Atom microscopy via two-photon spontaneous emission spectroscopy

PHYSICAL REVIEW A 79, 043814 共2009兲

Atom microscopy via two-photon spontaneous emission spectroscopy Sajid Qamar,1,2,* Jörg Evers,1,2,† and M. Suhail Zubairy1,3

1

Centre for Quantum Physics, COMSATS Institute of Information Technology, Islamabad, Pakistan 2 Max-Planck-Institut für Kernphysik, Saupfercheckweg 1, D-69117 Heidelberg, Germany 3 Department of Physics and Institute for Quantum Studies, Texas A&M University, College Station, Texas 77843-4242, USA 共Received 17 January 2009; published 15 April 2009兲 We study subwavelength position measurement via spontaneous emission spectroscopy with two photons. Our model systems are a single ⌳-type three-level atom, in which a dual interaction generates two independent photons, and an M-type five-level atom, which emits two photons in a double-Raman process. The difference is that the M-type system emits correlated pairs of photons, while the photon pairs from the ⌳ system are uncorrelated. In order to evaluate the performance of the position measurement, we introduce an analysis scheme motivated by a possible experimental measurement procedure. We find characteristic differences in the spontaneous emission spectra of the two systems arising from the photon correlation in the M-type system. But interestingly, the effect of this photon pair correlation on the conditional position probability distribution does not lead to an increase in the position information of the single atom. Nevertheless, both schemes allow for an efficient measurement of the particle position with unambiguous position determination over a wide range of parameters. DOI: 10.1103/PhysRevA.79.043814

PACS number共s兲: 42.50.Ct, 32.30.⫺r, 32.50.⫹d

I. INTRODUCTION

Spatial precision measurements are a topic of longstanding interest both for a number of important applications and from a fundamental point of view. For example, Heisenberg’s famous thought experiment in which he envisioned a microscope that could measure the position of an electron precisely helped in the development and understanding of the uncertainty principle. Modern applications arise, e.g., in life sciences, where both structuring and measuring at the nanoscale is one of the main technical challenges. A key limit for imaging is diffraction, such that typically the resolution is limited to a length scale given by the wavelength of the light field used for the measurement 关1兴. Tremendous success in optical imaging has been achieved with near-field techniques, where the imaging system is so close to the object that so-called evanescent waves can be picked up 关2兴. Complementary to this, far-field imaging techniques have been developed, e.g., by individually addressing only small parts of the sample 关3兴. In part, these schemes are based on the observation that quantum particles can be distinguished on a subwavelength scale if they are not fully identical 关4兴. In general, different scenarios have to be distinguished. For example, the localization of a single fixed particle has been considered, or the measurement of the distance of two fixed particles. Complementary effort aimed at measuring the position of flying particles while they pass through a measurement device. A large class of proposals is based on quantum optical ideas, and makes use of standing-wave driving fields for the position measurement. The standing wave acts as a ruler for the position measurement, and at the same time forms a spatial intensity pattern that can encode position in-

*[email protected] †

[email protected]

1050-2947/2009/79共4兲/043814共7兲

formation via an intensity-dependent dynamics of the atom. Suitable observables are the quadrature phases of light fields interacting off-resonantly with the particle 关5兴, long-lived electronic states 关6,7兴, resonance fluorescence light 关8兴, or combinations 关9兴. Schemes based on standing-wave fields, however, are restricted to the prediction of few potential positions of the atom within one wavelength. The reason for this is the periodicity of the intensity in the standing wave. This restriction was addressed in subsequent work, for example, via a control of the vacuum field-atom detuning 关10–12兴, via absorption spectroscopy of a four-level system 关13兴, via coherent population trapping in a three-level system 关14兴, via dark resonances in a four-level system 关15兴, or via multiple simultaneous quadrature measurements 关16兴. Recently some schemes have been proposed theoretically as well as experimentally to generate subwavelength structures beyond the diffraction limit using the phenomenon of coherent population trapping. These schemes use position-dependent dark states for nanoscale resolution fluorescence microscopy 关17–19兴 and for the generation of the subwavelength structures in interferometric optical lithography 关20,21兴. A qualitatively different approach has been used in schemes aiming at measuring the distance between two particles. It was found that correlations among the emitted photons can be used to enhance the distance measurement between two identical atoms beyond the diffraction limit 关22兴. As a matter of fact the spatial information between two identical atoms placed in a standing-wave field can be extracted from the collective resonance fluorescence spectrum of a two-atom system inside a standing-wave field 关23兴 and from the intensity-intensity correlation function of the light emitted by a collection of two- and three-level atom systems 关24兴. Such correlations among the spontaneously emitted photons can also be created in a single-particle system undergoing a two-photon emission process, for example, a three-level cascade system 关25兴. Thus, the question arises, whether also single-particle position measurements can benefit from cor-

