Precision in single atom localization via Raman-driven coherence: Role of detuning and phase shift

Physics Letters A 377 (2013) 1587–1592

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Physics Letters A www.elsevier.com/locate/pla

Precision in single atom localization via Raman-driven coherence: Role of detuning and phase shift Rahmatullah, Sajid Qamar ∗ Department of Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan

a r t i c l e

i n f o

Article history: Received 5 January 2013 Received in revised form 4 April 2013 Accepted 23 April 2013 Available online 29 April 2013 Communicated by P.R. Holland

a b s t r a c t Role of detuning and phase shift associated with the standing-wave driving fields is revisited for precision position measurement of single atom during its motion through two standing-wave fields. A four-level atomic system in diamond configuration is considered where the intermediate levels are coupled to upper and lower level via standing-wave driving fields and atomic decay channels, respectively. The former is responsible for the generation of quantum mechanical coherence via two-photon Raman transition while the latter leads to spontaneous emission of a photon. Due to standing-wave driving fields the atom–field interaction becomes position-dependent and measurement of the frequency of spontaneously emitted photon gives the position information of the atom. The unique position of the atom with much higher spatial resolution, i.e., of the order of λ/100 is observed using detuning and phase shift associated with the standing-wave driving fields. © 2013 Elsevier B.V. All rights reserved.

1. Introduction Precise position measurement of a single atom has been a question of considerable interest due to its fundamental nature [1] and involvement in certain applications like laser cooling [2], atom nanolithograghy [3] and Bose–Einstein condensation [4]. Earlier, atom imaging methods have been employed by Thomas and co-workers [5]. These were based on resonance imaging where a spatially varying potential shifted the resonance frequency of an atomic transition and the resonance frequency became positiondependent. The position distribution have been found by spectroscopic methods and a spatial resolution of 1.7 mm have been observed. Methods which involves light can be used for better precision in position measurement of single atom, however, the main restriction here is the diffraction limit [6]. It has been proposed that this limit could be overcome in microscopy techniques which are operating at the sub-wavelength scale based on near-field techniques [7] and in far-field optical nanoscopy [8]. In recent years, a large number of schemes for atom localization has been suggested where single atom which is moving through standing-wave field impart its position information via phase shift associated with the standing-wave field [9–11] or atomic state [12]. It has been observed that a relatively simple approach for atom microscopy could be via detection of spontaneously emit-

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ted photon [13–15] or absorption of the probe field [16]. Further, spatial measurement of an atom could also be possible via electromagnetic induced transparency [17,18], coherent population trapping [19] and interference of dark resonances [20]. It has also been observed that the spatial resolution in position information of the single atom could be improved via detuning between atomic transition frequency and driving field frequency [14,15]. However, schemes based on interaction of atom with standingwave field experience a common drawback of exhibiting four potential atom localization peaks within unit wavelength domain of the optical field for a single measurement. This is due to the periodic nature of the standing-wave field. This uncertainty in position information could be reduced to half by using the amplitude and phase control of the driving fields [17,21]. A further enhancement could be made via two standing-wave fields with slightly different wavelengths. In these schemes, it has been observed that unique position information of an atom could be done via dual measurement on quadrature field [22] or spontaneously emitted photon frequency measurement [23,24]. Another scheme, which gives similar results for spatial measurement, is based on coherent manipulation of the Raman gain process [25]. In parallel, the idea of atom microscopy has further been exploited for nanoscale resolution microscopy using electromagnetically induced transparency [26], coherent population trapping [27] and via running-wave fields [28]. Further, superposition of two standing-wave fields can also be used for precise localization of a two-level atom [29]. Recently, a scheme has been proposed to observe unique position information of a moving atom using a three-level atomic

