PHYSICAL REVIEW A 84, 063849 (2011)

Subwavelength atom localization via quantum coherence in a three-level atomic system

Fazal GhafoorCentre for Quantum Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan

(Received 17 June 2011; published 29 December 2011)

We propose a three-level atomic system where quantum coherence is generated by a classical standing-wave field coupled to the two upper excited decaying levels. Quantum coherence results in cancellation of thespontaneously emitted spectral lines depending on the choice of the phase of the standing wave. We exploitthis phenomenon for precision measurement of the atomic position in the standing wave. Measurement of theconditional position probability distribution shows one to eight peaks per unit wavelength of the standing wave.Only one controllable parameter, that is, the phase of the driving standing wave, is enough to control these atomicpositions. Along with the other results, the result of obtaining a single peak is remarkable as it enhances theefficiency of our system by a factor of 8.

DOI: 10.1103/PhysRevA.84.063849 PACS number(s): 42.50.Ct, 32.50.+d, 32.30.−r

I. INTRODUCTION

The localization of atoms has a long history of extensiveeffort. In this regard various techniques have been proposedfor precise measurement of the position of the moving atoms[1–4] . Earlier, some mechanical methods were also proposedfor this purpose [5]. However most of the contributionscome from optical techniques. For example, the concept ofoptical virtual slits has been proposed to localize the atomwithin the subwavelength domain of the optical field. In thisscheme the atom interacts with a standing-wave field. Duringthe interaction a phase shift is introduced in the field. Themeasurement of this phase shift provides information aboutthe atomic position [1]. Moreover, the idea of a phasequadrature measurement has been studied for the case when thetransverse motion of the atom cannot be neglected [2,3]. Forthis purpose the method of continuous homodyne detection [4]is employed to monitor the transverse motion of the atom.The interaction of the standing wave with the atom results inentanglement between the state of the radiation field and theatomic state. A measurement of the field or internal state of theatom leads to quantum state reduction of the center-of-massmotion. These ideas have been used to localize the position of apolarized atom by passing it through a classical standing-wavefield. The phases of the atomic dipole moments are thenmeasured in a Ramsey-type experiment [6]. Further, a coherentcavity field achieves a higher resolution than a classical field,and an almost perfect localization is possible when the atompasses through several identically prepared cavities [7–9].However, it is observed that a better spatial resolution is alwaysaccompanied by an increase in the number of peaks in theposition probability distribution of the atom.

A method based on the spontaneous emission spectrum ofa three-level atom interacting with a classical standing-wavefield has been suggested by Herkommer et al. [8] to giveinformation about the position of the atom during its flightthrough the standing-wave field. The method is based on thefact that the frequency of the spontaneously emitted photongives information about the atomic position due to its directrelation with the position-dependent Rabi frequency. Thismethod exhibits atom localization in real time and withinthe subwavelength domain of the optical field, without anycumbersome computation. Further, this technique has also

been extended in the context of the resonance fluorescencephenomenon [10]. However, in all these studies, due to theperiodic nature of the standing-wave field, the measurementof atomic position yields four equally probable but differentpositions in a unit wavelength domain of the optical field. Theperiodic nature of the atomic position is reduced in Ref. [12]by employing the phase and amplitude of the driving fieldin a four-level atomic system. In that paper, the number oflocalization peaks in the conditional position distribution isreduced to half as compared with the earlier related schemes[9–11]. The cancellation of the localization peaks in theconditional position probability distribution can be achieved bycontrolling the phase of the standing-wave field. This enhancesthe efficiency of the system and reduces the number of atomsrequired for a specific position measurement of the singleatom by half. The spatial resolution in the measurement ofthe position of the single atom can also be enhanced by upto λ/200 by controlling the amplitude of the Rabi frequency.However, this scheme also suffers from various drawbacks.First, the atom considered is multileveled and is driven bythree coherent driving fields leading to a very complex system.Second, some of the transitions are dipole forbidden. Third, theoptical standing-wave field which drives the dipole-forbiddentransition is required to be either a very strong coherent fieldor a strong magnetic field. Fourth, in the low-strength limit ofthe standing driving field one will not be able to localize theatom in the standing-wave field.

