Vector polarizability of atomic state induced by a linearly polarized vortex beam: External control of magic, tune-out wavelengths, and heteronuclear spin oscillations
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 $^{87}$Sr$^{+}$ ion to accurately calculate the magic wavelengths for the clock transitions and tune-out wavelengths for the clock states using relativistic coupled-cluster method. The induced vector component in the dynamic polarizability due to the linearly polarized focused vortex beam promotes fictitious magnetic field to the atomic state. We demonstrate that this fictitious magnetic field, depending on the focusing angle and OAM of the beam, improves the flexibility of the coherent heteronuclear spin oscillations in a spin-1 mixture of $^{87}$Rb and $^{23}$Na atoms.
We investigate oscillation frequencies for simultaneous trapping of more than one type of alkali atoms in a common optical lattice. For this purpose, we present numerical results for magic trapping conditions, where the oscillation frequencies for two different kind of alkali atoms using laser lights in the wavelength range 500-1200 nm are same. These wavelengths will be of immense interest for studying static and dynamic properties of boson-boson, boson-fermion, fermion-fermion, and boson-boson-boson mixtures involving different isotopes of Li, Na, K, Rb, Cs and Fr alkali atoms. In addition t…
Highly excited Rydberg states are usually extremely polarizable and exceedingly sensitive to electric fields. Because of this Rydberg ions confined in electric fields have state-dependent trapping potentials. We engineer a Rydberg state that is insensitive to electric fields by coupling two Rydberg states with static polarizabilities of opposite sign, in this way we achieve state-independent magic trapping. We show that the magically-trapped ion can be coherently excited to the Rydberg state without the need for control of the ion's motion.
We develop a theoretical understanding of trapping 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 wavefunctions. 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 usua…
Selection of "magic" trapping conditions with ultracold atoms or molecules, where pairs of internal states experience identical trapping potentials, brings substantial benefits to precision measurements and quantum computing schemes. Working at such conditions could ensure that detrimental effects of inevitable inhomogeneities across an ultracold sample are significantly reduced. However, this aspect of confinement remains unexplored for ultracold polar molecules. Here, we present means to control the AC Stark shift of rotational states of ultracold polar molecules, when subjected to both trap…
The dynamic dipole polarizabilities of the low-lying states of Ca$^{+}$ for linearly and circularly polarized light are calculated by using relativistic configuration interaction plus core polarization (RCICP) approach. The magic wavelengths, at which the two levels of the transitions have the same ac Stark shifts, for $4s$-$4p_{j,m}$ and $4s$-$3d_{j,m}$ magnetic sublevels transitions are determined. The present magic wavelengths for linearly polarized light agree with the available results excellently. The polarizability for the circularly polarized light has the scalar, vector and tensor com…
Ionic clocks exhibit as the most promising candidates for the frequency standards. Recent investigations show the profound advantages of interrogating two laser beams with different frequencies in developing the frequency standards. Here we present a scheme of a two-photon mechanism to calculate the dynamic polarizabilities for the clock states, 6$^2$S$_{\frac{1}{2}}$ and 5$^2$D$_{\frac{3}{2}, \frac{5}{2}}$, of Ba$^+$ by employing relativistic coupled-cluster method. We illustrate the Stark-shift cancellation between these clock states at the two-photon magic wavelengths. These magic wavelengt…
We report on the realization of an ultracold mixture of lithium atoms in the ground state and ytterbium atoms in the excited metastable 3P2 state. Such a mixture can support broad magnetic Feshbach resonances which may be utilized for the production of ultracold molecules with an electronic spin degree of freedom, as well as novel Efimov trimers. We investigate the interaction properties of the mixture in the presence of an external magnetic field and find an upper limit for the background interspecies two-body inelastic decay coefficient of K'2 < 3e-12 cm^3/s for the 3P2 m_J=-1 substate. We c…
We demonstrate a versatile, rotational-state dependent trapping scheme for the ground and first excited rotational states of $^{23}$Na$^{40}$K molecules. Close to the rotational manifold of a narrow electronic transition, we determine tune-out frequencies where the polarizability of one state vanishes while the other remains finite, and a magic frequency where both states experience equal polarizability. The proximity of these frequencies of only 10 GHz allows for dynamic switching between different trap configurations in a single experiment, while still maintaining sufficiently low scattering…
A generic scheme to trap atoms at the magic wavelengths ($λ_{\rm{magic}}$s) that are independent of vector and tensor components of the interactions of the atoms with the external electric field is presented. The $λ_{\rm{magic}}$s for the laser cooling D2 lines in the Rb and Cs atoms are demonstrated and their corresponding polarizability values without vector and tensor contributions are given. Consequently, these $λ_{\rm{magic}}$s are independent of magnetic sublevels and hyperfine levels of the atomic states involved in the transition, thus, can offer unique approaches to carry out many hig…
The zero crossing of the dynamic differential scalar polarizability of the $S_{1/2}-D_{5/2}$ clock transition in $^{138}$Ba$^+$ has been determined to be $459.1614(28)\,$THz. Together with previously determined matrix elements and branching ratios, this tightly constrains the dynamic differential scalar polarizability of the clock transition over a large wavelength range ($\gtrsim 700\,$nm). In particular it allows an estimate of the blackbody radiation shift of the clock transition at room temperature.
