## An atom-coupling cavity

Ensembles of atoms have emerged as powerful simulators of many-body dynamics. Engineering controllable interactions between the atoms is crucial, be it direct or through a mediator. Norcia *et al.* developed a flexible alternative to existing atomic simulators in a system consisting of strontium atoms placed in an optical cavity. Two atomic states connected by a clock transition each served as an effective spin, with long-range spin-exchange interactions mediated by the cavity photons. With improvements, the setup is expected to be amenable to simulating nonequilibrium quantum dynamics and to have applications in metrology.

*Science*, this issue p. 259

## Abstract

Laser-cooled and quantum degenerate atoms are being pursued as quantum simulators and form the basis of today’s most precise sensors. A key challenge toward these goals is to understand and control coherent interactions between the atoms. We observe long-range exchange interactions mediated by an optical cavity, which manifest as tunable spin-spin interactions on the pseudo spin-½ system composed of the millihertz linewidth clock transition in strontium. This leads to one-axis twisting dynamics, the emergence of a many-body energy gap, and gap protection of the optical coherence against certain sources of decoherence. Our observations will aid in the future design of versatile quantum simulators and the next generation of atomic clocks that use quantum correlations for enhanced metrology.

A crucial requirement for the development of atomic quantum simulators is the ability to create controllable coherent interactions between the atoms. Implementations of these interactions include direct atomic collisions (*1*), direct electric and magnetic dipole interactions (*2*–*7*), phonon-mediated couplings in trapped ions (*8*–*10*), and photon-mediated coupling in a driven optical cavity (*11*, *12*). Here, we add to this list spin-exchange interactions between ultra–long-lived optical dipoles mediated by photons in an undriven optical cavity (*13*, *14*). The effective spins are encoded in the ground and excited states of the millihertz linewidth strontium clock transition (Fig. 1, A and B). This optical transition currently forms the basis of the most precise atomic clocks (*15*, *16*) and is a promising candidate for the development of superradiant optical lasers with coherence times beyond 100 s (*17*, *18*).

The exchange interactions manifest in our system as a collective XX-Heisenberg spin model, which describes the behavior of a broad class of phenomena ranging from superconductivity (*19*) to quantum magnetism (*20*). We observe evidence of two of the main characteristic features of the collective XX-Heisenberg model dynamics (Fig. 1C): an orientation-dependent global spin precession of the collective Bloch vector, referred to as one-axis twisting (OAT) (*21*), and the emergence of a many-body energy gap between states of different symmetry (*22*). One-axis twisting can generate useful spin squeezing (*11*,* **21*, *23*), Schrodinger cat states (*24*), and quantum phase magnification (*25*), and also enables new measures of entanglement (*26*). A many-body energy gap can protect collective dynamics against dephasing that results from inhomogeneous shifts to the transition frequencies that vary slowly relative to the time scale set by the gap frequency, such as Zeeman shifts or spatially varying light shifts (*27*–*35*). The interplay between the many-body energy gap and inhomogeneity can stabilize new classes of dynamical phase transitions (*36*, *37*) and quantum phases forbidden at equilibrium (*38*, *39*). In particular, a system like ours may facilitate the study of nonequilibrium phases in quenched superconductors (*40*, *41*), which have so far been difficult to explore experimentally because of the fast time scales intrinsic to solid-state systems.

The cavity-mediated interactions lead both to the collective enhancement of photon emission and to unitary spin dynamics that emerge when the optical cavity is tuned off resonance from the radiating transition. In this work, we operate in a parameter regime where the observed behavior is well-described by a classical treatment of the dynamics. If technical sources of decoherence are reduced, the dynamics that we observe here at the classical level should enable the observation of interesting quantum effects applicable to simulation and metrology. For example, the collective interactions and multiple nuclear spin states of strontium may enable simulation of analogs to the so called Sachdev-Ye model (*42*). This model is believed to be dynamically equivalent to black holes in quantum gravity and has important implications for the scrambling of quantum information (*43*–*45*), which could be probed in a system like ours by measuring out-of-time order correlations via many-body echoes (*25*, *26*, *46*).

Our experimental system (*18*) consists of roughly Sr atoms cooled to 10 µK and tightly trapped by a deep one-dimensional optical lattice that is supported by an optical cavity and generates the same confinement for the ground ^{1}S_{0} and excited ^{3}P_{0} clock states. The atoms couple to the cavity via the clock transition with a single-photon Rabi frequency of up to Hz. Near the clock transition wavelength nm, the cavity has a finesse of and a linewidth of kHz. In practice, g varies between lattice sites, but this can be accounted for by renormalization of parameters (*47*). The cavity detuning from the atomic transition frequency, , is varied to generate exchange interactions.

