High-precision atom localization via controllable spontaneous emission in a cycle-configuration atomic system

Chunling Ding; Jiahua Li; Rong Yu; Xiangying Hao; Ying Wu

doi:10.1364/OE.20.007870

1. Introduction

In the quantum optics and quantum mechanics, studies of the precision position measurement of an atom are mainly motivated by the idea that it has many applications in laser cooling and neutral atoms trapping [1], Bose-Einstein condensation [2, 3], atom lithography [4, 5], the measurement of center-of-mass wave function of moving atoms [6], and so on. Considerable progress has been made in establishing precision position of an atom [7–11] both from theoretical and experimental points of view. For example, Paspalakis and Knight [12] proposed a scheme for the localization of an atom using the measurement of the population in the upper level. Zubairy and colleagues [13–16] have discussed atom localization via resonance fluorescence or phase and amplitude control of the absorption spectrum. Nha et al. [17] have studied the localization of atomic position via dual measurement when a three-level atom interacts with a quantized standing-wave field in the Ramsey interferometer setup. Qamar and coworkers [18] have presented a scheme of atom localization in a subwavelength domain via manipulation of Raman gain process. Also, controllable atom localization can be obtained via dark resonances [19, 20] or via coherent population trapping (CPT) [21]. Moreover, Thomas and coworkers have proposed and experimentally demonstrated subwavelength position localization of atoms using spatially varying energy shifts [22–24] in the early years. Recently, atomic localization using the technique of electromagnetically induced transparency (EIT) has also been experimentally observed in [25]. These schemes for realizing one-dimensional (1D) atom localization are mainly based on atomic coherence and quantum interference effects. Besides, quantum coherence and interference have led to the observation of many useful effects and techniques in atomic physics and quantum optics, including quantum multiphoton and quantum information processes [26–29], giant Kerr nonlinearities [30–32], hyper-Raman scattering [33, 34], four-wave mixing and EIT [35–38], control of spontaneous emission [39–41], as well as two-dimensional (2D) atom localization [42–46].

In addition to the above mentioned 1D atom localization, the 2D spatial localization behavior of an atom has also been developed in recent years. Because the 2D atom localization has unique properties and extensive applications, more and more attention and interest have been devoted to the study of the accurate measurement of the atomic position in 2D space. Evers et al. [42] have discussed the 2D localization of a quantum particle using multiple simultaneous quadrature measurement when the particle flying through the cavity field intersection area. Ivanov and Rozhdestvensky [43] have proposed a scheme for 2D subwavelength localization of a four-level tripod-type atom in laser fields and have found that the localization factors depend crucially on the atom-field coupling that results in such spatial structures of populations as spikes, craters, and waves. More recently, high-precision and high-resolution 2D atom localization has been demonstrated in Ref. [44] via controlled spontaneous emission from a driven tripod system. They also proposed other method to realize such 2D atom localization based on quantum interference in an inverted-Y atomic system driven by two orthogonal standing-wave laser fields [45]. Another related localization scheme based on double-dark resonances has been reported by Wan and coworkers [46] for a four-level N-type atom interacting with two orthogonal standing-wave fields. However, it should be noted that the maximum probability of finding the atom at a particular position is 1/2. It reminds us of another question: can we localize the atom at an expected position with a probability greater than the previous schemes [42–46]?

In order to address this question, we put forward a scheme for realizing 2D atom localization in a microwave-driven four-level atomic system with a closed-loop configuration. The results show that we can increase the probability of finding the atom at a particular position by employing the microwave-driven field when the two standing-wave fields drive simultaneously the same atomic transition. The new aspect of the present paper is the introduction of an external microwave field with respect to the model proposed by Wan et al. [44], which drives a hyperfine transition between the two ground-state hyperfine levels. Of particular interest is the application of a microwave-driven field, because the microwave source is easier to obtain and manipulate than other extra laser fields, the microwave field plays a crucial role in determining the position of the atom localization, and this is the situation considered in the context. Our motivation is to explore whether or not new localization phenomena arise when the microwave field is applied to drive the ground-state hyperfine transition. Two important results are found: first, when the two standing-wave fields are respectively used to drive the different atomic transitions, the maximum probability of finding the atom in one period is 50%; second, for our considered model, we find that the atom can be localized at a certain position with a probability of 100% when the two standing-wave fields couple the same atomic transition. This is the problem extensively explored. Additionally, our work has the following features: the interaction model considered here are most fundamental in the theoretical studies of dynamic behavior of the atom, and we do not need to assume any specific conditions for the structured environment. Thus the results of our study possess good adaptability. Moreover, our deductions are completely analytical and hence the physical explanation of the results is more transparent. These investigations have potential applications in laser cooling, Bose-Einstein condensation, and trapping of neutral atoms, etc.

The article is organized as follows. In Section 2, we present the physical model and its theoretical description, and then we derive an analytical expression of the conditional position probability distribution for the system in the process of atom-field interaction. In Section 3, we give a detailed analysis and explanation for the behavior of 2D atom localization. Finally, the main conclusions are presented in Section 4.

