# Dynamics of Feshbach Molecules in an Ultracold Three-Component Mixture

Alexander Y. Khramov  Anders H. Hansen  Alan O. Jamison  William H. Dowd
Subhadeep Gupta  Department of Physics, University of Washington, Seattle WA
98195

(August 27, 2024)

###### Abstract

We present investigations of the formation rate and collisional stability of
lithium Feshbach molecules in an ultracold three-component mixture composed of
two resonantly interacting fermionic 6Li spin states and bosonic 174Yb. We
observe long molecule lifetimes ($>\,100\,$ms) even in the presence of a large
ytterbium bath and extract reaction rate coefficients of the system. We find
good collisional stability of the mixture in the unitary regime, opening new
possibilities for studies and probes of strongly interacting quantum gases in
contact with a bath species.

Magnetic Feshbach resonances allow precise control of collisional properties,
making them a key tool in ultracold atom systems. They have been extensively
used to study ultracold molecules, as well as few- and many-body physics Chin
et al. (2010). Two-component Fermi gases near a Feshbach resonance provide
excellent opportunities to study strongly interacting quantum systems Giorgini
et al. (2008). This is possible due to the remarkable collisional stability of
the atom-molecule mixture on the positive scattering length side of the
resonance Jochim et al. (2003); Cubizolles et al. (2012), attributed largely
to Fermi statistics Esry et al. (2001); Petrov et al. (2004). Extending the
system to three-component mixtures in which only two are resonantly
interacting Spiegelhalder et al. (2009) offers the exciting possibility of
modifying or probing pairing dynamics by selective control of the third
component. A third-component may also be used as a coolant bath for exothermic
molecule-formation processes, provided that inelastic processes with the bath
are negligible. In the context of many-body physics, a third non-resonant
component can be useful as a microscopic probe of superfluid properties
Spiegelhalder et al. (2009); Targonska and Sacha (2010), as a stable bath for
studies of non-equilibrium phenomena Robertson and Galitski (2009), or as an
accurate thermometer of deeply degenerate fermions Nascimbène et al. (2011).

Collisional stability of Feshbach molecules in the absence of Fermi statistics
becomes a crucial question for multi-component mixtures D’Incao and Esry
(2008); Zirbel et al. (2008); Spiegelhalder et al. (2009). A recent
theoretical analysis of such mixtures suggests a possibility for enhanced
molecule formation rates with good collisional stability D’Incao and Esry
(2008). Enhanced atom loss has been observed near a 6Li _p_ -wave resonance in
the presence of a 87Rb bath Deh et al. (2008), while a small sample of the
probe species 40K has been found to be stable within a larger strongly
interacting 6Li sample Spiegelhalder et al. (2009).

In this paper we investigate a mixture composed of two resonantly interacting
spin states of fermionic 6Li immersed in a large sample of bosonic 174Yb
atoms. While the Li interstate interactions are arbitrarily tunable by means
of an $s$-wave Feshbach resonance at $834\,$G Dieckmann et al. (2002), the
interspecies interactions between Li and Yb are constant and small Ivanov et
al. (2011). We present the first observations of formation and evolution of
Feshbach molecules in a bath of a second atomic species. In the unitary
regime, we observe good collisional stability of the mixture with elastic
interactions dominating over inelastic losses. We extract the reaction rate
constants from a classical rate equations model of the system.

Our experimental procedure has been described in earlier work Hansen et al.
(2011). Briefly $3\\!\times\\!10^{6}$ atoms of 174Yb in the ${}^{1}S_{0}$
state and up to $4\times 10^{4}$ atoms of 6Li , distributed equally between
the two ${}^{2}S_{1/2}$, $F\\!=\\!\frac{1}{2}$ states (denoted Li$|1\rangle$,
Li$|2\rangle$), are loaded from magneto-optical traps into a crossed-beam
optical dipole trap. We then perform forced evaporative cooling on Yb to the
final trap depth $U_{{\rm Yb(Li)}}=15(55)\,\mu$K, with mean trap frequency
$\bar{\omega}_{{\rm Yb(Li)}}=2\pi\times 0.30(2.4)\,$kHz odt (2012), during
which Li is cooled sympathetically by Yb. Following evaporation, the mixture
is held at constant trap depth to allow inter-species thermalization. With a
time constant of $1\,$s the system acquires a common temperature $T_{{\rm
Yb}}\,=\,T_{{\rm Li}}\,=\,2\,\mu$K with atom number $N_{{\rm Yb(Li)}}=2\times
10^{5}(3\times 10^{4})$. This corresponds to $T_{{\rm Li}}/T_{F}\simeq 0.4$,
and $T_{{\rm Yb}}/T_{C}\simeq 2.5$, where $T_{F}$ is the Li Fermi temperature
and $T_{C}$ is the Yb Bose-Einstein condensation temperature bec (2012).

