Atomic Bose-Einstein condensate to molecular Bose-Einstein condensate transition

The starting point of our experiment is a BEC of $6\times 10^{4}$ cesium atoms prepared in a disk-shaped dipole trap with a radius of 18 $\mu$m in the $x-y$ horizontal plane. The disk-shaped potential is provided by a digital micromirror device (DMD) which projects 788 nm blue-detuned laser light on the plane with $1~{}\mu$m resolution. The atoms are loaded into a single layer of the optical lattice in the vertical direction with trap frequency $\omega_{z}/2\pi=400$ Hz CLthesis . A magnetic field gradient of 31 G/cm is applied to levitate the atoms and the magnetic field is tuned to 19.9 G.

We create the molecules with the $g-$wave narrow Feshbach resonance located at 19.87 G based on a procedure similar to Ref. Herbig2003 . Since atoms and molecules have different magnetic moments, they tend to separate vertically in the presence of a magnetic field gradient. To better confine both atoms and molecules in the molecular formation phase, we increase the magnetic field gradient to 42.5 G/cm in 2 ms before ramping the magnetic field to 19.76 G in 2 ms which creates molecules. After the formation of molecules, the magnetic field gradient is increased to 50 G/cm in 0.5 ms, which levitates the molecules and overlevitates the atoms.

To remove residual atoms after the molecular formation phase, a resonant light pulse of 20 $\mu$s illuminates and pushes atoms away from the imaging area in 4 ms. Molecules are detected by reversely ramping the magnetic field which dissociates the molecules back to atoms, and the atoms are detected by absorption imaging. The final value of the magnetic field and the hold time are selected to give a reliable image that reflects the distribution of the molecules. In our experiments, we set the final magnetic field to be 20.44 G and do the detection in 0.1 ms after the reverse ramp. We estimate that the atoms expand by 1 $\mu$m during the dissociation process, which is comparable with the imaging resolution of our experimental system.

The strong magnetic field gradient for levitating the molecules leads to an additional magnetic anti-trapping potential on the horizontal plane. We also apply a central potential barrier projected from a DMD to measure the density response of the molecules. A precise knowledge of both the magnetic anti-trapping potential and the optical potential barrier are needed in order to extract the equation of state of the molecular gas.

We load atomic BEC into the same trap as for molecules to calibrate the external potential. Since the magnetic moment and polarizability of the g-wave molecule are accurately known, the trapping potential for molecules can thus be obtained from the trapping potential for atoms.

The magnetic anti-trap frequency in the horizontal plane is given by

$\displaystyle\omega_{i}^{2}=\frac{\mu_{m}}{4mB_{0}}(B^{\prime 2}-4\epsilon_{i}B_{0}^{2}),$

(S1)

where $i=x,y$, $\mu_{m}$ is magnetic moment, $B_{0}$ and $B^{\prime}$ are magnetic field and magnetic field gradient, respectively, at the location of particles and $\epsilon_{i}$ is determined from the coil geometry CLthesis . We determine the offset field value $B_{0}$ with an accuracy of 2 mG using microwave spectroscopy. We prepare atomic BEC at 17.2 G where the $s-$wave scattering length $a_{\mathrm{S}}=4~{}a_{0}$. Because of the low chemical potential, the atomic density distribution is sensitive to the magnetic anti-trap and shows lower density at the center and higher density in the rim, see Fig. S1a. Since the vertical trap frequency $\omega_{z}/2\pi=400$ Hz is much larger than the chemical potential $\mu_{0}/h\approx 10$ Hz, the BEC is in the quasi-2D regime and the column density under Thomas-Fermi approximation is given by

$\displaystyle n(x,y)=\frac{m}{\hbar^{2}g_{\mathrm{2D}}}[\mu_{0}-V_{\mathrm{mag}}(x,y)],$

(S2)

