXba jba ihba j þ hxa aya aa 2m a¼1 a¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 n2 þ 1 Rb2 ;n1 ;n2 þ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qþn1 ;n2 qn1 ;n2 ; 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 n2 þ 1 T b2 ;n1 ;n2 þ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sþn1 ;n2 sn1 ;n2 ; 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ
ð9Þ
þ
2 X
hga ðaa jaihba j þ jba ihajaya Þ;
ð1Þ
a¼1
where m is the atomic mass and pz is the CM momentum operator of the atom. Here aa ðaya Þ are annihilation (creation) operators with a ¼ 1; 2 for the cavity field corresponding to mode 1 and 2, respectively. The resonance frequencies are xa ¼ Xa Xba . The parameters ga be the corresponding atom–field coupling constants that depend on spatial coordinate z through the mode function of the cavity. In the interaction picture, the Hamiltonian in Eq. (1) reduces to the form as [18]
HI ¼
2 X p2z hg ðaa jaihba j þ jba ihajaya Þ: þ 2m a¼1 a
ð2Þ
The mode function of the cavity, which is assumed to be a mesa function and is defined as
uðzÞ ¼ 1 for 0 < z < L; ¼ 0 elsewhere; with L being the length of the resonator.
ð3Þ
3. Dynamics
1 þ qn1 ;n2 þ qn1 ;n2 ; 2 1 þ ¼ sn1 ;n2 þ sn1 ;n2 ; 2
for the first case
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 n1 þ 1 Rb1 ;n1 þ1;n2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qþn1 ;n2 qn1 ;n2 ; 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ
ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 n1 þ 1 T b1 ;n1 þ1;n2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sþn1 ;n2 sn1 ;n2 ; 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ
ð7Þ
for the second case and
for the third case. Here q n1 ;n2 and sn1 ;n2 are the dressed-state reflection and transmission coefficients and can be calculated analytically using the mesa mod function (3) as
qn1 ;n2 ¼ iDn1 ;n2 sinðkn1 ;n2 LÞsn1 ;n2 ;
sn1 ;n2 ¼
1 P ; cos kn1 ;n2 L i n1 ;n2 sin kn1 ;n2 L
ð10Þ ð11Þ
with
! 1 kn1 ;n2 k ; 2 k kn1 ;n2 ! 1 kn1 ;n2 k ; Rn1 ;n2 ¼ þ 2 k kn1 ;n2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! u u 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ t 2 ; k j kn1 ;n2 ¼ g21 þ g22
Dn1 ;n2 ¼
where
j
ð12Þ ð13Þ
ð14Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m h
g21 þ g22 is the CM wave vector for which the ki-
netic energy ð hkÞ2 =2m of the atom equals the vacuum coupling enqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ergy h g21 þ g22 . Here we are interested in the regime where the
Fig. 1. Energy-level diagram of the three-level atomic system.
atomic wave number k is smaller than j as we are dealing with ultracold atoms. We consider a situation when the cavity field is initially in vacuum state, i.e., n1 ¼ n2 ¼ 0. The kinetic energy of the incident atoms is assumed to be so small that tunneling through the potential barrier plays no essential role in the mazer [12]. Under this
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condition, the transmission probability of the atom through the cavity without contributing any photon into the cavity modes, i.e., remaining in state jai is
Ptrans ða; 0; 0; kÞ ’
2 1 2 ½1 þ D0;0 sin k0;0 L 1 : 4
ð15Þ
Whereas the transmission probability that an atom will transmit after making a transition to the level jb1 i and emitting a photon in mode 1 of the cavity is given by
Ptrans ðb1 ; 1; 0; kÞ ’
2 g21 2 ½1 þ D0;0 sin k0;0 L 1 ; 2 þ g2 Þ
4ðg21
2 g22 2 ½1 þ D0;0 sin k0;0 L 1 : 4ðg21 þ g22 Þ
ð17Þ
It is clear from the Eqs. (15)–(17) along with (10)–(14) that the transmission shows resonances as a function of the scaled cavity length jL and the wave number of the atoms k=j. The resonances occur when the length of the cavity is an integral multiple of half of the de Broglie wave length of the atoms, i.e.,
k0;0 L ¼ mp;
ð18Þ
where m ¼ 1; 2; 3; . . . This is similar to the transmission of an excited two-level atom through a single-mode cavity. This behavior is just the transmission characteristic of the potential well as discussed in [11]. As is mentioned earlier that in the two photon transition process the dressed state analysis exhibits a dark state j/0n1 ;n2 i which is not present in one photon process. This dark state does not have any influence on the reflection and transmission coefficients when the initial state of the atom–field is ja; 0; 0i. However, its contribution appears in the probability amplitudes of reflection and transmission of the atom when the initial state of the atom is jb1 ; 1; 0i or jb2 ; 0; 1i [18]. This may be understand physically by considering the dressed state picture of the K system. The bare state of the atom–field system exhibits atomic transitions between levels jai and jb1 i and jai and jb2 i with the corresponding vacuum Rabi frequencies g1 and g2 . The dressed state analysis however, exhibits
i 1 h ja; 0; 0i ¼ pffiffiffi j/þ1;1 i þ j/1;1 i ; 2 1 g jb1 ; 1; 0i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1ffiffiffi j/þ1;1 i j/1;1 þ g2 j/01;1 i ; 2 g21 þ g22 1 g jb2 ; 1; 0i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2ffiffiffi j/þ1;1 i j/1;1 g1 j/01;1 i : 2 g21 þ g22
ð19Þ ð20Þ
ð21Þ
Considering the dressed state picture, we can observe that the atomic transitions from level jb1 i to jai follow the vacuum Rabi frequencies g2 and ð2g1 g2 Þ whereas the atomic transition from jb2 i to jai follow the vacuum Rabi frequencies g1 and ð2g2 g1 Þ. This shows that there is no atomic transition following the vacuum Rabi frequency g1 and g2 in the former and latter case, respectively. This clearly shows that if the atom is initially in one of the lower levels, i.e., jb1 i or jb2 i, the atomic transition exhibits a corresponding dark state. In the two-mode mazer, this dark state contributes to the enhancement of the probability of transmission of the atoms. The probability amplitude when the atom is transmitted in state jb1 ; 1; 0i comes out to be [18]
