The frequency of the ultranarrow ${^1\text{S}_0} - {^3\text{P}_2}$ transition in $^{87}\text{Sr}$

TL;DR

This paper precisely measures the ultranarrow ${^1 ext{S}_0} - {^3 ext{P}_2}$ transition frequency in $^{87} ext{Sr}$ using iodine reference spectroscopy, enabling its application in quantum simulation and computation.

Contribution

The study provides the first precise measurement of the $^{87} ext{Sr}$ transition frequency with 250 kHz uncertainty, referencing it to molecular iodine lines.

Findings

Transition frequency is 446648775(30) MHz.

Measurement uncertainty is 250 kHz.

Results facilitate quantum simulation applications.

Abstract

We determine the frequency of the ultranarrow $^{87}\text{Sr}$ ${^{1}\text{S}_{0}} - {^{3}\text{P}_{2}}$ transition by spectroscopy of an ultracold gas. This transition is referenced to four molecular iodine lines that are observed by Doppler-free saturation spectroscopy of hot iodine vapor. The frequency differences between the Sr and the I$_2$ transitions are measured with an uncertainty of 250 kHz. The absolute frequency of the $^{87}\text{Sr}$ ${^{1}\text{S}_{0}} - {^{3}\text{P}_{2}}$ ($\text{F}'=7/2$) transition is 446648775(30) MHz and limited in accuracy by the iodine reference. This work prepares the use of the Sr ${^{1}\text{S}_{0}} - {^{3}\text{P}_{2}}$ transition for quantum simulation and computation.

Tables

Table 1. Table 1: Frequencies and measured linewidths of the S 0 1 − P 2 3 superscript subscript S 0 1 superscript subscript P 2 3 {{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}} transition to three different hyperfine states in the P 2 3 superscript subscript P 2 3 {{}^{3}\text{P}_{2}} manifold. The frequencies are determined using a wavemeter, whereas the full-width half-maximum (FWHM) linewidths are obtained from Gaussian fits to the spectroscopy signals shown in Fig.  4 .

${\text{F}}^{\prime }$	frequency	FWHM linewidth
	($\frac{\mathrm{MHz}}{}$)	($\frac{\mathrm{MHz}}{}$)
$7/2$	$\mathrm{446\hspace{0.17em}648\hspace{0.17em}769}\left(30\right)\frac{}{}$	0.52(4)
$9/2$	$\mathrm{446\hspace{0.17em}647\hspace{0.17em}793}\left(30\right)\frac{}{}$	0.74(3)
$11/2$	$\mathrm{446\hspace{0.17em}646\hspace{0.17em}618}\left(30\right)\frac{}{}$	0.69(8)

Table 2. Table 2: Measured frequencies of the iodine transitions {a8, a9, a10, a11} within the manifold P(33) (9-9) and the S 0 1 − P 2 3 superscript subscript S 0 1 superscript subscript P 2 3 {}^{1}\text{S}_{0}-{{}^{3}\text{P}_{2}} ( F ′ = 7 / 2 superscript F ′ 7 2 \text{F}^{\prime}=7/2 ) transition frequency. The frequencies are reported with respect to the measured frequency of the a8 transition. The iodine transition frequencies are given for our conditions of the iodine cell (see text) and are shifted by -100(15) kHz with respect to iodine lines at zero pressure and temperature. The error of the Sr transition is dominated by drifts of the reference resonator.

transition	frequency
	($\frac{\mathrm{MHz}}{}$)
a8	$0.00\left(3\right)\frac{}{}$
a9	$25.88\left(3\right)\frac{}{}$
a10	$34.17\left(5\right)\frac{}{}$
a11	$48.39\left(4\right)\frac{}{}$
${}_{}{}^1\text{S}_0^{}-{}_{}{}^3\text{P}_2^{}$ (${\text{F}}^{\prime }=7/2$)	$93.27\left(25\right)\frac{}{}$

