# Photoassociative Spectroscopy of 87Sr

J.C. Hill  Department of Physics and Astronomy, Rice University, Houston, TX
77005, USA  Applied Physics Graduate Program, Smalley-Curl Institute, Rice
University, Houston, TX 77005, USA  W. Huie  P. Lunia  J. D. Whalen  S. K.
Kanungo  Y. Lu  Department of Physics and Astronomy, Rice University, Houston,
TX 77005, USA  T.C. Killian<EMAIL_ADDRESS>Department of Physics and
Astronomy, Rice University, Houston, TX 77005, USA  Applied Physics Graduate
Program, Smalley-Curl Institute, Rice University, Houston, TX 77005, USA

###### Abstract

We demonstrate photoassociation (PA) of ultracold fermionic 87Sr atoms. The
binding energies of a series of molecular states on the ${}^{1}\Sigma^{+}_{u}$
$5s^{2}\,{}^{1}$S${}_{0}+5s5p\,^{1}$P1 molecular potential are fit with the
semiclassical LeRoy-Bernstein model, and PA resonance strengths are compared
to predictions based on the known 1S${}_{0}+^{1}$S0 ground state potential.
Similar measurements and analysis were performed for the bosonic isotopes 84Sr
and 86Sr, allowing a combined analysis of the long-range portion of the
excited-state potential and determination of the $5s5p\,^{1}$P1 atomic state
lifetime of $5.20\pm 0.02$ ns. The results enable prediction of PA rates
across a wide range of experimental conditions.

Ultracold gases of the various isotopes of strontium are currently studied for
a wide range of applications, such as frequency metrology [1, 2, 3], quantum
simulation of many-body physics [4, 5], quantum information [6, 7], Rydberg
physics [8, 9], and cold collisions [10, 11, 12, 13]. The fermionic isotope
(87Sr) in particular attracts significant attention because it is used in
optical atomic clocks [14, 1] and the ground-state tenfold degeneracy arising
from the large nuclear spin ($I=9/2$) introduces novel magnetic phenomena [4].
For many of these experiments, ultracold samples are trapped in optical
lattices in order to prevent atom-atom interactions or introduce spatial
periodicity for quantum simulation of materials. Photoassociative (PA)
spectroscopy [15, 16], which is the optical formation of bound molecules from
the initial state of two colliding atoms, is a well-established and useful
technique for probing ultracold gases in optical lattices. It can be used to
detect double occupancy of lattice sites [17], which provides a measurement of
sample temperature and probe of quantum phase transitions [18]. This technique
has not been utilized for 87Sr, however, and no study on PA in this isotope
has been published. PA formation of molecules bound in the
$5s^{2}\,{}^{1}$S0+$5s5p\,^{3}$P1 molecular potential in 87Sr was mentioned in
[19], and this is more challenging than in bosonic isotopes [20, 21, 22]
because of the large nuclear spin and hyperfine splitting of the molecular
states.

Here, we report photoassociation in 87Sr to states on the
${}^{1}\Sigma^{+}_{u}$ $5s^{2}\,{}^{1}$S${}_{0}+5s5p\,^{1}$P1 molecular
potential, to the red of the principal transition at $\lambda=460.85$ nm.
Hyperfine splitting in the excited state is small ($\sim 60$ MHz) and
unresolved, producing a simple spectrum. We report line strengths in terms of
PA-collision-event rate constants for transitions across a wide range of
binding energies and provide parameters for a fit of the binding energies to
the semi-classical Leroy-Bernstein formula [23]. This can inform design of
experiments using PA spectroscopy as a probe of 87Sr in optical lattices. We
also report Leroy-Bernstein parameters describing molecular binding energies
for 84Sr and 86Sr. The extracted value of the $C_{3}$ coefficient and
associated 1P1 atomic lifetime are compared with previous results from PA
spectroscopy of 88Sr [24, 25] and 86Sr [26] and measurements of AC Stark
shifts [7]. PA-collision-event rate constants are compared to predictions [27]
based on the known ${}^{1}\Sigma^{+}_{g}$
$5s^{2}\,{}^{1}$S0+$5s^{2}\,{}^{1}$S0 potential [10, 28].