T b1 ;1;0 ¼
g21 g2 sþ0;0 þ s0;0 þ 2 2 2 ; 2 þ g2 Þ ðg1 þ g2 Þ
2ðg21
Ptrans ðb1 ; 1; 0; kÞ ¼
½g21 þ 2g22 cosðk0;0 LÞ2 þ 4g42 ðR0;0 Þ2 sinðk0;0 LÞ2 4ðg21 þ ½1 þ
ðD0;0 Þ2
2
g22 Þ2
sin ðk0;0 LÞ1 : ð23Þ
ð16Þ
and the transmission probability for the atom after making a transition to the level jb2 i and emitting a photon in mode 2 of the cavity can be written as
Ptrans ðb2 ; 0; 1; kÞ ’
where sþ 0;0 and s0;0 can be obtained from Eq. (11) and the term g22 =ðg21 þ g22 Þ is the contribution due to the dark state. The corresponding probability that the atom transmits through the cavity in the same initial state jb1 ; 1; 0; ki can be written as
ð22Þ
A similar expression, as in Eq. (22), for the transmitted atom when the initial and final atom–field state is jb2 ; 0; 1i can be obtained except that the coupling constant g1 is replaced with g2 and vice versa. 4. Discussion In the following, we discuss velocity selection for the ultracold atoms passing through the bimodal mazer cavity. We start with the analysis of the transmission probability Ptrans ðkÞ of the ultracold atoms with initial state ja; 0; 0i. That is, the initial state of the injected atom is assumed to be the excited state jai and the cavity fields in the two modes are considered to be initially in the vacuum state. We also assume that the atoms are injected inside the cavity with the same velocity which is the case in usual mazer. For all the analysis, we take the coupling constant g1 ¼ 3g2 . We study the behavior of the transmission probability Ptrans ðkÞ versus dimensionless atomic wave number k=j ranging from 0 to 0:5. We consider two different cavity lengths, i.e., jL is equal to (i) 20p (dashed line) and (ii) 200p (solid line) (jL is the dimensionless cavity length). We also consider three conditions, i.e., when atom exit the bimodal mazer cavity (a) without making any transition and remained in the initial state ja; 0; 0i, (b) making transition to the final state jb1 ; 1; 0i or (c) making transition to the final state jb2 ; 0; 1i. Figs. 2a–c show, that the maximum value of the transmission probability is 0.25, 0.22 and 0.025, respectively. This shows that the transmission probability depends on the condition whether the atom exits the cavity in state jai, jb1 i or jb2 i. If it remains in the same state then it does not contribute any photon inside the cavity, however, if it exits in any of the other two states then it contributes one photon in either of the two modes. The maximum value of the transmission probability depends upon the choice of the ratio g2 =g1 which is clear from Eqs. (15)–(17). For initial vacuum state P trans ða; 0; 0; kÞ is independent of this ratio, however, Ptrans ðb1 ; 1; 0; kÞ decreases while Ptrans ðb2 ; 0; 1; kÞ increases due to the increase in the ratio g2 =g1 . It is clear from the results that by increasing the length of cavity more resonances appear which become sharper and more closer to each other which is analogous to the behavior of a Fabry–Perot resonator. In order to study the velocity selection of ultracold atoms using the bimodal mazer cavity, we now consider that the atoms are injected inside the cavity with a certain velocity distribution. In the following, we assume that the atomic beam initially follows Maxwellian velocity distribution such that the number of atoms in the interval dk is given by 2
2 2 4 k Pi ðkÞdk ¼ pffiffiffiffi 3 ek =k0 dk; p k0
ð24Þ
where k0 corresponds to the most probable value for the wave number k. This form of Maxwellian velocity distribution is considered just to make our analysis in line with the earlier schemes for the velocity selection of ultracold atoms using micromazers [14–17]. It may be mentioned that the form of the Maxwellian distribution depends on whether one counts particles in a volume element or particles that enter the detector in a given time window [19]. After
A. Irshad et al. / Optics Communications 283 (2010) 54–59
Fig. 2. Transmission probability versus atomic wave number for ultracold atoms passing through a bimodal cavity in the vacuum state with g1 ¼ 3g2 whereas the cavity lengths are (i) jL ¼ 20p (dashed curve) and (ii) jL ¼ 200p (solid curve). For three different situations; (a) the atom exits the cavity in the excited state jai, (b) after making transition to state jb1 i and (c) after making transition to state jb2 i.
passing through the mazer cavity, the final k distribution of the atoms is given by
Pf ðkÞdk ¼ Ptrans ðkÞP i ðkÞdk:
ð25Þ
In Fig. 3, we plot the final wave number distribution for k0 =j ¼ 0:05 and jL ¼ 200p. We again consider three scenarios, i.e., atoms exit from the mazer cavity with final state (a) ja; 0; 0i, (b) jb1 ; 1; 0i and (c) jb2 ; 0; 1i. The inset in Fig. 3 shows the initial wave number distribution (Eq. (24)) with the most probable value for wave number k0 =j ¼ 0:05. We observe that the final k distribution in the main figure is dominated by a peak which corresponds to the single broad peak in the initial k distribution as shown in the
Fig. 3. Final wave number distribution for k0 =j ¼ 0:05, jL ¼ 200p and g1 ¼ 3g2 . Here, dashed line, dotted line and solid line correspond to the same conditions of the atom as mentioned in Figs. 2a–c, respectively. The inset shows the initial wave number distribution.