043814-1

©2009 The American Physical Society


QAMAR, EVERS, AND ZUBAIRY

FIG. 1. 共Color online兲 Level scheme of the M-type configuration. 共a兲 Bare-state representation. 共b兲 Dressed-state representation.

relations between photons emitted from the particle. Motivated by this, here, we investigate subwavelength position measurement via spontaneous emission spectroscopy based on a two-photon emission process. Our analysis is based on two different level schemes. First, we discuss the five-level M-type setup, where two photons are generated in a double-Raman process. Second, we discuss the three-level ⌳-type setup with two subsequent interactions, such that again two photons are emitted. The crucial difference between the two systems is that the photons emitted in the M-type system are correlated, whereas the photons emitted in the ⌳-type setup are not. In order to reveal the effects of the correlation, we first analyze the spontaneous emission spectra of the two systems, and find characteristic differences due to the photon correlations. In the main part, we analyze the microscopy power of the two schemes. To allow for an unbiased comparison of the two schemes to each other as well as to other setups suggested in the literature, we propose a fixed analysis procedure inspired by a possible experimental realization. It is based on the most probable detection events rather than on the most favorable events for microscopy in order to estimate the practical relevance of the scheme. As our main result, we show that due to the twophoton measurement, both systems allow for an efficient unambiguous position measurement with subwavelength precision. But somewhat intriguing, we find that the overall microscopy is not improved due to the photon correlation. A closer analysis reveals that this result can be understood from the properties of the assumed analysis procedure.

quency on transition 兩c典 ↔ 兩a典关兩b典 ↔ 兩d典兴 with dipole moment ជ ca关␮ ជ bd兴. The corresponding unit polarization vectors and ␮ amplitudes of the driving fields are denoted by ⑀ˆ 1关⑀ˆ 2兴 and E1关E2兴. The atom-vacuum field detuning corresponding to the spontaneous decay transition 兩a典 ↔ 兩b典关兩d典 ↔ 兩e典兴 is denoted by ␦k = ␯k − ␻ab关␦q = ␯q − ␻de兴, where ␻ab and ␻de are the corresponding atomic transition frequencies, and ␯k and ␯q are the frequencies of the emitted photons. We use the probability amplitude method to describe the dynamics of the system and the state vector for the complete system at time t is given by 兩⌿共t兲典 = Cc共t兲兩c,0k,0q典 + Ca共t兲兩a,0k,0q典 + 兺关Ckb 共t兲兩b,1k,0q典 k

+ Ckd 共t兲兩d,1k,0q典兴 + 兺 Ck,q e 共t兲兩e,1k,1q典,

共2兲

k,q

where the subindex i in the probability amplitude Ck,q i 共t兲 indicates the atomic state 共i = a , b , c , d , e兲 and k , q denote photons emitted on the two spontaneous emission transitions. B. Dressed-state model

In order to gain more insight into the system dynamics, we introduce the dressed atomic eigenstates 兩1典 =

冑2 共兩a典

1

+ 兩c典兲,

兩2典 =

冑2 共兩c典

1

− 兩a典兲,

共3a兲

兩3典 =

冑2 共兩d典

1

+ 兩b典兲,

兩4典 =

冑2 共兩b典

1

− 兩d典兲.