2. Theoretical model and basic formula

Let us start by considering a microwave-driven four-level atomic system, which consists of one excited level |3〉, and three ground levels |0〉, |1〉, and |2〉 as depicted in Fig. 1. The transition from the excited level |3〉 to the ground level |0〉 is coupled by the vacuum modes in the free space. An external microwave-driven field with a Larmor frequency 2Ω_m is used to resonantly couple the two hyperfine levels |1〉 and |2〉 through an allowed magnetic dipole transition. The excited level |3〉 is simultaneously coupled to the ground levels |1〉 and |2〉 by two coherent laser fields with Rabi frequencies G₁(x, y) and G₂(x, y), respectively. Here, we consider two cases for the atom interacts with the standing-wave laser fields. The first case is that G₁(x, y) and G₂(x, y) correspond respectively to the two orthogonal standing-wave fields that couple the different atomic transitions, i.e., G₁(x, y) = Ω₁ sin(k₁x) and G₂(x, y) = Ω₂ sin(k₂y) with k₁ = ω₁/c and k₂ = ω₂/c being the wave vectors of the two laser fields. The second case is that G₁(x, y) corresponds to the combination of two orthogonal standing-wave fields with the same frequency that drive simultaneously the transition |1〉 ↔ |3〉, while G₂(x, y) corresponds to a traveling-wave field, that is, G₁(x, y) = Ω₁[sin(k₁x)+ sin(k₁y)] and G₂(x, y) = Ω₂. An atom moves along the z direction and passes through the intersectant region of the two orthogonal standing-wave fields in the x–y plane. As a result, the interaction between the atom and the standing-wave fields is spatial dependent on the x–y plane. Here we assume that the center-of-mass position of the atom along the directions of the standing-wave fields is nearly constant and we can neglect the kinetic part of the atom in the Hamiltonian by applying the Raman-Nath approximation [47]. Under these conditions, the resulting interaction Hamiltonian which describes the dynamics of this system in the rotating-wave approximation (RWA) and the electric dipole approximation (EDA) can be written in the following form (taking h̄ = 1)

\begin{array}{l} H_{I} & = & G_{1} (x, y) e^{- Δ_{1} t} | 3 〉 〈 1 | + G_{2} (x, y) e^{- i Δ_{2} t} | 3 〉 〈 2 | + Ω_{m} e^{- i (Δ_{1} - Δ_{2}) t} | 2 〉 〈 1 | \\ + \sum_{k} g_{k} e^{- i δ_{k} t} | 3 〉 〈 0 | {\hat{b}}_{k} + H.c., \end{array}

where the quantities Δ₁ = ω₁ – E₃₁/h̄ and Δ₂ = ω₂ – E₃₂/h̄ stand for the frequency detunings of the coherent laser fields from the corresponding atomic resonance frequencies. Here Ω_m is one-half Larmor frequency for the relevant driven transition, i.e., Ω_m = μ₁₂B_m/(2h̄), with B_m being the amplitude of the microwave-driven field and μ₁₂ = μ⃗₁₂ ·e⃗_L (e⃗_L is the unit polarization vector of the corresponding laser field) denoting the dipole matrix element for the transition |1〉 ↔ |2〉. b̂_k and

{\hat{b}}_{k}^{†}

are interpreted as the annihilation and creation operators, respectively, corresponding to the kth vacuum mode with frequency ω_k. The coefficient g_k represents the coupling between the vacuum mode k and the atomic transition |3〉 ↔ |0〉, δ_k = ω_k – ω₃₀ is the corresponding frequency detuning. In the following calculations, we set Ω₁ and Ω₂ as real parameters, while Ω_m as a complex parameter, i.e., Ω_m = |Ω_m|e^iφ, here φ is the phase of the microwave-driven field and can also be called the relative phase. It is remarkable that there exist two possible transition pathways from level |1〉 to level |3〉, i.e., the direct one

| 1 〉 \overset{Ω_{1}}{\to} | 3 〉

and the indirect one

| 1 〉 \overset{Ω_{m}}{\to} | 2 〉 \overset{Ω_{2}}{\to} | 3 〉

, as can be seen from the atomic energy-level structure in Fig. 1. The influence of the relative phase φ on the spontaneous emission spectra in such a four-level atomic system with a closed-loop structure can be explained from quantum interference caused by these two excitation decay channels. As a consequence, we can investigate the behavior of 2D atom localization by modulating the relative phase φ, which can also be discussed in the following section.

Fig. 1 Schematic diagram of a four-level atomic system, which consists of one excited level |3〉, three ground levels |0〉, |1〉, and |2〉. The transitions $| 1 〉 \overset{G_{1} (x, y)}{\leftrightarrow} | 3 〉 \overset{G_{2} (x, y)}{\leftrightarrow} | 2 〉 \overset{Ω_{m}}{\leftrightarrow} | 1 〉$ form a cyclic configuration, in which G₁(x, y) is a standing-wave field or a composition of two orthogonal standing waves, G₂(x, y) is a standing-wave or traveling-wave field, and Ω_m is one-half Larmor frequency for the relevant transition. Δ₁ and Δ₂ are the frequency detunings of the corresponding standing-wave or traveling-wave fields. And the transition |3〉 ↔ |0〉 is coupled to vacuum modes in the free space.

