Electronic Interaction Anisotropy between Atoms in Arbitrary Angular Momentum States

Dipolar physics: a review of experiments with magnetic quantum gases

Since the achievement of quantum degeneracy in gases of chromium atoms in 2004, the experimental investigation of ultracold gases made of highly magnetic atoms has blossomed. The field has yielded the observation of many unprecedented phenomena, in particular those in which long-range and anisotropic dipole-dipole interactions (DDIs) play a crucial role. In this review, we aim to present the aspects of the magnetic quantum-gas platform that make it unique for exploring ultracold and quantum physics as well as to give a thorough overview of experimental achievements. Highly magnetic atoms distinguish themselves by the fact that their electronic ground-state configuration possesses a large electronic total angular momentum. This results in a large magnetic moment and a rich electronic transition spectrum. Such transitions are useful for cooling, trapping, and manipulating these atoms. The complex atomic structure and large dipolar moments of these atoms also lead to a dense spectrum of resonances in their two-body scattering behaviour. These resonances can be used to control the interatomic interactions and, in particular, the relative importance of contact over dipolar interactions. These features provide exquisite control knobs for exploring the few- and many-body physics of dipolar quantum gases. The study of dipolar effects in magnetic quantum gases has covered various few-body phenomena that are based on elastic and inelastic anisotropic scattering. Various many-body effects have also been demonstrated. These affect both the shape, stability, dynamics, and excitations of fully polarised repulsive Bose or Fermi gases. Beyond the mean-field instability, strong dipolar interactions competing with slightly weaker contact interactions between magnetic bosons yield new quantum-stabilised states, among which are self-bound droplets, droplet assemblies, and supersolids. Dipolar interactions also deeply affect the physics of atomic gases with an internal degree of freedom as these interactions intrinsically couple spin and atomic motion. Finally, long-range dipolar interactions can stabilise strongly correlated excited states of 1D gases and also impact the physics of lattice-confined systems, both at the spin-polarised level (Hubbard models with off-site interactions) and at the spinful level (XYZ models). In the present manuscript, we aim to provide an extensive overview of the various related experimental achievements up to the present.

Quantum Spin State Selectivity and Magnetic Tuning of Ultracold Chemical Reactions of Triplet Alkali-Metal Dimers with Alkali-Metal Atoms

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 NaLi(a^{3}Σ^{+}) producing singlet Na_{2}(^{1}Σ_{g}^{+}) and a Li atom. We find that the reaction rates are sensitive to the initial hyperfine states of the reactants. The chemical reaction of fully spin-polarized, high-spin states of rotationless NaLi(a^{3}Σ^{+},v=0,N=0) molecules with fully spin-polarized Na is suppressed by a factor of 10-100 compared to that of unpolarized reactants. We interpret these findings within the adiabatic state model, which treats the reaction as a sequence of nonadiabatic transitions between the initial nonreactive high-spin state and the final low-spin states of the reaction complex. In addition, we show that magnetic Feshbach resonances can similarly change reaction rate coefficients by several orders of magnitude. Some of these resonances are due to resonant trimer bound states dissociating to the N=2 rotational state of NaLi(a^{3}Σ^{+},v=0) and would thus exist in systems without hyperfine interactions.

On the formation of van der Waals complexes through three-body recombination

In this work, we show that van der Waals molecules X-RG (where RG is the rare gas atom) may be created through direct three-body recombination collisions, i.e., X + RG + RG → X-RG + RG. In particular, the three-body recombination rate at temperatures relevant for buffer gas cell experiments is calculated via a classical trajectory method in hyperspherical coordinates [Pérez-Ríos et al., J. Chem. Phys. 140, 044307 (2014)]. As a result, it is found that the formation of van der Waals molecules in buffer gas cells (1 K ≲ T ≲ 10 K) is dominated by the long-range tail (distances larger than the LeRoy radius) of the X-RG interaction. For higher temperatures, the short-range region of the potential becomes more significant. Moreover, we notice that the rate of formation of van der Walls molecules is of the same order of the magnitude independent of the chemical properties of X. As a consequence, almost any X-RG molecule may be created and observed in a buffer gas cell under proper conditions.