共3b兲

with corresponding eigenvalues ␭1 = ⍀1, ␭2 = −⍀1, ␭3 = ⍀2, and ␭4 = −⍀2, respectively. Now using the dressed states, we can write the interaction picture Hamiltonian as V共t兲 =

ប 兺 gkak共ei共⍀1−⍀2兲t兩1典具3兩 + ei共⍀1+⍀2兲t兩1典具4兩 2 k − e−i共⍀1+⍀2兲t兩2典具3兩 − ei共⍀2−⍀1兲t兩2典具4兩兲ei␦kt +

ប

gqaq共兩3典具e兩e 冑2 兺 q

i⍀2t

− 兩4典具e兩e−i⍀2t兲ei␦qt + H.c. 共4兲

II. SYSTEM HAMILTONIAN AND DYNAMICS A. Bare-state model

The schematic of the M-level atom-field system is shown in Fig. 1共a兲. The transitions 兩a典 ↔ 兩c典 and 兩b典 ↔ 兩d典 are driven on resonance by classical driving fields. The atomic decays take place from 兩a典 to 兩b典 and 兩d典 to 兩e典 with corresponding decay rates ␥1 and ␥2. The interaction picture Hamiltonian in the dipole and rotating-wave approximation can be written as

兩⌿共t兲典 = C1共t兲兩1,0,0典 + C2共t兲兩2,0,0典 + 兺关Ck3 共t兲兩3,1k,0典 k

+ Ck4 共t兲兩4,1k,0典兴 + 兺 Ck,q e 共t兲兩e,1k,1q典.

共5兲

k,q

From the Schrödinger equation iប⳵t兩⌿共t兲典 = V共t兲兩⌿共t兲典 we obtain the following equations of motion for the probability amplitudes:

HI共t兲 = ប⍀1兩c典具a兩 + ប⍀2兩b典具d兩 + ប 兺 gkei␦ktak兩a典具b兩 k

+ ប 兺 gqei␦qtaq兩d典具e兩 + H.c.,

The corresponding schematics of the dressed-state picture is shown in Fig. 1共b兲. The state vector 兩⌿共t兲典 in the dressedstate picture takes the form

共1兲

q

ជ ca⑀ˆ 1E1 / ប关⍀2 = −␮ ជ bd⑀ˆ 2E2 / ប兴 is the Rabi frewhere ⍀1 = −␮ 043814-2

˙ 共t兲 = 1 兺 g ei共␦k+⍀1兲t关Ck共t兲e−i⍀2t + Ck共t兲ei⍀2t兴, 共6a兲 C 1 k 3 4 2 k


ATOM MICROSCOPY VIA TWO-PHOTON SPONTANEOUS…

˙ 共t兲 = − 1 兺 g ei共␦k−⍀1兲t关Ck共t兲e−i⍀2t + Ck共t兲ei⍀2t兴, C 2 k 3 4 2 k

S共␯q兲 ⬀ 共6b兲

1 C˙k3 共t兲 = gkⴱ e−i共␦k−⍀2兲t关e−i⍀1tC1共t兲 − ei⍀1tC2共t兲兴 2 +

1

gqei共␦ +⍀ 兲tCk,q e 共t兲, 冑2 兺 q q

2

1

g qe 冑2 兺 q

i共␦q−⍀2兲t

Ck,q e 共t兲,

˙ k,q共t兲 = 1 gⴱ e−i␦qt关e−i⍀2tCk共t兲 − Ck共t兲ei⍀2t兴. C 3 4 e 冑2 q

The total fluorescence spectrum in this simple picture is then the sum of the two contributions, but in general also depends on the respective photon polarizations.

In this section, we calculate the filter function to find the conditional position probability distribution. For this, we assume that the two driving fields are standing-wave fields and therefore, the corresponding Rabi frequencies are position dependent, ⍀i共x兲 = 兩⍀i兩sin共kix + ␾i兲.

共6e兲

Here, ki = 2␲ / ␭i are the wave vectors corresponding to the driving field wavelengths ␭i, and ␾i are the phase shifts of the standing waves 共i 苸兵1 , 2其兲. Then, the Hamiltonian Eq. 共1兲 and the probability amplitudes become position dependent as well, such that the state vector can be written as

共7兲

k,q

兩⌿共t兲典 =

The final-state probability amplitude in the steady state follows from Eqs. 共6兲 as gkⴱ gqⴱ ⍀1⍀2 Ck,q , 共⬁兲 = 4 e D1共␦k + ␦q兲D2共␦q兲

D 1共 ␦ 兲 = i ␦ ␥ 1 +

共9a兲

D2共␦兲 = i␦␥2 + 2⍀22 − 2␦2 .