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The dynamics of this system can be described by using the probability amplitude equations. Then the wave function of our considered system at time t can be expressed as

\begin{array}{l} | Ψ (t) 〉 & = & \iint d x d y f (x, y) | x 〉 | y 〉 [A_{1, 0_{k}} (x, y; t) | 1, 0_{k} 〉 + A_{2, 0_{k}} (x, y; t) | 2, 0_{k} 〉 \\ + A_{3, 0_{k}} (x, y; t) | 3, 0_{k} 〉 + \sum_{k} A_{0, 1_{k}} (x, y; t) | 0, 1_{k} 〉], \end{array}

where A_{j,0_k} (x,y;t) ( j = 1 – 3) and A_{0,1_k} (x,y;t) give the probability amplitude to find the atom at time t. |j,0_k〉 denotes the atom in the level |j〉 with no photons present and |0,1_k〉 represents the atom in its ground level |0〉 with a single photon in the kth vacuum mode. Besides, f(x,y) is the center-of-mass wave function of the atom.

It should be explicitly pointed out that the 2D atom localization scheme in our system relies on the fact that the spontaneously emitted photon carries information about the position of atom in the x–y plane due to the spatial position-dependent interaction between atom and standing-wave fields. Therefore, the location of the atom can be determined by measuring the frequency of the spontaneously emitted photon. When we have detected a spontaneously emitted photon at time t in the vacuum mode of wave vector k, the atom is in its internal level |0〉 and the state vector of the system, by employing the following transformation over Ψ(t), is transformed into

| ψ_{0, 1_{k}} 〉 = 𝒩 〈 0, 1_{k} | Ψ (t) 〉 = 𝒩 \iint d x d y f (x, y) | x 〉 | y 〉 A_{0, 1_{k}} (x, y; t),

where 𝒩 is a normalization factor. Hence, the conditional position probability distribution, that is, the probability of finding the atom in the (x,y) position at time t is

P (x, y; t | 0, 1_{k}) = {| 𝒩 |}^{2} {| 〈 x | 〈 y | ψ_{0, 1_{k}} 〉 |}^{2} = {| 𝒩 |}^{2} {| f (x, y) |}^{2} {| A_{0, 1_{k}} (x, y; t) |}^{2},

which can be reduced to determine the probability amplitude A_{0,1_k} (x,y;t).

We now deduce an analytical expression for the probability amplitude A_{0,1_k} by substituting the interaction Hamiltonian [Eq. (1)] and the atomic wave function of our system [Eq. (2)] into the time-dependent Schrödinger wave equation i∂|Ψ(t)〉/∂t = H_I|Ψ(t)〉, and we can obtain the coupled equations of motion for the time evolution of the atomic probability amplitudes

i \frac{\partial A_{1, 0_{k}} (t)}{\partial t} = G_{1} (x, y) e^{i Δ_{1} t} A_{3, 0_{k}} (t) + Ω_{m}^{*} e^{i (Δ_{1} - Δ_{2}) t} A_{2, 0_{k}} (t),

i \frac{\partial A_{2, 0_{k}} (t)}{\partial t} = G_{2} (x, y) e^{i Δ_{2} t} A_{3, 0_{k}} (t) + Ω_{m} e^{- i (Δ_{1} - Δ_{2}) t} A_{1, 0_{k}} (t),

i \frac{\partial A_{3, 0_{k}} (t)}{\partial t} = G_{1} (x, y) e^{- i Δ_{1} t} A_{1, 0_{k}} (t) + G_{2} (x, y) e^{- i Δ_{2} t} A_{2, 0_{k}} (t) - i \frac{Γ_{0}}{2} A_{3, 0_{k}} (t),

i \frac{\partial A_{0, 1_{k}} (t)}{\partial t} = g_{k}^{*} e^{i δ_{k} t} A_{3, 0_{k}} (t),

where Γ₀ = 2π|g_k|²D(ω_k) is the spontaneous decay rate from level |3〉 to level |0〉, with D(ω_k) being the density of states (DOS) at frequency ω_k in the free space. Our calculations show that the decays of the excited level |3〉 to the ground levels |1〉 and |2〉 cannot affect the probability of finding the atom in the intersectant region of the standing-wave fields, but only slightly reduce the spatial resolution of the atom localization, and hence the two decay rates can be neglected here.

Making use of the Laplace transform method and the final value theorem, the probability amplitude A_{0,1_k} in the long time limit can be obtained as

A_{0, 1_{k}} (x, y; t \to \infty) = - i \int_{0}^{\infty} g_{k}^{*} e^{i δ_{k} t^{'}} A_{3, 0_{k}} (t^{'}) d t^{'} = - i g_{k}^{*} {\tilde{A}}_{3, 0_{k}} (s = - i δ_{k}),

here Ã_{3,0_k} (s) is the Laplace transform of A_{3,0_k} (t) with s = −iδ_k.

