Two-level atom dynamics induced by a spin-orbit coupled optical vortex: dressed states formulation

V. E. Lembessis; A. Lyras; O. M. Aldossary

doi:10.1364/OPTCON.488300

1. Introduction

Optical vortex is a term that describes any light beam where the photons carry an orbital angular momentum (OAM) along the propagation direction of the beam [1]. The twisted wavefronts and the quantized OAM of the photons are the main properties of these beams. In particular, the photons are endowed with an OAM given by $\ell \mathrm{\hbar }$ where $\ell $ could be a non-zero integer number or even a fractional number. When the optical vortex beam is circularly polarized, the photons carry also a non-zero spin angular momentum (SAM) which is associated with the two possible states of circular polarization, namely right- and left-hand polarization. Optical vortices, like Laguerre-Gaussian (LG) and Bessel beams are also characterized by highly spatially structured amplitudes and phases [2].

Lately we witness an intense theoretical and experimental work on the physics of tightly focused laser beams [3,4]. Such beams are characterized by a very small beam waist w₀ (of the order of or smaller than the wavelength of the beam) and the existence of longitudinal (or axial) electric and magnetic field components. Due to its small magnitude (in comparison with the transverse electric field), in the case of weak focusing, the longitudinal electric field component has been ignored from the analysis both in theoretical and experimental work [5]. But this component brings in new and very interesting physics; it is responsible for a coupling of the photon OAM and SAM bringing a chirality in the atom-light interaction and giving rise to very interesting physical effects [6]. Such effects have been observed in relativistic mechanics and recently obtained in the physics of optical vortices in the solution of Maxwell’s equations under tight focusing conditions. They are predicted to have considerable effects in the sub-wavelength regime in areas like nanophotonics and plasmonics [7].

The role of the spin-orbit coupling (SOC) in the mechanical effects of light on atoms and small particles has been known since 1996 when Allen and colleagues demonstrated that when a two-level atom interacts with a circularly polarized LG beam then the azimuthal component of the dissipative force depends on the coupling between the photon SAM and OAM. It was the first time in this research field to be shown that apart from the internal dynamics of an atom the SAM can affect also its translational motion [8]. As stated in that work, the authors did not consider whether this SOC contribution could be of considerable magnitude for tightly focused beams. Since then, there were no works concerning the mechanical effects of this SOC term except in Ref. [6], where the authors studied the modifications of the optical dipole potential on two-level atoms when irradiated by far-detuned monochromatic and bichromatic optical vortex fields, and in Ref. [9], where the authors investigated the effect SOC had on the rotational motion of small particles. Among other related works we could refer to that by Quinteiro and colleagues who investigated the role of the longitudinal electric field component in the quadrupole interaction of a Calcium ion situated at the dark core of an LG beam [10] and to that which investigated the existence of this field component by employing the interaction of a vortex beam with a single molecule possessing a permanent dipole moment [11]. More recently, the first-order SOC correction to paraxial beams has been employed in a model of photonic graphene to demonstrate experimentally its significance in controlling the angular momentum properties of a probe beam scattered from the photonic graphene lattice [12]. Tightly focused beams of light are characterized by a modified spatial structure due to the longitudinal field component. This effect is not limited to light beams but it is a common feature in relativistic electron vortices and gravitational vortex fields [13].

In this paper we aim to study fundamental aspects of the interaction of a vortex beam with SOC with a two-level system. We focus specifically on the internal dynamics of the two-level atom, within the context of the dressed-atom formalism, paying special attention to the modification of the well-known dynamical features by the presence and action of the SOC term. The dressed-atom formalism is a convenient and formally compact approach to study the dynamics of the field-atom interaction. However, any other semiclassical or fully quantum approach would reproduce the same results. We demonstrate that under specific conditions, clearly illustrated with realistic numerical examples, the effect of the SOC term could be significant and amenable to experimental observation.

The structure of this paper is the following. In Section 2 we give the exact form of the Hamiltonian which describes the atom-beam interaction. In section 3 we present the dressed states formalism and numerical examples based on experimentally accessible parameters. In section 4 we consider the modifications of the resonance fluorescence spectrum from a prototype two-level atom and we show that the chirality affects the characteristics of the well-known Mollow triplet. Our calculations are again accompanied by numerical examples. In section 4 we summarize the conclusions of this work.