Mobility of the Singly-Charged Lanthanide and Actinide Cations: Trends and Perspectives

The current status of gaseous transport studies of the singly-charged lanthanide and actinide ions is reviewed in light of potential applications to superheavy ions. The measurements and calculations for the mobility of lanthanide ions in He and Ar agree well, and they are remarkably sensitive to the electronic configuration of the ion, namely, whether the outer electronic shells are 6s, 5d6s or 6s². The previous theoretical work is extended here to ions of the actinide family with zero electron orbital momentum: Ac⁺ (7s², ¹S), Am⁺ (5f⁷7s ⁹S°), Cm⁺ (5f⁷7s^{2 8}S°), No⁺ (5f¹⁴7s ²S), and Lr⁺ (5f¹⁴7s^{2 1}S). The calculations reveal large systematic differences in the mobilities of the 7s and 7s² groups of ions and other similarities with their lanthanide analogs. The correlation of ion-neutral interaction potentials and mobility variations with spatial parameters of the electron distributions in the bare ions is explored through the ionic radii concept. While the qualitative trends found for interaction potentials and mobilities render them appealing for superheavy ion research, lack of experimental data and limitations of the scalar relativistic ab initio approaches in use make further efforts necessary to bring the transport measurements into the inventory of techniques operating in "one atom at a time" mode.

Cold Anisotropically Interacting van der Waals Molecule: TiHe

We have used laser ablation and helium buffer-gas cooling to produce titanium-helium van der Waals molecules at cryogenic temperatures. The molecules were detected through laser-induced fluorescence spectroscopy. Ground-state Ti(a^{3}F_{2})-He binding energies were determined for the ground and first rotationally excited states from studying equilibrium thermodynamic properties, and found to agree well with theoretical calculations based on newly calculated ab initio Ti-He interaction potentials, opening up novel possibilities for studying the formation, dynamics, and nonuniversal chemistry of van der Waals clusters at low temperatures.

Couplings and recouplings of four angular momenta: Alternative 9j symbols and spin addition diagrams

The Wigner 9j symbols of the first kind-also known as Fano X-coefficients-serve to connect different addition schemes of four angular momenta, widely known examples being the LS and the jj couplings in atomic, molecular, and nuclear spectroscopies. Here, we also consider alternative sequences of binary couplings of four angular momenta, which are dealt through the 9j symbols of the second kind, and are explicitly given by the pentagonal (or Biedenharn-Elliott) identity. These coefficients are essential ingredients in the quantum-mechanical treatments of rotational and polarization phenomena in reaction dynamics and photoinduced processes. We also emphasize the combinatorial structure underlying the extended construction of a previously introduced truncated icosahedral "abacus", and provide extensions useful for algebraical manipulations, semiclassical interpretations, and computational applications, including all the 120 addition schemes.

Molecular collisions and reactive scattering in external fields: Are field-induced couplings important at short range?

We use accurate quantum scattering calculations to elucidate the role of short-range molecule-field interactions in atom-molecule inelastic collisions and abstraction chemical reactions at low temperatures. We consider two examples: elastic and inelastic scattering of NH(Σ3) molecules with Mg(S1) atoms in a magnetic field; reactive scattering LiF + H → Li + HF in an electric field. Our calculations suggest that, for non-reactive collision systems and abstraction chemical reactions, the molecule-field interactions cannot generally be neglected at short range because the atom-molecule potential passes through zero at short range. An important exception occurs for Zeeman transitions in atom-molecule collisions at magnetic fields ≲1000 G, for which the molecule-field couplings need only be included at large ρ outside the range of the atom-molecule interaction. Our results highlight the importance of an accurate description of ρ-dependent molecule-field interactions in quantum scattering calculations on molecular collisions and chemical reactions at low temperatures.

Controlling interactions between highly magnetic atoms with Feshbach resonances

This paper reviews current experimental and theoretical progress in the study of dipolar quantum gases of ground and meta-stable atoms with a large magnetic moment. We emphasize the anisotropic nature of Feshbach resonances due to coupling to fast-rotating resonant molecular states in ultracold s-wave collisions between magnetic atoms in external magnetic fields. The dramatic differences in the distribution of resonances of magnetic (7)S3 chromium and magnetic lanthanide atoms with a submerged 4f shell and non-zero electron angular momentum is analyzed. We focus on dysprosium and erbium as important experimental advances have been recently made to cool and create quantum-degenerate gases for these atoms. Finally, we describe progress in locating resonances in collisions of meta-stable magnetic atoms in electronic P-states with ground-state atoms, where an interplay between collisional anisotropies and spin-orbit coupling exists.

