Operational Magic Intensity for Sr Optical Lattice Clocks

TL;DR

This paper experimentally investigates and determines the optimal lattice depth for Sr optical lattice clocks, achieving ultra-precise light shift cancellation at the $10^{-19}$ level by controlling atomic motion and polarizabilities.

Contribution

It provides the first precise measurement of lattice-induced polarizabilities and establishes an operational lattice depth that minimizes light shifts in Sr optical lattice clocks.

Findings

Operational lattice depth of 72(2) E_R for minimal light shift.

Light shift cancels to the $10^{-19}$ level within 30% intensity variation.

Controlled atomic motion enables accurate polarizability measurements.

Abstract

We experimentally investigate the lattice-induced light shift by the electric-quadrupole ($E2$) and magnetic-dipole ($M1$) polarizabilities and the hyperpolarizability in Sr optical lattice clocks. Precise control of the axial as well as the radial motion of atoms in a one-dimensional lattice allows observing the $E2$-$M1$ polarizability difference. Measured polarizabilities determine an operational lattice depth to be $72(2) E_R$, where the total light shift cancels to the $10^{-19}$ level, over a lattice-intensity variation of about 30%. This operational trap depth and its allowable intensity range conveniently coincide with experimentally feasible operating conditions for Sr optical lattice clocks.

Equations

$$h{\nu }_{\mathrm{LS}}(u,{\delta }_L,n_z)\approx \left(\frac{\partial {\tilde{\alpha }}^{E1}}{\partial \nu }{\delta }_L-{\tilde{\alpha }}^{qm}\right)\left(n_z+\frac{1}{2}\right)u^{1/2}$$

$$-\left[\frac{\partial {\tilde{\alpha }}^{E1}}{\partial \nu }{\delta }_L+\frac{3}{2}\tilde{\beta }\left(n_z^2+n_z+\frac{1}{2}\right)\right]u$$

$$+2\tilde{\beta }\left(n_z+\frac{1}{2}\right)u^{3/2}-\tilde{\beta }{u}^2,$$

$$\overline{u^j}=\int \rho (x,y){\left(\frac{{\alpha }^{E1}{I}_0{e}^{-2(x^2+y^2)/w^2}}{E_R}\right)}^{j}dxdy\equiv {\zeta }_j{u}^j,$$

$$h\Delta {\nu }_{\mathrm{LS}}^{\mathrm{vib}}(u,{\delta }_L)$$

$$=h[{\nu }_{\mathrm{LS}}(u,{\delta }_L,1)-{\nu }_{\mathrm{LS}}(u,{\delta }_L,0)]$$

$$=\left(\frac{\partial {\tilde{\alpha }}^{E1}}{\partial \nu }{\delta }_L-{\tilde{\alpha }}^{qm}\right)u^{1/2}+\tilde{\beta }u\left(2u^{1/2}-3\right).$$

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Recent progress of optical clocks has pushed their fractional uncertainty to the $10^{-18}$ level Chou et al. (2010); Ushijima et al. (2015); Nicholson et al. (2015); Huntemann et al. (2016), which opens up new applications of clocks, such as chronometric geodesy Lisdat et al. (2016); Takano et al. (2016), tests of fundamental constants Uzan (2003); Huntemann et al. (2014), detection of dark matter Derevianko and Pospelov (2014), or gravitational waves Kolkowitz et al. (2016). Triggered by these advances, a future redefinition of the second by optical clocks Targat et al. (2013); Grebing et al. (2016) is in range and its procedure is being discussed Riehle et al. (2018).

A better understanding and control of perturbations lies at the heart of the continued progress in atomic clocks. Isolating atoms from electromagnetic (EM) perturbations is of prime importance in designing ion clocks Dehmelt (1982) where ions are confined nearly free from EM perturbations. Optical lattice clocks have shown that cancellation of trap perturbation leads to stable and accurate clocks with uncertainties less than $10^{-17}$ Ushijima et al. (2015); Nicholson et al. (2015); Grebing et al. (2016); Brown et al. (2017a), the magic frequency aimed to equalize polarizabilities of the clock states so as to decouple the clock transition frequency from inhomogeneous trap perturbations Katori et al. (2003). Removal of perturbations by specifying the frequency is the essence of the optical lattice clock, which is based on the fact that the frequency is a precisely measurable quantity.