Using methods described in [29], ultracold strontium atoms are trapped in an
optical dipole trap ($1064$ nm wavelength) consisting of two crossed
elliptical beams propagating perpendicular to gravity with tight axis along
gravity. Atoms are initially loaded into the trap and then a short stage of
evaporation produces the sample used for PA spectroscopy. For 87Sr,
spectroscopy is performed in a trap with oscillation frequencies
($f_{x},f_{y},f_{z}$)=($68,85,433$) Hz. At the start of PA laser exposure,the
number of atoms is $N=2.3\times 10^{6}$, the temperature is $T=1.8\,\mu$K, and
the peak density is $n_{0}=2.1\times 10^{13}$ cm-3. The sample has
approximately equal population of the $10$ nuclear spin states [30]. For 84Sr,
these parameters are ($f_{x},f_{y},f_{z}$)=($81,30,406$) Hz, $N=4.5\times
10^{6}$, $T=1.9\,\mu$K, and $n_{0}=1.7\times 10^{13}$ cm-3. For 86Sr, these
parameters are ($f_{x},f_{y},f_{z}$)=($61,76,388$) Hz, $N=1.2\times 10^{6}$,
$T=1.8\,\mu$K, and $n_{0}=1.1\times 10^{13}$ cm-3. The PA laser intensity
($3.6-237$ mW/cm2) and exposure time ($10-1000$ ms) are varied depending upon
the sample and strength of the PA transition. Typical peak atom loss due to PA
is 10-50%. The sample temperature varies by no more than 25% for measurements
of the PA rate constant. The PA laser beam has $e^{-2}$ radii of
$w_{\textrm{horz}}=850\,\mu$m and $w_{\textrm{vert}}=440\,\mu$m on the atoms,
and it is treated as homogeneous over the sample.

Figure 1: Representative atom-loss spectra for excitation to molecular states
on the ${}^{1}\Sigma^{+}_{u}$ $5s^{2}\,{}^{1}$S${}_{0}+5s5p\,^{1}$P1 molecular
potential in 87Sr. Background atom number is normalized to one. PA laser
exposure times and intensities are adjusted to give comparable atom loss for
all spectra, so line intensity does not reflect the PA-collision-event rate
constant. Exposure times and intensities vary from 75-200ms and 3.6-11.5
mW/cm2 respectively. Inset is a representative fit to a Lorentzian.

After exposure, the PA laser and the dipole trap are extinguished, and the
atom number and sample temperature are measured with resonant time-of-flight
absorption imaging on the 461 nm transition. Photoassociation is indicated by
atom loss. Figure 1 shows representative atom-loss spectra for 87Sr. The PA
laser is locked to a wavemeter (Moglabs FZW600) that is calibrated against the
atomic $5s^{2}\,{}^{1}$S${}_{0}\rightarrow 5s5p\,^{1}$P1 transition, with a
frequency accuracy of $\sim\pm 30$ MHz.

The PA resonances, labeled by vibrational index $\nu$, are fit to a Lorentzian
lineshape. Typical linewidths range between $60-150$ MHz. This is consistent
with the lower limit given by twice the atomic linewidth
$\gamma_{atomic}=30.24$ MHz [25]. The dominant source of additional line
broadening is laser frequency jitter. Binding energies (defined as
$E_{\nu}>0$) are determined by taking the difference between PA resonance
positions and the wavenumber corresponding to excitation of the
$5s^{2}\,{}^{1}$S${}_{0}\rightarrow 5s5p\,^{1}$P1 atomic resonance.

The long-range form of the excited-state ${}^{1}\Sigma^{+}_{u}$ 1S0+1P1
molecular potential can be approximated as

$V_{e}(r)=D-\frac{C_{3}}{r^{3}}+\frac{\hbar^{2}[J(J+1)+2]}{2\mu\,r^{2}},\quad
C_{3}=\frac{3\hbar\lambda^{3}}{16\pi^{3}\tau}$ (1)

where D is the dissociation energy, $\mu$ the reduced mass, $r$ is the
internuclear separation, and $\tau=1/(2\pi\gamma_{atomic})$ is the lifetime of
the 1P1 atomic state. Because of the ultracold temperature, only $s$-wave
collisions occur and only $J=1$ molecular rotational states are excited. The
rotational energy is small and can be neglected, as can thermal energy of the
initial collisional state. The binding energies, $E_{\nu}$, can be fitted to
the semi-classical Leroy-Bernstein formula [23].

