State-dependent optical lattices for the strontium optical qubit

We demonstrate state-dependent optical lattices for the Sr optical qubit at the tune-out wavelength for its ground state. We tightly trap excited state atoms while suppressing the effect of the lattice on ground state atoms by more than four orders of magnitude. This highly independent control over the qubit states removes inelastic excited state collisions as the main obstacle for quantum simulation and computation schemes based on the Sr optical qubit. Our results also reveal large discrepancies in the atomic data used to calibrate the largest systematic effect of Sr optical lattice clocks.

The experimental implementation of innovative quantum simulation and quantum computation schemes based on the optical qubit in strontium [1][2][3][4][5][6][7] has been hindered by the presence of strong inelastic collisions between atoms in the excited qubit state [8]. Although these losses can be suppressed in deep three-dimensional optical lattices [9], such strong trapping precludes using tunneling and elastic collisions [10] to entangle atoms in different lattice sites. Controlled collisional phase gates [7] in particular require high-fidelity, independent control over atoms in either qubit state 1 S 0 (g) and 3 P 0 (e), shown in Fig. 1(a). Here, we provide a solution to these problems by demonstrating optical lattices at the so-called tuneout wavelength for the ground state [11,12], where its dipole polarizability vanishes, as shown in Fig. 1(b). At this tune-out wavelength, an e atom is tightly trapped, while a g atom is free to move. This condition shuts off the inelastic e-e collisions [8], while allowing the use of the elastic e-g and g-g collisions [10] to engineer novel systems for quantum simulation [13][14][15][16] and computation [7].
With a novel method, we measure an absolute frequency of 434, 972, 130(10) MHz for the tune-out wavelength in 88 Sr. At the tune-out wavelength, the differential AC Stark shift on the optical qubit transition is only due to the polarizability of the e state. We directly measure this polarizability with Stark shift spectroscopy, demonstrate trapping of e atoms in an optical lattice at the tune-out wavelength, and show that losses from light scattering are small. Given a moderate laser frequency stability corresponding to our measurement uncertainty, e atoms are tightly trapped while the trap's effect on g atoms is suppressed by more than four orders of magnitude, the highest level of suppression in any system to date [17][18][19][20][21][22].
We combine these measurements with high-precision Optical dipole traps in the red and infrared for the ground (g) and excited (e) qubit (clock) states of Sr are given by coupling to two low-lying singlet and triplet states, respectively. (b) The trap depth for each clock state as a function of wavelength is proportional to the dynamic dipole polarizability. At the magic wavelength (star), g and e experience the same trap depth. At the tune-out wavelength (circle), an atom in g is free to move, while an atom in e remains trapped. All quantities marked with an asterisk are measured in this work.
atomic structure theory and direct lifetime measurements of the 3 P 1 state [23] to extract new values for the 1 P 1 lifetime. We find a 7σ discrepancy to the currently accepted 1 P 1 lifetime from photoassociative spectroscopy [24]. Our polarizability measurements also improve the 3 S 1 lifetime by an order of magnitude and resolve discrepancies between prior measurements [25][26][27].
Our results highlight the importance of direct and precise atomic lifetime measurements to bring the accuracy of optical lattice clocks [28] into the 10 −19 regime.
Measuring the tune-out wavelength. We measure the tune-out wavelength by trapping g atoms in a magicwavelength optical lattice, overlapping an additional optical lattice beam close to the tune-out wavelength and  modulating its intensity. The modulation causes parametric heating and trap loss. Minimizing the induced loss allows us to precisely determine the tune-out wavelength. This method benefits from long integration times, has few systematic effects, and is readily applicable to atoms in excited states, molecules, or trapped ions.
We begin by loading 2 × 10 5 g atoms [29] into a deep near-vertical one-dimensional optical lattice, created by retro-reflecting 290 mW of magic-wavelength laser light at 813.4274(2) nm, as sketched in Fig. 2(a). The magicwavelength lattice has a longitudinal (transverse) trap frequency of ∼40 kHz (∼100 Hz), corresponding to a lattice depth of k B × 5.6 µK = h × 116 kHz and a 1/e 2 beam waist of 75 µm, where k B (h = 2πh) is the Boltzmann (Planck) constant. The atoms occupy a ∼60 µm (∼20 µm) long (wide) ellipsoid, shown in the absorption image in Fig. 2(a). From such in-situ and time-of-flight absorption images, we determine atom numbers and temperatures [29].