This magic frequency concept, however, becomes nontrivial for achieving inaccuracy of $10^{-18}$ because of non-negligible contribution of the higher-order light shifts than that given by the electric-dipole ($E1$) interaction. In a standing wave of light, a quarter-wavelength spatial mismatch between the $E1$ potential and the potential induced by the electric-quadrupole ($E2$) and magnetic-dipole ($M1$) interactions introduces an atomic-motion-dependent light shift Taichenachev et al. (2008); Katori et al. (2009). In addition, the hyperpolarizability effect introduces a light shift proportional to the square of lattice intensity Katori et al. (2003); Brusch et al. (2006). Different spatial dependence makes these light shifts difficult to eliminate. An operational magic frequency Katori et al. (2015) is proposed to compensate the higher order shifts by the $E1$ light shift and make the overall light shift insensitive to lattice-intensity variation around a “magic intensity.”

In order to find such an operational condition, precise knowledge of the higher-order polarizabilities is mandatory. Higher-order light shifts have been investigated theoretically Ovsiannikov et al. (2016); Porsev et al. (2018) and experimentally for Sr Westergaard et al. (2011); Targat et al. (2013); Nicholson et al. (2015), Yb Barber et al. (2008); Nemitz et al. (2016); Brown et al. (2017a), and Hg Yamanaka et al. (2015). Recently, the hyperpolarizability was measured for Yb to find the operational magic frequency Brown et al. (2017a) with the help of a theoretical calculation of the $E2$-$M1$ polarizability. As for Sr, in spite of significant efforts, discrepancies between reported polarizabilities are not yet solved.

In this Letter, we investigate the hyperpolarizability and the E2-M1 polarizability for Sr atoms in a one-dimensional (1D) lattice. From the nonlinear intensity dependence of the light shift, we derive the hyperpolarizability. The $E2$-$M1$ polarizability is evaluated by measuring the light shift difference by changing the vibrational state of atoms in the lattice. Using the obtained polarizabilities, we derive two distinctive operational conditions that make the total light shift insensitive to lattice intensity variation at the $10^{-19}$ level.

The lattice-induced light shift $\nu_{\rm LS}$ is given by the light shift difference between the ground and excited states on the clock transition. For a 1D optical lattice as shown in Fig. 1(a), the light shift depends on the vibrational state $n_{z}$ of atoms along the $z$ axis, the lattice laser intensity, and the detuning $\delta_{L}$ of lattice laser $\nu_{L}=\delta_{L}+\nu^{E1}$ from the $E1$ magic frequency $\nu^{E1}$ that makes the $E1$ polarizabilities $\alpha^{E1}$ for the clock states equal. Since the peak intensity $I_{0}$ of the lattice is proportional to the trap depth $U\approx\alpha^{E1}I_{0}$ (by neglecting the higher-order effects of less than $10^{-6}$), we rewrite the light shift formula Katori et al. (2015) in terms of a normalized trap depth $u=U/E_{R}$ with $E_{R}=(h\nu_{L}/c)^{2}/(2m)$ the lattice photon recoil energy as,

[TABLE]

where $\tilde{\alpha}^{E1}$, $\tilde{\alpha}^{qm}$, and $\tilde{\beta}$ are the difference (denoted by tildes) of $E1$ and $E2$-$M1$ polarizabilities, and hyperpolarizability on the clock transition. The conversion of these polarizabilities is summarized in the Supplemental Material SM (2). While the light shift model given in Ref. Katori et al. (2015) takes into account the anharmonicity of the lattice trap to ${\cal O}(z^{4})$ in the axial coordinate expansion, we verify that neglecting ${\cal O}(z^{6})$ terms is valid for describing the light shift with low $10^{-19}$ uncertainty for Sr SM (2).

