Determining the Magic Wavelength Without Modulation of the Trap Depth

Abstract

In this paper, the magic wavelength of the ⁸⁷Sr optical lattice clock is determined by a method that bypasses the need for lattice trap depth modulation. Instead, it relies on an additional AC Stark shift generated by a dipole beam operating near the frequency of the lattice light and oriented perpendicular to the optical lattice. The magic wavelength is inferred by measuring the AC Stark shift induced by the dipole beam as a function of its power under various frequency detunings. The effect of the dipole beam on the external states of the cold ensemble is evaluated through comparative analysis of the radial and axial sideband spectra, both with and without the dipole beam. Variations in density shift resulting from changes in external states are evaluated using comprehensive numerical calculations. By avoiding trap depth modulation, this method effectively suppresses the influence of the density shift, thereby offering a promising avenue for accurately determining the magic wavelength.

1. Introduction

Using an optical tweezer or lattice to trap neutral atoms in the Lamb–Dicke limit effectively suppresses the Doppler shift and the photon-recoil shift during the interrogation of specific transitions [1,2]. Operating the optical lattice near the magic frequency induces a low-level energy shift that is uniform between ground and excited states [2,3,4,5], thereby facilitating long coherent times between the cold ensemble and the optical field. This characteristic is pivotal for various quantum appliances and sciences reliant on light-trapped neutral atoms, including optical lattice clocks (OLCs) [6,7,8,9,10,11,12], quantum computing [13,14], and quantum simulation [15,16].

The magic wavelength of the trapped light for a specific transition can be roughly determined by observing the width and asymmetry of the transition spectra. Closer to the magic frequency, more homogeneous excitation is achieved across all motional states [3,4,5]. For precise determination, the typical method involves modulating the trap depth and measuring the differential AC Stark shift between high and low trap depths [17,18,19,20]. However, the density shift in a lattice-trapped system, which is contingent upon the trap depth and comparable to the AC Stark shift induced by the trap light, necessitates careful correction under varying trap depths [17,18,20]. Recently, the magic wavelength has also been determined with an uncertainty of 30 MHz by measuring the coherent time between two cold ensembles as a function of the lattice light wavelength [21]. This new method, however, requires employing the in situ imaging technique and interrogating the clock transition with a light pulse duration exceeding 10 s.

In this work, we present a method to determine the magic wavelength of the ⁸⁷Sr optical lattice clock by overlapping an additional dipole beam with the cold ensemble. The corresponding magic wavelength is deduced from the relationship between the differential AC Stark shift with and without the dipole beam and its frequency [22]. We carefully suppressed the influence of the dipole beam on the motional states and atomic temperature, using perpendicular polarization between the dipole and lattice beams. Additionally, we intentionally deviated the frequencies of the dipole and lattice beams by 5 MHz to further mitigate interference effects. The aforementioned possible effects were not considered in Ref. [22]. We thoroughly evaluated potential systematic errors from the dipole beam based on longitudinal and radial motional spectra. This comprehensive approach ensured that the impact of the additional dipole beam on the experimental results was properly accounted for and minimized, allowing for an accurate determination of the magic wavelength of the OLC.

