## Strongly Correlated Clocks

Optical lattice clocks with alkaline earth atoms provide one of the most stable time-keeping systems. Such clocks, in general, exhibit shifts in their transition frequencies as a consequence of interactions between atoms. Can this sensitivity be used to explore the dynamics of strongly correlated quantum systems? **Martin et al.** (p. 632) used a 1-dimensional optical lattice clock to study quantum many-body effects. Whereas the clock shift itself could be modeled within the mean field approximation, quantities such as spin noise required a full many-body treatment. This system may be useful for the quantum simulation of exotic magnetism.

## Abstract

Strongly interacting quantum many-body systems arise in many areas of physics, but their complexity generally precludes exact solutions to their dynamics. We explored a strongly interacting two-level system formed by the clock states in ^{87}Sr as a laboratory for the study of quantum many-body effects. Our collective spin measurements reveal signatures of the development of many-body correlations during the dynamical evolution. We derived a many-body Hamiltonian that describes the experimental observation of atomic spin coherence decay, density-dependent frequency shifts, severely distorted lineshapes, and correlated spin noise. These investigations open the door to further explorations of quantum many-body effects and entanglement through use of highly coherent and precisely controlled optical lattice clocks.

Strongly correlated quantum many-body systems have become a major focus of modern science. Researchers are using quantum-degenerate atomic gases (*1*–*6*), ultracold polar molecules (*7*–*9*), and ensembles of trapped ions (*10*, *11*) to realize previously unidentified quantum phases of matter and simulate complex condensed matter systems. Another promising system is optical atomic clocks that use fermionic alkaline earth atoms. The most stable of these clocks now operate near the quantum noise limit (*12*), with an accuracy surpassing that of the cesium standard (*13*). With atom-light coherence times reaching several seconds, permitting optical spectral resolution well below 1 Hz (Fig. 1A), even very weak interactions (such as fractional energy level shifts of order ≥1 × 10^{−16}) can dominate the dynamics of these systems, and the corresponding complex spectrum can be probed precisely. Atomic interactions in optical lattice clocks were first studied in the context of density-dependent frequency shifts, which were attributed to *s*-wave collisions allowed by inhomogeneous excitation (*14*–*17*); *p*-wave interactions were assumed to be suppressed because of the ∼1 μK sample temperature. More recently, in an optical clock based on ^{171}Yb atoms at ∼10 μK, *p*-wave interactions were reported to lead to two-body losses and density shifts (*18*, *19*). At the same time, even at ∼1 μK inelastic *p*-wave losses were observed in the ^{87}Sr system (*20*). The importance of many-body interactions in these clocks has been recognized theoretically (*15*–*17*), but measuring them experimentally has been challenging.

In this paper, we report the observation of quantum many-body effects in a high-density ^{87}Sr optical clock in a one-dimensional (1D) optical lattice. In a prior experiment (*21*), a strongly interacting regime (in which atom-laser and atom-atom interactions are energetically comparable) was reached by tightly confining the atoms in a 2D optical lattice, at the expense of reducing the occupancy to one or two atoms per site. In this work, we probed a strongly interacting system with an average of 20 atoms per disk-shaped 1D-lattice site and developed a detailed understanding of the complex many-body quantum dynamics. The role of *s*-wave collisions is suppressed by operating in the strongly interacting regime with highly homogeneous atom-laser coupling, making *p*-wave interactions, which operate collectively, dominant. The experimental observation of such quantum magnetic behavior at micro-Kelvin temperatures is made possible because the motional degrees of freedom are effectively frozen during the clock interrogation. Only the internal electronic degrees of freedom (pseudo-spin) are relevant, and these can be initialized in a pure state.

We considered an optical lattice clock that uses the ^{1}S_{0} → ^{3}P_{0} (henceforth ^{87}Sr. It comprises an array of quasi-2D trap sites loaded with atoms at micro-Kelvin temperatures. The tight lattice confinement along the longitudinal direction *Z* freezes the dynamics and the population distribution across the trap sites. A single site populated with *N* atoms is modeled as a slightly anharmonic 2D oscillator with radial (longitudinal) frequency ν_{R} = 450 Hz (ν* _{Z}* = 80 kHz).