In this paper we consider a three-level atomic system. Theupper two levels of the atom are driven by a classical standing-wave field. These upper two levels decay to a common groundlevel. Phase sensitivity is introduced into the system via aclassical standing driving field. The fluorescence spectrum isvery well controlled via control of the phase of the drivingfield. This control of the fluorescence spectrum is manifestin a reduction of the localization peaks of the atom in theconditional position probability distribution. Measurement ofthe conditional position probability distribution shows oneto eight peaks per unit wavelength of the standing-wavefield. Along with the other results, the result of obtaining asingle peak is remarkable as it enhances the probability ofthe conditional position distribution by a factor of 8. Onlyone controllable parameter, that is, the phase of the drivingstanding wave, is enough to control these atomic positions

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FAZAL GHAFOOR PHYSICAL REVIEW A 84, 063849 (2011)

FIG. 1. Level diagram of the atomic system: A standing-wavefield couples the transition |a〉-|b〉 having position-dependent Rabifrequency � sin(κx). The upper two levels decay to the ground level|c〉.

in the standing-wave field. Further, our proposed scheme isthe one used as a model for the quantum beat laser andcan be easily realized experimentally in the context of atomlocalization. This scheme has also been studied by Paspalakiset al. [13] in the context of spontaneous-emission control.Furthermore, this scheme can also be related to other schemesof coherent control, in particular the well-known pump-dumpscheme of atomic (molecular) ionization (dissociation) [14],the coherently driven three-level �-type atom of Martinezet al.[15], and the scheme of Quang et al. [16].

II. MODEL

The system under consideration is shown in Fig. 1. Thetwo excited states |a〉 and |b〉 are coupled by a classicalstanding-wave field having a frequency ωs and phase ϕs, andboth the states decay spontaneously to a common ground level|c〉. The atom during its motion in the z direction passesthrough the classical standing-wave field. The interactionbetween the atom and the classical standing-wave field isposition dependent. The corresponding Rabi frequency isdefined as �(x) = � sin(κx), where κ is the wave vector of thestanding-wave field and is defined as κ = 2π/λ. We furtherconsider that the atom is moving with a high enough velocity.Its interaction with the driving field does not affect its motionalong the z direction. We may treat the motion of the atomin the z direction classically. Furthermore, we assume that theinteraction of the atom with the standing-wave field is suffi-ciently small. Consequently, the Rabi frequency is also small.As a result the center-of-mass position of the atom throughoutthe standing-wave field does not change during the interactiontime. We thus may neglect the kinetic energy term of the atomin the interaction Hamiltonian under the Raman-Nath approx-imation [17]. Therefore, the resulting interaction Hamiltonianfor the off-resonant atom-field system can be written as

υ = � sin(κx)ei(t+ϕs )|a〉〈b| + � sin(κx)e−i(t+ϕs )|b〉〈a|+

∑k

g(1)k e−i(ωk−ωac)t |a〉

× 〈c|bk + g(1)∗k ei(ωk−ωac)t |c〉〈a|b†k

+∑

k

g(2)k e−i(ωk−ωbc)t |b〉

× 〈c|bk + g(2)∗k ei(ωk−ωbc)t |c〉〈b|b†k, (1)

where g(i)k (i = 1,2) is the coupling constant of the interaction

of the vacuum field with the atom = ωs − ωba , and bk andb†k are the annihilation and creation operators corresponding to

the reservoir modes k. Further, here in the detuning parameterωs applies to the frequency of the standing-wave field whileωij (i,j = a,b,c) is the frequency of the atomic transitionbetween the states i and j. The atom-field state vector for thecomplete system is given by the following equation:

|�(x; t)〉 =∫

dxf (x)|x〉[Aa,0(x; t)|a,0〉 + Bb,0(x; t)|b,0〉

+∑

k

Cc,1k (x; t)|c,1k〉], (2)

where Aa,0(x; t) and Bb,0(x; t) are the probability amplitudeswhen the atom is in state |a〉 and |b〉, respectively, such thatthere is no spontaneous emitted photon present in the reservoirmode k,while Cc,1k (x; t) is the probability amplitude whichshows that at time t the atom is in state |c〉 with one photonin the reservoir of mode k.

