Show Summary Details
More options …

# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

1 Issue per year

IMPACT FACTOR 2016 (Open Physics): 0.745
IMPACT FACTOR 2016 (Central European Journal of Physics): 0.765

CiteScore 2016: 0.82

SCImago Journal Rank (SJR) 2015: 0.458
Source Normalized Impact per Paper (SNIP) 2015: 1.142

Open Access
Online
ISSN
2391-5471
See all formats and pricing
More options …
Volume 16, Issue 1

# Three-dimensional atom localization via probe absorption in a cascade four-level atomic system

Wei Zeng
/ Li Deng
/ Aixi Chen
• Corresponding author
• Department of Physics, Zhejiang SCI-TECH University, Hangzhou, 310018, China
• Other articles by this author:
Published Online: 2018-03-20 | DOI: https://doi.org/10.1515/phys-2018-0009

## Abstract

For an atomic system with cascade four-level type, a useful scheme about three-dimensional (3D) atom localization is proposed. In our scheme the atomic system is coherently controlled by using a radio-frequency field to couple with two-folded levels under the condition of the existence of probe absorption. Our results show that detecting precision of 3D atom localization may be obviously improved by properly adjusting the frequency detuning and strength of the radio-frequency driving field. So our scheme could be helpful to realize 3D atom localization with high-efficiency and high-precision . In the field of laser cooling or the atom nano-lithography, our studies provide potential applications.

PACS: 42.50.-p; 42.50.Ct

## 1 Introduction

With the development of the technology of quantum coherent control, the research about atom localization has developed rapidly, and some researchers pay attention to this physical field. Via manipulation of laser, several pioneering works about atom localization has been proposed, where position of an atom going through a standard light field can be localized by carrying out a phase measurement on a probe field [1, 2, 3], and also be localized by using Ramsey interferometry [4], and other optical methods [5, 6]. Subsequently, in the papers of Zubairy and his coworkers, they have given some rather simple schemes in which they restrict the space of an atomic notion via applying the technology of Autler-Townes microscopy, detecting photons of spontaneous emission, and controlling phase and amplitude of the absorption spectra [7, 8, 9]. In Zubairy’s schemes, spontaneous emission need to be well controlled. We know it is difficult to effectively control spontaneous radiation in experiment, so some other new schemes [10, 11, 12, 13, 14, 15, 16] have been introduced, in which one can obtain atom localization by measuring atomic population of the upperstate or detecting absorption spectra of probe field, and by means of some coherent process, such as coherent population trapping, double-dark resonances, or coherent manipulation of the Raman gain. Besides these works in the field of theory, recently, atom localization in one-dimensional space has been achieved in a proof-of-principle experiment with the help of the technique of electromagnetically induced transparency [17].

Along with deep studies about atom localization, several theoretical schemes about atom localization in two-dimensional space have also been put forward by using two standing-wave light fields whose propagation direction is orthogonal to each other to couple with atomic system. For example, taking a measurement on the atom population of ground state or of upper state, Ivanov’s research group gave a new scheme where they obtained atom localization in two-dimensional space [18]. Later, a number of research papers [19, 20, 21, 22, 23, 24, 25] about atom localization in two-dimensional space have been published successively.

For an atom with motion, researchers prefer to limit its motion in three-dimensional space, so how to realize the atom localization in three-dimensional space begins to be concerned. Several schemes [26, 27, 28] about atom localization in three-dimensional (3D) space in various systems of atom have been reported. However, 3D atom localization based on different coupling mechanisms via taking a measurement on absorption and gain spectrum of the probe field is not considered. In this paper, we study a four-level atomic system with cascade type and we investigate its three-dimensional atom localization through detecting the absorption and gain spectra of the probe field. In our system, interaction between field and atom is dependent on space coordinates, so we can use the technology of detecting probe absorption and gain spectra to determine the position probability distribution of the atoms when they go through standing waves field. There are two channels of excitation in the closed interacting system: one is one-photon excitation and the other is three-photon excitation that depends on phase, and the quantum interference between two channels helps in achieving 3D atom localization. Through measuring absorption of probe field or its gain at a special frequency, we can find atom with 100% probability in three-dimensional space. By contrast with some schemes introduced in references [26, 27, 28], our scheme is added by a factor of 4 or 8.