We then overlap the deep magic-wavelength lattice with a shallow one-dimensional lattice created by retroreflecting 4.5 mW of light near the tune-out wavelength, as shown in Fig. 2(a). This geometry allows us to heat the atoms in the combined lattice by intensity-modulating the shallow lattice, as sketched in Fig. 2(b). Since the two lattices are incommensurate there will be lattice sites in which heating due to phase modulation dominates, while in others heating due to amplitude modulation dominates. By changing the modulation frequency and observing atom loss from the trap [30], we obtain spectra as shown in Fig. 2(c). We take a reference spectrum (top panel) by modulating the deep lattice intensity, while the shallow lattice is turned off. In this case, we observe a single minimum in the spectrum at f mod 80 kHz, corresponding to amplitude modulation and parametric heating [30] that results in transitions between lattice bands that are two motional quanta apart. The response of the combined lattice due to intensity-modulation of the shallow lattice (bottom panel) shows another minimum at ∼40 kHz, corresponding to phase modulation and transitions between adjacent lattice bands [30].
To compare the effect of heating at different wavelengths of the shallow lattice, we intensity-modulate it at f mod and measure the resulting exponential trap loss rate. Other loss mechanisms such as losses due to intensity noise of both lattices, collisions with background gas atoms, and photon scattering losses, also contribute to the measured heating rate. We determine the induced trap loss rate Γ(ω) by taking the difference between the measured loss rate without modulation and with modulation. Examples of such measurements are shown in Fig. 2(d) for 88 Sr (top) and 87 Sr (bottom). The 88 Sr data is well described by an exponential decay because of the isotope's vanishingly small scattering length. In contrast, the 87 Sr data shows additional superexponential two-body decay. This decay is due to elastic interactions [31,32] that lead to evaporative trap loss, which we fit with a two-body decay term [33]. The induced trap loss rate vanishes when the ground state polarizability α g crosses zero at the tune-out wavelength, and it is proportional to [33] where ω is the optical frequency of the tune-out laser, and I mod is the intensity modulation depth. To compensate for drifts in I mod and the trap frequency, we normalize the measured Γ(ω) according to Eq. (1). The wavelength of the shallow lattice laser is locked to a wavemeter but measured with a self-referenced femtosecond frequency comb, resulting in an absolute frequency error of 3 MHz. The normalized data for 88 Sr and 87 Sr are shown in Fig. 2(e) and (f), respectively, as a function of detuning from each isotope's 1 S 0 -3 P 1 transition [34]. The induced loss rate Γ for both isotopes shows a minimum at detuning ∆ t , corresponding to the tune-out wavelength for each isotope. For 87 Sr, the induced two-body loss coefficient b, given by the difference of the two-body coefficients extracted from the underlying atomic decay curves, shows the same behavior with respect to detuning as Γ. This behavior can be explained by an increased tunneling rate in the second lattice band, leading to increased evaporation, correlated exponential and two-body decay rates, and an increased uncertainty for ∆ 87 t . We model the induced loss rate as Γ(∆) = c 0 (1 − ∆ t /∆) 2 [33], where ∆ ≡ ω − ω3 P1 is the detuning from the isotopeshifted 1 S 0 -3 P 1 transition, and the unused fit parameter c 0 relates the parametric heating rate to the trap loss rate [30]. We find ∆ 88 t = 2π × 143.009(8) GHz and ∆ 87 t = 2π ×142.86(8) GHz for 88 Sr and 87 Sr, respectively. These numbers are in good agreement, considering the empirical two-body loss model for 87 Sr. In the Supplemental Material [33], we derive a conservative upper limit |∆ 88 t − ∆ 87 t | < 2π × 23 MHz due to hyperfine splitting, vector, and tensor shifts. In the following, we use the measured ∆ 88 t for 87 Sr and suppress the superscript for clarity. This choice leads to a residual α g = ±0.05 a.u. from the 2.4 a.u./GHz polarizability slope around ∆ t . Here 1 a.u. = 4π 0 a 3 0 is the atomic unit of polarizability, and 0 (a 0 ) is the vacuum permittivity (Bohr radius).