Next, after carrying out the Laplace transformations for Eqs. (5)–(7), we get the solution to the probability amplitude Ã_{3,0_k} (s) as

{\tilde{A}}_{3, 0_{k}} (s = - i δ_{k}) = \frac{C}{- i D},

where

\begin{array}{l} C & = & [{| Ω_{m} |}^{2} - (δ_{k} - Δ_{1}) (δ_{k} - Δ_{2})] A_{3, 0_{k}} (0) - [(δ_{k} - Δ_{2}) G_{1} (x, y) + Ω_{m} G_{2} (x, y)] A_{1, 0_{k}} (0) \\ - [(δ_{k} - Δ_{1}) G_{2} (x, y) + Ω_{m}^{*} G_{1} (x, y)] A_{2, 0_{k}} (0), \end{array}

\begin{array}{l} D & = & (δ_{k} + i \frac{Γ_{0}}{2}) [{| Ω_{m} |}^{2} - (δ_{k} - Δ_{1}) (δ_{k} - Δ_{2})] + (δ_{k} - Δ_{2}) G_{1}^{2} (x, y) + (δ_{k} - Δ_{1}) G_{2}^{2} (x, y) \\ + (Ω_{m} + Ω_{m}^{*}) G_{1} (x, y) G_{2} (x, y) . \end{array}

Finally, the conditional probability of finding the atom in its internal level |0〉 with a spontaneously emitted photon of frequency ω_k in the vacuum mode k is then given by

\begin{array}{l} P (x, y; t \to \infty | 0, 1_{k}) & = & {| 𝒩 |}^{2} {| f (x, y) |}^{2} {| A_{0, 1_{k}} (x, y; t \to \infty) |}^{2} \\ = & {| 𝒩 |}^{2} {| f (x, y) |}^{2} {| g_{k} |}^{2} \cdot {| \frac{C}{D} |}^{2} . \end{array}

Due to the center-of-mass wave function of the atom f(x,y) is assumed to be nearly constant over many wavelengths of the standing-wave fields in the x–y plane, the conditional position probability distribution P(x,y;t → ∞|0,1_k) is determined by the last term in Eq. (13). Therefore, we can define the filter function as $F (x, y) = {| \frac{C}{D} |}^{2}$ , which shows that the conditional position probability distribution depends upon the frequency detunings of the standing-wave driving fields and the population in the upper or lower levels, as well as the detuning of the spontaneously emitted photon. As a result, we can obtain the position information of the atom by measuring the frequency of spontaneously emitted photon under proper conditions.

Under the conditions A_{3,0_k} (0) = 1, A_{1,0_k} (0) = A_{2,0_k} (0) = 0, Δ₁ = Δ₂ = 0, and φ = 0, the filter function F(x,y) can be explicitly expressed in the following form

F (x, y) = \frac{1}{{[δ_{k} + \frac{δ_{k} [G_{1}^{2} (x, y) + G_{2}^{2} (x, y)] + 2 | Ω_{m} | G_{1} (x, y) G_{2} (x, y)}{{| Ω_{m} |}^{2} - δ_{k}^{2}}]}^{2} + \frac{Γ_{0}^{2}}{4}} .

3. Results and discussion

It has been reported that the spontaneous emission can be coherently manipulated by the microwave field in a cycle-configuration four-level atomic system [48, 49]. Some of the interesting phenomena involving spectral-line narrowing, spectral-line enhancement, and spectral-line suppression can be observed by adjusting the system parameters. Here we are interested in the precise position measurement of the atom when it passes through the standing-wave fields using the measurement of the frequency of the spontaneously emitted photon. As we mentioned earlier, the Rabi frequencies of the standing-wave fields are position dependent for our proposed scheme. Consequently, the spontaneous emission spectra become position dependent and thus the position of the atom as it passes through the standing-wave fields can be determined as soon as we monitor the frequency of the spontaneously emitted photon. The precise location of the atom can be given by those values of k₁x and k₂y (k₁y) when the filter function F(x, y) exhibits maxima. In the following discussion, we consider two different situations for the position measurement of the atom: (i) the two orthogonal standing-wave fields are respectively used to couple the different atomic transitions, i.e., G₁(x, y) = Ω₁ sin(k₁x) and G₂(x, y) = Ω₂ sin(k₂y); (ii) the two standing-wave fields with the same frequency are applied to drive the same atomic transition, i.e., G₁(x, y) = Ω₁[sin(k₁x) + sin(k₁y)] and G₂(x, y) = Ω₂. The spontaneous decay rate of the level |3〉 to level |0〉 is set as Γ₀ = 2γ. All the parameters used in this paper are in units of γ, which should be in the order of MHz for rubidium atoms.