57
inset of Fig. 3. The height of the main peak depends on the situation that whether the atom has contributed inside the cavity or not. The peak height reduces if the atoms exit the cavity in the state jb1 ; 1; 0i or jb2 ; 0; 1i. We also observe that the central peak in the final k distribution has about 1/12th of the width of the initial k distribution after passing through the bimodal cavity, see Fig. 3. Next we, change the most probable velocity and consider k0 =j ¼ 0:1 which corresponds to twice as fast atoms as in the previous case (Fig. 3). The behavior of the initial and final k wave number distributions versus k=j are shown in Fig. 4. Here, the final wave number distribution exhibits oscillatory behavior which has a comb-like structure with several peaks within the range 0 to 0.4 of k=j. This is due to the increase in the average velocity of the incoming atoms as more resonances appear for higher average velocity of the incoming atoms. This gives rise to the multiple peaks structure in the final k wave number distribution. The height of the peaks decrease with increase in k=j. The width of each peak in the final k distribution is now about 1/24th of the initial k wave number distribution. The amplitude of the peaks are also considerably high for k0 =j ¼ 0:1 as compared with k0 =j ¼ 0:05, see Figs. 3 and 4. This shows that we get a much higher magnitude of the velocity distribution with almost twice narrower width for k0 =j ¼ 0:1 as compared to the case when k0 =j ¼ 0:05. For a comparison, we again consider the cases when the atoms exit the cavity in state (a) ja; 0; 0i, (b) jb1 ; 1; 0i and (c) jb2 ; 0; 1i. A similar behavior of the final k distribution as in Fig. 3 is observed. That is, the heights of the peaks depend on the situation whether the atoms contribute inside the cavity or not and the height reduces if the atoms exit the cavity in the state jb1 ; 1; 0i or jb2 ; 0; 1i. Next, we consider the effect of dark states on the velocity selection of ultracold atoms using a bimodal mazer action. It has been shown in [18], that for atoms initially prepared in state jb1 i with one photon in mode 1 or in state jb2 i with one photon in mode 2, the effects of dark state appear in a two-mode mazer. It contributes to the enhancement of the probability of transmission of atoms. Therefore, it is interesting to consider the effect of the dark state on the velocity selection of the ultracold atoms. We discuss the case when the atoms with initial state jb1 i and following the Maxwellian velocity distribution enter the bimodal mazer cavity. The Fig. 5 shows the behavior of the transmission probability distribution Ptrans versus normalized wave number k=j. We present the results for jL ¼ 20p (dashed line) and jL ¼ 200p (solid line). The transmission probability exhibits sharp resonances as we increase the value of jL from 20p to 200p. However, the heights of peaks vary, i.e., having maximum values of 0.3 and 0.1. This is different as compared to the earlier case discussed in Fig. 2 where all peaks have same maximum value. To understand this different behavior we rewrite the expression for the transmission probability given in Eq. (23) in a simplified form using g1 ¼ 3g2 as
Fig. 4. Final wave number distribution for k0 =j ¼ 0:1 and other parameters are same as in Fig. 3. Here, dashed line, dotted line and solid line correspond to the same situation as mentioned in Fig. 3. The inset shows the initial wave number distribution.
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A. Irshad et al. / Optics Communications 283 (2010) 54–59
A comparison of our results with the earlier studies [14–17] show that the width of the final k distribution is much narrower then the schemes based on two-level atomic system. This is due to the fact that the three-level atomic system exhibits dark states which is not present in the case of two-level atomic systems. 5. Conclusion
Fig. 5. Transmission probability as a function of the atomic wave number for ultracold atoms encountering a bimodal cavity that is initially in the state jb1 ; 0; 1i . Here (i) dashed line corresponds to jL ¼ 20p while (ii) solid line corresponds to jL ¼ 200p.
e trans ðb1 ; 1; 0; kÞ P n o 1 2 ¼ 4 þ 9½9 þ 4 cosðk0;0 LÞ ½1 þ ðD0;0 Þ2 sin ðk0;0 LÞ1 : 400 ð26Þ This shows that the variation in height of the peaks in Fig. 5 is due to the presence of the dark state which contributes cosine term, i.e., cos k0;0 L in the transmission probability distribution Eq. (26). In Fig. 6, we show the final wave number distributions when atoms initially in the state jb1 i, enter the cavity having one photon in mode 1 and no photon in mode 2. In Fig. 6, (a) solid line and (b) dashed line correspond to k0 =j equals to 0.05 and 0.1, respectively. For k0 =j ¼ 0:05, the final k distribution is dominated by a sharp single peak at k=j ¼ 0:1 whose width is about 1/18th of the width of the initial distribution, compare Fig. 6 (solid line) with inset of Fig. 3. If we double the average velocity of the atoms, i.e., k0 =j ¼ 0:1 then the final wave number distribution has a comblike structure with multiple peaks of different heights, see Fig. 6 (dashed line). The wave number distribution exhibits that the height of the peaks varies however, the width of each peak remains almost the same. Here, width of the most dominant peak in the distribution, i.e., at k=j ¼ 0:14 is about 1/36th of the width of the initial wave number distribution peak, compare Fig. 6(dashed line) with inset of Fig. 4. This shows that the width of the velocity distribution peaks can be made narrower by the use of the dark state. Further, the magnitude of the probability distribution can also be made much larger by using the dark state. A comparison of the plots in the Figs. 3, 4 and 6 exhibits almost 8 times larger and 3 times narrower peaks in the wave number distribution when atoms which are initially prepared in the state jb1 ; 1; 0i are transmitted through the mazer cavity in the same initial state. This is due to the influence of the dark state. A similar effect of the dark state can be observed if the atoms are initially prepared in the state jb2 ; 0; 1i and transmitted in the same initial state.
In conclusion, we have studied the velocity selection of ultracold atoms through a bimodal cavity. We considered the atoms to be initially in their excited state and field in a vacuum state. If the incoming atoms have a Maxwellian velocity distribution then after passing through the mazer cavity, an order of magnitude narrow velocity distribution can be obtained. We have also discussed the case when atoms, initially prepared in their state jb1 i, are injected into the cavity which has one photon in mode 1 and zero photon in mode 2. In this case, further reduction in the width of the velocity distribution is obtained due to the presence of dark states. Our results show that significantly precise velocity selection of ultracold atoms can be achieved in this system. For the experimental realization we may quote the Garching experiment where the typical length of the cavity has been considered to be about 2.5 cm which corresponds to the values of jL about 105 . For such a geometry, sharp mazer resonances occur which are much close to each other to be resolved, if the atoms are injected with certain velocity spread. However, as discussed in [11], smaller values of jL can be obtained with a cavity of re-entrant type. In this cavity, the interaction length can be made considerably smaller than the wavelength of the microwaves and still much larger then the de-Broglie wavelength of the atoms. Experimental consideration regarding mazer are discussed in detail in Ref. [12]. There, a re-entrant type cavity has been discussed with which velocity selection of ultracold atoms should be possible. In order to see this, here we consider the parameters which corresponds to Figs. 4 and 6b. If we consider a coupling strength of 100 kHz and 85Rb atoms, we obtain j ¼ 1:6 107 m1 . In order to satisfy k0 =j ¼ 0:1, a mean velocity of 1.2 mm/s is required, that corresponds to a temperature of about 10nK. The de-Broglie wavelength of the atoms for parameters under consideration is in the range of lm, it should be possible to realize a mesa like potential with a TM mode [12]. The finite lifetime of the atomic states raises problem in realizing the proposed device. However, the use of circular Rydberg states, which provide life times of the order of approximately 0.1 s can overcome this problem. Acknowledgment The author Shahid Qamar would like to thank the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy for kind hospitality and support through Associate Scheme. He would also like to thank Prof. Joseph Niemela for some fruitful discussion. References
Fig. 6. Final wave number distribution for jL ¼ 200p and (a) k0 =j ¼ 0:05 (solid line) and (b) k0 =j ¼ 0:1 (dashed line). All the other parameters are the same as in Fig. 5.