## 2 The physical model and dynamic evolution

An atomic system with four cascade energy levels is studied and its concrete atomic structures are shown in Figure 1. State |1〉 is a ground state, |2〉 is an intermediate state, and the top two states |3〉 and |4〉 are two-folded levels. Where we use three standing-wave fields those are orthogonal to each other and whose Rabi frequency 2Ω(j)sin (kj) (j = x, y, z; k = ω/c) depends on position to couple with the transition between levels |2〉 and |3〉, and the total Rabi frequency 2Ωs(x,y,z) may be expressed as 2Ωs(x,y,z) = 2Ω(x)sin(kx)+2Ω(y)sin(ky)+2Ω(z)sin(kz). A weak probe field with angular frequency ωp and Rabi frequency 2Ωp interacts with the atomic system and excites the transition between levels |2〉 and |1〉. A radio-frequency field with angular frequency ωrf and Larmor frequency 2Ωrf is applied to drive transition between the two-folded levels |3〉 and |4〉.

Figure 1

Schematic diagram of a four-level atomic system

Here we neglect the kinetic part of the atom from the Hamiltonian according to the Raman-Nath approximation, and suppose that the center-of-mass coordinate of the atom almost does not change with time along with the directions of the laser when the intensity of Rabi frequency of the laser field is big enough. Under the condition of rotation wave approximation, the interaction Hamiltonian of this system is given by

$HI=Δxyz+Δp33+Δp22+Δrf+Δxyz+Δp44−(Ωsx,y,z32+Ωp21+Ωrf43+H.c.),$(1)

where we let ħ=0, Δrf=ω43ωrf, Δxyz=ω32ωxyz, and Δp=ω21ωp are corresponding to detuning of the radio-frequency field, the standing-wave field and the probe field, respectively, and ω43, ω32, and ω21 are corresponding to transition frequency between levels |4〉 and |3〉, between levels |3〉 and |2〉, and between levels |2〉 and |1〉, respectively.

According to the equation satisfied by the evolution of density matrix elements

$∂ρmn∂t=1iℏ∑kHmkρkn−ρmkHkn−12∑kΓmkρkn+ρmkΓkn,(k=1,2,3,4)$(2)

where Γmn stands for relaxation matrix elements, and it can be expressed as 〈n|Γ|m〉=γnδnm, and γn is the decay rate of level |n〉. Based on Eqs. (1) and (2), we can obtain the first derivative of density matrix elements with time:

$∂ρ11∂t=γ2ρ22−iΩpρ12+iΩp∗ρ21,$(3a)

$∂ρ22∂t=−γ2ρ22+γ3ρ33+γ4ρ44+iΩpρ12−iΩp∗ρ21−iΩsx,y,zρ23+iΩs∗(x,y,z)ρ32,$(3b)

$∂ρ33∂t=−γ3ρ33+iΩrf∗ρ43−iΩrfρ34+iΩsx,y,zρ23−iΩs∗(x,y,z)ρ32,$(3c)

$∂ρ44∂t=−γ4ρ44+iΩrfρ34−iΩrf∗ρ43,$(3d)

$∂ρ21∂t=−γ22+iΔpρ21+iΩs∗x,y,zρ31−iΩp(ρ22−ρ11),$(3e)

$∂ρ31∂t=−γ32+iΔxyz+Δpρ31+iΩsx,y,zρ21+iΩrf∗ρ41−iΩpρ32,$(3f)

$∂ρ41∂t=−γ42+iΔxyz+Δp+Δrfρ41+iΩrfρ31−iΩpρ42,$(3g)

$∂ρ32∂t=−γ2+γ32+iΔxyzρ32+iΩrf∗ρ42−iΩp∗ρ31−iΩs(x,y,z)(ρ33−ρ22),$(3h)

