Atomic circulation in evanescent of LG₁₀ mode with active dipole and quadrupole transitions

Abstract

The total internal reflection of a light with phase singularity, such as Laguerre–Gaussian (LG) light, can generate an evanescent wave. The distribution of this type of light is firmly localized in the vicinity above the dielectric surface and displays a rotational character. Any atom approaching the surface from the vacuum region interacts with evanescent wave and produces interesting rotational effects. Recent studies have established the coupling of LG beams to both electric atomic transitions; dipole and quadrupole according to the selection rules of each, permitted and prohibited. Due to the small magnitude of the quadrupole interaction compared to the dipole interaction, we chose the two counter-propagating beams configuration to generate the doubling Rabi frequencies with more stable surface optical forces. We describe the essential rotational features and discuss how they can be used to influence atoms. Atom confinement and dynamics in the interaction of surface optical vortices with the two different atomic transitions are illustrated using typical experimentally accessible parameters for the case of the Cs atom interacts with the $L{G}_{10}$ mode.

KEYWORDS:

• Active dipole
• quadrupole transition
• Laguerre–Gaussian

1. Introduction

With a combination of simple optical elements and appropriate conventional laser light, it is currently possible to generate a class of beams with rotational properties. In particular, the set of light modes known as Laguerre–Gaussian (LG) beams is distinguished by an optical torque that is determined by the beam structure and phase properties. The rotational property is directly related to the orbital angular momentum carried by the beams, which is in addition to any spin angular momentum associated with their wave polarization [1–6].

The mechanical effects of (LG) beams on atoms have been explored theoretically and experimentally, identifying the heating and cooling influences of translational, as well as rotational, motion. In most studies, the mixture of these two motions together is referred to as an optical vortex. The best-known application currently for such vortex beams is optical spanners, which are based on the same principle of optical tweezers, with the availability of turning round about the beam axis [7]. The vortex beam characteristic is mainly due to an azimuthal phase dependence of the form $\mathrm{ex}{p}^{\mathrm{i\ell \phi }}$, where $\ell$ is an integer, with $\ell \ge 1$. In this case, the (LG) beam carries angular momentum in discrete quantized units $\mathrm{\ell \hslash }$ [8–10].

In the last two decades, the mechanical effects of (LG) beams on atoms have received much attention in studies of quantum optics and have become a suitable tool in a number of important applications, especially those related to what can be called the taming of atoms. These investigations were diversified, including unbounded and bounded spaces with different situations of allowed electric atomic transitions, dipole and quadrupole. In all cases, the angular momentum of the quantized photon is responsible for the emergence on the atoms of a torque that strongly affects their rotational motion [11–13].

In this study, we carefully examine the dipole and quadrupole effects in the interaction of atoms with the transmitted evanescent field generated by $L{G}_{10}$ mode. The explanation required for the spatial dependence of the optical forces for both electric transitions – dipole and quadrupole – is presented. We also explore their influence on atomic systems near the surface and discuss the exploitation conditions in both cases.

