Sub-wavelength position measurements with running wave driving fields J¨org Evers1 and Sajid Qamar1,2 1Max-Planck-Institut fu¨r Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany 2Centre for Quantum Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan (Dated: January 29, 2009) A scheme for sub-wavelength position measurements of quantum particles is discussed, which operates with running-wave laser fields as opposed to standing wave fields proposed in previous setups. The position is encoded in the phase of the applied fields rather than in the spatially modulated intensity of a standing wave. Therefore, disadvantages of standing wave schemes such as cases where the atom remains undetected since it is at a node of the standing wave field are avoided. Reversing the directions of parts of the driving laser fields allows to switch between 9 different magnification levels, and thusto optimize thelocalization. 0 0 PACSnumbers: 42.50.Ct;42.50.Gy;42.30.-d;03.65.Wj 2 n a The localization of quantum particles is an intrigu- standing wave fields. The position is encoded in the J ing area of research that already in the early days of phases of the applied fields rather than in the spatially 9 quantum mechanics led to much discussion and an im- modulatedintensityofastandingwave. Therefore,cases 2 proved understanding of the underlying theory. Further where the atom remains undetected since it is at a node research is fueled by the need for effective structuring of the standing wave field are avoided. The phase sensi- ] h and measuring schemes at small length scales in many tivity arisessince the driving fields are appliedsuchthat p modern applications. Approaches involving light, how- they form a closed interaction loop [17, 18, 19, 20]. The - t ever, due to diffraction are typically restricted to an ac- number of applied laser fields determines the maximum n curacy of order of the involved wavelength λ [1]. This resolution of the measurement schemes, which can be a limit could be overcome in techniques operating at the improved by adding more fields. The higher resolution, u q sub-wavelengthscale which are based on near-field tech- however,comesatthecostofmorepotentialpositionsper [ niques[2]orrelyondistinguishablequantumobjects[3]. laserwavelength. Butthislimitationcanbeavoidedsince Alternatively, sub-wavelength measurements in the op- we find that for a given number of laser fields, changing 1 v tical far field have been suggested. A particular class the directions of individual driving laser fields allows to 5 of quantum optical localization schemes suitable to de- switch between different magnification levels. Thus, the 3 termine the position of a quantum particle on a sub- position can first be determined on a coarser scale with 6 wavelength scale makes use of standing wave driving fewpotentialpositions,andthenberefinedusingahigher 4 fields[4,5,6,7,8,9,10,11,12,13,14,15]. Theseonthe magnification. . 1 onehandactasarulerforthepositionmeasurement,and Westartbyconsideringafour-levelsystemindiamond 0 on the other hand encode position information into the configuration as a basic atomic level setup suitable for 9 atomic dynamics via their position dependent intensity. our localization scheme, see Fig. 1(a) [18]. In the final 0 : Standing wave based schemes, however, typically do part,wewillgeneralizeouranalysistogeneralclosed-loop v i not work equally well over the whole range of potential systems as indicated in Fig. 1(b). The diamond scheme X positions throughout one wavelength. In the worst case, consists of one ground state |1i, two intermediate states r the atom is located at a node of the standing wave field, |2i and |4i, and one excited state |3i. All four electric- a such that no direct detection is possible. Another dis- dipoleallowedtransitions|1i−|2i,|2i−|3i,|4i−|3iand advantage arises in recent schemes with improved local- ization that facilitate more than one position-dependent fields with different frequencies [15, 16]. This frequency (a) (b) other states |3i difference leads to a beat, and