Figure 1:  Li atom loss spectroscopy in the presence (filled circles) and
absence (open squares) of an Yb bath near the 6Li $834\,$G Feshbach resonance
(inset). We plot the number of Li atoms after $500\,$ms of evolution
normalized to that at $10\,$ms. The thick dashed line indicates the resonance
center and the thin dashed line indicates the magnetic field at which
$\epsilon_{B}=k_{B}T_{\rm Li}$ for the initial conditions.

After this initial preparation, we ramp up the magnetic field to a desired
value and observe the system after a variable hold time. For fields in the
vicinity of the Feshbach resonance, there is a field-dependent number loss and
heating for the Li cloud during the $20\,$ms ramp time, resulting in $T_{{\rm
Li}}$ rising to as high as $4.5\,\mu$K. At this point, the density-weighted
average density $\langle n_{{\rm Yb(Li)}}\rangle$ is $2.6(0.35)\times
10^{13}\,$cm-3. For interrogation in the absence of the bath, Yb is removed
from the trap with a $1\,$ms light pulse resonant with the
${}^{1}S_{0}\rightarrow{{}^{1}P_{1}}$ transition ybb (2012). Atom number and
temperature are monitored using absorption imaging for both species after
switching off the magnetic field.

We first present our results on atom loss spectroscopy near the Feshbach
resonance (see Fig. 1). The atom-loss maximum obtained in the absence of Yb
has been observed previously Dieckmann et al. (2002) and can be explained as a
result of the formation and subsequent decay of shallow lithium Feshbach
dimers Jochim et al. (2003); Cubizolles et al. (2012); Chin and Grimm (2004);
Kokkelmans et al. (2004); Zhang and Ho (2011) which form only on the positive
$a$ side of the resonance. Here $a$ denotes the Li$|1\rangle$-Li$|2\rangle$
scattering length. In the presence of the Yb bath, the loss feature is shifted
and broadened. We interpret the behavior of the mixture in terms of five
chemical processes:

$\displaystyle\mathrm{Li|1\rangle+Li|2\rangle+Li}\hskip 11.38109pt$
$\displaystyle\rightleftharpoons$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{s}}+Li\hskip
17.64069pt(+\epsilon_{B})}$ (I)
$\displaystyle\mathrm{Li_{2}^{\mathnormal{s}}+Li}\hskip 11.38109pt$
$\displaystyle\rightarrow$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{d}}+Li\hskip
17.64069pt(+\epsilon_{D})}$ (II)
$\displaystyle\mathrm{Li|1\rangle+Li|2\rangle+Yb}\hskip 11.38109pt$
$\displaystyle\rightleftharpoons$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{s}}+Yb\hskip
14.22636pt(+\epsilon_{B})}$ (III)
$\displaystyle\mathrm{Li_{2}^{\mathnormal{s}}+Yb}\hskip 11.38109pt$
$\displaystyle\rightarrow$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{d}}+Yb\hskip
14.22636pt(+\epsilon_{D})}$ (IV)
$\displaystyle\mathrm{Li|1\rangle+Li|2\rangle+Yb}\hskip 11.38109pt$
$\displaystyle\rightarrow$
$\displaystyle\mathrm{Li_{2}^{\mathnormal{d}}+Yb\hskip
14.22636pt(+\epsilon_{D})}$ (V)