3.1. Two standing-wave fields drive different atomic transitions

Initially, we consider the situation that the two orthogonal standing-wave fields drive the different atomic transitions, respectively. In such a case, we will analyze the behavior of 2D atom localization by adjusting the initial state preparation, the frequency detuning of two standing-wave fields, and the relative phase φ. In Fig. 2, we plot the filter function F(x,y) versus the normalized positions (k₁x, k₂y) by measuring the frequency of the spontaneously emitted photon under the condition of A_{3,0_k} (0) = 1 when the two orthogonal standing-wave fields are both tuned to the resonant interaction with their respective atomic transition and the relative phase φ = 0. Figure 3 shows the corresponding density plots of the filter function F(x,y) in the x–y plane. It can be seen from Fig. 2(a) that the peak maxima of the filter function exhibit a latticelike pattern when we detect the detuning of the spontaneously emitted photon is δ_k = 8γ, and the atom is localized at the second and fourth quadrants in the x–y plane [see Fig. 3(a)]. As the detuning δ_k increases, we observe that the localization peaks occur at k₁x + k₂y = 2mπ or k₁x – k₂y = (2n + 1)π (m,n are integers), which indicates that the atom is distributed on the diagonal in the second and fourth quadrants [see Figs. 2(b) and 3(b)]. And these localization peaks become very sharp due to constructive interference of quantum pathways. When the detuning is detected at δ_k = 16γ, the conditional position probability distribution of the atom is contrary to that shown in Figs. 2(a) and 2(b), the maxima of the filter function in Fig. 2(c) are situated in the first and third quadrants with a craterlike pattern, and the atom is localized at the circular edges of the craters [see Fig. 3(c)]. Moreover, when the frequency detuning of the spontaneously emitted photon is measured at an appropriate value [e.g., δ_k = 19.3γ in Fig. 2(d)], the resulting localization peaks display a spikelike pattern, which shows that the spatial resolution is greatly improved [see Fig. 3(d)]. As a result, we can achieve high-precision and high-resolution 2D atom localization by measuring the frequency of the spontaneously emitted photon under three-photon resonance conditions.

Fig. 2 The filter function F(x,y), which directly reflects the conditional position probability distribution, as a function of (k₁x, k₂y) in dependence on the detuning of spontaneously emitted photon δ_k. (a) δ_k = 8γ; (b) δ_k = 9.05γ; (c) δ_k = 16γ; (d) δ_k = 19.3γ. The other parameters used are Ω₁ = Ω₂ = 10γ, |Ω_m| = 9γ, Δ₁ = Δ₂ = 0, Γ₀ = 2γ, and φ = 0. The atom is initially prepared in level |3〉, i.e., A_{3,0_k} (0) = 1. All parameters are in units of γ.

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Fig. 3 Density plot of filter function F(x,y) in the x–y plane shown in Fig. 2.

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These interesting localization phenomena can be explained using quantum interference effect either in the bare-state picture or in the dressed-state picture. Under the condition that the atom is initially prepared in the excited level |3〉, there exist three spontaneous decay channels in the bare-state picture: |3〉 → |0〉, |3〉 → |1〉 → |2〉 → |3〉 → |0〉, and |3〉 → |2〉 → |1〉 → |3〉 → |0〉. Quantum interference among these three pathways results in the spectral-line narrowing and the quenching of spontaneous emission. Consequently, we can observe two sharp localization peaks in the x–y plane.

It is desirable to obtain the position of the atom when the atom passes through the standing-wave fields. However, how to extract the localization information, i.e., relatively to what point in space is the localization? We now discuss some possible solutions, which are similar to those reported in Refs. [13, 23, 50]. Our scheme is based on the fact that the conditional position probability P(x,y;t|0,1_k) carries the information about the atomic position. In Fig. 2, we plot the filter function F(x,y) for four cases of frequencies of the emitted photon ω_k recorded during measuring time t. It can be observed that if the detector records a larger frequency [e.g., ω_k = ω₃₀ + 19.3γ in Fig. 2(d)], two sharp probability distributions centered at the antinodes of the standing-wave fields in the first and third quadrants are expected. The appearance of two steep peaks originates from the combined effects of the microwave coupling field and spontaneously generated coherence between dressed levels, which has been demonstrated in Ref. [49]. We thus get a much precise position information due to a strong spectral-line narrowing effect when the frequency of the emitted photon is large enough. As a result, we can extract the immediate position information by making a measurement on the spontaneously emitted photon when the atom goes through the intersectant region of two standing-wave fields.

