Frequency Ratio of Hg and Sr Optical Lattice Clocks beyond the SI Limit
We report on a frequency ratio measurement of a Hg-based optical lattice clock referencing a Sr-based clock. Evaluations of lattice light shift, including atomic-motion-dependent shift, enable us to achieve a total systematic uncertainty of for the Hg clock. The frequency ratio is measured to be with a fractional uncertainty of , which is smaller than the uncertainty of the realization of the SI second, i.e., the SI limit.
pacs:06.30.Ft,37.10.Jk,32.60.+i, 42.62.Eh, 42.62.Fi
Rapid progress in optical lattice clocks Le Targat et al. (2013); Hinkley et al. (2013); Bloom et al. (2014); Falke et al. (2014); Ushijima et al. (2015) makes them potential candidates for the future redefinition of the second and new tools for testing the fundamental laws of physics Uzan (2003); Derevianko and Pospelov (2014). The absolute frequencies of Sr and Yb-based optical lattice clocks have already been measured with the uncertainty of the realization of the SI second, allowing them to be adopted as the secondary representations of the second by Comitè International des Poids et Mesures (CIPM) CIP (). Moreover, the systematic uncertainties of the Sr clock reach level Bloom et al. (2014); Ushijima et al. (2015), which is competitive to ion-based clocks Chou et al. (2010). Such clocks can be used as new references to investigate precise frequency measurements far beyond the SI second.
Hg is another promising candidate for optical lattice clocks Hachisu et al. (2008); McFerran et al. (2012), because its susceptibility to the blackbody radiation (BBR) is an order of magnitude smaller than that of Sr Bloom et al. (2014); Falke et al. (2014); Ushijima et al. (2015) and Yb Beloy et al. (2014). Furthermore, a large nuclear charge of makes the Hg clock a sensitive probe for testing the constancy of the fine-structure constant Angstmann et al. (2004). So far the absolute frequency of the clock transition of Hg has been reported with an uncertainty of McFerran et al. (2012), where the SI second is sufficient to describe the frequency.
In this Letter, we report on a frequency measurement of Hg by referencing to the Sr clock frequency Ushijima et al. (2015). We have determined the frequency ratio to be with a fractional uncertainty of . This ratio can be converted to via the recommended frequency for the Sr transition CIP () given by CIPM with uncertainty . To improve the systematic uncertainty of the Hg clock to , we investigated the lattice light shift taking into account the multipolar polarizabilities Katori et al. (2015), which affect the light shift more seriously than in Sr and Yb clocks.
Figure 1(a) shows the experimental setup for the frequency ratio measurements, which consists of a Hg clock, a Sr clock Ushijima et al. (2015), and a frequency link between them. The transition of Hg with a nuclear spin of [see Fig. 1(b)] is used as the clock transition and is investigated by the following experimental sequences with a cycle time of 1.5 s. Hg atoms are laser-cooled by a vapor-cell type magneto-optical trapping (MOT) on the transition at 254 nm with a natural linewidth Hachisu et al. (2008). Initially, to collect atoms, we apply a MOT laser detuning and an intensity per beam. After 860-ms-long atom loading time, we increase the gradient of quadrupole magnetic field from to to compress the atom cloud. We finally tune the laser frequency to and reduce the laser intensity to further cool the atoms to maximize the transfer efficiency into a lattice trap.
About 3 % of atoms are loaded into a vertically-oriented one-dimensional (1D) optical lattice operated at nm McFerran et al. (2012) with the maximum trap depth of , where is the lattice-photon recoil frequency, is the Planck constant, and is the mass of Hg atom. We then temporarily decrease the trap depth down to to release the atoms trapped in high-lying axial vibrational states with . This 1D lattice is formed inside a buildup cavity with a power enhancement factor of , which consists of two curved mirrors and a plane folding mirror [see Fig. 1(a)]. These cavity mirrors are placed outside the vacuum chamber to prevent vacuum-degradation of the mirror coating McFerran et al. (2012), while the Brewster windows provide optical access to the vacuum and selective enhancement of the -polarized light. Atoms in the lattice are spin-polarized by exciting the transition with the circularly polarized light. A bias magnetic field is applied during the spin polarization and clock excitation.