The lattice intensity is nonuniform in nature, as the spatial inhomogeneity itself is the essence of an optical trap. As the intensity critically affects the light shift as given in Eq. (1), precise control and evaluation of atomic distribution in the optical lattice is of particular importance. We consider atomic motion in the 1D lattice potential given by $\mathcal{U}(x,y,z)\approx-\alpha^{E1}I_{0}{e^{-2({x^{2}}+{y^{2}})/{w^{2}}}}\cos^{2}\left({2\pi z}/{\lambda_{L}}\right)$, where $I_{0}$, $w$, and $\lambda_{L}=c/\nu_{L}$ are the peak intensity, the radius, and the wavelength of the lattice laser with a Gaussian profile. The axial and radial oscillation frequencies of atoms are given by $\nu_{z}=2\sqrt{\alpha^{E1}I_{0}E_{R}}/h$ and ${\nu_{r}}={\nu_{z}}{\lambda_{L}}/(\sqrt{2}\pi w)(\approx{\nu_{z}}/320$ for our experiment). In contrast to the axial vibrational states with averaged occupation $\bar{n}_{z}\approx 0$ that require quantum treatment, the radial motion can be treated classically as the vibrational states typically occupy $\bar{n}_{r}=k_{B}T_{r}/(h\nu_{r})\approx 110$ with $T_{r}$ the radial temperature and $k_{B}$ the Boltzmann constant. Assuming a thermal distribution $\rho(x,y)=\frac{{m{(2\pi\nu_{r})^{2}}}}{{2\pi k_{B}T_{r}}}{e^{-\frac{1}{2}m{(2\pi\nu_{r})^{2}}({x^{2}}+{y^{2}})/(k_{B}{T_{r}})}}$ of atoms, the effective laser intensity experienced by the atoms is given by

[TABLE]

where we denote the thermal average by the bar and define a lattice-intensity reduction factor $\zeta_{j}(u)\approx 1-\frac{jk_{B}T_{r}}{uE_{R}}$. In the following, we evaluate the lattice light shifts of Eq. (1) by the effective intensity $\overline{u^{j}}=\zeta_{j}(u)u^{j}$.

To investigate the hyperpolarizability effect, we install a buildup cavity with a power enhancement factor of $\approx 20$ for the 1D optical lattice oriented vertically as shown in Fig. 1(a). The beam radius is chosen as $w\approx 60~{}\mu\rm{m}$ to moderate atomic collisions and allows a maximum trap depth of $u\sim 1200$. This cavity also works as a spatial filter to define a TEM00 Gaussian mode. We use a Ti:sapphire laser at $\nu_{L}\approx 368~{}\rm{THz}$ stabilized to a reference cavity that is calibrated by a frequency comb linked to the Sr clock. By applying a volume Bragg grating with a bandwidth of $\sim 20~{}\rm{GHz}$, we suppress amplified spontaneous emission of the lattice laser and reduce the relevant light shift Targat et al. (2013) to less than $10^{-19}$.

87Sr atoms are laser cooled to $\sim 5~{}\mu\rm{K}$ and loaded into the lattice with its depth of $u_{\rm ref}=272$ ($u_{\rm ref}E_{R}/k_{B}=45\,\mu$K). This loading condition is kept constant during measurements. A bias magnetic field of $|\mathbb{B}_{\rm{bias}}|=65~{}\rm{\mu T}$ is applied along the $x$ axis to define the quantization axis and to separate the Zeeman substates. Lattice, optical pumping, and clock laser are all polarized parallel to the bias field, while that of the cooling laser is perpendicular to the bias field so as to be decomposed into $\sigma^{\pm}$ components. Applying the $\pi$-polarized pumping laser resonant with the ${{}^{1}}S_{0}~{}(F=9/2)-{{}^{3}}P_{1}~{}(F=7/2)$ transition [see Fig. 1(b)], the atoms are optically pumped to the outermost Zeeman substates ${{}^{1}}S_{0}~{}(F=9/2,\ m_{F}=\pm 9/2)$ used for the clock interrogation. In the following, we discuss the case where we take the $m_{F}=9/2$ state as the clock state.

