We investigate Cs+Cs scattering in excited Zeeman and hyperfine states. We calculate the real and imaginary parts of the s-wave scattering length; the imaginary part directly provides the rate coefficient for 2-body inelastic loss, while the real part allows us to identify regions of magnetic field where 3-body recombination will be slow. We identify field regions where Cs in its $(f,m_f)=(3,+2)$ and $(3,+1)$ states may be stable enough to allow Bose-Einstein condensation, and additional regions for these and the $(3,0)$ and $(3,-3)$ states where high-density clouds should be long-lived.
Nonlinear magneto-optical Faraday rotation (NMOR) on the potassium D1 and D2 lines was used to study Zeeman relaxation rates in an antirelaxation paraffin-coated 3-cm diameter potassium vapor cell. Intrinsic Zeeman relaxation rates of $γ^{NMOR}/2π=2.0(6) {\rm Hz}$ were observed. The relatively small hyperfine intervals in potassium lead to significant differences in NMOR in potassium compared to rubidium and cesium. Using laser optical pumping, widths and frequency shifts were also determined for transitions between ground-state hyperfine sublevels of $^{39}$K atoms contained in the same paraf…
We demonstrate a technique for directly measuring the quadratic Zeeman shift using stimulated Raman transitions.The quadratic Zeeman shift has been measured yielding [delta][nju] = 1296.8 +/-3.3 Hz/G^{2} for magnetically insensitive sublevels (5S1/2, F = 2,mF = 0 -> 5S1/2, F = 3,mF = 0) of ^{85}Rb by compensating the magnetic eld and cancelling the ac Stark shift. We also measured the cancellation ratio of the differential ac Stark shift due to the imbalanced Raman beams by using two pairs of Raman beams ([sigma]^{+}, [sigma]^{+}) and it is 1:3.67 when the one-photon detuning is 1.5 GHz in the…
The role of the interaction with the nearest electronic state $^3Σ^+_{0^-}$ on the hyperfine structure and magnetic properties of the $a(1)[^3Σ^+_1]$ state of PbO is assessed. The accounting for this contribution leads to difference between $g$-factors of the $J=1$ $Ω$-doublet levels, $ Δg = 37\times10^{-4}$, that is in a good agreement with the experimental datum $ Δg = 30(8)\times10^{-4}$. The contribution of this interaction rapidly grows with $J$. For $J=30$ the difference of $g$-factors of $Ω$-doublet states reaches 100%; for hyperfine constants it is 18%. These differences also depend on…
The fully relativistic theory of the Zeeman splitting of the $1s$ hyperfine structure levels in hydrogenlike ions is considered for the magnetic field magnitude in the range from 1 to 10 T. The second-order corrections to the Breit -- Rabi formula are calculated and discussed. The results can be used for a precise determination of nuclear magnetic moments from $g$ factor experiments.
Sturmian basis set for the Dirac equation with finite nuclear size: Application to polarizability, Zeeman and hyperfine splitting, and vacuum polarization
The Zeeman hyperfine state dependent force in a Stern-Gerlach (SG) experiment has been exploited to separate and detect atoms having different Zeeman hyperfine states in a cold atom cloud. Utilizing this SG technique, we have made the quantitative estimate of atoms in different Zeeman hyperfine states in an atom cloud, which has been helpful in optimizing the optical pumping of atoms for efficient preparation of atomic state. Employing an optimized optical pumping, nearly $92 \% $ of cold $^{87}Rb$ atoms from a grey magneto-optical trap (G-MOT) on atom-chip have been optically pumped to the tr…
High-resolution Zeeman spectroscopy of electronic-nuclear hyperfine levels of ${^5\mathrm{I}_8}\rightarrow{^5\mathrm{I}_7}$ transitions in Ho$^{3+}$:Y$_2$SiO$_5$ is reported. Crystal-field parameters determined for the two $C_1$ symmetry sites in Er$^{3+}$:Y$_2$SiO$_5$ are successfully used to model the Zeeman-hyperfine data, including the prediction of avoided crossings between hyperfine levels under the influence of an external magnetic field. The two six- and seven-coordinate substitutional sites may be distinguished by comparing the spectra with crystal-field calculations.