Forward process (I) corresponds to a three-body collision event which produces
a shallow Feshbach dimer (denoted $\mathrm{Li_{2}^{\mathnormal{s}}}$)
accompanied by the release of the dimer binding energy
$\epsilon_{B}=\frac{\hbar^{2}}{2m_{{\rm Li}}a^{2}}$. Li denotes a 6Li atom in
either of the two spin states. Process (II) corresponds to two-body loss to a
deeply bound dimer (denoted $\mathrm{Li_{2}^{\mathnormal{d}}}$) with binding
energy $\epsilon_{D}$. Processes (III) and (IV) are similar to (I) and (II)
with the spectator atom being Yb rather than Li processiv (2012). Process (V)
corresponds to direct three-body loss to a deeply-bound molecule. Processes
(II, IV, V) always result in particle loss from the trap since
$\epsilon_{D}\gg U_{\rm Li}$. Vibrational relaxation due to collisions between
${\rm Li}_{2}^{\mathnormal{s}}$ Feshbach molecules may contribute at the
lowest fields, but has a negligible rate for the fields at which we perform
our analysis Jochim et al. (2003); Zhang and Ho (2011). We have experimentally
checked that direct three-body loss processes to deeply-bound states involving
three Li atoms as well as those involving one Li atom and two Yb atoms are
negligible for this work threebody (2012). Three-body losses involving Yb
atoms alone have a small effect Takasu et al. (2003) and are taken into
account in our analysis.

Figure 2:  Evolution of Li Feshbach molecule number at 709 G without (a) and
with (b) an Yb bath. The numbers are obtained by comparing Li atom numbers
(insets) ramped across resonance (diamonds) or not (open squares) as described
in the text. Lower inset also shows Yb number (filled squares). The curves are
fits with a rate equations-based model.

In the absence of Yb, only processes (I) and (II) contribute. If we neglect
loss process (II), the atom-molecule mixture approaches an equilibrium,
characterized by an equality of the forward and reverse rates and an
equilibrium molecule fraction $\frac{2N_{m}}{N_{\rm
Li}+2N_{m}}=\left(1+\frac{e^{-\epsilon_{B}/k_{B}T}}{\phi_{\mathrm{Li}}}\right)^{-1},$
where $N_{m}$ is the molecule number and $\phi_{{\rm Li}}$ is the phase space
density for each spin component in the ground state of the trap Chin and Grimm
(2004); Kokkelmans et al. (2004); Zhang and Ho (2011). The timescale for
achieving equilibrium depends on the three-body rate constant $L_{3}$ for
process (I), which scales with the scattering length as $a^{6}$, whereas rate
constant $L_{2}$ for process (II) scales as $a^{-3.3}$ Petrov (2003); D’Incao
and Esry (2005). The shape of the loss spectrum can thus be qualitatively
explained by noting that the dimer formation rate increases with magnetic
field while equilibrium dimer fraction and molecule decay rate decrease. The
large rate for process (I) at high fields close to resonance ensures
equilibrium molecule fraction at all times. Broadly speaking, the rate-
limiting step determining the system evolution is the molecule formation rate
at low fields and decay rate at high fields. The trap depth also affects the
loss spectrum shape, since it determines the magnetic field range over which
the formed shallow dimers remain trapped.

In the presence of Yb, the additional dimer formation (III), dimer decay (IV),
and 3-body loss (V) processes contribute. The observed loss spectrum is
broadened on the higher field side, suggesting that for our parameters,
processes (IV) and/or (V) play an important role while process (III) does not.
The rate constants $L^{\prime}_{3}$, $L^{\prime}_{2}$, and $L_{3}^{d}$, for
processes (III), (IV) and (V), have theoretical scalings $a^{4}$, $a^{-1}$,
and $a^{2}$, respectively D’Incao and Esry (2008); Petrov (2003). Overall, we
see two regimes of behavior - a lossy one where molecule formation is
energetically favored ($\epsilon_{B}>k_{B}T_{\rm Li}$) and a stable one closer
to resonance ($\epsilon_{B}<k_{B}T_{\rm Li}$). The criterion
$\epsilon_{B}=k_{B}T$ separating these two regimes is equivalent to $ka=1$
where $\frac{\hbar^{2}k^{2}}{2m_{\rm Li}}=k_{B}T_{\rm Li}$, i.e., the unitary
criterion.