2. Experimental Setup and Method

This work ws based on our ⁸⁷Sr optical lattice clock [23], and the experimental setup of measuring the differential AC Stark shift is illustrated in Figure 1a. The details of the laser cooling and the initial state preparation are elaborated in Ref. [23]; herein, we only introduce the key elements pertinent to this work. After reducing the atomic temperature to about 3 $\mathsf{\mu }\mathrm{K}$, using two-stage laser cooling, atoms are confined in a horizontal one-dimensional optical lattice with a depth of $U=132E_{\mathrm{r}}$. Here, $E_{\mathrm{r}}={\hslash }^2{k}^2/2m$ signifies the photon recoil energy of the lattice light, where $k=2\pi {\nu }_{\mathrm{L}}/c$ denotes the wave vector of the lattice light, defined by the lattice light frequency ${\nu }_{\mathrm{L}}$ and the speed of light c; ℏ represents the reduced Plank constant, while m denotes the atomic mass of ⁸⁷Sr. The lattice light frequency, stabilized to a 10 cm ultra-low expansion (ULE) cavity by the Pound–Drever–Hall (PDH) technique, is determined by an optical frequency comb (made by IMRA). The carrier–envelope offset frequency and repetition frequency of the optical frequency comb are referenced to the H-maser (VCH-1003M, No.5085). The cold atoms are optically repumped to the $|m_F=\pm 9/2\rangle$ states via the $|5s^2{}^{1}S_0,F=9/2\rangle \to |5s5p{}^{3}P_1,F=9/2\rangle$ transitions. To remove hotter atoms, an energy-filtering method is employed. We linearly reduce the trap depth of the optical lattice from $132E_{\mathrm{r}}$ to $60E_{\mathrm{r}}$ within 20 ms, using a voltage-controlled attenuator to adjust the diffraction efficiency of the acoustic optical modulator (AOM). This adjustment changes the power of the lattice light from approximately 310 mW to 140 mW. The trap depth is then maintained at $60E_{\mathrm{r}}$ for 10 ms before being linearly increased back to $132E_{\mathrm{r}}$. The entire process remains adiabatic, to avoid heating. During reduction of the trap depth, hotter atoms escape the lattice, resulting in approximately 1000 atoms being evenly distributed in 900 lattice sites. The atom number is determined by the fluorescence imaging method, with an uncertainty conservatively estimated at 30%. This uncertainty arises from limited knowledge of probe light intensity, influenced by the distribution of laser intensity, the central position of the laser beam profile, and the free-space Rabi frequency.

We eliminate the contributions from the vector Stark shift and first-order Zeeman shift via averaging the frequency of the $|5s^2{}^{1}S_0,m_F=\pm 9/2\rangle \to |5s5p{}^{3}P_0,m_F=\pm 9/2\rangle$ transitions at 698 nm, which corresponds to the output frequency of the ⁸⁷Sr OLC. The clock laser is stabilized in a 10 cm ULE cavity with a fineness of 300,000. To mitigate frequency drift caused by changes in cavity length, we linearly scan the driving frequency of the AOM, effectively reducing the drift below 1 mHz/s.

The measurement of the magic wavelength proceeds as follows: First, we fix the frequency of the lattice beam, to ensure a fixed trap depth. Next, we obtain the frequency of the clock transition by averaging the frequencies of the $|5s^2{}^{1}S_0,m_F=\pm 9/2\rangle \to |5s5p{}^{3}P_0,m_F=\pm 9/2\rangle$ transitions at 698 nm. We then sequentially turn on the dipole beam, as shown in Figure 1b, and we measure the frequency of the clock transition again, allowing us to infer the AC Stark shift resulting from dipole beam. Subsequently, we vary the frequency and power of the dipole beam and repeat the process. Ultimately, the frequency that eliminates the linear term of the AC Stark shift corresponds to the magic wavelength.

In this work, the lattice beam frequency was fixed at 368554839 MHz [24], corresponding to the magic wavelength when the polarization direction was perpendicular to the quantum axis. Importantly, the differential AC Stark shift is independent of the lattice light frequency, whether a dipole beam is present or not. Moreover, this method exhibits insensitivity to the drift and fluctuation of the lattice trap depth, because the differential AC Stark shift is solely determined by the power ($P_{\mathrm{d}}$) and frequency of the dipole beam. Additionally, the contribution from polarization fluctuations in the lattice beam is suppressed in a common-mode manner during the interleaved self-comparison process.