$E_{\nu}=[(\nu-\nu_{D})H_{3}]^{6},\quad
H_{3}=\frac{1}{C_{3}^{\frac{1}{3}}}\frac{\hbar\Gamma\left(\frac{4}{3}\right)}{2\Gamma\left(\frac{5}{6}\right)}\sqrt{\frac{2\pi}{\mu}}$
(2)

where $\nu=1$ corresponds to the least-bound state, $\Gamma$ is the gamma
function, and $\nu_{D}$ is a fit parameter ranging from 0 to 1 that takes on a
different value for each isotope. We perform a combined fit of all data to Eq.
2 using a common value of $C_{3}$ as a fit parameter and independent values of
$\nu_{D}$ for each isotope. The quantum numbers $\nu$ assigned to each level
are shifted to obtain $0<\nu_{D}<1$.

Figure 2 shows binding energies for 87Sr and residuals for all isotopes. Table
1 shows the fit parameters, including the value of $\tau$ extracted from the
$C_{3}$ coefficient. The statistical uncertainty in $\tau$ from this procedure
is very small, but the fit residuals for all three isotopes show a systematic
trend corresponding to $\sim 100$ MHz variation over a change of $\sim 100$
GHz in binding energy. This trend might indicate systematic wavemeter error,
variation in molecular-state AC Stark shifts from the optical dipole trap
laser fields, or the influence of additional terms in the molecular potential
not accounted for by the Leroy-Bernstein formula [31]. Adding an additional
fit parameter in the form of a binding energy offset removes the systematic
trends in the residuals and increases $\tau$ by $0.02$ ns or 0.4%, which we
take as our uncertainty. We thus quote a final value of $\tau=5.20\pm 0.02$
ns. The value of $\tau$ determined here differs by 1% from the most accurate
reported measurement of $\tau=5.263\pm 0.004$ ns, performed with
photoassociation in an optical lattice [25], and is in closer agreement with a
more recent value of $\tau=5.234\pm 0.008$ ns determined from AC Stark shifts
of Sr levels [7]. The binding energies for all observed transitions are
included in the supplementary material.

Figure 2: Fit of the binding energies of states on the ${}^{1}\Sigma^{+}_{u}$ $5s^{2}\,{}^{1}$S${}_{0}+5s5p\,^{1}$P1 molecular potential for various strontium isotopes. Plots (a-c) show the difference ($\delta$) between observed and fit values for the binding energies for 84Sr, 86Sr, and 87Sr respectively, using parameters from Tab. 1. Plot (d) shows the measured binding energies as a function of vibrational quantum number, $\nu$, for 87Sr. The dashed line is the best fit of the data to Eq. 2.  Isotope |  87Sr |  86Sr |  84Sr
---|---|---|---
$\nu_{D}$ | 0.87 $\pm$ 0.02 | 0.20 $\pm$ 0.02 | 0.87 $\pm$ 0.02
$\tau$ (ns) | 5.20 $\pm$ 0.02
Table 1: Parameters from a fit of binding energy for 84Sr, 86Sr, and 87Sr to
Eq.2.

The loss of atoms due to photoassociation is described by a local equation for
the evolution of the atomic density

$\dot{n}=-\beta(I,f)n^{2}-\Gamma_{1}n,$ (3)

where $\Gamma_{1}$ describes one-body loss due to light scattering on the
atomic transition or background gas collisions, and the two-body loss is
described by $\beta$, which depends on PA laser intensity, $I$, and frequency,
$f$. Assuming constant sample temperature, the number of atoms in the trap as
a function of the PA exposure time $t$ is given by

$N(t)={N_{0}\rm{e}^{-\Gamma_{1}t}\over 1+{n_{0}\beta\over
2\sqrt{2}\Gamma_{1}}(1-\rm{e}^{-\Gamma_{1}t})},$ (4)

where $N_{0}$ is the number at the beginning of the PA interaction time and
$n_{0}$ is the initial density. The density can be calculated from the number
and sample temperature, approximating the trap as an infinitely deep harmonic
potential with oscillation frequencies matching the measured frequencies of
the optical trap. Eq. 3. has been integrated over volume to yield the
evolution of sample number, which is solved by Eq. 4.