To minimize systematic shifts in ∆ t due to laser noise, unsuppressed longitudinal laser modes, and amplified spontaneous emission, we Fourier-filter [33] the shallow lattice laser and suppress light at the 1 S 0 -3 P 1 transition by >90 dB compared to the carrier. To avoid saturation of Γ, and to work with the same I mod throughout, we limit the measurement range to a few GHz. Reducing the measurement range further does not change ∆ t significantly, and we estimate saturation effects to be negligible.
For 87 Sr, we observe and fit a statistically significant offset of 2.3(4) × 10 −3 s −1 in the induced trap loss rate. While the offset for 87 Sr could be explained by contributions from vector and tensor polarizabilities, these cannot occur in 88 Sr. When fitting the 88 Sr data with an offset, we find a much smaller value of 2(1) × 10 −4 s −1 , which also causes a systematic shift of the tune-out frequency of 2 MHz. In conclusion, we find that the tune-out frequency ω t for the ground state is detuned from the 1 S 0 -3 P 1 transition at ω3 P1 by Measuring the excited state polarizability. Since the polarizability of the g state vanishes at ω t , the AC Stark shift on the clock transition induced by a laser beam at ω t is solely determined by the excited state polarizability. For this reason, we can directly measure the e state polarizability α e (ω t ) by Stark shift spectroscopy. For these measurements, we prepare a sample of 87 Sr in a one-dimensional magic wavelength lattice. We propagate a clock laser beam at 3.5 • with respect to the lattice axis and overlap it with the lattice at the position of the atoms. A typical spectrum of the clock transition is shown in Fig. 3(a), consistent with a ∼1 µK temperature. On the carrier transition, we observe damped Rabi oscillations, consistent with the mismatch between the lattice axis and the clock laser wave vector [35]. For the following measurements, we illuminate the atoms with the clock laser for 0.3 ms, corresponding to the maximum excited state fraction. For the AC Stark shift spectroscopy, we additionally apply a laser beam at ω t with a 1/e 2 waist of 300 µm, as illustrated in Fig. 2(a). As a function of clock laser detuning, we observe the spectra shown in Fig. 3(b), and fit them to extract the center frequencies. To calibrate the intensity, we load a sample of 88 Sr into the same magic-wavelength lattice at the same position as the 87 Sr sample. After diabatically switching off the lattice, we measure free-space Rabi oscillations on each of the three 1 S 0 -3 P 1 transitions in a small magnetic bias field, as shown in Fig. 3(c). We fit the Rabi oscillations with an analytic solution to the optical Bloch equations [36], and extract the Rabi frequency Ω ± for each σ ± polarization component, for an applied power P 0 = 112 µW. The Rabi frequency of the Stark-shifting beam with power P is calibrated as Ω 2 ≡ (Ω 2 + + Ω 2 − )P/P 0 , which allows expressing the AC Stark shift of the clock transition at the tune-out fre- in terms of measured quantities, where c is the speed of light. In Figure 3(d), we plot ∆ω eg as a function of the saturation parameter s 0 = 2Ω 2 τ 2 3 P1 , and use a linear fit to extract the excited state polarizability of 87 Sr at the tune-out frequency, where the systematic uncertainty includes the mismatch between the tuneout frequency for 88 Sr and 87 Sr, but is dominated by the effect of the laser spectrum. Our measured polarizability agrees well with our theoretical prediction of 1546 (14) a.u., based on Ref. [37].