where the 2D coupling strength $g_{\mathrm{2D}}=\sqrt{8\pi}a_{\mathrm{S}}/\sqrt{\hbar/m\omega_{z}}$, the magnetic anti-trap potential $V_{\mathrm{mag}}(x,y)$ is parametrized by the trap frequencies $\omega_{x}$ and $\omega_{y}$ as $V_{\mathrm{mag}}(x,y)=m\omega_{x}^{2}(x-x_{0})^{2}/2+m\omega_{y}^{2}(y-y_{0})^{2}/2$ and $(x_{0},y_{0})$ is the center position of the anti-trap. To determine the trap frequencies and the global chemical potential, we fit the in situ atomic density distribution using Eq. S2, see Fig. S1a. From the fit we get $\omega_{x}/2\pi=1.94(9)$ Hz, $\omega_{y}/2\pi=2.24(9)$ Hz and $\mu_{0}=h\times 9.19(7)$ Hz. In this way, we calibrate the geometric parameters to be $\epsilon_{x}=0.54(3)~{}\mathrm{cm}^{-2}$ and $\epsilon_{y}=0.45(3)~{}\mathrm{cm}^{-2}$, which we use to calculate the anti-trap frequencies for molecules based on Eq. S1, which gives $\omega_{x}^{mol}/2\pi=3.35(4)$ Hz and $\omega_{y}^{mol}/2\pi=3.48(4)$ Hz. For consistency check, we plot out the atomic density $n_{\mathrm{A}}$ versus the local chemical potential $\mu=\mu_{0}-V_{\mathrm{mag}}(x,y)$, which agrees with the equation of state of a pure 2D BEC $\mu=(\hbar^{2}g_{\mathrm{2D}}/m)n(x,y)$ (see Fig. S1b).

We calibrate the optical potential barrier projected by DMD using atomic BEC prepared at 19.2 G, where the atomic scattering length is $a_{\mathrm{S}}=127$ $a_{0}$ and the vertical trap frequency is $\omega_{z}/2\pi=409$ Hz. The intensity of the optical barrier is ramped up within 10 ms. After waiting for another 2 ms, absorption imaging is performed in the vertical direction to record the atomic column density, see Fig. S2a. Here the barrier height is controlled by the fraction of micromirrors $f_{\mathrm{DMD}}$ that are turned on. The fraction determines the intensity of the light projected onto the atom plane. In the region with higher light intensity, the atomic density is suppressed more, which in turn allows us to determine the light intensity. Because of the higher chemical potential of BEC in this case, the density depletion has a larger dynamical range that helps to calibrate larger range of barrier height. Since the chemical potential is comparable to the vertical trap frequency, the BEC is in 3D regime and the column density under Thomas-Fermi approximation is given by

$\displaystyle n(x,y)=[\alpha(\mu_{0}-V_{\mathrm{opt}}(x,y))]^{3/2}$

(S3)

where $\alpha=(4\sqrt{2}/3g\sqrt{m}\omega_{z})^{2/3}$, the 3D coupling strength $g=4\pi\hbar^{2}a_{\mathrm{S}}/m$ and the local optical potential $V_{\mathrm{opt}}(x,y)$ is proportional to the micromirror fraction as $V_{\mathrm{opt}}(x,y)=p(x,y)f_{\mathrm{DMD}}$. Thus for each pixel located at (x,y), we have $n^{2/3}(x,y)=\alpha[\mu_{0}-p(x,y)f_{\mathrm{DMD}}]$, from which the proportionality $p(x,y)$ can be extracted from a series of measurements with different $f_{\mathrm{DMD}}$, see Fig. S2b. Repeating the same procedure for all the pixels within the region of optical barrier, we can map out the spatial dependence of the proportionality $p(x,y)$, see Fig. S2c. The polarizability of weakly bound molecules is approximately twice as large as that of a free atom, thus the corresponding proportionality for the molecules is $2p(x,y)$.

After calibrating both the magnetic potential $V_{\mathrm{mag}}(x,y)$ and the optical potential $V_{\mathrm{opt}}(x,y)$ for molecules, we can get the local molecular density as a function of the total external potential $V(x,y)=V_{\mathrm{mag}}(x,y)+V_{\mathrm{opt}}(x,y)$ and follow the fitting procedure in Sec. III to extract the global chemical potential $\mu_{0}$. Then we obtain the corresponding local chemical potential $\mu$ and average the density in a certain spatial area with a proper range of local chemical potential to get the equation of state for molecules from the density profile and optical barrier measurements in Fig. 2. In addition, with the knowledge of the optical potential profile, we get the equation of state for the BEC in 3D regime, see the inset of Fig. 2c.

For a nondegenerate 2D ideal Bose gas, the phase space density is given by $n_{\phi}=-\ln(1-\zeta)$, where $\zeta=\exp(\beta\mu)$ is the fugacity, $\beta=1/k_{\mathrm{B}}T$ and $\mu=\mu_{0}-V(x,y)$ is the local chemical potential. If the gas is interacting, a mean field potential $2(\hbar^{2}g_{\mathrm{2D}}/2m)n(x,y)$ is added to the external potential based on the Hartree-Fock approximation Dalibard2011 . Then the equation of state for interacting 2D Bose gas becomes:

$\displaystyle n(x,y)=-\frac{1}{\lambda_{\mathrm{dB}}^{2}}\ln[1-e^{\beta\mu-g_{\mathrm{2D}}n(x,y)\lambda_{\mathrm{dB}}^{2}/\pi}],$

