Nature Physics  Letter
Manybody interferometry of a Rydbergdressed spin lattice
 Johannes Zeiher^{1}^{, }
 Rick van Bijnen^{2}^{, }
 Peter Schauß^{1}^{, }
 Sebastian Hild^{1}^{, }
 Jaeyoon Choi^{1}^{, }
 Thomas Pohl^{2}^{, }
 Immanuel Bloch^{1, 3}^{, }
 Christian Gross^{1}^{, }
 Journal name:
 Nature Physics
 Volume:
 12,
 Pages:
 1095–1099
 Year published:
 DOI:
 doi:10.1038/nphys3835
 Received
 Accepted
 Published online
Ultracold atoms in optical lattices are ideal to study fundamentally new quantum manybody systems^{1, 2} including frustrated or topological magnetic phases^{3, 4} and supersolids^{5, 6}. However, the necessary control of strong longrange interactions between distant ground state atoms has remained a longstanding goal. Optical dressing of ground state atoms via offresonant laser coupling to Rydberg states is one way to tailor such interactions^{5, 6, 7, 8}. Here we report the realization of coherent Rydberg dressing to implement a twodimensional synthetic spin lattice. Our singleatomresolved interferometric measurements of the manybody dynamics enable the microscopic probing of the interactions and reveal their highly tunable range and anisotropy. Our work marks the first step towards the use of lasercontrolled Rydberg interactions for the study of exotic quantum magnets^{3, 4, 9} in optical lattices.
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Neutral ultracold atoms in optical lattices are among the most promising platforms for the implementation of analog quantum simulators of condensed matter systems. However, the simulation of magnetic Hamiltonians, often emerging as an effective model in more complex manybody systems, is difficult with contact interactions due to the low energy scale of the associated superexchange process^{10}. Longrange interactions offer an alternative way to directly achieve strong effective spin–spin interactions. Such interactions emerge between magnetic atoms and between ultracold polar molecules^{11}, trapped ions^{12} or ground state atoms resonantly^{13} or offresonantly coupled (‘dressed’) to Rydberg states^{5, 6, 7, 8}. Rydberg dressing is especially appealing due to the simplicity of realizing atomic lattice systems with unity filling, combined with the great tunability of the interaction strength and shape, which might be exploited to explore exotic models of quantum magnetism^{3, 4}. While effects of longrange spin interactions have been observed in manybody systems of polar molecules^{14}, ions^{15, 16, 17} and resonantly excited Rydberg atoms^{18, 19}, none of these approaches combines the advantages of Rydberg dressing, which permits the realization of strong spin interactions in lattices with nearunity filling. So far, Rydberg dressing in a manybody system remains an experimental challenge, for which up to now only dissipative effects have been measured^{20, 21, 22, 23, 24}. For two atoms, first promising experimental results have been reported recently for nearresonant strong dressing^{25}, where, however, the assumption of a weak Rydbergstate admixture required for the realization of various manybody models^{5, 6, 9, 26, 27} does not hold^{28}.
Here we demonstrate Rydberg dressing in a twodimensional (2D) nearunityfilled atomic lattice with tailored extended range interactions between approximately 200 effective spins. In contrast to our previous experiments^{18, 29} on resonantly coupled Rydberg gases, all atoms participate here in the spin dynamics. We exploit the temporal control over such interactions to perform interferometric measurements on the manybody system that directly reveal the induced correlations via singlesitesensitive local detection^{30}. Our experiments illustrate the versatility of Rydberg dressing by tuning the range and isotropy^{31, 32} of the interaction potential induced by optical coupling to highlying Pstates. Firstprinciples potential calculations together with an exact solution of the manybody dynamics accurately reproduce these measurements. From this we conclude that stimulated Rydberg–Rydberg state transitions, recently observed as anomalous broadening^{24}, act collectively in a manybody system, implying that rare onebody decay events can globally affect the entire spin lattice.