The differential AC Stark shift can be expressed as follows [20]:

$${\displaystyle \frac{\Delta {\nu }_{\mathrm{AC}}}{{\nu }_{\mathrm{clock}}}}=-A_n{\alpha }_{E1}U-B_n{\alpha }_{QM}U-C_n\beta U^2,$$

(1)

where $A_n\equiv \langle \mathrm{n}|exp({\displaystyle \frac{-2({\mathrm{x}}^2+{\mathrm{y}}^2)}{{\omega }_0^2}}){cos}^2\left(\mathrm{kz}\right)|\mathrm{n}\rangle$, $B_n\equiv \langle \mathrm{n}|exp({\displaystyle \frac{-2({\mathrm{x}}^2+{\mathrm{y}}^2)}{{\omega }_0^2}}){sin}^2\left(\mathrm{kz}\right)|\mathrm{n}\rangle$, and $C_n\equiv \langle \mathrm{n}|exp({\displaystyle \frac{-4({\mathrm{x}}^2+{\mathrm{y}}^2)}{{\omega }_0^2}}){cos}^4\left(\mathrm{kz}\right)|\mathrm{n}\rangle$ represent the spatial average for the motional state $\mathrm{n}$, while ${\alpha }_{E1}$, ${\alpha }_{QM}$, and $\beta$ denote the differences in electric dipole (E1) polarizability, multipolarizability (which includes the electric quadrupole (E2) and magnetic dipole (M1) interaction), and hyperpolarizability on the clock transition. Considering that spatial modes scale with the trap depth, the differential AC Stark shift can be effectively expressed by [20]

$$\Delta {\nu }_{\mathrm{AC}}={\alpha }^{*}U+{\beta }^{*}{U}^2.$$

(2)

In Equation (2), ${\alpha }^{*}$ and ${\beta }^{*}$ are the coefficients determined through experimental measurements. Specifically, ${\alpha }_{E1}$ and ${\alpha }_{QM}$ are encompassed within ${\alpha }^{*}$, while $\beta$ is contained within ${\beta }^{*}$. Notably, this thermal model circumvents the need for precise knowledge of atomic coefficients or mode numbers. Since the trap depth is proportional to the light power, Equation (2) can be directly applied to the dipole beam case by substituting the trap depth with $P_{\mathrm{d}}$. Additionally, Equation (1) shows that three different lattice light-induced effects are identified. The first results from the atomic hyperfine structure, which induces small vector and tensor components in the atomic polarizability, making light shifts slightly dependent on the lattice polarization. Therefore, in our experiment, the dipole beam polarization along the quantization axis affected ${\alpha }_{E1}$, corresponding to ${\alpha }^{*}$ in Equation (2). However, according to the method described for determining the magic wavelength by analyzing how ${\alpha }^{*}$ changes with ${\nu }_{\mathrm{L}}$, dipole beam polarization does not impact the determination of the magic wavelength. Moreover, the size of the dipole beam is large enough to cover the atoms, ensuring that the atoms experience a uniform laser intensity.

3. Results and Discussion

The determination of the magic wavelength involved measuring the differential AC Stark shifts through the interleaved self-comparison method at $P_{\mathrm{d}}$ of 200 mW, 430 mW, 630 mW, and 830 mW, respectively. These measurements were repeated at dipole beam frequencies of 368544995.78 MHz, 368553979.18 MHz, and 368562962.58 MHz, while keeping the lattice frequency fixed at 368554839 MHz, respectively, as shown in Figure 2a. In other words, we measured additional differential AC Stark shifts only at the fixed trap depth. The Allan deviation of one AC Stark shift data is shown in Figure 2b, with an interleaved self-comparison stability of $7\times {10}^{-15}{(\tau /\mathrm{s})}^{-0.5}$. Applying least-squares fitting according to Equation (2) and weighted averaging across the three results, the coefficient ${\beta }^{*}$ was determined to be $-7.57\left(252\right)\times {10}^{-9}{\mathrm{MHz}\mathrm{W}}^{-2}$. Subsequently, with ${\beta }^{*}$ fixed, we fitted the linear coefficient ${\alpha }^{*}$ at different dipole beam frequencies. Due to ${\alpha }^{*}\left({\nu }_{\mathrm{L}}\right)=(\partial {\alpha }^{*}/\partial {\nu }_{\mathrm{L}})\times ({\nu }_{\mathrm{L}}-{\nu }_{\mathrm{zero}})$ with ${\nu }_{\mathrm{zero}}$ denoting the magic frequency, we could obtain $\partial {\alpha }^{*}/\partial {\nu }_{\mathrm{L}}=-6.75\left(22\right)\times {10}^{-4}{\mathrm{W}}^{-1}$ and the linear shift vanished at ${\nu }_{\mathrm{zero}}$ = 368554490(10) MHz, consistent with the values of Refs. [10,17,25,26,27]. It is crucial to emphasize that ${\nu }_{\mathrm{zero}}$ corresponded to the “magic frequency” considering solely the contributions from the E1 and E2/M1 interactions, due to ${\nu }_{\mathrm{zero}}$ being directly proportional to the ${\beta }^{*}\times P_{\mathrm{d}}(\partial {\alpha }^{*}/\partial {\nu }_{\mathrm{L}})$, and to its value being decreased by 9.3 MHz if hyperpolarizability and multipolarizability were not accounted for, given a maximum $P_{\mathrm{d}}$ of 0.83 W.