[1] M. Kasevich, S. Chu, Phys. Rev. Lett. 67 (1991) 181. [2] G. Timp, R.E. Behringer, D.M. Tennant, J.E. Cunningham, M. Prentiss, K.K. Berggren, Phys. Rev. Lett. 69 (1992) 1636; For a review, see,C.S. Adam, M. Sigel, J. Mlynek, Phys. Rep. 240 (1994) 143. [3] See, for example, E. Arimondo, W.D. Philips, F. Strumia (Eds.), Laser Manipulation of Atoms and Ions, North-Holland, Amsterdam, 1992. [4] For a review of early work on Rydberg atoms in microwave cavities, see S. Haroche, J.M. Raimond, Adv. At. Mol. Phys. 20 (1985) 350; A recent review is given byG. Raithel, C. Wagner, H. Walther, L.M. Narducci, M.O. Scully, in: P.R. Berman (Ed.), Cavity Quantum Electrodynamics, Academic, Boston, 1994, p. 57; D. Meschede, H. Walther, G. Muller, Phys. Rev. Lett. 54 (1985) 551; G. Rempe, F. Schmidt-Kaler, H. Walther, Phys. Rev. Lett. 64 (1990) 2783. [5] P. Filipowicz, J. Javanainen, P. Meystre, Phys. Rev. A 34 (1986) 3077; L.A. Lugiato, M.O. Scully, H. Walther, Phys. Rev. A 36 (1987) 740;
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P. Filipowicz, J. Javanainen, P. Meystre, Opt. Soc. Am. B 3 (1986) 906; P. Filipowicz, J. Javanainen, P. Meystre, Opt. Commun. 58 (1986) 327. P. Meystre, G. Rempe, H. Walther, Opt. Lett. 13 (1988) 1078; S. Qamar, K. Zaheer, M.S. Zubairy, Opt. Commun. 78 (1990) 341. M.O. Scully, H. Walther, G.S. Agarwal, T. Quang, W. Schleich, Phys. Rev. A 44 (1991) 5992. M. Weidinger, B.T.H. Varcoe, R. Heerlein, H. Walther, Phys. Rev. Lett. 82 (1999) 3795. B.G. Englert, J. Schwinger, A.O. Barut, M.O. Scully, Europhys. Lett. 14 (1991) 25; S. Haroche, M. Brune, J.M. Raimond, Europhys. Lett. 14 (1991) 19. M.O. Scully, G.M. Meyer, H. Walther, Phys. Rev. Lett. 76 (1996) 4144. G.M. Meyer, M.O. Scully, H. Walther, Phys. Rev. A 56 (1997) 4142. M. Loffler, G.M. Meyer, M. Schroder, M.O. Scully, H. Walther, Phys. Rev. A 56 (1997) 4153.
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[13] M. Schroder, K. Vogel, W.P. Schleich, M.O. Scully, H. Walther, Phys. Rev. A 56 (1997) 4164. [14] M. Löffler, G.M. Meyer, H. Walther, Europhys. Lett. 41 (1998) 593. [15] J. Martin, T. Bastin, Eur. Phys. D 29 (2004) 133. [16] Z.M. Zhang, S.W. Xie, Y.L. Chen, Y.X. Xia, S.K. Zhou, Phys. Rev. A 60 (1999) 3321; Z.M. Zhang, S.K. Zhou, Y.X. Xia, Y.L. Chen, S.W. Xie, Chin. Phys. Lett. 16 (1999) 568. [17] W.Q. Liag, Z.Q. Niu, Z.M. Zhang, S.W. Xie, J. Phys. B: At. Mol. Opt. Phys. 34 (2001) 4427. [18] R. Arun, G.S. Agarwal, M.O. Scully, H. Walther, Phys. Rev. A 62 (2000) 023809; Another reference byR. Arun, G.S. Agarwal, Phys. Rev. A 66 (2002) 043812. deals with the dark states and interferences in cascade transitions of ultracold atoms in a bimodal cavity. [19] N.F. Ramsey, Molecular Beams, Oxford University Press, Oxford, 1956; J.A. Eldridge, Phys. Rev. 30 (1927) 931.