2. The evanescent wave

Assume that an LG laser beam, coming from the $z<0$ half space, is totally internally reflected at a dielectric-vacuum at $z=0$. In this case, the electric field ${\mathbf{\text{E}}}_{\mathrm{k\ell p}}^{\mathrm{evan}}(x,y,z>0)={\mathbf{\text{E}}}_{\mathrm{k\ell p}}^{\mathrm{evan}}(\mathbf{\text{r}},z>0)$ of the transmitted evanescent wave beam is of the form [8–13] (1) $\begin{array}{rl}{\mathbf{\text{E}}}_{\mathrm{k\ell p}}^{\mathrm{evan}}(\mathbf{\text{r}},z>0)& ={\hat{e}}_y\frac{{\zeta }_{k00}{C}_{\mathrm{\ell p}}}{{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\\ & \times {\left[\frac{\sqrt{2(x^2{\cos }^2\theta +y^2)}}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right]}^{|\ell |}\\ & \times L_p^{|\ell |}\left(\frac{(x^2{\cos }^2\theta +y^2)}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right)\\ & \times \exp \left[-\frac{(x^2{\cos }^2\theta +y^2)}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right]\\ & \times \exp \left[-z{k}_0\sqrt{n^2{\sin }^2\theta -1}\right]\\ & \times \exp [-\mathrm{i\ell arctan}(y/\mathrm{xcos}\theta )]\\ & \times \exp (-i{k}_0\mathrm{nxsin}\theta )\end{array}$(1) In the above equation, ${\zeta }_{k00}$ is an electric field amplitude factor and $C_{\mathrm{\ell p}}$ is the standard LG beam coefficient (2) $\begin{array}{rl}{\zeta }_{k00}& =\sqrt{\frac{2I}{{\epsilon }_0{n}_0^2{n}^{2}c}}\end{array}$(2) (3) $\begin{array}{rl}{C}_{\mathrm{\ell p}}& =\sqrt{p!/(|\ell |+p)!}\end{array}$(3) where $I$ is the light intensity, $\theta >{\theta }_c$ is the incident angle, $n$ is the dielectric surface constant, $n_0=1$ is the dielectric constant of the vacuum region, $z_R$ is the Rayleigh range and $w_0$ is the minimum beam. The three lower indices $\mathrm{k\ell p}$ in the above equation indicate that the transmitted evanescent light of such a class of has a wave vector component $k=n{k}_0$ along the propagation direction and is characterized by two integer indices: a winding number $\ell$ and a radial number $p$. These three indices identify the order of the Laguerre polynomial $L_p^{|\ell |}(\dots )$. A winding number $\ell$ can take either positive or negative values, representing two senses of helical wavefront rotation. When both $\ell$ and $p$ are zero, the beam returns to an ordinary Gaussian distribution (i.e. no azimuthal dependence). Basically, equation (1) can be divided into three terms. The first is the azimuthal phase dependence term (4) $\begin{array}{r}\mathrm{\Phi }(\mathbf{\text{r}})=\left[-\mathrm{\ell arctan}\left(\frac{y}{\mathrm{xcos}\theta }\right)\right]-\left[k_0\stackrel{}{x}\mathrm{nsin}\theta \right]\end{array}$(4) It is responsible for generating the swirling feature from which LG light acquires its importance in modern studies. This feature and its effects on the atom with two types of electric interactions; dipole and quadrupole also represent the cornerstone of this study. The second term is the evanescent decay of laser intensity along the propagation direction (5) $\begin{array}{r}\psi (z)=\exp -z{k}_0\sqrt{n^2{\sin }^2\theta -1}\end{array}$(5) This shows that the evanescent wave has a maximum value at the center and gradually decreases away from it. In contrast to the case of a pure plane wave, the evanescent field depends on two directions. The decay magnitude is just controlled by the light incident angle $\theta$ and the refractive index of the dielectric surface $n$. For example, some studies in the atomic mirror based in the evanescent field have shown that when $n=1.5$, the intensity peak occurs at $\theta ={42.5}^{\circ }$ [14, 15]. It is also well known that the length scale of the intensity decay at $z>0$ spans a small fraction $\lambda$. Once the laser source is turned on with sufficient intensity, the light–atom interaction region can be free from the influence of the van der Waal sticking that will dominate when the laser source is turned off [16]. Thus, an effective evanescent field requires exceeding a certain threshold of incident light intensity, which is fortunately easy to cross. The decay term has no role at all, neither in the vortex property nor in the trapping processes perpendicular to the surface. The third term is the beam amplitude (or the standard envelope function) (6) $\begin{array}{rl}{F}_{\mathrm{k\ell p}}(\mathbf{\text{r}})& ={\hat{e}}_y\frac{\sqrt{p!/(|\ell |+p)!}\sqrt{2I/(n^2{n}_0^{2}c)}}{{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\\ & \times {\left[\frac{\sqrt{2(x^2{\cos }^2\theta +y^2)}}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right]}^{|\ell |}\\ & \times L_p^{|\ell |}\left(\frac{(x^2{\cos }^2\theta +y^2)}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right)\\ & \times \exp \left[-\frac{(x^2{\cos }^2\theta +y^2)}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right]\end{array}$(6) It is considered the main factor in Rabi frequency calculation, which is the cornerstone of all optical atomic trapping processes. In fact, the formation of surface optical vortices and their effects on atoms is the result of solidarity of all three terms together and each of them needs to cross a threshold of its own to complete the task [14–16].

3. Light–atom interaction

Normally, the total Hamiltonian $H$ of the system is given as a sum of three terms as follows [17] (7) $\begin{array}{r}H=H_F+H_A+H_{\mathrm{int}}\end{array}$(7) where $H_F$ is the Hamiltonian for the field in the absence of the atom, $H_A$ is the Hamiltonian for atom in the absence of the field and $H_{\mathrm{int}}$ represents the interaction term that appears from the coupling between the atom and the field. These terms can be given as (8) $\begin{array}{rl}{H}_F& =\mathrm{\hslash \omega }{a}^{†}a\end{array}$(8) (9) $\begin{array}{rl}{H}_A& =({\mathbf{\text{P}}}^2/2M)+\hslash {\omega }_0{\pi }^{†}\pi \end{array}$(9) (10) $\begin{array}{rl}{H}_{\mathrm{int}}& =H_{\mathrm{ED}}+H_{\mathrm{EQ}}+H_{\mathrm{MD}}+H_{\mathrm{NL}}\end{array}$(10)

In the above equations, the operators $a$ and $a^{†}$ entering $H_F$ are the creation and destruction operators of the electromagnetic field, respectively, and $\omega$ is its frequency while the operators $\pi$ and ${\pi }^{†}$ entering $H_A$ are the ladder operators for the two-level system. In such an atomic system, we have $\{|g⟩,|e⟩\}$ with corresponding energy levels ${\epsilon }_1$ and ${\epsilon }_2$, where ${\omega }_0=({\epsilon }_2-{\epsilon }_1)/\hslash$. The term between the brackets in equation (9) represents the kinetic mechanical energy, where $\mathbf{\text{P}}$ is the center-of-mass momentum operator and $M$ is the atomic mass. $H_{\mathrm{int}}$ contains four types of interactions: electric–dipole $H_{\mathrm{ED}}$, electric–quadrupole $H_{\mathrm{EQ}}$, magnetic–dipole $H_{\mathrm{MD}}$ and diamagnetic $H_{\mathrm{NL}}$. Primarily, it should be noted that $H_{\mathrm{NL}}$ is negligible compared with the first three interactions, so $H_{\mathrm{NL}}$ is always completely ignored. $H_{\mathrm{EQ}}$ and $H_{\mathrm{MD}}$ have almost the same magnitude, but they are much less than the magnitude of $H_{\mathrm{ED}}$. Hence $H_{\mathrm{EQ}}$ and $H_{\mathrm{MD}}$ cannot be achieved and exploited in normal circumstances but only in cases where $H_{\mathrm{ED}}$ is prohibited. Here, we are only concerned with $H_{\mathrm{ED}}$ and $H_{\mathrm{EQ}}$ to compare the swirling features produced by them with the Laguerre–Gaussian laser beam.

