Precision Feshbach spectroscopy of ultracold ${\mathrm{Cs}}_2$

We have observed and located more than 60 magnetic field-induced Feshbach resonances in ultracold collisions of ground-state ${}^{133}\mathrm{Cs}$ atoms. Multiple extremely weak Feshbach resonances associated with $g$-wave molecular states are detected through variations in the radiative collision cross sections. The Feshbach spectroscopy allows us to determine the interactions between ultracold cesium atoms and the molecular energy structure near the dissociation continuum with unprecedented precision. Our work not only represents a very successful collaboration of experimental and theoretical efforts, but also provides essential information for cesium Bose-Einstein condensation, ${\mathrm{Cs}}_2$ molecules, and atomic clock experiments.

Figure 1

Illustration of radiative, elastic, and inelastic Feshbach resonance. The molecular state is supported by the closed channel.

Figure 2

Apparatus for Feshbach spectroscopy of ${}^{133}\mathrm{Cs}$.

Figure 3

Resonant and off-resonant time evolution of the atomic density. Atoms are prepared in $\mid F=4,m_F=-4⟩$ at $T=5.3\mu \mathrm{K}$ and $\overline{n}=2.1\times {10}^{12}{\mathrm{cm}}^{-3}$ at $B=14\mu \mathrm{T}$ (solid circles) and $B=20.5\mathrm{mT}$ (open circles), where a strong Feshbach resonance is located. See also Table .

Figure 4

Binary loss coefficient in a gas of $\mid F=3,m_F=-3⟩$ atoms as a function of external magnetic field. The initial mean atomic density is $\overline{n}=5\times {10}^{12}{\mathrm{cm}}^{-3}$ at a temperature of $\sim 5\mu \mathrm{K}$. Populations in the $\mid F=3,m_F=-3⟩$, $\mid F=3,m_F=-2⟩$, and all other states are 95%, $\sim 5\%$, and $<1\%$, respectively. Magnetic field resolution is $10\mu \mathrm{T}$.

Figure 5

Atom loss as a function of magnetic field for an initial mean density $\overline{n}=5\times {10}^{12}{\mathrm{cm}}^{-3}$, temperature of $\sim 5\mu \mathrm{K}$, and interaction time of $300\mathrm{ms}$. Populations in the $\mid F=3,m_F=-3⟩$, $\mid F=3,m_F=-2⟩$, and all other states are 95%, $\sim 5\%$, and $<1\%$ for the lower curve, 85%, $\sim 10\%$, and $\sim 5\%$ for the middle curve, and $\sim 70\%$, $\sim 20\%$, and $\sim 10\%$ for the upper curve, respectively. Comparison between the curves facilitates the identification of the incident channel responsible for the Feshbach resonances. Magnetic field resolution is $10\mu \mathrm{T}$.

Figure 6

A detailed radiative loss spectrum. The probe beam at wavelength $\lambda =844\mathrm{nm}$ has an intensity of $50\mathrm{W}∕{\mathrm{cm}}^2$. Here 85% (15%) of the atoms are in the $\mid 3,3⟩$ $\left(\mid 3,2⟩\right)$ state. The temperature and mean density of the sample are $3.5\mu \mathrm{K}$ and $1\times {10}^{13}{\mathrm{cm}}^{-3}$, respectively. “S” indicates a shape resonance; the stars indicate Feshbach resonances in the $(3,3)+(3,2)$ channel. Magnetic field resolution near the resonances is $5\mu \mathrm{T}$.

Figure 7

Total angular momentum projection $M=6$, $s$-, $d$-, and $g$-wave bound state energies as a function of magnetic field. The zero of energy corresponds to the $(F=3,m_F=3)+(F=3,m_F=3)$ dissociation limit. Dotted, dashed, and solid lines correspond to $l=s$, $d$, and $g$ states, respectively. Furthermore, each curve is labeled by the quantum numbers $fl$—i.e., $fl=6s$. The molecular spin $f$ is a zero-field quantum number. For the $fl=6d$ states the magnetic quantum number $m_f$ is also indicated. The solid circles represent the observed threshold resonances.

Figure 8

Expanded view of the $M=6$, $s$-, $d$-, and $g$-wave bound states shown in Fig. 7. An avoided crossing between $g$-wave $m_f=3$ and 5 bound states occurs around $B=1.4\mathrm{mT}$.

Figure 9

Position of two $M=6$ $g$-wave resonances as a function of $C_8$ or equivalently $C_6$ for a zero-energy collision of two Cs atoms in the lowest hyperfine state. The $C_6$ and $C_8$ are expressed in units of $E_H{a}_0^6$ and $E_H{a}_0^8$, respectively. The dotted lines correspond to the peak of the experimental radiative Feshbach spectroscopy signal.

Figure 10

Scattering length in the $(F=3,m_F=3)+(F=3,m_F=3)$ scattering channel. Resonances resulting from the $M=6$, $s$-, $d$-, and $g$-wave bound states are assigned with quantum number $(l,f,m_f)$, where $l$ is the orbital angular momentum, $f$ is the total internal angular momentum, and $m_f$ is the magnetic quantum number. Calculations are done with a magnetic field grid size of $50\mu \mathrm{T}$ for off-resonance regions and $100\mathrm{nT}$ near the narrow resonances. The inset shows the resonance structure near $17\mathrm{G}$ in detail.

Figure 11

Collision rate constants in the $(F=3,m_F=-3)+(F=3,m_F=-3)$ scattering channel at $E∕k_B=1\mathrm{nK}$. Inelastic collision rates (a) and elastic collision rates (b) are calculated with a magnetic field grid size of $10\mu \mathrm{T}$ for off-resonance regions and $1\mu \mathrm{T}$ near the resonances. Resonances from the $M=-6$, $s$-, $d$-, and $g$-wave bound states are included and assigned with quantum number $(l,f,m_f)$, where the notation is the same as in Fig. 10.