## Keeping Time

Optical lattice clocks are comprised of atoms placed in an optical lattice formed by opposing laser beams and can be more precise than traditional microwave atomic clocks because of the higher frequency at which they operate, and the number of atoms available for interrogation. However, interactions between the atoms may lead to shifts in the frequency of the clock transition, usually proportional to the atomic density. **Swallows et al.** (p. 1043, published online 3 February) demonstrate an opposite and unexpected effect of interactions: For sufficiently strongly interacting systems, the frequency shift is suppressed. Indeed, in a strontium-based fermionic lattice clock, the shift and its associated spread were reduced by an order of magnitude.

## Abstract

Optical lattice clocks with extremely stable frequency are possible when many atoms are interrogated simultaneously, but this precision may come at the cost of systematic inaccuracy resulting from atomic interactions. Density-dependent frequency shifts can occur even in a clock that uses fermionic atoms if they are subject to inhomogeneous optical excitation. However, sufficiently strong interactions can suppress collisional shifts in lattice sites containing more than one atom. We demonstrated the effectiveness of this approach with a strontium lattice clock by reducing both the collisional frequency shift and its uncertainty to the level of 10^{−17}. This result eliminates the compromise between precision and accuracy in a many-particle system; both will continue to improve as the number of particles increases.

Strongly interacting quantum systems can exhibit counterintuitive behaviors. For example, frequency shifts of a microwave transition in a quantum gas remain finite close to a Feshbach resonance (*1*–*3*). In particular, the effective interaction strength is enhanced in low-dimensional systems, resulting in particles that avoid each other so as to minimize their total energy. This tendency can lead to behavior that in many respects resembles that of noninteracting systems. One such example is the Tonks-Girardeau regime of an ultracold Bose gas, in which the strong repulsion between particles mimics the Pauli exclusion principle, causing the bosons to behave like noninteracting fermions (*4*–*7*). Here we show that the enhancement of atomic interactions in a strongly interacting, effectively one-dimensional (1D) fermionic system suppresses collisional frequency shifts in an optical atomic clock.

A primary systematic effect of state-of-the-art optical lattice clocks is the density-dependent frequency shift (*8*, *9*). This shift arises from collisions between fermionic atoms that are subject to slightly inhomogeneous optical excitations (*10*, *11*); several theories of the shift mechanism have been proposed (*12*–*14*). By tightly confining atoms in an array of quasi-1D potentials formed by a 2D optical lattice, we increase the strength of atomic interactions to the point where the thermally averaged mean interaction energy per particle becomes the largest relevant energy scale other than the temperature, making the system strongly interacting in effect. In this regime, collisions are suppressed because evolution into a many-particle state in which *s*-wave scattering can occur is energetically unfavorable. This mechanism was first suggested in (*13*).

Collisional frequency shifts could also be suppressed by confining atoms in a 3D lattice with filling factor less than or equal to 1 per lattice site. However, vector and tensor shifts of the optical clock transition are a serious concern with a 3D fermionic lattice clock (*15*). A 3D lattice clock using bosonic ^{88}Sr has been demonstrated (*16*), and its collisional shift was characterized at the level of 7 × 10^{−16}, but the state-mixing techniques that are used to enable the ^{1}*S*_{0} → ^{3}*P*_{0} clock transition in bosonic isotopes result in sizable systematic shifts of the clock frequency that must be carefully controlled. The work presented here will allow operation of a fermionic lattice clock with a filling factor much greater than 1 and a greatly reduced sensitivity to collisional effects.