A. Electric–dipole interaction

In cases where the dipole transition is active, the standard electric dipole interaction $H_{\mathrm{ED}}$ between the two-level atom and the electric field is given by (11) $\begin{array}{r}{H}_{\mathrm{ED}}=-\mathit{\mu }.\mathbf{\text{E}}(\mathbf{\text{r}},z)\end{array}$(11) where $\mathit{\mu }=q\mathbf{\text{R}}$ is the electric dipole moment operator, $\mathbf{\text{R}}$ is the internal position vector and $\mathbf{\text{E}}$ is the electric field vector. The main significant property of the atomic trapping process is the Rabi frequency $\mathrm{\Omega }$. It governs the interaction of an atom that has an electric dipole moment $\hat{\mathit{\mu }}$ when it touches the evanescent wave from the vacuum region $z>0$. The position-dependent Rabi frequency ${\mathrm{\Omega }}_{\mathrm{k\ell p}}^{\mathrm{ED}}(\mathbf{\text{r}},z)$ due to the dipole transition is defined as (12) $\begin{array}{r}\hslash {\mathrm{\Omega }}_{\mathrm{k\ell p}}^{\mathrm{ED}}(\mathbf{\text{r}},z)=|H_{\mathrm{ED}}|=|\mu F_{\mathrm{k\ell p}}(x,y)\psi (z)|\end{array}$(12) Thus, the Rabi frequency due to the electric field of the transmitted evanescent LG beam is (13) $\begin{array}{rl}\frac{{\mathrm{\Omega }}_{\mathrm{k\ell p}}^{\mathrm{ED}}(\mathbf{\text{r}},z)}{{\mathrm{\Omega }}_0^{\mathrm{ED}}}& =\frac{1}{\hslash {[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\\ & \times {\left[\frac{\sqrt{2(x^2{\cos }^2\theta +y^2)}}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right]}^{|\ell |}\\ & \times L_p^{|\ell |}\left(\frac{(x^2{\cos }^2\theta +y^2)}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right)\\ & \times \exp \left[-\frac{(x^2{\cos }^2\theta +y^2)}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right]\\ & \times \exp \left[-z{k}_0\sqrt{n^2{\sin }^2\theta -1}\right]\end{array}$(13) where (14) $\begin{array}{r}{\mathrm{\Omega }}_0^{\mathrm{ED}}=\frac{\mu C_{\mathrm{\ell p}}{\zeta }_{k00}}{\hslash }\end{array}$(14)