The fully relativistic theory of the Zeeman splitting of the $(1s)^2 2s$ hyperfine-structure levels in lithiumlike ions with $Z=6 - 32$ is considered for the magnetic field magnitude in the range from 1 to 10 T. The second-order corrections to the Breit -- Rabi formula are calculated and discussed including the one-electron contributions as well as the interelectronic-interaction effects of order 1/Z. The 1/Z corrections are evaluated within a rigorous QED approach. These corrections are combined with other interelectronic-interaction, QED, nuclear recoil, and nuclear size corrections to obtai…
We present the results of an experimental and theoretical study of the electronically excited $\tripletex$ state of $^{87}$Rb$_2$ molecules. The vibrational energies are measured for deeply bound states from the bottom up to $v'=15$ using laser spectroscopy of ultracold Rb$_2$ Feshbach molecules. The spectrum of each vibrational state is dominated by a 47\,GHz splitting into a $\cog$ and $\clg$ component caused mainly by a strong second order spin-orbit interaction. Our spectroscopy fully resolves the rotational, hyperfine, and Zeeman structure of the spectrum. We are able to describe to first…
Ultracold molecules with both electron spin and an electric dipole moment offer new possibilities in quantum science. We use density-functional theory to calculate hyperfine coupling constants for a selection of molecules important in this area, including RbSr, LiYb, RbYb, CaF and SrF. We find substantial hyperfine coupling constants for the fermionic isotopes of the alkaline-earth and Yb atoms. We discuss the hyperfine level patterns and Zeeman splittings expected for these molecules. The results will be important both to experiments aimed at forming ultracold open-shell molecules and to thei…
Computational crystal-field models have provided consistent models of both electronic and Zeeman-hyperfine structure for several rare earth ions. These techniques have not yet been applied to the Zeeman-hyperfine structure of Eu$^{3+}$ because modeling the structure of the $J=0$ singlet levels in Eu$^{3+}$ requires inclusion of the commonly omitted lattice electric quadrupole and nuclear Zeeman interactions. Here, we include these terms in a computational model to fit the crystal field levels and the Zeeman-hyperfine structure of the $^7F_0$ and $^5D_0$ states in three Eu$^{3+}$ sites: the C$_…
We demonstrate coherent control of both the rotational and hyperfine state of ultracold, chemically stable $^{87}$Rb$^{133}$Cs molecules with external microwave fields. We create a sample of ~2000 molecules in the lowest hyperfine level of the rovibronic ground state N = 0. We measure the transition frequencies to 8 different hyperfine levels of the N = 1 state at two magnetic fields ~23 G apart. We determine accurate values of rotational and hyperfine coupling constants that agree well with previous calculations. We observe Rabi oscillations on each transition, allowing complete population tr…
We produce ultracold dense trapped samples of 87Rb133Cs molecules in their rovibrational ground state, with full nuclear hyperfine state control, by stimulated Raman adiabatic passage (STIRAP) with efficiencies of 90%. We observe the onset of hyperfine-changing collisions when the magnetic field is ramped so that the molecules are no longer in the hyperfine ground state. A strong quadratic shift of the transition frequencies as a function of applied electric field shows the strongly dipolar character of the RbCs ground-state molecule. Our results open up the prospect of realizing stable bosoni…
Anomalously large linear and quadratic Zeeman shifts are measured for weakly bound ultracold $^{88}$Sr$_2$ molecules near the intercombination-line asymptote. Nonadiabatic Coriolis coupling and the nature of long-range molecular potentials explain how this effect arises and scales roughly cubically with the size of the molecule. The linear shifts yield nonadiabatic mixing angles of the molecular states. The quadratic shifts are sensitive to nearby opposite $f$-parity states and exhibit fourth-order corrections, providing a stringent test of a state-of-the-art \textit{ab initio} model.