Full text

I Introduction

Atoms with two valence electrons, such as the alkaline-earth metals or ytterbium, possess ultra-narrow intercombination transitions from their singlet ground state to metastable triplet states. The ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{0}}$ transition, which connects two states that are free of electronic magnetic moment, is used as frequency reference in optical atomic clocks Ludlow et al. (2015) and is of interest for quantum simulation Martin et al. (2013); Livi et al. (2016); Bromley et al. (2018); Cooper and Rey (2015); Covey et al. (2018), computation Daley et al. (2008); Shibata et al. (2009); Gorshkov et al. (2009), and gravitational wave detection Yu and Tinto (2011); Graham et al. (2013); Kolkowitz et al. (2016). The ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition is equally narrow Porsev and Derevianko (2004), but it connects the ground state to an excited state with electronic magnetic moment. This property has enabled high-resolution imaging of an Yb quantum gas in a magnetic field gradient Kato et al. (2012); Shibata et al. (2014), a method that could also provide selective access to qubits in a quantum computer Daley et al. (2008); Shibata et al. (2009). Isotopes with nuclear spin exhibit hyperfine structure in the ${{}^{3}\text{P}_{2}}$ state, which will make it possible to induce nuclear spin state specific AC Stark shifts and Raman couplings using the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition. This property might allow the creation of artificial gauge fields that are significantly less hampered by off-resonant scattering of photons or collisions between metastable state atoms compared to schemes exploiting broader transitions Goldman et al. (2014); Lin et al. (2009); Mancini et al. (2015); Song et al. (2018); Leroux et al. (2018) or using metastable atoms Gerbier and Dalibard (2010); Cooper and Rey (2015); Livi et al. (2016). Ultracold mixtures containing ${{}^{3}\text{P}_{2}}$ atoms have been obtained from quantum gases of ground state atoms by excitation on the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition, leading to the discovery of Feshbach resonances between Yb ${{}^{1}\text{S}_{0}}$ and ${{}^{3}\text{P}_{2}}$ atoms Kato et al. (2013). These resonances are interesting for quantum information processing Daley et al. (2008); Shibata et al. (2009); Gorshkov et al. (2009), are predicted to show signatures of quantum chaos Green et al. (2016) and have been exploited to form Feshbach molecules Taie et al. (2016); Takasu et al. (2017). Also mixtures of ${{}^{3}\text{P}_{2}}$ Yb with Li have been created Khramov et al. (2014); Hara et al. (2014) and their collisional stability investigated Dowd et al. (2015); Konishi et al. (2016); González-Martínez and Hutson (2013); Petrov et al. (2015); Chen et al. (2015). The ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition might also be useful to induce optical Feshbach resonances Fedichev et al. (1996); Ciuryło et al. (2005); Enomoto et al. (2008), or to create quantum gases with quadrupole interactions Derevianko (2001); Santra et al. (2004); Bhongale et al. (2013); Lahrz et al. (2014); Huang et al. (2014).

Many of these applications require quantum degenerate gases and so far three two-valence-electron elements have been cooled to quantum degeneracy: Yb Takasu et al. (2003), Ca Kraft et al. (2009), and Sr Stellmer et al. (2009). The frequency of the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition is only well known for Yb. Strontium has properties that significantly distinguish it from Yb, offering different opportunities. It enables higher phase space densities directly by laser cooling, which makes it possible to create quantum gases with large atom number or with high repetition rate Stellmer et al. (2013). Its fermionic isotope 87Sr has a nuclear spin of 9/2, which should enable better Pomeranchuck cooling Taie et al. (2012); Ozawa et al. (2018) or larger synthetic dimensions Boada et al. (2012). In order to combine these favorable properties with the possibilities offered by the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition, the frequency of this transition needs to be determined to at least the MHz level.

In this article we report the measurement of the ultra-narrow ${}^{87}\text{Sr}$ ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition by direct optical excitation. We perform loss spectroscopy of an ultracold strontium sample and determine the resonance frequency by comparison to four spectral lines of molecular iodine, which serves as a natural and documented reference. The iodine lines are identified by comparing a $\frac{\mathrm{GHz}}{}$-wide iodine spectrum around the Sr lines with the spectra calculated by the IodineSpec5 software Iod . The accuracy of the measurement is limited by the uncertainty in iodine transition frequencies, whereas the precision is limited by frequency drifts of an optical resonator used for spectroscopy laser stabilization. The relative frequency between the Sr transition and specific iodine lines is obtained with an accuracy of $250\text{\,}\frac{\mathrm{kHz}}{}$ and the absolute frequency is limited by the iodine line accuracy of $30\text{\,}\frac{\mathrm{MHz}}{}$. These measurements open the door to using the Sr ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition for important applications, such as the creation of artificial gauge fields or quantum computation.