To gain insight into the origin of the collisional frequency shift and the interaction-induced suppression, we consider a model system: two fermionic atoms, each of whose electronic degrees of freedom form a two-level, pseudospin-½ system (|*g*〉 and |*e*〉), confined in a 1D harmonic oscillator potential [for a full many-body treatment of an arbitrary number of atoms, see (*17*)]. The collective pseudospin states of these two identical fermions can be expressed with a basis comprising three pseudospin-symmetric triplet states and an antisymmetric singlet state (*12*, *13*). Because the atoms are initially prepared in the same internal state (|*g*〉), with their internal degrees of freedom symmetric with respect to exchange, the Pauli exclusion principle requires that their spatial wave function be antisymmetric and thus they experience no *s*-wave interactions. If the atoms are coherently driven with the same Rabi frequency, their electronic degrees of freedom remain symmetric under exchange. Here, *n _{i}* represents axial vibrational modes in each 1D tube-shaped optical trap, and is the mode-dependent Rabi frequency, which is proportional to the bare Rabi frequency . Consequently, these atoms will not experience any

*s*-wave interactions during the excitation of the clock transition. However, if is not zero, the optical excitation inhomogeneity can transfer atoms with a certain probability to the antisymmetric spin state (singlet) that is separated from the triplet states by an interaction energy

*U*, because in the singlet state the atoms do interact.

*U*, which is inversely proportional to the atomic confinement volume, gives rise to a frequency shift during clock interrogation (

*12*,

*13*).

Figure 1 contrasts the current 2D lattice experiment with prior studies carried out in a 1D lattice (*10*, *11*). In a 1D lattice, *U* is typically smaller than (the energy spread of the driven triplet states at zero detuning). Consequently, any small excitation inhomogeneity ∆Ω can efficiently populate the singlet state. By tightly confining atoms in a 2D lattice, one can reach the limit where *U* >>, inhibiting the evolution into the singlet state; as a result, the collisional frequency shift of the clock transition is suppressed. In this regime, the singlet state can only participate as a “virtual” state in second-order excitation processes and the frequency shift scales as ∆Ω^{2}/*U*. Such behavior is reminiscent of the dipolar blockade mechanism in a Rydberg atom gas (*18*). In effect, the singlet resonance has been shifted so far from the triplet resonances that it is completely resolved from them, and any associated line pulling is negligible.

This simple spin model can be extended to the finite temperature regime with a thermal average over vibrational modes *n _{i}*. Figure 2 shows the calculated fractional frequency shift as a function of the temperature-independent interaction parameter [

*u*is related to the thermally averaged quantity

*U*; see (

*17*) for detailed derivations]. Here, is the geometric mean of the transverse trapping frequencies, is the singlet

*g*-

*e*scattering length, and

*a*

_{ho}is the harmonic oscillator length along . The suppression becomes less effective if becomes comparable to

*u*, or when increases at larger temperatures. These considerations imply that clock experiments based on Ramsey interrogation will not easily satisfy the suppression conditions outlined here, because the short pulses applied in the Ramsey scheme generally have a Rabi frequency much larger than those used in Rabi spectroscopy.

Our experiment uses ultracold fermionic ^{87}Sr atoms that are nuclear spin-polarized (*17*). We determined the nuclear spin purity of the atomic sample to be greater than 97%. An ultranarrow optical clock transition, whose absolute frequency has been precisely measured (*19*), exists between the ground ^{1}*S*_{0} (|*g*〉) and excited metastable ^{3}*P*_{0} (|*e*〉) states. Atoms are trapped in a deep 2D optical lattice at the magic wavelength where the ac Stark shifts of |*g*〉 and |*e*〉 are matched (*20*). The 2D lattice provides strong confinement along two directions ( and ) and relatively weak confinement along the remaining dimension (). Using Doppler and sideband spectroscopy, we determined that the lattice-confined atoms are sufficiently cold (*T _{X}* ≈

*T*≈ 2 μK) that they primarily occupy the ground state of the potentials along the tightly confined directions, with trap frequencies ω

_{Y}*/2π ≈ 75 to 100 kHz and ω*

_{X}*/2π ≈ 45 to 65 kHz. This creates a 2D array of isolated tube-shaped potentials oriented along , which have trap frequencies ω*