In order to further show the influence of the system parameters on the behavior of 2D atom localization, we give the filter function F(x,y) versus the normalized positions (k₁x, k₂y) by monitoring the spontaneously emitted photon under the condition of $Ψ (0) = (| 1 〉 + | 2 〉) / \sqrt{2}$ when the two standing-wave fields are tuned to nonresonant with the corresponding atomic transitions and the relative phase φ = π, as shown in Fig. 4. The corresponding density plots are illustrated in Fig 5. When the spontaneously emitted photon with detuning δ_k = 8.5γ is detected, the corresponding filter function in Figs. 4(a) and 5(a) is distributed in the first and third quadrants with a craterlike pattern. We find that only when the frequency of the spontaneously emitted photon is detected at an appropriate value [see Figs. 4(b) and 5(b)], that is, the quantum interference between |1〉 → |2〉 → |3〉 → |0〉 and |2〉 → |1〉 → |3〉 → |0〉 is so strong that we can observe two localization peaks with a spikelike pattern in the x–y plane. Under this situation, the high-spatial-resolution and high-precision localization of the atom can be achieved, and the probability of finding the atom within one period is 50%. However, when the detuning is increased to δ_k = 17.2γ, the peak maxima of the filter function in Figs. 4(c) and 5(c) are mostly distributed in the second and fourth quadrants with a lotus-like structure and little in the first and third quadrants. But, it is accompanied with a lower localization precision. With further increase of the detuning of the spontaneously emitted photon, it can be seen from Figs. 4(d) and 5(d) that the localization peaks in the first and third quadrants are completely vanished due to the destructive quantum interference in such a four-level atomic system with a closed-loop configuration. As can be seen from these figures, large detunings of the standing-wave fields do not alter qualitatively features revealed in our paper. That is to say, this kind of mismatch can not change the probability of finding the atom in the subwavelength regime and the precision of the atom localization. For simplicity, but without loss of generality, we focus our discussion on the resonant case in the following subsection.

Fig. 4 The filter function F(x,y) as a function of (k₁x, k₂y) in dependence on the detuning of spontaneously emitted photon. (a) δ_k = 8.5γ; (b) δ_k = 12.2γ; (c) δ_k = 17.2γ; (d) δ_k = 20γ. The system parameters used are the same as Fig. 2 except that Δ₁ = Δ₂ = 5γ, φ = π, and the atom is initially in $Ψ (0) = (| 1 〉 + | 2 〉) / \sqrt{2}$ .

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Fig. 5 Density plot of filter function F(x,y) shown in Fig. 4.

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3.2. Two standing-wave fields couple one atomic transition

As a matter of fact, we are particularly interested in the case that the two orthogonal standing-wave fields are applied to couple the same atomic transition. This is mainly because we can get a higher probability of finding the atom at a particular position in the x–y plane, thus realizing the 2D atom localization, indeed. Although the values of the system parameters are the same as in Fig. 2, the behavior of the atom localization in Fig. 6 is different from those observed in the previous subsection. Similarly, the corresponding density plot of the filter function F(x,y) is given in Fig. 7. When the detuning of the spontaneously emitted photon is detected at δ_k = 7γ, the peak maxima of the filter function are mostly distributed in the third quadrant with a bicycli-clike pattern, and little in the second and fourth quadrants, as can be seen from Figs. 6(a) and 7(a). For the case that δ_k = 8.9γ, the localization peaks with a bicycliclike pattern in the third quadrant evolve into a craterlike pattern, and the localization peaks distributed in the second and fourth quadrants become very sharp [see Figs. 6(b) and 7(b)]. This means that the spatial resolution of atomic position is greatly improved. When the detuning of the spontaneously emitted photon is measured at δ_k = 13.5γ, it can be seen from Figs. 6(c) and 7(c) that the peak maxima of F(x,y) are situated on the diagonal in the second and fourth quadrants, and display a cross-shaped structure, while the localization peak in the third quadrant has a craterlike pattern. For a suitable detuning of the emitted photon, e.g., δ_k = 20γ, the filter function in this case exhibits different patterns in one period, that is, the localization peaks distributed in the first quadrant has a craterlike pattern, while the localization peaks in the third quadrant display a spikelike pattern, as shown in Figs. 6(d) and 7(d). In particular, it can also be seen that, the localization peaks in the third quadrant have a higher precision and resolution than that shown in the first quadrant, which originated from the constructive quantum interference induced by the microwave, standing-wave and traveling-wave fields.

Fig. 6 The filter function F(x,y) as a function of (k₁x, k₁y) in dependence on the detuning of spontaneously emitted photon. (a) δ_k = 7γ; (b) δ_k = 8.9γ; (c) δ_k = 13.5γ; (d) δ_k = 20γ. The system parameters used are the same as Fig. 2.

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Fig. 7 Density plot of filter function F(x,y) in the x–y plane shown in Fig. 6.