共9b兲

Thus the probability to measure one photon in mode k and one photon in mode q is proportional to 16兩gk兩2兩gq兩2兩⍀1兩2兩⍀2兩2 2 . 兩Ck,q e 共⬁兲兩 = 兩D1共␦k + ␦q兲兩2兩D2共␦q兲兩2

再

dxf共x兲兩x典 C1共x;t兲兩1,0k,0q典 + C2共x;t兲兩2,0k,0q典

冎

k

+ 兺 Ck,q e 共x;t兲兩e,1k,1q典 , k,q

− 2␦ , 2

冕

The spontaneous emission spectrum S共␯兲 is proportional to the Fourier transform of the field correlation function 关25兴具⌿共t兲兩E共−兲共r,t + ␶兲E共+兲共r,t兲兩⌿共t兲典t→⬁ .

共11兲

The spectrum decomposes into two parts related to the k photons and the q photons emitted on the two respective transitions. The single-photon spectra can be obtained from the full final-state probability by tracing out one of the photons, i.e., integrating either ␦k or ␦q over all possible frequencies. Using Eqs. 共7兲, 共8兲, 共9a兲, 共9b兲, 共10兲, and 共11兲, we find S共␯k兲 ⬀

冕

2 兩Ck,q e 共⬁兲兩 d␯q ,

共12a兲

共15兲

with the filter function 2 F共x;t兩e,1k,1q兲 ⬅ 兩N兩2兩Ck,q e 共x;t兲兩 ,

C. Fluorescence spectrum

共14兲

where f共x兲 be the center-of-mass wave function of the atom. This allows us to calculate the conditional position distribution of the atom, which is the probability W共x ; t 兩 e , 1k , 1q兲 of finding the atom at position x under the condition that the atom is in state 兩e典 and that two photons are emitted spontaneously in modes k and q 关8兴. For the M-level configuration, we obtain W共x;t兩e,1k,1q兲 = F共x;t兩e,1k,1q兲兩f共x兲兩2 ,

共10兲

共13兲

+ 兺关Ck3 共x;t兲兩3,1k,0q典 + Ck4 共x;t兲兩4,1k,0q典兴

共8兲

where we have defined 2⍀21

共12b兲

共6d兲

In the long-time limit, 兩⌿共t → ⬁兲典 = 兺 Ck,q e 共⬁兲兩e,1k,1q典.

2 兩Ck,q e 共⬁兲兩 d␯k .

D. Filter function

共6c兲

˙ k共t兲 = 1 gⴱ e−i共␦k+⍀2兲t关e−i⍀1tC 共t兲 − ei⍀1tC 共t兲兴 C 1 2 4 2 k −

冕

共16兲

where N is a normalization constant. This shows that in the long-time limit, the conditional position distribution is di2 rectly proportional to the probability 兩Ck,q e 共⬁兲兩 in Eq. 共10兲 with position-dependent Rabi frequencies defined in Eq. 共13兲. E. ⌳-type system

In order to estimate the influence of potential correlations in the M-type system, we make use of a three-level reference system in ⌳ configuration 共see Fig. 2兲. Initially, the atom is in state 兩c典. Due to the driving field ⍀1, it undergoes a Raman transition to 兩b典, emitting a photon with frequency ␦k. The final steady-state amplitude for this process is

043814-3



FIG. 2. 共Color online兲 Level scheme of the ⌳-type configuration. The two subpanels indicate the two steps in the dual interaction. 共a兲 First interaction. The initial atomic state is 兩c典, the final state is 兩b典. 共b兲 Second interaction. The initial atomic state is 兩b典, the final state is 兩c典.

k C⌳ 共⬁兲 = 2

gkⴱ ⍀1 . D 1共 ␦ k兲

共17兲

In a second step, we reverse role of driving and emitting transition, drive the atom with Rabi frequency ⍀2 on transition 兩b典 ↔ 兩a典, and obtain a photon with frequency ␦q, with final steady-state amplitude q C⌳ 共⬁兲 = 2

gqⴱ ⍀2 . D 2共 ␦ q兲

共18兲

The two photons are uncorrelated, and the total probability for the emission of a pair of photons with frequencies k and q in the ⌳ system is q k 共⬁兲兩2兩C⌳ 共⬁兲兩2 = 兩C⌳