$∂ρ42∂t=−γ2+γ42+iΔxyz+Δrfρ42+iΩrfρ32−iΩp∗ρ41−iΩs(x,y,z)ρ43,$(3i)

$∂ρ43∂t=−γ3+γ42+iΔrfρ43+iΩrf(ρ33−ρ44)−iΩs∗(x,y,z)ρ42,$(3j)

where ρ11+ρ22+ρ33+ρ44=1, γ2, γ3 and γ4 are decay rates for |2〉→|1〉, |3〉→|2〉and |4〉→|2〉, respectively. Based on the theory of light propagating in atomic medium, we get the expression of the complex susceptibility for the probe field

$χp=Nμ212ϵ0ℏΩpρ21$(4)

where physical parameter ε0 is the free space permittivity and N is the atom number density of four-level atomic system.

According to the theory about light propagating in medium, we know the imaginary part Im(χp) of the susceptibility can be applied to describe absorption of probe field. To simplify the formula, we let δ2= $\begin{array}{}\frac{{\gamma }_{2}}{2}\end{array}$ +p, δ3= $\begin{array}{}\frac{{\gamma }_{3}}{2}\end{array}$ +i(Δxyz+Δp), and δ4= $\begin{array}{}\frac{{\gamma }_{4}}{2}\end{array}$ +i(Δxyz+Δp+Δrf). So from Eqs. (3) and (4), the expression about Im(χp) is given by

$Imχp=Imδ3δ4+Ωrf2δ2Ωrf2+δ4Ωs2x,y,z+δ4δ3δ2i=Aγ4γ34−Δ4Δ3+Ωrf2−B(γ4Δ32+γ3Δ42)A2+B2$(5)

where

Δ3=Δxyz+Δp,

Δ4=Δp+Δrf+Δxyz,

$\begin{array}{}{\mathit{\Delta }}_{3}={\mathit{\Delta }}_{xyz}+{\mathit{\Delta }}_{p},\\ {\mathit{\Delta }}_{4}={\mathit{\Delta }}_{p}+{\mathit{\Delta }}_{rf}+{\mathit{\Delta }}_{xyz},\\ A=\left({\gamma }_{4}{\mathit{\Omega }}_{s}^{2}\left(x,y,z\right)+{\gamma }_{2}{\mathit{\Omega }}_{rf}^{2}-{\gamma }_{4}{\mathit{\Delta }}_{p}{\mathit{\Delta }}_{3}-{\mathit{\Delta }}_{3}{\mathit{\Delta }}_{4}{\gamma }_{2}-{\mathit{\Delta }}_{4}{\mathit{\Delta }}_{p}{\gamma }_{3}+\\ \frac{{\gamma }_{4}{\gamma }_{3}{\gamma }_{2}}{4}\right)/2,\\ B={\mathit{\Delta }}_{4}{\mathit{\Omega }}_{s}^{2}\left(x,y,z\right)+{\mathit{\Delta }}_{p}{\mathit{\Omega }}_{rf}^{2}-{\mathit{\Delta }}_{4}{\mathit{\Delta }}_{3}{\mathit{\Delta }}_{p}+\left({\mathit{\Delta }}_{3}{\gamma }_{2}{\gamma }_{4}+{\mathit{\Delta }}_{p}{\gamma }_{3}{\gamma }_{4}+\\ {\mathit{\Delta }}_{4}{\gamma }_{3}{\gamma }_{2}\right)/4.\end{array}$

Eq. (5) tells us that the imaginary part Im(χp) depends on not only space coordinates (x,y,z) and the detunings Δp, Δxyz, Δrf between natural frequency of atom and light fields, but also the Rabi frequencies Ωrf and Ωs(x,y,z). Accordingly, we can use absorption of the probe field to detect distribution of atoms in space. In the process of numerical calculations, we will determine probe absorption Im(ρ21/Ωp) and measure the probe absorption depending on space position. Then informations about the atomic space distribution are obtained, and finally realize atom localization in three dimensional. Through choosing appropriate parameters of system, we can control the localization behavior of the atom in 3D space and improve the precision of detection of atomic position.

