Production of cold molecules via magnetically tunable Feshbach resonances

Magnetically tunable Feshbach resonances were employed to associate cold diatomic molecules in a series of experiments involving both atomic Bose and two-spin-component Fermi gases. This review illustrates theoretical concepts of both the particular nature of the highly excited Feshbach molecules produced and the techniques for their association from unbound atom pairs. Coupled-channels theory provides a rigorous formulation of the microscopic physics of Feshbach resonances in cold gases. Concepts of dressed versus bare energy states, universal properties of Feshbach molecules, and the classification in terms of entrance- and closed-channel-dominated resonances are introduced on the basis of practical two-channel approaches. Their significance is illustrated for several experimental observations, such as binding energies and lifetimes with respect to collisional relaxation. Molecular association and dissociation are discussed in the context of techniques involving linear magnetic-field sweeps in cold Bose and Fermi gases and pulse sequences leading to Ramsey-type interference fringes. Their descriptions in terms of Landau-Zener, two-level mean-field, as well as beyond mean-field approaches are reviewed in detail, including the associated ranges of validity.

Figure 1

(Color online) Order-of-magnitude variation of the scattering length $a$ as a function of the magnetic-field strength $B$ in the vicinity of the $907\mathrm{G}$ $\left(90.7\mathrm{mT}\right)$ zero-energy resonance of ${}^{23}\mathrm{Na}$. Circles refer to measurements using sodium Bose-Einstein condensates (109), while the solid curve indicates associated theoretical predictions (226). The scattering length is normalized to its asymptotic value $a_{\mathrm{bg}}$ away from the singularity at $B_0=907\mathrm{G}$. Adapted by permission from Macmillan Publishers Ltd: Nature (London) (109), copyright, 1998.

Figure 2

(Color online) Ramsey fringes between atomic and molecular components produced from a Bose-Einstein condensate of 17 100 ${}^{85}\mathrm{Rb}$ atoms (59) via a sequence of magnetic-field pulses in the vicinity of the $155\mathrm{G}$ zero-energy resonance. Circles and squares are the number of particles in the remnant condensate and in an atomic burst consisting of correlated pairs with a comparatively high relative velocity, respectively. Diamonds indicate the total amount of particles in both of these components. The difference with respect to the total number of atoms (dotted line) indicates the production of undetected Feshbach molecules. Fringes are shown as a function of the delay time of the interferometer $t_{\mathrm{evolve}}$, in which the atomic and molecular states acquire their phase difference. The fringe frequency provides an accurate measure of the binding energy (44). Adapted by permission from Macmillan Publishers Ltd: Nature (London) (59), copyright, 2002.

Figure 3

(Color online) A molecular component of about 3000 dimers is out-coupled from a dilute cloud of $25000$ ground-state cesium atoms using Stern-Gerlach separation in an inhomogeneous magnetic field (103). Left and middle images are situations in which the magnetic field is calibrated such that it exactly compensates for the gravitational force acting on atoms in gases without and with a molecular component, respectively. Due to their magnetic moment difference of $-0.57{\mu }_{\mathrm{Bohr}}$ (${\mathrm{\mu }}_{\mathrm{Bohr}}=9.27400949\times {10}^{-24}\mathrm{J}∕\mathrm{T}$ is the Bohr magneton) with respect to separated atoms, Feshbach molecules in the middle image drop below the atomic cloud, which is levitated and centered at the same position as in the left reference image. Conversely, the right image shows levitation of molecules and upward acceleration of separated atoms using a suitably adjusted inhomogeneous magnetic field.

Figure 4

(Color online) Density distributions of dilute thermal (left image) and partially Bose-Einstein condensed (right image) gases of ${}^{40}\mathrm{K}_2$ Feshbach molecules (98). Both molecular vapors were produced using linear magnetic-field sweeps across the $202\mathrm{G}$ zero-energy resonance. The initial two-spin-component clouds of $470000$ and $250000$ Fermi atoms were prepared above $\left(250\mathrm{nK}\right)$ and below $\left(79\mathrm{nK}\right)$ the associated critical temperatures for condensation, respectively. The condensate fraction in the right image is $12\%$. Adapted by permission from Macmillan Publishers Ltd: Nature (London) (98), copyright, 2003.

