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Magic wavelengths for trapping of Sr$^+$ with focused optical vortex

Anal Bhowmik, Sonjoy Majumder

Published Mar 14, 2018• 0 citations
OpenAlex

Abstract

A theory of dynamic polarizability for trapping relevant states of Sr$^+$ is presented here when the ions interact with a focused optical vortex. The coupling between the orbital and spin angular momentum of the optical vortex varies with focusing angle of the beam and is studied in the calculation of the magic wavelengths for $5s_{{1}/{2}}\rightarrow 4d_{{3}/{2}, {5}/{2}}$ transitions of Sr$^+$. The initial state of our interest here is $5s_{{1}/{2}}$ with $m_J = -1/2$ of which is different possible trapping state compare to our recent work on Sr$^+$ [Phys. Rev. A \textbf{97}, 022511 (2018)]. We find variation in magic wavelengths and the corresponding polarizabilities with different combinations of orbital and spin angular momentum of the vortex beam. The variation is very significant when the wavelengths of the beam are in the infrared region of electromagnetic spectrum. The calculated magic wavelengths will help the experimentalists to trap the ion for performing the high precision spectroscopic measurements.

Related papers

Vector polarizability of an atomic state induced by a linearly polarized vortex beam: External control of magic, tune-out wavelengths, and heteronuclear spin oscillations

Anal Bhowmik, Narendra Nath Dutta, Sonjoy Majumder

Experiments with vortex beams have shown a surge of interest in controlling cold atoms. Most of the controlling protocols are dominated by circularly polarized light due to its ability to induce vector polarization at atoms, which is impossible for paraxial linearly polarized light. Here we develop a theory for frequency dependent polarizability of an atomic state interacting with a focused linearly polarized vortex beam. The naturally induced spin-orbit coupling in this type of linearly polarized beam produces vector component of the valence polarizability to an atomic state, obeying the total angular-momentum conservation of the beam. The theory is employed on $^{87}\mathrm{Sr}^{+}$ to accurately calculate the magic wavelengths for the clock transitions and tune-out wavelengths for the clock states using the relativistic coupled-cluster method. The induced vector component in the dynamic polarizability due to the linearly polarized focused vortex beam promotes a fictitious magnetic field to the atomic state. We demonstrate that this fictitious magnetic field, depending on the focusing angle and orbital angular momentum of the beam, improves the flexibility of the coherent heteronuclear spin oscillations in a spin-1 mixture of $^{87}\mathrm{Rb}$ and $^{23}\mathrm{Na}$ atoms.

arXiv:1909.03484•Physical Review A•Dec 21, 2020•6 citations

Tunable magic wavelengths for trapping with focused Laguerre-Gaussian beams

Anal Bhowmik, Narendra Nath Dutta, Sonjoy Majumder

We present in this paper a theory of dynamic polarizability for an atomic state due to an external field of nonparaxial Laguerre-Gaussian (LG) beam using the sum-over-states technique. A highly correlated relativistic coupled-cluster theory is used to evaluate the most important and correlation-sensitive parts of the sum. The theory is applied on ${\mathrm{Sr}}^{+}$ to determine the magic wavelengths for $5{s}_{1/2}\ensuremath{\rightarrow}4{d}_{3/2},4{d}_{5/2}$ transitions. Results show the variation of magic wavelengths with the choice of orbital and spin angular momenta of the incident LG beam. Also, the tunability of the magic wavelengths is studied by using the focusing angle of the LG beam and its efficiency in the near-infrared region is observed. Evaluations of the wide spectrum of magic wavelengths from infrared to ultraviolet have substantial importance to experimentalists for carrying out high-precision measurements in fundamental physics. These magic wavelengths can be used to confine the atom or ion at the dark central node or at the high-intensity ring of the LG beam.

arXiv:1712.06059•Physical Review A•Feb 21, 2018•13 citations

Magic wavelengths and triple magic trapping conditions for $5s^2~^1\!S_0$ and $5s5p~^3\!P_{0,2}$ states of Sr atoms