Trapping excited atoms at the tune-out wavelength. We prepare a sample of e state 87 Sr atoms in the magicwavelength lattice and transfer them to the tune-out lattice [33]. In Fig. 4, we compare the number of e atoms in the tune-out lattice as a function of hold time to the case where we trap e atoms in the magic-wavelength lattice. In both cases, the atoms decay superexponentially via e-e collisions [8]. In the magic wavelength lattice, this inelastic loss dominates, while the atoms in the tune-out lattice experience additional exponential loss with a 1/e lifetime ∼1.2 s. The intensity of the tune-out lattice is chosen to match the trap frequencies of the magic-wavelength and tune-out lattices at ∼40 kHz, corresponding to a lattice depth of ∼17 recoil, confirmed by parametric heating. The measured lifetime agrees well with the theoretically predicted loss due to photon scattering for each lattice axis of 24 s per recoil of lattice depth. Depending on the application, a compromise between lattice depth and tunneling rate needs to be found. For instance, a twodimensional tune-out lattice trapping e atoms in a Mott insulator state would have a lifetime ∼1 s.
Determination of atomic lifetimes. Our measurement of the tune-out wavelength strongly constrains the relationship between the 1 S 0 -1 P 1 and 1 S 0 -3 P 1 matrix elements [33] allowing us to cross-check the 1 P 1 and 3 P 1 lifetimes τ1 P1 and τ3 P1 . While τ3 P1 has been recently measured directly [23], the currently accepted value for τ1 P1 comes from photoassociative spectroscopy [24,38]. Using our tune-out wavelength and τ3 P1 [23], we find τ1 P1 = 5.234(8) ns [33], a 7σ discrepancy with the currently accepted value [24]. The e state polarizability α e at the Number of e atoms versus hold time in a one-dimensional optical lattice at the magic wavelength (squares) and at the tune-out wavelength (circles). Both lattices exhibit the expected losses due to inelastic collisions, while the tune-out lattice additionally exhibits exponential one-body decay due to light scattering.
tune-out wavelength is dominated (87%) by the 3 P 0 -3 S 1 transition. We determined all other contributions with a total uncertainty of 4 a.u. using a high-precision relativistic method [37]. Combining these theoretical values with our measurement of α e determines the 3 P 0 -3 S 1 matrix element. We extract τ3 S1 from a calculation of the 3 S 1 -3 P J branching ratios with 0.1% accuracy. This accurate prediction of branching ratios is possible due to very similar electronic correlation effects for these transitions, which largely cancel for their ratios. We find τ3 S1 = 13.92(11) ns, an improvement of an order of magnitude over prior measurements that ranged from 10.9(1.1) ns to 15.0(8) ns [25][26][27]. We note in passing that the currently best values for τ3 P1 and the lifetime of the 3 D 1 state are correlated because they are extracted from a single data set [23]. The 3 D 1 lifetime directly determines the dynamic contribution to the Sr lattice clock blackbody radiation shift [37], its currently largest systematic uncertainty [39]. This uncertainty can be directly improved by a new direct measurement of τ1 P1 in combination with our results and Ref. [23]. Our measurements show that direct measurements of atomic lifetimes and improvements to atomic structure calculations will be essential in bringing optical frequency standards to the 10 −19 level.
In conclusion, we have demonstrated state-dependent optical lattices for the clock states of strontium at the tune-out wavelength for its ground state. With a new spectroscopic method, we achieved a record suppression of the lattice depth for the ground state of more than four orders of magnitude. Our method can be applied to thermal gases of atoms, molecules [40], or to trapped ions. As a modulation technique for trapped particles, the method benefits from suppression of systematic errors and long integration times. Using our technique in three-dimensional optical lattices in combination with band mapping [41] will enable measurements of excited state tune-out wavelengths, such as the tune-out wavelength for the 3 P 0 state around 633 nm [12], even in the presence of interactions. We have demonstrated high-fidelity, state-dependent control of the strontium optical qubit and remove the main obstacle for the realization of quantum computation schemes with two-electron atoms [7]. Finally, our work creates new opportunities to use state-dependent optical lattices for quantum simulations of nanophotonics [13][14][15] and quantum chemistry [16].