Motivated by compelling advances in manipulating cold Rydberg (Ry) atoms in optical traps, we consider the effect of large extent of Ry electron wave function on trapping potentials. We find that when the Ry orbit lies outside inflection points in laser intensity landscape, the atom can stably reside in laser intensity maxima. Effectively, the free-electron AC polarizability of Ry electron is modulated by intensity landscape and can accept both positive and negative values. We apply these insights to determining magic wavelengths for Ry-ground-state transitions for alkali atoms trapped in infr…
Recently, we studied the magic wavelength for the atomic hydrogen 1S-2S transition [A.K., Phys. Rev. A 92, 042507 (2015)]. An explicit summation over virtual atomic states of the discrete part of the hydrogen spectrum was performed to evaluate the atomic polarizability. In this paper, we supplement the contribution of the continuum part of the spectrum and add the reduced-mass correction. The magic wavelength, at which the lowest-order ac Stark shifts of the 1S and 2S states are equal, is found to be equal to 514.6 nm. The ac Stark shift at the magic wavelength is -221.6 Hz / (kW/cm^2), and th…
For the first time, we experimentally determine the infrared magic wavelength for the $^{40}$Ca$^{+}$ $4s\, ^{2}\!S_{1/2} \rightarrow 3d\,^{2}\!D_{5/2}$ electric quadrupole transition by observation of the light shift canceling in $^{40}$Ca$^{+}$ optical clock. A "magic" magnetic field direction is chosen to make the magic wavelength insensitive to both the linear polarization purity and the polarization direction of the laser. The determined magic wavelength for this transition is 1056.37(9)~nm, which is not only in good agreement with theoretical predictions but also more precise by a factor…
We have calculated ac polarizabilities of the $2 \ ^3S$ and $2 \ ^1S$ states of both $^4$He and $^3$He in the range 318 nm to 2.5 $μ$m and determined the magic wavelengths at which these polarizabilities are equal for either isotope. The calculations, only based on available ab initio tables of level energies and Einstein A coefficients, do not require advanced theoretical techniques. The polarizability contribution of the continuum is calculated using a simple extrapolation beyond the ionization limit, yet the results agree to better than 1% with such advanced techniques. Several promising ma…
We characterize the anisotropic differential ac-Stark shift for the Dy $626$ nm intercombination transition, induced in a far-detuned $1070$ nm optical dipole trap, and observe the existence of a "magic polarization" for which the polarizabilities of the ground and excited states are equal. From our measurements we extract both the scalar and tensorial components of the dynamic dipole polarizability for the excited state, $α_E^\text{s} = 188 (12)\,α_\text{0}$ and $α_E^\text{t} = 34 (12)\,α_\text{0}$, respectively, where $α_\text{0}$ is the atomic unit for the electric polarizability. We also p…
We demonstrate that the differential ac-Stark shift of a ground-state hyperfine transition in an optical trap can be eliminated by using properly polarized trapping light. We use the vector polarizability of an alkali-metal atom to produce a polarization-dependent ac-Stark shift that resembles a Zeeman shift. We study a transition from the |2S1/2,F=2,mF=-2> to the |2S1/2,F=1,mF=-1> state of 7Li to observe 0.59+-0.02 Hz linewidth with interrogation time of 2 s and 0.82+-0.06 s coherence time of a superposition state. Implications of the narrow linewidth and the long coherence time for precision…
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 present in this paper a theory of dynamic polarizability for an atomic state due to an external field of non-paraxial 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$^+$ to determine the magic wavelengths for $5s_{{1}/{2}}\rightarrow 4d_{{3}/{2}, {5}/{2}}$ transitions. Results show the variation of magic wavelengths with the choice of orbital and spin angular momenta of the incident LG beam. Also, the tunabili…