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system in V-configuration [30]. The author reports a unique atom localization peak within unit wavelength of the optical field which according to the claim enhances the efficiency of their system. However, it is clear from the results and comparison with earlier work [13–15,17,21] that the single atom localization peak suffers a bad resolution and experiences a large full width at half maximum (FWHM), i.e., for two potential localization peaks the best FWHM of each peak is ≈ λ/32 which, however, increases to ≈ 3λ/32 for the case where the single localization peak has been observed. In fact, in this scheme, any effort to increase the spatial resolution can lead to more than one localization peaks which is also noticed by Zia et al. [31]. It could also be noticed that in the proposed model [30] one has to incorporate a dipole forbidden transition and the system also requires initial coherent superposition of upper two atomic states, these are experimentally difficult to achieve. In another scheme, continuously tunable sub-half-wavelength localization could be done via the coherent control of the spontaneous emission of a four-level Y-type atomic system, which is coupled to three strong coupling fields including a standing-wave field together with a weak probe field [32]. In the present proposal, a four-level atomic system with diamond configuration is considered for localization of a moving atom. Here, two intermediate levels are coupled to a common upper level via two standing-wave driving fields having slightly different wavelengths. The quantum mechanical coherence is generated through two-photon Raman transition. Under long-time limit, the atom decays from any one of the intermediate levels to the ground level and a photon is emitted spontaneously. The measurement of the frequency of the spontaneously emitted photon, due to its direct relation with position-dependent Rabi frequency [13–15], leads to potential position information of the atom. Here, the role of detuning and phase shift associated with the standingwave driving fields are investigated. Using a similar approach as has been described earlier in [22], a unique position information of the atom with high spatial resolution, i.e., of the order of λ/100 can be obtained. This scheme is experimentally more viable because there is no dipole forbidden transition involved and also no superposition of two atomic states, as an initial condition, is required. This scheme can also be used for two-dimensional atom localization [38–42] as has been done in [43] using a four-level atomic system in tripod configuration. 2. Model We consider that an atom with energy levels |a, |b, |c  and |d is moving through two standing-wave driving fields, i.e., E 1 and E 2 having frequencies ν1 and ν2 , respectively. The atom–field interaction is position-dependent and the levels |b and |c  are coupled to level |a via two optical standing-wave fields, E 1 and E 2 , respectively. The corresponding Rabi frequencies are positiondependent and are denoted by Ω1 (x) and Ω2 (x). The coherence between atomic levels |b and |c  is therefore generated using two-photon Raman transition via energy level |a and the atomic system does not require an initial superposition state of energy levels |b and |c  as is needed in [30]. The atom due to reservoir modes can decay from levels |b or |c  to level |d with rates γ1 and γ2 , respectively. We can observe that the atom–field configuration is diamond shaped, see Fig. 1. As mentioned above, the atom during its motion passes through the standing-wave fields and the interactions between atom and standing-wave fields are position-dependent. The corresponding position-dependent Rabi frequencies are defined as

Ω1 (x) = Ω1 sin(k1 x + η), Ω2 (x) = Ω2 sin(k2 x + ξ ),

Fig. 1. (Color online.) Four-level Raman-driven atomic system. The upper -type configuration is governed by two standing-wave driving optical fields with corresponding position-dependent Rabi frequencies Ω1 (x) and Ω2 (x) whereas the lower V-type configuration corresponds to two atomic decays at the rates γ1 and γ2 .

where k1 = 2π /λ1 and k2 = 2π /λ2 are the wave-vectors whereas η and ξ are the phase shifts associated with the standing-wave driving fields. We assume, that the center-of-mass position of the atom along the standing wave does not change during the interaction time and thus neglect the kinetic energy of the atom under Raman–Nath approximation. As a result, the interaction Hamiltonian in dipole and rotating-wave approximation can be written as

 V = h¯ Ω1 (x)e −i 1 t |ab| + Ω2 (x)e −i 2 t |ac | +



(1)

g k (x)e −i δk t |bd|bk (1 )

k

+



(2 )

g k (x)e

(2)

−i δk t

 |c d|bk + H.c. ,

(1)

k

where 1 = ν1 − ωab ( 2 = ν2 − ωac ) is the detuning of the standing-wave driving field having frequency ν1 (ν2 ) with the (1) (2) atomic transition |a ↔ |b (|a ↔ |c ). The parameter g k (g k ) is the coupling constant between the reservoir mode k and the atomic transition from level |b (|c ) to level |d. The annihilation operator corresponding to the reservoir mode is represented by bk (1) (2) whereas δk = νk − ωbd (δk = νk − ωcd ) are the corresponding atom–vacuum field detuning, with ωbd (ωcd ) is the atomic transition frequency between level |b (|c ) to |d, νk is the frequency of the spontaneously emitted photon corresponding to the reservoir mode k. The atom–field state-vector for our system can be written as