In order to expand upon this qualitative picture, we study the time evolution
of the three-component mixture at representative magnetic fields in the above
two regimes. We are then able to extract quantitative information for the
above processes from a rate-equations model of the system.

Fig. 2 shows the Li atom and molecule number evolution at $709\,$G
($\epsilon_{B}=k_{B}\times 8.3\,\mu$K) a field value where modifications due
to the Yb bath are apparent in Fig. 1. The number of Feshbach molecules at a
particular field is determined by using a procedure similar to earlier works
Jochim et al. (2003); Cubizolles et al. (2012). After variable evolution time,
we ramp the magnetic field with a speed of $40\,$G/ms either up to $950\,$G,
which dissociates the molecules back into atoms that remain in the trap, or to
$506\,$G, which does not. We then rapidly switch off the magnetic field and
image the atomic cloud. The molecule number is obtained from the number
difference in the two images (see insets in Fig. 2).

We see that the presence of Yb alters the molecule decay rate while the
formation rate is unchanged. The Feshbach molecules appear to coexist for a
long time ($>100\,$ms) with the Yb bath, even in the absence of Pauli blocking
D’Incao and Esry (2008). We adapt the recent rate-equations analysis of
Feshbach losses in a Fermi-Fermi mixture Zhang and Ho (2011) to incorporate a
third component, temperature evolution, and trap inhomogeneity. $T_{{\rm
Li}}/T_{F}>0.5$ is satisfied throughout the measurement range, allowing a
classical treatment of the Li cloud. We model the density evolutions due to
processes (I-V) using:

$\displaystyle\dot{n}_{m}$ $\displaystyle=$ $\displaystyle
R_{m}+R^{\prime}_{m}-L_{2}n_{m}n_{{\rm Li}}-L^{\prime}_{2}n_{m}n_{{\rm Yb}}$
(1) $\displaystyle\dot{n}_{{\rm Li}}$ $\displaystyle=$
$\displaystyle-2R_{m}-2R^{\prime}_{m}-L_{2}n_{m}n_{{\rm Li}}-2L_{3}^{d}n_{{\rm
Li}}^{2}n_{{\rm Yb}}$ (2) $\displaystyle\dot{n}_{{\rm Yb}}$ $\displaystyle=$
$\displaystyle-L^{\prime}_{2}n_{m}n_{{\rm Yb}}-L_{3}^{d}n_{{\rm
Li}}^{2}n_{{\rm Yb}}.$ (3)

Here $n_{m}$, $n_{{\rm Li}}$ and $n_{{\rm Yb}}$ are the densities of shallow
dimers $\mathrm{Li_{2}^{\mathnormal{s}}}$, Li atoms and Yb atoms,
respectively. $R_{m}(R^{\prime}_{m})=\frac{3}{4}L_{3}(L^{\prime}_{3})n_{{\rm
Li}}^{2}n_{{\rm Li(Yb)}}-qL_{3}(L^{\prime}_{3})n_{m}n_{{\rm Li(Yb)}}$ is the
net-rate for molecule production via process (I)((III)). We determine $q$
through the constraints on the molecule fraction at equilibrium
($R_{m}(R^{\prime}_{m})=0$). We obtain an upper bound for $L_{3}^{d}$ by
observations at large negative $a$ (described below) which indicates a
negligible effect for the data in Fig. 2, allowing us to set $L_{3}^{d}=0$ for
the analysis at $709\,$G.

Figure 3:  The evolution of temperature and number at $810\,$G for Li atomic
cloud with Yb (filled circles) and without (empty circles) and also for Yb in
the presence of Li (filled squares). The curves are fits with a rate
equations-based model.