16兩gk兩2兩gq兩2兩⍀1兩2兩⍀2兩2 . 兩D1共␦k兲兩2兩D2共␦q兲兩2

共19兲

Comparing with the corresponding result for the M-type system 关Eq. 共10兲兴, we find that the only difference is in the argument of the function D1 in the denominator. For the M system, the two-photon frequencies appear as a sum, indicating their correlation as in the case of the ladder-type system 关25兴. In contrast, the two photons from the ⌳ system are uncorrelated as expected. For the ⌳-type system, the individual spectra of each of the two photons emitted are again proportional to the finalk q 共⬁兲兩2 or 兩C⌳ 共⬁兲兩2, respectively. The filter state probability, 兩C⌳ function in the ⌳ system with dual interactions is proportional to q k 兩C⌳ 共⬁兲兩2兩C⌳ 共⬁兲兩2 ,

共20兲

and follows from Eq. 共19兲 after introducing the positiondependent Rabi frequencies Eq. 共13兲.

FIG. 3. 共Color online兲 Single-photon spontaneous emission spectra. Dashed lines show S共␯k兲, and the solid lines stand for S共␯q兲. 共a兲 depicts results for the ⌳-type system with dual interactions, 共b兲 for the M-level system. The parameters are ␥1 = ␥2 = 0.1⌫, 兩⍀1兩 = ⌫, and 兩⍀2兩 = 0.5⌫.

ing laser field. Examples for k and q spectra are shown in Fig. 3共a兲. The dashed line shows the spectrum of the k photon with peak positions ⫾⍀1. The solid line shows the spontaneous emission spectrum of the q photon with peaks at ⫾⍀2. Due to the correlation between the photons, the results look different in the M-type scheme 关see Fig. 3共b兲兴. The spectrum for the k photon 共dashed line兲 has four peaks, while the spectrum for the q photon 共solid line兲 has two peaks only. This qualitatively different result can easily be understood from the dressed-state representation in Fig. 1共b兲. The first photon can be emitted from 兩1典 or 兩2典 to 兩3典 or 兩4典, giving rise to four different possible emission frequencies, which depend on both ⍀1 and ⍀2. The second photon can decay from 兩3典 or 兩4典 to 兩e典, such that two peaks appear. It is interesting to note, however, that not all combinations for the k and q decays are possible. For example, if the k decay is such that the atom evolves into 兩3典, then the q photon can only be emitted from 兩3典 to 兩e典. Therefore, each peak in the q spectrum corresponds to two potential peaks in the k spectrum, clearly demonstrating the correlation between the photons. Mathematically speaking, this correlation follows from the contribution D1共␦k + ␦q兲 in Eq. 共8兲. Physically speaking, constraints on the two-photon frequencies arise from energy conservation and from the two possible intermediate states 兩3典 or 兩4典. Thus, we conclude that the emission spectra of the M-type system are different from the spectra of a ⌳ setup with dual measurement, in that the photons emitted by the M-type system are correlated. The photon correlation makes the k and the q spectra dependent, and leads to a richer structure of the k spectrum. B. Atom microscopy

III. RESULTS A. Spontaneous emission spectrum

We start our analysis with the three-level ⌳ system 共see Fig. 2兲. For our numerical calculations, we introduce a scaling parameter ⌫ by which we scale all parameters of our system. For simplicity, we assume ␥1 = ␥2 in the following. Since the two photons emitted after dual interaction are independent, the spectrum of each of the photons is of AutlerTownes type due to the ac Stark splitting caused by the driv-

In this section, we compare the microscopy power of the M-type system to that of the ⌳ system with dual measurement. Both, to drive the atom, and as the reference for the position measurement, we apply standing-wave fields to each of the two driven transitions. The corresponding positiondependent Rabi frequencies are in Eq. 共13兲. In Sec. II D we found that the potential positions of the atom are given by the filter function Eq. 共16兲. Examples for the filter function of both the M-type and the dual measurement ⌳-type atoms are given in Fig. 4 for identical param-