Figure 5

(Color online) Helium dimer interaction potential (dashed curve) (210) and radial probability density of the ${}^4\mathrm{He}_2$ molecule (solid curve). Dotted and dot-dashed lines are the bound-state energy $E_b=-4.2\times {10}^{-9}$ a.u. ($1\mathrm{a}.\mathrm{u}.=4.35974417\times {10}^{-18}\mathrm{J}$ is the atomic unit of energy) and classical turning point $r_{\mathrm{classical}}=27a_{\mathrm{Bohr}}$ ($a_{\mathrm{Bohr}}=0.052917\mathrm{nm}$ is the Bohr radius), respectively. The interatomic distance $r$ is given on a logarithmic scale.

Figure 6

(Color online) The ${}^1\Sigma _g^{+}$ and ${}^3\Sigma _u^{+}$ Born-Oppenheimer potential-energy curves for the ${\mathrm{Rb}}_2$ dimer molecule correlating with two separated ${}^{2}S_{1∕2}\mathrm{Rb}$ atoms. An energy $E$ associated with the wave-number unit $E∕hc=1{\mathrm{cm}}^{-1}$ corresponds to $E∕h=29.979\mathrm{GHz}$. Inset: The long-range adiabatic potentials (150, 153) on a much smaller energy scale with the zero field $(B=0)$ hyperfine structure of the ${}^{85}\mathrm{Rb}$ isotope with $E_{\mathrm{hf}}∕h=3.035\mathrm{GHz}$. Arrows show the van der Waals length $l_{\mathrm{vdW}}$ of Eq. (7) and the uncoupling distance $r_u$, where the hyperfine energy $E_{\mathrm{hf}}$ is the difference between the ${}^3\Sigma _u^{+}$ and ${}^1\Sigma _g^{+}$ Born-Oppenheimer potential curves. The ${}^{87}\mathrm{Rb}$ isotope has the same Born-Oppenheimer potential-energy curves, but the long-range curves would be different with atomic levels $f=1$ and 2 and $E_{\mathrm{hf}}∕h=6.835\mathrm{GHz}$.

Figure 7

(Color online) Zeeman levels of the ${}^{85}\mathrm{Rb}{}^{2}S_{1∕2}$ atom vs the magnetic-field strength $B$. Solid and dashed curves are the $f=2$ and $3$ multiplets, respectively. The projection quantum numbers $m_f$ and the alphabetic labels $a$, $b$, $c,\dots$ are shown.

Figure 8

(Color online) Magnetic-field-dependent $M=-4$ $s$-wave energy levels of the ${}^{85}\mathrm{Rb}_2$ dimer. Dotted curves labeled by $ee$, $df$, $eg$, $fh$, and $gg$ show the energies of the five separated atom channels of Table . Solid curves indicate the calculated $s$-wave coupled-channel bound-state energies labeled by the vibrational quantum number $v$. Their symmetry labels refer to the set of quantum numbers $\left(f_1{f}_2\right)F$. The $ee$ limit gives rise to a single vibrational progression with $F=4$; the $df$ and $eg$ limits give rise to two $\left(23\right)F$ series with $F=4$ and 5; and the $fh$ and $gg$ limits give rise to two $\left(33\right)F$ series with $F=4$ and 6. In all cases, the $F=4$ level has lower energy than the $\left(f_1{f}_2\right)$ pairs. At $B=0$, the $v=-1$ and $-2$, $\left(23\right)F$ levels and the $v=-1$, $-2$, and $-3$, $\left(33\right)F$ levels represent metastable states with $E>0$ embedded in the $\left(22\right)4$ $ee$ scattering continuum. These metastable levels are represented by dashed curves, which give approximate positions of scattering resonances in the $ee$ channel. The metastable levels with $F=4$ are coupled to the $ee$ $s$-wave scattering continuum through the exchange interaction. The $F=5$ and 6 metastable levels are not exchange coupled to the $ee$ entrance channel at $B=0$, but become weakly coupled at higher fields where $F$ is no longer a good quantum number.