2. Atom-beam interaction

For a tightly focused paraxial LG beam of wavelength λ, the electric field is given by [14]:

(1)$$\textbf{E} = \frac{1}{2}\left[ {\alpha U\hat{\textbf{x}} + \beta U\hat{\textbf{y}} + \frac{1}{k}\left[ {i\left( {\alpha \frac{{\partial U}}{{\partial x}} + \beta \frac{{\partial U}}{{\partial y}}} \right) - U\left( {\alpha \frac{{\partial \mathrm{\Theta }}}{{\partial x}} + \beta \frac{{\partial \mathrm{\Theta }}}{{\partial y}}} \right)} \right]\hat{\textbf{z}}} \right]{e^{ikz + i\mathrm{\Theta } - i\omega t}}$$

with

(2)$$U = \frac{{{E_0}{C_{p,|\ell |}}}}{{\sqrt {1 + {z^2}/z_R^2} }}{\left( {\frac{{\rho \sqrt 2 }}{{{w_0}\sqrt {1 + {z^2}/z_R^2} }}} \right)^{|\ell |}}exp\left[ { - \frac{{2{\rho^2}}}{{w_0^2({1 + {z^2}/z_R^2} )}}} \right]L_p^{|\ell |}\left( {\frac{{2{\rho^2}}}{{w_0^2({1 + {z^2}/z_R^2} )}}} \right), $$

(3)$$\mathrm{\Theta } = \ell \phi + \arctan \left( {\frac{z}{{{z_R}}}} \right) + \frac{{kz{\rho ^2}}}{{2({{z^2} + z_R^2} )}}, $$

where ${w_0}$ is the beam waist, ${z_R}$ the Rayleigh range, $k = 2\pi /\lambda $ is the wave number of the beam, ${C_{p,|\ell |}} = \sqrt {p!/(p! + |\ell |!} )$, $L_p^{|\ell |}$ the associated Laguerre polynomial and ${E_0}$ the field amplitude. The quantity $\rho $ is the radial distance $\rho = \sqrt {{x^2} + {y^2}} $ in cylindrical coordinates, while $\alpha $ and $\beta $ are in general complex numbers defining the transverse beam polarization with ${|a |^2} + {|\beta |^2} = 1$. The intensity of the beam is proportional to the modulus squared of this field given by:

(4)$$\begin{array}{l} {|\textbf{E} |^2} = \textbf{E} \cdot {\textbf{E}^{\boldsymbol \ast }} = \frac{1}{4}({{{|a |}^2} + {{|\beta |}^2}} ){U^2} - \frac{{i\sigma \ell U}}{{4{k^2}\rho }}\frac{{\partial U}}{{\partial \rho }} + \frac{1}{{4{k^2}}}{|a |^2}co{s^2}\phi {\left( {\frac{{\partial U}}{{\partial \rho }}} \right)^2} + {|\beta |^2}si{n^2}\phi {\left( {\frac{{\partial U}}{{\partial \rho }}} \right)^2}\\ + {|a |^2}{U^2}{\left( {\frac{{ - \ell sin\phi }}{\rho }} \right)^2} + {|\beta |^2}{U^2}{\left( {\frac{{\ell cos\phi }}{\rho }} \right)^2} \end{array}$$

where the partial derivatives with respect to Cartesian coordinates x and y have been replaced by partial coordinates with respect to $\rho $. In Eq. (4) the quantity $\sigma $ is defined as $\sigma = ({a{\beta^\ast } - {\alpha^\ast }\beta } )$ and it is actually the spin (SAM) of the beam and for a circularly polarized light is purely imaginary. As we may see the second term in the RHS of Eq. (4) involves a coupling of the spin of the beam with the OAM of its photons. The sign of this term can be reversed either by reversing the circular polarization sense or the OAM winding sense; reversing both OAM and SAM simultaneously will leave the sign of the SOC term unchanged. Moreover, since this term depends on $\frac{{\partial U}}{{\partial \rho }}$ it clearly originates from the longitudinal part of the field defined in (1). The interaction of the beam with the atom is described by a Rabi frequency $\mathrm{\tilde{\Omega }}$ which is proportional to the square root of the intensity of the beam and so ${\mathrm{\tilde{\Omega }}^2} \propto {|\textbf{E} |^2}$. The expression for ${\mathrm{\tilde{\Omega }}^2}$ is given by:

(5)$${\mathrm{\tilde{\Omega }}^2} = {\mathrm{\Omega }^2} - \frac{{i\sigma \ell \mathrm{\Omega }}}{{{k^2}\rho }}\frac{{\partial \mathrm{\Omega }}}{{\partial \rho }} + \frac{1}{{2{k^2}}}\left\{ {{{\left( {\frac{{\partial \mathrm{\Omega }}}{{\partial \rho }}} \right)}^2} + \frac{{{\mathrm{\Omega }^2}{\ell^2}}}{{{\rho^2}}}} \right\}$$

where, assuming we work close to the beam focus, z = 0, and defining $\mathrm{\Omega } = {\mathrm{\Omega }_0}{C_{p,|\ell |}}{\left( {\frac{{\rho \sqrt 2 }}{{{w_0}}}} \right)^{|\ell |}}$$exp\left[ { - \frac{{2{\rho^2}}}{{w_0^2}}} \right]L_p^{|\ell |}\left( {\frac{{2{\rho^2}}}{{w_0^2}}} \right)$, with ${\mathrm{\Omega }_0} = \frac{{d{E_0}}}{\hbar },$ d being the induced dipole moment, we also took into account the fact that for a circularly polarized beam $|a |= |\beta |= 1/2$. The above relation can be written in another way if we recall that for circularly polarized light $\sigma ={\pm} i$ thus

(6)$${\mathrm{\tilde{\Omega }}^2} = {\mathrm{\Omega }^2} + {\left[ {\frac{1}{{k\sqrt 2 }}\left( {\frac{{\ell \mathrm{\Omega }}}{\rho } \pm |\sigma |\frac{{\partial \mathrm{\Omega }}}{{\partial \rho }}} \right)} \right]^2}$$

From Eq. (5) we also see that the contribution of the spin-orbit coupling is given by the term $\frac{{i\sigma \ell \mathrm{\Omega }}}{{{k^2}\rho }}\frac{{\partial \mathrm{\Omega }}}{{\partial \rho }}$, that is, directly proportional to the term $\partial \mathrm{\Omega }/\partial \rho $, which, for p = 0, is given by:

(7)$$\; \frac{{\partial \mathrm{\Omega }}}{{\partial \rho }} = \mathrm{\Omega }\left( {\frac{{|\ell |}}{\rho } - \frac{{2\rho }}{{w_0^2}}} \right).$$

This quantity becomes zero when $\rho = {\rho _0} = {w_0}\sqrt {|\ell |/2} $. At such radial distances the spin-orbit coupling vanishes locally. It is interesting that in the radial distance ${\rho _0}$ the intensity of a weakly (not a tightly) focused LG beam maximizes. In general, the contribution of the SOC term in the overall interaction is ρ-dependent.

3. Dressed states picture

We consider now the interaction of a tightly focused LG beam with a two-level atom, of transition frequency ${\omega _0}$, in the context of dressed states. The diagram presented in Fig. 1 shows the dressed states of the field-atom system and their characteristic quantities.

Fig. 1. The dressed states of a two-level atom interacting with the coherent field of a light beam with angular frequency ${\omega _L}$.

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The energies of the two dressed states in any manifold are given by:

(8)$${E_{1,2}} = \frac{{ - \hbar \mathrm{\Delta } \pm \hbar \sqrt {{\mathrm{\Delta }^2} + {{\mathrm{\tilde{\Omega }}}^2}} }}{2}$$

where $\mathrm{\Delta } = {\omega _0} - {\omega _L}$ is the detuning. As it is obvious the term ${\mathrm{\tilde{\Omega }}^2}$ contributes a SOC term in the energies of the dressed states. The expressions for the two-dressed states are given by:

(9a)$$|{1(N)\rangle= sin\theta } |g,N + 1\rangle + cos\theta |e,N\rangle, $$

(9b)$$|{2(N)\rangle= cos\theta } |g,N + 1\rangle - sin\theta |e,N\rangle, $$

where

(10)$$cos2\theta ={-} \frac{\mathrm{\Delta }}{{\sqrt {{\mathrm{\Delta }^2} + {{\mathrm{\tilde{\Omega }}}^2}} }}. $$