Yan-Min Wang, Qing-Yi Liu, Yong-Bo Tang +4 more

The static and dynamic electric dipole polarizabilities of the $5s^2~^1\!S_0$ and $5s5p~^3\!P_{0,2}$ states of Sr atoms are calculated using the relativistic configuration interaction plus the many-body perturbation theory (RCI+MBPT) method. Magic wavelengths are determined for the transitions $5s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{0}$, $5s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{2}$, and $5s5p~^3\!P_0\rightarrow 5s5p~^3\!P_{2}$. A comprehensive study is conducted on the dependence of magic wavelengths on the angle between the laser polarization and the magnetic field. Furthermore, the conditions for realizing triple magic trapping at 813.4~nm for the $5s^2~^1\!S_0$, $5s5p~^3\!P_{0}$ and $5s5p~^3\!P_{2}$ states are investigated. In the case of linearly polarized light, when the angle ($θ_p$) between the laser polarization direction and the magnetic field is $79.1(0.7)^\circ$, triple magic trapping for the $5s^2~^1\!S_0$, $5s5p~^3\!P_{0}$, and $5s5p~^3\!P_{2}~M=0$ states can be achieved. This result agrees well with the recent experimental measurement (78.49(3)$^\circ$)[Phys. Rev. Lett. 135, 143401 (2025)]. Meanwhile, triple magic trapping involving the $5s5p~^3\!P_{2}~M=2$ state can be achieved when $θ_p= 37.4(0.3)^\circ$. The conditions for achieving triple magic trapping with circularly and arbitrarily elliptically polarized light are also presented.

arXiv:2512.23326•Dec 29, 2025•0 citations

Magic wavelengths in the alkaline-earth-metal ions

Jasmeet Kaur, Sukhjit Singh Sehra, Bindiya Arora +1 more

We present magic wavelengths for the $n{S}_{1/2}\text{\ensuremath{-}}n{P}_{1/2,3/2},$ and $n{S}_{1/2}\text{\ensuremath{-}}m{D}_{3/2,5/2}$ transitions in ${\mathrm{Mg}}^{+},$ ${\mathrm{Ca}}^{+},$ ${\mathrm{Sr}}^{+},$ and ${\mathrm{Ba}}^{+}$ alkaline-earth-metal ions, for the principal quantum numbers $n$ and $m$ for the ground and first excited $D$ states, respectively. These wavelengths are presented for linearly polarized light by plotting the dynamic polarizabilities of the $nS,n{P}_{1/2,3/2},$ and $m{D}_{3/2,5/2}$ states of the considered ions. Required dynamic polarizabilities are evaluated by employing a relativistic all-order perturbative method and their accuracies are ratified by comparing their static values with available high-precision experimental or other theoretical results. Moreover, some of the magic wavelengths identified by us in ${\mathrm{Ca}}^{+}$ concur with the recent measurements reported by Liu et al. [Phys. Rev. Lett. 114, 223001 (2015)]. Knowledge of these magic wavelengths is propitious to carry out many proposed high-precision measurements trapping the above ions in electric fields with the corresponding frequencies.

arXiv:1508.00653•Physical Review A•Sep 14, 2015•30 citations

Magic wavelengths for the State-insensitive Optical Trapping of Cs Atoms: case studies using the linearly and circularly polarized lights

Sukhjit Singh Sehra, Kiranpreet Kaur, B. K. Sahoo +1 more

We demonstrate magic wavelengths for the $6s ~ {^2}S_{1/2}-6p ~ {^2}P_{3/2,1/2}$ transitions in the Cs atom due to the circularly polarized light. In addition, we also obtain magic wavelengths using the linearly polarized light in order to verify the previously reported values and make a comparative study with the values obtained for the circularly polarized light. Most of these wavelengths are found to be in the optical region and could be of immense interest to the experimentalists for carrying out high precision measurements. To obtain these wavelengths, we have calculated dynamic dipole polarizabilities of the ground, $6p ~{^2}P_{1/2}$ and $6p ~{^2}P_{3/2}$ states. We use the available precise values of the electric dipole (E1) matrix elements for the most dominantly contributing transitions from the lifetime measurements of the above excited states and obtain other relevant important E1 matrix elements by employing a relativistic coupled-cluster method approximating at the singles and doubles level. Accuracies in these quantities are substantiated by comparing our evaluated scalar polarizability values with the corresponding experimental results.