043814-4



FIG. 4. 共Color online兲 Filter function or steady-state population of the final state as a function of the photon frequencies. 共a兲 Equation 共19兲 for the ⌳ system with dual interaction. 共b兲 Equation 共10兲 for the M-level system. The parameters are as in Fig. 3.

eters. It can be seen that filter function of the M-type system is sheared as compared to the corresponding results of the ⌳-type system. This shear arises from the photon correlation in the M-type system. Thus also the filter function is modified by the presence of the photon correlation. In the following we analyze whether this difference also has consequences for the position measurement. Most previous work focused on the question whether it is in principle possible to localize a particle at a single position or to determine the position of a fixed particle unambiguously. This leaves the question how likely such a favorable measurement is, and thus, how practical the localization scheme is in general. In order to analyze this question, we proceed as follows. First, we choose parameters for both standing-wave fields. This choice has to be made without prior knowledge of the position, and remains constant throughout the whole analysis outlined below. Second, we assume that the particle is located at a fixed position x0 and apply the standing-wave laser fields as chosen in the first step. Third, fixing x0 enables one to calculate the most probable emission frequencies for both photons. In our analysis, we assume that the detectors measure the photons with the most probable frequency combination. Fourth, assuming the most probable frequency combination as the measurement outcome, the conditional position probability distribution can be calculated. Finally, the obtained probability distribution can be analyzed. In the best case, it contains only a single narrow peak exactly at the position x0 assumed in the first step. In this case, an experiment would lead with high probability to a correct measurement of the particle position with laser field parameters chosen as in step one if the atom is at x0. This sequence is repeated for different positions x0 within one wavelength of the driving fields. Using this analysis sequence, the laser field parameters should be optimized such that they lead to single-peak localization for all positions x0. If this can be achieved, then the microscopy setup is powerful in the sense that a fixed measurement scheme without prior knowledge about the particle position enables one to measure the position unambiguously up to the spatial periodicity of the standing wave. Augmented by a classical measurement as also required in previous microscopy schemes based on standing waves, then the true particle position becomes accessible. It is important to note that the above analysis sequence focuses on the most probable detection events. Other detection events can in principle lead to better localization, however, at the cost of detection probability.

In Fig. 5, we show a possible outcome of this analysis. The maximum Rabi frequencies are chosen as 兩⍀1兩 = 兩⍀2兩 = ⌫, and we set k1 = ␩k2 as the ratio of the wave vectors with ␩ = 1.2. In subpanel 共a兲, the phases are chosen as ␾1 = 0 and ␾2 = ␲ / 4. We observe nearly unique position determination of the single atom over a wide range of positions of the atom. Notable exceptions are x0 = ⫾ 0.25␭, where there are two peaks in the conditional position probability. The reason for this is a symmetry in the driving fields ⍀i共x兲 with respect to the position x. Due to this symmetry, the most probable photon emission frequencies are the same for x0 = 0.25␭ and x0 = −0.25␭, and therefore, the detection of these frequencies also does not allow to distinguish between the two symmetric positions x0 = ⫾ 0.25␭. Thus, there are two comparable peaks in the conditional position probability. Also, no measurement is possible if the atom is at one of the nodes of the driving fields. This occurs, for example, at positions x0 = 0 and x0 = ⫾ ␭ / 2, and therefore these positions are left out in Fig. 5. If a measurement with the parameters in Fig. 5共a兲 leads to a conditional position probability which does not predict a unique position, then it is possible to improve the measurement by switching to a second set of laser field parameters as shown in Fig. 5共b兲. For these parameters, unique position information is obtained at positions complementary to the unique cases in Fig. 5共a兲. In particular, close to singlepeak localization is achieved for x0 = ⫾ 0.25␭. It should be noted that only the phases of the standing-wave fields were changed, i.e., their positions in space. This leads to different node positions of the two standing-wave fields, such that now also the positions x0 = 0 and x0 = ⫾ ␭ / 2 can be predicted with high probability. The corresponding conditional position probabilities are shown in Fig. 6 for parameters as in Fig. 5共b兲. Thus we conclude that a suitable sequence of measurements with at most two sets of laser field parameters in the M-level scheme gives unique position information within one wavelength with high probability. Repeating the same analysis with the dual measurement in ⌳-type atoms, we obtain virtually identical results as in the M-type system. This at first may seem surprising since the involved emission spectra for the M-type and the ⌳-type systems are different. But a closer analysis reveals that by choosing the emission frequencies with maximum probability as outlined in our measurement scheme, this difference vanishes from the final conditional position probability. Therefore, we conclude that using the above outlined simple measurement scheme motivated by a potential experimental procedure, photon correlation do not improve the results of the position determination. Rather, the single-peak localization arises from the dual measurement inherent to both schemes. The two measurements provide independent position information, which in combination allows obtaining good position estimates. IV. DISCUSSION AND SUMMARY