Evaluation of the systematic error induced by quadratic Zeeman effect using hyperfine ground state exchange method in a long-baseline dual-species atom interferometer
We study the combination of hyperfine and Zeeman structure in the spin-orbit coupled $A^1Σ_u^+-b^3Π_u$ complex of $^{87}\textrm{Rb}_2$. For this purpose, absorption spectroscopy at a magnetic field around $B=1000\:\textrm{G}$ is carried out. We drive optical dipole transitions from the lowest rotational state of an ultracold Feshbach molecule to various vibrational levels with $0^+$ symmetry of the $A-b$ complex. In contrast to previous measurements with rotationally excited alkali-dimers, we do not observe equal spacings of the hyperfine levels. In addition, the spectra vary substantially for…
We show that Zeeman excitations of ultracold Dy atoms trapped in an optical lattice can be used to engineer extended Hubbard models with tunable inter-site and particle number-non-conserving interactions. We show that the ratio of the hopping amplitude and inter-site interactions in these lattice models can be tuned in a wide range by transferring the atoms to different Zeeman states. We propose to use the resulting controllable models for the study of the effects of direct particle interactions and particle number-non-conserving terms on Anderson localization.
We report the full control over the internal states of ultracold $^{23}$Na$^{87}$Rb molecules, including vibrational, rotational and hyperfine degrees of freedom. Starting from a sample of weakly bound Feshbach molecules, we realize the creation of molecules in single hyperfine levels of both the rovibrational ground and excited states with a high-efficiency and high-resolution stimulated Raman adiabatic passage. Starting from the rovibrational and hyperfine ground state, we demonstrate rotational and hyperfine control with one- and two-photon microwave spectroscopy. This achievement of fully …
We investigate magnetically tunable Feshbach resonances in ultracold collisions between ground-state Yb and Cs atoms, using coupled-channel calculations based on an interaction potential recently determined from photoassociation spectroscopy. We predict resonance positions and widths for all stable isotopes of Yb, together with resonance decay parameters where appropriate. The resonance patterns are richer and more complicated for fermionic Yb than for spin-zero isotopes, because there are additional level splittings and couplings due to scalar and tensorial Yb hyperfine interactions. We exami…
Ultracold molecular collisions in magnetic fields: Efficient incorporation of hyperfine structure in the total rotational angular momentum representation
We report on hyperfine-resolved spectroscopic measurements of the electric-dipole forbidden 5$p_{3/2} \to 8p_{1/2}$ transition in a sample of ultracold $^{87}$Rb atoms. The hyperfine selection rules enable the weak magnetic-dipole (M1) contribution to the transition strength to be distinguished from the much stronger electric-quadrupole (E2) contribution. An upper limit on the M1 transition strength is determined that is about 50 times smaller than an earlier experimental determination. We also calculate the expected value of the M1 matrix element and find that it is less than the upper limit …
We investigate the effect of hyperfine interactions on ultracold molecular collisions in magnetic fields, using ^{24}Mg(^1S) + ^{14}NH(^3Sigma^-) as a prototype system. We explore the energy and magnetic field dependence of the cross sections, comparing the results with previous calculations that neglected hyperfine interactions (Phys. Rev. Lett. 103, 183201 (2009)). The main effect of hyperfine interactions for spin relaxation cross sections is that the kinetic energy release of the dominant outgoing channels does not reduce to zero at low fields. This results in reduced centrifugal suppressi…
Compared to purely atomic collisions, ultracold collisions involving molecules have the potential to support a much larger number of Fano-Feshbach resonances due to the huge amount of ro-vibrational states available. In order to handle such ultracold atom-molecule collisions, we formulate a theory that incorporates the ro-vibrational Fano-Feshbach resonances in a statistical manner while treating the physics of the long-range scattering, which is sensitive to such things as hyperfine states, collision energy and any applied electromagnetic fields, exactly within multichannel quantum defect the…
We report the production of ultracold heteronuclear Cs$^*$Yb molecules through one-photon photoassociation applied to an ultracold atomic mixture of Cs and Yb confined in an optical dipole trap. We use trap-loss spectroscopy to detect molecular states below the Cs($^{2}P_{1/2}$) + Yb($^{1}S_{0}$) asymptote. For $^{133}$Cs$^{174}$Yb, we observe 13 rovibrational states with binding energies up to $\sim$500 GHz. For each rovibrational state we observe two resonances associated with the Cs hyperfine structure and show that the hyperfine splitting in the diatomic molecule decreases for more deeply …
We implement a 4-level semiclassical model of a single pulse interacting with the hyperfine structure in ultracold rubidium aimed at control of population dynamics and quantum state preparation. We discuss a method based on pulse chirping to achieve population inversion between hyperfine states of the 5S shell. The results may prove useful for quantum operations with ultracold atoms.