Simultaneously with the optical pumping, we apply Doppler cooling for the radial motion with the $\sigma^{+}$ component of the cooling laser on the ${{}^{1}}S_{0}~{}(F=9/2,\ m_{F}=9/2)-{{}^{3}}P_{1}~{}(F=11/2,\ m_{F}=11/2)$ transition. Consequently, the radial temperature is reduced to $T_{r}\approx 2~{}\mu\rm{K}$ (correspondingly $\zeta_{1}(u_{\rm ref})\approx 0.96$), as measured by time-of-flight (TOF) thermometry, and the linewidth of the blue sideband on the clock transition is reduced to $\sim 8$ kHz as shown in the inset of Fig. 1(c). The atoms remaining in the $m_{F}=-9/2$ state are heated out of the lattice by the $\sigma^{-}$ component of the cooling laser. Subsequently, we apply sideband cooling to reduce axial vibrational states to $\bar{n}_{z}<0.01$, as measured by the ratio of red and blue sidebands, using the $\sigma^{+}$-polarized cooling laser propagating along the lattice axis.

In order to purify the $m_{F}$ state, we excite the atoms to the ${{}^{3}}P_{0}~{}(m_{F}=9/2)$ state with a 22-ms-long clock $\pi$ pulse so as to resolve the Zeeman substates and to select a single $m_{F}$ state. Atoms in the other Zeeman substates remain unexcited and are subsequently blown away by a laser pulse tuned to the ${{}^{1}}S_{0}-{{}^{1}}P_{1}$ transition. For the preparation of atoms in the ${{}^{3}}P_{0}~{}(m_{F}=-9/2)$ state, we apply the similar procedure with the $\sigma^{-}$ component of the cooling laser.

Finally, in order to evaluate the lattice light shift dependence on the trap depth, we adiabatically ramp up or down the lattice depth from $u_{\rm ref}$ to $u$ over 80 ms. Symbols in Fig. 1(c) show reduction factors determined by the TOF measurements, which reasonably follow those assuming adiabatic temperature changes, i.e., $\zeta^{\rm ad}_{j}(u)=1-\frac{{1-\zeta_{j}({u_{\rm ref}})}}{{\sqrt{u/{u_{\rm ref}}}}}$ as shown by dashed lines with corresponding colors. As the reduction factor after the adiabatic ramp is in the range of $0.95<\zeta_{1}(u)<0.99$ for $150<u<1150$, we approximate $\overline{{u^{j}}}\approx{({\zeta_{1}}u)^{j}}$, which is valid within 0.2% error. The axial vibrational number $\bar{n}_{z}<0.01$ is measured unchanged after the adiabatic ramp.

We operate two Sr clocks, Sr1 and Sr2, to evaluate the light shift: Sr1 measures the light shift by varying the lattice depth $u$ or vibrational state $n_{z}$ of atoms, while Sr2 serves as a frequency anchor. Sr1 and Sr2 simultaneously interrogate the clock transition at $\nu_{0}\approx 429$ THz with a common laser to cancel out the Dick effect noise introduced by the clock laser, which improves the Allan deviation for the light shift measurements Takamoto et al. (2011).

Figure 2 shows the intensity-dependent light shift $\Delta\nu_{\rm LS}^{\bar{u}}=\nu_{\rm LS}(\bar{u},\delta_{L},0)-\nu_{\rm LS}(\bar{u}_{\rm ref},\delta_{L},0)$ as a function of the effective depth $\bar{u}=\zeta_{1}(u)u$ by taking $\bar{u}_{\rm ref}=\zeta_{1}(u_{\rm ref})u_{\rm ref}=263$ as a reference. We change the lattice laser frequency $\nu_{L}$ every 30 MHz, which is measured with uncertainties less than $100~{}\rm{kHz}$. The detunings $\delta_{L}$ given in the legend are calculated after determining the $E1$ magic frequency $\nu^{E1}$ as described below. The hyperpolarizability effect introduces the nonlinear dependence for higher intensity, where we correct the density shift of low $10^{-18}$ by measuring the density-dependent shift SM (2).