In Ref. [22], the density shift caused by interaction between atoms at a lattice site was overlooked throughout the measurement process. This phenomenon was extensively investigated in OLCs [6,7,8,9,10,11,12,17,18,20], as the density shift competes significantly with the lattice AC Stark effect and necessitates meticulous mitigation, especially in determining the differential AC Stark shift through trap depth modulation. Despite endeavors to suppress density shift variation through dipole beam detuning and perpendicular polarization, subtle changes in radial trap frequency (${\nu }_r$) and temperature ($T_r$) exist, as evidenced by Figure 3a,b, illustrating motional sideband spectra without and with radial components. The asymmetric lineshape of the longitudinal sideband spectrum arises from the longitudinal anharmonic confinement and the transverse spatial extent of the atomic wave function, where the trap confinement is weaker. This configuration results in most atoms occupying the ground state ( $n_z=0$), suppressing the red sideband ($n_z\to n_z-1$) relative to the blue sideband ($n_z\to n_z+1$). In the transverse direction, the weaker confinement results in almost equal probabilities for the $\Delta n=\pm 1$ transitions, leading to a symmetric distribution in the radial sideband spectrum. With the maximum $P_{\mathrm{d}}$ utilized in this work, the variations in axial trap frequency (${\nu }_z$), ${\nu }_r$ and $T_r$ were determined to be −0.5(10) kHz, 13(2) Hz, and −0.6(2) $\mathsf{\mu }\mathrm{K}$, respectively. Notably, no discernible variation in longitudinal temperature ($T_z$) was detected. As the factors influencing the density shift encompassed trap depth, atom number, excitation fraction, and atomic temperature [28], variations in trap frequency and atomic temperature could lead to non-negligible error in the measurement result. However, we failed to observe the variation of the density shift as a function of $P_{\mathrm{d}}$, due to its minute fraction, estimated at ${10}^{-17}$.

To evaluate potential errors stemming from density shift accounting for atomic temperature, we started with the system Hamiltonian ${\widehat{H}}_s$ , which includes the scattering Hamiltonian [28,30,31], and which can be expressed as ${\widehat{H}}_s={\sum }_{\overrightarrow{\mathit{n}}}{\widehat{H}}_{\overrightarrow{\mathit{n}}}$, where