We show that by averaging over transitions to multiple hyperfine levels, quadrupole shifts and dominant Zeeman effects exactly cancel whenever the nuclear spin, I, is at least as large as the total electronic angular momentum, J. The average frequency thus defines a frequency reference which is inherently independent of external magnetic fields and electric field gradients. We use Lu+ to illustrate the method although the approach could be readily adapted to other atomic species. This approach practically eliminates the quadrupole and Zeeman shift considerations for many clock candidates.
While attaining external field control of bimolecular chemical reactions has long been a coveted goal of physics and chemistry, the role of hyperfine interactions and dc magnetic fields in achieving such control has remained elusive. We develop an extended coupled-channel statistical theory of barrierless atom-diatom chemical reactions, and apply it to elucidate the effects of magnetic fields and hyperfine interactions on the ultracold chemical reaction Li($^2\text{S}_{1/2}$) + CaH($^2Σ^+$) $\to$ LiH($^1Σ^+$) + Ca($^1\text{S}_{0}$) on a newly developed set of ab initio potential energy surface…
Positronium is an ideal system for the research of the quantum electrodynamics (QED) in bound state. The hyperfine splitting (HFS) of positronium, $Δ_{\mathrm{HFS}}$, gives a good test of the bound state calculations and probes new physics beyond the Standard Model. A new method of QED calculations has revealed the discrepancy by 15\,ppm (3.9$σ$) of $Δ_{\mathrm{HFS}}$ between the QED prediction and the experimental average. There would be possibility of new physics or common systematic uncertainties in the previous all experiments. We describe a new experiment to reduce possible systematic unc…
We demonstrate coherent microwave control of rotational and hyperfine states of trapped, ultracold, and chemically stable $^{23}$Na$^{40}$K molecules. Starting with all molecules in the absolute rovibrational and hyperfine ground state, we study rotational transitions in combined magnetic and electric fields and explain the rich hyperfine structure. Following the transfer of the entire molecular ensemble into a single hyperfine level of the first rotationally excited state, $J{=}1$, we observe collisional lifetimes of more than $3\, \rm s$, comparable to those in the rovibrational ground state…
We have studied hetero- and homonuclear excited state/ground state collisions by loading both $^{85}$Rb and $^{87}$Rb into a far off resonant trap (FORT). Because of the relatively weak confinement of the FORT, we expect the hyperfine structure of the different isotopes to play a crucial role in the collision rates. This dependence on hyperfine structure allows us to measure collisions associated with long range interatomic potentials of different structure: such as long and short ranged; or such as purely attractive, purely repulsive, or mixed attractive and repulsive. We observe significantl…
Simple and efficient "λ-method" and "λ/2-method" (λis the resonant wavelength of laser radiation) based on nanometric-thickness cell filled with rubidium are implemented to study the splitting of hyperfine transitions of 85Rb and 87Rb D_1 line in an external magnetic field in the range of B = 0.5 - 0.7 T. It is experimentally demonstrated from 20 (12) Zeeman transitions allowed at low B-field in 85Rb (87Rb) spectra in the case of σ+ polarized laser radiation, only 6 (4) remain at B > 0.5 T, caused by decoupling of the total electronic momentum J and the nuclear spin momentum I (hyperfine Pasch…
Starting from weakly bound Feshbach molecules, we demonstrate a two-photon pathway to the dipolar ground state of bi-alkali molecules that involves only singlet-to-singlet optical transitions. This pathway eliminates the search for a suitable intermediate state with sufficient singlet-triplet mixing and the exploration of its hyperfine structure, as is typical for pathways starting from triplet dominated Feshbach molecules. By selecting a Feshbach state with a stretched singlet hyperfine component and controlling the polarization of the excitation laser, we assure coupling to only a single hyp…