B. Electric–quadrupole interaction

In cases where the quadrupole transition is active, the standard electric quadrupole interaction $H_{\mathrm{EQ}}$ between the two-level atom and the electric field is given by (15) $\begin{array}{r}{H}_{\mathrm{EQ}}=-\frac{1}{2}\sum _{\mathrm{ij}}{\hat{Q}}_{\mathrm{ij}}\frac{\mathrm{\partial }{\hat{E}}_{}^j(\mathbf{\text{r}})}{\mathrm{\partial }{\mathbf{\text{r}}}_i}\end{array}$(15) where ${\hat{Q}}_{\mathrm{ij}}=-q{X}_i{X}_j$ are the elements of the quadrupole tensor operator and $\mathbf{\text{X}}=(X,Y,Z)$ are the components of the internal position vector. The two-level atom is given by (16) $\begin{array}{r}{\hat{Q}}_{\mathrm{ij}}=Q_{\mathrm{ij}}(\pi +{\pi }^{†})\end{array}$(16) where $Q_{\mathrm{ij}}=⟨1|{\hat{Q}}_{\mathrm{ij}}|2⟩$ are quadrupole matrix elements between the two atomic levels. If the polarization of the electric field $\mathbf{\text{E}}(\mathbf{\text{r}})$ is chosen to be in the $y$ direction as assumed earlier in equation (1), then $H_{\mathrm{EQ}}$ becomes (17) $\begin{array}{r}{H}_{\mathrm{EQ}}=\frac{1}{2}\left[{\hat{Q}}_{\mathrm{xx}}\frac{\mathrm{\partial }{E}^{\mathrm{evan}}}{\mathrm{\partial }x}+{\hat{Q}}_{\mathrm{xy}}\frac{\mathrm{\partial }{E}^{\mathrm{evan}}}{\mathrm{\partial }y}+{\hat{Q}}_{\mathrm{xz}}\frac{\mathrm{\partial }{E}^{\mathrm{evan}}}{\mathrm{\partial }z}\right]\end{array}$(17) If the atom is forced to move in the $\mathrm{xy}$ plane and the quadrupole transition is such that ${\hat{Q}}_{\mathrm{xy}}={\hat{Q}}_{\mathrm{xz}}=0$, the position-dependent Rabi frequency ${\mathrm{\Omega }}_{\mathrm{k\ell p}}^{\mathrm{EQ}}(\mathbf{\text{r}},z)$ at the vacuum region $z>0$ is given by (18) $\begin{array}{r}\hslash {\mathrm{\Omega }}_{\mathrm{k\ell p}}^{\mathrm{EQ}}(\mathbf{\text{r}},z)=|H_{\mathrm{EQ}}|=\frac{1}{2}\left|{\hat{Q}}_{\mathrm{xx}}\frac{\mathrm{\partial }{F}_{\mathrm{k\ell p}}(x,y)\psi (z)}{\mathrm{\partial }x}\right|\end{array}$(18) where (19) $\begin{array}{r}\frac{\mathrm{\partial }{F}_{\mathrm{k\ell p}}(\mathbf{\text{r}})}{\mathrm{\partial }x}=F_{\mathrm{k\ell p}}(r)\left[\frac{|\ell |y}{r^2}-\frac{2y}{w_0^2}-\frac{\mathrm{i\ell x}}{r^2}+\frac{1}{L_p^{|\ell |}}\frac{\mathrm{\partial }{L}_p^{|\ell |}}{\mathrm{\partial }x}\right]\end{array}$(19) It is well-known that for any doughnut mode, $\mathrm{\partial }{L}_p^{|\ell |}/\mathrm{\partial }x=0$. Fortunately, this mode, in its lower and higher orders, is the most widely used in LG beam studies because it has a central zero-intensity point caused by phase singularity. In this case, the quadrupole Rabi frequency is reduced to (20) $\begin{array}{rl}\frac{{\mathrm{\Omega }}_{\mathrm{k\ell p}}^{\mathrm{EQ}}(\mathbf{\text{r}},z)}{{\mathrm{\Omega }}_0^{\mathrm{EQ}}}& =\frac{1}{\hslash {[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\\ & \times {\left[\frac{\sqrt{2(x^2{\cos }^2\theta +y^2)}}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right]}^{|\ell |}\\ & \times L_p^{|\ell |}\left(\frac{(x^2{\cos }^2\theta +y^2)}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right)\\ & \times \exp \left[-\frac{(x^2{\cos }^2\theta +y^2)}{w_0^2{[1+x^2{\sin }^2\theta /z_R^2]}^{1/2}}\right]\\ & \times \left[\frac{|\ell |y}{{\rho }^2}-\frac{2y}{w_0^2}-\frac{\mathrm{i\ell x}}{{\rho }^2}\right]\\ & \times \exp \left[-z{k}_0\sqrt{n^2{\sin }^2\theta -1}\right]\end{array}$(20) where (21) $\begin{array}{r}{\mathrm{\Omega }}_0^{\mathrm{EQ}}=\frac{C_{\mathrm{\ell p}}{\zeta }_{k00}}{\hslash }\frac{Q_{\mathrm{xx}}}{w_0}\end{array}$(21)

