1. Introduction
In recent decades, considerable attention and research efforts have been directed towards precisely localizing atoms, a field known as atomic microscopy, from both experimental and theoretical perspectives. This focus has arisen from the diverse range of applications that atomic microscopy offers across various domains such as Bose–Einstein condensation, atomic nanolithography and neutral atom trapping, as highlighted in previous studies [1–9]. Early investigations conducted by specific research groups achieved atomic localization within a cavity using optical slits. In these studies, the probability distribution of atomic localization was assessed by monitoring the phase shift of the optical field induced by atom–field interactions [10, 11]. Notably, phenomena such as quantum interference and coherence play pivotal roles in various aspects, including giant Kerr nonlinearity, electromagnetically induced transparency, modulation of emission, four-wave mixing and spontaneous optical bistability [12–23].
Several methodologies have been proposed for localizing atoms in one-dimensional (1D) systems, leveraging atomic coherence and quantum interference phenomena. Herkommer et al utilized the Autler–Townes spontaneous spectrum methodology to investigate atomic localization within a 1D system [24]. Paspalakis et al achieved atomic localization in a three-level atomic medium using classical standing waves and a weak probe field [25, 26]. Quantum interference phenomena were employed by Jin et al to achieve atomic localization within a ladder-type four-level arrangement [27]. Kapale and Zubairy, as well as Sahrai et al, utilized absorption spectroscopy to modulate both amplitude and phase for 1D atomic localization [28, 29]. Additionally, alternative techniques such as coherent population dark resonance or trapping, superposition of two standing waves and surface plasmons have been employed for the localization of 1D atomic systems [30–34].
In recent years there has been a significant increase in scientific investigations aimed at the localization of atoms within two-dimensional (2D) spatial domains [35–44]. Wan et al demonstrated atomic localization by employing a four-level Y-type atomic system, achieving confinement within an area measuring half the wavelength in both dimensions with a probability of 25% [35]. Li et al implemented a Λ-type four-level atomic arrangement to achieve 2D atomic localization through the utilization of radiofrequency-driven fields and phase-sensitive absorption spectroscopy [36]. Ding et al investigated atomic localization within a 2D four-level atomic setup, effectively confining atoms within an area measuring half the wavelength in both dimensions, and achieving a 100% probability of localization through probe absorption [37]. Various other methods for localizing atoms in two dimensions have been explored, including spontaneous emission, level population measurements and absorption spectra of gain [38–44]. Furthermore, advances in atomic localization have been observed across diverse sources employing various methodologies within the domains of quantum optics and quantum plasmonics [45–59].
We introduce a three-level λ-type atomic configuration for the investigation of precise ultrahigh-resolution atomic localization in 2D space. This investigation explores the symmetric and asymmetric superposition of standing wave fields. Our research emphasizes the capability to localize an atom within an small area of λ/35 × λ/35, achieving exceptional precision and resolution in atomic localization with the highest probability. Importantly, we observe the pivotal roles played by symmetric and asymmetric superposition in the emergence of localized peaks within the 2D space bounded by −π ≤ ky ≤ π and −π ≤ kx ≤ π. As the atom propagates within the 2D standing waves, changes in the spatial interaction between light and atom occur due to the position-dependent Rabi frequencies within the standing waves. Consequently, absorption probability peaks occur within the regions traversed by the atom, precisely within the locality of λ/35 × λ/35, while remaining negligibly low elsewhere in the 2D space. The structure of this manuscript is organized as follows: section 2 elucidates our proposed model system and its dynamics, while section 3 presents the numerical findings and discusses their implications for 2D atomic localization. Finally, section 4 provides a summary of our key conclusions.
