Spectroscopy of ultracold trapped cesium Feshbach molecules

We explore the rich internal structure of ${\text{Cs}}_2$ Feshbach molecules. Pure ultracold molecular samples are prepared in a ${\text{CO}}_2$-laser trap, and a multitude of weakly bound states is populated by elaborate magnetic-field ramping techniques. Our methods use different Feshbach resonances as input ports and various internal level crossings for controlled state transfer. We populate higher partial-wave states of up to eight units of rotational angular momentum ($l$-wave states). We investigate the molecular structure by measurements of the magnetic moments for various states. Avoided level crossings between different molecular states are characterized through the changes in magnetic moment and by a Landau-Zener tunneling method. Based on microwave spectroscopy, we present a precise measurement of the magnetic-field-dependent binding energy of the weakly bound $s$-wave state that is responsible for the large background scattering length of Cs. This state is of particular interest because of its quantum-halo character.

Figure 1

(Color online) Molecular energy structure below the threshold of two free Cs atoms in the absolute ground state $|F=3,m_F=3⟩$. Molecular state labeling is according to the quantum numbers $f\ell \left(m_f\right)$, $m_{\ell }=6-m_f$. The quantum number $m_f$ is omitted for states with $m_f=f$ and $m_{\ell }=\ell$. The solid lines represent the $s$-, $d$-, and $g$-wave states included in the NIST model [23]. The intersections of the $d$- and $g$-wave states with the threshold cause narrow Feshbach resonances that can be used for molecule production. The curvature of the $6s$ state arises from a large avoided crossing between two states of the same $f\ell$ quantum numbers. The NIST model does not take into account weak avoided crossings between bound molecular states mediated by the relativistic spin-spin dipole and second-order spin-orbit interactions. If these interactions are taken into account, the crossings between bound molecular states become avoided as illustrated in the inset for the example of the $4g\left(4\right)∕6g\left(6\right)$ crossing. The dashed lines represent $l$-wave states $\left(\ell =8\right)$ obtained from extended calculations based on the NIST model.

Figure 2

(Color online) Successive stages of optical dipole traps. (a) We first realize a large volume “reservoir” trap by crossing a ${\text{CO}}_2$-laser beam and a 1070 nm fiber-laser beam in the presence of a levitating gradient. (b) We ramp up a tightly focused ${\text{CO}}_2$-laser beam, (c) switch off the 1070 nm reservoir beam, and then evaporate along the vertical direction ($z$ axis) by lowering the gradient. We obtain typically $4\times {10}^5\text{Cs}$ atoms at a temperature of 200 nK.

Figure 3

(Color online) The molecular energy structure for very small binding energies in the region of the 19.8, 47.9, and 53.4 G Feshbach resonances, not resolved in Fig. 1. Above 45 G two avoided crossings are present in the energy structure. We use all three Feshbach resonances to associate molecules by ramping or switching the magnetic field. The arrows indicate the pathway after molecule association as the magnetic field is ramped down to allow for the optical removal of the atoms from the molecules. For details see the text.

Figure 4

(Color online) Illustration of the experimental toolbox for the preparation of molecules in various internal states. As an example the creation of molecules in the $6l\left(4\right)$ state is shown. (a) First, we produce $4g\left(4\right)$ molecules at the $g$-wave Feshbach resonance at 19.8 G and remove the remaining atoms by a short blast pulse, indicated by the rippled arrow. (b) The $4g\left(4\right)$ state is transferred into state $6g\left(6\right)$ by slowly lowering the magnetic field. (c) To overcome the avoided level crossing between state $4g\left(4\right)$ and $6g\left(6\right)$ a very fast magnetic-field ramp is applied. The target state $6l\left(4\right)$ is then accessed by using a second adiabatic ramp. (d) To remove the residual $4g\left(4\right)$ molecules from the $6l\left(4\right)$ dimers we ramp the magnetic field slightly above 20 G, dissociating the $4g\left(4\right)$ molecules into atoms while not affecting the $l$-wave molecules. (e) The magnetic field is lowered again to increase the binding energy of the target state molecules. A second blast pulse removes the remaining atoms.