We thank J. Ye, A. González-Tudela, and J. I. Cirac for stimulating discussions. The optical dipole potential of two retroreflected laser beams with different wavelengths λ i = 2π/k i along the z-direction is u(z) = u 1 cos 2 (k 1 z) + u 2 (t)cos 2 (k 2 z). (S1) Here u i is the lattice depth induced on the atomic state via the AC Stark effect. In the parametric-heating experiments described in the main text, the first lattice is the deep magic-wavelength lattice with a fixed depth u 1 . The second lattice is the shallow tune-out lattice with a variable depth u 2 (t) = u 0 2 + u mod 2 cos(ω mod t) that is sinusoidally modulated around its mean value u 0 2 with a modulation amplitude u mod 2 and a modulation frequency ω mod . Since we use shallow tune-out lattices with u 2 (t) u 1 , we treat the effect of the shallow lattice in perturbation theory by expressing its effect as a variation in the trap position and the trap frequency of each lattice site [S1-S3]. To this end, we expand the combined lattice potential around the position z j = (2j + 1)π/(2k 1 ) of the j-th minimum of the deep lattice and find By completing the squares and dropping constant terms, we can bring the above expression into the standard form [S1-S3] for parametric heating in a harmonic oscillator where m is the mass of the atom, ω j (z j ) is the unmodulated trap frequency (position of the minimum) of lattice site j, and (δ j ) is the amplitude (position) modulation causing transitions between the harmonic oscillator lev-els. Explicitly, we find From these expressions we see that the trap frequency of the deep lattice mω 2 1 /2 = u 1 k 2 1 is very weakly influenced by the presence of a weak lattice with bare trap frequency mω 2 2 /2 = u 0 2 k 2 2 . These trap frequencies scale with the ratio of the ground state polarazibilites α g (λ i ) at the magic and tune-out wavelengths and the lattice powers P i . For our parameters, we find For this reason, the trap frequency of the combined lattice is unchanged when modulating the deep lattice by itself or when modulating the shallow lattice in the presence of the deep lattice. In the latter case, the modulation spectrum simply acquires a second minimum at the trap frequency, corresponding to position modulation of the lattice sites.

II. TRAP LOSS FITTING
To analyze the trap loss curves throughout the main text, we fit the atom number data N (t) as a function of time t with exponential decay curves N (t) = N 0 e −Γt resulting from a single-body decay processṄ = −ΓN . We decide whether to fit with superexponential decay homogeneously without changing the sample temperature. Because we do not take the temperature effects of evaporation and anti-evaporation into account, the results of the superexponential decay curves are not used to derive any of the important quantities in the main text and are meant simply as a guide to the eye.

III. TRAP LOSS RATE RESCALING
If we modulate with ω mod = 2πf mod = 2ω 1 (at twice the deep lattice trap frequency), the differential trap loss rate is proportional to the heating rate caused by the intensity modulation alone [S1-S3] where S (f mod ) is the power spectrum of the fractional intensity noise at the modulation frequency f mod = ω mod /2π. The ratio of shallow modulation amplitude to deep lattice trap depth, u mod where I mod is the intensity modulation amplitude of the shallow lattice and I 1 is the intensity of the deep lattice. Thus, where we have used I 1 ∝ ω 2 1 ∝ f 2 mod . To compensate for possible experimental variations in ω 1 and I mod , we scale the measured induced loss rate with respect to the reference values and work with these scaled induced loss rates throughout the main text.

IV. THE 1 S0 GROUND STATE TUNE-OUT WAVELENGTH
The polarizability of an electronic state consists of contributions from the core electrons and valence electrons. The valence part of the ground state 1 S 0 (g) polarizability is determined by summing over all the contributions of excited states dipole-coupled to g, dominated by the 1 P 1 and 3 P 1 contributions. For this reason, we write the g polarizability using the four components α g (ω) = α g ( 1 P 1 ; ω) + α g ( 3 P 1 ; ω) + α g (v; ω) + α g (c; ω), (S10) where α g (j; ω) are the contributions from the excited state j, α g (v; ω) is the sum of contributions from all other valence states, α g (c; ω) is the contribution of the core electrons, and ω = 2πc/λ is the optical frequency for wavelength λ.
