Feshbach resonances are the essential tool to control the interaction between atoms in ultracold quantum gases. They have found numerous experimental applications, opening up the way to important breakthroughs. This review broadly covers the phenomenon of Feshbach resonances in ultracold gases and their main applications. This includes the theoretical background and models for the description of Feshbach resonances, the experimental methods to find and characterize the resonances, a discussion of the main properties of resonances in various atomic species and mixed atomic species systems, and an overview of key experiments with atomic Bose-Einstein condensates, degenerate Fermi gases, and ultracold molecules.

Feshbach resonances in ultracold gases

A self-consistent system of additive covalent radii, R(AB)=r(A) + r(B), is set up for the entire periodic table, Groups 1-18, Z=1-118. The primary bond lengths, R, are taken from experimental or theoretical data corresponding to chosen group valencies. All r(E) values are obtained from the same fit. Both E-E, E-H, and E-CH 3 data are incorporated for most elements, E. Many E-E' data inside the same group are included. For the late main groups, the system is close to that of Pauling. For other elements it is close to the methyl-based one of Suresh and Koga [J. Phys. Chem. A 2001, 105, 5940] and its predecessors. For the diatomic alkalis MM' and halides XX', separate fits give a very high accuracy. These primary data are then absorbed with the rest. The most notable exclusion are the transition-metal halides and chalcogenides which are regarded as partial multiple bonds. Other anomalies include H 2 and F 2 . The standard deviation for the 410 included data points is 2.8 pm.

Molecular single-bond covalent radii for elements 1-118.

Feshbach Resonances in Ultracold Gases

A quantum gas of ultracold polar molecules, with long-range and anisotropic interactions, not only would enable explorations of a large class of many-body physics phenomena but also could be used for quantum information processing We report on the creation of an ultracold dense gas of potassium-rubidium (40K87Rb) polar molecules Using a single step of STIRAP (stimulated Raman adiabatic passage) with two-frequency laser irradiation, we coherently transfer extremely weakly bound KRb molecules to the rovibrational ground state of either the triplet or the singlet electronic ground molecular potential The polar molecular gas has a peak density of 1012 per cubic centimeter and an expansion-determined translational temperature of 350 nanokelvin The polar molecules have a permanent electric dipole moment, which we measure with Stark spectroscopy to be 0052(2) Debye (1 Debye = 3336 × 10–30 coulomb-meters) for the triplet rovibrational ground state and 0566(17) Debye for the singlet rovibrational ground state

A High Phase-Space-Density Gas of Polar Molecules

Have you ever felt that the very title, Progress in Optics, conjured an image in your mind? Don’t you see a row of handsomely printed books, bearing the editorial stamp of one of the most brilliant members of the optics community, and chronicling the field of optics since the invention of the laser? If so, you are certain to move the bookend to make room for Volume 16, the latest of this series. It contains seven chapters on widely diverging topics, written by well-known authorities in their fields. These are: 1) Laser Selective Photophysics and Photochemistry by V. S. Letokhov, 2) Recent Advances in Phase Profiles (sic) Generation by J. J. Clair and C. I. Abitbol, 3 ) Computer-Generated Holograms: Techniques and Applications by W.-H. Lee, 4) Speckle Interferometry by A. E. Ennos, 5 ) Deformation  Invariant, Space-Variant Optical  Pattern Recognition by D. Casasent and D. Psaltis, 6) Light Emission from High-Current Surface-Spark Discharges by R. E. Beverly, and 7) Semiclassical Radiation  Theory within a  QuantumMechanical Framework by I. R. Senitzkt. The  breadth of topic  matter  spanned  by  these  chapters makes it impossible, for this reviewer at least, to pass judgement on the comprehensiveness, relevance, and completeness of every chapter. With an editorial board as prominent as that of Progress in Optics, however, it seems hardly likely that such comments should be necessary. It should certainly be possible to take the authority of each author as credible. The only remaining judgment to be  made on these chapters is their readability. In short, what are they like to read? The first sentence of the first chapter greets the eye with an obvious typographical error: “The creation of coherent laser light source, that have tunable radiation, opened the . . . .” Two pages later we find: “When two types of atoms or molecules of different isotopic composition ( A and B ) have even one spectral line that does not overlap with others, it is pos-

Progress in optics

A comprehensive study of the electronic states at the $4s+5s$ asymptote in $\mathrm{KRb}$ is presented. Abundant spectroscopic data on the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ state were collected by Fourier-transform spectroscopy, which allows one to determine an accurate experimental potential energy curve up to $14.8\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$. The existing data set [C. Amiot et al., J. Chem. Phys. 112, 7068 (2000)] on the ground state $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ was extended by several additional levels lying close to the atomic asymptote. In a coupled channels fitting routine complete molecular potentials for both electronic states were fitted. Along with the line frequencies of the molecular transitions, recently published positions of Feshbach resonances in $^{40}\mathrm{K}$ and $^{87}\mathrm{Rb}$ mixtures [F. Ferlaino et al., Phys. Rev. A 74, 039903 (2006)] were included in the fit. This makes the derived potential curves capable for an accurate description of observed cold collision features so far. Predictions of scattering lengths and Feshbach resonances in other isotopic combinations are reported.