This article has the following structure: Sec. II describes the spectroscopy laser system, the iodine spectroscopy setup, and Sr sample preparation; Sec. III introduces relevant Sr transitions, presents initial coarse and final precise determination of the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition frequency and analyses the measurement error. Conclusions are given in Sec. IV.

II Experimental details

II.1 ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ spectroscopy laser setup

Light for the spectroscopy of Sr and iodine is produced by an external cavity diode laser (ECDL; wavelength: 671 nm; power: $24\text{\,}\frac{\mathrm{mW}}{}$; diode: Toptica LD-0670-0035-AR-1), see Fig. 1. The ECDL is locked to a resonator (free spectral range $1\text{\,}\frac{\mathrm{GHz}}{}$, length tunable by piezo) by the Pound-Drever-Hall (PDH) method Black (2001). A frequency shift of $550\text{\,}\frac{\mathrm{MHz}}{}$ to $850\text{\,}\frac{\mathrm{MHz}}{}$ is introduced by an AOM between the ECDL and the light used for locking. Spectroscopy scans are performed by slowly varying the AOM frequency so that the lock follows. The feedback loop uses a fast PID controller (Toptica FALC 110), providing feedback of $1.9\text{\,}\frac{\mathrm{MHz}}{}$ bandwidth to the ECDL current and $10\text{\,}\frac{\mathrm{kHz}}{}$ bandwidth to the ECDL grating. Based on the error signal, we estimate the laser linewidth to be at most $85\text{\,}\frac{\mathrm{kHz}}{}$. The light is sent through polarization-maintaining single-mode optical fibers to the Sr sample and to the iodine spectroscopy setup. The absolute frequency of the spectroscopy laser can be obtained with a wavemeter (Toptica HighFinesse WSU-30, accuracy of 30 MHz) calibrated to the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{1}}$ transition frequency of ${}^{88}\text{Sr}$, known to within $10\text{\,}\frac{\mathrm{kHz}}{}$ Ferrari et al. (2003).

II.2 Iodine spectroscopy setup

Spectra of iodine molecule vapor (natural sample, essentially 100% ${}^{127}\text{I}_{2}$ de Laeter et al. (2003)) contained in a heated quartz cell are recorded using Doppler-free saturated absorption spectroscopy Hänsch et al. (1971); Huang et al. (2013). We will now briefly describe the $\text{I}_{2}$ spectroscopy setup, see Fig 1. The quartz cell is $60\text{\,}\frac{\mathrm{cm}}{}$ long and kept at approximately $530\text{\,}\frac{\mathrm{\SIUnitSymbolCelsius}}{}$ (not stabilized by feedback) in order to populate the higher vibrational levels of the iodine molecule Huang et al. (2013). A cold finger, stabilized at $20.0(3)\text{\,}\frac{\mathrm{\SIUnitSymbolCelsius}}{}$, is used to set the iodine partial pressure. The collimated spectrocopy beam entering the setup (waist $0.7\text{\,}\frac{\mathrm{mm}}{}$, power $5\text{\,}\frac{\mathrm{mW}}{}$) is split into a pump and a probe beam. The probe beam (power $0.5\text{\,}\frac{\mathrm{mW}}{}$) is sent through the cell onto a photodiode that records the spectroscopy signal. The pump beam is frequency shifted by acousto-optic modulator AOM${}_{\text{I}}$, after which it has approximately $3\text{\,}\frac{\mathrm{mW}}{}$ of power, and gets sent through the iodine cell in a counterpropagating manner with respect to the probe End (a). We enhance the weak Doppler-free signal by lock-in detection. We use AOM${}_{\text{I}}$ to chop the pump beam at $50\text{\,}\frac{\mathrm{kHz}}{}$ (square wave) and we demodulate the detected probe signal at that frequency on a lock-in amplifier (EG & G Instruments Model 7265) using a time constant of $100\text{\,}\frac{\mathrm{ms}}{}$. A few things are worth mentioning about this approach: first of all, the lock-in method is necessary, because the bare Lamb dips in the Doppler spectrum are too weak to be seen directly with the available power; secondly, the method is quite forgiving in terms of the chopping frequency, and in particular, the chopping frequency can be increased if one wants to reduce the time constant of the lock-in for faster scans, at the expense of signal-to-noise End (b); thirdly, the method is forgiving to slight misalignment in the overlap of the pump and probe beams, imperfect collimation of the beams, and slight power fluctuations of pump and probe.