  Ψ (x; t ) =





dx f (x)|x





C a,0k (x; t )|a, 0k  + C b,0k (x; t )|b, 0k 

+ C c,0k (x; t )|c , 0k  +



 C d,1k (x; t )|d, 1k  ,

(2)

k

with C i ,0k (x, t ) (i = a, b, c) being the probability amplitudes which represent the state of the atom at time t when there is no spontaneously emitted photon in the kth vacuum mode and C d,1k (x, t ) is the probability amplitude of the atom to be in level |d with one photon in mode k and f (x) is the center of mass wave function of the atom. Our atom localization scheme is based on the fact that the interaction between the atom and field is position-dependent therefore, the spontaneously emitted photons carries the information about the position of an atom. The conditional position probability distribution W (x; t /d; 1k ) is defined as the probability of finding the atom at position x under the condition that the spontaneously

Rahmatullah, S. Qamar / Physics Letters A 377 (2013) 1587–1592

emitted photon is detected at time t when the atom is in the state |d. The reduced state-vector after making appropriate projection can be written as

  |Ψd,1k  = N 1k , dΨ (x; t ) = N



dx f (x)C d,1k (x; t )|x,

(3)

where N is a normalization factor. So, we can write the conditional position probability distribution as

  2 W (x) = W (x; t |d; 1k ) = | N | xΨ (x; t )   2 = F (x; t |d, 1k ) f (x) , 2

(4)

where F (x; t )|d, 1k ) is the filter function defined as



2

F (x; t )|d, 1k ) = | N |2 C d,1k (x; t ) .

(5)

Eq. (5) shows that the filter function depends on the probability amplitude C d,1k (x; t ). We can derive analytical expression for the probability amplitude C d,1k (x; t ) by solving the Schrödinger wave equation with interaction Hamiltonian in Eq. (1) and state-vector in Eq. (2). We obtain rate equations for the corresponding probability amplitudes as follows





C˙ a,0k (t ) = −i Ω1 (x)e −i 1 t C b,0k (t ) + Ω2 (x)e −i 2 t C c ,0k (t ) ,



C˙ b,0k (t ) = −i Ω1 (x)e i 1 t C a,0k (t ) +



(1)

g k (x)e −i δk (1 )

k

(7)



C˙ c ,0k (t ) = −i Ω2 (x)e i 2 t C a,0k (t ) +



(2 )

g k (x)e

(2)

−i δk

 t C d,1k (t ) ,

k

(8)



∗(1)

C˙ d,1k (t ) = −i g k

(x)e

(1)

∗(2)

i δk t

C b,0k (t ) + g k

(x)e

(2)

i δk t

C c ,0k (t ) . (9)

∗(1)

t

(1)  t

e i δk

(x)

 C b,0k t  dt 

∗(2)

− i ∗ gk

(2)  t

e i δk

(x)

 C c ,0k t  dt  .

(10)

0

Substituting Eq. (10) in Eqs. (7) and (8), and applying Weisskopf– Wigner theory, we get

C˙ b,0k (t ) = −i Ω1 (x)e i 1 t C a,0k (t ) −

γ1 2

C b,0k (t ),

(11)

C c ,0k (t ).

(12)

and

C˙ c ,0k (t ) = −i Ω2 (x)e i 2 t C a,0k (t ) −

γ2 2

By applying the Laplace transform method, we obtain C d,1k in the long-time limit as ∗(1) ˜

C d,1k (x; t → ∞) = −ig k

(1 ) 

C b,0k s = −i δk

 − ig k∗(2) C˜ c,0k s = −i δk(2) ,

(13)

where C˜ b,0k (s) and C˜ c ,0k (s) are the Laplace transforms of C b,0k (t ) (1) (2) and C c ,0k (t ) with s = −i δk and s = −i δk , respectively. After performing the Laplace transformation, we obtain

(1 ) 

(2 ) 



γ2 2

γ1

 