The contribution from the 1 P 1 and 3 P 1 states further splits into scalar, vector, and tensor components as [S4, S5] α g (j; ω) = α s g (j; ω) S11) where F is the hyperfine quantum number associated with the ground state g, m F is the corresponding magnetic quantum number, and j indicates one of the excited states 1 P 1 and 3 P 1 . The 1 S 0 (g) state of the fermionic isotope 87 Sr has a single hyperfine state F = 9/2, since the electronic angular momentum J and the nuclear spin I are 0 and 9/2, respectively (|J − I| ≤ F ≤ |J + I|). For the bosonic isotope 88 Sr where I = 0, J replaces F (F = J). Therefore, any discussion involving F j throughout this text only applies to 87 Sr. The quantization axis is assumed to be along e z , and is the complex polarization vector of the applied laser at frequency ω. The vector and tensor components depend on the polarization of the applied beam, while the scalar component does not.
The scalar, vector, and tensor parts themselves can be written as sums over all hyperfine states F j in the excited fine structure state [S4, S5] α s g (j; ω) = (S12) All polarizability terms scale in exactly the same way as a function of laser detuning, which is encapsulated in the detuning factor where ω j,Fj is the transition frequency from the ground to the excited hyperfine state F j , and τ j is the excited state lifetime. Note that ω j,Fj can be expressed in terms of the (hypothetical) hyperfine-free transition frequencyω j and the hyperfine shift ∆ j,Fj as ω j,Fj =ω j + ∆ j,Fj . The hyperfine shift can be calculated from the magnetic dipole interaction constant A j and the electric quadrupole interaction constant Q j as [S6, S7] where J j is the electronic angular momentum of the excited state. We use the hyperfine constants A1 P1 = −2π × 3.4(4) MHz, Q1 P1 = 2π × 39(4) MHz, A3 P1 = −2π × 260.084(2) MHz, and Q3 P1 = −2π × 35.658(6) MHz, summarized in Ref. [S6]. The isotope shift ofω3 P1 was measured recently in Ref. [S8]: the hyperfine-free transition frequency of 1 S 0 -3 P 1 in 87 Sr is red detuned by 62.1865(123) MHz from the transition frequency in 88 Sr, 434, 829, 121, 311(10) kHz [S9]. The 1 S 0 -1 P 1 transition frequency does not need to be as precise and is obtained from Ref. [S10]. The isotope shift on this transition does not influence the calculations, as will be shown below.
Because we work ∼10 7 linewidths detuned from the 1 S 0 -3 P 1 transition, we can ignore the imaginary term in the denominator of Eqn. (S13), which we have confirmed numerically. All polarizability terms also scale with the modulus-squared of the reduced matrix element associated with the corresponding dipole transition (S15) Since the 1 S 0 state of 87 Sr has a single hyperfine state, the reduced matrix element can be expressed in terms of the excited state lifetime τ j as (S16) The two excited states of interest 1 P 1 and 3 P 1 both have J j = 1, and thus have the same three hyperfine states F j = 7/2, 9/2, and 11/2. For this reason, the scalar, vector, and tensor polarizabilities only differ in the grouptheoretic numerical prefactors associated with the excited states.
The situation simplifies strongly for the case of 88 Sr. If we replace F (F j ) with J (J j ) in Eqn. (S12), the vector and tensor polarizabilities vanish. Only the scalar component remains in the absence of hyperfine structure.