A straightforward comparison of Eq. (10) with Eqs. (9) shows clearly that SOC contributes also in the dressed states wavefunctions, and thus it will also contribute in the transition rates between dressed states of different manifolds. For example:

(11)$${\mathrm{\Gamma }_{12}} \propto \mathrm{\Gamma }co{s^4}\theta , $$

with

(12)$$co{s^2}\theta = \frac{{1 + cos2\theta }}{2} = \frac{{1 - \; \frac{\mathrm{\Delta }}{{\sqrt {{\mathrm{\Delta }^2} + {{\mathrm{\tilde{\Omega }}}^2}} }}}}{2}. $$

Moreover the SOC will also affect the steady-state populations of the two levels given by:

\pi _1^{\textrm{st}} = \frac{{si{n^4}\theta }}{{si{n^4}\theta + co{s^4}\theta }}, \qquad \pi _2^{\textrm{st}} = \frac{{co{s^4}\theta }}{{si{n^4}\theta + co{s^4}\theta }}

with

(14)$$\textrm{co}{\textrm{s}^4}\theta = {\left( {\frac{{\sqrt {{\mathrm{\Delta }^2} + {{\mathrm{\tilde{\Omega }}}^2}} - \mathrm{\Delta }}}{{2\sqrt {{\mathrm{\Delta }^2} + {{\mathrm{\tilde{\Omega }}}^2}} }}} \right)^2}. $$

Consider the case of Na atom and the two-level transition ${3^2}{\textrm{S}_{1/2}} - {3^2}{\textrm{P}_{3/2}}$ ($\mathrm{\Gamma }/2\pi = 10.01\; \textrm{MHz}$), $\mathrm{\Delta } = 1000\mathrm{\Gamma }$, ${\mathrm{\Omega }_0} = 2500\mathrm{\Gamma }$, $\lambda = 589.16\; nm$. The reason we choose so large detuning is that we want to study an interaction which is not sensitive to the details of the atomic transition since the longitudinal field component actually turns our LG beam from circularly to elliptically polarized. We assume tight focusing conditions with a beam waist reduced to half the wavelength ${w_0} = \lambda /2$. In this case, we get the Figs. 2 to 4 for the energies of the dressed states, the relaxation rate between the dressed states and the steady-state populations of the dressed states, respectively. In each figure there are three different curves. The solid one represents the quantity in the case of the ordinary dressed state without taking into account the terms due to the small longitudinal electric field component, i.e. the SOC term and any additional term depending on $\frac{{\partial \mathrm{\Omega }}}{{\partial \rho }}$. The dashed and dotted lines represent the cases where the longitudinal component is taken into account for winding numbers $\ell ={+} 1$ and $\ell ={-} 1$, respectively. The LG beams have a counter-clockwise circular polarization with $\alpha = 1/\sqrt 2 $ and $\beta ={-} i/\sqrt 2 $.

Fig. 2. The energies of the two dressed states as a function of the radial distance from the beam axis. The solid one represents the quantity in the case of the ordinary dressed states without taking into account the terms due to the small longitudinal electric field component. The dashed and dotted lines represent the case where the longitudinal component is taken into account for winding numbers $\ell ={+} 1$ and $\ell ={-} 1$, respectively. The LG beams have a counter-clockwise circular polarization with $\alpha = 1/\sqrt 2 $ and $\beta ={-} i/\sqrt 2 $.

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The effect of the S-O coupling term on eigenergies, relaxation rates and dressed states populations is clearly shown in Figs. 2, 3 and 4. In the case of $\ell ={-} 1$ it is even more striking since it gives a robust difference from the ordinary behavior for the on-axis region (ρ= 0) where, as we know, the intensity of a LG beam is zero. This is easily understood from Eq. (6) when $\ell ={-} 1$ and the circular polarization is right-handed so that the term $+ |\sigma |\frac{{\partial \mathrm{\Omega }}}{{\partial \rho }}$ applies. In this case, the term $\frac{{\ell \mathrm{\Omega }}}{\rho }$ becomes a constant as $\rho \to 0$ resulting in a significant on-axis contribution. This characteristic on-axis behavior is present in all the quantities we have plotted and disappears when we consider values for $\ell $ larger than one as we can see in Figs. 5, 6 and 7. This is easily understood from Eq. (6), once more, since $\mathrm{\Omega }\sim {\rho ^l}$ and for $l \ge 2$, the term $\frac{{\ell \mathrm{\Omega }}}{\rho }$ vanishes as $\rho \to 0$.