2. System model and dynamics
We have investigated a three-level system with the λ-type configuration, as illustrated in figure 1. In this atomic configuration, the quantum state denoted by $\left|3\right\rangle $ indicates an excited state, while $\left|1\right\rangle $ and $\left|2\right\rangle $ represent the ground states. The αp (the Rabi frequency of the probe field) facilitates the transition between quantum states $\left|1\right\rangle $ and $\left|3\right\rangle $. The Rabi frequency αc(x, y), which drives the transition between quantum states $\left|2\right\rangle $ and $\left|3\right\rangle $, describes a standing wave field. In our model, the standing waves have components along both the x- and y-axes, but their full wave vectors k1 and k2 can have non-zero components in other directions [60]. For clarity, we are only considering the projections of these wave vectors along the x- and y-directions for the purpose of 2D localization. The Rabi frequencies in this configuration are given by3 ) and (4 ), we derive the equations governing the dynamic transitions within the defined atomic medium. To investigate atomic localization, specifically concerning the density matrix element ${\mathop{\widetilde{\rho }}\limits^{\cdot }}_{13}$, we apply the following rate equations:
$\begin{eqnarray}{\rm{\Omega }}(x,y)={{\rm{\Omega }}}_{0}\sin ({k}_{1x}x+{\phi }_{1})+{{\rm{\Omega }}}_{0}\sin ({k}_{2y}y+{\phi }_{2}),\end{eqnarray}$
where ${k}_{1x}={k}_{1}\cos {\theta }_{1}$ and ${k}_{2y}={k}_{2}\cos {\theta }_{2}$, with θ1 and θ2 being the respective angular separation of the wave vector from the x-(y-)direction. To simplify the notation, we introduce the parameters ${\eta }_{1}=\cos {\theta }_{1}$ and ${\eta }_{2}=\cos {\theta }_{2}$, leading to $\begin{eqnarray}{\rm{\Omega }}(x,y)={{\rm{\Omega }}}_{0}\sin ({\eta }_{1}{k}_{1}x+{\phi }_{1})+{{\rm{\Omega }}}_{0}\sin ({\eta }_{2}{k}_{2}y+{\phi }_{2}),\end{eqnarray}$
where the wave vector ki = 2π/λi (where i = 1, 2), the phase shift introduced by the formation of standing wave components is denoted by φi (where i = 1, 2) and η1 and η2 represent the fractions of the wave vectors k1 and k2 aligned with the x- and y-axes, respectively. We note that symmetric superposition occurs when η1 and η2 equal 1 or −1, respectively, while asymmetric superposition arises for −1 < η1,2 < 1, which significantly influences the evolution of the peaks of localization in 2D space [61–63]. To experimentally control and vary η1 and η2 independently, we can adjust the angular separation θ1 and θ2 of the standing wave fields using beam-steering devices. In the context of 2D atomic localization, the decay rate of the excited state denoted as $\left|3\right\rangle $ is ascertained to be γ = 2π × 6.07 MHz, indicative of the natural linewidth of the D2 line in 87Rb. In subsequent numerical analyses, all parameters are scaled with respect to γ. Further, the rotating wave approximation yields the resultant Hamiltonian that defines the behavior of the atomic medium as [64] $\begin{eqnarray}{H}_{I}=-\displaystyle \frac{{\hslash }}{2}[{{\rm{\Omega }}}_{p}{{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{p}t}\left|3\right\rangle \left\langle 1\right|+{{\rm{\Omega }}}_{c}(x,y){{\rm{e}}}^{-{\rm{i}}{{\rm{\Delta }}}_{c}t}\left|3\right\rangle \left\langle 2\right|+{\rm{H}}.{\rm{C}}.],\end{eqnarray}$