The time evolution of $T_{{\rm Li}}$ and $T_{{\rm Yb}}$ are modeled
considering the energy deposition from processes (I) and (III) as well as
heating from the density-dependent loss processes (II), (IV) and (V) Weber et
al. (2003). In addition, our model also takes into account the effects of
evaporative cooling O’Hara and Thomas (2001), inter-species thermalization
Ivanov et al. (2011), one-body losses from background gas collisions, and Yb
three-body losses Takasu et al. (2003); Weber et al. (2003). The Li scattering
length at 709 G is $a=1860\,a_{0}$, ensuring rapid thermalization ($<1\,$ms)
in the lithium atom-Feshbach molecule mixture Petrov (2003). This allows the
assumption of equal temperature $T_{{\rm Li}}$ for lithium atoms and Feshbach
molecules. The heating from molecule-formation at $709\,$G dominates over
inter-species thermalization, maintaining $T_{\rm Li}\simeq 4.5\,\mu$K and
$T_{\rm Yb}\simeq 2\,\mu$K, as observed in both experiment and model.

The best-fit rate coefficients extracted from the atom data (shown in the
insets) are $L_{3}=(1.4\pm 0.3)\times 10^{-24}\,{\rm cm}^{6}/$s,
$L_{2}=(1.3\pm 0.3)\times 10^{-13}\,{\rm cm}^{3}/$s, and
$L^{\prime}_{2}=(2.3\pm 0.2)\times 10^{-13}\,{\rm cm}^{3}/$s. $L^{\prime}_{3}$
is consistent with 0. All reported uncertainties are statistical. The $L_{3}$
value is consistent with that obtained in Jochim et al. (2003), after
accounting for the slight differences in experimental parameters. Using
$L^{\prime}_{2}\langle n_{{\rm Yb}}\rangle$ as a measure of the dimer decay
rate, we get $170\,$ms as the lifetime of a Li Feshbach molecule in the Yb
bath.

We now turn to the unitary regime, where we choose 810 G ($ka=+6$,
$\epsilon_{B}=k_{B}\times 0.11\,\mu$K) as our representative field to study
the mixture properties. It is difficult to reliably observe the molecule
number using our earlier method in this regime, so we only monitor the atoms
(see Fig. 3). Starting with an inter-species temperature differential as
before, we observe a fast drop in $T_{{\rm Li}}$ in the presence of Yb and
clear evidence of inter-species thermalization. The Li number in the three-
component mixture exhibits a long $1/e$ lifetime of $2\,$s, far larger than at
$709\,$G. However this is still an order of magnitude shorter than that
obtained in the absence of Yb. The interpretation of the decay is not
straightforward as both two-body (process (IV)) and three-body (process (V))
inelastic loss can contribute Du et al. (2009); Spiegelhalder et al. (2009).
The large rate for process (I) in this regime ensures equilibrium molecule
fraction at all times. By fitting to data taken at $935\,$G where $ka=-2$ and
process (V) is expected to dominate inelastic loss, we obtain
$L_{3}^{d}=(4.3\pm 0.3)\times 10^{-28}\,{\rm cm}^{6}/$s. This sets a lower
bound for $L_{3}^{d}$ at $810\,$G. We fit the first $2.5\,$s of data in Fig. 3
after fixing $L^{\prime}_{2}$ to its value scaled from $709\,$G and find
$L_{3}^{d}=(9.5\pm 0.5)\times 10^{-28}\,{\rm cm}^{6}/$s at $810\,$G. The
slight disagreement in Li atom number at long times may be due to a small
($<10\%$) inequality in our spin mixture composition, which the model does not
take into account.

The qualitative features of both spectra in Fig. 1 can be theoretically
reproduced by using field-dependent reaction coefficients scaled from our
measured values at $709$ and $810\,$G. However, a full quantitative comparison
will need to take into account the theoretical deviations from scaling
behavior in the unitary regime as well as experimental variations in the
initial temperature, and is open to future investigation.

By extending the forced evaporative cooling step, lower temperature mixtures
can be produced where bosonic 174Yb shrinks to a size smaller than the Fermi
diameter of the 6Li cloud. Such experiments at $834\,$G yield $T_{{\rm
Li}}/T_{F}\simeq 0.25$ with $N_{{\rm Yb}}=N_{{\rm Li}}=2.5\times 10^{4}$.
Here, the estimated volume of the Yb sample is $\simeq 0.3$ of the Li sample
volume, compared to 3.3 in the classical regime. The mixture is thus also
capable of achieving the opposite regime of a second species being immersed
inside a strongly interacting quantum degenerate Fermi gas, similar to earlier
studies in the K-Li mixture Spiegelhalder et al. (2009).