III. LOCALIZATION SCHEME

The localization scheme we propose in this paper is basedon the fact that the spontaneously emitted photon cariesposition information about the atom in the standing wave dueto the dependence of its frequency on the position-dependentRabi frequency of the atom (see Fig. 2). As a result the positionmeasurement of the atom is conditioned on the measurementof the spontaneously emitted photon. The conditional positionprobability distribution W (x; t | c,1k) is therefore defined asthe probability of finding the atom at position x in thestanding-wave field such that a spontaneous emitted photonis detected at a time t in the reservoir mode of wave vector k.

To determine the conditional position probability distributionW (x; t | c,1k), we start by reducing the state vector as follows:

|ψc,1k〉 = N〈1k,c | �(x; t)〉

= N

∫dxf (x)Cc,1k (x; t)|x〉, (3)

FIG. 2. Schematic for the localization of an atom. A three-levelatom is allowed to interact along the z direction with a standing-wavefield aligned along the x axis. The spontaneously emitted photon isdetected by the detector.

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where N represents the normalization constant. Using thedefinition of the conditional position probability distributionW (x; t | c,1k), as described above, we may write its expressionas

W (x; t | c,1k) = |〈x | ψc,1k〉|2

= |f (x)|2|N |2|Cc,1k (x; t)|2, (4)

where the normalization constant is defined as

N =[

1∫dx|f (x)|2|Cc,1k (x; t)|2

]1/2

, (5)

where |f (x)|2 is the initial position distribution of the atom.Now we introduce a filter function �(x; t | c,1k), which isdefined as

�(x; t | c,1k) = |N |2|Cc,1k (x; t)|2. (6)

The conditional position probability distribution then takes theform

W (x) = W (x; t | c,1k) = �(x; t | c,1k)|f (x)|2. (7)

Equation (7) shows that the conditional position probabilitydistribution is directly proportional to |Cc,1k (x; t)|2 which isfurther proportional to the steady-state spontaneous-emissionspectrum. To determine the conditional position probabilitydistribution we need to determine the probability amplitudeCc,1k (x; t).

IV. EQUATIONS OF MOTION

To find this probability amplitude Cc,1k (x; t) we need tofind the rate equations for our system. These rate equationscan be calculated from the Schrodinger wave equation usingthe interaction picture Hamiltonian (1). We then obtain

iAa(x; t) = � sin(κx)eit+iϕs Bb(x; t)

+∑

k

g(1)k Cc,1k

(x; t)e−i(ωk−ωac)t , (8)

iBb(x; t) = � sin(κx)eit+iϕs Aa(x; t)

+∑

k

g(2)k Cc,1k

(x; t)e−i(ωk−ωbc)t , (9)

iCc,1k(x; t) = g

(1)k Aa(x; t)ei(ωk−ωac)t

+ g(2)k Bb(x; t)ei(ωk−ωbc)t . (10)

We perform time integration of Eq. (10) and substi-tute the result in Eqs. (8) and (9). We further ex-tend the calculation using Weisskopf-Wigner theory [18]and make the transformationAa(x; t) = Aa(x; t), Bb(x; t) =Bb(x; t)eit ,Cc,1k (x; t) = Cc,1k (x; t) to get the following rateequation for our system:

˙Aa(x; t) = − 1

2Aa(x; t) − i� sin(κx)eϕs Bb(x; t)

−p

√ 1 2

2e−i(ωba+)t Bb(x; t), (11)

˙Bb(x; t) = −i� sin(κx)e−iϕs Aa(x; t) − p

√ 1 2

2eiωba t

× Aa(x; t) +(

i − 2

2

)Bb(x; t), (12)

˙Cc,1k(x; t) = −ig

(1)k Aa(x; t)ei(δk+ωba/2)t

−ig(2)k Bb(x; t)ei(δk−ωba/2−)t , (13)

where i = |μca |2ω3cj

6π2ε0hc3 (i = 1,2 and j = a,b) and√

1 2

2 =|μbc||μca |ω3

ca

6π2ε0hc3 while δk = ωk − (ωac + ωbc)/2. The term

p√

1 2

2 e±ωbat is a common term that arises if quantuminterference from both the spontaneous-emission channelsfrom the two closely spaced upper levels is involved [18]. Theparameter p denotes the alignment of the two matrix elementsand is defined as p = μbc · μca/|μbc||μca|. For orthogonalmatrix elements p = 0 and there is no interference, whilefor parallel matrix elements p = 1 and there is maximuminterference. In the context of our system, however, we neglectthese terms under the approximation ωba � 1, 2. We furtherassume that the atom is initially prepared in a coherentsuperposition of the two upper excited states. The initial statefunction is then read as