Figure 5

(Color online) Example of a detection scheme for $l$-wave molecules. (a) First the $6l\left(4\right)$ molecules are adiabatically transferred into the state $4g\left(4\right)$ via the state $6g\left(6\right)$ by ramping down the magnetic field. (b) The avoided crossing at 13.6 G is passed by using a fast magnetic-field ramp. When ramping up to a field of $\sim 21\text{G}$, the molecules are brought above threshold and dissociate. The resulting atom cloud is detected using the standard absorption imaging technique.

Figure 6

(Color online) Magnetic moment of Cs dimers across the $6s\text{–}6g\left(6\right)\text{–}6l\left(4\right)$ molecular path. (a) The measured magnetic moments (open squares) are compared to the NIST calculations (dashed lines). The fast changes of the magnetic moment at 18.5 and 16 G result from the $6l\left(4\right)∕6g\left(6\right)$ and $6s∕6g\left(6\right)$ avoided crossings, respectively. (b) Molecular binding energies of the $6s$, $6g\left(6\right)$, and $6l\left(4\right)$ levels calculated from the NIST model; see also Fig. 1. The molecular path followed in the measurement is indicated by the black arrow.

Figure 7

(Color online) Energy spectrum of weakly bound Cs molecules as a function of the magnetic field. The binding energies for the different molecular states are derived from magnetic moment spectroscopy (open circles). The solid lines are the molecular binding energies resulting from the extended NIST model (for details see Sec. 2).

Figure 8

(Color online) Magnetic moments of Cs dimers across the $6g\left(6\right)∕4g\left(4\right)$ avoided crossing. Both the change in magnetic moment from the $6g\left(6\right)$ to the $4g\left(4\right)$ state (circles) and the one from the $4g\left(4\right)$ to the $6g\left(6\right)$ state (triangles) are shown. The measured magnetic moments are fitted using Eq. (7) (solid lines).

Figure 9

Coupling strengths $V∕h$ between the $6g\left(6\right)$ state and the intersecting $f\ell \left(m_f\right)$ molecular states as a function of the difference of their orbital angular momentum quantum numbers, $\left|\Delta \ell \right|$. The data refer to the values obtained via the magnetic moment spectroscopy (filled circles), a magnetic levitation method (empty circle) [14], an interferometer method (square) [40], and the Landau-Zener method (triangle).

Figure 10

(Color online) Conversion efficiency on the $6g\left(6\right)∕6l\left(3\right)$ avoided crossing as a function of the ramp speed. A pure sample of $6l\left(3\right)$ molecules is partially transferred into the $6g\left(6\right)$ state at different ramp speeds. We measure either the fraction of transferred $6g\left(6\right)$ molecules (open circles) or the fraction of nonconverted $6l\left(3\right)$ molecules (filled circles). The solid line refers to the Landau-Zener formula for $p$ given by Eq. (8), while the dashed line is $1-p$.

Figure 11

(Color online) Microwave spectroscopy of Cs dimers. (a) A bound-bound transition is driven from the $6s$ state to a $7s$ state slightly below the $|F=3,m_F=3⟩+|F=4,m_F=4⟩$ scattering channel, as illustrated by the longer arrow. The $7s$ state is offset from the scattering channel for clarity. The frequency corresponding to the $|F=3,m_F=3⟩\to |F=4,m_F=4⟩$ atomic transition at zero field is ${\nu }_{\text{at}}\approx 9.193\text{G}\text{Hz}$. (b) Microwave spectrum of atoms (triangles) and $6s$ molecules (dots) at $B\approx 18.7\text{G}$ as a function of the frequency offset $\nu -{\nu }_{\text{at}}$. The molecular transition corresponds to a sharp loss resonance. We determine the center position to be 4.9545(3) MHz and the resonance width to 12(3) kHz from a Gaussian fit (solid line).

Figure 12

(Color online) Binding energy of the $6s$ molecules as a function of the magnetic field (triangles). The binding energies correspond to the measured frequency shift from the expected $|F=3,m_F=3⟩\to |F=4,m_F=4⟩$ atomic transition [see Eq. (9)]. The statistical errors are about 1 kHz, i.e., much smaller than the data symbol size. The solid line is the result of the NIST model. The inset is an expanded view of the binding energy in the proximity of the $6s∕6g\left(6\right)$ avoided crossing.