During the last several decades, the high-precision position measurement of moving atoms has been investigated extensively due to its potential applications in laser cooling and trapping,^[1–3] Bose–Einstein condensation,^[4–6] atom lithography,^[7,8] and so on. Of particular interest, the optical method is an efficient way to measure the position of the moving atoms. Earlier schemes include measurement of the phase shift of the standing wave,^[9–11] the entanglement between the atom’s position and its internal states,^[12,13] and resonance imaging methods.^[14,15] On the other hand, in quantum optics and laser physics, atomic coherence and quantum interference can lead to a lot of interesting phenomena such as coherent population trapping (CPT),^[16] electromagnetically induced transparency (EIT),^[17–23] the modification of spontaneous emission,^[24–26] giant Kerr nonlinearity,^[27,28] quantum entanglement, etc.^[29–31]

In recent years, varieties of atomic systems were studied to realize one-dimensional (1D) atomic localization based on atomic coherence and quantum interference. For example, Zubairy et al. proposed several schemes to investigate the position probability of the atoms based on resonance fluorescence from a simple two-level system^[32] as well as the coherence control of the spontaneous emission spectrum in a three-level system.^[33] Paspalakis and Knight put forward a scheme to obtain atom localization by measuring the probe absorption or the population of the upper state.^[34] In these works, four equally localization peaks distribute in a unit wavelength domain of the standing wave with 25% detecting probability. In order to enhance the detecting precision, one employed a four-level closed system to realize sub-half-wavelength localization by probing the absorption spectrum,^[35] wherein the detecting probability can reach 50%. Subsequently, it is reported that the atoms can be localized at the nodes of the standing-wave field with a relatively high spatial precision due to the interference of double dark resonances.^[36] In addition, the coherent manipulation of the Raman gain process,^[37] and the polychromatic excitation^[38,39] have also been proposed to obtain 1D atomic localization.

More recently, two-dimensional (2D) atom localization has attracted a great deal of attention. Ivanov et al. reported initially 2D localization in a four-level tripod system.^[40] By adjusting the parameters appropriately, such spatial periodic structures of populations as spikes, craters, and waves were displayed. After that, high-precision and high-resolution 2D atom localization were attained based on the interference of double-dark resonances^[41] and the controlled spontaneous emission.^[42] In a coherently driven multi-level atomic system,^[43] the localization precision was significantly improved by the spontaneously generated coherence and the dynamically induced quantum interference. In particular, the 100% detecting probability was realized in a cycle-configuration system using the controllable spontaneous emission spectra.^[44] In brief, in the above-mentioned schemes, one mainly utilizes quantum interference and atomic coherence either to realize the atomic localization or to enhance the position precision and spatial resolution.

On the other hand, it is well known that the optical properties of the light-atom interaction system are strongly influenced by the environment. Since Gardiner studied a direct effect of squeezing in a two-level system,^[45] it becomes an intense field in quantum optics. A large number of classical phenomena were proved to be modified by the squeezed vacuum effect including the two-photon absorption rate,^[46,47] resonance fluorescence,^[48] and hole burning.^[49] The broadband squeezed vacuum can greatly modify the absorption, dispersion and nonlinearity of the three-level systems.^[50–52] Based on these properties, Calderón et al. investigated the effect of a squeezed vacuum on the resonance fluorescence spectrum,^[53] optical bistablility and multistability,^[54] transient higher order correlations,^[55] and absorption profile.^[56] Up to now, the atomic localization is realized in the systems bathed in an ordinary vacuum. Naturally a question arises: Would the atomic localization be obtained or improved by the squeezed vacuum effect? Here we take into account a scheme for 2D atom localization, wherein the three-level V-type atoms interact with two standing-wave fields in a broadband squeezed vacuum. By calculating the resonance fluorescence spectrum, the 2D atomic localization is demonstrated in the following three cases. First, when the environment in which the atoms are bathed is changed from an ordinary vacuum to a squeezed vacuum, the 2D atom localization emerges at the nodes of the standing wave fields. Second, we find that the 2D atomic localization is also obtained based on the quantum interference effect. However, the spatial structures are completely distinct from the first case. Finally, we consider the combined effects of quantum interference and a squeezed vacuum on atomic localization. The localization precision and spatial resolution are shown to be inferior to the first case but superior to the second one. The remainder of this article is organized as follows. In Section 2, we present the model and master equation. The numerical results and discussion are given in Section 3, followed by Section 4 that concludes the paper.