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Optics Communications journal homepage: www.elsevier.com/locate/optcom
Velocity selection for ultracold atoms using mazer action in a bimodal cavity Afshan Irshad a, Sajid Qamar b, Shahid Qamar a,c,* a
Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan Centre for Quantum Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan c The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34151 Trieste, Italy b
a r t i c l e
i n f o
Article history: Received 22 April 2009 Received in revised form 21 August 2009 Accepted 10 September 2009
a b s t r a c t In this paper, we discuss the velocity selection of ultracold three-level atoms in K configuration using a mazer. Our model is the same as discussed by Arun et al. [R. Arun, G.S. Agarwal, M.O. Scully, H. Walther, Phys. Rev. A 62 (2000) 023809] for mazer action in a bimodal cavity. We show that the initial Maxwellian velocity distribution of ultracold atoms can be narrowed due to the presence of resonances in the transmission through dressed-state potential. When the atoms are initially prepared in one of the two lower atomic states then significantly better velocity selectivity is obtained due to the presence of dark states. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction Certain applications in atom optics, like matter-wave interferometry [1] and using light as lenses for submicron neutral atom lithography [2] desire production of atomic beams and the control of their motion. These require long coherence length, i.e., a small velocity spread of the atomic beam. It is known that in most cases the velocity of the atoms follow a Maxwellian distribution with a velocity spread equivalent to the most probable velocity. It is also possible to get ultracold atoms by using laser cooling techniques [3]. From some earlier studies, it has been realized that micromaser systems are of considerable interest in the field of quantum optics [4]. This is due to the fact that a number of features of the radiation field could be predicted in the micromaser systems. For example, sub-Poissonian photon statistics [5], trapping states, number states [6], and others [7]. The idea of trapping state has been realized in experiments [8]. When the micromaser is injected with ultracold atoms such that their kinetic energy is smaller than the atom–field interaction energy then their motion has to be described quantum mechanically [9]. It has been shown by Scully et al. [10] that the quantization of the center of mass motion leads to a completely new kind of induced emission called mazer (microwave amplification via z-induced emission of radiation). In ordinary maser, stimulated emission prevails as the mechanism for amplification of radiation, but in case of ultracold atoms the physics of the induced emission process is associated with the quantization of the center of mass mo-
* Corresponding author. Address: Department of Physics and Applied Mathematics, Pakistan Institute of Engineering and Applied Sciences, Nilore, Islamabad 45650, Pakistan. Tel.: +92 51 2207381; fax: +92 51 2208070. E-mail address: [email protected] (S. Qamar). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.09.022
tion. The physical mechanism responsible for the induced emission is the abrupt change in the atom–field coupling strength, which an atom experiences upon passing through the cavity. The quantum theory of mazer has been established by Scully and co-workers [11–13] and a number of interesting features, like emission probability, photon statistics and transmission of the atoms through the mazer cavity has been studied therein. Later, the idea of mazer was used for velocity selection of ultracold atoms [14]. It was shown that the velocity distribution of the two-level atoms passing through a resonant single mode microwave cavity can be significantly narrowed. The velocity selection appears due to resonances in the transmission through dressedstate potential. Through this passive device, the coherence length of the atomic beam can be made macroscopic. In fact, it is an atom-optical analogue of a Fabry–Perot resonator. In another study the detuning effects on the velocity selection of two-level atoms has also been discussed [15]. The point to be mentioned here is that all these mazer studies are based on a single photon resonance process. The idea has been extended to two photon resonance case either for ultracold atoms with cascade three-level configuration using two micromaser cavities [16] or using V-type three-levelatom mazer with atomic coherence [17]. Former proposal exhibits that the velocity selectivity is better for ultracold three-level atoms using two-cavity system than for ultracold two-level atoms using one-cavity system. However, the latter scheme shows that the atomic coherence does not play a favorable role on the velocity selectivity. In both these schemes the two photon resonance corresponds to a single mode of the micromaser cavities. Recently, the operation of a two-mode mazer with particular reference to the interesting question of mode–mode correlation has been studied [18] using three-level K configuration where the atom–field system is a two photon resonance process. It has
A. Irshad et al. / Optics Communications 283 (2010) 54–59
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been observed that the degree of anticorrelation between the cavity modes increases when the micromaser is pumped by ultracold atoms instead of fast atoms. This exhibits much stronger sub-Poissonian photon statistics for each mode. Motivated from this work, we study velocity selection of the ultracold atoms using mazer action in a bimodal cavity with three-level atoms in the K configuration. Here, instead of considering the same velocity for the injected atoms, we consider an initial Maxwellian velocity distribution. Our results show that the initial velocity distribution of the atoms can be narrowed down to many orders in magnitudes after passing through the cavity. The dressed state analysis exhibits dark state when the atoms are initially prepared in one of the two lower energy levels [18]. Therefore, we also discuss the effect of dark states on the velocity selection in this system.
We use the atom–field dressed states j/0n1 þ1;n2 þ1 i, j/ n1 þ1;n2 þ1 i, detailed analysis is available in [18], which reduces the center of mass motion to a scattering problem. We consider that the atom with initial state jai and center of mass momentum h k enters into the mazer cavity containing n1 and n2 photons in mode 1 and mode 2, respectively, and is reflected or transmitted during its passage through the cavity. There are three possible choices for the atom during its motion through the mazer cavity, first; it does not change its initial state ja; n1 ; n2 i, second; it makes a transition to the lower level jb1 ; n1 þ 1; n2 i, third; it makes a transition to the lower level jb2 ; n1 ; n2 þ 1i. The corresponding probability amplitudes for the reflected or transmitted atom can be written as [18]
2. Model
Ra;n1 ;n2 ¼
ð4Þ
We consider a beam of ultracold, monoenergetic three-level atoms with K-type configuration (as shown in Fig. 1) which are injected in high-Q two-mode mazer cavity of length L [18]. It is assumed that the atomic flux is adjusted such that there is only one atom at a time inside the cavity. The transition between the two lower states jb1 i and jb2 i is dipole forbidden, however, the transition from upper to lower levels jb1 i and jb2 i is allowed. The modes 1 and 2 of the cavity are assumed to be resonant with the transition frequencies of the levels jai ! jb1 i and jai ! jb2 i. The Hamiltonian for the atom-cavity system including the effects of quantized center of mass (CM) motion is given by [18]
T a;n1 ;n2
ð5Þ
2 2 X X p2 H ¼ z þ hXa jaihaj þ hXba jba ihba j þ hxa aya aa 2m a¼1 a¼1
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 n2 þ 1 Rb2 ;n1 ;n2 þ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qþn1 ;n2 qn1 ;n2 ; 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ
ð8Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g2 n2 þ 1 T b2 ;n1 ;n2 þ1 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sþn1 ;n2 sn1 ;n2 ; 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ
ð9Þ
þ
2 X
hga ðaa jaihba j þ jba ihajaya Þ;
ð1Þ
a¼1
where m is the atomic mass and pz is the CM momentum operator of the atom. Here aa ðaya Þ are annihilation (creation) operators with a ¼ 1; 2 for the cavity field corresponding to mode 1 and 2, respectively. The resonance frequencies are xa ¼ Xa Xba . The parameters ga be the corresponding atom–field coupling constants that depend on spatial coordinate z through the mode function of the cavity. In the interaction picture, the Hamiltonian in Eq. (1) reduces to the form as [18]
HI ¼
2 X p2z hg ðaa jaihba j þ jba ihajaya Þ: þ 2m a¼1 a
ð2Þ
The mode function of the cavity, which is assumed to be a mesa function and is defined as
uðzÞ ¼ 1 for 0 < z < L; ¼ 0 elsewhere; with L being the length of the resonator.