the relative light field in- |2i tensities are not periodic in space on a length scale of a |2Ni typical wavelength λ. Thus, the localization is not pe- |4i |2i riodic with λ, which so far has been neglected in the |3i theoretical analysis. Third, current standing-wavebased schemes usually require an additional classical measure- |1i=|5i |1i=|2N +1i ment to determine one out of several potential positions due to the periodicity of the standing wave. FIG.1: (Coloronline)(a)Thediamondschemewith2N =4 InthisLetter,wediscusssub-wavelengthpositionmea- states that is considered as specific example. (b) Generic surementofquantumparticlesusingrunningwavefields, atomicloopsystemwith2N states. Redarrowsindicatedriv- whichallowstocircumventthe problemsassociatedwith ing laser fields, spontaneous emission is not depicted. 2 |1i−|4iaredrivenbycoherentlaserfields. Thus,starting 1.0 from the state with lowest energy |1i, the system can 0.8 (iii) evolveina non-trivialloopsequenceof laserinteractions via |2i, |3i and |4i back to the initial state. We denote 0.6 (ii) the atomic levels with state indices increasing along the R closedlooppathfrom1to2N,andidentifystate|1iwith 0.4 |2N +1i for the sake of simpler analytical expressions. The spontaneous decay rates from level |ii to the levels 0.2 (i) |ji are denoted as 2γ . The Hamiltonian in dipole and ji rotating wave approximation is given by 0.0 0.0 0.5 1.0 1.5 2.0 4 H =X EjAjj +~(g21A21eiα21 ++g32A32eiα32 Φ (units of π) j=1 +g A eiα34 +g A eiα41 +H.c.), (1) FIG.2: (Color online) RatioR of thefluorescenceintensities 34 34 41 41 ontransitions|3i→|2iand|2i→|1i. (i)Ω=γ,(ii)Ω=5γ, whereA =|iihj|,themodulioftheRabifrequenciesare (iii) Ω=10γ. ij g ,andtheenergiesoftheinvolvedstatesaredenotedby ij E (j ∈{1,...,4}). The parameters α =ω t−~k ~r+ j ij ij ij Here, ǫ ∈ {−1,1} and eˆ is the unit vector in z direc- ij z φ contain ω as the laser frequencies, the wave vectors ij ij tion. Then, the closed-loop phase Φ evaluates to ~k , absolute phases φ arising from both the laser field ij ij andthedipolemoment,and~rasthepositionoftheatom. Φ=(k +k −ǫ k −ǫ k )z+φ . (4) 21 32 34 34 41 41 0 We further define the transition frequencies ω¯ =(E − ij i E )/~ and laser field detunings ∆ = ω −ω¯ . In a Eq. (4) is the origin of our central results. It demon- j ij ij ij suitable interaction picture, the dynamics of the system strates that it is possible to choose a laser setup such density matrixρ is determinedby∂ ρ=−i[V,ρ]/~+Lρ, that the multiphoton detuning does not contribute to t where the Liouvillian L describes spontaneous emission the closed-loop phase, whereas the dependence on the and the Hamiltonian V is wavevectorsleads to apositiondependence ofthe phase Φ controllable by the propagationdirections of the laser V =−~∆ A −~(∆ +∆ )A −~∆ A fields. In order to analyze this control further, we ap- 21 22 21 32 33 41 44 +~(g A +g A +g A +g e−iΦA +H.c.). proximate kij ≈k =2π/λ, and obtain 21 21 34 34 41 41 32 32 ξ Asitiswellknownforclosed-loopsystems,thedynamics Φ=2π z+φ , (5) 0 λ depends on the phase [18, 19] where ξ =(2−ǫ −ǫ ). FromEq.(5), we findthat the 34 41 Φ=∆t−K~~r+φ0, (2) closed loop phase Φ changes by 2π over a position range of λ/ξ. It is interesting to note that in closed-loop sys- 2N where ∆ = Pi=1∆i+1,i is known as the multiphoton tems studiedbefore,typicallythe phase-matchingcondi- detuning,K~ = 2N ~k asthewavevectormismatch, tion K~~r ≈ 0 was assumed in order to avoid a position Pi=1 i+1,i andφ = 2N φ astherelativephaseoftheinvolved dependence on a sub-wavelength scale. In contrast, for 0 Pi=1 i+1,i driving fields. Here, X =−X for X ∈{∆,~k,φ}. sub-wavelength localization, this dependence is exactly ij ji what is required. In order to proceed with the analysis, we make cer- In order to make use of Eq. (4) in a realistic setup, an tainassumptionsonthesetupandonthelevelstructure. easilyaccessibleobservableisrequiredwhichdepends on First, we adjust all laser fields to be on resonance, i.e., the closed-loop phase Φ. A number of potential observ- ∆ = 0. Further, we note that the phase