Photons leaking out of the cavity lead to dissipation in the form of collective atomic decay (superradiance), which can be described by the collective jump operator , with . The collective spin operators characterize the atomic coherence between the ground and excited states. Here and are Pauli operators acting on the and clock state of atom i.

As we detune the cavity from resonance, cavity-mediated unitary dynamics emerge in addition to the dissipative dynamics associated with collective emission. The dynamics can be described by using an effective XX-Heisenberg Hamiltonian written in terms of collective operators as

(1)Here , corresponds to atomic inversion, and is the total spin operator. The last term in , proportional to , can be safely ignored in the large N limit where our experiment operates. The term realizes one-axis twisting (Fig. 1C); the term induces a many-body energy gap as large as between adjacent states with total spin J and .

The behavior of the Bloch vector components can be described geometrically at the mean-field level by treating the cavity optical field as a self-generated effective magnetic field lying in the *x*-*y* plane of the Bloch sphere. This effective field induces a rotation of the Bloch vector about the field’s axis at a frequency proportional to the field’s magnitude (Fig. 1D). The field can be written in complex notation as . The effective field’s azimuthal angle follows the azimuthal angle of the Bloch vector’s projection onto the *x*-*y* plane ϕ up to an offset of that is set by the cavity detuning: . By detecting the field leaking out of the cavity, we obtain a real-time nondestructive measure of the phase, frequency, and magnitude of the transverse component of the Bloch vector (*48*).

At resonance (), the relative phase is . The cavity field C causes a rotation of the Bloch vector from the north pole (all atoms in ) to the south pole (all atoms in ). This is the manifestation of collective or superradiant decay in this picture.

In the large detuning limit, , the relative azimuthal angle of the effective field is now or π, depending on the sign of the detuning. In this limit, the cavity field generated by the atoms drives a rotation of the Bloch vector that changes its azimuthal angle, but not its polar angle. We interpret this additional precession of the Bloch vector’s azimuthal angle as an inversion-dependent frequency shift of the atomic coherence: . This same frequency shift is inherited by the light emitted from the cavity.

To generate a spin-coherent atomic state with arbitrary inversion, we optically pump the atoms to ^{1}S_{0}, , and coherently drive the ^{1}S_{0} to ^{3}P_{0} transition through the optical cavity with light polarized to maintain spin projection (*47*). After switching off the coherent drive, we overlap the subsequently emitted superradiant light with a stable reference laser to form a heterodyne beat note to determine the light’s phase and its frequency . We extract from only the first 8 ms of the superradiant pulse during which changes in inversion are small.

First, we explore the variation of with cavity detuning (Fig. 2A) at fixed initial inversion. On each trial, we prepare the atoms in the same state below the equator of the Bloch sphere. We observe the expected dispersive behavior of the frequency shift with detuning where is an arbitrary offset, as can be seen by the fitted dispersive curve with the cavity linewidth held fixed to its independently measured value.

Figure 2B displays the measured change in frequency of the emitted light when the cavity is detuned by kHz versus an effective population inversion that accounts for inhomogeneous coupling of atoms to the cavity (*47*). As one would expect for OAT dynamics, this change in frequency depends linearly on inversion.

When noncollective effects are present that would modify J, the term in the Hamiltonian substantially modifies the dynamics. To observe these modifications, we prepare half of the atoms in (described by a Bloch vector ) and the other half in (described by a Bloch vector ) (*49*, *50*) (Fig. 3A). The two ensembles experience a differential Zeeman energy shift proportional to an applied magnetic field. The Hamiltonian can be written as , where the sum and difference operators are defined as and .

The mean-field equations of motion for the expectation values and (neglecting Γ for simplicity) are given by

(2)(3)The detuning δ converts the amplitude of into and back as it causes a relative rotation between the two Bloch vectors and . In general, varies with time (*47*), complicating the interpretation of these equations. However, when the total Bloch vector is prepared near a pole of the Bloch sphere with initial (a case that we consider here both for simplicity and for comparison with experimental observations), energy conservation requires that remain small , and to an excellent approximation can be set to zero in Eq. 3. The normal modes of the system are then and , with corresponding frequencies of (Fig. 3B). The mode mixing angle α is . The frequency splitting between the two modes when is equal to . This energy gap suppresses the interconversion of and when and is a classical manifestation of the energy gap between states of and .

In the experiment, we detect the field radiated into the cavity, which is proportional to (and independent of ). For , we expect that the mode will be bright and the mode will be dim, whereas for large δ, the two modes should be equally bright. The imbalance occurs because the mode both radiates more strongly and is preferentially populated by our state preparation process when . In addition to exhibiting an imbalance in radiated power, the two modes should undergo an avoided-crossing type of behavior. These features are clearly apparent in the output of a simulation in which (Fig. 3B).