The transition is excited by a clock laser at 266 nm generated by two-stage frequency doubling of a fiber laser at 1063 nm stabilized to an Er-doped fiber optical frequency comb by a linewidth transfer method Inaba et al. (2013). The carrier envelope offset frequency of the comb is stabilized by using a self-referencing interferometer. The repetition rate of the comb is then stabilized by referencing a sub-harmonic (1397 nm) of a Sr clock laser at 698 nm, which is prestabilized to a stable reference cavity with instability at s and stabilized to the Sr clock transition for s Ushijima et al. (2015).
The clock laser is superimposed on the lattice laser with the same polarization. Figure 1(c) shows a clock spectrum with a Fourier-limited Rabi linewidth of , corresponding to a -factor of , which is obtained for the clock interrogation time . For the data presented below, we operate the Hg clock with so that the frequency stabilization to the atomic transition becomes robust against the variations of experimental conditions for several hours. The atom population in the state is determined from the fluorescence by operating the MOT for 20 ms. We then optically pump the atoms in the state back to the state by exciting the transition at 405 nm to determine of atoms in the state. The excited atom fraction is used to stabilize the clock laser frequency. We alternately interrogate the two transitions to average out the 1st-order Zeeman shift and the vector light shift Takamoto et al. (2006).
The lattice light shift has been the primary source of the uncertainty of Hg clock McFerran et al. (2012). To evaluate the lattice light shift , we measure the intensity-dependent clock shift in successive measurement cycles by varying the lattice laser intensity , while we keep constant. Here the intensities are given in terms of the peak intensity of a single traveling-wave laser that creates the lattice potential depth of . Figure 2 (a) shows the data taken at 7 different lattice laser frequencies , which are stabilized to the optical frequency comb within . Each data point is measured with an uncertainty of .
In the standing wave field, a spatial mismatch of the light shift due to (i) the electric-dipole (E1) interaction and (ii) the electric-quadrupole (E2) and magnetic-dipole (M1) interactions introduces an atomic-motion-dependent light shift Taichenachev et al. (2008), which we refer to as the multipolar effect. In addition, the light shift due to the hyperpolarizability, coupled with the atomic motion in a lattice potential, introduces non-linear light shift Ovsiannikov et al. (2013). Consequently, the total light shift shows intricate non-linear response to the light intensity .
We estimate the leading hyperpolarizability shift to be Katori et al. (2015) at our maximum lattice laser intensity of , which is smaller than our measurement uncertainties of . Therefore we neglect the other hyperpolarizability effects than the leading term and approximate the clock shift as Katori et al. (2015),
Here, is the vibrational state for the axial motion, and are the differences of E1 and combined E2+M1 polarizabilities of the two clock states, and is the lattice laser detuning from the “E1-magic frequency” that equalizes E1 polarizabilities for the ground (g) and excited (e) states, i.e., . We note that the Gaussian intensity profile of the lattice laser confines atoms radially, where is the beam radius and is the radial displacement. For the typical lattice potential depth , the radial and the axial vibrational frequencies are and . Since the typical kinetic energy of the lattice trapped atoms is about times higher than the radial vibrational energy separation , we treat the radial motion classically. In addition, when calculating an energy eigenvalue for the -th axial vibrational state, the axial and radial motion may be decoupled as the adiabaticity condition is satisfied.
To evaluate the light shift given in Eq. (1), atomic motion in the lattice plays a crucial role, as the axial motion determines the averaged motional state and the radial motion determines the effective lattice intensity via the averaged atomic distribution in the Gaussian intensity profile. The axial vibrational population in the state is measured to be by the difference of the total area of the red and blue motional sideband spectra Leibfried et al. (2003). As atoms in are removed in the state preparation, the average occupation is estimated to be by assuming the atoms are Boltzmann-distributed among vibrational states. The radial atomic distribution can be inferred from the inhomogeneously broadened sideband lineshapes Blatt et al. (2009), as the sideband frequency becomes smaller as increases due to the Gaussian intensity distribution of the lattice laser. The blue-sideband frequency is given by,
Using the sideband spectrum, we determine .
We experimentally measure from the axial motional sideband frequency, where the uncertainty is given by the measurement uncertainty of the lattice laser intensity. We use calculated value and Katori et al. (2015). The entire data are then fitted to Eq. (1) (see Fig. 2), where we employ a multiple regression analysis taking and as explanatory variables and as a response variable. This determines the E1-magic frequency of and .