Ω1 (x) , (δk + 0.5ωcb ) + 1

Ω2 (x) , 2 (δk − 0.5ωcb ) + 2 

 1 − γ1 − 2i (δk + 0.5ωcb ) (δk + 0.5ωcb ) B=

C = −i (δk − 0.5ωcb ) +

4

 2Ω12 (x)(γ2 − 2i (δk + 0.5ωcb )) × i γ2 + 2(δk + 0.5ωcb ) − (δk + 0.5ωcb )2 − 21  2Ω22 (x)(γ1 − 2i (δk + 0.5ωcb )) , − (δk + 0.5ωcb )2 − 22 

 1 − γ1 − 2i (δk − 0.5ωcb ) (δk − 0.5ωcb ) D= 4

 2Ω12 (x)(γ2 − 2i (δk − 0.5ωcb )) × i γ2 + 2(δk − 0.5ωcb ) − (δk − 0.5ωcb )2 − 21  2Ω22 (x)(γ1 − 2i (δk − 0.5ωcb )) , − (δk − 0.5ωcb )2 − 22 with (1 )

(2 )

δk = νk − 0.5(ωbd + ωcd ) = δk − 0.5ωcb = δk + 0.5ωcb , is the spontaneously emitted photon frequency corresponding to the vacuum mode k and atomic transition from the middle of the two upper levels |b and |c  to lower level |d with ωcb = ωc − ωb is the frequency difference of the two upper levels. Finally, the filter function in the steady state limit (t → ∞) can be defined as



2

(16)

3. Results and discussion

0

t



A = −i (δk + 0.5ωcb ) +

F (x; t )|d, 1k ) = | N |2 C d,1k (x; t → ∞) .

The integration of Eq. (9) yields

C d,1k (t ) = −ig k

where

(6)

 t C d,1k (t ) ,

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C˜ b,0k s = −i δk

= A/B ,

(14)

C˜ c ,0k s = −i δk

= C /D,

(15)

In some earlier work, it has been observed that detuning of the driving field with the atomic transition plays an important role in precision position measurement of the single atom [14,15] and much narrower localization peaks could be observed for certain choices of the detuning. In the following, we investigate the behavior of conditional position probability distribution of the single atom (4) for different choices of parameters, i.e., the detunings ( 1 and 2 ) and the difference between the phase shifts (ξ and η ). In all of our analysis, we consider an initially broad wave packet, i.e., | f (x)|2 which is remained constant over the range −0.5λ to 0.5λ of kx. We also consider that the wave-vectors, i.e., k1 and k2 corresponding to the two standing-wave fields (having slightly different wavelengths) are related to each other via the ratio k2 = αk1 with α being some constant. As a starting point we consider the case when the driving fields are resonant with the corresponding atomic transitions, i.e.,

1 = 2 = 0. We also assume that the two standing-waves are in phase, i.e., η = ξ = 0 and have the same wavelengths, i.e., α = 1. Fig. 2(a) shows the plots of conditional position probability distribution W (x) versus normalized position kx for |Ω1 | = |Ω2 | = 3Γ , δ = 1.5Γ , ωcb = Γ , and γ1 = γ2 = 0.1Γ where Γ is the scaling parameter which is selected for simplicity sake. The plot shows four broad atom localization peaks (dashed line) which are overlapped to each other due to their large full width at half maximum and thus exhibiting no spatial resolution. Keeping in view earlier observations [14,15] that detuning of the driving fields with the coherent atomic transition plays an

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Fig. 2. (Color online.) The conditional position probability distribution W (x) is plotted versus normalized position kx with |Ω1 | = |Ω2 | = 3Γ , δ = 1.5Γ , ωcb = Γ , η = 0 and γ1 = γ2 = 0.1Γ for (a) α = 1.0, ξ = 0, 1 = 2 = 0 (dashed line) and 1 = 2 = 2Γ (solid line). For (b) α = 0.85, ξ = π /8 and 2 = 2Γ whereas 1 is equal to 1.2Γ (solid line) and 2Γ (dashed line). In (c) 1 = 0.5Γ ; η = 0, ξ = π /8 (solid line) and η = π /8, ξ = 0 (dashed line) remaining parameters are same as in (b).