II.3 Strontium sample preparation and spectroscopy principle

Spectroscopy of the Sr ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition is done on an ultracold cloud of 87Sr in an equal mixture of all nuclear spin states contained in an optical dipole trap (ODT). The ODT consists of two horizontally propagating, linearly-polarized $1064\text{\,}\frac{\mathrm{nm}}{}$ beams crossing at right angles and having waists of approximately $60\text{\,}\frac{\mathrm{\SIUnitSymbolMicro m}}{}$ and $95\text{\,}\frac{\mathrm{\SIUnitSymbolMicro m}}{}$ and powers of $2.2\text{\,}\frac{\mathrm{W}}{}$ and $1.2\text{\,}\frac{\mathrm{W}}{}$; the beams have a $160\text{\,}\frac{\mathrm{MHz}}{}$ frequency difference in order to avoid mutual interference. To prepare the sample, a magneto-optical trap (MOT) is loaded from a Zeeman-slowed atomic beam and then transferred into the ODT using the techniques described in Ref. Stellmer et al. (2013). We obtain a cloud of $2\times 10^{5}$ Sr atoms at $360\text{\,}\frac{\mathrm{nK}}{}$, which has a $1/e$-width of \sim$$18\text{\,}\frac{\mathrm{\SIUnitSymbolMicro m}}{} in the vertical direction and \sim$$27\text{\,}\frac{\mathrm{\SIUnitSymbolMicro m}}{} in the horizontal direction. We reduce the residual magnetic field to less than $30\text{\,}\frac{\mathrm{mG}}{}$ at the location of the atomic cloud. The Sr spectroscopy beam is focused to a waist of about $60\text{\,}\frac{\mathrm{\SIUnitSymbolMicro m}}{}$ at the sample position. Spectroscopy is performed time sequentially and measures frequency dependent loss of ground state atoms. A sample is prepared, exposed to spectroscopy light, which leads to atom loss, and the remaining ground-state atom number is detected by absorption imaging on the ${{}^{1}\text{S}_{0}}-{{}^{1}\text{P}_{1}}$ transition.

III Determination of the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition frequency

Strontium levels and transitions that are relevant for this work are shown in Fig. 2. The transitions ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{0,2}}$ are dipole forbidden in isotopes with pure spin-orbit (LS) coupling because of spin and total angular momentum selection rules Boyd et al. (2007). A small dipole matrix element can be induced by mixing of the ${{}^{3}\text{P}_{0,2}}$ states with ${{}^{1}\text{P}_{1}}$ through the application of a magnetic field or through hyperfine coupling in the case of ${}^{87}\text{Sr}$, the only stable Sr isotope with nuclear spin. For the bosonic ${{}^{88}\text{Sr}}$ the observation of the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{0}}$ clock transition has been reported with an external mixing field as low as $13\text{\,}\frac{\mathrm{G}}{}$ Taichenachev et al. (2006); Poli et al. (2009). Most Sr optical lattice clocks use fermionic Sr in order to exploit hyperfine mixing to enable the clock transition.

In this work we use fermionic ${}^{87}\text{Sr}$, which allows dipole transitions between ${{}^{1}\text{S}_{0}}\leavevmode\nobreak\ (\text{F}=9/2)$ and ${{}^{3}\text{P}_{2}}\leavevmode\nobreak\ (\text{F}^{\prime}=\{7/2,9/2,11/2\})$ with a linewidth of approximately $1\text{\,}\frac{\mathrm{mHz}}{}$ by hyperfine mixing Porsev and Derevianko (2004). The transitions ${{}^{1}\text{S}_{0}}\leavevmode\nobreak\ (\text{F}=9/2)-{{}^{3}\text{P}_{2}}\leavevmode\nobreak\ (\text{F}^{\prime}=\{5/2,13/2\})$ have $\Delta\text{F}=2$ and the ground and excited states have opposite parity, which makes them dipole forbidden. They are however still accessible as magnetic quadrupole transitions (M2) Yamaguchi (2008) and we observe the ${{}^{1}\text{S}_{0}}\leavevmode\nobreak\ (\text{F}=9/2)-{{}^{3}\text{P}_{2}}\leavevmode\nobreak\ (\text{F}^{\prime}=5/2)$ transition.