_{Y}*/2π ≈ 0.55 to 0.75 kHz. We estimate that 20 to 30% of the populated lattice sites are occupied by more than one atom. At a typical axial temperature*

_{Z}*T*of a few μK, various axial vibrational modes

_{Z}*n*are populated in each tube. In our clock experiment, the |

*g*〉→ |

*e*〉 transition is interrogated via Rabi spectroscopy with the use of a narrow-linewidth laser propagating along . The clock laser and both lattice beams are linearly polarized along . As described in (

*10*,

*21*), any small projection of the probe beam along leads to a slightly different Rabi frequency Ω

*for each mode , where is the Lamb-Dicke parameter and*

_{n}*k*represents a small component of the probe laser wave vector along , resulting in a typical η

_{Z}*≈ 0.05.*

_{Z}Spectroscopy of the clock transition is performed with an 80-ms pulse, resulting in a Fourier-limited linewidth of ~10 Hz. The laser power is adjusted to produce a π-pulse on resonance, and the clock laser is locked to the atomic resonance by probing two points on either side of the resonance, with a frequency separation corresponding to the resonance full width at half maximum. The high-finesse Fabry-Perot cavity (*22*) used to narrow the clock laser’s linewidth is sufficiently stable over short time scales that it can be used as a frequency reference in a differential measurement scheme (*23*). A single experimental cycle (e.g., cooling and trapping atoms, preparing the 2D lattice, and interrogating the clock transition) requires about 1.5 s, and we modulate the sample density every two cycles. The corresponding modulation of the atomic resonance frequency relative to the cavity reference is a measurement of the density shift.

We performed measurements at several trap depths to directly observe the interaction-induced suppression of the collisional frequency shift. To access different interaction energies, we varied the intensity of the horizontal lattice beam (*I _{X}*), which resulted in changed values mainly for ω

*but also for ω*

_{X}*and ω*

_{Y}*. The change in ω*

_{Z}*arises from the fact that the laser beams that create the two lattices are not orthogonal but instead meet at an angle of 71°. The change in ω*

_{Y}*results from the Gaussian profile of the beams. Because , an increase of the horizontal beam power leads to a monotonic increase of*

_{Z}*u*. We observe a significant decrease of the collisional shift with increasing horizontal lattice power, as shown by the data points (solid black squares and blue triangles) in Fig. 3 (inset); squares and triangles indicate data taken with slightly different beam waists. We have also studied the dependence of the collisional shift on the Rabi frequency used to drive the clock transition. was increased by a factor of 2 and the interrogation time was decreased by a factor of 2, yielding a constant Rabi pulse area. Under these conditions, we observed that the collisional shift under similar temperature and trapping conditions increases sharply (green open square and green open triangle in Fig. 3, inset), which confirms that the shift suppression mechanism will not operate effectively for short, higher Rabi-frequency pulses.

In (*17*), we accounted for the combined effects of tunneling, energy offset between wells, and interactions and estimated a fractional error due to tunneling on the order of 1%. Also shown in the inset of Fig. 3 is the shift predicted by the spin model, assuming two atoms per lattice site. The theoretical points are scaled by the fraction of the atomic population in doubly occupied lattice sites. The pink rectangles are the theory results, with *i* = 1, …, 9, obtained at different temperatures, trapping frequencies, and Rabi frequencies corresponding to the actual experimental conditions under which the data were taken. The spread of theory results (indicated by the vertical extent of the pink rectangles) corresponds to a range of effective scattering lengths (where *a*_{0} is the Bohr radius). Here we neglect the variation of trap depths across lattice sites (*17*) by assuming that all sites are equivalent to those at the center; the true magnitude of is therefore larger than these effective values.