ð3Þ
3. Dynamics
1 þ qn1 ;n2 þ qn1 ;n2 ; 2 1 þ ¼ sn1 ;n2 þ sn1 ;n2 ; 2
for the first case
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 n1 þ 1 Rb1 ;n1 þ1;n2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qþn1 ;n2 qn1 ;n2 ; 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ
ð6Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffi g1 n1 þ 1 T b1 ;n1 þ1;n2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sþn1 ;n2 sn1 ;n2 ; 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ
ð7Þ
for the second case and
for the third case. Here q n1 ;n2 and sn1 ;n2 are the dressed-state reflection and transmission coefficients and can be calculated analytically using the mesa mod function (3) as
qn1 ;n2 ¼ iDn1 ;n2 sinðkn1 ;n2 LÞsn1 ;n2 ;
sn1 ;n2 ¼
1 P ; cos kn1 ;n2 L i n1 ;n2 sin kn1 ;n2 L
ð10Þ ð11Þ
with
! 1 kn1 ;n2 k ; 2 k kn1 ;n2 ! 1 kn1 ;n2 k ; Rn1 ;n2 ¼ þ 2 k kn1 ;n2 vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! u u 2 g21 ðn1 þ 1Þ þ g22 ðn2 þ 1Þ t 2 ; k j kn1 ;n2 ¼ g21 þ g22
Dn1 ;n2 ¼
where
j
ð12Þ ð13Þ
ð14Þ
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2m h
g21 þ g22 is the CM wave vector for which the ki-
netic energy ð hkÞ2 =2m of the atom equals the vacuum coupling enqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ergy h g21 þ g22 . Here we are interested in the regime where the
Fig. 1. Energy-level diagram of the three-level atomic system.
atomic wave number k is smaller than j as we are dealing with ultracold atoms. We consider a situation when the cavity field is initially in vacuum state, i.e., n1 ¼ n2 ¼ 0. The kinetic energy of the incident atoms is assumed to be so small that tunneling through the potential barrier plays no essential role in the mazer [12]. Under this
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condition, the transmission probability of the atom through the cavity without contributing any photon into the cavity modes, i.e., remaining in state jai is
Ptrans ða; 0; 0; kÞ ’
2 1 2 ½1 þ D0;0 sin k0;0 L 1 : 4
ð15Þ
Whereas the transmission probability that an atom will transmit after making a transition to the level jb1 i and emitting a photon in mode 1 of the cavity is given by
Ptrans ðb1 ; 1; 0; kÞ ’
2 g21 2 ½1 þ D0;0 sin k0;0 L 1 ; 2 þ g2 Þ
4ðg21
2 g22 2 ½1 þ D0;0 sin k0;0 L 1 : 4ðg21 þ g22 Þ
ð17Þ
It is clear from the Eqs. (15)–(17) along with (10)–(14) that the transmission shows resonances as a function of the scaled cavity length jL and the wave number of the atoms k=j. The resonances occur when the length of the cavity is an integral multiple of half of the de Broglie wave length of the atoms, i.e.,
k0;0 L ¼ mp;
ð18Þ
where m ¼ 1; 2; 3; . . . This is similar to the transmission of an excited two-level atom through a single-mode cavity. This behavior is just the transmission characteristic of the potential well as discussed in [11]. As is mentioned earlier that in the two photon transition process the dressed state analysis exhibits a dark state j/0n1 ;n2 i which is not present in one photon process. This dark state does not have any influence on the reflection and transmission coefficients when the initial state of the atom–field is ja; 0; 0i. However, its contribution appears in the probability amplitudes of reflection and transmission of the atom when the initial state of the atom is jb1 ; 1; 0i or jb2 ; 0; 1i [18]. This may be understand physically by considering the dressed state picture of the K system. The bare state of the atom–field system exhibits atomic transitions between levels jai and jb1 i and jai and jb2 i with the corresponding vacuum Rabi frequencies g1 and g2 . The dressed state analysis however, exhibits
i 1 h ja; 0; 0i ¼ pffiffiffi j/þ1;1 i þ j/1;1 i ; 2 1 g jb1 ; 1; 0i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p1ffiffiffi j/þ1;1 i j/1;1 þ g2 j/01;1 i ; 2 g21 þ g22 1 g jb2 ; 1; 0i ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2ffiffiffi j/þ1;1 i j/1;1 g1 j/01;1 i : 2 g21 þ g22
ð19Þ ð20Þ
ð21Þ
Considering the dressed state picture, we can observe that the atomic transitions from level jb1 i to jai follow the vacuum Rabi frequencies g2 and ð2g1 g2 Þ whereas the atomic transition from jb2 i to jai follow the vacuum Rabi frequencies g1 and ð2g2 g1 Þ. This shows that there is no atomic transition following the vacuum Rabi frequency g1 and g2 in the former and latter case, respectively. This clearly shows that if the atom is initially in one of the lower levels, i.e., jb1 i or jb2 i, the atomic transition exhibits a corresponding dark state. In the two-mode mazer, this dark state contributes to the enhancement of the probability of transmission of the atoms. The probability amplitude when the atom is transmitted in state jb1 ; 1; 0i comes out to be [18]
T b1 ;1;0 ¼
g21 g2 sþ0;0 þ s0;0 þ 2 2 2 ; 2 þ g2 Þ ðg1 þ g2 Þ
2ðg21
Ptrans ðb1 ; 1; 0; kÞ ¼
½g21 þ 2g22 cosðk0;0 LÞ2 þ 4g42 ðR0;0 Þ2 sinðk0;0 LÞ2 4ðg21 þ ½1 þ
ðD0;0 Þ2
2
g22 Þ2
sin ðk0;0 LÞ1 : ð23Þ
ð16Þ
and the transmission probability for the atom after making a transition to the level jb2 i and emitting a photon in mode 2 of the cavity can be written as
Ptrans ðb2 ; 0; 1; kÞ ’
where sþ 0;0 and s0;0 can be obtained from Eq. (11) and the term g22 =ðg21 þ g22 Þ is the contribution due to the dark state. The corresponding probability that the atom transmits through the cavity in the same initial state jb1 ; 1; 0; ki can be written as
ð22Þ
A similar expression, as in Eq. (22), for the transmitted atom when the initial and final atom–field state is jb2 ; 0; 1i can be obtained except that the coupling constant g1 is replaced with g2 and vice versa. 4. Discussion In the following, we discuss velocity selection for the ultracold atoms passing through the bimodal mazer cavity. We start with the analysis of the transmission probability Ptrans ðkÞ of the ultracold atoms with initial state ja; 0; 0i. That is, the initial state of the injected atom is assumed to be the excited state jai and the cavity fields in the two modes are considered to be initially in the vacuum state. We also assume that the atoms are injected inside the cavity with the same velocity which is the case in usual mazer. For all the analysis, we take the coupling constant g1 ¼ 3g2 . We study the behavior of the transmission probability Ptrans ðkÞ versus dimensionless atomic wave number k=j ranging from 0 to 0:5. We consider two different cavity lengths, i.e., jL is equal to (i) 20p (dashed line) and (ii) 200p (solid line) (jL is the dimensionless cavity length). We also consider three conditions, i.e., when atom exit the bimodal mazer cavity (a) without making any transition and remained in the initial state ja; 0; 0i, (b) making transition to the final state jb1 ; 1; 0i or (c) making transition to the final state jb2 ; 0; 1i. Figs. 2a–c