important role in enhancement of the spatial resolution in spatial measurement of single atom, we select detuning parameters as 1 = 2 = 2Γ while the remaining parameters are the same as in Fig. 2(a). The plot of W (x) versus kx exhibits four narrow peaks (solid line) where each localization peak depicts potential position information of the atom with high spatial resolution within a unit wavelength domain. The comparison of the two plots in Fig. 2(a) (dashed and solid lines) clearly shows the role of detuning 1 and 2 in precision position measurement of the single atom. However, due to the periodicity associated with standing-wave fields, we observe four localization peaks with equal probability [13–15]. Thus in general the system exhibits four equally probable potential positions of the single atom. Our next objective here is to reduce these four atom localization peaks and to obtain a unique potential position information of the single atom. It has already been noticed that the number of localization peaks can be reduced and one can get either a unique position information via multiple field quadrature measurement [22] and two-photon spontaneous emission processes [23,24] or a most probable atom localization can be observed using Raman gain process [25]. For this purpose, in the following analysis, we use detuning and phase shifts associated with the driving standing-wave fields to reduce the number of localization peaks within a unit wavelength domain of the optical field. We now assume that the two standing-wave driving fields have slightly different wavelengths and therefore α = 0.85. We also consider a phase shift associated with the corresponding standingwave fields and choose η = 0 and ξ = π /8. In Fig. 2(b), we plot conditional position probability distribution W (x) versus normalized position kx for 1 = 2Γ (dashed line) and for 1 = 1.2Γ (solid line) while keeping other parameters remained unchanged as in Fig. 2(a). We observe that the heights of the localization peaks and position of their maxima depend on the value of 1 , however, in both cases, the plot shows three peaks in the conditional position probability distribution with the most probable position is at ≈ 0.015λ. The spatial resolution is very high in these cases, i.e., of the order of λ/100. This shows that a much higher probability of finding the atom at a particular position with high spatial resolution could be achieved with suitable choices of the detunings and the phase shift associated with the two optical standing-wave fields. As observed above, by using the detunings and phase associated with the driving standing-wave fields we are able to reduce the number of potential positions of the atom from four (equally probable) to three (with different probabilities). Next, we try to obtain a single position information of the atom via change in detuning

1 while keeping remaining parameters same as in Fig. 2(b). In Fig. 2(c), we plot conditional position probability distribution W (x) versus normalized position kx for 1 = 0.5Γ and observe a unique position information at −0.43λ with a spatial resolution of around

λ/100, see solid line. For the case when η = π /8 and ξ = 0 we do not obtain single localization peak, see dashed line, however, still one position information at ≈ 0.39λ is more probable than the other position at ≈ 0.45λ. From above analysis it is now clear that precision in single atom localization is sensitive to the selection of detunings and phases associated with the driving fields. In the following, we further analyse the role of different choices of detunings and phases on single atom localization. In Fig. 3, we plot conditional position probability distribution W (x) versus normalized position kx for different combinations of

1 and 2 . Initially, we fix the value of 2 (equal to 2Γ ) and select three different values of 1 , i.e., (a) Γ , (b) 0.75Γ and (c) 0. The remaining parameters are same as in Fig. 2(c). We observe unique position information of the single atom with high spatial resolution, i.e., λ/100 in each case. We also notice that the position of the peak maximum depends on 1 and therefore the peak maximum is shifted from −0.48λ when 1 =≈ Γ to −0.46λ and then to −0.425λ for 1 = 0.75Γ and 0, respectively. Next, we fix 1 (equal to 2Γ ) and choose three different values of 2 , i.e., (d) Γ , (e) 0.75Γ and 0. Here, again the conditional position probability distribution exhibits single atom localization peak (except in (d) where still one position information is most probable) but with low spatial resolution, i.e., of the order of λ/20. Finally, in Fig. 4, we plot the conditional position probability distribution versus kx for different combinations of the two phases η and ξ . For η = 0 and ξ is equal to (a) π /6, (b) π /4 and (c) π /2, we observe single, three and four atom localization peaks, respectively, with different heights. Similarly, for ξ = 0 and η is equal to (a) π /6, (b) π /4 and (c) π /2, we observe two, four and four atom localization peaks, respectively, with different heights. 4. Conclusion In this Letter, we suggested a scheme to obtain precision in position measurement of single atom. We considered that an atom is moving perpendicular to the direction of propagation of the standing-wave fields with high enough velocity such that the interaction time of the atom with the standing-wave fields is sufficiently small. As a result, the center-of-mass position of the atom along the standing-wave fields does not change during the interaction and thus we may neglect the kinetic-energy term in the interaction Hamiltonian, i.e., Raman–Nath approximation. The atom– field configuration is similar to a four-level system with Ramandriven coherence which has been used to investigate role of phase coherence and decoherence in the quantum beat laser [33] and also employ for the entanglement generation [34]. Here we investigated the role of detunings and phases associated with the standing-wave driving fields for precision position measurement of the single atom within a unit wavelength domain of the optical field.