Because the temperature and trapping conditions substantially varied for different experimental data points, some scaling is required to make direct comparisons between the data in Fig. 3 (inset) and the behavior predicted in Fig. 2. To aid visualization of the experimental confirmation of the interaction suppression mechanism, we rescaled the experimentally measured shift values by a factor , which is extracted from the theoretical model. Figure 3 shows that after rescaling, all data points lie very close to the theoretical curve of fractional frequency shift versus *u* at constant kHz, μK, and . The data confirm three important features of the theoretical prediction: (i) The collisional shift ∆ν decreases with increasing *u* at similar *T _{Z}* and trapping conditions, (ii) ∆ν increases with increasing at similar

*T*and trapping conditions, and (iii) ∆ν decreases with smaller

_{Z}*T*. The sign of the observed shift is negative (i.e., an increased sample density shifts the atomic resonance to lower frequencies). Previous studies of the collisional shift in a 1D optical lattice (

_{Z}*8*,

*10*) are consistent with this observation. From the present data set, we can unambiguously conclude that is negative.

We have made an extensive series of collisional shift measurements at the largest trap depths available to us (Fig. 4). The free-running clock laser has a stability of ~1.5 × 10^{−15} at time scales of 1 to 10 s (*22*). Therefore, a substantial integration time is required to determine the collisional shift with an uncertainty of 10^{−17}. Frequency drifts are minimized by measuring the long-term drift in the resonance frequency (relative to the ultrastable reference cavity) and applying a feedforward correction to the clock laser. The correlation between the atomic resonance frequencies and the density of trapped atoms was calculated by analyzing overlapping sequences of four consecutive measurements and eliminating frequency drifts of up to second order (*24*). Approximately 60 hours of data were acquired at *T _{Z}* = 7 μK over a ~2-month time period for the record shown in Fig. 4A. Each data point represents a period during which the clock was continuously locked, with error bars determined from the standard error of the measurements in that data set (

*17*). At an axial temperature

*T*≈ 7 μK, the collisional shift in our 2D lattice clock was measured to be 5.6 (±1.3) × 10

_{Z}^{−17}in fractional units, with . At a lower

*T*of 3.5 μK, the collisional shift is reduced to 0.5 (±1.7) × 10

_{Z}^{−17}, with (Fig. 4C). The corresponding Allan deviations of both data sets are shown in Fig. 4, B and D.

Relative to previous measurements of collisional shifts in a 1D optical lattice (*8*, *10*), the atomic density in our 2D lattice is an order of magnitude higher. Hence, given a similar level of excitation inhomogeneity, if the collisional shift in a 2D lattice were not suppressed, we would expect a larger shift than seen in earlier results, even after accounting for the fact that only 20 to 30% of lattice sites are contributing.

The advance presented here removes an important obstacle to further increasing the precision and accuracy of neutral atom–based optical clocks. Increasing the number of atoms loaded into our 2D lattice system will enable us to improve the stability of our clock without imposing an onerous systematic effect. As clock lasers become more stable, we will increase the duration of the Rabi interrogation pulse, thus decreasing the Rabi frequency, further reducing the collisional shift systematic well into the 10^{−18} domain. This, together with the fact that in the strongly interacting regime the collisional shift will remain suppressed (*25*) as more atoms are loaded into individual lattice sites, should enable neutral atom clocks to operate with the large sample sizes needed to achieve the highest possible stability.

## Supporting Online Material

www.sciencemag.org/cgi/content/full/science.1196442/DC1

Materials and Methods

Figs. S1 to S4

References

## References and Notes

- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵See supporting material on
*Science*Online. - ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵
- ↵We recently became aware of a conference proceedings report that contains a brief discussion on suppression of collisional shifts under strong interactions (
*26*). - ↵
- We thank C. Benko for technical assistance, and A. D. Ludlow and A. Gorshkov for useful discussions. Supported by a National Research Council postdoctoral fellowship (M.D.S.), a National Defense Science and Engineering Graduate fellowship (M.B.), and a grant from the Army Research Office with funding from the Defense Advanced Research Projects Agency OLE program, the National Institute of Standards and Technology, the NSF Physics Frontier Center at JILA, and the Air Force Office of Scientific Research.