A three-level V-type atomic system with two upper states |1〉 and |2〉 and a single ground state |0〉 is considered as shown in Fig. 1. Two standing-wave fields are applied to drive the two transitions |1〉 – |0〉 and |2〉 – |0〉, respectively. The atoms are assumed to be bathed in a broadband squeezed vacuum reservoir.

Generally, the correlation functions for the field operators a(ω_kλ) and a^†(ω_kλ) are written in the forms^[53–56]

Fig. 1.

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	Fig. 1. Energy level structure of the three-level V-type system. Two standing-wave fields Ω1d = Ω1 sin(kx), Ω2d = Ω2 sin(ky) are applied to two dipole-allowed transitions |1〉 → |0〉 and |2〉 → |0〉, respectively.

Following the standard approach of Weisskopf and Wigner,^[57,58] in the Born and Markov approximation, we can obtain the master equation for the reduced density matrix ρ. In an appropriate rotating frame the master equation reads

The spectrum of the spontaneously emitted photons or scattered light mimics the position probability of the center-of-mass motion of the atoms. Hence we can localize the atoms at a particular position of the standing wave fields in the x–y plane via probing the resonance fluorescence spectrum,^[33]

When the scattered lights are measured at a fixed frequency, the conditional position probability becomes only position-dependent, which is denoted by W(x,y). In our numerical calculations, the Rabi frequencies, the decay rates, the frequencies of the fluorescence spectrum, and the detunings are always scaled in units of decay rate γ₁. We always set γ₂ = 1 throughout this paper. Firstly, the squeezed vacuum effect is considered and the two dipole-transitions are assumed to be perpendicular, i.e., β = 0. The squeezed vacuum effect on atomic localization is mainly shown in Fig. 2, wherein a three-dimensional distribution of the filter function W(x,y) as a function of the normalized positions kx and ky is plotted at the central frequency ω = 0. The parameters are chosen as Δ₁ = −Δ₂ = 1, Ω₁ = Ω₂ = Ω = 10, and Φ = 0. As shown in Fig. 2(a), at N = 0, it is clear that the atomic localization is not obtained in a period of [−π,π]. However, we note that some dips appear at the nodes of the standing-wave fields. As a result, it can be deduced that the emitted photons cannot be detected at kx = ky = 0, ±π when the atoms are bathed in the ordinary vacuum. Once the squeezed vacuum is introduced, in Fig. 2(b), the localization peaks arise by taking N = 0.1. With increasing the squeezing parameter N from 0.1 to 0.2, the localization peaks rise remarkably with a spike-like pattern. During this process, the width of the localization peaks becomes narrower, implying higher precision of atom localization. When we further increase the squeezed parameter to N = 0.6, the atoms can be localized at the nodes of the standing wave fields.

Fig. 2.

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	Fig. 2. Two-dimensional atomic localization induced by squeezed vacuum effect for β = 0 and Φ = 0. Other parameters are chosen as γ1 = γ2 = 1, Ω1 = Ω2 = 10, (a) N = 0; (b) N = 0.1; (c) N = 0.2; (d) N = 0.6.

It has been shown that the relative phase between the applied fields and the squeezed field can be used to control the resonance fluorescence spectrum.^[45] Accordingly, we would like to explore the phase control of atomic localization in the present scheme. In Fig. 3, the relative phase is chosen for (a) Φ = 0 and (b) Φ = π. Other parameters are the same as those in Fig. 2(d). The fluorescence is assumed to be detected at the sideband ω = 0.5. From these figures, it is found that the spike-like localization peaks are obtained at Φ = 0 and split at Φ = π. This implies that the detecting probability decreases when the phase is changed from Φ = 0 to π. The phase-sensitive atomic localization can be attributed to the phase control of the resonance fluorescence spectrum, which was first reported in a two-level atomic system damped by a squeezed vacuum.^[45]

Fig. 3.

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	Fig. 3. Phase control of atomic localization for (a) Φ = 0; (b) Φ = π. The parameters are the same as those in Fig. 2 except for N = 0.6, ω = 0.5.

Figure 4 exhibits the effect of quantum interference on atom localization. The parameters are chosen as γ₂ = 1, Ω₁ = Ω₂ = 10, Δ₁ = −Δ₂ = 1, N = 0, (a) β = 0; (b) β = 0.5; (c) β = 0.9; (d) β = 0.99. When β = 0, in Fig. 4(a), the atomic localization is not obtained at the central frequency ω = 0. However, at β = 0.5 (Fig. 4(b)), the values of the function W(x,y) are reduced in the second and fourth quadrants. As the parameter β increases to 0.9 (see Fig. 4(c)), the atomic localization emerges slowly. Furthermore, for β = 0.99, we can see that the symmetrical localization peaks are mainly distributed in the first and third quadrants. Obviously, the spatial precision and resolution induced by the quantum interference are inferior to those induced by the squeezed vacuum.

