Nowadays, the Laguerre–Gaussian (LG) beam [1] has engendered tremendous advanced applications [2–7]. For example, it is widely used in the superresolution fluorescence microscopy such as the stimulated emission depletion [8, 9] and minimal photon fluxes [10, 11] in order to overcome the diffraction limit. Another approach to this target is utilizing the spatially dependent coherent light–matter interaction in atom-light coupling systems [12–14], which essentially depends on a spatially modulated atom-light coupling. By the detection of spontaneously emitted photons [15–18], level population [19–25], absorption [26–28], and gain [29, 30], subwavelength-scale atom localization can be obtained.

As far as we know, atom localization with LG beams can exhibit a large number of advantages [31, 32]. For example, the LG beam has a donut intensity spot naturally, which may avoid the need of two orthogonal standing wave (SW) fields for generating spatially modulated atom-light coupling in a two-dimensional (2D) atom localization system. That fact largely reduces the complexity of experimental implementation. Moreover, it is easier to create a single excitation spot in its core by a LG beam. In traditional SW-based localization schemes, due to the periodicity of the SW field intensity there may exist more than one localization spots within single optical wavelength. Therefore after one-time measurement, the probability of finding atoms at a certain position can be deeply reduced. So far, some approaches have been proposed to break this periodicity of SW fields, via utilizing the sensitivity of light–matter interactions to the light phase in a closed-loop atomic system [33, 34] and the interference of multiple SW fields with different wavelengths and phases [35, 36]. These methods, however, will increase the complexity of experimental setup. Although the LG beam has the aforementioned advantages in localization, it can only localize atoms in the vicinity of its beam core where the laser intensity is close to zero. Off-axis atom localization must be accompanied by the movement of the LG beam itself, which undoubtedly adds to extra complexity.

In traditional SW localization schemes, the superposition of multiple SW lasers with different wavelengths and phases is commonly adopted for reaching a single excitation point [35, 36]. In addition this effect between two LG beams can show interesting patterns such as optical Ferris wheels where the light intensity can be modulated to be zero in certain positions [37]. Inspired by these contributions, in the present work, we study a 2D off-axis atom localization in a three-level Λ system, in which a Gaussian beam serves as the probe field and a LG beam together with a Gaussian beam as the hybrid coupling field. The quantum interference effect between these two beams (LG and Gaussian) can achieve a unique zero-intensity spot at arbitrary position. We show that by appropriately tuning the ratio of peak amplitudes between the LG and any Gaussian beams, atoms can be localized at arbitrary position, with a certain distance to the beam core. Both the spatial resolution and radial distance of localization can be flexibly manipulated via tuning laser Rabi frequencies. Depending on the numerical simulation with experimental parameters our scheme enables the realization of an efficient off-axis 2D atom localization, accompanied by a best spatial resolution ∼ 200 nm and a radial distance of a few micrometers. Our scheme provides a more convenient route to the target of ultraprecise off-axis 2D atom localization.

To describe the scheme mechanism we consider a simple three-level Λ system as displayed in Figure 1, where states 1 and 3 are resonantly coupled by a weak probe field Ω _p and states 3 and 2 are connected by another coupling field Ω _c2, with zero detuning. In order to realize an off-axis atom excitation, we have assumed that the probe and coupling beams as Gaussian beams, which are Ω i r = Ω i 0 e − r 2 W i 2 , (1) where i = p, c2 and Ω _i0 represent the peak amplitude and W _i the Gaussian spot size. Remarkably, states 3 and 2 are also coupled by a second LG field Ω _c1 at the same time, as [38]. Ω c 1 r , θ = Ω c10 r W c 1 | l | e − r 2 W c 1 2 e i l θ + θ c 1 , (2) where Ω _c10 , W _c1, θ _c1, and l are the peak amplitude, the beam waist, the initial phase, and the winding number, respectively. r and θ are the cylindrical radius and the azimuthal angle, respectively. Here, we take the winding number l = 1, which enables a single-spot excitation. Other higher-order modes with l > 1 would lower the localization precision by redistributing the atomic population among multiple azimuthal nodes. To our knowledge, this three-level model can be experimentally realized by the D1 line of ultracold ⁸⁷Rb atoms with energy levels | 1 〉 = 5 S 1 / 2 , F = 1 , | 2 〉 = 5 S 1 / 2 , F = 2 , and | 3 〉 = 5 P 1 / 2 , F = 2 . Based on Ref. [14], we assume the decay rates from |3⟩ → |1⟩ and |3⟩ → |2⟩ are equal, typically calculated by Γ₃₁ = Γ₃₂ = 2π × 5.75 MHz. The decay rate between two hyperfine ground states |1⟩ and |2⟩ is Γ₂₁ = 5 kHz, satisfying Γ₂₁ ≪Γ₃₁, Γ₃₂ [39] so the lifetime of 2 is about 200μs. The beam width is W _i = W _c1 = W estimated to be same for simplicity. Under the frozen-gas limit where the atomic center of mass is unvaried we can take a measurement for the population on state |2⟩ by collecting its fluorescence signals with a CCD camera and a well-localized position distribution of atoms could facilitate this measurement [14].