Collisions with chemically inert atoms or molecules change the hyperfine coupling of an alkali-metal atom through the hyperfine-shift interaction. This interaction is responsible for the pressure shifts of the microwave resonances of alkali-metal atoms in buffer gases, is an important spin interaction in alkali-metal--noble-gas van der Waals molecules, and is anticipated to enable the magnetoassociation of ultracold molecules such as RbSr. An improved estimate is presented for the long-range asymptote of this interaction for Na, K, Rb, and Cs. To test the results, the change in hyperfine coupl…
We demonstrate that it is possible to efficiently control ultracold chemical reactions of alkali-metal atoms colliding with open-shell alkali-metal dimers in their metastable triplet states by choosing the internal hyperfine and rovibrational states of the reactants as well as by inducing magnetic Feshbach resonances with an external magnetic field. We base these conclusions on coupled-channel statistical calculations that include the effects of hyperfine contact and magnetic-field-induced Zeeman interactions on ultracold chemical reactions of hyperfine-resolved ground-state Na and the triplet…
A semiclassical, four-level model of a nanosecond, chirped pulse interacting with all optically accessible hyperfine states in the ultracold Rb atom is analyzed aiming at population inversion within $5S_{1/2}$ electronic state. The nature of two-photon adiabatic passage performed by such a single pulse having a bandwidth smaller than the hyperfine splitting of $5S_{1/2}$ state is investigated in the framework of the dressed state picture. It is shown that two dressed states are involved in the adiabatic dynamics of population inversion. The excited state manifold appeared to play an important …
We report a detailed molecular spectroscopy study on the lowest excited electronic states of $^{23}\rm{Na}^{87}\rm{Rb}$ for producing ultracold $^{23}\rm{Na}^{87}\rm{Rb}$ molecules in the electronic, rovibrational and hyperfine ground state. Starting from weakly-bound Feshbach molecules, a series of vibrational levels of the $A^{1}Σ^{+}-b^{3}Π$ coupled excited states were investigated. After resolving, modeling and interpreting the hyperfine structure of several lines, we successfully identified a long-lived level resulting from the accidental hyperfine coupling between the $0^+$ and $0^-$ com…
Through a combination of spin-exchange collisions in a magnetic field and optical pumping, it is possible to cool a gas of atoms without requiring the loss of atoms from the gas. We investigated this technique, Collision Assisted Zeeman Cooling, using 85Rb and 87Rb. We experimentally confirmed that our measured two isotope CAZ cooling rate agreed with a cooling rate predicted though a simple analytic model. As part of the measured cooling rate, we quantitatively characterized the heating rates associated with our actual implementation of this cooling technique and found hyperfine-changing coll…
Spin-polarized metastable atoms of ultracold ytterbium are trapped at high density and their inelastic collisional properties are measured. We reveal that in collisions of Yb(3P2) with Yb(1S0) there is relatively weak inelastic loss, but with a significant spin-dependence consistent with Zeeman sublevel changes as being the dominant decay process. This is in strong contrast to our observations of Yb(3P2)-Yb(3P2) collisional loss, which are, at low field, much more rapid and have essentially no spin dependence. Our results give a guideline to use the 3P2 states in many possible applications.