Dressed atomic states emerge as new eigenstates of an atom driven by a laser field with a Rabi frequency Ω and frequency detuning Δ. Considering two levels g and e in the weak dressing regime, Ω Δ, the dressed ground state contains a small admixture β = Ω/(2Δ) of the state e. As a result, it acquires a finite lifetime τ_{r}/β^{2}, which, however, greatly exceeds the lifetime τ_{r} of the bare excited state. Choosing e to be a Rydberg state, the laser coupling also induces effective interactions between two dressed atoms in state (Fig. 1a). At large interatomic distances R this interaction, U(R) ≈ β^{4}V (R), corresponds to the Rydberg–Rydberg atom interaction potential V (R) reduced by the probability β^{4} to excite both atoms at once. At short distances, however, the strong interaction between Rydberg atoms blocks this simultaneous excitation within a critical distance, R_{c}, determined by V (R_{c}) = 2ℏΔ. As a result, the induced interaction acquires a softcore shape and saturates to a value of U_{0} = ℏΩ^{4}/(8 Δ ^{3}) (Fig. 1c)^{5, 6, 8}. Involving other atomic ground states in the dynamics, one naturally obtains various kinds of lattice models of interacting spins that have been proposed for metrology applications^{8, 33} or the exploration of exotic quantum magnetism^{3, 4}. In the simplest case, a single additional ground state that is not coupled to the Rydberg state (Fig. 1a, b) results in a system described by a 2D Ising Hamiltonian
Here, is a spin1/2 operator with eigenstates ↑ and ↓, corresponding to the Rydbergdressed and uncoupled atomic ground state, respectively. The longitudinal field δ arises from the singleatom light shift, δ ≈ Ω^{2}/(4Δ) and U_{i, j} = U_{0}/(1 + (R/R_{c})^{6}) denotes the dressinginduced interaction between spins located on lattice sites i and j at a distance R/a_{lat} = i − j, where a_{lat} denotes the lattice constant. The collective contribution Δ_{i}^{(coll)} = ∑ _{j≠i}^{N}(U_{i, j}/2) results from the transformation from the original atomicstate representation to effective spin operators^{18, 34}. The interactioninduced energy shift resulting from this additional longitudinal field is measurable as a frequency shift in an interferometric Ramsey sequence (Supplementary Figs 1 and 2). Furthermore, due to the extended range of the interactions, this collective field depends on the spins nearby; hence, spins near the edge of a finite system evolve differently compared to spins in the centre. We directly observe this as an inhomogeneous spin distribution significantly contributing to dephasing of the Ramsey fringe (Fig. 1d).
The optically switchable spin interactions U_{i, j} ∝ Ω(t)^{4} enable such interferometric measurements of the manybody system by sequential application of the interaction Hamiltonian equation (1) and a transverse magnetic field induced by microwave coupling of the two ground states^{33, 35}.
Our experiments started with a 2D degenerate gas of rubidium87 in the F, m_{F} = 1, −1 hyperfine state, confined in a single antinode of a vertical (z axis) optical lattice. In this single x–y plane, we then switched on a square optical lattice with a_{lat} = 532 nm spacing and prepared about 190 atoms in a unityfilling Mott insulator with a defect fraction of about 3%. The chosen atom number ensured a negligibly small number of doubly occupied sites. Transitions from the state 1, −1 (‘spin down’, ↓) to 2, −2 (‘spin up’, ↑) were driven globally via microwave pulses. To introduce longrange interactions, the state ↑ was lasercoupled to the 31P_{J} state (J = 1/2 or J = 3/2), which has a lifetime of approximately τ_{r} = 27 μs (see ref. 36). The excitation beam at a wavelength of 297 nm propagated in the plane of the atoms along the diagonal of the cubic lattice (Fig. 1b). A static magnetic field was used to set the quantization axis either along the zdirection or aligned with the laser beam wavevector k, which allowed selective coupling to different Rydberg states, depending on the polarization of the excitation beam (Supplementary Fig. 3 and Supplementary Information). The positions of all the atoms were then detected with singlelatticesite resolution and singleatom sensitivity^{37}. By optically removing atoms in state ↑ before imaging, we could also perform spinresolved detection and, in particular, measurements of spin–spin correlations.