where H.C. represents the Hermitian conjugate. In this system, the characterization of field detunings becomes paramount, with Δc signifying the difference between the atomic transition frequency ω23 and the control field frequency ωc, and Δp denoting the difference between the atomic transition frequency ω13 and the probe field frequency ωp. These parameters play a pivotal role in delineating the interactions inherent within the atomic system. The density matrix formalism, which examines the dynamics and evolution of the system, is expressed as [65] $\begin{eqnarray}{\dot{\rho }}_{ij}=-\displaystyle \frac{{\rm{i}}}{\hslash }[{H}_{I},\rho ]-\displaystyle \frac{1}{2}\sum {\gamma }_{j}({\sigma }^{\dagger }\sigma \rho +\rho {\sigma }^{\dagger }\sigma -2\sigma \rho {\sigma }^{\dagger }),\end{eqnarray}$
where HI denotes the interaction Hamiltonian that governs the interactions within atomic systems. The decay rates of the atomic transitions are represented by γj (where j = 1, 2). Furthermore, σ† and σ correspond to the raising and lowering operators, respectively. Utilizing equations ( $\begin{eqnarray}{\mathop{\mathop{\rho }\limits^{\unicode{x0007E}}}\limits^{\cdot }}_{13}=A{\mathop{\rho }\limits^{\unicode{x0007E}}}_{13}+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{\mathop{\rho }\limits^{\unicode{x0007E}}}_{33}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{\mathop{\rho }\limits^{\unicode{x0007E}}}_{11}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{c}(x,y){\mathop{\rho }\limits^{\unicode{x0007E}}}_{12},\end{eqnarray}$
$\begin{eqnarray}{\mathop{\mathop{\rho }\limits^{\unicode{x0007E}}}\limits^{\cdot }}_{12}=B{\mathop{\rho }\limits^{\unicode{x0007E}}}_{12}+\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{p}{\mathop{\rho }\limits^{\unicode{x0007E}}}_{32}-\displaystyle \frac{{\rm{i}}}{2}{{\rm{\Omega }}}_{c}^{\ast }(x,y){\mathop{\rho }\limits^{\unicode{x0007E}}}_{13},\end{eqnarray}$
where $\begin{eqnarray}A={\rm{i}}{{\rm{\Delta }}}_{p}-\displaystyle \frac{1}{2}{\gamma }_{1},\end{eqnarray}$
$\begin{eqnarray}B={\rm{i}}({{\rm{\Delta }}}_{p}-{{\rm{\Delta }}}_{c})-\displaystyle \frac{1}{2}({\gamma }_{1}+{\gamma }_{2}).\end{eqnarray}$
Figure 1. (a) Schematic diagram of the λ-type three-level atomic system using 87Rb. The symbols αc(x, y) and αp denote the magnitudes of the Rabi frequencies of the standing wave and probe fields, respectively. Additionally, γ1 and γ2 indicate the decay rates associated with the atomic transitions, reflecting the probabilities of spontaneous emission from the excited atomic states. (b) Field configurations for two-dimensional localization. |
In our investigation we observed that the strength of the probe field is significantly weaker than that of the other laser field (i.e. αp ≪ αc(x, y)). Furthermore, we assume that most atoms predominantly occupy the ground state ∣1⟩, leading to the density matrix element ${\widetilde{\rho }}_{11}$ being approximately equal to unity (i.e. ${\widetilde{\rho }}_{11}\approx 1$). Under these conditions, we employ a perturbative approach to derive the solution for the steady-state density matrix element ${\mathop{\widetilde{\rho }}\limits^{\cdot }}_{13}$ by solving equations (5 ) and (6 ), given as9 ) and (10 ) we arrive at a final expression for the optical susceptibility of the proposed λ-type system as follows:11 ) represents the two-dimensional distribution of susceptibility, χ(x, y), for a three-level atomic system arranged in a λ-type configuration. The term αc(x, y) describes the spatial variation of the Rabi frequency. This equation illustrates how the susceptibility varies across a 2D domain as a function of position.