$${\displaystyle \frac{{\widehat{H}}_{\overrightarrow{\mathit{n}}}}{\hslash }}=-2\pi \delta \sum _i^N{\widehat{S}}_{{\mathit{n}}_{\mathit{i}}}^z+2\pi \sum _i^N{\Omega }_{{\mathit{n}}_{\mathit{i}}}{\widehat{S}}_{{\mathit{n}}_{\mathit{i}}}^x-\sum _{i\ne j}^N{C}_{{\mathit{n}}_{\mathit{i}},{\mathit{n}}_{\mathit{j}}}{\displaystyle \frac{{\widehat{S}}_{{\mathit{n}}_{\mathit{i}}}^z+{\widehat{S}}_{{\mathit{n}}_{\mathit{j}}}^z}{2}}-\sum _{i\ne j}^N{X}_{{\mathit{n}}_{\mathit{i}},{\mathit{n}}_{\mathit{j}}}{\widehat{S}}_{{\mathit{n}}_{\mathit{i}}}^z{\widehat{S}}_{{\mathit{n}}_{\mathit{j}}}^z-\sum _{i\ne j}^N{J}_{{\mathit{n}}_{\mathit{i}},{\mathit{n}}_{\mathit{j}}}{\widehat{\mathit{S}}}_{{\mathit{n}}_{\mathit{i}}}·{\widehat{\mathit{S}}}_{{\mathit{n}}_{\mathit{j}}}.$$

(3)

Here, N is the atom number within a lattice site. The subscript $\overrightarrow{\mathit{n}}$ of ${\widehat{H}}_{\overrightarrow{\mathit{n}}}$ refers to the external mode configuration, with $\overrightarrow{\mathit{n}}=\{{\mathit{n}}_{\mathbf{1}},{\mathit{n}}_{\mathbf{2}},{\mathit{n}}_{\mathbf{3}},...,{\mathit{n}}_{\mathit{n}}\}$ being an array of length N; ${\mathit{n}}_{\mathit{i}}=\{n_{i_x},n_{i_y},n_{i_z}\}$ represents a three-dimensional harmonic mode. The effective spin operator is defined as ${\widehat{\mathit{S}}}_{{\mathit{n}}_{\mathit{i}}}={\displaystyle \frac{1}{2}}{\sum }_{\alpha ,\beta =e,g}{\widehat{c}}_{\alpha {\mathit{n}}_{\mathit{i}}}^{†}{\sigma }_{\alpha \beta }{\widehat{c}}_{\beta {\mathit{n}}_{\mathit{i}}}$, where ${\sigma }_{\alpha \beta }^{x,y,z}$ represents the Pauli matrices, and where ${\widehat{c}}_{\alpha {\mathit{n}}_{\mathit{i}}}$ is the fermionic annihilate operator that annihilates an ⁸⁷Sr atom with external mode ${\mathit{n}}_{\mathit{i}}$ and electronic state $\alpha$. Here, $\delta$ denotes the laser detuning from the resonant clock transition frequency, and ${\Omega }_{\mathit{i}}$ stands for the mode-dependent Rabi frequency. The coefficients $J_{{\mathit{n}}_{\mathit{i}},{\mathit{n}}_{\mathit{j}}}$, $C_{{\mathit{n}}_{\mathit{i}},{\mathit{n}}_{\mathit{j}}}$, and $X_{{\mathit{n}}_{\mathit{i}},{\mathit{n}}_{\mathit{j}}}$ describe the scattering processes between the external modes ${\mathit{n}}_{\mathit{i}}$ and ${\mathit{n}}_{\mathit{j}}$. Note that the scattering Hamiltonian ${\widehat{H}}_s$ is conserved with external configuration, meaning that atom–light interaction and scattering do not lead to transitions between different external configurations. This symmetry results from the frozen mode approximation [28], which assumes that the collisional energy scale is too small to induce transitions between mode configurations. Due to the conservation of the external configurations, we could address each external mode configuration independently. By diagonalizing the subspace Hamiltonian ${\widehat{H}}_{\overrightarrow{\mathit{n}}}$, we could simulate the Rabi spectrum for the external configuration $\overrightarrow{\mathit{n}}$. The Rabi spectrum of the whole atomic ensemble was obtained by the weighted average of the Rabi spectrum of each configuration according to the thermal distribution of $\overrightarrow{\mathit{n}}$. The density shift could then be extracted from the simulated Rabi spectrum. However, the number of possible external configurations was extremely large. Instead of calculating the spectrum for each $\overrightarrow{\mathit{n}}$, we employed multiband sampling exact diagonalization (MBSED) [31], which combined the Monte Carlo sampling of the motional states of the ensemble with the exact diagonalization algorithm, allowing for high-precision simulations of the collisional dynamics during clock interrogation. Specifically, we first sampled the external mode configuration $\overrightarrow{\mathit{n}}$ according to the Fermion–Dirac distribution; then, we calculated the Rabi spectrum by diagonalizing the subspace Hamiltonian ${\widehat{H}}_{\overrightarrow{\mathit{n}}}$. Finally, we averaged all the spectrum lineshapes to determine the density shift.