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The main purpose of the present paper is to localize the atom at an expected position in the x–y plane by modulating the system parameters. Under the conditions that the two orthogonal standing-wave fields couple the same atomic transition and the atom is initially prepared in a coherent superposition of two ground levels, i.e., $Ψ (0) = (| 1 〉 - | 2 〉) / \sqrt{2}$ , we can obtain a 100% probability of finding the atom at a particular position for different sets of system parameters. Figures 8 and 9 display the different localization patterns of the filter function F(x,y) and the corresponding density plots, respectively, as varying the combination values of the detuning δ_k and the relative phase φ. More precisely, in the case of φ = π/2 and δ_k = 14.5γ, the maxima of the filter function F(x,y) are distributed in the four quadrants but with different probability, which shows that the spatial resolution is very poor [see Figs. 8(a) and 9(a)]. While, when the relative phase is tuned to φ = π/4 and the detuning of the spontaneously emitted photon is detected at δ_k = 19.5γ, the filter function exhibits a craterlike pattern as shown in Fig. 8(b) and its maxima are situated in the third quadrant [see Fig. 9(b)]. Furthermore, if the relative phase φ is increased by a factor of π/2 compared with Fig. 8(b) and the detuning of the emitted photon is increased to δ_k = 26γ, the atom is completely localized in the third quadrant, and the localization peak becomes very sharp, with a spikelike pattern [see Figs. 8(c) and 9(c)]. Therefore, high-precision and high-resolution of 2D atom localization is realized. In addition, the application of microwave field leads to an improvement of the probability of finding the atom at a particular position by a factor of up to 4 or 2 compared to the previous proposed schemes without the microwave-driven field [42–46]. More importantly, we can not only make the atom localized at a particular position, but we can also localize the atom at an expected spatial position. It is further demonstrated that when the relative phase is adjusted to φ = π and the spontaneously emitted photon with detuning δ_k = 6.75γ is monitored, the peak maxima of the filter function in Fig. 8(d) are confined within the first quadrant with a spikelike pattern. In this case, the atom is localized at the position (k₁x, k₁y) = (π/2,π/2) with very high spatial resolution during one period of the standing-wave fields [see Fig. 9(d)]. It is thereby possible to rigorously determine the position of the atom is localized when it passes through the standing-wave fields. It should be noted that a sharp single localization peak can be obtained in a subwavelength region [see, for instance, Figs. 8(c) and 8(d)]. However, because of the mechanical action of the standing-wave fields on the atom, it is inevitably accompanied by a wide momentum spread. Yet we find that the position-momentum uncertainty results from the mechanical action does not affect the precision position measurement of an atom such as that proposed by Storey et al. [7,8]. Moreover, under certain conditions, the uncertainty can be minimized.

Fig. 8 The filter function F(x,y) as a function of (k₁x, k₁y) for different combinations of the detuning δ_k and the phase φ. (a) δ_k = 14.5γ, φ = π/2; (b) δ_k = 19.5γ, φ = π/4; (c) δ_k = 26γ, φ = 3π/4; (d) δ_k = 6.75γ, φ = π. The system parameters used are the same as Fig. 2 except that the atom is initially in $Ψ (0) = (| 1 〉 - | 2 〉) / \sqrt{2}$ .

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Fig. 9 Density plot of filter function F(x,y) shown in Fig. 8.

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Before concluding this section, we now turn our attention to estimate the influence of the perturbations from the perfect microwave standing configurations of the two driving fields on the atom localization behavior. This is mainly because the perturbations of intensity and detuning are unavoidable in the process of experimental realization. The resulting localization patterns by considering an intensity fluctuation (for instance 0.1γ) are plotted in Figs. 10(a) and 10(b), which correspond to the localization profiles without the intensity fluctuation as shown in Figs. 2(c) and 2(d), respectively. By contrast, we found that the intensity perturbation does not affect the most probable positions of finding the atom in the subwavelength regime, but the resolution of atom localization is slightly reduced. Similarly, the perturbations of detuning may cause a small change of the atomic resolution, but cannot affect the precise location of the atom, these can be clearly seen by comparing the localization peaks with a small fluctuation of Figs. 10(c) and 10(d) to that shown in Figs. 8(c) and 8(d) without any perturbation. It can be concluded that the behavior of the atom localization does not vary with the fluctuations of laser intensity and detuning, therefore, we use the exact values of the intensity and detuning in the above-mentioned calculation and analysis is reasonable.

Fig. 10 The filter function F(x,y) as a function of the normalized positions for different system parameters. (a) and (b) denote the cases that the above Fig. 2(c) and Fig. 2(d) which are added a fluctuation 0.1γ, respectively, i.e., Ω₁ = Ω₂ = 10γ + 0.1γ. Other parameters are the same as that in Fig. 2(c) and Fig. 2(d). (c) and (d) correspond to Fig. 8(c) and Fig. 8(d) which are added a detuning fluctuation 0.05γ, i.e., Δ₁ = Δ₂ = 0.05γ. The other parameters used are the same as Fig. 8(c) and Fig. 8(d), respectively.