$\begin{eqnarray}{\mathop{\widetilde{\rho }}\limits^{\cdot }}_{13}=-\displaystyle \frac{2{\rm{i}}B{{\rm{\Omega }}}_{p}}{4{AB}+{{\rm{\Omega }}}_{c}^{2}(x,y)}.\end{eqnarray}$
The optical susceptibility of a material arises from its interaction with an external electromagnetic field. This interaction is characterized by the transition element ${\mathop{\widetilde{\rho }}\limits^{\cdot }}_{13}$, which signifies the connection between the material’s ground state ∣1⟩ and an excited state ∣3⟩. More precisely, the optical susceptibility χ(x, y) is directly related to the transition element ${\mathop{\widetilde{\rho }}\limits^{\cdot }}_{13}$, expressed as a proportional relationship given by [64] $\begin{eqnarray}\chi (x,y)=\displaystyle \frac{| {\mu }_{13}{| }^{2}2N}{{\epsilon }_{0}{\hslash }{{\rm{\Omega }}}_{p}}{\mathop{\widetilde{\rho }}\limits^{\cdot }}_{13},\end{eqnarray}$
where N represents the atomic density, μ13 signifies the dipole moment between the states ∣1⟩ and ∣3⟩, ε0 denotes the permittivity of a vacuum and ℏ symbolizes the reduced Planck constant. Utilizing equations ( $\begin{eqnarray}\chi (x,y)=\displaystyle \frac{-2N| {\mu }_{13}{| }^{2}}{{\epsilon }_{0}{\hslash }}\left[\displaystyle \frac{2{\rm{i}}B}{4{AB}+{{\rm{\Omega }}}_{c}^{2}(x,y)}\right].\end{eqnarray}$
Equation (3. Findings and discussion
We have investigated 2D atomic localization within a one-wavelength domain by utilizing equation (11 ) and adjusting multiple system parameters. These parameters include fundamental constants such as Planck’s constant (ℏ = 1.05 × 10−34 J s), vacuum permittivity (ε0 = 1.054 × 10−12 N m2 C−2) and the number density of atoms (N = 2 × 1017 m−3) [66–69]. Figure 2 illustrates the probability distribution ($W(x,y)\propto \mathrm{Im}[\chi ]$) as a function of (kx, ky) within a single-wavelength domain. This distribution facilitates an understanding of atomic localization by analyzing the absorption profile of the probe spectrum. Our investigation elucidates how the resolution and precision of 2D atomic positioning are influenced by both asymmetric and symmetric combinations of standing wave fields. Under symmetric superposition conditions (η1 = −1 and η2 = 1), two localized peaks are observed in quadrants II and IV (figures 2(a) and (b)). These absorption maxima correspond to regions of maximum probability of presence of an atom. Moreover, asymmetric superposition (−1 < η1,2 < 1) notably enhances the precision and spatial resolution of atomic localization. When asymmetric superposition occurs (η1 = −0.6 and η2 = 0.4) in 2D space, a single narrow localized peak emerges, indicating probe absorption at a specific region (figures 2(c) and (d)), predominantly in quadrant IV of the 2D space. Figures 2(a) and (c) present intensity plots in 2D space, while figures 2(b) and (d) depict corresponding density plots, providing a clear visualization of atomic localization. Different configurations of standing waves impact the resolution of atomic localization. We explore the complex relationship between interference patterns, atom–light interactions and spectral characteristics. Specific parameters, such as wave intensity ratios and phase differences, generate unique interference patterns within the atomic medium. These patterns influence atom–light interactions and observed absorption profiles. Symmetric standing waves (η1 = −1 and η2 = 1) create nodes and antinodes, enhancing resolution by concentrating atomic probability in specific spatial regions. Asymmetric configurations (e.g. η1 = −0.6 and η2 = 0.4) produce sharper peaks due to constructive interference in some areas and destructive interference elsewhere. Peak width and shape correlate directly with resolution.