In conclusion, we analyzed two-photon measurement of the position of a quantum particle both in the ⌳-type system with dual measurement and in the M-type system. In order to estimate the microscopy power of our schemes, and to en-

043814-5



FIG. 5. 共Color online兲 Conditional position probability distribution for the M-type configuration. The different subpanels show results for various positions x0 of the particle within the standing wave as discussed in Sec. III B. The standing-wave phases are chosen as 共a兲 ␾1 = 0, ␾2 = ␲ / 4 and 共b兲 ␾1 = ␲ / 4, ␾2 = ␲ / 4. The other parameters are 兩⍀1兩 = 兩⍀2兩 = ⌫, ␥1 = ␥2 = 0.1⌫, and ␩ = 1.2.

able quantitative comparison to other schemes, we introduced a strict analysis procedure. It is based on fixed laser parameters, such that no prior knowledge about the position is assumed. In contrast to previous work, it assumes detection of the most probable emission events, rather than the most favorable, but potentially unlikely, detection events. As the main result, we could show that both systems enable one to unambiguously determine the position of the particle with high precision and high detection probability. Regarding the photon correlation, we showed that the spontaneous emission spectra for the individual photons as well as the filter function are affected by the correlations to a measurable degree. Nevertheless, these differences do not

appear in the localization following the fixed analysis scheme. This difference is somewhat intriguing, but can be understood from the fact that our analysis scheme makes use of a maximization of the photon emission probability, such that the individual emission frequencies itself do not matter. Still, the M-type system may have advantages over the ⌳-type scheme, depending on the candidate system. For example, in a practical implementation the measurement is likely to be more rapid in the M-type system. There is always the possibility that the detector misses an emitted photon. In the ⌳-type scheme, one therefore has to wait rather long on a scale given by the lifetime of the excited states until a photon in mode k is emitted. Only then one can conclude that the

043814-6



FIG. 6. 共Color online兲 Conditional position probability distribution for the M-type configuration. The parameters are as in Fig. 5共b兲. The figure shows results for particle positions that are not accessible with the standing-wave configuration in Fig. 5共a兲.

photon was missed, and proceed with the second photon. In the M-type system, both laser fields are applied at the same time, such that even with a missed first or second photon the waiting time can be reduced. The reduced interaction time in particular becomes important if the localization of flying particles is considered. However, single-peak localization is likely to violate the Raman-Nath assumption such that the momentum distribution of the flying particle will be perturbed. Also, the photon correlation leads to different emission frequencies in the M-type system. Only certain combinations of frequencies for the two photons are possible, such that a consistency check of the two frequencies allows eliminating unwanted measurement events.

We expect that more advanced detection schemes could make use of the photon correlation. Such schemes could be based on higher-order correlation functions or on ideas similar to the two-photon spectroscopy on pairs of atoms 关22兴.

We thank COMSTECH for their support. The research of M.S.Z. was supported by a grant from Qatar National Research Fund 共QNRF兲. He is also grateful for support from Alexander von Humboldt Foundation.

