Feshbach resonances, weakly bound molecular states, and coupled-channel potentials for cesium at high magnetic fields

We explore the scattering properties of ultracold ground-state Cs atoms at magnetic fields between 450 G (45 mT) and 1000 G. We identify 17 previously unreported Feshbach resonances, including two very broad ones near 549 and 787 G. We measure the binding energies of several different dimer states by magnetic field modulation spectroscopy. We use least-squares fitting to these experimental results, together with previous measurements at lower field, to determine a six-parameter model of the long-range interaction potential, designated M2012. Coupled-channels calculations using M2012 provide an accurate mapping between the $s$-wave scattering length and the magnetic field over the entire range of fields considered. This mapping is crucial for experiments that rely on precise tuning of the scattering length, such as those on Efimov physics.

Figure 1

Molecular potential energy curves $V_0\left(R\right)$ and $V_1\left(R\right)$ for the singlet and triplet states of Cs${}_2$. The inset shows an expanded view of the long-range potentials separating to the two different $f=3$ and 4 hyperfine states of the atoms at magnetic field $B=0$.

Figure 2

Scattering length and energy levels versus magnetic field $B$ for Cs collisions in the lowest energy aa spin channel. (a) Scattering length, calculated including only $s$ basis functions. The vertical lines indicate the pole positions. (b) Cs${}_2$ $s$-wave bound-state energies below the aa threshold with $M_{\mathrm{tot}}=+6$, $M_F=+6$ in the range $B=-200$ to 1000 G, calculated with $s$ basis functions only. (c) Scattering length, calculated including $s$ and $d$ basis functions and including all matrix elements of the spin-dipolar coupling. (d) Cs${}_2$ $s$-wave and $d$-wave bound-state energies below the aa threshold with $M_{\mathrm{tot}}=+6$ and all allowed values of $M_F$ in the range $B=-200$ to 1000 G. The legend shows the $L\left(M_F\right)$ labels. Levels with different $L$ or $M_F$ cross since small off-diagonal spin-dipolar matrix elements coupling them were not included in the calculation. Properties in the aa channel at negative values of $B$ apply to the gg channel with reversed $M_{\mathrm{tot}}=-6$ (see text). The lines in the upper left corner of panel (d) show the energies of the fg, ff, and eg atomic channels.

Figure 3

(a) The real (red, solid) and imaginary (blue, dashed) parts of the complex scattering length (see Sec. 4), calculated with an $sd$ basis set. For $B<0$, two-body relaxation is possible to channels fg, ff, or eg. The inset shows an expanded view of the pole region of the resonance near $-12$ G, indicating a maximum variation in scattering length of $\approx$$\pm {10}^5$ $a_0$ due to resonance decay. Panel (b) shows the bound states below threshold, calculated as in Fig. 2 so that levels of different $L$ or $M_F$ cross instead of showing avoided crossings. The $M_F$ labels in the negative $B$ regions are shown with reversed sign.

Figure 4

Low-field near-threshold bound levels of $s$, $d$, and $g$ symmetry. As for Figs. 2 and 3, the avoided crossings between states of different $L$ or $M_F$ are not calculated because spin-dipolar coupling is omitted in this calculation. The low-field $g$-wave levels are of $-2\left(33\right)$ character, whereas the level marked $x2g\left(2\right)$ is of mixed $n\left(f_1{f}_2\right)=-2\left(33\right)$, $-6\left(34\right)$, and $-7\left(44\right)$ character.

Figure 5

Timing sequence for the magnetic field ramps for trap-loss spectroscopy and binding energy measurements. The magnetic field strength $B$ is linearly ramped from the final evaporation field $B_{\mathrm{evap}}$ within the ramping time $t_{\mathrm{ramp},1}$ to the probe field $B_{\mathrm{probe}}$. After an experimentally optimized hold time $t_{\mathrm{hold}}$ ranging between 0.2 and 1 s, the magnetic field strength is (linearly) ramped to the imaging field $B_{\mathrm{imag}}$. For Scheme A, $B_{\mathrm{evap}}$ and $B_{\mathrm{imag}}$ are in the magnetic low-field region. The magnified segment visualizes $B_{\mathrm{probe}}$, which is constant in time for trap-loss spectroscopy (solid line) and sinusoidally modulated for the determination of binding energies (dashed line).

Figure 6

Results of trap-loss spectroscopy in the magnetic field region between 450 and 1000 G, performed with Scheme B. The enhancements in losses result from broad Feshbach resonances centered at 494, 549, and 787 G. The resonance positions, which are derived from our theoretical model (see Sec. 4), are indicated by dashed lines, labeled according to the quantum numbers $FL\left(M_F\right)$ of the molecular states that cause the resonances. The measurements are performed with a hold time of 500 ms. Narrow Feshbach resonances are not visible in this scan because of the large step size of about 2 G. A remaining atom fraction of $1.0$ corresponds to $8\times {10}^4$ atoms.

Figure 7

Detailed results of trap-loss spectroscopy in the magnetic high-field region. We observe 15 Feshbach resonances, stemming from $d$-, $g$-, and $i$-wave molecular states. Measurements indicated by $\left(▴\right)$ symbols are obtained by Scheme A, whereas the $(•)$ symbols refer to data points acquired with Scheme B. The poles of the Feshbach resonances, omitting the $s$-wave resonances, are marked with an arrow. The loss features at 498.1, 553.3, 554.7, 818.9, and 853.1 G, which are indicated by “E”, are related to Efimov loss resonances as reported in Refs. [18, 52]. Note that the data in the intervals [728,729.5] and [980, 1000] G, which are measured for different $t_{\mathrm{hold}}$, are multiplied by scaling factors of 0.5 and 0.9, respectively, to reproduce the overall behavior shown in Fig. 6. A remaining atom fraction of $1.0$ corresponds to $8\times {10}^4$ atoms.