The presence of dissipation Γ in the experimentally accessible regime makes quantitative comparison difficult. However, the qualitative signatures of the case, especially the imbalanced brightness of the two modes, are still clearly present in the data of Fig. 3C. Notably, a pure Hamiltonian leads only to an overall frequency shift of both modes and cannot explain the apparent curvature near or the dimming of one mode relative to the other, as is shown by simulation that includes dissipation in Fig. 3C.

To measure the frequency splitting associated with the many-body gap, we compare the rate at which bright and dark atomic states accumulate phase. We populate only the state and prepare the total Bloch vector near the south pole of the Bloch sphere in a bright state that radiates light. We can convert this bright state into a dark state by applying inhomogeneous phase shifts to the atoms using a pulse of laser light that is tuned off-resonance from the 7.5-kHz linewidth S to P transition and whose intensity varies over the spatial extent of the atoms. The pulse greatly reduces the magnitude of the atomic coherence , and thus superradiance, while leaving unchanged. The system is then allowed to evolve for a time , after which it is reconverted into a bright state by applying a second pulse from the same laser with opposite detuning from the ^{1}S_{0} to ^{3}P_{1} transition, causing the atoms to rephase, and superradiance to be restored.

To access the frequency of the dark state, we perform two phase measurements and of the light emitted from the cavity before dephasing and after rephasing, respectively. We measure how the difference between the two phases changes when the cavity is alternately detuned by kHz, and label this quantity (Fig. 4).

If the system evolves as a bright state during (i.e., if we do not apply the dephasing and rephasing pulses), the measured linear slope of versus implies that the bright state experiences a frequency shift of Hz. When the system evolves as a dark state during (i.e., if we apply the dephasing and rephasing steps), the slope of versus is consistent with no frequency shift. The difference in frequency between the bright and dark state phase evolutions during is the direct manifestation of the many-body gap for a system near the south pole of the Bloch sphere [for details, see (*47*)].

Instead of the prior procedure, we can instead partially dephase the system and measure the frequency of the residual emission. We find that the shift in frequency of the emitted light when the cavity detuning is toggled is independent of the magnitude of the residual coherence, within experimental error (Fig. 4C).

Even though dephasing populates a variable combination of the bright and dark modes of the system, we measured a frequency shift consistent with the bright mode. This is to be expected, as only the bright mode radiates. Because J remains fixed during the portion of the experiment over which we measure frequency, we only see the effects of a pure Hamiltonian, which do not depend on the degree of atomic coherence.

Our work demonstrates the suitability of ensembles of Sr atoms interacting with an optical cavity via long-lived transitions for the simulation of long-range quantum magnetism. We benchmarked the experimental realization of a collective XX-Heisenberg model, finding agreement with the predictions of mean-field theory.

Our observations pave the way for accessing rich phenomena, including new classes of dynamical phase transitions (*36*, *37**, **40*) and phases of matter forbidden at equilibrium (*38*). In the future, the system may be able to operate in a regime where unitary dynamics dominate over collective dissipation, and the many-body energy gap suppresses dephasing from slow, noncollective frequency shifts such as light and Zeeman shifts. This may be beneficial for entanglement-enhanced metrology (*13*) and for quantum simulation of fast scrambling and dynamics that saturate the quantum chaos bound (*43*–*45*).

## Supplementary Materials

This is an article distributed under the terms of the Science Journals Default License.

## References and Notes

**Acknowledgments:**We thank the members of Jun Ye’s lab for the use of their reference light, and A. Kaufman and S. Kolkowitz for useful comments on the manuscript.

**Funding:**This work is supported by DARPA QuASAR and the DARPA Extreme Sensing award W911NF-16-1-0576 through ARO, ARO, NSF PFC grant no. PHY 1734006, AFOSR FA9550-13-1-0086, and AFOSR MURI Advanced Quantum Materials, and NIST. J.R.K.C. acknowledges financial support from NSF GRFP.

**Author contributions:**M.A.N., J.R.K.C., and J.K.T. contributed to the experiments. R.J.L.-S., B.Z., and A.M.R. contributed to the development of the theoretical model. All authors discussed the results and contributed to the data analysis and to the manuscript.

**Competing interests:**The authors declare that there are no competing interests.

**Data and materials availability:**All data needed to evaluate the conclusions in the paper are present in the figures of the paper and/or the supplementary materials. The data can be accessed at doi:10.5061/dryad.k6tp614Dryad (

*51*).