Mobility of singly-charged lanthanide cations in rare gases: theoretical assessment of the state specificity

High quality, ab initio calculations are reported for the potential energy curves governing the interactions of four singly-charged lanthanide ions (Yb(+), Eu(+), Lu(+), and Gd(+)) with the rare gases (RG = He-Xe). Scalar-relativistic coupled cluster calculations are used for the first three S-state ions, but for Gd(+)((10)D°) it is necessary to take the interaction anisotropy into account with the help of the multi-reference technique. The potential energy curves are used to determine the ion mobility and other transport properties describing the motion of the ions through the dilute RG, both as functions of the temperature, T, in the low-field limit, and at fixed T as functions of the ratio of the electrostatic field strength to the gas number density, E/N. The calculated mobilities are in good agreement with the very limited experimental data that have become available recently. The calculations show a pronounced dependence of the transport properties on the electronic configuration of the ion, as well as a significant effect of the spin-orbit coupling on the transport properties of the Gd(+) ion, and predict that state-specific mobilities could be detectable in Gd(+)-RG experiments.

Control of resonant interaction between electronic ground and excited states

We observe magnetic Feshbach resonances in a collision between the ground and metastable states of two-electron atoms of ytterbium (Yb). We measure the on-site interaction of doubly occupied sites of an atomic Mott-insulator state in a three-dimensional optical lattice as a collisional frequency shift in a high-resolution laser spectroscopy. The observed spectra are well fitted by a simple theoretical formula, in which two particles with an s-wave contact interaction are confined in a harmonic trap. This analysis reveals a wide variation of the interaction with a resonance behavior around a magnetic field of about 1.1 G for the energetically lowest magnetic sublevel of 170Yb, as well as around 360 mG for the energetically highest magnetic sublevel of 174Yb. The observed Feshbach resonance can only be induced by an anisotropic interatomic interaction. This scheme will open the door to a variety of studies using two-electron atoms with tunable interaction.

Anisotropy-induced Feshbach resonances in a quantum dipolar gas of highly magnetic atoms

We explore the anisotropic nature of Feshbach resonances in the collision between ultracold highly magnetic submerged-shell dysprosium atoms in their energetically lowest magnetic sublevel, which can only occur due to couplings to rotating bound states. This is in contrast to well-studied alkali-metal atom collisions, where broadest (strongest) Feshbach resonances are hyperfine induced and due to rotationless bound states. Our first-principle coupled-channel calculation of the collisions between these spin-polarized bosonic dysprosium atoms reveals a strong interplay between the anisotropies in the dispersion and magnetic dipole-dipole interaction. The former anisotropy is absent in alkali-metal and chromium collisions. We show that both types of anisotropy significantly affect the Feshbach spectrum as a function of an external magnetic field. Effects of the electrostatic quadrupole-quadrupole interaction are small. Over a 20 mT magnetic field range, we predict about 10 Feshbach resonances and show that the resonance locations depend on the dysprosium isotope.

Bose-Einstein condensation of erbium

We report on the achievement of Bose-Einstein condensation of erbium atoms and on the observation of magnetic Feshbach resonances at low magnetic fields. By means of evaporative cooling in an optical dipole trap, we produce pure condensates of 168Er, containing up to 7×10(4) atoms. Feshbach spectroscopy reveals an extraordinary rich loss spectrum with six loss resonances already in a narrow magnetic-field range up to 3 G. Finally, we demonstrate the application of a low-field Feshbach resonance to produce a tunable dipolar Bose-Einstein condensate and we observe its characteristic d-wave collapse.

Anisotropy of the static dipole polarizability induced by the spin–orbit interaction: the S-state atoms N–Bi, Cr, Mo and Re

A systematic ab initio study is performed on the ground state static dipole polarizabilities of the 2 S +1 S, S >1/2, atoms from N to Bi, Cr, Mo, Mn, Tc and Re. The benchmark scalar-relativistic values of the scalar polarizability components are obtained using the coupled cluster method. The spin–orbit configuration interaction calculations are carried out for the anisotropic (tensor) polarizability components of these atoms (except Mn and Tc) that arise from the second-order spin–orbit interaction. The tensor polarizabilities are calculated for the first time and found to increase from 10 −5 (N) to 3.8 atomic units (Bi) approximately as the fourth power of the nuclear charge. The simple correlations and implication to magnetic trapping of cold atoms are discussed.