Figure 9

(Color online) Scattering length (upper panel) and bound-state energy levels (lower panel) vs $B$ for two ${}^{85}\mathrm{Rb}$ atoms in the $e$ state. The separated atom energy of two $e$-state atoms is taken to be zero. A singularity of the scattering length occurs where an energy level becomes degenerate with the scattering threshold at $E=0$. The resolution of the figure is not sufficient to show the variation of the bound-state energies with $B$ just below threshold; this variation is discussed in Fig. 12. The broad resonance is due to the threshold crossing of the strongly coupled $\left(f_1{f}_2\right)F\ell =\left(33\right)4s$, $v=-3$ level, whereas the narrow resonance is due to the crossing of the weakly coupled $\left(f_1{f}_2\right)F\ell =\left(33\right)6s$, $v=-3$ level.

Figure 10

(Color online) Scheme of the entrance- (solid curve) and closed-channel (dashed curve) potentials associated with a model representation of the $155\mathrm{G}$ zero-energy resonance of ${}^{85}\mathrm{Rb}$ (126). Horizontal dashed and vertical dot-dashed lines are the magnetic-field-dependent energy $E_{\mathrm{res}}\left(B\right)$ of the Feshbach resonance level ${\varphi }_{\mathrm{res}}\left(r\right)$ and the van der Waals length of $l_{\mathrm{vdW}}=82a_{\mathrm{Bohr}}$, respectively. In this model, the outer classical turning point of the resonance level ${\varphi }_{\mathrm{res}}\left(r\right)$ associated with the energy $E_{\mathrm{res}}\left(B\right)$ in the closed-channel potential is located at $r_{\mathrm{classical}}=72a_{\mathrm{Bohr}}$. Its vibrational quantum number $v=-1$ with respect to the dissociation threshold of $V_{\mathrm{cl}}(B,r)$ is chosen arbitrarily. Horizontal solid lines indicate the energies $E_{-1}$ and $E_{-2}$ of bare vibrational bound states associated with the entrance-channel potential. This bare interaction is adjusted such that it mimics the highest excited $\left(f_1{f}_2\right)F=\left(22\right)4$, $v=-1$ and $-2$ levels of Fig. 8. In this model, the off-diagonal spin exchange coupling is $W\left(r\right)=\beta \exp (-r∕\alpha )$, with $\alpha =5a_{\mathrm{Bohr}}$ and $\beta ∕k_B=38.5\mathrm{mK}$ used to match the measured Feshbach resonance parameters (44).

Figure 11

(Color online) Entrance- and closed-channel components of the highest excited vibrational bound state associated with the $155\mathrm{G}$ zero-energy resonance of ${}^{85}\mathrm{Rb}$ vs the interatomic distance $r$ (126). The wave functions were determined using the model in Fig. 10 for magnetic-field strengths of $160\mathrm{G}$ (upper panel) and $155.5\mathrm{G}$ (lower panel), respectively. Nodes of the wave functions at short distances are associated with the vibrational states supported by the bare potentials. The long-range behavior of the entrance-channel wave functions, beyond the van der Waals length of $82a_{\mathrm{Bohr}}$ (dot-dashed line), is largely determined just by the binding energy of the Feshbach molecule (125). Note that the interatomic distance is given on a logarithmic scale.

Figure 12

(Color online) Bound-state energy of the ${}^{85}\mathrm{Rb}_2$ Feshbach molecule as a function of the magnetic field strength in the vicinity of the $155\mathrm{G}$ zero-energy resonance. Circles indicate a full coupled-channels calculation (129). The solid curve results from a two-channel approach (95). For comparison, the dotted curve represents the universal estimate of $E_b$, while the dashed curve includes the leading correction to Eq. (1) due to the van der Waals tail of the background scattering potential (99; 125) given by Eq. (66). We note the pronounced shift of about $9\mathrm{G}$ between the measurable resonance position $B_0$ and the magnetic-field strength $B_{\mathrm{res}}$ at which the resonance energy (dot-dashed line) crosses the dissociation threshold of the entrance channel. Inset: Theoretical approaches compared with measurements (44) indicated by squares.