To study the contributions from each component in detail, we dig deeper into the scalar, vector, and tensor polarizabilities of the 87 Sr 1 S 0 state. By specializing the prefactors, we re-write the reduced matrix elements as where the numerical factors in the braces correspond to F j = 7/2, 9/2, and 11/2, respectively. As expected, this means that the lifetime τ proportionally scales up scalar, vector, and tensor polarizabilities. Bringing it all together, we find (S18) To get an intuitive picture of what happens in the fardetuned regime compared to the hyperfine structure, we approximate |D j,Fj | 2 ≈ 3π 0hc 3 j,F j τj out of the sum in Eqn. (S18). After this approximation, we see that both vector and tensor polarizabilities sum to zero when we are in the far-detuned regime compared to the hyperfine structure, where the detuning factor contributes equally, and can be pulled out of the sum. In this regime, the only contribution that survives is the scalar polarizability. For the same reason, we also expect negligible differences in α s between the two isotopes at this regime. The scalar polarizabilities of the two isotopes are illustrated in Fig. S1(a) where we see that they become indistinguishable as the detuning from the 1 S 0 -3 P 1 transition increases (the plot was generated based on Eqn. (S18) without the approximation on |D j,Fj | 2 ). Note that the detuning for 87 Sr is referenced with respect toω j , which takes the isotope shift into account.
Next, we take a closer look at each contribution at the detuning range relevant to our experimental data. Comparing Fig. S1(b)-(c) and (e)-(f), the 1 P 1 tensor and vector contributions are more than two orders of magnitude smaller than the contributions from the 3 P 1 state, since the frequency of the shallow lattice is more than a few hundred THz detuned from the 1 S 0 -1 P 1 transition. For this reason, we can ignore the vector and tensor contributions to α g ( 1 P 1 ; ω) and set α g ( 1 P 1 ; ω) = α s g ( 1 P 1 ; ω). With this simplification, we analyze the difference of the 87 Sr and 88 Sr tune-out detuning ∆ t due to the hyperfine structure. At the tune-out detuning, α g ( 1 P 1 ; ω) balances α g ( 3 P 1 ; ω) and the remaining valence and core contributions to the polarizability, α rg = α g (v; ω) + α g (c; ω).
We calculated the valence part of α g solving the inhomogeneous equation as described in Refs. [S11, S12]. Then using a sum-over-states formula, Eq. (S12), we extracted the contributions of the 1 P 1 and 3 P 1 states and determined the remaining valence contributions, α g (v, ∆t) = 6.57 (14) a.u.. The core part of the polarizability, α g (c; ∆ t ), was calculated in the single-electron approximation including random-phase approximation corrections [S13] to be α g (c; ∆ t ) = 5.30(5) a.u.. Table I shows the shifts due to scalar, vector, and tensor polarizabilities. For the calculations due to the vector FIG. S1. (color online). (a) The 3 P1 scalar contribution as a function of detuning from the 1 S0-3 P1 transition. For the case of 87 Sr, the detuning is referenced with respect toω3 P 1 , which includes the isotope shift. The inset shows α s 3 P 1 at the detuning range relevant to our data. (b) the 3 P1 vector contribution as a function of detuning from the 1 S0-3 P1 transition for different mF states (Eqn. (S12)). We assume the beam ellipticity of 1 to show the upper limit. (c) the 3 P1 tensor contribution (Eqn. (S12)) as a function of detuning from the 1 S0-3 P1 transition for different mF states. Here, we assume a linear polarization to show the upper limit. (c)-(f) same as (a)-(c), but from the 1 P1 contributions. polarizabilities, we used an upper limit on the beam ellipticity of 2%, measured by a polarimeter after transmission through our vacuum chamber, and assumed perfect alignment of · e z = 1 to evaluate the tensor polarizability. The shifts depend on the m F states (Eq. (S12)), and we show the maximum ranges of the shift as a worst case estimate. In the experiment, we do not spin-polarize the sample, and we likely work with an equal population among all m F states. As a conservative estimate, we use the full span due to the vector shift to estimate differential shifts between the measured tune-out detuning ∆ 88 t for 88 Sr and the tune-out detuning ∆ 87 t for 87 Sr. By choosing to work with ∆ 88 t , we thus estimate a mismatch of ±23 MHz, corresponding to a residual ground state polarizability of ±0.05 a.u. for the AC Stark shift measurements presented in the main text.