Near resonance with the transition to a molecular state with vibration quantum
number $\nu$ and center frequency $f_{\nu}$, the PA loss is described with a
Lorentzian lineshape [26],

$\beta=\frac{2K_{\nu}\gamma_{mol}}{\gamma}\frac{1}{1+4(f-f_{\nu})^{2}/\gamma^{2}},$
(5)

where $\gamma_{mol}=2\gamma_{atomic}$ is the natural linewidth of the PA
transition due to radiative decay of the molecular state and $\gamma$ is the
observed linewidth of the transition. $K_{\nu}$, which is proportional to
laser intensity $I$, is the resonant collision-event rate constant that would
be observed in the absence of any broadening beyond the natural linewidth.
Thermal broadening is much smaller than $\gamma_{mol}$ and the laser intensity
is low enough that saturation effects are negligible.

Figure 3: Atom loss spectra for PA to the $\nu=105$ state of the
${}^{1}\Sigma^{+}_{u}$ $5s^{2}\,{}^{1}$S${}_{0}+5s5p\,^{1}$P1 molecular
potential for 86Sr. PA laser exposure time is indicated in the legend. A
combined fit of all data to Eqs. 4 and 5 determines the the peak PA collision-
event rate constant $K_{\nu}$. The PA laser intensity is 3.6 mW/cm2 and the
initial density is $n_{0}=1.3\times 10^{13}$ cm-3. The fit result is
$K_{\nu}/I=2.8\times 10^{13}$ cm5/(s$\cdot$mW).

Figure 3 shows a fit of typical atom-loss spectra to Eqs. 4 and 5 for
different PA laser exposure times for 86Sr. All fit parameters are well
determined, but systematic uncertainty in laser intensity, atom density, and
sample temperature lead to systematic uncertainty of about a factor of 3 in
the fundamental quantity for comparison with theory, $K_{\nu}/I$.

$K_{\nu}/I$ is proportional to $F^{\nu}_{eg}(E,\Delta_{\nu})$, the free-bound
Franck-Condon factor for excitation to the excited state $\nu$ from the ground
${}^{1}\Sigma^{+}_{g}$ $5s^{2}\,{}^{1}$S0+$5s^{2}\,{}^{1}$S0 potential, where
$E$ is the initial collision energy, and $\Delta_{\nu}<0$ is the detuning from
atomic resonance. Through the relation

$h\Delta_{\nu}=V_{e}(r^{\nu}_{C})-D-V_{g}(r^{\nu}_{C})$ (6)

the detuning defines the Condon radius, $r^{\nu}_{C}$, which is the
internuclear separation at which the photon energy (via the Plank constant
$h$) is resonant with the difference in molecular potentials. $r^{\nu}_{C}$
can be interpreted as the classical separation at which excitation occurs. At
long range and ultracold temperatures, the ground potential can be
approximated by $V_{g}(r)=-{C_{6}}/{r^{6}}-{C_{8}}/{r^{8}}-{C_{10}}/{r^{10}}$.

Using the reflection approximation [27], the Franck-Condon factor can be
related to the energy-normalized ground-state wave function at the Condon
point, $\Psi_{g}(r^{\nu}_{C},E)$ through

$F^{\nu}_{eg}(E,\Delta_{\nu})=\frac{\partial
E_{\nu}}{\partial\nu}\frac{1}{d_{c}}|\Psi_{g}(r^{\nu}_{C},E)|^{2}$ (7)

where ${\partial E_{\nu}}/{\partial\nu}$ is the spacing between adjacent
vibrational levels in the excited state at level $\nu$, which can be found
from Eq. 2. Also,

$d_{c}=\left|\frac{d}{dr}\left[V_{e}(r)-V_{g}(r)\right]\right|_{r=r^{\nu}_{C}}$
(8)

is the difference in slopes of the excited and ground potentials.

We calculate ground-state wavefunctions for a collision energy of $2$ $\mu$K
by numerically integrating the Schrödinger equation [32]. We use the full
ground-state potential from [28] in the form from [33] with recommended values
of $C_{6}=1.525\times 10^{7}\,\mathrm{cm}^{-1}{\mathrm{\AA}}^{6}$,
$C_{8}=5.159\times 10^{8}\mathrm{cm}^{-1}{\mathrm{\AA}}^{8}$, and
$C_{10}=1.91\times 10^{10}\,\mathrm{cm}^{-1}{\mathrm{\AA}}^{10}$.

Figure 4:  Intensity-normalized resonant collision-event rate constant
($K_{\nu}$) for strontium PA transitions. Solid lines are predictions based on
Franck-Condon factors derived from numerically calculated ground state wave
functions (Eq. 7) as described in the text.