�(t = 0) = eiϕp sin θ |a,{0}〉 + cos θ |b,{0}〉. (14)

For convenience we assume ϕp = 0. Using the initial condi-tion, the above rate equations are solved exactly for the steady-state probability amplitude Cc,1k

(t → ∞). The expression forthis probability amplitude gets a very simple and interpretableform given by the following expression:

Cc,1k(x; t → ∞) = g

(1)k

2

[sin θ − eiϕs cos θ

δk + ωba

2 + � sin(κx) + i 2

+ sin θ − eiϕs cos θ

δk + ωba

2 − � sin(κx) + i 2

]

+ g(2)k

2

[sin θ − eiϕs cos θ

δk − ωba

2 − + � sin(κx) + i 2

+ sin θ − eiϕs cos θ

δk − ωba

2 − − � sin(κx) + i 2

], (15)

where we assume that 1 = 2 = . It is worthwhile to notethat we have neglected the terms proportional to

√ 1 2

2 e±ωbat

in the rate equations of our system under the condition of

large frequency between the upper two states as compared tothe decay rates from the two excited states. These terms arethe sources of quantum interference and if they are neglected

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there is no quantum interference in the system. We, however,consider orthogonal dipole moments, which are easy to findin nature as compared to parallel dipole moments [19]. In thecontext of our system, quantum coherence is generated by thestanding-wave field coupled to the two upper excited states,which results in cancellation of the spontaneously emittedspectral lines from either side of the central line on thefrequency axis depending on the choice of the phase of thestanding-wave field. We exploit this phenomenon for preciseposition information about the atom in the standing-wave field.

By inspection of Eq. (15) we note that the probabilityamplitude Cc,1k

(x; t → ∞) consists of two major parts. Each

part further contains two terms. The spontaneous-emissionspectrum can be calculated from |Cc,1k

(x; t → ∞)|2 whichcontains cross or interference terms along with absolutesquared terms. There is negligible contribution from theinterference or cross terms to the spontaneous emissionspectrum and consequently to the conditional positiondistribution. For convenience we neglect these cross termsin the analysis and interpretation of our system as theyrequire cumbersome computation and the results may notbe displayed in readable and interpretable form. Therefore,under this assumption the conditional position probabilitydistribution gets the following shape:

W (x) = W (x; t | c,1k)

= |f (x)|2|N |2∣∣g(1)k

∣∣2

4

[1 − 2 cos(θ ) sin(θ ) cos(ϕs)

[sin(κx) − λ1][sin(κx) − λ2] + (δk + ωba

2

)2 + 1 + 2 cos(θ ) sin(θ ) cos(ϕs)

[sin(κx) − ξ1][sin(κx) − ξ2] + (δk + ωba

2

)2

]

+ |f (x)|2|N |2∣∣g(2)k

∣∣2

4

[1 − 2 cos(θ ) sin(θ ) cos(ϕs)

[sin(κx) − μ1][sin(κx) − μ2] + (δk + ωba

2 − )2

+ 1 + 2 cos(θ ) sin(θ ) cos(ϕs)

[sin(κx) − η1][sin(κx) − η2] + (δk + ωba

2 − )2

], (16)

where

λi(i = 1,2) = 1

�

[(δk + ωba

2

)±

√(δk + ωba

2

)2

− 2

4

], (17)

ξi(i = 1,2) = − 1

�

[(δk + ωba

2

)∓

√(δk + ωba

2

)2

− 2

4

], (18)

μi(i = 1,2) = 1

�

[(δk − ωba

2−

)±

√(δk − ωba

2−

)2

− 2

4

], (19)

ηi(i = 1,2) = − 1

�

[(δk − ωba

2−

)∓

√(δk − ωba

2−

)2

− 2

4

]. (20)