We determine the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition frequency in two steps. The first step, described in Sec. III.1, determines the transition indirectly to within \sim$$100\text{\,}\frac{\mathrm{MHz}}{}. This is sufficiently precise to find the transition with direct spectroscopy, see Sec. III.2. In Sec. III.3 we discuss the error of our measurement.

III.1 Coarse, indirect determination

The ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition frequency has only been measured for the most abundant isotope 88Sr with an accuracy of 120 MHz Sansonetti and Nave (2010). The 87Sr ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition frequencies can be estimated by adding the 87Sr ${{}^{3}\text{P}_{2}}$ hyperfine shifts, which have been determined by radiofrequency spectroscopy in hot Sr Heider and Brink (1977), and the isotope shift. Here we assume that the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ isotope shift is the same as the measured ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{0}}$ and ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{1}}$ isotope shifts Bender et al. (1984); Buchinger et al. (1985); Takano et al. (2017), which are both within  $1\text{\,}\frac{\mathrm{MHz}}{}$ of $f_{88}-f_{87}=$ $62\text{\,}\frac{\mathrm{MHz}}{}$. We verify the estimated transition frequency by performing a simple, coarse and indirect frequency determination. We determine f(${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$) using conservation of energy: we measure f(${{}^{3}\text{P}_{2}}-{{}^{3}\text{S}_{1}}$) and use the well-known transition frequencies f(${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{0}}$) and f(${{}^{3}\text{P}_{0}}-{{}^{3}\text{S}_{1}}$) Courtillot et al. (2005) to calculate f(${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$), see Fig. 2. The ${{}^{3}\text{P}_{2}}-{{}^{3}\text{S}_{1}}$ transition is dipole-allowed, which makes it much broader and easier find than the doubly-forbidden mHz-linewidth ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition. Similar schemes were used to determine the Sr ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{0}}$ transition Courtillot et al. (2005) and the Yb ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition Yamaguchi et al. (2010).

To determine $f({{}^{3}\text{P}_{2}}-{{}^{3}\text{S}_{1}})$ we use reservoir spectroscopy Stellmer and Schreck (2014). This technique relies on the fact that atoms in the ${{}^{1}\text{S}_{0}}-{{}^{1}\text{P}_{1}}$ MOT cycle can decay through the ${{}^{1}\text{D}_{2}}$ state into the metastable and magnetic ${{}^{3}\text{P}_{2}}$ state, the low field seeking $m_{\text{F}}$ substates of which are captured in the magnetic quadrupole field of the MOT. These atoms can be pumped back into the MOT cycle with light on a transition from the ${{}^{3}\text{P}_{2}}$ state to some higher-lying state that has a high chance of decaying into the ground state. We use the ${{}^{3}\text{S}_{1}}$ state as the higher-lying state, from which atoms decay to the ground state through the short-lived ${{}^{3}\text{P}_{1}}$ state. The ${{}^{3}\text{P}_{2}}-{{}^{3}\text{S}_{1}}$ transition corresponds to a repump laser operating around $707\text{\,}\frac{\mathrm{nm}}{}$. Thus, when the repump laser is tuned to a resonance, the ${{}^{3}\text{P}_{2}}$ atoms from the magnetically trapped reservoir are quickly brought back into the ${{}^{1}\text{S}_{0}}-{{}^{1}\text{P}_{1}}$ MOT cycle, rapidly increasing the number of ground state atoms and causing a MOT fluorescence flash. There are nine repump resonances due to the number of hyperfine states in both ${{}^{3}\text{P}_{2}}$ and ${{}^{3}\text{S}_{1}}$, but we do not need to measure all of those transitions to determine $f({{}^{3}\text{P}_{2}}-{{}^{3}\text{S}_{1}})$.