4. Optical forces and atom dynamics

The steady-state light forces ${\mathbf{\text{F}}}_{\mathrm{opt}}(\mathbf{\text{r}},z,\mathbf{\text{v}})$ on the two-level atom are conveniently divided into the scattering force ${\mathbf{\text{F}}}_{\mathrm{scat}}(\mathbf{\text{r}},z,\mathbf{\text{v}})$ and the gradient force ${\mathbf{\text{F}}}_{\mathrm{grad}}(\mathbf{\text{r}},z,\mathbf{\text{v}})$, where [8–13] (22) $\begin{array}{rl}{\mathbf{\text{F}}}_{\mathrm{scat}}(\mathbf{\text{r}},z,\mathbf{\text{v}})& =2\hslash \mathrm{\Gamma }(\mathbf{\text{r}}){\mathrm{\Omega }}_{}^2(\mathbf{\text{r}},z)\\ & \times \left(\frac{\mathrm{\nabla \Phi }(\mathbf{\text{r}})}{{\mathrm{\Delta }}^2(\mathbf{\text{r}},\mathbf{\text{v}})+2{\mathrm{\Omega }}^2(\mathbf{\text{r}},z)+{\mathrm{\Gamma }}^2(\mathbf{\text{r}})}\right)\end{array}$(22) and (23) $\begin{array}{rl}{\mathbf{\text{F}}}_{\mathrm{grad}}(\mathbf{\text{r}},z,\mathbf{\text{v}})& =-2\hslash \mathrm{\Omega }(\mathbf{\text{r}},z)\mathrm{\Delta }(\mathbf{\text{r}},\mathbf{\text{v}})\\ & \times \left(\frac{\mathrm{\nabla \Omega }(\mathbf{\text{r}})}{{\mathrm{\Delta }}^2(\mathbf{\text{r}},\mathbf{\text{v}})+2{\mathrm{\Omega }}^2(\mathbf{\text{r}})+{\mathrm{\Gamma }}^2(\mathbf{\text{r}})}\right)\end{array}$(23) Corresponding to the gradient force ${\mathbf{\text{F}}}_{\mathrm{grad}}(\mathbf{\text{r}},z,\mathbf{\text{v}})$ is a potential $U(\mathbf{\text{r}},z,\mathbf{\text{v}})$, which has the form (24) $\begin{array}{r}U(\mathbf{\text{r}},z,\mathbf{\text{v}})=\frac{1}{2}\hslash \mathrm{\Delta }(\mathbf{\text{r}},\mathbf{\text{v}})\ln \left(1+\frac{2{\mathrm{\Omega }}^2(\mathbf{\text{r}},z)}{{\mathrm{\Delta }}^2(\mathbf{\text{r}},\mathbf{\text{v}})+{\mathrm{\Gamma }}^2(\mathbf{\text{r}})}\right)\end{array}$(24) Here $\mathrm{\Gamma }$ is the decay rate and $\mathrm{\Delta }(\mathbf{\text{r}},\mathbf{\text{v}})={\mathrm{\Delta }}_0-\mathbf{\text{v}}\cdot \mathrm{\nabla \Phi }(\mathbf{\text{r}})$ is the dynamic detuning, including the Doppler shift $\delta =-\mathbf{\text{v}}\cdot \mathrm{\nabla \Phi }(\mathbf{\text{r}})$, where $\mathbf{\text{v}}$ is the atomic velocity and ${\mathrm{\Delta }}_0=\omega -{\omega }_0$ is the static detuning, with $\omega$ as the frequency of the light field. Here, we should recognize some key points: the gradient force ${\mathbf{\text{F}}}_{\mathrm{grad}}$ acts in the direction of the gradient of the field intensity $\mathrm{\nabla \Omega }$, that is, either toward high field intensities when ${\mathrm{\Delta }}_0<0$ or toward low field intensities when ${\mathrm{\Delta }}_0>0$. Accordingly, the potential exhibits a minimum in the high-intensity region of the field in the negative case, while the trapping process takes place in the low-intensity regions in the positive case. From Equation (2), the gradient phase $\mathrm{\nabla \Phi }(\mathbf{\text{r}})$ can be written as (25) $\begin{array}{rl}\mathrm{\nabla \Phi }(\mathbf{\text{r}})& =\left[\frac{-\mathrm{\ell y}}{x^2\cos \theta \left[1+{\left(\frac{y}{\mathrm{xcos}\theta }\right)}^2\right]}-(k_0\mathrm{nsin}\theta )\right]\widehat{\mathbf{\text{x}}}\\ & +\left[\frac{\ell }{\mathrm{xcos}\theta \left[1+{\left(\frac{y}{\mathrm{xcos}\theta }\right)}^2\right]}\right]\widehat{\mathbf{\text{y}}}\end{array}$(25) At this stage, according to the selection rules, activating one of the two atomic transitions; dipole and quadrupole seems enough to acquire atom dynamics with rotating properties. This can be done simply by solving the classical equation of motion (26) $\begin{array}{r}M\left(\frac{d^2\mathbf{\text{r}}}{d{t}^2}\right)={\mathbf{\text{F}}}_{}^{\mathrm{scat}}+{\mathbf{\text{F}}}_{}^{\mathrm{grad}}\end{array}$(26) subject to a given set of initial conditions of atom; position and velocity when the laser is switched on. In fact, this is not practicable because by carefully checking the gradient phase $\mathrm{\nabla \Phi }$, we find that the gradient of the second term gives rise to a component of the scattering force purely along the $x$ direction. This is in addition to the angular force arising from the gradient of the first term. This means that the atom is subsequently forced in a direction parallel to the surface, preventing it from achieving roundness. To confine the atom to an angular path in the plane parallel to the surface and eliminate undesirable motion, two internally incident beams should be used instead of one. This type of optical construction is well-known in quantum optics studies as two co-axial counter-propagating beams [10]. Both beams; 1 and 2 are incident at angles ${\theta }_1=-{\theta }_2=\theta$, and the total electric field in the vacuum region results from their interference. To explore the new situation, we need to rewrite the phase gradient Equation (25) as (27) $\begin{array}{r}\mathrm{\nabla \Phi }(\mathbf{\text{r}})=f\widehat{\mathbf{\text{x}}}+g\widehat{\mathbf{\text{y}}}\end{array}$(27) In this case, the dynamic detuning of beam 1 is (28) $\begin{array}{r}{\mathrm{\Delta }}_{1\theta }^{{\ell }_1}(\mathbf{\text{r}},\mathbf{\text{v}})={\mathrm{\Delta }}_0-\mathbf{\text{v}}\cdot \mathbf{\nabla }{\mathrm{\Phi }}_{1\theta }^{{\ell }_1}={\mathrm{\Delta }}_0-f_{1\theta }^{{\ell }_1}(\mathbf{\text{r}})v_x-g_{1\theta }^{{\ell }_1}(\mathbf{\text{r}})v_y\end{array}$(28) It can be seen that beam 2 arises as a result of a circulation about the $y-\mathrm{axis}$ by an angle $(-\theta )$; hence (29) $\begin{array}{r}{\mathrm{\Omega }}_{2(-\theta )}^{{\ell }_{2}p}(\mathbf{r},z)={\mathrm{\Omega }}_{1(-\theta )}^{{\ell }_{2}p}(\mathbf{r},z)\end{array}$(29) and the dynamic detuning of beam 2 is (30) $\begin{array}{rl}{\mathrm{\Delta }}_{2(-\theta )}^{{\ell }_2}(\mathbf{\text{r}},\mathbf{\text{v}})& ={\mathrm{\Delta }}_0-\mathbf{\text{v}}\cdot \mathbf{\nabla }{\mathrm{\Phi }}_{1(-\theta )}^{{\ell }_2}\\ & ={\mathrm{\Delta }}_0-f_{1(-\theta )}^{{\ell }_2}(\mathbf{\text{r}})v_x-g_{1(-\theta )}^{{\ell }_2}(\mathbf{\text{r}})v_y\end{array}$(30) In this case, the total scattering force is (31) $\begin{array}{rl}& {\mathbf{\text{F}}}_{1+2}^{\mathrm{sact}}(\mathbf{\text{r}},z,\mathbf{\text{v}})=2\hslash \mathrm{\Gamma }(\mathbf{\text{r}})\\ & \times \left\{\begin{array}{c}{\left({\displaystyle \frac{{\mathrm{\Omega }}_{1\theta }^2(\mathbf{\text{r}}).(f_{1\theta }\widehat{\mathbf{\text{x}}}+g_{1\theta }\widehat{\mathbf{\text{y}}})}{{\mathrm{\Delta }}_{1\theta }^2(\mathbf{\text{r}},\mathbf{\text{v}})+2{\mathrm{\Omega }}_{1\theta }^2(\mathbf{\text{r}})+{\mathrm{\Gamma }}_{}^2(\mathbf{\text{r}})}}\right)}_{{\ell }_1}+{\left({\displaystyle \frac{{\mathrm{\Omega }}_{1(-\theta )}^2(\mathbf{\text{r}}).(f_{1(-\theta )}\widehat{\mathbf{\text{x}}}+g_{1(-\theta )}\widehat{\mathbf{\text{y}}})}{{\mathrm{\Delta }}_{1(-\theta )}^2(\mathbf{\text{r}},\mathbf{\text{v}})+2{\mathrm{\Omega }}_{1(-\theta )}^2(\mathbf{\text{r}})+{\mathrm{\Gamma }}_{}^2(\mathbf{\text{r}})}}\right)}_{{\ell }_2}\end{array}\right\}\end{array}$(31)