Figure 2. The distribution of probability (W(x, y)) versus kx and ky, with specific parameters including γ = 2π × 6.07 MHz, γ1 = γ2 = Γ = 0.3γ, α0 = 10γ, Δc = 5γ, Δp = 10γ, φ1 = 0 and φ2 = π/4. In parts (a) and (b), η1 = −1 and η2 = 1, while in parts (c) and (d), η1 = −0.6 and η2 = 0.4. The left panels display intensity plots, while density plots are depicted in the right panels. |
We further investigate the effects of varying intensities of control fields (α0) and probe field detuning (Δp) on the observed phenomena in 2D atomic localization. As the intensity of the control fields decreases (α0 = 9.9γ), our observations reveal a phenomenon characterized by the broadening of localized peaks that exhibit spike-like patterns, particularly located in quadrant IV (see figures 3(a) and (b)). Concurrently, there is a reduction in the probability of localization within a given wavelength range, indicating a decrease in precision and resolution in 2D atomic localization. Further analysis, depicted in figures 3(c) and (d), explores the influence of probe field detuning on localization behavior. Specifically, when the probe detuning is set at Δp = 9.9γ, a distinct crater-like pattern emerges, prominently situated within quadrant IV. Notably, atoms tend to localize along the rings surrounding these crater-like structures. Furthermore, there is an observable widening of the localized peak with a decrease in probe field detuning. Consequently, there is a decrease in the probability of finding the atom within a specific wavelength range, highlighting compromised accuracy and resolution inherent in 2D atomic localization under these circumstances. It is noticed that lower control field intensities lead to broader and less sharply defined peaks in the spatial localization and distribution of atoms within the standing wave field due to the fundamental principles of quantum mechanics, specifically the Rabi oscillation phenomenon. Rabi oscillations occur when a two-level quantum system (such as an atom) interacts with an external oscillating field, such as the control field in this case. At lower intensities of the control field, the Rabi frequency (the rate at which the system oscillates between its two states) is reduced. This reduction in Rabi frequency leads to slower and less efficient population transfer between the two states of the atom. As a result, atoms experience weaker coupling with the control field and undergo less pronounced transitions between energy levels. Consequently, the spatial localization and distribution of atoms within the standing wave field become broader and less sharply defined. In essence, the lower intensity of the control field reduces the precision of the manipulation of atomic states, leading to reduced spatial resolution in the distribution of atoms within the field. Similarly, changes in probe field detuning impact the formation of distinct spatial patterns, affecting the precision and accuracy of atomic localization.
Figure 3. The distribution of probability (W(x, y)) versus kx and ky: (a), (b) α0 = 9.9γ and (c), (d) Δp = 9.9γ. The other parameters have the same values as in figure 2. |
The modulation of decay rate (Γ) significantly influences the 2D atomic localization phenomenon. Decreasing decay rates from 0.3γ to 0.03γ induce a substantial reduction in spontaneous decay losses relative to the amplitude of the probe detuning and position-dependent Rabi frequency. Consequently, a distinct localized peak emerges, as depicted in figure 4(a) as an intensity plot within quadrant IV, accompanied by the corresponding density plot in figure 4(b). This decline in decay rates increases the probability of precise atomic localization within a single-wavelength range, thereby facilitating highly accurate 2D atomic localization. Further reducing decay rates from 0.03γ to 0.003γ achieves ultrahigh precision and resolution in atomic localization in 2D space, particularly within quadrant IV, as supported by the probe absorption spectra (illustrated in figures 4(c) and (d)). The conditional position probability peaks around 1, indicating complete localization of the atom in a region less than λ/35 × λ/35. Modulating decay rates (γ1,2) in 2D atomic localization is crucial in quantum optics and atomic physics. Higher rates, such as 0.3γ, shorten atomic coherence and broaden response spectra. Lower rates, for example 0.03γ or 0.003γ, extend coherence, reducing spontaneous decay losses. This decrease minimizes broadening effects in absorption spectra, preserving features of atomic localization. Lower rates also enhance precise atomic localization, resulting in sharper absorption peaks and higher resolution. Control over decay rates finds applications in nanolithography and laser cooling, enabling ultrahigh precision in atomic manipulation and technological innovation.