The key parameters for this numerical calculation included a free-space coupling strength of $3.9\times 2\pi$ Hz and a 6 mrad misalignment angle between the clock laser and the lattice. Additionally, other significant parameters were ${\nu }_z$ = 79.5 kHz, ${\nu }_r$ = 386 Hz, $T_r=6.8\mathsf{\mu }\mathrm{K}$, and $T_z=3.4\mathsf{\mu }\mathrm{K}$, all without the dipole beam. We meticulously calculated the variation of the density shift by the dipole beam, as illustrated in Figure 4a,b. The density shift was calculated using the MBSED method, with $P_{\mathrm{d}}$ set to 830 mW across a wide parameter range. We corrected the change of density shift due to the dipole beam, based on the motional sideband spectra and MBSED method, point by point, as depicted in Figure 2a. The uncertainty of the density shift corrections was conservatively estimated to be less than 2 mHz, accounting for 30% uncertainty in the atom number measurement and 15% uncertainty in the motional states and atomic temperature. This 2 mHz uncertainty translated to a deviation of 3.7 MHz in the measurement of the magic wavelength. Considering the uncertainties of 9.5 MHz from the fitting results, the overall uncertainty associated with the measurement of the magic wavelength in our experimental setup was determined to be 10 MHz. This level of precision surpassed that reported in Ref. [22], where the effect of density shift on the determination of the magic wavelength was not considered. Obviously, the density shift cannot be ignored when aiming for the magic wavelength with precision at MHz level, even when the average atom in each lattice approximates to 1. To further reduce the influence of the density shift, operating the lattice in the shallow lattice region or even at the "magic trap depth" becomes necessary [32]. For an optical lattice clock, atoms within one lattice site experience the same probe light field, leading to highly collective internal dynamics. Consequently, only a few anti-symmetric components exist for atoms within one lattice site, suppressing s-wave interactions and allowing only p-wave interactions. However, atoms located at different sites can exhibit anti-symmetric components, due to variations in probe laser phases caused by mismatches between the lattice laser and the probe laser wavelengths. This leads to the possibility of s-wave interactions between atoms at different sites. In shallow lattices aligned with gravity, atoms are localized in Wannier–Stark states, which are spread across multiple lattice sites. As the lattice becomes shallower, these Wannier–Stark states become more delocalized, increasing the likelihood of off-site (s-wave) interactions. According to Ref. [32], at specific lattice depths the on-site p-wave interaction achieves a "magic balance" with the off-site s-wave interaction, effectively eliminating the collisional shift. Therefore, the density shift can be further suppressed by approximately 3 orders of magnitude, reducing its influence due to the dipole beam to below 0.1 MHz.

4. Conclusions

In summary, our study presents a convenient method for accurately measuring the AC Stark shift by overlapping an 813 nm dipole beam, ultimately determining the magic wavelength to be 368554490(10) MHz. However, subtle changes in radial trap frequency and temperature are revealed through motional sideband spectra. Our analysis, based on the MBSED method, indicates that when the measurement precision of the magic wavelength is at the MHz level, the impact of the dipole beam on the density shift requires careful evaluation. To enhance the measurement precision of this method to 0.1 MHz, it is essential to integrate the "magic trap depth", which can effectively mitigate density shifts. Thereby, except for the determination of the magic wavelength, this convenient method can also offer avenues for measuring the atomic polarizations and hyperpolarizability [5,20].