With the development of microwave atomic clocks, the Zeeman shifts for the spectral lines of black-body radiation need to be investigated carefully. In this Letter, the frequency shifts of hyperfine splittings of atomic ground states due to the magnetic field of black-body radiation are reported. The relative frequency shifts of different alkali atoms and alkali-like ions, which could be candidates of microwave atomic clocks, were calculated. The results vary from $-0.977\times10^{-17}[T(K)/300]^{2}$ to $-1.947\times10^{-17}[T(K)/300]^{2}$ for different atoms considered. These results are cons…
We investigate the effect of far-off-resonant trapping light on ultracold bosonic RbCs molecules. We use kHz-precision microwave spectroscopy to measure the differential AC Stark shifts between the ground and first excited rotational levels of the molecule with hyperfine-state resolution. We demonstrate through both experiment and theory that coupling between neighboring hyperfine states manifests in rich structure with many avoided crossings. This coupling may be tuned by rotating the polarization of the linearly polarized trapping light. A combination of spectroscopic and parametric heating …
Alkali-metal atomic magnetometers suffer from heading errors in geomagnetic fields as the measured magnetic field depends on the orientation of the sensor with respect to the field. In addition to the nonlinear Zeeman splitting, the difference between Zeeman resonances in the two hyperfine ground states can also generate heading errors depending on initial spin polarization. We examine heading errors in an all-optical scalar magnetometer that uses free precession of polarized $^{87}\text{Rb}$ atoms by varying the direction and magnitude of the magnetic field at different spin polarization regi…
In this paper we investigate the feasibility of Zeeman slowing calcium monofluoride (CaF) molecules originating from a cryogenic buffer gas cell. We measure the $A^2Π_{1/2} (v=0, J=\frac{1}{2}) - X^2Σ_{1/2} (v=0, N=1)$ hyperfine spectrum of CaF in the Paschen-Back regime and find excellent agreement with theory. We then investigate the scattering rate of the molecules in a molecular Zeeman slower by illuminating them with light from a 10mW broad repumper and a 10mW multi-frequency slowing laser. By comparing our results to theory we can calculate the photon scattering rate at higher powers, le…
We present accurate ab initio and quantum scattering calculations on a prototypical hybrid ion-atom system Yb$^+$-Rb, recently suggested as a promising candidate for the experimental study of open quantum systems, quantum information processing, and quantum simulation. We identify the second-oder spin-orbit (SO) interaction as the dominant source of hyperfine relaxation and decoherence in cold Yb$^+$-Rb collisions. Our results are in good agreement with recent experimental observations [L. Ratschbacher et al., Phys. Rev. Lett. 110, 160402 (2013)] of hyperfine relaxation rates of trapped Yb$^+$…
We report on the development and application of a parametrized crystal-field model for both C$_{1}$ symmetry centers in trivalent erbium-doped Y$_{2}$SiO$_{5}$. High resolution Zeeman and temperature dependent absorption spectroscopy was performed to acquire the necessary experimental data. The obtained data, in addition to the ground ($^{4}$I$_{15/2}$Z$_{1}$) state and exited ($^{4}$I$_{13/2}$Y$_{1}$) state Zeeman and hyperfine structure, was simultaneously fitted in order to refine an existing crystal-field interpretation of the Er$^{3+}$:Y$_{2}$SiO$_{5}$ system. We demonstrate that it is po…
We prepare mixtures of ultracold $^{39}$K atoms in various hyperfine spin states and $^{23}$Na$^{39}$K molecules in an optical dipole trap at a fixed magnetic field and study inelastic two-body atom-molecule collisions. We observe atom-molecule losses that are hyperfine dependent with a two-body loss rate far below the universal limit. We analyze the two-body loss dynamics based on the derivation of general and easy applicable analytic solutions for the differential equations describing the loss of an arbitrary number $γ$ of particles in a single collisional event.
We report on the creation of an ultracold dipolar gas of fermionic $^{23}$Na$^{40}$K molecules in their absolute rovibrational and hyperfine ground state. Starting from weakly bound Feshbach molecules, we demonstrate hyperfine resolved two-photon transfer into the singlet ${\rm X}^1Σ^+ |v{=}0,J{=}0\rangle$ ground state, coherently bridging a binding energy difference of 0.65 eV via stimulated rapid adiabatic passage. The spin-polarized, nearly quantum degenerate molecular gas displays a lifetime longer than 2.5 s, highlighting NaK's stability against two-body chemical reactions. A homogeneous …
We characterize the Europium (Eu$^{3+}$) hyperfine interaction of the excited state ($^5$D$_0$) and determine its effective spin Hamiltonian parameters for the Zeeman and quadrupole tensors. An optical free induction decay method is used to measure all hyperfine splittings under weak external magnetic field (up to 10 mT) for various field orientations. On the basis of the determined Hamiltonian we discuss the possibility to predict optical transition probabilities between hyperfine levels for the $^7$F$_{0} \longleftrightarrow ^5$D$_{0}$ transition. The obtained results provide necessary infor…