•Jan 4, 2016•0 citations

Divalent Rydberg atoms in optical lattices: Intensity landscape and magic trapping

Türker Topçu, Andrei Derevianko

We develop a theoretical understanding of the trapping of divalent Rydberg atoms in optical lattices. Because the size of the Rydberg electron cloud can be comparable to the scale of spatial variations of laser intensity, we pay special attention to averaging optical fields over the atomic wave functions. The optical potential is proportional to the ac Stark polarizability. We find that in the independent-particle approximation for the valence electrons, this polarizability breaks into two contributions: the singly ionized core polarizability and the contribution from the Rydberg electron. Unlike the usually employed free-electron polarizability, the Rydberg contribution depends both on the laser intensity profile and on the rotational symmetry of the total electronic wave function. We focus on the $J=0$ Rydberg states of Sr and evaluate the dynamic polarizabilities of the $5sns$(${}^{1}{S}_{0}$) and $5snp$(${}^{3}{P}_{0}$) Rydberg states. We specifically chose the Sr atom for its optical-lattice clock applications. We find that there are several magic wavelengths in the infrared region of the spectrum at which the differential Stark shift between the clock states [$5{s}^{2}$(${}^{1}{S}_{0}$) and $5s5p$(${}^{3}{P}_{0}$)] and the $J=0$ Rydberg states [$5sns$(${}^{1}{S}_{0}$) and $5snp$(${}^{3}{P}_{0}$)] vanishes. We tabulate these wavelengths as a function of the principal quantum number $n$ of the Rydberg electron. We find that because the contribution to the total polarizability from the Rydberg electron vanishes at short wavelengths, magic wavelengths below $\ensuremath{\sim}$1000 nm are ``universal'' as they do not depend on the principal quantum number $n$.