(S4)

which we use for fitting the data from the optical barrier measurement in Fig. 2c for density $n_{\mathrm{M}}<1~{}\mathrm{\mu m}^{-2}$. On the other hand, the density of 2D superfluid outside the fluctuation region is Svistunov2001 :

$\displaystyle n(x,y)=\frac{2\pi\beta}{g_{\mathrm{2D}}\lambda_{\mathrm{dB}}^{2}}\mu+\frac{1}{\lambda_{\mathrm{dB}}^{2}}\ln[2n(x,y)\lambda_{\mathrm{dB}}^{2}g_{\mathrm{2D}}/\pi-2\beta\mu],$

(S5)

which we use for fitting the data from the density profile measurement in Fig. 2c for density $n_{\mathrm{M}}>4~{}\mathrm{\mu m}^{-2}$.

We perform a global fit to the data points within the range $n_{\mathrm{M}}<1~{}\mathrm{\mu m}^{-2}$ and $n_{\mathrm{M}}>4~{}\mathrm{\mu m}^{-2}$ using Eq. S4 and Eq. S5, respectively, with temperature T, global chemical potential $\mu_{0}$ and 2D coupling constant $g_{\mathrm{2D}}$ as fitting parameters. Since the experimental condition drifted, the global chemical potential between the optical barrier and density profile measurements are different, and the chemical potential difference $\delta\mu$ is also set as an free parameter in the global fit. The fit gives $T=11(1)$ nK, $g_{\mathrm{2D}}=0.19(3)$ and the global chemical potential for the optical barrier and density profile measurements as $h\times 45(7)$ Hz and $h\times 61(7)$ Hz, respectively. We also performed independent fits to the data at low density $n_{\mathrm{M}}<1~{}\mathrm{\mu m}^{-2}$ and high density $n_{\mathrm{M}}>4~{}\mathrm{\mu m}^{-2}$. The resulting temperatures are $10(3)$ nK and $11(1)$ nK, in agreement with each other and with the global fit.

With the extracted 2D coupling constant $g_{\mathrm{2D}}$, the critical phase density for BKT superfluid transition is evaluated as $\ln(\xi/g_{\mathrm{2D}})\approx 7.5$, where the coefficient $\xi=380(3)$ Svistunov2001 . On the other hand, the BEC transition in our 2D box potential occurs at critical phase space density of $\ln(4\pi R^{2}/\lambda_{\mathrm{dB}}^{2})\approx 7.5$ Dalibard2011 , which coincides with the BKT transition.

For BECs in 3D regime as shown in the inset of Fig. 2c, the low density part where the column density $n_{\mathrm{A}}<10~{}\mathrm{\mu m}^{-2}$ is fitted using the classical gas formula $n(x,y)=(2\pi l_{z}^{2}/\lambda_{\mathrm{dB}}^{4})\exp(\beta\mu)$, where the harmonic oscillator length $l_{z}=\sqrt{\hbar/m\omega_{z}}$. The high density part is fitted based on Eq. S3.

In order to study the lifetime of g-wave molecules, we hold the molecules in different traps and monitor the decay of particle number as a function of the hold time. The two traps we used have horizontal radius $R_{1}=12.5~{}\mathrm{\mu m}$, $R_{2}=9~{}\mathrm{\mu m}$ and vertical trap frequency $\omega_{z1}/2\pi=400$ Hz, $\omega_{z2}/2\pi=167$ Hz, respectively. The molecular density distribution in these traps are approximately uniform in the horizontal direction and Gaussian in the vertical direction, given by

$\displaystyle n(\vec{r})=\frac{N_{\mathrm{M}}}{\pi^{3/2}R_{i}^{2}l_{zi}}e^{-z^{2}/l_{zi}^{2}}\theta(R_{i}-\rho),$

(S6)

where $i$ = 1,2, $\rho=\sqrt{x^{2}+y^{2}}$ and $\theta(x)$ is the heaviside step function.

Even though the 1064 nm light intensity in the vertical direction of the two traps differ by a factor of $\omega^{2}_{z1}/\omega^{2}_{z2}\approx 6$, the decay rate of molecular number are similar, see Fig. 3a. This suggests that the one-body loss process due to the off-resonant laser light is negligible. In fact, since the g-wave molecules are in a highly excited rovibrational state, two-body loss process dominates which is modelled by $\partial_{t}n(\vec{r},t)=-L_{2}n^{2}(\vec{r},t)$. The molecular number decay corresponding to the density profile in Eq. S6 is thus given by

$\displaystyle N_{\mathrm{M}}(t)=\frac{N_{\mathrm{M}}(0)}{1+L^{\prime}_{2}N_{\mathrm{M}}(0)t},$

(S7)

where $L^{\prime}_{2}=L_{2}/\sqrt{2}\pi^{3/2}R_{i}^{2}l_{zi}$. We use Eq S7 to fit the data of molecular number decay and extract the inelastic loss coefficient $L_{2}$ in Fig. 3b.