All the data in Fig. 2 are fitted using the light shift model given in Eq. (1), where we take $\nu^{E1}$, $\frac{\partial\tilde{\alpha}^{E1}}{\partial\nu}$, and $\tilde{\beta}$ as free parameters. As $\tilde{\alpha}^{qm}$ scarcely contributes to this fitting, we conduct another measurement to determine $\tilde{\alpha}^{qm}$ and apply the results to this fitting. We repeat these two fittings until the fitting parameters converge. Finally, the solid fitting curves determine $\nu^{E1}=368~{}554~{}465.1(1.0)~{}\rm{MHz}$, $(\partial\tilde{\alpha}^{E1}/\partial\nu)/h=1.735(13)\times 10^{-11}$, and $\tilde{\beta}/h=-0.461(14)~{}\rm{\mu Hz}$.

As the light shift arising from the multipolar polarizability $\tilde{\alpha}^{qm}$ is sensitive to the vibrational states $n_{z}$ Katori et al. (2009), we measure the differential light shift between $n_{z}=1$ and $n_{z}=0$ vibrational states given by

[TABLE]

This eliminates the otherwise dominating contributions from $\tilde{\alpha}^{E1}$ and $\tilde{\beta}$, and allows extracting $\tilde{\alpha}^{qm}$.

For this measurement, we excite the atoms to the $n_{z}=0$ or $1$ vibrational state in the ${{}^{3}}P_{0}~{}(m_{F}=9/2)$ state by applying a rapid adiabatic passage (RAP) Melinger et al. (1992) by frequency sweeping the $\pi$-polarized clock laser across the carrier and blue sideband in 6 ms. The Rabi frequency of the clock laser is about 50 kHz (10 kHz) for the carrier (the blue sideband). This RAP allows transferring more than 90% of the atoms to the desired vibrational states. The atoms remaining in the ground state are heated out of the trap by driving the ${{}^{1}}S_{0}-{{}^{1}}P_{1}$ transition.

Figure 3 shows the differential light shift $\Delta\nu_{\rm LS}^{\rm{vib}}(\bar{u},\delta_{L})$ measured for the lattice detuning $\delta_{L}=0.4~{}\rm{MHz}$. A green line fits the measurements by taking $\tilde{\alpha}^{qm}$ as a free parameter, while $\tilde{\beta}$, $\nu^{E1}$, and $\partial\tilde{\alpha}^{E1}/\partial\nu$ are fixed with the values obtained with the data in Fig. 2. The updated result of $\tilde{\alpha}^{qm}$ is recursively used for deriving the hyperpolarizability. We determine the differential multipolar polarizability to be $\tilde{\alpha}^{qm}/h=-0.962(40)~{}\rm{mHz}$. The black dashed line shows $\Delta\nu_{\rm LS}^{\rm{vib}}(\bar{u},0)$ at the $E1$ magic frequency $\nu^{E1}$. By setting $\tilde{\beta}=0$ and $\tilde{\alpha}^{qm}=0$, we obtain red and blue lines, which indicate that $\Delta\nu_{\rm LS}^{\rm{vib}}(\bar{u},0)$ is mainly determined by the multipolar polarizability for $\bar{u}<200$ and the hyperpolarizability starts to contribute for higher intensity. Note that the two lines divide the plot into 3 sections indicated by different colors depending on the signs of these polarizabilities.