φ is a rela- 0 ables such as state populations, fluorescence spectra, or tive phase which can be controlled independent of the light propagation dynamics have been identified in the position of the atom by phase-locking the different laser literature [17, 18, 19, 20]. In the following, we choose fields to each other [20]. Next, to fix a definite setup, we the fluorescence intensity of light emitted on the differ- assume that all transitions couple to circularly polarized ent transitions as the simplest observable accessible in light with polarizationvector in the x-y-plane and prop- the optical far field. The fluorescence intensity on the agationdirectionsofthelaserfieldsparalleltothez-axis, different transitions is proportional to the spontaneous decay rate on the transition times the population in the ~k =k eˆ , ~k =ǫ k eˆ , (3a) upper state of the respective transition. 21 21 z 34 34 34 z Since the intensities also depend on experimental con- ~k =k eˆ , ~k =ǫ k eˆ . (3b) 32 32 z 41 41 41 z ditions such as the distance of the detector to the atom, 3 1.0 0.18 0.17 (a) 0.8 0.16 λ / 0.15 z 0.6 0.14 λ / 0.13 z 0.4 0.12 0.70 0.75 0.80 0.85 0.90 0.2 R 0.154 0.0 (b) 0.0 0.2 0.4 0.6 0.8 1.0 0.152 λ / 0.150 R z 0.148 FIG.3: (Coloronline)Potentialpositionsz dependingonthe 0.146 measured ratio R. Solid blue lines show ξ = 4, dashed red 0.45 0.50 0.55 0.60 0.65 lines indicate ξ=2, and thedash-dottedmagenta line shows ξ = 0.25. The horizontal green line indicates the assumed R particle position z=0.15λ. 0.154 (c) 0.152 λ theareaofthedetectororitsefficiency,itisconvenientto / 0.150 z consider the ratio of two intensities as the observable in 0.148 order to reduce the dependence on these quantities. We 0.146 defineRastheratioofthefluorescenceintensityontran- sition |3i→|2i to the intensity on transition |2i→|1i. 0.30 0.35 0.40 0.45 0.50 Themeasurementthencouldproceedasfollows. After R applying the driving fields, the intensities of the light emitted on the two transitions is measured. The light FIG. 4: (Color online) Particle position z depending on the from the different transitions can be distinguished via measured ratio R, with parameters as in Fig. 3. The shaded the polarizations or the frequencies. From the ratio R of areaindicatesuncertaintiesduetoanimperfectmeasurement ofR. Thehorizontalgreenlineindicatestheassumedparticle the two intensities, the phase Φ can be determined. Via position z = 0.15λ. (a) Laser configuration ξ = 2, relative Eqs.(4)or(5),Φisrelatedtothepositionz oftheatom. phase φ0 = 0; (b) Laser configuration ξ = 4, relative phase Morespecific,assumingforthesakeofsimpleranalytical φ0 = 0; (c) Laser configuration ξ = 2, relative phase φ0 = results all spontaneous decay rates to be equal to γ, and π/4. Notethe different position axis scale in (a). all Rabi frequencies equal to xγ with x ∈ R, we obtain R=N(x)/D(x), where N(x)=2x2cos(Φ/2)2[−(3+2x2)2+4x4cosΦ], (6a) could be used to determine the true position out of sev- D(x)=−18−51x2−28x4−2x6+ eral potential positions [6, 10, 15]. However, the dash- dottedmagentacurveinFig.3suggestshowthisclassical x2(9+4x2)cosΦ+2x6cos(2Φ). (6b) measurementcanbeavoided. Forthis,ξ ≪1isrequired. This can be achieved with ǫ =ǫ =1 in Eq. (4) if the ThisratioRisplottedfordifferentdrivingfieldstrengths 34 41 magnitudes of the four wave vectors slightly differ due Ω=xγ inFig.2. Ideally,theratioshouldbechosensuch to unequal transition frequencies. Alternatively, a small that there is a highslope |∂R/∂Φ|throughoutthe whole mismatch of the propagation directions from ±eˆ allows rangeofphasesΦ. FromFig.2,itcanbeseenthatx≈5 z to tune ξ to small non-zero values. In the latter case, offers a good compromise of high maximum slope and however, the driving fields acquire polarization compo- low minimum slope over the whole range of 0≤Φ≤2π. nents that may drive unwanted transitions. Small val- Then,therelativephasecanbedeterminedwellfromthe ues of ξ allow to measure the approximate position of measured intensity ratio. the particle on a coarse scale, because the phase Φ then