Considering a frozen atomic gas the time evolution of the systematic density-matrix elements can be described by (ℏ = 1) [40]. ρ ̇ 11 = Γ 31 ρ 33 + Γ 21 ρ 22 − i Ω p ρ 31 − ρ 13 , ρ ̇ 33 = − Γ 31 + Γ 32 ρ 33 − i Ω p ρ 13 − ρ 31 − i Ω c ρ 23 − ρ 32 , ρ ̇ 12 = − γ 12 ρ 12 − i Ω p ρ 32 + i Ω c ρ 13 , ρ ̇ 13 = − γ 13 ρ 13 − i Ω p ρ 33 − ρ 11 + i Ω c ρ 12 , ρ ̇ 23 = − γ 23 ρ 23 + i Ω p ρ 21 − i Ω c ρ 33 − ρ 22 , (3) where ρ n m = ρ m n * and ∑ n = 1 3 ρ n n = 1 mean the conservation. The population on state |2⟩ is solved by ρ ₂₂ = 1 − ρ ₁₁ − ρ ₃₃. In deriving Eq. 3 we have defined Ω c = Ω c 1 + Ω c 2 , (4) representing the superposition of two coupling fields. Γ _nm denotes the spontaneous decay from n to m and γ _nm is defined as γ 12 = Γ 21 / 2 , γ 13 = Γ 31 + Γ 32 / 2 , γ 23 = Γ 32 + Γ 31 + Γ 21 / 2 . (5)

The steady solutions of Eq. 3 can be obtained by assuming ρ ̇ n m = ρ ̇ n n = 0 . Due to the presence of decay Γ₂₁, it is intuitive that ρ ₂₂ decreases with Γ₂₁. Luckily, accounting for the condition of Γ₂₁ ≪ Γ₃₁₍₃₂₎ that makes the effect of Γ₂₁ negligible [23, 39], then ρ ₂₂ takes a simple form of ρ 22 r , θ = 1 1 + I c r , θ / I p r , (6) where Γ₃₁ = Γ₃₂, Γ₂₁ = 0 are used. I c = Ω c 1 + Ω c 2 2 and I p = Ω p 2 stand for the laser intensities. Note that ρ ₂₂ (r, θ) reveals a position-dependent feature due to the use of several structured fields. From Eq. 6, it is apparent that the condition I _c (r _loc , θ) ≪ I _p (r _loc ) will cause ρ ₂₂ → 1, which means a perfect atomic confinement can be achieved at arbitrary position r _loc in our scheme.

According to Eq. 4 together with the definitions in Eqs 1, 2, the intensity of the hybrid coupling field can be written as I c r , θ = | Ω c 1 + Ω c 2 | 2 = Ω c10 W 2 e − 2 r 2 W 2 r + κ c W e − i θ + θ c 1 2 , (7) where the peak ratio is κ _c = Ω _c20 /Ω _c10 , which can be tuned by Ω _c20 if Ω _c10 is fixed. Note that this hybrid coupling field is composed by one LG beam and one Gaussian beam, which resonantly couple states |2⟩ and |3⟩ at the same time. Finally we can arrive at an analytical solution to the equation I _c (r, θ) = 0, that is, the perfect condition of localization can be reached at r loc , θ loc = κ c W , π − θ c 1 , (8) where the population ρ ₂₂ attains 1.0 in principle. That means atoms can be precisely placed at any desired position (r _loc , θ _loc ) with a very high probability. While in fact, owing to the influence from intrinsic noises in the experimental setup, the observed localization resolution is quite limited. In Section 5 we will discuss the fluctuation of laser intensities, the steady time as well as the noise from atomic thermal motion, in order to present a practical estimation for the experimental observation. Moreover, we have to point out that benefiting from the interference between two hybrid coupling fields Ω _c1 and Ω _c2 [41], the localization position (r _loc , θ _loc ) can be widely adjusted by the beam parameters, which is not restricted merely at the beam core as in most previous works [31, 32].