V. RESULTS AND DISCUSSION

We proceed with the analysis of the conditional positionprobability distribution. There are two parts in the conditionalposition probability distribution (16). Each part further con-tains two terms. The denominator of the first term of thefirst part of the conditional position probability distributioncontains two factors which are functions of sin(κx). Wenote that two localization peaks appear in the domain of0–π (generally, depending on the choice of spectroscopicparameters) at the following normalized positions:

κx = sin−1(λi) ± nπ, (21)

where λi (i = 1,2) is given by Eq. (17). Similarly, thedenominator of the second term of the first part of theconditional position distribution contains two factors whichare functions of sin(κx). The contribution of the conditional

position probability distribution to the localization peaks in thedomain −π–0 appears at the following normalized positions:

κx = sin−1(ξi) ± nπ, (22)

where ξi (i = 1,2) is given by Eq. (18). Similarly, thedenominator of the first term of the second part of theconditional position distribution again contains two factorswhich are functions of sin(κx). We note that two localizationpeaks appear in the domain of 0–π at the following normalizedpositions:

κx = sin−1(μi) ± nπ, (23)

where μi (i = 1,2) is given by Eq. (19). Also the denominatorof the second term of the second part of the conditionalposition distribution again contains two factors which arefunctions of sin(κx). The contribution of the conditional

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position distribution to the localization peaks in the samedomain (−π–0) appears at the following normalized positions:

κx = sin−1(ηi) ± nπ, (24)

where ηi (i = 1,2) is given by Eq. (20). From this discussion itis concluded that for a specific phase there appear eight peaksper unit wavelength of the standing wave for the measurementof the conditional position probability distribution of the atom.

It has already been discussed that the measurement ofspontaneously emitted photons determines the conditional po-sition distribution via the position-dependent Rabi frequency.It is therefore important to determine the frequencies ofthe spontaneously emitted photons that correspond to therequired maxima of the localization peaks. The frequency ofthe spontaneously emitted photons can be measured as

ωk = (ωac + ωbc)

2−ωba

2+ 1

2� sin(κx)

[ 2

4+ �2 sin2(κx)

],

(25)

when the maxima of the localization peaks correspond to thecase of sin(κx) − λi = 0 (i = 1,2), while for the maxima ofthe localization peaks corresponding to the case sin(κx) −ξi = 0(i = 1,2), the frequency of the spontaneously emittedphotons can be measured from the following equation:

ωk = (ωac + ωbc)

2−ωba

2− 1

2� sin(κx)

[ 2

4+ �2 sin2(κx)

].

(26)

Similarly, the frequency of the spontaneously emitted photonscan be measured as

ωk = (ωac + ωbc)

2+ + ωba

2

+ 1

2� sin(κx)

[ 2

4+ �2 sin2(κx)

](27)

when the maxima of the localization peaks correspond to thecase of sin(κx) − μi = 0 (i = 1,2) while for the maxima of thelocalization peaks corresponding to the case sin(κx) − ηi =0(i = 1,2) the frequency of the spontaneously emitted photonscan be measured from the following equation:

ωk = (ωac + ωbc)

2+ + ωba

2

− 1

2� sin(κx)

[ 2

4+ �2 sin2(κx)

]. (28)

A detailed analysis of the conditional position distribution re-veals various facts, that is, for which spectroscopic parametersthere is a minimum number of localization peaks and for whichvalues the number is maximum. The factors in the numeratorsof the first terms of the first and second parts of the conditionalposition probability distribution are similar, while the factorsin the numerators of the second terms of the first and secondparts of the conditional position probability distribution arealso similar. These factors are respectively given by

1 − 2 cos(θ ) sin(θ ) cos(ϕs) (29)

and

1 + 2 cos(θ ) sin(θ ) cos(ϕs). (30)

Initially, for coherent superposition of the atom in the uppertwo excited states, i.e., cos(θ ) = sin(θ ) = 1/

√2, and for a

phase ϕs = π/2, both the above terms survive and thereforeeight peaks appear in the conditional position distributionper unit wavelength. Similarly, for cos(θ ) = sin(θ ) = 1/