show, that the maximum value of the transmission probability is 0.25, 0.22 and 0.025, respectively. This shows that the transmission probability depends on the condition whether the atom exits the cavity in state jai, jb1 i or jb2 i. If it remains in the same state then it does not contribute any photon inside the cavity, however, if it exits in any of the other two states then it contributes one photon in either of the two modes. The maximum value of the transmission probability depends upon the choice of the ratio g2 =g1 which is clear from Eqs. (15)–(17). For initial vacuum state P trans ða; 0; 0; kÞ is independent of this ratio, however, Ptrans ðb1 ; 1; 0; kÞ decreases while Ptrans ðb2 ; 0; 1; kÞ increases due to the increase in the ratio g2 =g1 . It is clear from the results that by increasing the length of cavity more resonances appear which become sharper and more closer to each other which is analogous to the behavior of a Fabry–Perot resonator. In order to study the velocity selection of ultracold atoms using the bimodal mazer cavity, we now consider that the atoms are injected inside the cavity with a certain velocity distribution. In the following, we assume that the atomic beam initially follows Maxwellian velocity distribution such that the number of atoms in the interval dk is given by 2
2 2 4 k Pi ðkÞdk ¼ pffiffiffiffi 3 ek =k0 dk; p k0
ð24Þ
where k0 corresponds to the most probable value for the wave number k. This form of Maxwellian velocity distribution is considered just to make our analysis in line with the earlier schemes for the velocity selection of ultracold atoms using micromazers [14–17]. It may be mentioned that the form of the Maxwellian distribution depends on whether one counts particles in a volume element or particles that enter the detector in a given time window [19]. After
A. Irshad et al. / Optics Communications 283 (2010) 54–59
Fig. 2. Transmission probability versus atomic wave number for ultracold atoms passing through a bimodal cavity in the vacuum state with g1 ¼ 3g2 whereas the cavity lengths are (i) jL ¼ 20p (dashed curve) and (ii) jL ¼ 200p (solid curve). For three different situations; (a) the atom exits the cavity in the excited state jai, (b) after making transition to state jb1 i and (c) after making transition to state jb2 i.
passing through the mazer cavity, the final k distribution of the atoms is given by
Pf ðkÞdk ¼ Ptrans ðkÞP i ðkÞdk:
ð25Þ
In Fig. 3, we plot the final wave number distribution for k0 =j ¼ 0:05 and jL ¼ 200p. We again consider three scenarios, i.e., atoms exit from the mazer cavity with final state (a) ja; 0; 0i, (b) jb1 ; 1; 0i and (c) jb2 ; 0; 1i. The inset in Fig. 3 shows the initial wave number distribution (Eq. (24)) with the most probable value for wave number k0 =j ¼ 0:05. We observe that the final k distribution in the main figure is dominated by a peak which corresponds to the single broad peak in the initial k distribution as shown in the
Fig. 3. Final wave number distribution for k0 =j ¼ 0:05, jL ¼ 200p and g1 ¼ 3g2 . Here, dashed line, dotted line and solid line correspond to the same conditions of the atom as mentioned in Figs. 2a–c, respectively. The inset shows the initial wave number distribution.
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inset of Fig. 3. The height of the main peak depends on the situation that whether the atom has contributed inside the cavity or not. The peak height reduces if the atoms exit the cavity in the state jb1 ; 1; 0i or jb2 ; 0; 1i. We also observe that the central peak in the final k distribution has about 1/12th of the width of the initial k distribution after passing through the bimodal cavity, see Fig. 3. Next we, change the most probable velocity and consider k0 =j ¼ 0:1 which corresponds to twice as fast atoms as in the previous case (Fig. 3). The behavior of the initial and final k wave number distributions versus k=j are shown in Fig. 4. Here, the final wave number distribution exhibits oscillatory behavior which has a comb-like structure with several peaks within the range 0 to 0.4 of k=j. This is due to the increase in the average velocity of the incoming atoms as more resonances appear for higher average velocity of the incoming atoms. This gives rise to the multiple peaks structure in the final k wave number distribution. The height of the peaks decrease with increase in k=j. The width of each peak in the final k distribution is now about 1/24th of the initial k wave number distribution. The amplitude of the peaks are also considerably high for k0 =j ¼ 0:1 as compared with k0 =j ¼ 0:05, see Figs. 3 and 4. This shows that we get a much higher magnitude of the velocity distribution with almost twice narrower width for k0 =j ¼ 0:1 as compared to the case when k0 =j ¼ 0:05. For a comparison, we again consider the cases when the atoms exit the cavity in state (a) ja; 0; 0i, (b) jb1 ; 1; 0i and (c) jb2 ; 0; 1i. A similar behavior of the final k distribution as in Fig. 3 is observed. That is, the heights of the peaks depend on the situation whether the atoms contribute inside the cavity or not and the height reduces if the atoms exit the cavity in the state jb1 ; 1; 0i or jb2 ; 0; 1i. Next, we consider the effect of dark states on the velocity selection of ultracold atoms using a bimodal mazer action. It has been shown in [18], that for atoms initially prepared in state jb1 i with one photon in mode 1 or in state jb2 i with one photon in mode 2, the effects of dark state appear in a two-mode mazer. It contributes to the enhancement of the probability of transmission of atoms. Therefore, it is interesting to consider the effect of the dark state on the velocity selection of the ultracold atoms. We discuss the case when the atoms with initial state jb1 i and following the Maxwellian velocity distribution enter the bimodal mazer cavity. The Fig. 5 shows the behavior of the transmission probability distribution Ptrans versus normalized wave number k=j. We present the results for jL ¼ 20p (dashed line) and jL ¼ 200p (solid line). The transmission probability exhibits sharp resonances as we increase the value of jL from 20p to 200p. However, the heights of peaks vary, i.e., having maximum values of 0.3 and 0.1. This is different as compared to the earlier case discussed in Fig. 2 where all peaks have same maximum value. To understand this different behavior we rewrite the expression for the transmission probability given in Eq. (23) in a simplified form using g1 ¼ 3g2 as
Fig. 4. Final wave number distribution for k0 =j ¼ 0:1 and other parameters are same as in Fig. 3. Here, dashed line, dotted line and solid line correspond to the same situation as mentioned in Fig. 3. The inset shows the initial wave number distribution.