Figure 13

(Color online) Occupation of the closed-channel resonance level relative to the number of atom pairs in a balanced mixture of two spin components of ${}^6\mathrm{Li}$ vs magnetic-field strength in the vicinity of the $834\mathrm{G}$ zero-energy resonance (168). On the low-field side of the resonance position $B_0$ (dot-dashed line), the gas was prepared as a Bose-Einstein condensate of ${}^6\mathrm{Li}_2$ Feshbach molecules. Experimental data (circles) indicate the suppression of the closed-channel admixture to the dressed molecular state, i.e., $Z\left(B\right)={\mathcal{N}}_{\mathrm{b}}^{-2}$, over three orders of magnitude in agreement with Eq. (68) (solid curve). The dashed curve shows the universal estimate of Eq. (59) for comparison. Note that the bond length ofthe Feshbach molecules given by Eq. (3) reaches the order of magnitude of the average interatomic spacing of the gas ${10}^4{a}_{\mathrm{Bohr}}$ at a magnetic-field strength of about $800\mathrm{G}$. In this strongly interacting regime, the purely two-body theories are bound to break down. On the high-field side of $B_0$, the remnant small resonance state population indicates pairing phenomena in a cold, strongly correlated two-spin-component Fermi gas with a negative scattering length (168).

Figure 14

(Color online) Loss rates of diatomic Feshbach molecules (upper panel) and remnant atoms (lower panel) vs magnetic-field detuning $B-B_0$ in the vicinity of the $202\mathrm{G}$ zero-energy resonance in a cold gas of ${}^{40}\mathrm{K}$ with a peak density of about $1.5\times {10}^{13}\mathrm{atoms}∕{\mathrm{cm}}^3$ (181). Dimers were produced by adiabatic sweeps of the magnetic-field strength described in Sec. 4. Their number is denoted by $N_d$, while $N_a$ refers to the number of remnant unbound atoms. Circles in the upper panel indicate the measured dimer loss, whose general trend is well fit by an $a^{-2.3\pm 0.4}$ power law as a function of the scattering length (solid curve). Significant deviations from this scattering length dependence occur in a small region of magnetic-field strengths where the bond length of the dimers given by Eq. (3) is comparable to the average distance of the atoms in the gas. The lower panel shows the loss rate of remnant unbound atoms in gases with (circles) or without (squares) deliberate production of Feshbach molecules. The associated trends suggest that the atomic loss is largely unaffected by the presence of a dimer component in the gas (181).

Figure 15

(Color online) Lifetime of ${}^{85}\mathrm{Rb}_2$ dimers vs magnetic-field strength in the vicinity of the $155\mathrm{G}$ zero-energy resonance. Circles indicate measurements in a cold thermal gas with a peak density of $6.6\times {10}^{11}\mathrm{atoms}∕{\mathrm{cm}}^3$ (215), while squares refer to exact coupled-channels calculations for spontaneous dissociation of Feshbach molecules due to spin relaxation (128). The solid curve represents an asymptotic estimate of the lifetime based on the universal halo wave function given by Eq. (2) and the spin relaxation loss rate constant $K_2\left(B\right)$ associated with a nondegenerate Bose gas in the limit of zero collision energy. The two-body theory only fails close to resonance when the molecular bond length becomes comparable to the average interatomic distance of the gas.