√2

and for the phase ϕs = 0, the first term of the first partand the second term of the second part of the conditionalposition distribution contribute to the domain 0–π , and thesecond term of the first part and first term of the secondpart of the conditional position distribution contribute to thedomain −π–0 for the phase ϕ = π. One of the numeratorsin these coupled terms vanishes, resulting in cancellation oftwo peaks in each domain, and we are left with four local-ization peaks appearing per unit wavelength. Furthermore,with the choice of specific spectroscopic parameters withlow strength of the standing-wave field, the denominator ofthe two parts of the conditional distribution is adjusted insuch a way that the first term of the first part and the firstterm of the second part contribute to the domain 0–π. Thenumerators in these coupled terms are similar. The numerator1 − 2 cos(θ ) sin(θ ) cos(ϕs) = 0 when the phase ϕ = 0, whichcorresponds to the cancellation of the localization peaks inthe domain of 0–π , while for cos(θ ) = sin(θ ) = 1/

√2 and for

the phase ϕ = π , the factor 1 + 2 cos(θ ) sin(θ ) cos(ϕs) = 0,

which results in the cancellation of the localization peaks in thedomain −π–0. This analysis shows that the correlation of thecorresponding terms in the denominators of the first and secondparts of the conditional position distribution results in furthercancellation of peaks. Two of the localization peaks disappearin either domain of the normalized position of the conditionalposition probability distribution. We are left with two peaksper unit wavelength of the standing-wave field, enhancingthe efficiency of our system by a factor of 4. This result ofachieving two localizations per unit wavelength agrees withthe work reported in Ref. [12] but with a simple scheme andunder relaxed experimental conditions as discussed earlier. Thecancellation of the peaks is not limited to these values, and thecorrelation in the corresponding terms in the denominator alsoresults in generating a single peak. This is due to the fact thatthe two roots of sin(κx) of the coupled terms become equalunder specific chosen parameters, leading to two single peakseach appearing in the two domains at the phase ϕs = π/2.However, choice of the phase ϕs = 0(π ) results in cancellationof one peak from either side in the two domains, and we are leftwith a single peak for the two phases and in the two domainsof unit wavelength. This result is very important as it enhancesthe efficiency of our system by a factor of 8.

Next we discuss the graphical results for the conditionalposition distribution with a normalized position of the atom in astanding-wave field for different experimental parameters. Theplot of the conditional position distribution with normalizedposition shows a maximum number of peaks when the phaseof the standing wave is π/2 for larger strength of the standing-wave field. Eight peaks appear per unit wavelength of thestanding wave, four at the normalized position in each domain0–π and −π–0, respectively, when θ = π/4, ϕs = π/2, � =15 , = 5 ,δk = 0.3 , and ωba = 20 . However, when thephase is varied from π/2 to 0 two of the localization peaksdisappear in the range 0–π , while two of the localization peaksdisappear from the range −π–0 when the phase is varied from

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π/2 to −π , as shown in Fig. 3. This is in accord with whatwe expect from the analysis of our system, as the terms in thedenominators of two parts of the expression for the conditionalposition distribution are correlated in such a way that the neteffect on the system of the two terms in the two domains iscompensated, resulting in the cancellation of the peaks.

Choosing the strength of the standing-wave field small andchoosing appropriate values of the other parameters, i.e., θ =π/4, ϕs = π/2, � = 15 , = 9 , δk = 0.3 , and ωba =20 , we find four peaks per unit wavelength for the phaseπ/2. However, when the phase is varied from π/2 to 0, whilekeeping the other parameters the same, two of the four peaksdisappear in the domain 0–π , while two of the four peaks dis-appear in the domain −π–0 when the phase is varied from π/2to π , as shown in Fig. 4. The decrease in the number of peaksis not limited to these two peaks, and we can observe a single

(a)

(b)

(c)

FIG. 3. (Color online) The conditional position probability distri-bution plotted against the normalized position κx for θ = π/4,� =15 , = 5 , δk = 0.3 , and ωba = 20 . (a) ϕ = π/2, (b) ϕ = 0,and (c) ϕ = π .

localization peak. For example, when the parameters are cho-sen in such a way that θ = π/4, ϕs = π/2, � = 10 , = 5 ,