We observe three ${{}^{3}\text{P}_{2}}-{{}^{3}\text{S}_{1}}$ repump resonances, which we can attribute to specific transitions between hyperfine states in the ${{}^{3}\text{P}_{2}}$ and ${{}^{3}\text{S}_{1}}$ manifolds using knowledge of the ${{}^{3}\text{P}_{2}}$ hyperfine structure Heider and Brink (1977), knowledge of the ${{}^{3}\text{S}_{1}}$ hyperfine structure Courtillot et al. (2005), and selection rules. The absolute frequencies of these transitions are obtained with the wavemeter that is also part of the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ spectroscopy setup. These measurements, combined with the known transition frequencies $f({{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{0}})$ and $f({{}^{3}\text{P}_{0}}-{{}^{3}\text{S}_{1}})$ Courtillot et al. (2005), provides estimates of the individual transition frequencies $f({{}^{1}\text{S}_{0}}\leavevmode\nobreak\ (\text{F}=9/2)-{{}^{3}\text{P}_{2}}\leavevmode\nobreak\ (\text{F}^{\prime}=\{7/2,9/2,11/2\}))$. This determination has an accuracy of \sim$$100\text{\,}\frac{\mathrm{MHz}}{} and confirms the estimated transition frequencies. Using this good starting point we now expect to find the transitions quickly in a direct spectroscopy search.

III.2 Precise, direct determination

We perform direct spectroscopy of the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transitions using Sr samples in an ODT. When the spectroscopy laser is tuned into resonance, ground-state atoms are excited to the ${{}^{3}\text{P}_{2}}$ state, making them transparent to the ${{}^{1}\text{S}_{0}}-{{}^{1}\text{P}_{1}}$ absorption imaging beam. Return of excited state atoms to the ground state while remaining trapped is improbable since atoms in the ${{}^{3}\text{P}_{2}}$ state are not trapped in the 1064-$\frac{\mathrm{nm}}{}$ ODT and inelastic collisions involving ${{}^{3}\text{P}_{2}}$ atoms also lead to loss from the trap. To find a resonance we measure the fraction of ${{}^{1}\text{S}_{0}}$ atoms remaining in the ODT as a function of spectroscopy laser frequency, while keeping other parameters, such as illumination time and laser power, constant. During the first search for the transition we use the full power of the spectroscopy beam ($4.5\text{\,}\frac{\mathrm{mW}}{}$). We repeatedly prepare Sr samples and use each to scan a $1\text{\,}\frac{\mathrm{MHz}}{}$ frequency interval over $1\text{\,}\frac{\mathrm{s}}{}$. An example for the resulting spectrum is shown in Fig. 3 and determines all ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transitions to within $10\text{\,}\frac{\mathrm{MHz}}{}$, using the known ${{}^{3}\text{P}_{2}}$ hyperfine splittings.

In order to determine the transition frequency with more precision we zoom in on the detected broad spectroscopy signals by recording spectra with $0.45\text{\,}\frac{\mathrm{mW}}{}$ of spectroscopy laser power and reduced frequency interval and illumination time per sample. The narrowest spectroscopy signals for $\text{F}^{\prime}=\{7/2,9/2,11/2\}$ are shown in Fig. 4 together with Gaussian fits, the parameters of which are summarized in Table 1. The values of the transition frequencies measured with the wavemeter are also reported in Table 1, and the uncertainty is dominated by wavemeter inaccuracy.

Next we determine the frequency difference between four iodine transitions and the ${}^{87}\text{Sr}$ ${{}^{1}\text{S}_{0}}\leavevmode\nobreak\ (\text{F}=9/2)-{{}^{3}\text{P}_{2}}\leavevmode\nobreak\ (\text{F}^{\prime}=7/2)$ transition. This allows us to determine the frequency of the Sr transition with the accuracy of the known iodine transition frequencies, which currently is the same as the accuracy of the wavemeter ($30\text{\,}\frac{\mathrm{MHz}}{}$), but can be improved in the future with iodine spectroscopy alone Huang et al. (2013). More importantly the accuracy of the frequency difference ($250\text{\,}\frac{\mathrm{kHz}}{}$) is much higher than the accuracy of the absolute frequency, which makes it possible to find the Sr transition with simple iodine spectroscopy and to lock the Sr laser to iodine lines.