## 3 Localization structures

In this section, we will give some results of numerical calculations based on the imaginary part of the probe absorption under the condition of choosing different parameters of the system. Numerical analyses show measuring probe absorption can help one to achieve high-efficiency and high-precision 3D atom localization under the condition of choices of appropriate parameters. In the following process of numerical calculations, we keep γ2, γ3, and γ4 at the same values. Here, an atomic structure can be realized in cold 87Rb atoms using the transitions 5S1/2 − 5P1/2 − 5D3/2. In our system, the selections of atomic levels are: |1〉 = |5S1/2,F = 2〉, let |5P1/2,F = 2〉 = |2〉, |5D3/2,F = 1〉is set to |3〉, and |5D3/2,F = 2〉 is arranged as level |4〉. In this real atomic system, the decay rates γ2 = 5.3Mhz and γ3 = γ4 = 0.67Mhz, respectively.

First of all, when the detuning of the probe field has different values, we consider the influence of the absorption of probe field Im(χp) along with the position (kx, ky, kz). Numerical analyses are shown In Figure 2. From this figure, we can see that, when we vary Δp, the precision of the 3D localization changes a lot. When Δp = 5γ2, space distribution of the absorption of the probe field only presents one large sphere in the coordinate space (−1≤kj/Π≤1, j = x, y, z) see Figure 2(a), which means that the possibility of detecting the atom can reach 1 in this space but with low precision. In the cases that Δp = 5.5γ2 (Figure 2(b)) and Δp = 6γ2 (Figure 2(c)), the spheres become more and more smaller. We continue to increase the value of Δp until detuning Δp = 6.5γ2 (Figure 2(d)], the sphere occupies a very small volume in space becomes very small, and here our spatial resolution is about 0.1λ. From Figure 2(d), we find we only carry out measurement on absorption of probe field in a very small space, that is position of atoms is confined in this narrow space. Figure 2 shows that the precision of detecting position of atoms can be improved if we increase gradually the value of detuning of weak probe light. We can use the quantum coherence to explain the phenomenon of atom localization. Under the quantum coherent manipulation, one can build a quantum correlation between the detuning and absorption for the weak probe light. This type of strong correlation evidently affects the properties of absorption of medium when changing the detuning of the weak probe field, and then distribution of absorption of probe light in three-dimensional space will be influenced. So choosing appropriate parameters and by means of quantum coherence, various structure diagrams about atom localization in three-dimensional space are obtained in Figure 2.

Figure 2

Iso-surfaces for the probe absorptionIm(χ) = 0.1versus position (−1 ≤ kj/Π ≤ 1, j = x, y, z) for different values of Δp: (a) Δp = 5γ2 (b) Δp = 5.5γ2, (c) Δp = 6γ2, and (d) Δp = 6.5γ2, While the other parameters are N|μ21|2/ε0ħ = 1, Δxyz = 0, Δrf = 0.5γ2, Ωp = 0.2γ2, Ωrf = γ2, γ2 ≌ 5.3Mhz, γ3 ≌ 0.67Mhz, γ4 ≌ 0.67Mhz

Numerical results about atom localization in three-dimensional space is shown in Figure 3 when we consider the case with changing detuning of standing-wave field Δxyz. Firstly we let Δxyz=γ2, and structure diagram about distribution of the probe absorption is plotted in the three-dimensional space (−1≤kj/Π≤1, j=x, y, z) [see Figure 3(a)]. The shape of structure diagram is a sphere and its volume is biggest in Figure 3, which means we do not obtain atom localization. Next, we increase the values of Δxyz, and its values are set to be 2γ2 [Figure 3(b)] and 4γ2 [Figure 3(c)], respectively. we can still plot a sphere in the three-dimensional space. However, comparing with Figure 3(a), the size of spheres grow smaller. Which means, increasing the detuning of standing-wave field, the probability of detecting the atom can approach 1 all the time and the precision has been improved. When values of Δxyz reaches 5γ2 [Figure 3(d)], a very small sphere appears in the whole three-dimensional space, that is motion of atoms is confined in the tight range. Here the spatial resolution is still about 0.1λ, so atom localization in three-dimensional is realized with high probability and high precision.