Fig. 3. The transition rate between the two dressed states as a function of the radial distance from the beam axis. The solid one represents the quantity in the case of the ordinary dressed states without taking into account the terms due to the small longitudinal electric field component. The dashed and dotted lines represent the case where the longitudinal component is taken into account for winding numbers $\ell ={+} 1$ and $\ell ={-} 1$, respectively. The LG beams have a counter-clockwise circular polarization with $\alpha = 1/\sqrt 2 $ and $\beta ={-} i/\sqrt 2 $.

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Fig. 4. The population $\pi _1^{\textrm{st}}$ of the first dressed state as a function of the radial distance from the beam axis. The solid one represents the quantity in the case of the ordinary dressed states without taking into account the terms due to the small longitudinal electric field component. The dashed and dotted lines represent the case where the longitudinal component is taken into account for winding numbers $\ell ={+} 1$ and $\ell ={-} 1$, respectively. The LG beams have a counter-clockwise circular polarization with $\alpha = 1/\sqrt 2 $ and $\beta ={-} i/\sqrt 2 $.

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Fig. 5. The energies of the two dressed states as a function of the radial distance from the beam axis. The solid one represents the quantity in the case of the ordinary dressed states without taking into account the terms due to the small longitudinal electric field component. The dashed and dotted lines represent the case where the longitudinal component is taken into account for winding numbers $\ell ={+} 2$ and $\ell ={-} 2$, respectively. The power is the same as in the case presented in Fig. 2. The LG beams have a counter-clockwise circular polarization with $\alpha = 1/\sqrt 2 $ and $\beta ={-} i/\sqrt 2 $.

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Fig. 6. The transition rate between the two dressed states as a function of the radial distance from the beam axis. The solid one represents the quantity in the case of the ordinary dressed states without taking into account the terms due to the small longitudinal electric field component. The dashed and dotted lines represent the case where the longitudinal component is taken into account for winding numbers $\ell ={+} 2$ and $\ell ={-} 2$, respectively. The power is the same as in the case presented in Fig. 3. The LG beams have a counter-clockwise circular polarization with $\alpha = 1/\sqrt 2 $ and $\beta ={-} i/\sqrt 2 $.

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Fig. 7. The population $\pi _1^{\textrm{st}}$ of the first dressed state as a function of the radial distance from the beam axis. The solid one represents the quantity in the case of the ordinary dressed states without taking into account the terms due to the small longitudinal electric field component. The dashed and dotted lines represent the case where the longitudinal component is taken into account for winding numbers $\ell ={+} 2$ and $\ell ={-} 2$, respectively. The power is the same as in the case presented in Fig. 4. The LG beams have a counter-clockwise circular polarization with $\alpha = 1/\sqrt 2 $ and $\beta ={-} i/\sqrt 2 $.

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The increase in the value of the winding number makes the SOC term stronger (at off-axis distances) and this is clearly reflected in the difference in the values of the plotted quantities when $\ell ={\pm} 2$ compared to the corresponding quantities when $\ell ={\pm} 1$. A special comment is warranted for Fig. 7 as compared to Fig. 4. In both figures the population of the lower dressed state in the steady state is depicted vs. the radial distance from the axis. Since the two-level system is closed the population of the upper state is given by $\pi _2^{\textrm{st}} = 1 - \pi _1^{\textrm{st}}$. It is evident from Fig. 7 that the steady-state depopulation of the lower state (consequently, the population of the upper state) is overall smaller for $\ell ={\pm} 2$ than for $\ell ={\pm} 1$ (Fig. 4). This result is obviously not related to the SOC terms as it also obtained when this term is ignored. It is simply due to the fact that for lower values of $\ell $ the intensity of the beam is larger since the beam energy is distributed over a smaller area around the symmetry axis of the LG beam.