For the measurement of Hg clock frequency, we take the lattice frequency to be so that the light shift becomes insensitive, i.e., , to the variation of the lattice intensity around . The relevant light shift (black curve) and its uncertainty (gray shaded area) are shown in Fig. 2(b). The lattice light shift is estimated to be corresponding to the fractional frequency shift of . In order to further investigate the lattice light shift, experimental determinations of and are crucial.
|Hg clock systematic total|
|Sr clock systematic|
As listed in Table 1, effects other than the lattice light shifts give relatively minor contributions to the systematic uncertainty of the Hg clock. The BBR shift is estimated to be for the ambient temperature of 297(3) K, where the uncertainty mainly comes from the theoretical polarizabilities with 10% uncertainty Hachisu et al. (2008). The atom density shift is measured by varying the number of atoms by . A linear fit to the data points infer the collisional shift of for atoms in the lattice, corresponding to the atom density of or an average single-lattice-site occupation of 1.2 atoms. The second-order Zeeman shift is investigated by varying in successive measurement cycles, where the magnetic field is measured through the first-order Zeeman shift and the g-factor difference in the clock transition Lahaye and Margerie (1975). Fitting to yields , giving a shift of at our operating condition of . The light shift induced by the clock laser is estimated to be Wexler et al. (1980). The servo error is estimated to be smaller than the statistical uncertainty of the measurement for an averaging time by analyzing the error signal of the frequency stabilization to the clock transition.
Sources of uncertainties for the frequency ratio measurement are also listed in Table 1. The Sr clock achieves the systematic uncertainty of Ushijima et al. (2015). The gravitational redshift is estimated from the height difference of between the two clocks. The first-order Doppler shift between the Hg and the Sr clocks is estimated to be less than , which is due to the temperature drift in the uncompensated optical path of about .
Figure 3(a) displays the Allan standard deviation for the ratio measurement , which shows a trend for an averaging time of 10 s, mainly responsible to the Dick-effect-limited instability Santarelli et al. (1998) of . The quantum projection noise (QPN) Itano et al. (1993) only contributes to the instability of . Application of a clock laser with smaller instability or synchronous operation of two clocks to reject frequency noise of the clock laser Takamoto et al. (2011) will allow approaching the QPN-limited instability.
Figure 3(b) summarizes the frequency ratio measurement . For the evaluation of the statistical uncertainty of , we construct a histogram of all the data with the bin size of for a fractional frequency. The histogram is fitted to the normal distribution function, where the reduced chi-squared yields . We thus conservatively evaluate the statistical uncertainty to be by inflating the standard uncertainty of the mean by . A weighted mean of the result is , where the fractional uncertainty of is essentially given by the systematic uncertainty of the Hg clock (see Table 1). The ratio , both of which are measured by referencing Cs primary standards at LNE-SYRTE McFerran et al. (2012); Le Targat et al. (2013), is shown by a triangle in Fig. 3(c). This result is consistent with our measurements to about uncertainty. Applying the recommended value of Sr as a secondary representation of the second CIP (), the absolute frequency of Hg is given by in the unit of SI second, where an uncertainty of is given by that of .
In summary, we have investigated an optical lattice clock based on Hg and achieved a total systematic uncertainty of . We determine the magic frequency by including the multipolar effect that seriously affects the uncertainty budget at low . Similar strategy will be applied for Sr and Yb-based clocks, where the multipolar and hyperpolarizability effects become relevant for the systematic uncertainty at the level Katori et al. (2015). We have determined the frequency ratio between optical lattice clocks with an uncertainty of . Accurate determinations of such ratios allow the investigation of the constancy of the fine-structure constant . Taking the Sr clock as an -insensitive anchor, the fractional change of the Hg clock reveals Angstmann et al. (2004), which will be competitive to the previous constraints Rosenband et al. (2008) if is investigated over a year.
Acknowledgements.We thank F.-L. Hong, H. Inaba, Y. Kaneda, and P. Thoumany for laser developments, and T. Akatsuka, M. Das, N. Nemitz, T. Pruttivarasin, T. Takano, and A. Yamaguchi for useful comments and discussions. This work was partly supported by the FIRST Program of the JSPS and by the Photon Frontier Network Program of the MEXT, Japan. K. Y. acknowledges financial support from Grant-in-Aid for JSPS Fellows and ALPS.
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