δk = 0.3 , and ωba = 20 , there appears a single peak in eachof the two domains as shown in Fig. 5. One of these two peaksin the two domains disappears when the phase of the standingwave is varied from π/2 to π (0). It is worthwhile to note thatthere is no significant change in this scenario of the singlelocalization peak when the vacuum field detuning is varied.This result is different from the one reported in Ref. [12], wherethe decrease in the vacuum field detuning results in movementof the localization peaks. The initial broad wave packet lyingat antinodes of the standing wave splits into a doublet and thetwo parts move toward the antinodes. However, in this systemthe single localization peak maintains its shape with decreaseor increase in the value of the vacuum field detuning. By choiceof different experimental parameters it is possible to localizean atom at one particular position in the standing-wave field.

(a)

(b)

(c)

FIG. 4. (Color online) The conditional position probability dis-tribution W (x) plotted against the normalized position κx for θ =π/4,� = 15 , = 9 , δk = 0.3 , and ωba = 20 . (a) ϕ = π/2,(b) ϕ = 0, and (c) ϕ = π .

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SUBWAVELENGTH ATOM LOCALIZATION VIA QUANTUM . . . PHYSICAL REVIEW A 84, 063849 (2011)

(a)

(b)

(c)

FIG. 5. (Color online) The conditional position probability dis-tribution W (x) plotted against the normalized position κx for θ =π/4,� = 10 , = 5 , δk = 0.3 , and ωba = 20 . (a) ϕ = π/2,(b) ϕ = 0, and (c) ϕ = π .

The results of a single peak in Figs. 5(b) and 5(c) are themost important in this paper, and it is desirable to know howwe can extract the position information from the given details.We have discussed already that the first terms of the first andsecond parts of the conditional position probability distributioncontribute to the domain 0–π, while the second terms of thefirst and the second parts contribute to the domain −π–0. Theroots of the factors in the denominators for each case, whichare functions of sin(κx), approximately become equal underspecific chosen parameters. The corresponding calculated

frequencies of the photons are then given by Eqs. (27) and(28), respectively. Choosing the phase ϕs as 0 (π ), we are leftwith a single peak per unit wavelength of the standing-wavefield in each case. Once the frequency of the photon is detectedit is then related to the position-dependent Rabi frequency� sin(κx). This position-dependent Rabi frequency is furtherrelated to the conditional position probability distribution. Inthis way we can calculate the exact location of the single peakin space once the frequency of the photon is measured by thedetector for the given specific chosen parameters.

VI. CONCLUSION

In conclusion, we propose an atomic scheme, based onthe spontaneous-emission spectrum of a three-level atominteracting with a classical standing-wave field, to givethe position information of the atom during its motionthrough the standing-wave field. The method is based on thefact that the frequency of the spontaneously emitted photongives information about the atomic position due to its directrelation with the position-dependent Rabi frequency. Thismethod exhibits atom localization in real time and withinthe subwavelength domain of the optical field, without anycumbersome computation. We further consider orthogonaldipole moments which are easy to find in nature as comparedto parallel dipole moments [19]. In the context of our system,quantum coherence is generated by the standing wave coupledto the two upper excited states. The fluorescence spectrum isvery well controlled via control of the phase of the standingwave. This results in cancellation of the spontaneously emittedspectral lines from either side of the central line on thefrequency axis depending on the choice of the phase of thestanding-wave field. We exploit this phenomenon to obtainprecise position information of the atom in the standing-wavefield. The control of the fluorescence spectrum is manifestin reduction of the number of localization peaks of the atomin the conditional position distribution. The response of theconditional position distribution to the phase of the standingwave is enormous, and we observe localization of the atomfrom a single peak to eight peaks per unit wavelength of thestanding wave. Only one controllable parameter, that is, thephase of the driving standing-wave field, is enough to controlthe atomic position in the standing-wave field. In our systemthe single localization peak maintains its shape with decreaseor increase in the value of the vacuum field detuning. Bychoice of different experimental parameters, it is possible tolocalize an atom at one particular position in the standing-wave field. The atom therefore restrict its oscillation to oneparticular position in this atom-field interaction. This result isremarkable as it enhances the probability of the conditionalposition distribution by a factor of 8 in the measurementof atomic position per unit wavelength of the standing-wavefield.

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