Inelastic collisions in optically trapped ultracold metastable ytterbium

We report measurement of inelastic loss in dense and cold metastable ytterbium (Yb[3P2]). Use of an optical far-off-resonance trap enables us to trap atoms in all magnetic sublevels, removing m-changing collisional trap loss from the system. Trapped samples of Yb[3P2] are produced at a density of 2 x 10(13) cm(-3) and temperature of 2 microK. We observe rapid two-body trap loss of Yb[3P2] and measure the inelastic collision rate constant 1.0(3) x 10(-11) cm3 s(-1). The existence of the fine-structure changing collisions between atoms in the 3P2 state is strongly suggested.

Spin-exchange collisions of submerged shell atoms below 1 Kelvin

Angular momentum changing collisions can be suppressed in atoms whose valence electrons are submerged beneath filled shells of higher principle quantum number. To determine whether spin-exchange collisions are suppressed in these "submerged shell" atoms, we measured collisional rates for six hyperfine states of Mn at T < 1 K. Although the 3d valence electrons in Mn are submerged beneath a filled 4s orbital, we find spin-exchange rate coefficients similar to Na and H (both nonsubmerged shell atoms).

Electronic anisotropy between open shell atoms in first and second order perturbation theory

The interaction between two atoms in states with nonzero electronic orbital angular momenta is anisotropic and can be represented by a spherical tensor expansion. The authors derive expressions for the first order (electrostatic) and second order (dispersion and induction) anisotropic interaction coefficients in terms of the multipole moments and dynamic polarizabilities of the atoms and show that a complete description of the second order interaction requires odd rank or "out-of-phase" polarizabilities. The authors relate the tensorial expansion coefficients to the adiabatic Born-Oppenheimer potentials of the molecule and show that there are linear, and in some cases nonlinear, constraints on the van der Waals coefficients of these potentials.

van der Waals interactions and dipole polarizabilities of lanthanides: Tm(F2)–He and Yb(S1)–He potentials

Anisotropic dipole polarizabilities of Tm(2F), Tm+2(2F), and Yb(1S) are calculated using the finite-field multireference averaged quadratic coupled cluster (MR-AQCC) (Tm and Tm+2) and RCCSD(T) (Yb) methods with small-core relativistic pseudopotentials ECP28MWB combined with the augmented ANO basis sets. The lanthanide atoms are strongly polarizable with the scalar part originating from the 6s electrons and the tensorial part from the open 4f shells. The adiabatic interaction potentials 2Sigma+, 2Pi, 2Delta, and 2Phi of Tm(2F)-He and Tm+2(2F)-He were examined by the multireference approaches, multireference configuration interaction and MR-AQCC, using the basis sets designed in the polarizability calculations. A closed-shell lanthanide system Yb(1S)-He was included for comparison. The Tm-He 2Sigma+, 2Pi, 2Delta, and 2Phi interaction potentials are very shallow and nearly degenerate (within 0.01 cm(-1)), with the well depths in the range of 2.35-2.36 cm(-1) at R=6.17 A. The basis-set saturated well depths are expected to be larger by ca. 25%, as estimated using the bond-function augmented basis set. The interactions of lanthanide atoms with He are one order of magnitude less anisotropic than those involving first-row transition metal atoms. The suppression of anisotropy is chiefly attributed to the screening effected by the 6s shell. When these electrons are removed as in the di-cation complex Tm+2(2F)-He, the potentials deepen to a thousand wave number range and their anisotropy is enhanced 500-fold.

Interactions of 2P Atoms with Closed-Shell Diatomic Molecules: Alternative Diabatic Representations for the Electronic Anisotropy

The matrices of electrostatic and spin-orbit Hamiltonians for the system of a 2P atom interacting with a closed shell diatomic molecule in uncoupled, coupled, and complex-valued representations for electronic diabatic basis functions are rederived, and the unitary transformations connecting them are given explicitly. The links to previous derivations are established and existing inconsistencies are identified and eliminated. It is proven that the block-diagonalization of a 6 x 6 matrix of the electronic Hamiltonian is a result of using the basis functions with well-defined properties with respect to time reversal. Consideration of time-reversal symmetry also enforces phase consistency relevant for applications to multisurface reactive scattering and photodetachment spectroscopy calculations, as well as for perspective studies of inelastic effects in cold and ultracold environments. These and further developments are briefly sketched.