Figure 3

Iso-surfaces for the probe absorption Im(χ) = 0.1versus position (−1 ≤ kj/Π ≤ 1, j = x, y, z) for different values of Δxyz: (a)Δxyz = γ2, (b)Δxyz = 2γ2, (c)Δxyz = 4γ2, and (d)Δxyz = 5γ2, while the other parameters are N|μ21|2/ε0ħ = 1, Δp = 4γ2,\ Δrf = 0.5γ2,Ωp = 0.2γ2, Ωrf = γ2, γ2 ≌ 5.3Mhz, γ3 ≌ 0.67Mhz, γ4 ≌ 0.67Mhz

Figure 4

Iso-surfaces for the probe absorption Im(χ) = 0.1versus position(−1 ≤ kj/Π ≤ 1, j = x, y, z) for different values ofΩrf (a) Ωrf = γ2, (b) Ωrf = 3γ2, (c) Ωrf = 5γ2, and (d) Ωrf = 6γ2 while the other parameters are N|μ21|2/ε0ħ = 1 Δp = 4.5γ2, Δxyz = 4γ2,Δrf = γ2, Ωp = 0.2γ2, γ2 ≌ 5.3Mhz, γ3 ≌ 0.67Mhz, γ4 ≌ 0.67Mhz

## 4 Conclusion

To sum up, through detecting the size of spatial region of probe absorption, the three-dimensional atom localization in a coherently driven atomic system with cascade four levels is investigated in detail. Through analyzing situations with different physical parameters, we find one can significantly improve the precision and efficiency of atom localization in three-dimensional space under the condition with appropriate physical parameters. Our results show that we can confine the motion of atom in a narrow space, and we can find atom in this region with 100% probability. Our scheme is more precise than that in other research [28]. In our atomic system, we use a radio-frequency field to couple with a transition between two hyperfine levels, and numerical results show weak radio-frequency field can help us to get efficient 3D atom localization. Compared with laser, the radio-frequency is easier to operate in experiment, so our scheme has feasibility in the experiment. In addition, atom localization with high precision and high efficiency has some potential practical value in some fields such as atom lithography, laser cooling and so on.

## Acknowledgement

This work is supported by the National Natural Science Foundation of China (Grant Nos. 11365009 and 11775190), and by Science Foundation of Zhejiang SCI-TECH University under Grant No. 17062071-Y.

## References

• [1]

Storey P., Collett M., Walls D., Measurement-induced diffraction and interference of atoms, Phys. Rev. Lett. 2007, 68, 472. Google Scholar

• [2]

Kunze S., Dieckmann K., Rempe G., Diffraction of atoms from a measurement induced grating, Phys. Rev. Lett. 1997, 78, 2038.

• [3]

Quadt R., Collett M., Walls D.F., Measurement of atomic motion in a standing light field by homodyne detection, Phys. Rev. Lett., 1995, 74, 351.

• [4]

Kien F.L., Rempe G., Schleich W.P., Zubairy M.S., Atom localization via ramsey interferometry: a coherent cavity field provides a better resolution, Phys. Rev. A, 1997, 56, 2972.

• [5]

Thomas J. E., Uncertainty-limited position measurement of moving atoms using optical fields, Opt. Lett., 1989, 14,1186.

• [6]

Stokes K.D., Schnurr C., Gardner J.R., Marable M., Welch G.R., Thomas J. E., Precision position measurement of moving atoms using optical fields, Phys. Rev. Lett., 1991, 67, 1997.

• [7]

Qamar S., Zhu S.Y., Zubairy M.S., Precision localization of single atom using Autler–Townes microscopy, Opt. Commun., 2000, 176, 409.

• [8]

Qamar S., Zhu S.Y., Zubairy M.S., Atom localization via resonance fluorescence, Phys. Rev. A, 2000, 61, 063806.