arXiv:1312.1738•Physical Review A•Feb 10, 2014•18 citations
Magic wavelengths for trapping of Sr+^++ with focused optical vortex
A theory of dynamic polarizability for trapping relevant states of Sr+^++ is presented here when the ions interact with a focused optical vortex. The coupling between the orbital and spin angular momentum of the optical vortex varies with focusing angle of the beam and is studied in the calculation of the magic wavelengths for 5s1/2→4d3/2,5/25s_{{1}/{2}}\rightarrow 4d_{{3}/{2}, {5}/{2}}5s1/2​→4d3/2,5/2​ transitions of Sr+^++. The initial state of our interest here is 5s1/25s_{{1}/{2}}5s1/2​ with mJ=−1/2m_J = -1/2mJ​=−1/2 of which is different possible trapping state compare to our recent work on Sr+^++ [Phys. Rev. A \textbf{97}, 022511 (2018)]. We find variation in magic wavelengths and the corresponding polarizabilities with different combinations of orbital and spin angular momentum of the vortex beam. The variation is very significant when the wavelengths of the beam are in the infrared region of electromagnetic spectrum. The calculated magic wavelengths will help the experimentalists to trap the ion for performing the high precision spectroscopic measurements.
Experiments with vortex beams have shown a surge of interest in controlling cold atoms. Most of the controlling protocols are dominated by circularly polarized light due to its ability to induce vector polarization at atoms, which is impossible for paraxial linearly polarized light. Here we develop a theory for frequency dependent polarizability of an atomic state interacting with a focused linearly polarized vortex beam. The naturally induced spin-orbit coupling in this type of linearly polarized beam produces vector component of the valence polarizability to an atomic state, obeying the total angular-momentum conservation of the beam. The theory is employed on 87Sr+^{87}\mathrm{Sr}^{+}87Sr+ to accurately calculate the magic wavelengths for the clock transitions and tune-out wavelengths for the clock states using the relativistic coupled-cluster method. The induced vector component in the dynamic polarizability due to the linearly polarized focused vortex beam promotes a fictitious magnetic field to the atomic state. We demonstrate that this fictitious magnetic field, depending on the focusing angle and orbital angular momentum of the beam, improves the flexibility of the coherent heteronuclear spin oscillations in a spin-1 mixture of 87Rb^{87}\mathrm{Rb}87Rb and 23Na^{23}\mathrm{Na}23Na atoms.
We present in this paper a theory of dynamic polarizability for an atomic state due to an external field of nonparaxial Laguerre-Gaussian (LG) beam using the sum-over-states technique. A highly correlated relativistic coupled-cluster theory is used to evaluate the most important and correlation-sensitive parts of the sum. The theory is applied on Sr+{\mathrm{Sr}}^{+}Sr+ to determine the magic wavelengths for $5{s}_{1/2}\ensuremath{\rightarrow}4{d}_{3/2},4{d}_{5/2}$ transitions. Results show the variation of magic wavelengths with the choice of orbital and spin angular momenta of the incident LG beam. Also, the tunability of the magic wavelengths is studied by using the focusing angle of the LG beam and its efficiency in the near-infrared region is observed. Evaluations of the wide spectrum of magic wavelengths from infrared to ultraviolet have substantial importance to experimentalists for carrying out high-precision measurements in fundamental physics. These magic wavelengths can be used to confine the atom or ion at the dark central node or at the high-intensity ring of the LG beam.
Magic wavelengths and triple magic trapping conditions for 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​ and 5s5p 3 ⁣P0,25s5p~^3\!P_{0,2}5s5p 3P0,2​ states of Sr atoms
The static and dynamic electric dipole polarizabilities of the 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​ and 5s5p 3 ⁣P0,25s5p~^3\!P_{0,2}5s5p 3P0,2​ states of Sr atoms are calculated using the relativistic configuration interaction plus the many-body perturbation theory (RCI+MBPT) method. Magic wavelengths are determined for the transitions 5s2 1 ⁣S0→5s5p 3 ⁣P05s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{0}5s2 1S0​→5s5p 3P0​, 5s2 1 ⁣S0→5s5p 3 ⁣P25s^2~^1\!S_0\rightarrow 5s5p~^3\!P_{2}5s2 1S0​→5s5p 3P2​, and 5s5p 3 ⁣P0→5s5p 3 ⁣P25s5p~^3\!P_0\rightarrow 5s5p~^3\!P_{2}5s5p 3P0​→5s5p 3P2​. A comprehensive study is conducted on the dependence of magic wavelengths on the angle between the laser polarization and the magnetic field. Furthermore, the conditions for realizing triple magic trapping at 813.4~nm for the 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​, 5s5p 3 ⁣P05s5p~^3\!P_{0}5s5p 3P0​ and 5s5p 3 ⁣P25s5p~^3\!P_{2}5s5p 3P2​ states are investigated. In the case of linearly polarized light, when the angle (θpθ_pθp​) between the laser polarization direction and the magnetic field is 79.1(0.7)∘79.1(0.7)^\circ79.1(0.7)∘, triple magic trapping for the 5s2 1 ⁣S05s^2~^1\!S_05s2 1S0​, 5s5p 3 ⁣P05s5p~^3\!P_{0}5s5p 3P0​, and 5s5p 3 ⁣P2 M=05s5p~^3\!P_{2}~M=05s5p 3P2​ M=0 states can be achieved. This result agrees well with the recent experimental measurement (78.49(3)∘^\circ∘)[Phys. Rev. Lett. 135, 143401 (2025)]. Meanwhile, triple magic trapping involving the 5s5p 3 ⁣P2 M=25s5p~^3\!P_{2}~M=25s5p 3P2​ M=2 state can be achieved when θp=37.4(0.3)∘θ_p= 37.4(0.3)^\circθp​=37.4(0.3)∘. The conditions for achieving triple magic trapping with circularly and arbitrarily elliptically polarized light are also presented.