关1兴 M. Born and E. Wolf, Principles of Optics 共Cambridge University Press, Cambridge, England, 1999兲. 关2兴 A. Lewis, M. Isaacson, A. Harootunian, and A. Muray, Ultramicroscopy 13, 227 共1984兲; A. Lewis, H. Taha, A. Strinkovski, A. Manevitch, A. Khatchatouriants, R. Dekhter, and E. Ammann, Nat. Biotechnol. 21, 1378 共2003兲. 关3兴 S. Hell and E. H. K. Stelzer, J. Opt. Soc. Am. A 9, 2159 共1992兲; S. W. Hell and J. Wichmann, Opt. Lett. 19, 780 共1994兲; T. A. Klar, S. Jakobs, M. Dyba, A. Egner, and S. W. Hell, Proc. Natl. Acad. Sci. U.S.A. 97, 8206 共2000兲. 关4兴 E. Betzig, Opt. Lett. 20, 237 共1995兲. 关5兴 P. Storey, M. Collett, and D. F. Walls, Phys. Rev. Lett. 68, 472 共1992兲; Phys. Rev. A 47, 405 共1993兲; R. Quadt, M. Collett, and D. F. Walls, Phys. Rev. Lett. 74, 351 共1995兲. 关6兴 S. Kunze, K. Dieckmann, and G. Rempe, Phys. Rev. Lett. 78, 2038 共1997兲. 关7兴 F. Le Kien, G. Rempe, W. P. Schleich, and M. S. Zubairy, Phys. Rev. A 56, 2972 共1997兲. 关8兴 A. M. Herkommer, W. P. Schleich, and M. S. Zubairy, J. Mod. Opt. 44, 2507 共1997兲; To determine center-of-mass atomic wave function using Autler-Townes spectroscopy, see K. T. Kapale, S. Qamar, and M. S. Zubairy, Phys. Rev. A 67, 023805 共2003兲. 关9兴 H. Nha, J.-H. Lee, J.-S. Chang, and K. An, Phys. Rev. A 65, 033827 共2002兲. 关10兴 S. Qamar, S.-Y. Zhu, and M. S. Zubairy, Opt. Commun. 176, 409 共2000兲. 关11兴 S. Qamar, S.-Y. Zhu, and M. S. Zubairy, Phys. Rev. A 61, 063806 共2000兲. 关12兴 E. Paspalakis and P. L. Knight, Phys. Rev. A 63, 065802

共2001兲. 关13兴 M. Sahrai, H. Tajalli, K. T. Kapale, and M. S. Zubairy, Phys. Rev. A 72, 013820 共2005兲; K. T. Kapale and M. S. Zubairy, ibid. 73, 023813 共2006兲. 关14兴 G. S. Agarwal and K. T. Kapale, J. Phys. B 39, 3437 共2006兲. 关15兴 C. Liu, S. Gong, D. Cheng, X. Fan, and Z. Xu, Phys. Rev. A 73, 025801 共2006兲. 关16兴 J. Evers, S. Qamar, and M. S. Zubairy, Phys. Rev. A 75, 053809 共2007兲. 关17兴 S. Bretschneider, C. Eggeling, and S. W. Hell, Phys. Rev. Lett. 98, 218103 共2007兲. 关18兴 D. D. Yavuz and N. A. Proite, Phys. Rev. A 76, 041802共R兲共2007兲. 关19兴 A. V. Gorshkov, L. Jiang, M. Greiner, P. Zoller, and M. D. Lukin, Phys. Rev. Lett. 100, 093005 共2008兲. 关20兴 M. Kiffner, J. Evers, and M. S. Zubairy, Phys. Rev. Lett. 100, 073602 共2008兲. 关21兴 H. Li, V. A. Sautenkov, M. M. Kash, A. V. Sokolov, G. R. Welch, Y. V. Rostovtsev, M. S. Zubairy, and M. O. Scully, Phys. Rev. A 78, 013803 共2008兲. 关22兴 A. Muthukrishnan, M. O. Scully, and M. S. Zubairy, J. Opt. B: Quantum Semiclassical Opt. 6, S575 共2004兲; M. O. Scully and C. H. Raymond Ooi, ibid. 6, S816 共2004兲. 关23兴 J.-T. Chang, J. Evers, M. O. Scully, and M. S. Zubairy, Phys. Rev. A 73, 031803共R兲共2006兲. 关24兴 J.-T. Chang, J. Evers, and M. S. Zubairy, Phys. Rev. A 74, 043820 共2006兲. 关25兴 M. O. Scully and M. S. Zubairy, Quantum Optics 共Cambridge University Press, Cambridge, England, 1997兲.

ACKNOWLEDGMENTS

043814-7