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In our previous work [51], the microwave field is used to resonantly couple the two excited-state hyperfine levels, we can obtain a 100% probability of finding the atom at a particular position within one period of the standing-wave fields only for a suitable set of parameters. In other words, we can only localize the atom at a certain position. In the present paper, we have employed the probability amplitude approach rather than the density-matrix approach in the above calculations and discussions, mainly due to its simplicity in the mathematical treatment and its transparency in the physical explanation of the results obtained. Most interestingly, the results of the present paper clearly show that if we use a microwave field to drive the ground-state hyperfine transition, the atom can be localized at the expected position by altering the tunable system parameters. While, although the atom can also be localized at a desired position by tuning the phase of the driving fields in Refs. [52, 53]. The hyperfine transition which is driven by the radio-frequency or microwave field lies within the excited states. In our scheme, the two hyperfine levels of the ground-state structure are coupled by a microwave field. Due to the ground states are stable with respect to the excited states. The collisional dephasing rate of the ground-level coherence and the radiative decay rate of the two ground levels are very small and can be neglected, accordingly, the loss of the system is relatively small. On the other hand, both the phase of the microwave field and the frequency of the spontaneously emitted photon can be used to change the location of the atom. In addition, Wang et al. [54] have investigated the atom localization behaviors via spontaneous emission in a coherently driven five-level atomic system by means of a radio-frequency field driving a ground-state hyperfine transition. However, as far as we know, the atomic transitions cannot form a closed-loop configuration. And we can easily manipulate the behavior of the atom localization by adjusting the collective phase of the closed loop. Hence, our proposed scheme in this paper provides more degrees of freedom to realize the 2D atom localization. Most importantly, our scheme is based on the fact that the frequency of the spontaneously emitted photon provides information about the atomic position owing to its direct relation with the position-dependent Rabi frequency. Our method displays 2D atom localization in real time and within a subwavelength range of the standing-wave fields, without any tedious calculations. As a matter of fact, this optical technique has also been extended from the text of the resonance fluorescence phenomenon [13].

Finally, it should be pointed out that such a four-level atomic configuration can be realized in cold ⁸⁷Rb atoms [47,48,55] using the D₂ line structure. The designated states can be chosen as follows: |0〉 = |5S_1/2, F = 2, m_F = 2〉, |1〉 = |5S_1/2, F = 1, m_F = 0〉, |2〉 = |5S_1/2, F = 2, m_F = 0〉, and |3〉 = |5P_3/2, F = 2, m_F = 1〉, respectively. The spontaneous decay rate of the state |3〉 = |5P_3/2, F = 2, m_F = 1〉 in this system is 6 MHz. In practical experiments, the transition between the states 5S_1/2 and 5P_3/2 is driven by standing-wave or traveling-wave laser fields at a wavelength of 780.2 nm. The hyperfine transition |5S_1/2, F = 1, m_F = 0〉 ↔ |5S_1/2, F = 2, m_F = 0〉 is resonantly coupled by a microwave field with frequency around 6.8 GHz, which can greatly enhance the localization precision and improve the spatial resolution. These fields can be obtained from the external cavity diode lasers [56]. The spontaneously emitted photon at appropriate frequencies is position dependent. Such position-dependent spontaneous emission can be reflected by a standard spectroscopic method or the heterodyne measurement of fluorescence, which may be realized via the experiment proposed in [25, 50]. Simultaneously, we should point out that the induced transitions between the magnetic sublevels by the two laser fields can be combined into a manifold of four-level systems according to the selection rules. It is a good approximation to say that the coupled Rb system can be regarded as equivalent to a generic four-level system depicted in Fig. 1. The validity of such a simplification has been supported by several previous researches [46, 48, 49, 52, 53].

4. Conclusion

In summary, we have demonstrated the scheme for 2D atom localization of a four-level atomic system with a cyclic configuration based on the controllable spontaneous emission. In particular, the behavior of 2D atom localization in this kind of system in the presence of microwave field has been discussed in detail, giving rise to some interesting localization phenomena such as latticelike, craterlike, spikelike, bicycliclike patterns and lotus-like structure. These results show that the conditional position probability distribution of the atom is very sensitive to the tunable system parameters. Furthermore, the maximum probability of finding the atom in one period of the standing-wave fields is reached to 100%, that is, we can localize the atom at a particular position. Also, according to our predictions, the atom can be localized at an expected position for different sets of parameters due to the quantum interference effects. Finally, our proposal may have potential applications in improving the performance of laser cooling and neutral atoms trapping, Bose-Einstein condensation, atom lithography, etc. With the development of optical techniques, we hope that this 2D atom localization scheme proposed here will be implemented by actual experiments in the near future.

Our scheme may be extended to the subwavelength localization of an ensemble of atoms. In other words, the localization properties of a region containing multiple atoms can be treated using our single-atom localization approach if the atoms evolve independently. The assumption of independent atoms is reasonable, if it satisfies the following conditions: (i) The atoms do not interact strongly with one another in the process of atoms traveling through the standing-wave fields. (ii) These atoms must be separated far enough from each other that we can neglect the quantum interference from the two atoms interacting with the same mode of the reservoir. (iii) The applied fields are classical.

Acknowledgments

We would like to thank Professor Xiaoxue Yang for her encouragement and helpful discussion. This research was supported in part by the National Natural Science Foundation of China under Grants No. 11004069, No. 11104210, and No. 91021011, by the Doctoral Foundation of the Ministry of Education of China under Grant No. 20100142120081, by the National Basic Research Program of China under Contract No. 2012CB922103, and by the Fundamental Research Funds from Huazhong University of Science and Technology (HUST) under Grant No. 2010MS074.

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High-precision atom localization via controllable spontaneous emission in a cycle-configuration atomic system

Abstract

1. Introduction

2. Theoretical model and basic formula

3. Results and discussion

3.1. Two standing-wave fields drive different atomic transitions

3.2. Two standing-wave fields couple one atomic transition

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (10)

Equations (14)

Optics Express