Putting everything together, the scalar parts can be explicitly written as , (S19) where we have neglected the hyperfine splitting in α s g ( 1 P 1 ; ω). These approximations result in an additional shift of ∆ t by a few hundred kHz, which is more than two orders of magnitude smaller than our experimental precision. As a consequence, we can express τ1 P1 directly in terms of the tune-out frequency For 88 Sr where the 3 P 1 hyperfine splittings are absent, and no approximations are necessary. Thus, α s g ( 3 P 1 , ω t ) reduces to To derive a fitting function to model the induced loss rate Γ mod ∝ (α s g ) 2 , we first need to express g3 P1,F3 P 1 explicitly in terms of the laser detuning ∆ = ω −ω3 P1 , (S22) where we use ω j,Fj =ω j + ∆ j,Fj and 2ω3 P1 ∆ + ∆3 P1,F3 P 1 . For the case of 88 Sr,ω j is replaced by ω j . The last approximation on g3 P1 shifts ∆ t by only several kHz. Then, α s g (j; ω) can be written in terms of detuning as ω3 P1 ∆ − ∆3 P1,F3 P 1 .
(S23) From this equation, we see that α s g ( 3 P 1 ; ∆) scales inversely proportional to detuning. As shown in Fig. S1(c), α s g ( 1 P 1 ; ∆) varies at the 10 −3 level in the detuning range of our experimental data. Thus, we treat α s g ( 3 P 1 ; ∆) as a constant α s g ( 3 P 1 ; ∆)=α s g ( 3 P 1 ; ∆ t ), which shifts ∆ t by less than a kHz. Focusing on 88 Sr where the hyperfine structure is absent, we arrive at the following expression when treating α bg also as a constant in the vicinity of ∆ t , where we used the fact that α g vanishes at the tune-out wavelength. We use the function above to fit the induced loss rates, Γ mod ∝ α 2 g (∆3 P1 ).

V. FOURIER FILTERING
To suppress amplified spontaneous emission (ASE) that would induce systematic shifts in our measurements we took great care in filtering it. Of particular importance is the amount of ASE near the 1 S 0 -3 P 1 transition where the polarizability diverges. Even very low light levels near this transition lower the lifetime of atoms in the magic lattice and introduce an unknown heating mechanism that is not easily accounted for.
To minimize the amount of light away from the carrier we Fourier filter it with a grating as the dispersive element. We expand the beam to a 1/e 2 waist of 2 cm and diffract it from a holographic grating with 2400 lines/mm, propagate it over ∼6 m, before finally coupling it into a single-mode polarization-maintaining fiber.
We determine the achieved resolution by measuring the transmission through the fiber while changing the input wavelength, at the same time keeping the grating angle fixed. The Gaussian suppression line shape has a FWHM of 13 GHz giving a suppression of more than 40 dB at the 1 S 0 -3 P 1 frequency when the input light is at the tune-out frequency. Before filtering we measure an ASE level of -50 dB compared to the carrier with a spectrum analyzer with a 0.05 nm resolution. Together with the achieved Fourier filtering suppression of 40 dB we conclude that the ASE level is brought down to levels below -90 dB compared to the carrier.

VI. LOADING EXCITED STATE ATOMS IN THE TUNE-OUT LATTICE
To load excited state 87 Sr atoms into the tune-out lattice the experimental procedure is as follows. We i) load a sample of 87 Sr g atoms into the magic wavelength lattice; (ii) after 50 ms, we transfer ∼70% of the atoms to e using a 10-ms-long adiabatic-rapid-passage pulse; (iii) after another 2 ms, we diabatically switch on the tune-out lattice using 64 mW of power; (iv) after an additional 1 ms, we ramp down the magic wavelength lattice over 10 ms and retain ∼80% of the e atoms in the tune-out lattice; (v) we hold the atoms in the tune-out lattice for a given time, diabatically turn it off, repump the atoms to g over 2 ms [S14], and take an absorption image to determine the final atom number. In addition to matching the trapping frequencies of the tune-out and magic wavelength lattices, care was taken to achieve good mode matching between the two lattice beams. As a result the size and position of the area occupied by the trapped e atoms in the tune-out and the magic lattice differs on the single micron level. This is also supported by a measured moderate increase of the atom temperature to 3 µK after the transfer into the tune-out lattice.