Figure 4. The distribution of probability (W(x, y)) versus kx and ky: (a), (b) γ1 = γ2 = 0.03γ and (c), (d) γ1 = γ2 = 0.003γ. The other parameters have the same values as in figure 2. |
To explicitly demonstrate the effect of varying control field intensities (α0) and decay rates (Γ) on atomic localization, we discuss figures 5(a)–(d). Figures 5(a) and (b) show the effects of different control field intensities (α0) on 2D atomic localization in a symmetric superposition of standing wave fields. We observe that decreasing the control field intensities broadens both the two and the single localized peaks. Furthermore, the atomic localization shows behavior identical to that in figures 5(a) and (b) with the variation of probe field detuning (Δp). Conversely, figures 5(c) and (d) illustrate the effects of varying decay rates (Γ) on 2D atomic localization in an asymmetric superposition of standing wave fields. Lower control field intensities result in less precession in the two and single localized peaks. This leads to broader, less-defined peaks in the spatial localization of atoms due to quantum mechanics, particularly the Rabi oscillation phenomenon. With reduced control field intensity, the Rabi frequency decreases, causing slower and less efficient population transfer between atomic states. Consequently, atoms experience weaker coupling with the control field, resulting in broader and less sharply defined spatial distributions. Essentially, a lower control field intensity reduces the precision of atomic state manipulation, decreasing spatial resolution. However, lower decay rates enhance precise atomic localization, resulting in sharper absorption peaks and higher resolution. Thus, we have introduced a method to achieve precise and efficient 2D atomic localization within the superposition of standing wave fields.
Figure 5. (a), (b) The distribution of probability (W(x, y)) versus kx, ky and α0. (c), (d) The distribution of probability (W(x, y)) versus kx, ky and Γ. The other parameters have the same values as in figure 2. |
Before summarizing our work, the experimental realization of our proposed three-level λ-type atomic system to achieve high-resolution atomic localization involves utilizing cold 87Rb atoms and the transition of the D2 line (5S1/2 → 5P3/2). In our envisioned setup, we focus on two ground states: $\left|1\right\rangle =\left|5{{\rm{S}}}_{1/2},\,F=1\right\rangle $ and $\left|2\right\rangle =\left|5{{\rm{S}}}_{1/2},\,F=2\right\rangle $, along with one excited state $\left|3\right\rangle =\left|5{\rm{P}}_{3/2},\,F=2\right\rangle $, as depicted in figure 6. The excitation process involves two specific transitions, such that $\left|5{S}_{1/2},F=1\right\rangle \leftrightarrow \left|5{P}_{3/2},F=2\right\rangle $ driven by a weak probe field (αp), and $\left|5{{\rm{S}}}_{1/2},\,F=2\right\rangle \leftrightarrow \left|5{{\rm{P}}}_{3/2},\,F=2\right\rangle $ excited by a strong standing wave field (αc(x, y)). The atom’s precise localization beyond optical resolution is determined when it traverses through the intersection of these orthogonal fields. The spatial information encoded within this region’s nonlinear response enables high-resolution atomic localization. For experimental validation, fluorescence imaging or atom probe microscopy can be employed to detect and visualize atomic positions with enhanced resolution. These techniques will substantiate the feasibility of our proposed theoretical framework using cold 87Rb atoms.
Figure 6. Possible experimental schematic of a three-level λ-type medium (87Rb). |
4. Summary
In summary, our investigation concentrated on accurately determining the spatial distribution of atoms within a 2D ensemble governed by a three-level λ-type atomic configuration. This localization was achieved through interactions with both asymmetric and symmetric superpositions of standing waves. Our findings represent a significant advance in the accuracy and resolution of 2D atomic localization, achieving an exceptionally high standard. Notably, we observed a marked influence of both symmetric and asymmetric standing waves on the variation of localized peaks within the spatial boundaries defined by −π ≤ ky ≤ π and −π ≤ kx ≤ π. Additionally, we explored the effects of various parameters, including the intensity of the control field, probe field detuning and decay rates, on atomic localization. Within a single-wavelength domain, we noted a more effective generation of sharply localized peaks, both single and double. Theoretically, we attained ultrahigh precision and resolution in atomic localization, pinpointing the atom to an area smaller than λ/35 × λ/35. These results hold considerable promise for applications in the trapping and laser cooling of neutral atoms, atomic nanolithography, the measurement of center-of-mass wave functions and both fundamental and applied research in physics.
Acknowledgments
Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2024R8), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Declarations
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Competing interests
The author has no relevant financial or non-financial interests to disclose.
Author’s contributions
Muhammad Idrees: conceptualization, writing—original draft, software, methodology. Ahmed S Hendy: writing—review and editing, formal analysis. Zareen A Khan: investigation, visualization, project administration, supervision.