Settingx=5,wenowproceedtoextractpotentialpo- changesonascaleλ/ξ ≫λ. Usingthismethod,theneed sitionsoftheatomfromthemeasuredfluorescenceinten- for an additional classical measurement common to the sity ratio R. For this, we calculate the positions leading localization schemes suggested so far is eliminated. to the measured rations R using Eqs. (5) and (6). The results for ξ ∈{2,4} are shown in Fig. 3. Like in We now turn to a discussion of measurements on a standing wave localization schemes, each ratio R corre- sub-wavelength scale with higher ξ. In the following, we sponds to several potential positions {z } [15]. As sug- assume as a concrete example that a particle is located i gested previously, an additional classical measurement atpositionz =0.15λ. This positionis indicated as solid 4 green line in Figs. 3 and 4. We further assume that for tions of the corresponding laser beams. Approximating ξ =2, a measurement of the ratio R obtained a value of again k ≈2π/λ, one obtains ij R=0.81·(1±0.05)withanoveralluncertaintyof5%. Fi- nally,weassumethatthecorrectbranchisidentifiedbya ξ Φ≈2π z+φ , (8a) measurement with small ξ determining the appxoximate λ 0 positionoftheparticle. Fig.4(a)thendemonstrateshow ξ ∈{0,2,...,2N}. (8b) the result for R together with its overall error is trans- lated into position information. It can be seen that the Thevalueofξcanbecontrolledbythechoiceoftheprop- slope of the curve essentially determines the uncertainty agationdirectionsǫ . Thus we findthat the prefactorξ i,j in the position measurement. In Fig. 4(a), a localization already present in Eq. (5) that enables one to determine up to an uncertainty of about 20% is achieved. the magnification of the localization scheme can be cho- This uncertainty can be reduced using two methods. seninawiderange. Sinceforagivenξ,therelativephase First,changingthepropagationdirectionthelaserbeams Φchangesby2π overapositionrangeofλ/ξ,amoreex- andthusξ leadstoanimprovementofthemeasurement. tendedclosedloopschemeenablesonetoachieveabetter Fig. 4(b) shows the example for ξ =4. Again, an overall resolution for the position determination, and to switch error of 5% in the measurement of R is assumed. It can betweenmoremagnificationlevels. Potentialrestrictions be seen that due to the smaller slope than in the case arisefromtherequirementthattheindividualtransitions ξ = 2, the position error is reduced by about one order should ideally be addressed individually by laser fields. ofmagnitude to about2%. Thus,by reversingthe direc- This, however,is not a strict requirement,as long as the tion of one of the laser fields, the localization is greatly multiphotonresonanceconditionisfulfilled, butrestricts improved. Second, the relative phase φ in Eq. (2) can 0 thepossiblesettingsforξ. Inparticularinmoreextended beusedtoimprovethemeasurement. Itturnsoutthata systems, other observables than the simple ratio of two change ofφ shifts all curvesin Fig.3 alongthe position 0 fluorescenceintensitiescanbeexpectedtoleadtofurther axis. Thus the phase φ can be optimized such that the 0 improvement for the position determination. true position of the particle leads to a measured ratio R inarangewheretheslopeofthecurvesinFig.3issmall. ThisisdemonstratedinFig.4(c)forξ =2. Inthisfigure, a relative phase φ = π/4 was used to shift the ratio R 0 corresponding to the true position from about R = 0.81 [1] M. Born and E. Wolf, Principles of Optics (Cambridge in Fig. 4(a) to about R = 0.4 in Fig. 4(c). At this ra- University Press, Cambridge, England, 1999). tio, the slope |∂z/∂R| is much smaller, and this reduces [2] A. Lewis et al.,Nature Biotechnology 21, 1378 (2003). [3] E. Betzig, Opt.Lett. 20, 237 (1995). the position error by almost one order of magnitude to [4] P. Storey, M. Collett, and D. F. Walls, Phys. Rev. Lett. about 2.5%. In general, the best localization results are 68, 472 (1992); Phys. Rev. A 47, 405 (1993); R. Quadt, obtainediftheoptimizationviathephaseφ0 iscombined M. Collett, and D. F. Walls, Phys. Rev. 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