Figure 4 shows the measured values of $K_{\nu}/I$ and theoretically expected
values based on the calculated wavefunctions. A common scaling factor is
applied to theoretical predictions to account for the proportionality between
$K_{\nu}/I$ and the Franck-Condon factors. Predicted rates for 87Sr are
further reduced with respect to the bosons by the ratio of the pair
correlation functions $g^{(2)}_{f}/g^{(2)}_{b}$, where $g^{(2)}_{b}=2$
reflects bunching for bosons and $g^{(2)}_{f}=0.9$ reflects Pauli exclusion
for a gas of identical fermions in ten equally populated internal states [30].
Measurements for 86Sr agree within a factor of two with previously reported
values [26], which is reasonable given systematic uncertainties.

The rate coefficients go to zero when the Condon radius for the transition is
near a node of the ground-state wave function (Fig. 4). For 87Sr and 84Sr, the
node interrogated by measurements reported here corresponds to internuclear
spacing equal to the $s$-wave scattering lengths, $a_{84}=123$ $a_{0}$ or
$a_{87}=96$ $a_{0}$ [10], where $a_{0}$ is the Bohr radius. For 86Sr,
$a_{86}=823$ $a_{0}$ is a larger lengthscale than probed here, and the node
corresponding to $\Delta=-500$ GHz is the second pre-asymptotic node [26].

For 87Sr and 84Sr, the two isotopes most often used for quantum-gas research
[1, 2, 5, 4, 9], the PA rate constants are relatively small at convenient
detunings for experiments. A useful figure of merit is the ratio of the
number-loss-rate for PA of two atoms in a single lattice site ($\dot{N}_{PA}$)
to the off-resonant, single-atom, photon-scattering rate ($R\approx 2\pi
s_{0}\gamma^{3}_{atomic}/\Delta^{2}$). The ratio of laser intensity to the
atomic transition’s saturation intensity ($I_{sat}=40\,$mW/cm2) is indicated
by $s_{0}$.

For two 87Sr atoms in a single site of an optical lattice, each of mass $m$,
$\dot{N}_{PA}\approx 2K_{\nu}\int\,d^{3}r\,n^{2}(r)=2K_{\nu}/(2\pi
a_{HO}^{2})^{3/2}$, where we have assumed the atoms are both in the ground
state of a single site in a deep optical lattice in different internal spin
states [34]. $a_{HO}=\sqrt{\hbar/m\omega}$ is the harmonic oscillator length
for $\omega=\sqrt{4V_{0}E_{R}}/\hbar$ and lattice depth $V_{0}$.
$E_{R}=2\pi^{2}\hbar^{2}/(m\lambda_{lat}^{2})$ is the recoil energy for
lattice laser wavelength $\lambda_{lat}$. For $\lambda_{lat}=1064$ nm and
$V_{0}=16E_{R}$, $1/(2\pi a_{HO}^{2})^{3/2}=1\times 10^{14}$ cm-3. For PA at
small detuning ($\Delta=-10$ GHz), $\dot{N}_{PA}\approx 3\times R$. For
detuning beyond the wave-function node ($\Delta=-300$ GHz),
$\dot{N}_{PA}\approx 13\times R$, which is more favorable. At $\Delta=-300$
GHz, a PA laser intensity of 20$I_{sat}$ yields $\dot{N}_{PA}\approx
10/\mathrm{s}$.

In summary, we have measured and characterized photoassociation resonances up
to $\sim 314$ GHz red detuned from the atomic asymptote of the
${}^{1}\Sigma^{+}_{u}$ $5s^{2}\,{}^{1}$S${}_{0}+5s5p\,^{1}$P1 molecular
potential in fermionic 87Sr. Similar measurements were made in bosonic 86Sr
and 84Sr, and a combined fit to the semiclassical LeRoy-Bernstein model
allowed determination of the 1P1 atomic state lifetime and other spectroscopic
parameters. Resonance intensities were compared with predictions from a
reflection approximation and the ground-state wavefunction calculated with the
best available ground-state potential. We find that, to within experimental
uncertainties, the resonance frequencies and intensities are reasonably well-
described by theoretical predictions. This work will enable accurate
prediction of photoassociative transition frequencies and rates for
experiments with the strontium isotopes most commonly used in quantum gas
experiments, including experiments with fermionic 87Sr.

## Acknowledgments

This work was supported by the National Science Foundation (PHY-1607665),
Welch Foundation C-1844 and the AFOSR (FA9550-14-1-0007).

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