4. Mollow triplet

We anticipate that the changes brought by the longitudinal field component and the concomitant SOC term in all the above quantities will have observable consequences in well-known effects from two-level atom spectroscopy. Such an effect is the Mollow triplet that appears in the fluorescence spectrum of a two-level atom when it is irradiated by an intense laser beam of frequency ${\omega _L}$. In this case, the fluorescence spectrum is peaked at the following three different frequencies (expressed with respect to the atomic resonance frequency):

- \sqrt {{\mathrm{\Delta }^2} + {{\mathrm{\tilde{\Omega }}}^2}} ,\; \; \; 0,\; \; \; \; \sqrt {{\mathrm{\Delta }^2} + {{\mathrm{\tilde{\Omega }}}^2}}

It is obvious that if we consider two LG beams which have opposite winding numbers and the same circular polarization, or the same winding number and opposite circular polarizations then the two sideband frequencies will be shifted at different positions. We know that in a two-level atom the resonance fluorescence spectrum gets the form [15]:

(16)$$\begin{array}{c} g({{\omega_{sc}} - {\omega_0}} )= \frac{{{s_0}}}{{8\pi }}\frac{s}{{({1 + s} )}} \times \\ \frac{{1 + \frac{{{s_0}}}{4} + {{\left( {\frac{\mathrm{\Delta }}{\mathrm{\Gamma }}} \right)}^2}}}{{{{\left[ {\frac{1}{4} + \frac{{{s_0}}}{4} + {{\left( {\frac{\mathrm{\Delta }}{\mathrm{\Gamma }}} \right)}^2} - 2{{\left( {\frac{{{\omega_{sc}} - {\omega_0}}}{\mathrm{\Gamma }}} \right)}^2}} \right]}^2} + {{\left( {\frac{{{\omega_{sc}} - {\omega_0}}}{\mathrm{\Gamma }}} \right)}^2}{{\left[ {\frac{5}{4} + \frac{{{s_0}}}{2} + {{\left( {\frac{\mathrm{\Delta }}{\mathrm{\Gamma }}} \right)}^2} - {{\left( {\frac{{{\omega_{sc}} - {\omega_0}}}{\mathrm{\Gamma }}} \right)}^2}} \right]}^2}}} \end{array}$$

where ${\omega _{sc}}$ is the frequency of the emitted (scattered) light and ${\omega _0}$ the atomic resonance frequency. The saturation parameter is defined as ${s_0} = 2{\mathrm{\tilde{\Omega }}^2}/{\mathrm{\Gamma }^2}$ and $s = {s_0}/({1 + 4{\mathrm{\Delta }^2}/{\mathrm{\Gamma }^2}} )$ [16]. We consider now the case of the Na transition used in the previous section and assume a detuning $\mathrm{\Delta } = 1000\mathrm{\Gamma }$, ${\mathrm{\Omega }_0} = 400\mathrm{\Gamma }$, ${w_0} = 0.6\lambda $ and $\ell ={\pm} 10$. The beam intensity (at the plane $z = 0$) becomes maximum at a distance $\rho \approx 2.3{w_{0\; }}.$ We assume that the atom is located at the plane $z = 0$ and at a radial distance $\rho = 2{w_0}$. In Fig. 8 we present the fluorescence spectrum for two beams with $\ell ={\pm} 10$, respectively, together with the “conventional” spectrum for the case where the longitudinal component of the electric field is omitted from the calculations. The spectrum is symmetric around ${\omega _{sc}} - {\omega _0} = 0$, so we only present the central part and the high-energy side-band of the spectrum. We must note that for higher values of detuning we need to take into account the non-RWA terms in the Hamiltonian and the symmetry of the spectrum is destroyed. We clearly see the different maximum values in the different spectra as well as the different positions of the maxima in the “wing”. The effect of the SOC term is evident as the frequency shift of the sideband is affected by the sign of the beam winding parameter. It is also evident that the positive-winding beam induces a stronger departure from the conventional spectrum both in terms of intensity and in terms of frequency shift. Similar results can be obtained if we keep $\ell $ fixed and simply reverse the circular polarization of the beam.