• [9]

Sahrai M., Tajalli H., Kapale K. T., Zubairy M. S., Subwavelength atom localization via amplitude and phase control of the absorption spectrum Phys. Rev. A, 2005, 72, 013820.

• [10]

Paspalakis E., Knight P.L., Localizing an atom via quantum interference, Phys. Rev. A, 2001, 63, 065802.

• [11]

Kapale K.T., Zubairy M.S., Subwavelength atom localization via amplitude and phase control of the absorption spectrum. II, Phys. Rev. A, 2006, 73, 023813.

• [12]

Liu C.P., Gong S.Q., Cheng D.C., Fan X.J., Xu Z.Z., Atom localization via interference of dark resonances, Phys. Rev. A 2006, 73, 025801.

• [13]

Xu J., Hu X. M., Sub-half-wavelength atom localization via phase control of a pair of bichromatic fields, Phys. Rev. A, 2007, 76, 013830.

• [14]

Agarwal G.S., Kapale K.T., Subwavelength atom localization via coherent population trapping, J. Phys. B: At. Mol. Opt. Phys., 2006, 39, 3437.

• [15]

Macovei M., Evers J., Keitel C.H., Zubairy M.S., Localization of atomic ensembles via superfluorescence, Phys. Rev. A, 2007, 75, 033801.

• [16]

Qamar S., Mehmood A., Qamar S., Subwavelength atom localization via coherent manipulation of the Raman gain process, Phys. Rev. A, 2009, 79, 033848.

• [17]

Proite N.A., Simmons Z.J., Yavuz D.D., Observation of atomic localization using electromagnetically induced transparency, Phys. Rev. A, 2011, 83, 041803.

• [18]

Ivanov V., Rozhdestvensky Y., Two-dimensional atom localization in a four-level tripod system in laser fields Phys. Rev. A, 2010, 81, 033809.

• [19]

Li J.H., Yu R., Liu M., Ding C.L., Yang X.X., Efficient two-dimensional atom localization via phase-sensitive absorption spectrum in a radio-frequency-driven four-level atomic system, Phys. Lett. A, 2011, 375, 3978.

• [20]

Jin L.L., Sun H., Niu Y.P., Jin S.Q., Gong S.Q., Two-dimension atom nano-lithograph via atom localization, J. Mod. Opt., 2009, 56, 805.

• [21]

Ding C.L., Li J.H., Zhan Z.M., Yang X.X., Two-dimensional atom localization via spontaneous emission in a coherently driven five-level M-type atomic system, Phys. Rev. A, 2011, 83, 063834.

• [22]

Ding C.L., Li J.H., Yang X.X., Zhang D., Xiong H., Proposal for efficient two-dimensional atom localization using probe absorption in a microwave-driven four-level atomic system, Phys. Rev. A, 2011, 84, 043840.

• [23]

Wan R.G., Zhang T.Y., Kou J., Two-dimensional sub-half-wavelength atom localization via phase control of absorption and gain, Phys. Rev. A, 2013, 87, 043816.

• [24]

Rahmatullah, Qamar S., Two-dimensional atom localization via probe-absorption spectrum, Phys. Rev. A, 2013, 88, 013846.

• [25]

Rahmatullah, Qamar S., Two-dimensional atom localization via Raman-driven coherence, Phys. Lett. A, 2014, 378, 684.

• [26]

Wang Z., Chen J., Yu B., High-dimensional atom localization via spontaneously generated coherence in a microwave-driven atomic system, Opt. Express, 2017, 25,3358.

• [27]

Ivanov V.S., Rozhdestvensky Y.V., Suominen K., Three-dimensional atom localization by laser fields in a four-level tripod syste, Phys. Rev. A, 2014, 90, 063802.

• [28]

Zhu Z., Yang W.X., Xie X.T., Liu S., Lee R.K., Three-dimensional atom localization from spatial interference in a double two-level atomic system, Phys. Rev. A, 2016, 94, 013826.

Accepted: 2018-02-05

Published Online: 2018-03-20

Citation Information: Open Physics, Volume 16, Issue 1, Pages 46–51, ISSN (Online) 2391-5471,

Export Citation