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A comparison of our results with the earlier studies [14–17] show that the width of the final k distribution is much narrower then the schemes based on two-level atomic system. This is due to the fact that the three-level atomic system exhibits dark states which is not present in the case of two-level atomic systems. 5. Conclusion
Fig. 5. Transmission probability as a function of the atomic wave number for ultracold atoms encountering a bimodal cavity that is initially in the state jb1 ; 0; 1i . Here (i) dashed line corresponds to jL ¼ 20p while (ii) solid line corresponds to jL ¼ 200p.
e trans ðb1 ; 1; 0; kÞ P n o 1 2 ¼ 4 þ 9½9 þ 4 cosðk0;0 LÞ ½1 þ ðD0;0 Þ2 sin ðk0;0 LÞ1 : 400 ð26Þ This shows that the variation in height of the peaks in Fig. 5 is due to the presence of the dark state which contributes cosine term, i.e., cos k0;0 L in the transmission probability distribution Eq. (26). In Fig. 6, we show the final wave number distributions when atoms initially in the state jb1 i, enter the cavity having one photon in mode 1 and no photon in mode 2. In Fig. 6, (a) solid line and (b) dashed line correspond to k0 =j equals to 0.05 and 0.1, respectively. For k0 =j ¼ 0:05, the final k distribution is dominated by a sharp single peak at k=j ¼ 0:1 whose width is about 1/18th of the width of the initial distribution, compare Fig. 6 (solid line) with inset of Fig. 3. If we double the average velocity of the atoms, i.e., k0 =j ¼ 0:1 then the final wave number distribution has a comblike structure with multiple peaks of different heights, see Fig. 6 (dashed line). The wave number distribution exhibits that the height of the peaks varies however, the width of each peak remains almost the same. Here, width of the most dominant peak in the distribution, i.e., at k=j ¼ 0:14 is about 1/36th of the width of the initial wave number distribution peak, compare Fig. 6(dashed line) with inset of Fig. 4. This shows that the width of the velocity distribution peaks can be made narrower by the use of the dark state. Further, the magnitude of the probability distribution can also be made much larger by using the dark state. A comparison of the plots in the Figs. 3, 4 and 6 exhibits almost 8 times larger and 3 times narrower peaks in the wave number distribution when atoms which are initially prepared in the state jb1 ; 1; 0i are transmitted through the mazer cavity in the same initial state. This is due to the influence of the dark state. A similar effect of the dark state can be observed if the atoms are initially prepared in the state jb2 ; 0; 1i and transmitted in the same initial state.
In conclusion, we have studied the velocity selection of ultracold atoms through a bimodal cavity. We considered the atoms to be initially in their excited state and field in a vacuum state. If the incoming atoms have a Maxwellian velocity distribution then after passing through the mazer cavity, an order of magnitude narrow velocity distribution can be obtained. We have also discussed the case when atoms, initially prepared in their state jb1 i, are injected into the cavity which has one photon in mode 1 and zero photon in mode 2. In this case, further reduction in the width of the velocity distribution is obtained due to the presence of dark states. Our results show that significantly precise velocity selection of ultracold atoms can be achieved in this system. For the experimental realization we may quote the Garching experiment where the typical length of the cavity has been considered to be about 2.5 cm which corresponds to the values of jL about 105 . For such a geometry, sharp mazer resonances occur which are much close to each other to be resolved, if the atoms are injected with certain velocity spread. However, as discussed in [11], smaller values of jL can be obtained with a cavity of re-entrant type. In this cavity, the interaction length can be made considerably smaller than the wavelength of the microwaves and still much larger then the de-Broglie wavelength of the atoms. Experimental consideration regarding mazer are discussed in detail in Ref. [12]. There, a re-entrant type cavity has been discussed with which velocity selection of ultracold atoms should be possible. In order to see this, here we consider the parameters which corresponds to Figs. 4 and 6b. If we consider a coupling strength of 100 kHz and 85Rb atoms, we obtain j ¼ 1:6 107 m1 . In order to satisfy k0 =j ¼ 0:1, a mean velocity of 1.2 mm/s is required, that corresponds to a temperature of about 10nK. The de-Broglie wavelength of the atoms for parameters under consideration is in the range of lm, it should be possible to realize a mesa like potential with a TM mode [12]. The finite lifetime of the atomic states raises problem in realizing the proposed device. However, the use of circular Rydberg states, which provide life times of the order of approximately 0.1 s can overcome this problem. Acknowledgment The author Shahid Qamar would like to thank the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy for kind hospitality and support through Associate Scheme. He would also like to thank Prof. Joseph Niemela for some fruitful discussion. References
Fig. 6. Final wave number distribution for jL ¼ 200p and (a) k0 =j ¼ 0:05 (solid line) and (b) k0 =j ¼ 0:1 (dashed line). All the other parameters are the same as in Fig. 5.
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