The unitarity limit of the two-body loss coefficient is $U_{2}(k)=4h/2mk$, where $k$ is the magnitude of the relative wavevector $\vec{k}$ between two colliding molecules, associated with the relative kinetic energy $E=\hbar^{2}\vec{k}^{2}/2m$ Chin2010 . Due to the finite temperature in our experiment, the relative kinetic energy obeys Bolzmann distribution as $p(E)=A\exp(-E/k_{\mathrm{B}}T)$, where the coefficient $A=(1/4)(\hbar^{2}/\pi mk_{\mathrm{B}}T)^{3/2}$. The distrbution of the wavenumber $k$ is then given by

$\displaystyle p(k)=4\pi Ak^{2}e^{-\hbar^{2}k^{2}/2mk_{\mathrm{B}}T}.$

(S8)

The unitarity limit that we evaluate in Fig. 3b is $U_{2}=\int_{0}^{\infty}U_{2}(k)p(k)dk=(4h/2m)\langle k^{-1}\rangle$, where the thermal average of $k^{-1}$ with respect to the distribution $p(k)$ is $\langle k^{-1}\rangle=\sqrt{\hbar^{2}/\pi mk_{\mathrm{B}}T}$. For comparison with the loss coefficients, we evaluate the interaction scale as $\mu_{0}/\hbar n_{\mathrm{3D}}$, where the 3D mean density $n_{\mathrm{3D}}=\int_{-\infty}^{+\infty}n^{2}(\vec{r})d^{3}\vec{r}/\int_{-\infty}^{+\infty}n(\vec{r})d^{3}\vec{r}=N_{\mathrm{M}}/\sqrt{2}\pi^{3/2}R_{1}^{2}l_{z1}^{2}$.

After preparing a pure molecular BEC below the Feshbach resonance, if the magnetic field is then switched to a value high above the resonance, the molecules quickly dissociate into a continuum of free atoms. The dissociation rate follows Fermi’s golden rule as $\Gamma=(2\pi/\hbar)|V_{\mathrm{MA}}|^{2}\rho(E)=2m^{1/2}a_{\mathrm{bg}}\Delta\mu\Delta BE^{1/2}/\hbar^{2}$, where $V_{\mathrm{MA}}$ is the coupling matrix element between molecular and atomic states and is independent of the energy E above the continuum to leading order, the density of state $\rho(E)\propto E^{1/2}$ and $a_{\mathrm{bg}}$ is the background scattering length. In this high field limit, our measured dissociation rate is consistent with Fermi’s golden rule $\gamma=\alpha\Gamma$, where the coefficient $\alpha=0.4(1)$. The dissociation rate in Fig. 4b is extracted by fitting the data in Fig. 4a using the formula $N_{\mathrm{M}}(t)=N_{\mathrm{M}}(t_{0})\{1-\exp[-\gamma(t-t_{0})]\}\theta(t-t_{0})$, where $t_{0}$ is the time when the molecules start to dissociate.

On the other hand, when the magnetic field is ramped to near the resonance where $\rho(E)\approx 0$, we still observe a finite dissociation rate of $8~{}\mathrm{ms}^{-1}$. This is because the molecular state can couple to a band of scattering states that are Lorentzian distributed Fano1961 . We thus define an effective density of state $\rho_{\mathrm{eff}}$ to be a convolution between $\rho(E)$ and a Lorentzian distribution. Thus the effective dissociation rate becomes

$\displaystyle\Gamma_{\mathrm{eff}}=\Gamma\sqrt{(\sqrt{1+\Omega^{2}/4E^{2}}+1)/2},$

(S9)

where $\Omega$ is the full width of the Lorentzian distribution. We use Eq. S9 to fit the dissociation rate as a function of the magnetic field we measured, shown as the blue solid line in the upper panel of Fig. 4b.

The dissociated molecular fraction drops when the magnetic field is ramped back closer to the resonance, which we attribute to the inelastic collision loss between atoms and molecules near the resonance. The data of the fraction in Fig. 4b is fitted using an empirical function $f=N_{s}\{1/2+(1/\pi)\arctan[\Delta\mu(B-B_{0})/V_{s}]\}$, where $N_{s}$ and $V_{s}$ are set as free parameters.