Figure 16

(Color online) Avoided crossing of the highest excited vibrational levels of ${}^{87}\mathrm{Rb}_2$ (upper panel) and the magnetic moment of the Feshbach molecule (lower panel) vs the magnetic-field strength in the vicinity of the $1007\mathrm{G}$ zero-energy resonance. The solid curve in the upper panel indicates the bound-state energy $E_b\left(B\right)$ of the Feshbach molecule, while the dashed curve refers to the next more tightly bound dressed vibrational level. Dotted and dot-dashed lines are associated with the energies $E_{-1}∕h=-24\mathrm{MHz}$ of the bare highest excited vibrational level of $V_{\mathrm{bg}}\left(r\right)$ and $E_{\mathrm{res}}\left(B\right)$ of the closed-channel resonance state, respectively. The crossing between the bare levels at $1001.7\mathrm{G}$ leads to the measured variation in the magnetic moment of the Feshbach molecule indicated by circles in the lower panel (66). For comparison, the solid and dashed curves refer to exact coupled-channels calculations (229) and a two-channel approach (96), respectively.

Figure 17

(Color online) The Zeeman multiplets of ${}^{87}\mathrm{Rb}$ associated with the total angular momentum quantum numbers $f=1$ (solid curves) and $f=2$ (dashed curves). Inset: The magnetic moments $\partial E_a∕\partial B$ determined by Eq. (79) and the orientation quantum numbers $m_f$ of those Zeeman states that are relevant to Fig. 16.

Figure 18

(Color online) Schematic illustration of molecular association of ground-state ${}^{87}\mathrm{Rb}$ atoms via a downward magnetic-field sweep in a spherical harmonic atom trap with an oscillator frequency ${\nu }_{\mathrm{ho}}=39\mathrm{kHz}$ (212). The bare vibrational levels $(v=0,\dots ,6)$ associated with the background scattering quasicontinuum and the Feshbach resonance energy $E_{\mathrm{res}}\left(B\right)$ are indicated by dotted and dashed lines, respectively. Solid curves refer to the magnetic-field dependence of dressed energy levels in the vicinity of the zero-energy resonance position $B_0=1007.4\mathrm{G}$.

Figure 19

(Color online) Efficiency of Feshbach molecular association (upper panel) and dissociation into the lowest energy band of an optical lattice (lower panel) via linear magnetic-field sweeps across the $1007\mathrm{G}$ zero-energy resonance of ${}^{87}\mathrm{Rb}$ vs the ramp speed (212). Solid curves refer to the Landau-Zener formula given by Eq. (124) and the coefficient for a harmonic oscillator given by Eq. (128) using the measured frequency ${\nu }_{\mathrm{ho}}=39\mathrm{kHz}$. The dashed curve indicates the same Landau-Zener prediction scaled by a factor of 0.95, which accounts for possible deficits in the ideal diatomic filling, for instance, due to tunneling between lattice sites (212).

Figure 20

(Color online) Loss of condensate atoms in upward sweeps of the Feshbach resonance level across the 853 and $907\mathrm{G}$ zero-energy resonances of ${}^{23}\mathrm{Na}$ vs the inverse ramp speed $1∕\mid \dot{B}\mid$ (202). Experimental data are compared to theoretical predictions using the Landau-Zener (153; 96) and two-level mean-field (227) approaches. The theoretical Feshbach resonance parameters employed refer to the values of $a_{\mathrm{bg}}$ and $\mathrm{\Delta }B$ given in Tables . Note the differences in ramp speeds of three orders of magnitude between the upper and lower panels, which reflect the different widths of the 853 and $907\mathrm{G}$ zero-energy resonances.

Figure 21

(Color online) Fraction of remnant atoms $1-2N_d^f∕N$ at the end of asymptotic magnetic-field sweeps across the closed-channel-dominated $543\mathrm{G}$ zero-energy resonance of ${}^6\mathrm{Li}$ vs the inverse ramp speed (206). The solid curve refers to a Landau-Zener estimate for the onset of molecule production in the fast sweep limit (41) using the local-density approximation. Inset: The molecular conversion $2N_d^f∕N$ observed in a thermal Bose gas with a mean density of $1.3\times {10}^{11}\mathrm{atoms}∕{\mathrm{cm}}^3$ and a temperature of $40.6\mathrm{nK}$ using magnetic-field sweeps across the $155\mathrm{G}$ zero-energy resonance of ${}^{85}\mathrm{Rb}$ (105). For comparison, the dashed curve indicates the pairwise ensemble average over the transition probabilities $2p_{\mathrm{ass}}\left(\mathbf{k}\right)$ with respect to the Maxwell distribution of relative velocities. This semiclassical estimate for the onset of molecule production in Bose gases is based on Eq. (142) and is therefore consistent with the limit of Eq. (145). Both solid and dashed curves are associated with the parameters $a_{\mathrm{bg}}$ and $\mathrm{\Delta }B$ of Tables , respectively.