The $\text{F}^{\prime}=7/2$ state is used for iodine comparison because it is within the spectroscopy AOM tuning range of the strong iodine transitions $(J^{\prime}-J^{\prime\prime}=32-33)(\nu^{\prime}-\nu^{\prime\prime}=9-9)$ {a8, a9, a10, a11} End (c). The frequencies corresponding to the other hyperfine states in the ${{}^{3}\text{P}_{2}}$ manifold can be found by using this frequency and the known ${{}^{3}\text{P}_{2}}$ hyperfine splittings Heider and Brink (1977). Figure 5 presents an example of a recorded iodine spectrum, fitted with Lorentzians, combined with the Sr spectrum, fitted with a Gaussian. Table 2 lists the fitted central frequency values of the iodine transitions, where each value is an averaged result from the fits to four measured spectra. Using the a8 frequency calculated by IodineSpec5 we obtain $446\,648\,775(30)\text{\,}\frac{\mathrm{MHz}}{}$ for the ${}^{87}\text{Sr}$ ${{}^{1}\text{S}_{0}}\leavevmode\nobreak\ (\text{F}=9/2)-{{}^{3}\text{P}_{2}}\leavevmode\nobreak\ (\text{F}^{\prime}=7/2)$ transition frequency, which is consistent with the frequency determined by the wavemeter.

The frequency of the ${}^{1}\text{S}_{0}-{{}^{3}\text{P}_{2}}$ ($\text{F}^{\prime}=5/2,13/2$) M2 transitions can be determined from the previous measurement and the ${{}^{3}\text{P}_{2}}$ hyperfine splittings. Guided by this calculation we observe the ${}^{1}\text{S}_{0}-{{}^{3}\text{P}_{2}}$ ($\text{F}^{\prime}=5/2$) transition by direct optical excitation. Since it is an M2 transition it is expected to be much weaker than the HFM-E1 lines, and we indeed must use about 10 times larger intensity and a 20 times longer illumination time to induce observable atom loss on this transition compared to the case of the dipole transitions. Fig. 6 shows a spectrum of this line recorded with a spectroscopy beam power of $4.5\text{\,}\frac{\mathrm{mW}}{}$ and an illumination time of $10\text{\,}\frac{\mathrm{s}}{}$.

Based on our measurement of the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{2}}$ transition and the previously reported results for the ${{}^{1}\text{S}_{0}}-{{}^{3}\text{P}_{1}}$ and ${{}^{3}\text{P}_{1}}-{{}^{3}\text{S}_{1}}$ transition frequencies and hyperfine splittings of all mentioned states Courtillot et al. (2005); Heider and Brink (1977), we can also give a more accurate value for the ${{}^{3}\text{P}_{2}}(F=7/2)-{{}^{3}\text{S}_{1}}(F^{\prime}=7/2)$ repumping transition frequency, which evaluates to $423\,914\,969(30)\text{\,}\frac{\mathrm{MHz}}{}$. The corresponding values involving any other hyperfine states can be easily calculated from the known hyperfine splittings Courtillot et al. (2005); Heider and Brink (1977).

III.3 Error analysis

We measure absolute frequencies in two ways, using the wavemeter or using iodine lines as reference Iod . Both methods have an uncertainty of 30 MHz, which dominates all other sources of error. We also determine the relative frequency between the Sr transitions and the iodine lines. The error in the relative frequency is much smaller than the absolute error and will be discussed in the following.

The error in the relative frequency measurement has statistical and systematic components. Statistical errors arise from the drifts of the Fabry-Perot resonator to which the laser is locked, from errors in the fits used to determine the center of spectral lines, and from iodine line shifts by iodine temperature and pressure changes Dareau et al. (2015); End (d). Systematic errors are the collisional shifts of the iodine lines, which effectively move our frequency reference point from its literature value Dareau et al. (2015), and the AC Stark shift of the Sr transitions by the dipole trap light.