Our experiments with the Yb-Li mixture near a Feshbach resonance demonstrate
effects of an additional species on chemical reaction rates in the microKelvin
regime. We observe a long lifetime for Feshbach molecules, even in the absence
of Pauli blocking. Our demonstrated stability of the mixture near the unitary
regime of the resonance opens various possibilities of studying strongly
interacting fermions immersed in a bath species or being interrogated by a
small probe species. Future experimental opportunities include realizations of
non-equilibrium states, and studies of superfluid properties, for instance by
controlled relative motion between the two species. Finally, our results
constitute an advance in the manipulation of ultracold mixtures of alkali and
alkaline-earth-like atoms Hara et al. (2011); Hansen et al. (2011); Baumer et
al. (2011). These mixtures may be used to produce quantum gases of
paramagnetic polar molecules which are promising for diverse applications such
as quantum simulation of lattice spin models Micheli et al. (2006), tests of
fundamental symmetries Hudson et al. (2011), and probes of time variations in
fundamental constants Kajita et al. (2011).

We thank Ben Plotkin-Swing for experimental assistance, and J.P. D’Incao,
T.-L. Ho and M.W. Zwierlein for helpful discussions. We gratefully acknowledge
support from the National Science Foundation and the Air Force Office of
Scientific Research. A.K. thanks the NSERC.

