Two-photon photoassociative spectroscopy of ultracold ${}^{88}\mathrm{Sr}$

We present results from two-photon photoassociative spectroscopy of the least-bound vibrational level of the $X{}^1\Sigma _g^{+}$ state of the ${}^{88}\mathrm{Sr}_2$ dimer. Measurement of the binding energy allows us to determine the $s$-wave scattering length $a_{88}=-1.4\left(6\right)a_0$. For the intermediate state, we use a bound level on the metastable ${}^{1}S_0\text{−}{}^{3}P_1$ potential, which provides large Franck-Condon transition factors and narrow one-photon photoassociative lines that are advantageous for observing quantum-optical effects such as Autler-Townes resonance splittings.

Figure 1

Two-photon PAS diagram. The energy of two well-separated ${}^{1}S_0$ atoms at rest is taken as zero. $E_g$ is the kinetic energy of the colliding atom pair. $E_{b_1}$ is the unperturbed energy of the bound state of the excited molecular potential that is near resonance with the free-bound laser. $E_{b_2}$ $(<0)$ is the unperturbed energy of the bound state of the ground molecular potential. The photon of energy $h{f}_1$ is detuned from $E_{b1}$ by $h{\delta }_1$, while the photon of energy $h{f}_2$ is detuned from $E_{b2}$ by $h{\delta }_2$. The decay rate of $b_1$ is ${\gamma }_1$. Stark shifts of the levels due to trapping laser fields are neglected in this schematic.

Figure 2

(Color online) Atomic Sr energy levels involved in the two-photon PAS experiments. Decay rates $\left({\mathrm{s}}^{-1}\right)$ and excitation wavelengths are given for selected transitions. Laser light used for the experiment is indicated by solid lines. Atoms decaying to the ${}^{3}P_2$ level may be repumped by $3\mu \mathrm{m}$ light.

Figure 3

(Color online) Photoassociation lasers. The master laser that provides light for the intercombination-line MOT is frequency stabilized via saturated absorption spectroscopy to the atomic transition, and it also provides the photoassociation lasers. The one-photon PAS beam, with frequency $f_1$, is generated directly from the master with an AOM. The two-photon PAS beam, with frequency $f_2$, is formed by injection-locking a slave diode with a double-passed deflected beam from an AOM in a cat’s eye configuration.

Figure 4

(Color online) Left: Atom number versus free-bound laser detuning from the one-photon ${}^{1}S_0\text{−}{}^{3}P_1$ atomic transition. Spectra shown here are for 6 and $13\mathrm{W}$ ODT single-beam powers with sample temperatures of 6 and $13\mu \mathrm{K}$, respectively. Right: Collision-event rate constant $K_{\mathrm{eff}}$ derived from the atom loss. The ODT at $1064\mathrm{nm}$ causes an ac Stark shift of the excited molecular state compared to the ground state, which shifts and broadens the line. The solid lines are fits using Eqs. (8, 9). A peak shift of $480\mathrm{kHz}$ is measured for a single-beam power of $13\mathrm{W}$. The dashed line marks the position of our measured unperturbed resonance frequency at $-222.25\left(15\right)\mathrm{MHz}$, which is in reasonable agreement with a previous measurement of $-222.161\left(35\right)\mathrm{MHz}$ [21].

Figure 5

(Color online) Area under one-photon PAS spectra versus free-bound laser intensity $I_1$. The area can be related to molecular and experimental parameters to determine the stimulated linewidth ${\gamma }_s\left(E_g\right)$ of the PAS transition due to $I_1$. For low free-bound laser intensities the area is independent of temperature and linearly dependent on $I_1$.

Figure 6

(Color online) Collision-event rate constant $K_{\mathrm{eff}}$ versus free-bound laser detuning from ${}^{1}S_0\text{−}{}^{3}P_1$ atomic resonance for different bound-bound laser intensities. These Autler-Townes doublets are measured with the bound-bound laser frequency fixed such that ${\delta }_2\approx 0$ while scanning $f_1$. The splitting of the spectra is given by the Rabi frequency ${\Omega }_{12}∕2\pi$ and varies as $\sqrt{I_2}$ (shown by the lines; spectra offset is proportional to $\sqrt{I_2}$), where the bound-bound laser intensity $I_2$ is indicated in the legend. The asymmetry in the line shapes arises from the bound-bound laser frequency being slightly off resonance from the bound-bound transition. The free-bound intensity $I_1$ is constant for all four spectra at $0.05\mathrm{W}∕{\mathrm{cm}}^2$. The sample temperature is $8\mu \mathrm{K}$.

Figure 7

(Color online) Collision-event rate constant $K_{\mathrm{eff}}$ versus frequency difference between free-bound and bound-bound lasers for spectroscopy of the $J=0$, $v=62$ level of the $X{}^1\Sigma _g^{+}$ potential. The free-bound laser frequency is fixed close to the one-photon PAS resonance and the intensity is $I_1=0.05\mathrm{W}∕{\mathrm{cm}}^2$. The bound-bound laser frequency is scanned, and its intensity is indicated in the legend. On two-photon resonance, PAS loss is suppressed due to quantum interference. The solid lines are fits using Eqs. (8, 3), which yield $E_{b2}∕h=-136.7\left(2\right)\mathrm{MHz}$. The atom temperature is $8\mu \mathrm{K}$.

Figure 8

(Color online) PAS suppression spectra for the $J=2$, $v=62$ level of the ground molecular potential, as described in Fig. 7. The sample temperature is $9\mu \mathrm{K}$. The free-bound laser intensity is $I_1=0.04\mathrm{W}∕{\mathrm{cm}}^2$, and the bound-bound intensity is indicated in the legend. The solid line is a fit using Eqs. (8, 3), which yields $E_{b2}∕h=-66.6\left(2\right)\mathrm{MHz}$.

Figure 9

(Color online) Dependence of elastic-scattering cross sections on collision energy $\left(E\right)$ for selected Sr isotopes. The thick lines are cross sections including partial waves up to $l=4$. Shape resonances are indicated. Thin lines indicate cross section contributions from $l=0$ only. For the plot, a potential is used that gives $a_{88-88}=-1.2a_0$ at $E=0$. The cross sections are given by the usual expressions $\sigma =(8\pi ∕k^2){\sum }_{l=0,2,\dots }^{\infty }(2l+1){\sin }^2{\delta }_l\left(k\right)\stackrel{k\to 0}{\to }8\pi a^2$ for identical bosons and $\sigma =(4\pi ∕k^2){\sum }_{l=0}^{\infty }(2l+1){\sin }^2{\delta }_l\left(k\right)\stackrel{k\to 0}{\to }4\pi a^2$ for distinguishable atoms. The phase shift, ${\delta }_{l=0}\left(k\right)$, depends on $k=\sqrt{2\mu E}∕\hslash$ and is related to the scattering length $a$ and effective range $r_e$ at low $k$ by $k\cot {\delta }_{l=0}\left(k\right)=-\frac{1}{a}+\frac{1}{2}{r}_e{k}^2$. The data symbols are cross section measurements from thermalization experiments [48], and the respective collision energies are set to $E=k_{B}T$, where $T$ is the sample temperature.