Similarly, the total gradient force is (32) $\begin{array}{rl}& {\mathbf{\text{F}}}_{1+2}^{\mathrm{grad}}(\mathbf{\text{r}},z,\mathbf{\text{v}})\\ & =-2\hslash \left\{\begin{array}{c}{\left({\displaystyle \frac{{\mathrm{\Delta }}_{1\theta }^{}(\mathbf{\text{r}},\mathbf{\text{v}})\cdot {\mathrm{\Omega }}_{1\theta }^{}(\mathbf{\text{r}})\cdot \mathrm{\nabla }{\mathrm{\Omega }}_{1\theta }^{}(\mathbf{\text{r}})}{{\mathrm{\Delta }}_{1\theta }^2(\mathbf{\text{r}},\mathbf{\text{v}})+2{\mathrm{\Omega }}_{1\theta }^2(\mathbf{\text{r}})+\mathrm{\Gamma }(\mathbf{\text{r}})}}\right)}_{{\ell }_1}+{\left({\displaystyle \frac{\begin{array}{c}{\mathrm{\Delta }}_{1(-\theta )}^{}(\mathbf{\text{r}},\mathbf{\text{v}})\cdot {\mathrm{\Omega }}_{1(-\theta )}^{}(\mathbf{\text{r}})\\ \cdot \mathrm{\nabla }{\mathrm{\Omega }}_{1(-\theta )}^{}(\mathbf{\text{r}})\end{array}}{\begin{array}{c}{\mathrm{\Delta }}_{1(-\theta )}^2(\mathbf{\text{r}},\mathbf{\text{v}})\\ +2{\mathrm{\Omega }}_{1(-\theta )}^2(\mathbf{\text{r}})+\mathrm{\Gamma }(\mathbf{\text{r}})\end{array}}}\right)}_{{\ell }_2}\end{array}\right\}\end{array}$(32) Now, it is clear that the combined effect of the two beams is to generate the total force acting on the center of mass of the atom. This influence appears as the sum of the forces carried by the surface optical vortices in the regime of permitted; dipole or quadrupole atomic transitions, according to their selection rules. The dynamics again follow from solutions of equation (26), subject to a sum of the forces delivered by each beam (33) $\begin{array}{r}M\left(\frac{d^2\mathbf{\text{r}}}{d{t}^2}\right)={\mathbf{\text{F}}}_{1+2}^{\mathrm{scat}}+{\mathbf{\text{F}}}_{1+2}^{\mathrm{grad}}\end{array}$(33) The numerical solutions of the dynamic equation with related potentials will be illustrated by considering a $\mathrm{Cs}$ atom interacts with two counter-propagating $L{G}_{10}$ beams whose parameters are shown in Table 1 [8–19].

Table 1. Parameters used for the illustration of the trapping and atom dynamics.