As illustrated in Figure 2A, we show that atoms denoted as the steady population ρ ₂₂ on state |2⟩, can be confined in any spatial position (r _loc , θ _loc ) by changing the parameters (κ _c , θ _c1). For example when (κ _c , θ _c1) = (0.1, π), (0.5, π), (0.5, π/4), (0.5, − π/4), Figure 2A explicitly shows the off-axis atom localization at different positions as labeled by A ∼D. Figures 2B1–B4 amplify the distribution of atoms at different localized places. It is apparent that the spatial resolution of atom localization remains unchanged for different κ _c and θ _c1. Therefore, thanks to the zero-intensity point (I _c (r, θ) = 0) created by the interference between two light beams Ω _c10 and Ω _c20 , our scheme can realize an effective off-axis localization at arbitrary position in a 2D space.

The quality of localization also depends on a high spatial resolution, which is characterized by the full width at half maximum (FWHM) of the steady distribution ρ ₂₂ (r, θ). A narrower linewidth indicates that the position of atoms can be well-resolved within a smaller range. By replacing the profiles of light fields (Eqs 1, 2) the expression of ρ ₂₂(r) takes an explicitly Lorentz form ρ 22 r = 1 1 + r − κ c W 2 κ p 2 W 2 , (9) where we have omitted the azimuth angle by letting θ = π − θ _c1 and paid attention to the variation of ρ ₂₂(r) along the radial direction. We treat the FWHM of function ρ ₂₂(r) as a measurement to localization, which can also be analytically solved, a r = 2 κ p W . (10)

In Figure 3, we plot the steady distribution ρ ₂₂(r) vs. r for different peak ratios κ _p . Clearly a weaker probe field leads to the atomic population more confined in the vicinity of the localization point r = r _loc = 0.5W, promising for a higher resolution localization. For example, we find that a _r = 0.02W when κ _p = 0.01, but this value is decreased by one order of magnitude, which is a _r = 0.004W as κ _p reduces to 0.002. From Eq. 10, it is intuitive that a _r → 0 if κ _p ≪ 1, enabling an ultraprecise localization under a sufficiently weak probe field. However in a realistic system, the fact that the time for a steady state becomes much longer in the weak probe limit, results in the atomic motion non-negligible. We will discuss this point in Section 5.2.

The numbers presented in this work are considered from ⁸⁷Rb where the lifetime of state |3⟩( = |5P _1/2, F = 2⟩) is 27.7 ns [42], leading to the decay rates Γ₃₁ = Γ₃₂ = 2π × 5.75 MHz, and the lifetime of |2⟩( = |5S _1/2, F = 2⟩) is 200 μs, leading to Γ₂₁ = 5 kHz. We assume that the co-propagating probe and coupling lasers are overlapping in space and have a same beam width W = 5 μm. As explicitly presented in section 3 and section 4, our scheme can achieve an ultraprecise off-axis atom localization due to the flexible manipulation of peak ratios κ _c and κ _p , together with the azimuth angle θ _c1. Due to the rotational invariance we ignore θ _c1 by focusing on the radial distance r. However, as for an experimental implementation these parameters are also restrained. In this section, we numerically solve the spatial resolution a _r and the peak value of ρ ₂₂(r) by evolving the motional Eq. 3 under more realistic conditions coming from measurement.

To conclude, our scheme presents a novel 2D atom localization, having both ultraprecise and off-axis features. Differing from the previous works using a single LG field we adopt a LG beam together with a Gaussian beam as the hybrid coupling field. The previous contributions can only localize atom in the beam core where the intensity of coupling field is zero. While our protocol shows that atoms can be localized at arbitrary position due to the effect of quantum interference between these two coupling beams that leads to a zero-intensity spot in space. Our numerical simulation confirms that with an appropriate adjustment for the peak ratios of laser Rabi frequencies a wider off-axis localization range and higher quality spatial resolution can be achieved at the same time. Under experimentally feasible parameters an estimation for the implementation of realistic off-axis atom localization is predicted, promising for a resolution of ∼ 200 nm and a localized radius of a few μm. In addition, we also discuss the weakness of our scheme when some intrinsic quantum noises from imperfect measurement, including laser intensity noise, limited steady time, and atomic thermal motion, are considered. Our approach may provide unique application to atomic lithography with more flexibility and better resolution [46]. An extension to the 3D off-axis atom localization is possible by implementing a spatial modulation to the probe detuning which is our next-step work [32].