We present magic wavelengths for the $n{S}_{1/2}\text{\ensuremath{-}}n{P}_{1/2,3/2},$ and $n{S}_{1/2}\text{\ensuremath{-}}m{D}_{3/2,5/2}$ transitions in Mg+,{\mathrm{Mg}}^{+},Mg+, Ca+,{\mathrm{Ca}}^{+},Ca+, Sr+,{\mathrm{Sr}}^{+},Sr+, and Ba+{\mathrm{Ba}}^{+}Ba+ alkaline-earth-metal ions, for the principal quantum numbers nnn and mmm for the ground and first excited DDD states, respectively. These wavelengths are presented for linearly polarized light by plotting the dynamic polarizabilities of the nS,nP1/2,3/2,nS,n{P}_{1/2,3/2},nS,nP1/2,3/2​, and mD3/2,5/2m{D}_{3/2,5/2}mD3/2,5/2​ states of the considered ions. Required dynamic polarizabilities are evaluated by employing a relativistic all-order perturbative method and their accuracies are ratified by comparing their static values with available high-precision experimental or other theoretical results. Moreover, some of the magic wavelengths identified by us in Ca+{\mathrm{Ca}}^{+}Ca+ concur with the recent measurements reported by Liu et al. [Phys. Rev. Lett. 114, 223001 (2015)]. Knowledge of these magic wavelengths is propitious to carry out many proposed high-precision measurements trapping the above ions in electric fields with the corresponding frequencies.
We demonstrate magic wavelengths for the 6s 2S1/2−6p 2P3/2,1/26s ~ {^2}S_{1/2}-6p ~ {^2}P_{3/2,1/2}6s 2S1/2​−6p 2P3/2,1/2​ transitions in the Cs atom due to the circularly polarized light. In addition, we also obtain magic wavelengths using the linearly polarized light in order to verify the previously reported values and make a comparative study with the values obtained for the circularly polarized light. Most of these wavelengths are found to be in the optical region and could be of immense interest to the experimentalists for carrying out high precision measurements. To obtain these wavelengths, we have calculated dynamic dipole polarizabilities of the ground, 6p 2P1/26p ~{^2}P_{1/2}6p 2P1/2​ and 6p 2P3/26p ~{^2}P_{3/2}6p 2P3/2​ states. We use the available precise values of the electric dipole (E1) matrix elements for the most dominantly contributing transitions from the lifetime measurements of the above excited states and obtain other relevant important E1 matrix elements by employing a relativistic coupled-cluster method approximating at the singles and doubles level. Accuracies in these quantities are substantiated by comparing our evaluated scalar polarizability values with the corresponding experimental results.
We develop a theoretical understanding of the trapping of divalent Rydberg atoms in optical lattices. Because the size of the Rydberg electron cloud can be comparable to the scale of spatial variations of laser intensity, we pay special attention to averaging optical fields over the atomic wave functions. The optical potential is proportional to the ac Stark polarizability. We find that in the independent-particle approximation for the valence electrons, this polarizability breaks into two contributions: the singly ionized core polarizability and the contribution from the Rydberg electron. Unlike the usually employed free-electron polarizability, the Rydberg contribution depends both on the laser intensity profile and on the rotational symmetry of the total electronic wave function. We focus on the J=0J=0J=0 Rydberg states of Sr and evaluate the dynamic polarizabilities of the 5sns5sns5sns(1S0{}^{1}{S}_{0}1S0​) and 5snp5snp5snp(3P0{}^{3}{P}_{0}3P0​) Rydberg states. We specifically chose the Sr atom for its optical-lattice clock applications. We find that there are several magic wavelengths in the infrared region of the spectrum at which the differential Stark shift between the clock states [5s25{s}^{2}5s2(1S0{}^{1}{S}_{0}1S0​) and 5s5p5s5p5s5p(3P0{}^{3}{P}_{0}3P0​)] and the J=0J=0J=0 Rydberg states [5sns5sns5sns(1S0{}^{1}{S}_{0}1S0​) and 5snp5snp5snp(3P0{}^{3}{P}_{0}3P0​)] vanishes. We tabulate these wavelengths as a function of the principal quantum number nnn of the Rydberg electron. We find that because the contribution to the total polarizability from the Rydberg electron vanishes at short wavelengths, magic wavelengths below $\ensuremath{\sim}$1000 nm are ``universal'' as they do not depend on the principal quantum number nnn.