Motivated by recent interest in their applications, we report a systematic study of Cs atomic properties calculated by a high-precision relativistic all-order method. Excitation energies, reduced matrix elements, transition rates, and lifetimes are determined for levels with principal quantum numbers $n \leq 12$ and orbital angular momentum quantum numbers $l \leq 3$. Recommended values and estimates of uncertainties are provided for a number of electric-dipole transitions and the electric dipole polarizabilities of the $ns$, $np$, and $nd$ states. We also report a calculation of the electric …
We report calculations designed to assess the ultimate precision of an atomic clock based on the 578 nm $6 ^1S_0 --> 6 ^3P^o_0$ transition in Yb atoms confined in an optical lattice trap. We find that this transition has a natural linewidth less than 10 mHz in the odd Yb isotopes, caused by hyperfine coupling. The shift in this transition due to the trapping light acting through the lowest order AC polarizability is found to become zero at the magic trap wavelength of about 752 nm. The effects of Rayleigh scattering, higher-order polarizabilities, vector polarizability, and hyperfine induced e…
Improvements in both theory and frequency metrology of few-electron systems such as hydrogen and helium have enabled increasingly sensitive tests of quantum electrodynamics (QED), as well as ever more accurate determinations of fundamental constants and the size of the nucleus. At the same time advances in cooling and trapping of neutral atoms have revolutionized the development of increasingly accurate atomic clocks. Here, we combine these fields to reach the highest precision on an optical tranistion in the helium atom to date by employing a Bose-Einstein condensate confined in a magic wavel…
The frequency dependent polarizabilities of the francium atom are calculated from the available data of energy levels and transition rates. Magic wavelengths for the state insensitive optical dipole trapping are identified from the calculated light shifts of the $7s~^2S_{1/2}$, $7p~^2P_{1/2, 3/2}$ and $8s~^{2}S_{1/2}$ levels of the $7s~^{2}S_{1/2}-7p~^{2}P_{1/2,3/2}$ and $7s~^{2}S_{1/2}-8s~^{2}S_{1/2}$ transitions, respectively. Wavelengths in the ultraviolet, visible and near infrared region is identified that are suitable for cooling and trapping. Magic wavelengths between 600-700~nm and 700…
We demonstrate magic wavelengths, at which external electric field produces null differential Stark shifts, for the $6s ~ {^2}S_{1/2}-6p ~ {^2}P_{1/2,3/2}$ transitions in the Cs atom due to circularly polarized light. In addition, we also obtain magic wavelengths using linearly polarized light, in order to verify the previously reported values, and make a comparative study with the values obtained for circularly polarized light. A number of these wavelengths are found to be in the optical region and could be of immense interest to experimentalists for carrying out high precision measurements. …
The present work determines the precise values of magic wavelengths corresponding to the clock transitions 5$^2S$-4$^2D$ of Y$^{2+}$ ion both at the levels of fine- and hyperfine-structures due to the external light beams having linear as well as circular polarization. To calculate the dynamic polarizabilities of the associated states of the transitions, we employ the sum-over-states technique, where the dominating and correlation sensitive part of the sum is evaluated using a highly correlated relativistic coupled-cluster theory. The estimated magic wavelengths of the light beams have substan…
Using first-principles calculations, we identify magic wavelengths for the 2s-2p and 2s-3p transitions in lithium. The ns and np atomic levels have the same ac Stark shifts at the corresponding magic wavelength, which facilitates state-insensitive optical cooling and trapping. Tune-out wavelengths for which the ground-state frequency-dependent polarizability vanishes are also calculated. Differences of these wavelengths between 6Li and 7Li are reported. Our approach uses high-precision, relativistic all-order methods in which all single, double, and partial triple excitations of the Dirac-Fock…
Magic-wavelength matter-wave interferometry with optical clock states
Nanyang Technological University（南洋理工大学）；CNRS（法国国家科学研究中心）；Université Côte d’Azur（蔚蓝海岸大学）；Sorbonne Université（索邦大学）；National University of Singapore（新加坡国立大学）；Center for Quantum Technologies（量子技术中心）