In a first experiment, we aim to reveal the characteristic spin correlations in the manybody system that emerge over time as a result of the longranged spin interactions. To this end, we employ an interferometric spinecho sequence (Supplementary Fig. 1), which is sensitive to interactioninduced phase shifts, whereas the influence of singleparticle effects including the collective longitudinal field (Fig. 1d) is suppressed. Starting with all atoms in state ↓ we first applied a π/2 microwave pulse on the ↓ − ↑ transition to generate an equal superposition of the two spin states. This was followed by two identical Rydbergdressing pulses of duration t/2 each, separated by a π microwave pulse. After closing the interferometer with another π/2 pulse, in the absence of interactions the system returns to its initial state with all the population in ↓, such that any deviations from this are expected to provide a precise probe of the interactioninduced dynamics described by the Hamiltonian equation (1). Optical dressing to the 31P_{1/2} Rydberg state was performed with a detuning Δ/2π = 6 MHz and a Rabi frequency of Ω_{s}/2π = 1.33(7) MHz, determined independently by Ramsey interferometry (Supplementary Fig. 2 and Supplementary Information). These laser parameters ensure the system is in the weak dressing regime with a small Rydbergstate admixture of β = 0.11(1), corresponding to a population probability of β^{2} = 0.012(2). The resulting longrange spin interactions with U_{0}/2π ≈ 1.8 kHz induce correlated phase rotations during the two dressing stages peaking at , and ultimately lead to measurable spin correlations at the end of the interferometric echo sequence. The time dependence of U_{0} arises from the finite rise time of Ω. Our spinresolved detection scheme provides direct access to longitudinal correlations , where measures the ↓population at site i. Measurements versus time or, equivalently, versus the interaction phase ϕ_{0} permit the dynamical growth of the spin–spin correlations to be traced in the regime of small interaction phase, whereas for large ϕ_{0} the correlation signal is expected to decrease. Figure 2a shows the measured spin correlation function g^{(2)}(R) = ∑ _{i≠j}δ_{ij, R} g_{i, j}^{(2)}/∑ _{i≠j}δ_{ij, R}, which is obtained from a translational average constrained to a given distance R by the Kronecker symbol δ_{ij, R}. The observed correlation functions resemble the softcoreshaped interaction potential shown in Fig. 1c. This behaviour is readily understood in the shorttime limit for N_{eff}ϕ_{0}^{2} 1, where N_{eff} denotes the number of spins within the range R_{c} of the interaction potential. In this shorttime limit, one obtains a direct proportionality g^{(2)}(R) = ϕ_{0}^{2}/(4(1 + (R/R_{c})^{6})^{2}) between the induced correlations and the square of the softcore potential. However, for all nonzero interaction times probed in the experiment, the correlations deviate from this simple expression due to the presence of surrounding spins (Fig. 2c), as reproduced by the analytic theory (see Supplementary Information). This directly highlights the importance of manybody effects in our longrange interacting system despite the seemingly small interaction phases.
Next to the optical switchability, Rydberg dressing also enables designing the extent and anisotropy of the interactions between the spins. In the following, we demonstrate this control capability by selecting different coupled Rydberg states. As shown in Fig. 3a, changing the Rydberg state from 31P_{1/2} to 31P_{3/2} leads to notable modifications of the measured correlation function and, hence, the underlying spin interactions. In contrast to 31P_{1/2}, the 31P_{3/2} state features repulsive interactions with a tenfold larger magnitude (Supplementary Fig. 4), which implies a 50% larger interaction radius R_{c}/a_{lat} ≈ 3 that is reflected in the enhanced correlation range shown in Fig. 3a. In both cases, the angular symmetry of the induced interactions is dictated by the strong applied magnetic field, which causes isotropic interactions to emerge for the B_{z}configuration (Fig. 1b) used in the measurements of Fig. 2. Rotating the magnetic field permits tailoring of the anisotropy of the dressinginduced interactions, which is maximized by aligning the magnetic field with the wavevector k of the circularly polarized dressing laser in the plane of the spin lattice (Supplementary Fig. 4). The correlation measurements confirm the expected anisotropy, and we observe an aspect ratio of ~3/2, in quantitative agreement with the theory (Fig. 3b).