Dipole transition	Quadrupole transition
$(6S_{1/2}\to 6P_{3/2})$	$(6S_{1/2}\to 5D_{5/2})$
$\lambda =852.0\text{nm}$	$\lambda =685\text{nm}$
$\nu =3.52\times {10}^{15}\text{Hz}$	$\nu =4.37\times {10}^{14}\text{Hz}$
${\epsilon }_{\mathrm{re}}=1.362\times {10}^{-30}\text{J}$	${\epsilon }_{\mathrm{re}}=2.184\times {10}^{-28}\text{J}$
${\mathrm{\Gamma }}_0^{\mathrm{ED}}=2\pi \times 5.22\text{MHz}$	${\mathrm{\Gamma }}_0^{\mathrm{EQ}}=2\pi \times 124\text{kHz}$
${\mathrm{\Delta }}_0={10}^2{\mathrm{\Gamma }}_0^{\mathrm{ED}}$	${\mathrm{\Delta }}_0={10}^2{\mathrm{\Gamma }}_0^{\mathrm{EQ}}$
$\mu =5.0e{a}_B$	$Q_{\mathrm{xx}}=10e{a}_B^2$
${\mathrm{\Omega }}_0^{\mathrm{ED}}=1.8\times {10}^3{\mathrm{\Gamma }}_{6P6S}$	${\mathrm{\Omega }}_0^{\mathrm{EQ}}=93.0{\mathrm{\Gamma }}_{5D6P}$
$U_0^{\mathrm{ED}}=1.73\times {10}^{-27}J$	$U_0^{\mathrm{EQ}}=4.108\times {10}^{-29}J$
$w_0=0.5\lambda$
$I={10}^9\text{W }{\text{m}}^{-\text{2}}$
$n=1.5$
$\theta ={42.5}^{\circ }$

Finally, we point out that gravity is ignored in the classical equation of motion. This is because its effect is negligible, as the laser intensity is high and thus has strong potential. With the gradual decrease in the laser intensity, a turning point is reached, where gravity begins to increase until the atom sticks to the surface. In general, the minimum intensity used to produce the optical vortex is so far from the turning point that it can simply be avoided [10, 14–16].

5. Result and discussion

There are three cases of atomic transitions resulting from an interaction with an electric field. In the first case, the dipole transitions are active but the quadrupole transitions are forbidden, while in the second case, the reverse is the case. Despite the two previous cases, they can also be simultaneously allowed together, as is the case when an external field is present. Normally, quadrupole effects are insignificant compared to fluctuations in the forces corresponding to the dipole transition. In fact, each of the three possibilities has its own selection rules, which are explained in books of atomic spectra [18]. Out of the complexities associated with the third case, the dipole interaction examines when quadrupole transitions are prohibited and the quadrupole interaction examines when dipole transitions are not allowed. Before starting the calculations for both types of interaction; dipole and quadrupole a necessary condition should be observed. The bottom of the resulting optical potential must be greater or at least of the order of the recoil energy ${\epsilon }_{\mathrm{re}}={\hslash }^2{k}^2/2M$ to confine an atom.

We now explore the variations of the total trapping potential and atom dynamics by selecting the appropriate atomic transition in the $\mathrm{Cs}$ atom from experimental reality. The transition between the $(6S_{1/2}\to 6P_{3/2})$ states is only a dipole-permitted transition with $\lambda =852.0\text{nm}$ and $\nu =3.52\times {10}^{15}\text{Hz}$, so the recoil energy for this transition is ${\epsilon }_{\mathrm{re}}=1.362\times {10}^{-30}\text{J}$. On the other hand, the transition between the $(6S_{1/2}\to 5D_{5/2})$ states is only a quadrupole-permitted transition with $\lambda =685\text{nm}$ and $\nu =4.37\times {10}^{14}\text{Hz}$, which means that ${\epsilon }_{\mathrm{re}}=2.184\times {10}^{-28}\text{J}$. Since the last transition is dipole-prohibited, the decay rate is ${\mathrm{\Gamma }}_{5D6S}=2\pi \times 3.5\text{Hz}$, which is very small compared with the decay rate of the dipole transition line, which is about ${\mathrm{\Gamma }}_{6P6S}=2\pi \times 5.22\text{MHz}$. Due to the low decay rate of the electric quadrupole transition, the excited atoms will not often decay directly back to the ground state $6S_{1/2}$ but will rather decay to the $6P_{3/2}$ state because this dipole transition has a much higher decay rate of ${\mathrm{\Gamma }}_{5D6P}=2\pi \times 124\text{kHz}$ [19]. All previous values were for the atomic decay rate of the cesium atom in unbounded space and not in bounded space, as is the case here. In fact, the value of the decay emission rate near a dielectric surface is too close to its value in an unbounded space (i.e. ${\mathrm{\Gamma }}_{6P6S}\approx {\mathrm{\Gamma }}_0^{\mathrm{ED}}$ and ${\mathrm{\Gamma }}_{5D6P}\approx {\mathrm{\Gamma }}_0^{\mathrm{EQ}}$). Therefore, this value can be taken as a good approximation in the interaction region, which is actually relatively far from the surface [14, 15].

A. Active dipole transition

The distribution shown in Figure 1 is the variation of the dipole potential (in units of $U_0^{\mathrm{ED}}=(1/2)\hslash {\mathrm{\Gamma }}_{6P6S}$), which generated two identical counter-propagating $L{G}_{10}$ modes with ${\theta }_1=-{\theta }_2={42.5}^{\circ }$ for positive detuning case. Figure 2 shows the same type of potential with the same parameters except that the detuning is negative. It can be seen that the shapes of the two potential wells are not perfectly round (almost elliptical), as they are in the case of free space. In fact, the cause of the distortion is the oblique incidence of laser light. It can simply be realized that the bottom of both potential wells is much greater than the recoil energy ($U_0/{\epsilon }_{\mathrm{re}}=1.27\times {10}^{3}J$), so the atom can be trapped.