As shown in Fig. 1B, atoms within a given trap site can elastically interact with one another through the *p*-wave channel. Because all trap frequencies are much greater than the characteristic *p*-wave interaction energy, the motional degrees of freedom are effectively frozen, and interactions thus manifest themselves in the electronic *22*). Fermi statistics guarantee that no two atoms within a given trap site occupy the same motional state. We initialized our nuclear spin–polarized gas with all atoms in the ground state *S* = *N*/2, where *S*(*S* + 1) is the eigenvalue of the observable *S* = *N*/2 manifold (*22*). The second is the relatively small spread in mode-dependent interaction parameters with respect to the gap. Using the collective spin operators, we can thus describe the spin dynamics with the following Hamiltonian:*V _{gg}* +

*V*− 2

_{ee}*V*)/2, and

_{eg}*C*= (

*V*−

_{ee}*V*)/2.

_{gg}*V*,

_{gg}*V*, and

_{ee}*V*represent

_{eg}*p*-wave interaction parameters between the three possible electronic symmetric states,

*J*

^{⊥}is responsible for the energy gap (

*22*). We find that the weak modification of the motional degrees of freedom by interactions can be accounted for as a term of order (

*S*)

^{z}^{3}. Equation 1 links the spin dynamics of interacting thermal fermions at micro-Kelvin temperatures to those of two-mode Bose-Einstein condensates (BECs), and it has been shown both theoretically (

*23*,

*24*) and experimentally (

*25*–

*27*) to give rise to nontrivial many-body correlations and quantum noise–squeezed states. The validity of the collective model has been tested against the full multimode model with good agreement (

*22*).

In the presence of excited-state inelastic loss, which has been observed in ^{87}Sr (*20*), our system becomes a many-body open quantum system. To capture the full many-body dynamics observed in the experiment, we solved a master equation in the presence of a two-body decay that is largely independent of the thermal occupation. The mode-independent losses preserved the collective nature of the system to leading order (Fig. 1C) and allowed us to solve the master equation efficiently for up to 50 atoms (*22*).

To determine the interaction parameters that characterize our spin Hamiltonian, Eq. 1, we measured the density-dependent frequency shift of the clock transition using a modified Ramsey spectroscopy sequence. The initial pulse area θ_{1} = Ω*T _{R}*, chosen so that 0 < θ

_{1}< π, controls the initial value of

*N*

_{tot}is the total number of atoms loaded into the lattice. In the presence of two-body losses,

_{dark}, was fixed at 80 ms, and the final pulse area was set to π/2. We measured the shift by modulating the density by a factor of ∼2 (Fig. 2).

Simple mean-field analysis of Eq. 1 (neglecting cubic terms and losses), in which the time-dependent operators are replaced by their expectation values, reveals that the average interaction experienced by a single atom behaves as an effective magnetic field along *N*) = *B*(*N*)/(2π) scales linearly with the excitation fraction and agrees with experimental observations (Fig. 2). Additionally, we fit an exact solution of Eq. 1 to the data. Both fits are shown in Fig. 2. To compare with the experiment, we always performed an average over the atom number distribution across the lattice sites. From this measurement, we extracted χ = 2 π × 0.20(4) Hz and *C* = −0.3χ.

As a further step, we directly measured the spectrum of the many-body Hamiltonian with subherz spectral resolution and as a function of the drive strength, parameterized by the Rabi frequency Ω. We found that for Ω >> *N* χ, the lineshapes are perturbatively shifted. However, for Ω ≲ *N* χ, the lineshapes become significantly distorted, and the onset of an interaction blockade mechanism is observed, reflecting the dominant effect of strong interactions on the many-body spectrum. The observed Rabi lineshapes can be fully reproduced with the mean-field treatment by using the interaction parameters extracted from the density shift measurements. In this case, a full many-body treatment of the master equation agrees with the mean-field predictions.

To explore the development of many-body correlations during the full many-body dynamical evolution, we measured the Ramsey fringe contrast, which can undergo a periodic series of collapses and revivals, reflecting the quantized structure of the many-body spectrum. The results require a beyond–mean-field treatment. The mean-field model at the single-site level (with fixed *N*) predicts no decay of the Ramsey fringe contrast because when correlations are neglected, the interactions lead only to a pure precession of the collective Bloch vector (*22*). By taking the average over atom distributions among lattice sites and properly treating two-body loss during the Ramsey dark time, the mean-field model does show a decay of the contrast. However, this decay is associated mainly with dephasing arising from different precession rates exhibited by sites with different *N*.