The lattice-induced light shifts $\nu_{\rm LS}(\bar{u},\delta_{L},0)$ predicted by the obtained polarizabilities are shown in Fig. 4. In addition to making the light shift insensitive to the trap depth $\bar{u}$, i.e., ${{\textstyle{{\partial{\nu_{{\rm{LS}}}}}\over{\partial u}}}}|_{u=\bar{u}^{\rm op}}=0$, the Sr clock transition offers two distinctive operational conditions $(\bar{u}^{\rm op},\delta_{L}^{\rm op})$, as it has the same sign for $\tilde{\beta}$ and $\tilde{\alpha}^{qm}$ SM (2) as indicated by the green area in Fig. 3: (i) by taking $\delta_{L}^{\rm op}=5.3(2)~{}\rm{MHz}$ and $\bar{u}^{\rm{op}}=72(2)$, the total light shift can be reduced to less than $1\times 10^{-19}$ over the trap depth $60<u<83$ as indicated by a red line. Alternatively, (ii) by taking $\delta_{L}^{\rm op}=4.1(1)~{}\rm{MHz}$ and $\bar{u}^{\rm{op}}=28(1)$, an inflection point determined by ${{\textstyle{{\partial^{2}{\nu_{{\rm{LS}}}}}\over{\partial u^{2}}}}}|_{u=\bar{u}^{\rm op}}=0$ offers the light shift variation less than $1\times 10^{-19}$ over the trap depth $17<u<43$ as shown by a blue line. Orange and sky-blue shaded areas indicate the uncertainties of $4\times 10^{-19}$ and $2\times 10^{-19}$ given by those of measured polarizabilities. The $E1$ magic frequency uncertainty of 1.0 MHz for the present measurements, including the tensor-shift contribution as discussed in the Supplemental Material SM (2), gives an overall light-shift uncertainty $3\times 10^{-18}$ at $\bar{u}^{\rm{op}}=72$ (hatched area) and $1\times 10^{-18}$ at $\bar{u}^{\rm{op}}=28$, which can be reduced by improving the statistics of the clock measurements. For the lattice depth of $72E_{R}$ and $28E_{R}$, the off-resonant lattice-photon scattering rate Dörscher et al. , including Raman scattering in the ${}^{3}P_{0}$ state and Rayleigh scattering, is estimated to be $0.1$ and $0.04\,{\rm s}^{-1}$, allowing a sufficient clock interrogation time over multiple seconds.

Figure 5 summarizes reported polarizabilities for the ${}^{1}S_{0}-{}^{3}P_{0}$ clock transition of Sr. The hyperpolarizability $\tilde{\beta}$ determined in this work agrees with the previous results Westergaard et al. (2011); Nicholson et al. (2015) within their uncertainties and is close to a recent theory Porsev et al. (2018). Our multipolar polarizability $\tilde{\alpha}^{qm}$ deviates from the previous experiment Westergaard et al. (2011) that indicates zero within the uncertainty, and from two theories Ovsiannikov et al. (2016); Porsev et al. (2018) that give opposite signs with each other.

In summary, we have determined the differential multipolar $(\tilde{\alpha}^{qm})$ and hyper $(\tilde{\beta})$ polarizabilities for Sr optical lattice clocks by precisely controlling the atomic motion. These polarizabilities predict two distinctive operational conditions: the lattice depth and frequency $\delta_{L}^{\rm op}$ of $(72E_{R},5.3\,{\rm MHz})$ allows canceling out the lattice light shift and $(28E_{R},4.1\,{\rm MHz})$ allows using the inflection point, both of which coincide with typical operating conditions for Sr clocks Ushijima et al. (2015); Dörscher et al. . These operational lattice depths are conveniently described by magic sideband frequencies of $\nu_{z}^{\rm op}=59(1)/\sqrt{\zeta_{1}}$ kHz and $29(1)/\sqrt{\zeta_{1}}$ kHz for the axial motion, respectively, with $\zeta_{1}$ the intensity reduction factor to be measured. A narrow-line cooling Katori et al. (1999) allows $\zeta_{1}\approx 0.91$ or better, which well meets the predicted lattice intensity tolerance of more than 30% around the magic intensity. Combined with cryogenic clocks that reduce the blackbody radiation shift Ushijima et al. (2015), the clock uncertainty at the level of $10^{-19}$ falls within the scope.

We thank M. Das for his contribution to the experiments in the early stage, N. Ohmae for the operation of the comb, and N. Nemitz for careful readings of the manuscript and valuable comments. This work is supported by JST ERATO Grant No. JPMJER1002 10102832 (Japan), by JSPS Grant-in-Aid for Specially Promoted Research Grant No. JP16H06284, and by the Photon Frontier Network Program of the Ministry of Education, Culture, Sports, Science and Technology, Japan (MEXT).