Coupling of the X Σ + 1 and a Σ + 3 states of KRb

Coupling of theXΣ+1andaΣ+3states ofKRb

The experimental characterization of the electronic states correlated to the asymptote of ground state Na (3S) and Rb (5S) atoms was expanded by spectroscopic data on $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ state levels using a high resolution Fourier transform spectroscopy technique. The hyperfine splitting of the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ state levels was partially resolved and analyzed for both $\mathrm{Na}\phantom{\rule{0.2em}{0ex}}^{85}\mathrm{Rb}$ and $\mathrm{Na}\phantom{\rule{0.2em}{0ex}}^{87}\mathrm{Rb}$ isotopomers. Transitions to high lying levels of the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ and $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ states were recorded simultaneously which enables one to determine long range parameters of the molecular potentials. Coupled channels calculations based on the Fourier grid method were finally applied for deriving accurate potential energy curves of the $a\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Sigma}^{+}$ and $X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ states capable of a reliable description of cold collisions between Na and Rb atoms in their ground states. Scattering lengths and Feshbach resonances were calculated for some quantum states.

Potentials for modeling cold collisions between Na (3S) and Rb (5S) atoms

Laser-induced fluorescence (LIF) spectra $(4)\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}\ensuremath{\rightarrow}A\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}--b\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Pi}$ and collisionally enhanced $A\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}--b\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Pi}\ensuremath{\rightarrow}X\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ LIF spectra measured by Fourier transform spectrometer with the resolution of $0.03--0.05\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$ provided about 1160 term values of the $e$-symmetry rovibronic levels of the fully mixed $A\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}$ and $b\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Pi}$ states of a NaCs molecule. Direct deperturbation treatment of the experimental data field, covering rotational quantum numbers $J∊[5,151]$ of the $A\text{\ensuremath{-}}b$ complex in the energy region $E∊[10577,13668]\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, was accomplished in the framework of the inverted channel-coupling approach by means of the $4\ifmmode\times\else\texttimes\fi{}4$ Hamiltonian constructed on Hund's coupling case $(a)$ basis functions. The nonequidistant spin-orbit splitting of the $b\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Pi}_{\ensuremath{\Omega}=0,1,2}$ substates and the indirect coupling $A\phantom{\rule{0.2em}{0ex}}^{1}\ensuremath{\Sigma}^{+}--b\phantom{\rule{0.2em}{0ex}}^{3}\ensuremath{\Pi}_{\ensuremath{\Omega}=1}$ matrix element were introduced in the Hamiltonian phenomenologically to account for the regular perturbations by remote states manifold. The expanded Morse oscillator model was used to approximate both potential energy curves of the mutually perturbed states and spin-orbit coupling matrix elements as an analytical function of internuclear distance. Overall 31 fitting parameters have been required to reproduce 98% of experimental term values with a standard deviation of $0.006\phantom{\rule{0.3em}{0ex}}{\mathrm{cm}}^{\ensuremath{-}1}$, which is consistent with the uncertainty of the experiment. The term values deperturbation analysis was confirmed by a calculation of relative intensity distributions in the $A--b\ensuremath{\rightarrow}X$ LIF progressions. The predicted probabilities for both weakly and strongly perturbed levels agree with their experimental counterparts within the accuracy of measurements. The evaluated nonadiabatic $A\text{\ensuremath{-}}b$ wave functions were applied for a prediction of radiative lifetimes of the $A\text{\ensuremath{-}}b$ complex as well as transition probabilities of the $a\ensuremath{\rightarrow}A--b\ensuremath{\rightarrow}X$ cycle proposed in Stwalley, Eur. Phys. J. D 31, 221 (2004), for efficient transformation of ultracold molecules to their absolute ground level ${v}_{X}=0$; ${J}_{X}=0$.

Solution of the fully-mixed-state problem: Direct deperturbation analysis of the A Σ + 1 – b Π 3 complex in a NaCs dimer

The states X1?+ and a3?+ correlated to the ground-state asymptote of Na (3s) and Cs (6s) atoms have been experimentally investigated using high resolution Fourier-transform spectroscopy. The hyperfine splitting of the a3?+ state levels is partially resolved. Transitions to asymptotic vibrational levels of the a3?+ and X1?+ states were recorded simultaneously. The joint evaluation of the data of both the a3?+ and the X1?+ states allows us to determine accurate potential energy curves of both electronic states. Coupled-channels calculations are finally applied for deriving long range dispersion parameters and the exchange contribution of the molecular potentials, yielding a reliable description of cold collisions between Na and Cs atoms in their two different hyperfine ground states. Scattering lengths and Feshbach resonances are predicted for selected quantum states.

http://iopscience.iop.org/article/10.1088/0953-4075/39/19/S08/pdf

M. Tamanis

Papers

Coupling of the X Σ + 1 and a Σ + 3 states of KRb

Coupling of theXΣ+1andaΣ+3states ofKRb

Potentials for modeling cold collisions between Na (3S) and Rb (5S) atoms

Solution of the fully-mixed-state problem: Direct deperturbation analysis of the A Σ + 1 – b Π 3 complex in a NaCs dimer

The coupling of the X1Σ+ and a3Σ+ states of the atom pair Na + Cs and modelling cold collisions