For the Ramsey sequence designed to measure the fringe contrast effects, the pulse durations are <6 ms, satisfying Ω >> *N* χ, to suppress interaction effects during the pulses. We applied the final π/2 readout pulse with a variable relative optical phase of 0° to 360° and recorded the fraction of excited atoms as a function of the readout phase. The contrast of the resulting fringe was extracted in a manner that was insensitive to the frequency noise of the ultrastable clock laser (*22*).

We explored three distinct experimental conditions in order to rule out single-particle decoherence mechanisms and thoroughly test the model. The first condition represents the typical operating parameters of the lattice clock, with *N*_{tot} = 4 × 10^{3} and ν* _{Z}* = 80 kHz. In the second case, we reduced the lattice intensity so that ν

*= 65 kHz, which results in a reduction of the density by a factor of ∼1.8. Last, we maintained ν*

_{Z}*= 80 kHz but reduced the atom number to*

_{Z}*N*

_{tot}= 1 × 10

^{3}. Under all conditions, the full many-body density matrix model reproduces the experimental observations well (Fig. 3, A, C, and E). The inclusion of the

_{1}> π/2 and for the high-density conditions (

*22*). We also observed a striking breakdown of the mean-field model for θ

_{1}≳ π/2, where many-body corrections are dominant (Fig. 3, B, D, and F).

The frequency shift, lineshape, and Ramsey fringe contrast are quantities that all depend on the first-order expectation values of the spin operators *23*), the distribution of the spin noise becomes a compelling measurement to probe many-body correlations beyond the mean field.

To minimize single-particle dephasing effects (for example, arising from the distribution of site occupancies), we added a spin-echo pulse to the Ramsey sequence. As a result, the sensitivity to low-frequency laser noise was reduced at the expense of increased sensitivity to high-frequency laser noise. With atoms initialized in ^{2} is important for defining the ultimate stability of lattice clocks (*12*). For an ideal coherent spin state of the entire ensemble, the standard quantum limit (SQL) of σ^{2} is given by *p* is the probability of finding an atom in the excited state and can be estimated as

We performed measurements for different *N*_{tot} and τ_{dark}—the total atom number and Ramsey free evolution time, respectively—in order to probe the time evolution of the spin noise distribution. Long π pulses were used to reduce the sensitivity to spurious high-frequency components of laser noise. For *N*_{tot} = 1 × 10^{3}, the quantum noise contribution to the spin noise is comparable with that of the laser noise (Fig. 4); however, with *N*_{tot} = 4 × 10^{3} the laser noise is responsible for a larger fraction of the noise in repeated measurements of

There are qualitative differences between the low– and high–atom-number cases; for example, for *N*_{tot} = 4 × 10^{3} with τ_{dark} = 20 and 40 ms we observed a phase shift for the minimum of the spin noise. To compare the predictions of the full many-body master equation with the experiment, we added the effect of laser noise in quadrature with the calculated spin quantum noise. In the absence of laser noise, the theory predicts a small degree of sub-SQL squeezing. This effect is masked by laser noise in both the theoretical prediction for the total spin noise and in our experimental observations but gives rise to a shift of the spin noise minimum with respect to measurement quadrature. We additionally treated the effects of interactions during the laser pulses. The theory predicts the direction and magnitude of the phase shift of the noise minimum in agreement with the experimental observations (Fig. 4), in addition to significantly enhanced spin noise for rotations near ±90°. Despite the presence of laser noise, the measurements of the total spin noise are consistent with the many-body spin model.

Although the investigation described here is restricted to nuclear-spin–polarized gases, exploration of similar many-body effects in a clock making use of additional nuclear spin degrees of freedom with SU(*N*) symmetry may allow investigation of unconventional frustrated quantum magnetism (*28*–*30*).

## Supplementary Materials

www.sciencemag.org/cgi/content/full/341/6146/632/DC1

Materials and Methods

Supplementary Text

Figs. S1 to S5

## References and Notes

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**Acknowledgments:**We thank S. Blatt, J. Thomsen, W. Zhang, T. Nicholson, J. Williams, B. Bloom, and S. Campbell for technical help and A. D. Ludlow, K. R. A. Hazzard, M. Foss-Feig, A. J. Daley, and J. K. Thompson for discussions. The work is supported by the National Institute of Standards and Technology, Defense Advanced Research Projects Agency Optical Lattice Emulator Program administered by Army Research Office, NSF, and Air Force Office of Scientific Research. M.B. acknowledges support from the National Defense Science and Engineering Graduate fellowship program. A.V.G. acknowledges support from NSF IQIM, the Lee A. DuBridge Foundation, and the Gordon and Betty Moore Foundation.