Figure 22

(Color online) Molecular production efficiency $2N_d^f∕N$ vs peak phase-space density of cold Bose as well as two-spin-component Fermi gases. Circles are dimer populations measured at the end of adiabatic magnetic-field sweeps across the zero-energy resonances of ${}^{85}\mathrm{Rb}$ at $155\mathrm{G}$ (inset) and of ${}^{40}\mathrm{K}$ at $202\mathrm{G}$ (105). Curves indicate predictions (243) based on coupled Boltzmann equations including quantum statistical effects (solid curve) as well as their associated classical gas limit (dashed curves). For comparison, the square and diamond refer to molecular production efficiencies observed for the lowest ramp speeds in earlier sweep experiments using zero-energy resonances of ${}^{40}\mathrm{K}$ at $224\mathrm{G}$ (185) and of ${}^6\mathrm{Li}$ at $543\mathrm{G}$ (206), respectively.

Figure 23

(Color online) Mean energies of the atomic fragments of ${}^{23}\mathrm{Na}_2$ Feshbach molecules dissociated by a linear upward sweep of the $907\mathrm{G}$ resonance level vs the ramp speed. Circles indicate experimental data (157) while the solid curve refers to the predictions of Eq. (149) using the parameters $a_{\mathrm{bg}}$ and $\mathrm{\Delta }B$ of Tables , respectively.

Figure 24

(Color online) Typical magnetic-field pulse sequence (solid lines) used in experiments producing coherent oscillations between final atomic and molecular components in a ${}^{85}\mathrm{Rb}$ Bose-Einstein condensate (59). A pair of pulses with a minimum field strength $B_{\min }=155.5\mathrm{G}$ is separated by an evolution period of variable duration $t_{\mathrm{evolve}}$ and field strength $B_{\mathrm{evolve}}$. The zero-energy resonance position $B_0\approx 155\mathrm{G}$ is indicated by the dashed line.

Figure 25

(Color online) Schematic illustration of the Ramsey interferometry with ${}^{85}\mathrm{Rb}$ atoms and Feshbach molecules associated with the magnetic-field pulse sequence of Fig. 24 (59; 259). The first pulse drives the initial Bose-Einstein condensate into a coherent superposition of bound and unbound atom pairs. Between the pulses, the individual orthogonal components evolve independently. The second pulse forces the atomic and molecular states to overlap and thereby probes their phase difference. Inset: A typical interference fringe pattern at the end of the pulse sequence as a function of the evolution time $t_{\mathrm{evolve}}$. Circles refer to measurements of the remnant Bose-Einstein condensate (44), while squares indicate theoretical predictions (95).

Figure 26

(Color online) Predicted Ramsey fringes between three components of a dilute gas of ${}^{85}\mathrm{Rb}$ (95) at the end of the magnetic-field pulse sequence of Fig. 24 vs the evolution time $t_{\mathrm{evolve}}$. Circles are the remnant Bose-Einstein condensate, while squares indicate the final population of correlated atom pairs constituting the burst component. The remnant condensate and burst add up to the number of detectable atoms (diamonds). Its difference to the total number of $N=17100$ atoms (dotted line) in the experiments (59) determines the population of Feshbach molecules. The magnetic-field strength in the evolution period of $B_{\mathrm{evolve}}=159.84\mathrm{G}$ associated with these predictions refers to the conditions of the measured fringes of Fig. 2.