## References

  * Chin et al. (2010) C. Chin, R. Grimm, P. S. Julienne, and E. Tiesinga, Rev. Mod. Phys. 82, 1225 (2010).
  * Giorgini et al. (2008) S. Giorgini, L. P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 80, 1215 (2008).
  * Jochim et al. (2003) S. Jochim, M. Barternstein, A. Altmeyer, G. Hendl, C. Chin, J. Hecker Deschlag, and R. Grimm, Phys. Rev. Lett. 91, 240402 (2003).
  * Cubizolles et al. (2012) J. Cubizolles, T. Bourdel, S. J. J. M. F. Kokkelmans, and C. Salomon, Phys. Rev. Lett. 108, 043201 (2012).
  * Esry et al. (2001) B. D. Esry, C. H. Greene, and H. Suno, Phys. Rev. A. 65, 010705(R) (2001).
  * Petrov et al. (2004) D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. Lett. 93, 090404 (2004).
  * Spiegelhalder et al. (2009) F. Spiegelhalder, A. Trenkwalder, D. Naik, G. Hendl, F. Schreck, and R. Grimm, Phys. Rev. Lett. 103, 223203 (2009).
  * Targonska and Sacha (2010) K. Targonska and K. Sacha, Phys. Rev. A 82, 033601 (2010).
  * Robertson and Galitski (2009) A. Robertson and V. M. Galitski, Phys Rev A 80, 063609 (2009).
  * Nascimbène et al. (2011) S. Nascimbène, N. Navon, K. J. Jiang, F. Chevy, and C. Salomon, Nature 463, 1057 (2010).
  * D’Incao and Esry (2008) J. P. D’Incao and B. D. Esry, Phys. Rev. Lett. 100, 163201 (2008).
  * Zirbel et al. (2008) J. J. Zirbel, K.-K. Ni, S. Ospelkaus, J. P. D’Incao, C. E. Wieman, J. Ye, and D. S. Jin, Phys. Rev. Lett. 100, 143201 (2008).
  * Deh et al. (2008) B. Deh, C. Marzok, C. Zimmermann, and P. W. Courteille, Phys Rev A 77, 010701(R) (2008).
  * Dieckmann et al. (2002) K. Dieckmann, C. A. Stan, S. Gupta, Z. Hadzibabic, C. H. Schunck, and W. Ketterle, Phys. Rev. Lett. 89, 203201 (2002).
  * Ivanov et al. (2011) V. V. Ivanov, A. Khramov, A. H. Hansen, W. H. Dowd, F. M$\ddot{\rm u}$nchow, A. O. Jamison, and S. Gupta, Phys. Rev. Lett. 106, 153201 (2011).
  * Hansen et al. (2011) A. H. Hansen, A. Khramov, W. H. Dowd, A. O. Jamison, V. V. Ivanov, and S. Gupta, Phys. Rev. A. 84, 011606(R) (2011).
  * odt (2012) The optical trap wavelength $=1064\,$nm, initial power $=10\,$W per beam, waist $=26\,\mu$m, crossing angle $=20^{\circ}$. During the evaporative cooling step, the power is reduced by a factor of $16$ over $3\,$s. At the highest optical trap powers, $\bar{\omega}_{\rm Yb}/\bar{\omega}_{\rm Li}=0.12$ and $U_{\rm Yb}/U_{\rm Li}=0.45$. At the lower final power, gravity reduces the effective trap depth of the heavier component more, but has neglible effect on the frequency ratio.
  * bec (2012) By continuing the forced evaporation to lower trap depths, our apparatus can produce pure Yb BECs with $>2\times 10^{5}$ atoms in single-species experiments and deeply Fermi degenerate gases $T_{\rm Li}/T_{F}\simeq 0.1$ in two-species experiments.
  * ybb (2012) This step is carried out either immediately before or after the field ramp, with no discernible difference in lithium number or temperature between the two.
  * Chin and Grimm (2004) C. Chin and R. Grimm, Phys. Rev. A 69, 033612 (2004).
  * Kokkelmans et al. (2004) S. J. J. M. F. Kokkelmans, G. V. Shlyapnikov, and C. Salomon, Phys. Rev. A. 69, 031602(R) (2004).
  * Zhang and Ho (2011) S. Zhang and T.-L. Ho, New J. Phys. 13, 055003 (2011).
  * processiv (2012) Since we observe and model process (IV) by its loss signature, we cannot distinguish it from the analogous process resulting in a vibrationally relaxed LiYb molecule, $\mathrm{Li_{2}^{\mathnormal{s}}+Yb}\rightarrow\mathrm{LiYb+Li\hskip 2.84526pt(+\epsilon_{D})}$. Thus $L^{\prime}_{2}$ is the combined rate constant for both processes.
  * threebody (2012) We obtain an upper limit on the rate of processes involving one Li atom and two Yb atoms from lifetime of mixtures of one spin state of Li with Yb at a magnetic field near the Li-Li Feshbach resonance. We obtain an upper limit on the rate of direct three body decay of Li from lifetimes of Li cloud without Yb taken on the BCS side of the resonance.
  * Takasu et al. (2003) Y. Takasu, K. Maki, K. Komori, T. Takano, K. Honda, M. Kumakura, T. Yabuzaki, and Y. Takahashi, Phys. Rev. Lett. 91, 040404 (2003).
  * Petrov (2003) D. S. Petrov, Phys. Rev. A. 67, 010703(R) (2003).
  * D’Incao and Esry (2005) J. P. D’Incao and B. D. Esry, Phys. Rev. Lett. 94, 213201 (2005).
  * Weber et al. (2003) T. Weber, J. Herbig, M. Mark, H.-C. Nägerl, and R. Grimm, Phys. Rev. Lett. 91, 123201 (2003).
  * O’Hara and Thomas (2001) K. M. O’Hara and J. E. Thomas, Phys. Rev. A. 64, 051403(R) (2001).
  * Du et al. (2009) X. Du, Y. Zhang, and J. E. Thomas, Phys. Rev. Lett. 102, 250402 (2009).
  * Hara et al. (2011) H. Hara, Y. Takasu, Y. Yamaoka, J. M. Doyle, and Y. Takahashi, Phys. Rev. Lett. 106, 205304 (2011).
  * Baumer et al. (2011) F. Baumer, F. Münchow, A. Görlitz, S. E. Maxwell, P. S. Julienne, and E. Tiesinga, Phys. Rev. A 83, 040702(R) (2011).
  * Micheli et al. (2006) A. Micheli, G. K. Brennen, and P. Zoller, Nature Physics 2, 341 (2006).
  * Hudson et al. (2011) J. J. Hudson, D. M. Kara, I. J. Smallman, B. E. Sauer, M. R. Tarbutt, and E. A. Hinds, Nature 473, 493 (2011).
  * Kajita et al. (2011) M. Kajita, G. Gopakumar, M. Abe, and M. Hada, Phys. Rev. A 84, 022507 (2011).