Fig. 4.

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	Fig. 4. The quantum interference effect on two-dimensional atomic localization by taking N = 0. The other parameters are chosen as γ2 = 1, Ω1 = Ω2 = 10, Δ1 = −Δ2 = 1, (a) β = 0; (b) β = 0.5; (c) β = 0.9; (d) β = 0.99.

Finally, the combined effects of the squeezed vacuum effect and quantum interference are shown in Fig. 5. The parameters are chosen as β = 0.99, N = 0.6 for (a) Φ = 0, (b) Φ = π. We note that the atom localization is different from those in Fig. 2(d) and Fig. 4(d). In Fig. 5(a), the values of the filter function W(x,y) in the second and fourth quadrants are smaller than those in the first and third quadrants. The detecting precision is evidently inferior to that in Fig. 2(d) but superior to Fig. 4(d). When the relative phase is changed into Φ = π, as shown in Fig. 5(b), the star-like patterns appear in the first and third quadrants.

Fig. 5.

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	Fig. 5. Two-dimensional atomic localization for (a) Φ = 0 and (b) Φ = π with β = 0.99, N = 0.6. Other parameters are the same as those in Fig. 2.

Due to the position-dependent Rabi frequencies Ω₁ sin(kx) and Ω₂ sin(ky), the structure of the resonance fluorescence spectrum is periodical. In this regard, in Fig. 6, we plot the resonance fluorescence spectra by taking weak field Ω = 0.1 and strong field Ω = 10 for different cases, i.e., N = 0, β = 0 (solid line); N = 0, β = 0.99 (dashed line); N = 0.6, β = 0 (dotted line); N = 0.6, β = 0.99 (dash dotted line).

Fig. 6.

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	Fig. 6. The resonance fluorescence spectra by choosing different Rabi frequencies: (a) Ω = 0.1; (b) Ω = 10 for N = 0, β = 0 (solid line); N = 0, β = 0.99 (dashed line); N = 0.6, β = 0 (dotted line); N = 0.6, β = 0.99 (dash dotted line). Other parameters are the same as those in Fig. 2.

In Fig. 6(a), the fluorescence is suppressed for the cases of N = 0, β = 0 and N = 0, β = 0.99, which indicates the absence of fluorescence at the nodes of the standing wave fields. However, a central peak arises when N = 0.6, β = 0 and splits into two for N = 0.6, β = 0.99, resulting in the emergence of the localization peak(s). Differing from the weak-field case, in Fig. 6(b), when the quantum interference and squeezed vacuum effects are introduced, the fluorescence at the central frequency is quenched because the height of the fluorescence decreases. In comparison to quantum interference, the squeezed vacuum has a more remarkable effect on the fluorescence. As a consequence, the atomic localization induced by the squeezed effect is superior to that induced by quantum interference. To gain insight into the effect of the squeezed vacuum, we plot the 1D atomic localization at central frequency ω = 0 in Fig. 7 by considering different cases. It can be seen clearly that the atoms are localized at the nodes of the standing wave field when the squeezing parameter is N = 0.6.

Physically, for the weak-field cases, the atoms would stay in the ground state without fluorescence emission in the ordinary vacuum. Nevertheless, for N ≠ 0, the atoms would be transferred into excited states and then the fluorescence happens due to the phase-dependent squeezed reservoir. On the other hand, for the strong-field cases, the atomic population is transferred by the squeezed vacuum reservoir and the dynamically induced coherence, which gives rise to the spectral structures shown in Fig. 6(b); this is responsible for the appearance of the atomic localization induced by the squeezed vacuum. In particular, when the maximal quantum interference is considered (i.e., β = 1), it is well known that the spontaneous emission is canceled due to the destructive interference between the two decay passages. The population would be trapped into a superposition state and the fluorescence is completely quenched in an ordinary vacuum under the two-photon resonance conditions. If the atoms are bathed into a squeezed vacuum, however, the CPT would be spoiled. This leads to the fact that the fluorescence emission becomes possible. Besides, since the steady state atomic population and the fluorescence spectrum are dependent on the phase Φ, the 2D atomic localization in the present scheme is phase-sensitive.