Electronic interaction anisotropy between open-shell lanthanide atoms and helium from cold collision experiment

Based on measurements of the Zeeman relaxation in a cold gas of (3)He [C. I. Hancox, S. C. Doret, M. I. Hummon, L. Luo, and J. M. Doyle, Nature (London) 431, 281 (2004)], we show that the electronic interaction anisotropy between rare-earth atoms with nonzero electronic orbital angular momenta and helium is extremely small. The interaction of the rare-earth atoms with He gives rise to several adiabatic potentials with different electronic symmetries. It is demonstrated that the energy splitting between these potentials does not exceed 0.09 cm(-1) at interatomic distances larger than the turning point for collisions at 0.8 K, including the region of the van der Waals interaction minima.

Anisotropic dipole polarizability of transition metal atoms: Sc(D2), Ti(F3,P3), V(F4,P4,D6), Ni(F3) and ions: Sc2+(D2), Ti2+(F3,P3)

Dipole polarizability tensor components and quadrupole moments of transition-metal atoms Sc, Ti, V, Ni, and Cu and ions Sc2+ and Ti2+ are computed using finite field complete active space self-consistent field and multireference configuration interaction ab initio methods. Perpendicular components of the dipole polarizability tensor are calculated from equations involving only parallel components of the polarizability tensor and its average value. Mean polarizability and polarizability anisotropy decrease in the Sc-Ni series. Relativistic effects are accounted for with the Douglas-Kroll Hamiltonian. The consequences of the anisotropic properties of these atoms to their interactions with spherically symmetric rare gases are also discussed.

Suppression of angular momentum transfer in cold collisions of transition metal atoms in ground States with nonzero orbital angular momentum

The Zeeman relaxation rate in cold collisions of Ti(3d(2)4s(2) 3F2) with He is measured. We find that collisional transfer of angular momentum is dramatically suppressed due to the presence of the filled 4s(2) shell. The degree of electronic interaction anisotropy, which is responsible for Zeeman relaxation, is estimated to be about 200 times smaller in the Ti-He complex than in He complexes with typical non-S-state atoms.

Suppression of angular forces in collisions of non-S-state transition metal atoms

Angular momentum transfer is expected to occur rapidly in collisions of atoms in states of nonzero angular momenta due to the large torque of angular forces. We show that despite the presence of internal angular momenta transition metal atoms interact in collisions with helium effectively as spherical atoms and angular momentum transfer is slow. Thus, magnetic trapping and sympathetic cooling of transition metal atoms to ultracold temperatures should be readily achievable. Our results open up new avenues of research with a broad class of ultracold atoms.

Magnetic trapping of rare-earth atoms at millikelvin temperatures

The ability to create quantum degenerate gases has led to the realization of Bose-Einstein condensation of molecules, atom-atom entanglement and the accurate measurement of the Casimir force in atom-surface interactions. With a few exceptions, the achievement of quantum degeneracy relies on evaporative cooling of magnetically trapped atoms to ultracold temperatures. Magnetic traps confine atoms whose electronic magnetic moments are aligned anti-parallel to the magnetic field. This alignment must be preserved during the collisional thermalization of the atomic cloud. Quantum degeneracy has been reached in spherically symmetric, S-state atoms (atoms with zero internal orbital angular momentum). However, collisional relaxation of the atomic magnetic moments of non-S-state atoms (non-spherical atoms with non-zero internal orbital angular momentum) is thought to proceed rapidly. Here we demonstrate magnetic trapping of non-S-state rare-earth atoms, observing a suppression of the interaction anisotropy in collisions. The atoms behave effectively like S-state atoms because their unpaired electrons are shielded by two outer filled electronic shells that are spherically symmetric. Our results are promising for the creation of quantum degenerate gases with non-S-state atoms, and may facilitate the search for time variation of fundamental constants and the development of a quantum computer with highly magnetic atoms.