Fig. 8. The inelastic resonance fluorescence spectrum for the Na ${3^2}{S_{1/2}} - {3^2}{P_{3/2}}$ transition. To the left we see the central part of the spectrum and to the right we see the high-energy sideband of the triplet. The low-energy sideband is symmetric to the high-energy one. The red line represents the conventional spectrum which does not take into account the contribution of the longitudinal component of the electric field. The dashed and dotted lines represent the spectrum we get if we take into account the longitudinal component of the field. The dashed line corresponds to an interaction of the atom with an LG beam of counter-clockwise circular polarization with $\alpha = 1/\sqrt 2 $ and $\beta ={-} i/\sqrt 2 $, and $\ell = 10$ while the dotted line with a similarly polarized LG beam of $\ell ={-} 10$. Identical results are obtained if the value of $\ell $ is kept constant and we reverse the polarization of the beam.

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A very interesting case arises when the winding number gets values of $\ell ={\pm} 1$. When $\ell ={+} 1$ and $\sigma ={-} i$ or $\ell ={-} 1$ and $\sigma = i$ the on-axis intensity vanishes due to the contribution of SOC term. This is very important because it could help us to demonstrate in a resonance fluorescence experiment the effect of SOI interaction. In Fig. 9 we give the on-axis ($\rho = 0$) resonance fluorescence spectrum for the Na ${3^2}{S_{1/2}} - {3^2}{P_{3/2}}$ transition in the case where $\ell ={+} 1$ and $\sigma ={+} i$, for $\mathrm{\Delta } = 1000\mathrm{\Gamma }$, ${\mathrm{\Omega }_0} = 400\mathrm{\Gamma }$, ${w_0} = 0.6\lambda $. If the experimenter reverses the winding this Mollow triplet will just vanish.

Fig. 9. The on-axis ($\rho = 0)$ inelastic resonance fluorescence spectrum for the Na ${3^2}{S_{1/2}} - {3^2}{P_{3/2}}$ transition in the case where $\sigma = i$ and $\ell = 1\; \; $ for $\mathrm{\Delta } = 1000\mathrm{\Gamma }$, ${\mathrm{\Omega }_0} = 400\mathrm{\Gamma }$, ${w_0} = 0.6\lambda $. In the numerical work the longitudinal component of the electric field has been taken into account. If it is omitted an LG beam cannot give on-axis resonance fluorescence. For the case where we reverse either the polarization or the winding number we do not get an on-axis spectrum because the on-axis intensity becomes zero.

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5. Conclusions

In this paper we have investigated effects in the dressed states dynamics which arise when a prototype two-level system interacts with a tightly focused paraxial LG beam with circular polarization. In this case, the interaction involves a term which arises from the coupling of the spin of the photon (polarization) with the photon angular momentum, a coupling enabled by taking into account a longitudinal component of the electric field that becomes significant for tight focusing conditions. This coupling introduces a type of chirality in the interaction in the sense that observables of the beam-atom interaction depend on the sign of the beam winding parameter and/or its sense of circular polarization. The particular form of the light beam SOC term, a first-order correction resulting from taking into account the longitudinal component of the electric field, is insensitive to the simultaneous flipping of both SAM and OAM. Moreover, any spatial properties of the atomic system, like the ones encoded in its total angular momentum, have not been taken into account because of the large detuning employed in our calculations. It should be noted that most if not all demonstrated cases of chiral effects, e.g. enantioselectivity or circular dischroism, are resulting from the interaction of polarized beams with targets possessing chiral spatial structure. Therefore, it is not clear if a more general chiral dependence will obtained if a more detailed model of the light-atom interaction is employed.

The SOC dependence is present in all the quantities related to the dynamics of the dressed states namely their energies, populations, coherences and relaxation rates. As such it is anticipated that it will also be present in the various effects that arise in the interaction of an optical vortex beam with a two-level atom. We discussed one of them, namely the Mollow triplet, appearing in the resonance fluorescence spectrum. Our numerical examples with experimentally accessible values of relevant parameters confirmed our theoretical analysis. The analysis presented here can be extended to other physical effects like, for example, the Autler-Townes effect where in a three-level configuration there are two connected transitions, one driven by a strong field and another be a weak probe [17]. In this case, the absorption line of the weak probe beam is split into two components. The experimental observation of the chiral-like character of the dressed states and the physical quantities associated with them requires a strong localization of the atom in certain radial distances on the transverse plane. This can be achieved with the help of optical vortex beams that can provide ring-shaped optical dipole traps on the transverse plane. It is very interesting to mention that, as it can be easily verified, for the numerical values of the parameters we have used in our examples the atom is deeply trapped at regions of maximum intensity by the same light that induces the fluorescence. But what is even more interesting is the considerable on-axis magnitude of the observable quantities when the winding number gets values of $\ell ={\pm} 1$. This interesting feature gives more opportunities for experimental observation of the resonance fluorescence if we irradiate with a tightly focused LG beam atoms that already have been trapped on-axis, for example, with the help of an ordinary two-dimensional optical lattice or optical tweezers. Comparing resonance fluorescence spectra obtained with weakly focused beams vs. the ones from tightly focused beams will reveal the effects predicted from our numerical calculations. Both the intensity maxima and the frequency shifts at the spectral sidebands are sufficiently sizeable in order to be recorded with modern spectroscopy setups.