Figure 8

Expanded view of Fig. 7, showing the Feshbach resonances arising from the $g$(3) state (a) and the $-6\left(34\right)6d\left(6\right)$ state (b), which overlap with the broad $s$-wave resonances. The solid and open arrows mark the poles and zero crossings of the scattering length, respectively. The remaining atom fraction at the zero crossing in (b) is limited due to losses during the magnetic field ramps.

Figure 9

Typical signals for magnetic field modulation spectroscopy. (a) Frequency scan at a fixed magnetic field strength of 911.69 G. The asymmetric shape of the signal is fitted by a model that takes the temperature of the sample into account [59]. (b) Magnetic field scan at a constant modulation frequency of 425 kHz, revealing a loss resonance at 557.02 G. This resonance stems from the $s$-wave molecular state. The line represents a Gaussian fit. A remaining atom fraction of 1.0 corresponds to (a) $4.5\times {10}^4$ and (b) $2.5\times {10}^4$ atoms.

Figure 10

Results of experimental binding energy measurements. (a) The $-6\left(34\right)6d\left(6\right)$ and $-6\left(34\right)5d\left(5\right)$ states (dashed lines) and the $-6\left(34\right)6s\left(6\right)$ state (solid line) between 820 and 950 G. The lines are guides for the eye. The inset shows a binding energy measurement at the crossing of the $-6\left(34\right)6s\left(6\right)$ and $-6\left(34\right)5d\left(5\right)$ states at 509 kHz. At fields above 900 G the $s$-wave state takes on the character of the least-bound state $-1\left(33\right)6s\left(6\right)$, which has a binding energy around $h\times 50$ kHz away from avoided crossings. The strong variation of the magnetic moment follows from the avoided crossing with this threshold $s$-wave state. (b) The binding energies of the $g$(3) state (dashed line), an $i$-wave state (dot-dash line), and the $-6\left(34\right)7s\left(6\right)$ state (solid line). The positions of the resonance poles are marked by a ($⧫$) symbol, with values taken from Table .

Figure 11

(a) The scattering length in the region between 875 and 950 G. The solid lines show results calculated for the M2012 potential with an $sdg$ basis set. The diamond near 880 G indicates the position of the zero crossing. (b) The calculated bound-state energies with the same basis set. The diamonds show the measured energies in this region.

Figure 12

The scattering length and bound-state energy in the pole region of the $-6\left(34\right)6d\left(6\right)$ resonance. The solid lines show results from model M2012, calculated with an $sdg$ basis set. The points in panel (a) show the measured pole position and loss minimum. The points in panel (b) show the measured binding energies. The dashed line in (a) and the arrow in (b) show the calculated pole position. Error bars refer to $1\sigma$ uncertainties.

Figure 13

(a) The scattering length between 546 and 564 G. The solid lines show results calculated for the M2012 potential with an $sdg$ basis set. The diamonds show the measured pole position and loss minimum near 554 G and the zero crossing near 556 G. The dashed line shows the calculated pole position. (b) The calculated binding energies with the same basis set as for panel (a). The diamonds show the measured binding energies.

Figure 14

Expanded view of the scattering length in the pole region of the $-3\left(33\right)6g\left(3\right)$ resonance. The solid line shows results calculated for the M2012 potential with an $sdg$ basis set. The points show the measured bound-state energies. The calculated and measured pole positions are indicated by solid and open arrows, respectively. Error bars refer to $1\sigma$ uncertainties.

Figure 15

The bound-state energies between 35 and 60 G. The solid lines show results calculated for the M2012 potential with an $sdg$ basis set, showing one side of the avoided crossings of the $-7\left(44\right)6s\left(6\right)$ bound state with the $-1\left(33\right)6s\left(6\right)$ least-bound state in the threshold entrance channel. The points show the previously measured bound-state energies [14, 37]. The agreement is much better with the new M2012 model than with the old M2004 model, shown as light dashed lines.

Figure 16

Comparison between experimental results for $s$-wave levels (diamonds [14, 37]) and $g$- and $l$-wave levels (open circles [14, 15]) with the results of the M2012 model (solid lines). Only levels for which experimental results exist are shown. The $s$-wave and $g$-wave levels are calculated with the full $sdg$ basis set. The $l$-wave levels are calculated with an $l$ basis only. The light dashed lines show the $l$-wave levels calculated with the M2004 model. The arrows show the separation between equivalent levels in the M2004 and M2012 models. The additional quantum numbers $n\left(f_1{f}_2\right)$ are $-2\left(33\right)$ for the two $4g$ levels and $-3\left(33\right)$ for the three $6l$ levels.

Figure 17

The scattering length (a) and bound-state energies (b) between 490 and 508 G, calculated for the M2012 potential with an $sdg$ basis set. The solid diamonds in (a) show the positions of experimental loss maxima and the open circles show the positions of loss minima. The four ramping $-6\left(34\right)7d\left(x\right)$ levels, labeled collectively by $x$ since their $M_F$ components are mixed, undergo avoided crossings with the $-2\left(33\right)6g\left(6\right)$ level that passes through this region with $-E/h\approx 2$ MHz. The avoided crossings with the least-bound level $-1\left(33\right)6s\left(6\right)$ near $-E/h\approx 0.01$ MHz are too close to the $E=0$ line to be seen on the figure.

Figure 18

The scattering length between 460 and 1000 G, calculated for the M2012 potential with an $sdg$ basis set. The dashed lines indicate the pole positions of the broad $s$-wave resonances near 549 and 787 G and the $d$-wave one near 820 G. Note that some narrow resonances appear as spikes because of the limited grid spacing of 0.1 G, and some of the narrowest are not visible at all.