In addition to probing the coherent spin interactions, the applied interferometric technique allows an indepth characterization and understanding of decoherence in the present system, which is of great importance for future applications of Rydberg dressing. The measured spin correlations (Fig. 2c) reveal an unexpected homogeneous offset, in addition to the characteristic spatial dependence. This requires a process affecting the system globally, and can be explained in terms of an additional dissipation channel triggering the loss of all particles in the dressed ↑ state. Such a loss process is also consistent with the measured atom number distributions that steadily develop a bimodal structure with increasing dressing time (Fig. 4 and Supplementary Figs 5 and 6). Incoherent transitions to other Rydberg states due to blackbody radiation present a plausible mechanism^{24}, since such transitions project the ↑ state onto a nearby Rydberg state, and thereby produce a real Rydberg excitation (Supplementary Fig. 8). The production of such atoms with opposite parity with a small rate β^{2}γ_{BB} can induce strong dipolar exchange interactions, and thereby trigger a fast avalanchelike atom loss due to strong resonance broadening^{24, 38, 39}. We can incorporate this picture into our theoretical description by assuming a stochastically triggered instantaneous loss of all ↑state atoms as a simplification, still permitting an analytical solution of the manybody dynamics (Supplementary Information). This loss process, being spatially uniform, does not modify the shape of the correlations, but leads to an overall scaling and can account for a uniform offset. Indeed, the measured correlations shown in Fig. 2 are well explained by this model and the offset is reproduced to within 20%. We extract a value of γ_{BB}/2π = 1.6 kHz, approximately half the literature value for 31P_{1/2}states^{36}.
Furthermore, we observe a decreasing atom number N with increasing total dressing time t (Fig. 4), consistent with the predicted exponential loss N(t) = N(0)e^{−(N(0)/4)β2γbbt} (Supplementary Information). Due to the dependence on system size via N(0), the extracted value of 130(20) μs is significantly lower than the anticipated value of τ_{r}/β^{2} = 2.2 ms in the absence of additional loss processes; however, it surpasses the bare Rydberg state lifetime by a factor of five. The spinresolved measurement agrees equally well with the predicted dynamics, strongly supporting the developed understanding of the manybody dynamics.
The demonstrated spatiotemporal control over longrange spin interactions in a unityfilled lattice holds promise for a new generation of quantum simulation platforms. We anticipate that significant improvement in the current coherence time is possible in designed lattice systems with fewer spins or reduced dimensionality^{18}, an appropriately adjusted Rydbergstate detuning to avoid coupling to broadened resonances, or via stroboscopic dressing that allows detrimental impurity Rydberg atoms to decay or be laserquenched^{29} before triggering the avalanche loss. Our results pave the way towards experimental explorations of more complex quantum magnets^{3, 4, 9} and the study of novel phenomena in Rydbergdressed atomic lattices^{26, 27}.
Data availability
The data that support the plots within this paper and other findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
We thank A. Glätzle, P. Zoller, M. Cheneau and N. Henkel for discussions. We acknowledge funding by MPG, EU (UQUAM, SIQS, RYSQ, HAIRS, Marie Curie Fellowship to J.y.C.) and the Körber Foundation.
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Affiliations

MaxPlanckInstitut für Quantenoptik, HansKopfermannStraße 1, 85748 Garching, Germany
 Johannes Zeiher,
 Peter Schauß,
 Sebastian Hild,
 Jaeyoon Choi,
 Immanuel Bloch &
 Christian Gross

MaxPlanckInstitut für Physik komplexer Systeme, Nöthnitzer Straße 38, 01187 Dresden, Germany
 Rick van Bijnen &
 Thomas Pohl

LudwigMaximiliansUniversität, Fakultät für Physik, Schellingstraße 4, 80799 München, Germany
 Immanuel Bloch
Contributions
J.Z. performed the measurements; J.Z. and P.S. planned the experiment under the supervision of C.G. and I.B.; R.v.B. and T.P. developed the analytical model and the ab initio potential calculation. All authors contributed to interpreting the data and writing the manuscript.
Competing financial interests
The authors declare no competing financial interests.
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Johannes Zeiher
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Rick van Bijnen
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Peter Schauß
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Sebastian Hild
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Jaeyoon Choi
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Thomas Pohl
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Immanuel Bloch
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Christian Gross
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