Fig. 7.

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	Fig. 7. One-dimensional distribution function as a function of kx with different cases. The parameters are the same as those in Fig. 6.

To understand the above-mentioned results, one can resort to the dressed state analysis as usual. For simplicity, we take Δ₁ = −Δ₂ = Δ, Ω₁ = Ω₂ = Ω. By diagonalizing the Hamiltonian, the dressed states can be obtained as^[62]

Case 1 N = 0, β = 0

Case 2 N = 0, β ≠ 0

Case 3 N≠0, β = 0

Fig. 8.

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	Fig. 8. The evolution of dressed state population versus Rabi frequencies Ω and the squeezed parameters N by taking β = 0 for (a) and (c) ρ0̃0̃; (b) and (d) ρ++ = ρ− −. Other parameters are the same as those in Fig. 2.

From Eqs. (16) and (17), it is noticed that the dressed state populations are prominently dependent on the squeezing parameters N, |M|, Φ, and the quantum interference effect β.^[53] In Fig. 8 and Fig. 9, the three-dimensional (3D) evolution of the dressed-state population at a steady state are numerically plotted by considering the effect of a squeezed vacuum and quantum interference respectively. In Figs. 8(a) and 8(c), the population of ρ_0̃0̃ increases in the weak-field region but decreases in the strong-field region as the parameter N increases. In contrast, the variation tendencies of ρ₊₊ and ρ_{− −} in Figs. 8(b) and 8(d) are reversed owing to the closure relation. As a matter of fact, it implies a raise of the fluorescence at the nodes and a suppression at other positions, which is responsible for the realization of atomic localization as shown in Fig. 2. Nevertheless, the evolutions of dressed state populations induced by quantum interference (see Fig. 9) are different. In the weak-field region, the populations for ρ_0̃0̃, ρ₊₊, and ρ_{− −} always keep unchanged. For the strong-field cases, the dressed state populations first change slowly and then vary sharply as the quantum interference factor β increases. When the quantum interference becomes maximal, that is, β = 1, the atoms would be trapped into a dark state again. These phenomena result in the atom localization induced by the quantum interference effect. In order to observe the evolution of the dressed state population more clearly, the 2D demonstration is plotted in Fig. 10 by taking Ω = 0.1 and Ω = 10 respectively.

Fig. 9.

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	Fig. 9. The evolution of dressed state population versus Rabi frequencies Ω and the quantum interference effect β by taking N = 0 for (a) and (c) ρ0̃0̃; (b) and (d) ρ++ = ρ− −. Other parameters are the same as those in Fig. 2.

Fig. 10.

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	Fig. 10. The demonstration of the evolution of dressed state population versus the squeezed parameter N in panels (a) and (b) and the parameter β in panels (c) and (d). The Rabi frequencies are chosen as: (a) and (c) Ω = 0.1; (b) and (d) Ω = 10. Other parameters are the same as those in Fig. 2.

Before ending this section, we would like to emphasize the main innovative points in the present work. First, by comparing with the previous works, we demonstrate a novel avenue for the localization in a three-level V-type system. In this system, when the atoms are bathed in a standard vacuum, the atomic localization is hard to attain by detecting the resonance fluorescence at a fixed frequency. Nevertheless, once the system is bathed into a broadband squeezed vacuum, it is found that the 2D atomic localization is realized at the nodes of the standing wave fields with good spatial resolution, which may find potential application in experiments. Second, we note that the atomic localization is also possible to obtain based on the quantum interference between the two decay channels. However, the spatial structures as well as the localization precision are distinct from those induced by the squeezed vacuum effect. Third, we demonstrate that various 2D spatial structures can be controlled by changing the squeezed parameters, the detunings, and the relative phase appropriately. In terms of the dressed-state analysis, the resonance fluorescence spectra are closely related to the coherent population trapping, which is essentially determined by the squeezed vacuum and quantum interference effects.

In conclusion, we exhibit that the atom localization can be induced by the squeezed vacuum effect. For a three-level V-type system bathed into a broadband squeezed vacuum reservoir, two standing-wave fields are used to drive two dipole-allowed transitions. By detecting the resonance fluorescence spectrum, the 2D atom localization is obtainable and the localization precision becomes higher as the squeezed parameters increase. On the other hand, we find that the atom localization induced by the quantum interference is inferior to that induced by the squeezed vacuum. Furthermore, the combined effects of a squeezed vacuum and quantum interference have also been discussed under proper conditions. The present scheme provides a novel idea for the realization of atomic localization.