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Fig. 3. (Color online.) The plot of conditional position probability distribution W (x) versus kx with 2 = 2Γ and 1 is equal to (a) Γ , (b) 0.75Γ and (c) 0, whereas for

1 = 2Γ , 2 is equal to (d) Γ , (e) 0.75Γ and (f) 0. The remaining parameters same as in Fig. 2(c).

Fig. 4. (Color online.) Plot shows the conditional position probability distribution W (x) versus normalized position kx with α = 0.85. We select (b) π /4 and (c) π /2. Next, we choose ξ = 0 and η is (d) π /6, (e) π /4 and (f) π /2. The remaining parameters are same as in Fig. 2(c).

We observed that for certain choices of detunings and phases associated with the standing-wave driving fields and corresponding to the atomic transitions |a to |b and |c  unique position information of the single atom with high spatial resolution could be achieved. It is also observed that for some other choices of these parameters the conditional position probability distribution exhibited more than one localization peaks, however, a most probable position information could still be obtained. Thus the proposed scheme leads to a better precision in position measurement of single atom during its flight through the standing-wave fields for arbitrary choice of the detunings and phase shifts.

η = 0 and ξ is (a) π /6,

We also would like to mention that the position distributions are conditioned on the measurement of the frequency and the direction of the spontaneously emitted photon. The position information of an atom moving through the standing-wave fields is based on measurement of the spontaneous photon frequency by the detector. As spontaneous emission is an isotropic process therefore it appears that 4π detectors must be required in an experimental setup. Further, the calculation of |C d,1k (x; t → ∞)|2 Eq. (16) does not mean that an atom should spent a long enough time interacting with the field. However, in fact, it is not necessary to measure every atom and it would be sufficient to be able

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to detect only those atoms whose spontaneous photons have been detected. Therefore, taking time t → ∞ only means that a spontaneous photon must have been emitted whose frequency is then measured. As far as the measurement of spontaneous emission frequency δk is concerned it could be done via spectrum analyzer whose detailed can be found in [35]. Here, we mention that the present technique is based on assumption that most favorable emission events are detected as has been considered in all the relevant schemes [14–18], however, in order to estimate the microscopy power we can introduce a strict analysis procedure as has been done in [23] which assumes detection of the most probable emission events, rather than the most favorable, but potentially unlikely, detection events. Here, we emphasize that the conditional position probability distribution is narrowed to domains smaller than the wavelength of the standing-wave fields, however, we expect that this must accompanied by a widening of the conditional momentum distribution as has been demonstrated in [13] which is in complete agreement with the complementarity principle. We feel that this scheme not only can give much precise position informations of the atom but also experimentally more viable as compared to the scheme presented in [30]. This is due to the fact that no dipole forbidden transition is involved in our proposed scheme and further, there is no requirement for initial superposition of atomic states. We also would like to mention that this proposal is a theoretical one, however, for a possible experiment realization 87 Rb may be used in a diamond configuration, see some recent experiments [36,37]. Further, we suggest that this scheme can also be extended to two-dimensional atom localization, the topic which has gained considerable interest in recent years [38–43]. Acknowledgement Rahmatullah would like to thank Ziauddin for valuable discussion. References [1] Heisenberg microscope; a gedanken experiment proposed by W. Heisenberg to verify uncertainty principle. [2] W.D. Phillips, Rev. Mod. Phys. 70 (1998) 721. [3] K.S. Johnson, J.H. Thywissen, N.H. Dekker, K.K. Berggren, A.P. Chu, R. Younkin, M. Prentiss, Science 280 (1998) 1583.

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