Figure 1. The distribution of the evanescent dipole potential generated by two counter-propagating $L{G}_{10}$ beam for ${\mathrm{\Delta }}_0>0$.

Figure 2. The distribution of the evanescent dipole potential generated by two counter-propagating $L{G}_{10}$ beam for ${\mathrm{\Delta }}_0<0$.

Figure 3 shows the in-plane trajectory of the $\mathrm{Cs}$ atom for the positive detuning case. It is expected that the atom is confined to a central potential well and spirals outward in an elliptically shaped path. Figure 4 displays the trajectory for the negative case. Here, the atomic motion combines two types of confinement in an elliptical concentric valley defined by the intensity distribution and radial confinement, leading to vibrational motion in a radial direction. In both detuning, the regions with lower bottoms are well suited to allow for a very good number of quasi-rotational quantum states. These states can be easily calculated using the numerical solution of the two-dimensional Schrödinger equation.

Figure 3. The trajectory of the atom $\mathrm{Cs}$ inside the potential for the case shown in Figure 1 The initial in-plane velocity is $v_x(0)=v_y(0)=(0,0)$ and the initial position indicated by red dot is $(x_0,y_0)=(0,0)$.

Figure 4. The trajectory of the atom $\mathrm{Cs}$ inside the potential for the case shown in Figure 2 The initial in-plane velocity is $v_x(0)=v_y(0)=(0,0)$ and the initial position indicated by the dot is $(x_0,y_0)=(1.5,1.5)w_0$.

B. Active quadrupole transition

Once again, assuming the two identical counter-propagating $L{G}_{10}$ modes with ${\theta }_1=-{\theta }_2={42.5}^{\circ }$, but now the electric quadrupole is active. Figures 5 and 6 show the distribution of the quadrupole potential (in units of $U_0^{\mathrm{EQ}}=(1/2)\hslash {\mathrm{\Gamma }}_{5D6P}$). The ratio $U_0/{\epsilon }_{\mathrm{re}}$ is equal to $0.184$, and this refers to the bottom of the potential needs additional support to be exploited in atomic trapping processes. It is well-known that the bottom of potential in LG light gives considerable enhancement as the twisting number $\ell$ increases. In fact, there is no obstacle, because a twisting number about $\ell =300$ has been experimentally achieved. In reality, from a theoretical point of view, much less is required to start the atomic trap. The bottom of the quadrupole potential slightly above $\approx 17U_0$ (i.e. $U_0/{\epsilon }_{\mathrm{re}}\approx 1$) is sufficient and can be achievable with $\ell \ge 11$. In free space, it has been demonstrated theoretically that it is possible to achieve an appropriate quadrupole potential bottom for atomic confinement at $\ell \ge 10$ [12].

Figure 5. The distribution of the evanescent quadrupole potential generated by two counter-propagating $L{G}_{10}$ beam at a planar dielectric interface for ${\mathrm{\Delta }}_0>0$.

Figure 6. The distribution of the evanescent quadrupole potential generated by two counter-propagating $L{G}_{10}$ beam at a planar dielectric interface for ${\mathrm{\Delta }}_0<0$.

7. Conclusion

This work aims to study the coupling of the surface optical vortex generated by $L{G}_{01}$ to the two types of electric atomic transitions: dipole and quadrupole. The cesium atom was chosen for the two transitions: $(6S_{1/2}\to 6P_{3/2})$ for dipole-active and $(6S_{1/2}\to 5D_{5/2})$ for quadrupole-active. We have shown that the coupling between the dipole and quadrupole moments with the electric field is inconsistent in the two transitions. In contrast to the dipole-permitted transition, in which the atomic motion evolves with the optical field strength, in the quadrupole-permitted transition, it is driven by the field gradients of optical beams. Thus, it became interesting because they can occur even where there is no light intensity at the center-of-mass position of the atom but only a field gradient. The beam phase and Rabi frequency required to calculate the optical forces responsible for the formation of atom traps and their dynamics in the interaction in the two cases have been derived.

We have confirmed that the rotational motion property of the atom interacting with the evanescent field resulting from $L{G}_{01}$ is possible and exploitable only when the dipole transition is active. The striking point is that the quadrupole potential due to $L{G}_{01}$, as we have seen, did not reach the stage where it would be effective, despite the advantages provided by the configuration of two counter-propagating optical vortex beams. On the other hand, the quadrupole potential bottom, which has been obtained with $L{G}_{01}$, can be enhanced. It is expected that a good enhancement source will be achieved by placing the atoms near the planar surface of a dielectric prism coated with multilayer dielectric thin films of different thicknesses to form a waveguide and then arranging the creation of an evanescent field [20–22]. Work towards obtaining a rotational action resulting from $L{G}_{01}$ with active quadrupole transition in different configurations is currently under consideration and we hope to achieve acceptable results.

Disclosure statement

No potential conflict of interest was reported by the author(s).