The dominant contribution to the statistical error comes from changes of the cavity resonance to which the spectroscopy laser is locked while the measurements are performed. This error is estimated by recording iodine spectra several times before, during, and after the one-hour timespan during which the Sr spectra are recorded, and then analyzing the drift of the relative frequency between the cavity resonance and the iodine lines. We assume that the iodine lines do not change significantly over this timespan (the validity of that assumption will be analyzed below), therefore the drift is due to changes of the resonator frequency. The maximum drift we observe is $250\text{\,}\frac{\mathrm{kHz}}{}$.

Errors also originate from pressure and temperature shifts of the iodine lines. Compared to a zero temperature and pressure gas the iodine lines are shifted by $\delta f_{I_{2}}=\alpha_{S}P_{\text{I}_{2}}T^{-7/10}$ at pressure $P$ and temperature $T$, where $\alpha_{S}=$-400(60)\text{,}\mathrm{k}\mathrm{Hz}\leavevmode\nobreak\ \mathrm{K}^{7/10}\mathrm{/}\mathrm{Pa} is an empirically determined proportionality constant Dareau *et al.* ([2015](#bib.bib69)). $P_{\text{I}_{2}}$ is set by the cold finger temperature, and the relevant iodine vapor pressure equation is given in Ref. Huang *et al.* ([2013](#bib.bib53)), whereas $T$ is set by the iodine cell body temperature. Statistical errors arise from uncertainties in $P$ and $T$. The cold finger temperature uncertainty of $0.3\text{\,}\frac{\mathrm{K}}{}$ translates into a pressure uncertainty below $1\text{\,}\frac{\mathrm{Pa}}{}$. The body temperature has an uncertainty below $20\text{\,}\frac{\mathrm{K}}{}$. These uncertainties lead to a statistical error of $3\text{\,}\frac{\mathrm{kHz}}{}$ in the iodine line frequency. The systematic shift of the iodine lines is $\delta f_{I_{2}}=$-100(15)\text{\,}\frac{\mathrm{kHz}}{}, where the dominant contribution to the error arises from uncertainties in $\alpha_{S}$. The values reported in Table 2 are given in presence of this shift.

Another source of error is the light shift induced on the Sr transition by the ODT. We obtain an upper limit for this shift by recording spectra using ODT depths up to a factor two higher or lower than the depth used usually. We do not observe a correlation of the transition frequency with the ODT depth, which we attribute to drifts of the reference resonator during the few hours that we spent to record this data. We conclude that the light shifts are insignificant compared to the resonator drifts of $250\text{\,}\frac{\mathrm{kHz}}{}$. The Sr spectroscopy lines are broadened by ODT light shifts, the Doppler effect, Zeeman shifts of the unresolved $m_{F}$ levels, and collisional effects.

IV Conclusion

We have determined the frequency of the ${}^{87}\text{Sr}$ ${}^{1}\text{S}_{0}-{{}^{3}\text{P}_{2}}$ transition with an accuracy of $30\text{\,}\frac{\mathrm{MHz}}{}$ and the frequency difference of that transition to molecular iodine lines with an accuracy of $250\text{\,}\frac{\mathrm{kHz}}{}$. This knowledge enables the use of simple iodine spectroscopy to find the ${}^{87}\text{Sr}$ ${}^{1}\text{S}_{0}-{{}^{3}\text{P}_{2}}$ transition frequency or to lock a Sr ${}^{1}\text{S}_{0}-{{}^{3}\text{P}_{2}}$ laser. Our work prepares the use of this Sr transition for applications, such as quantum simulation or computation.

V Acknowledgements

We thank Jan Matthijssen for technical assistance in the early stages of the project and H. Knöckel, B. Bodermann, and E. Tiemann for the software IodineSpec5 Iod . This project has received funding from the European Research Council (ERC) under the European Union’s Seventh Framework Programme (FP7/2007-2013) (Grant agreement No. 615117 QuantStro). We thank the Netherlands Organisation for Scientific Research (NWO) for funding through Vici grant No. 680-47-619 and Gravitation grant No. 024.003.037, Quantum Software Consortium. G. S. thanks the European Commission for Marie Curie grant SYMULGAS, No. 661171. C.-C. C. thanks the Ministry of Education of the Republic of China (Taiwan) for a MOE Technologies Incubation Scholarship.