Funding

National Plan for Science, Technology and Innovation (15-MAT5110-02).

Acknowledgments

This project was funded by the National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award Number (15-MAT5110-02).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. D. L. Andrews and M. Babiker (eds), The Angular Momentum of Light (Cambridge University Press, 2013).

2. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A 45(11), 8185–8189 (1992). [CrossRef]

3. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91(23), 233901 (2003). [CrossRef]

4. V. V. Kotlyar, S. S. Stafeev, and A. G. Nalimov, Sharp Focusing of Laser Light (CRC Press, 2019).

5. M. Babiker, D. L. Andrews, and V. E. Lembessis, “Atoms in complex twisted light,” J. Opt. 21(1), 013001 (2019). [CrossRef]

6. V. E. Lembessis, K. Koksal, J. Yuan, and M. Babiker, “Chirality enabled optical dipole potential energy for two-level atoms,” Phys. Rev. A 103(1), 013106 (2021). [CrossRef]

7. K. Y. Bliokh, F. J. Rodríguez-Fortuño, F. Nori, and A. V. Zayats, “Spin–orbit interactions of light,” Nat. Photonics 9(12), 796–808 (2015). [CrossRef]

8. L. Allen, V. E. Lembessis, and M. Babiker, “Spin-orbit coupling in free-space Laguerre-Gaussian light beams,” Phys. Rev. A 53(5), R2937–R2939 (1996). [CrossRef]

9. Y. Cao, T. Zhu, H. Lv, and W. Ding, “Spin-controlled orbital motion in tightly focused high-order Laguerre-Gaussian beams,” Opt. Express 24(4), 3377–3384 (2016). [CrossRef]

10. G. F. Quinteiro, F. Schmidt-Kaler, and C. T. Schmiegelow, “Twisted-light–ion interaction: the role of longitudinal fields,” Phys. Rev. Lett. 119(25), 253203 (2017). [CrossRef]

11. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal field modes probed by single molecules,” Phys. Rev. Lett. 86(23), 5251–5254 (2001). [CrossRef]

12. Z. Zhang, S. Liang, F. Li, S. Ning, Y. Li, G. Malpuech, Y. Zhang, M. Xiao, and D. Solnyshkov, “Spin-orbit coupling in photonic graphene,” Optica 7(5), 455–462 (2020). [CrossRef]

13. M. Krenn and A. Zeilinger, “On small beams with large topological charge: II. photons, electrons and gravitational waves,” New J. Phys. 20(6), 063006 (2018). [CrossRef]

14. L. Allen, M. J. Padgett, and M. Babiker, “The orbital angular momentum of light,” Prog. Opt. 39, 291–372 (1999). [CrossRef]

15. L. Ortiz-Gutiérrez, R. C. Teixeria, A. Eloy, D. F. da Silva, R. Kaiser, R. Bachelard, and M. Foussché, “Mollow triplet in cold atoms,” New J. Phys. 21(9), 093019 (2019). [CrossRef]

16. H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping, (Springer-Verlag, 1999), p.25.

17. S. H. Autler and C. H. Townes, “Stark effect in rapidly varying fields,” Phys. Rev. 100(2), 703–722 (1955). [CrossRef]

Two-level atom dynamics induced by a spin-orbit coupled optical vortex: dressed states formulation

Abstract

1. Introduction

2. Atom-beam interaction

3. Dressed states picture

4. Mollow triplet

5. Conclusions

Funding

Acknowledgments

Disclosures

Data availability

References

Data availability

Cited By

Figures (9)

Equations (17)

Optics Continuum