We present an experimental study of the lattice induced light shifts on the 1S_0-3P_0 optical clock transition (v_clock~518 THz) in neutral ytterbium. The ``magic'' frequency, v_magic, for the 174Yb isotope was determined to be 394 799 475(35)MHz, which leads to a first order light shift uncertainty of 0.38 Hz on the 518 THz clock transition. Also investigated were the hyperpolarizability shifts due to the nearby 6s6p 3P_0 - 6s8p 3P_0, 6s8p 3P_2, and 6s5f 3F_2 two-photon resonances at 759.708 nm, 754.23 nm, and 764.95 nm respectively. By tuning the lattice frequency over the two-photon resonan…
We present precise values of the dipole polarizabilities ($α$) of the ground $\rm [4f^{14}6s] ~ ^2S_{1/2}$ and metastable $\rm [4f^{14} 5d] ~ ^2D_{3/2}$ states of Yb$^+$, that are %vital {\bf important} in reducing systematics in the clock frequency of the $\rm[4f^{14}6s] ~ ^2S_{1/2} \rightarrow [4f^{14}5d] ~ ^2D_{3/2}$ transition. The static values of $α$ for the ground and $\rm [4f^{14} 5d] ~ ^2D_{3/2}$ states are estimated to be $9.8(1) \times 10^{-40} \,\,\rm Jm^2V^{-2}$ and $17.6(5) \times 10^{-40}\,\, \rm Jm^2V^{-2}$, respectively, while the tensor contribution to the $\rm [4f^{14} 5d] ~…
With direct polarizability measurements we demonstrated extremely low sensitivity of the inner-shell clock transition at $1.14\,μ$m in Tm atoms to external dc electric fields and black-body radiation (BBR). We measured differential polarizabilities of clock levels in Tm at wavelengths of 810--860\,nm and at 1064\,nm and inferred the static scalar differential polarizability of the inner-shell clock transition of $-0.047(18)$ atomic units corresponding to only $2\times10^{-18}$ fractional frequency shift from BBR at the room temperature. This is a few orders of magnitude smaller compared to the…
We carry out a systematic study of the static and dynamic polarizabilities of the potassium atom using a first-principles high-precision relativistic all-order method in which all single, double, and partial triple excitations of the Dirac-Fock wave functions are included to all orders of perturbation theory. Recommended values are provided for a large number of electric-dipole matrix elements. Static polarizabilities of the 4s, 4p_j, 5s, 5p_j, and 3d_j states are compared with other theory and experiment where available. We use the results of the polarizability calculations to identify magic …
We report on the observation and coherent excitation of atoms on the narrow inner-shell orbital transition, connecting the erbium ground state $[\mathrm{Xe}] 4f^{12} (^3\text{H}_6)6s^{2}$ to the excited state $[\mathrm{Xe}] 4f^{11}(^4\text{I}_{15/2})^05d (^5\text{D}_{3/2}) 6s^{2} (15/2,3/2)^0_7$. This transition corresponds to a wavelength of 1299 nm and is optically closed. We perform high-resolution spectroscopy to extract the $g_J$-factor of the $1299$-nm state and to determine the frequency shift for four bosonic isotopes. We further demonstrate coherent control of the atomic state and ext…
We present a scheme for entangling the atoms of an optical lattice to reduce the quantum projection noise of a clock measurement. The divalent clock atoms are held in a lattice at a ``magic'' wavelength that does not perturb the clock frequency -- to maintain clock accuracy -- while an open-shell J=1/2 ``head'' atom is coherently transported between lattice sites via the lattice polarization. This polarization-dependent ``Archimedes' screw'' transport at magic wavelength takes advantage of the vanishing vector polarizability of the scalar, J=0, clock states of bosonic isotopes of divalent atom…
We carry out relativistic many-body calculations of the static and dynamic dipole polarizabilities of the ground $6s^2 ^1S_0$ and the first excited $6s6p ^3P^o_0$ states of Yb. With these polarizabilities, we compute several properties of Yb relevant to optical lattice clocks operating on the $6s^2 ^1S_0 - 6s6p ^3P^o_0$ transition. We determine (i) the first four {\em magic} wavelengths of the laser field for which the frequency of the clock transition is insensitive to the laser intensity. While the first magic wavelength is known, we predict the second, the third and the forth magic waveleng…
