Quantum spin dynamics and entanglement generation with hundreds of trapped ions
Justin G. Bohnet, Brian C. Sawyer, Joseph W. Britton, Michael L. Wall, Ana Maria Rey, Michael Foss-Feig, John J. Bollinger
TTitle: Quantum spin dynamics and entanglementgeneration with hundreds of trapped ions
Authors: Justin G. Bohnet ∗ , Brian C. Sawyer , Joseph W. Britton ,Michael L. Wall , Ana Maria Rey , Michael Foss-Feig , John J. Bollinger Affiliations: NIST, Boulder, Colorado 80305, USA, GTRI, Atlanta, Georgia 30332, USA JILA, NIST and University of Colorado, Boulder, Colorado, 80309, USA JILA, NIST and Department of Physics, University of Colorado, Boulder, Colorado, 80309, USA Joint Quantum Institute and NIST, Gaithersburg, Maryland, 20899, USA ∗ To whom correspondence should be addressed; E-mail: [email protected]
Abstract: Quantum simulation of spin models can provide insight into com-plex problems that are difficult or impossible to study with classical comput-ers. Trapped ions are an established platform for quantum simulation, butonly systems with fewer than 20 ions have demonstrated quantum correlations.Here we study non-equilibrium, quantum spin dynamics arising from an en-gineered, homogeneous Ising interaction in a two-dimensional array of Be + ions in a Penning trap. We verify entanglement in the form of spin-squeezedstates for up to 219 ions, directly observing 4.0 ± a r X i v : . [ qu a n t - ph ] J a n ntractable.One Sentence Summary: We report experimental measurements of open-system Ising spindynamics in two-dimensional arrays of more than 200 trapped ions, demonstrating entangle-ment by observing spin-squeezing and acquiring full counting statistics, both of which showquantitative agreement with theory that accounts for decoherence.
Main Text:
Quantum simulation, where one well-controlled quantum system emulates an-other system to be studied, anticipates solutions to intractable problems in fields includingcondensed-matter and high-energy physics, cosmology, and chemistry, before the developmentof a general purpose quantum computer ( ). Of particular interest are simulations of thetransverse-field Ising spin model ( ), described by the Hamiltonian ˆ H T = ˆ H I + ˆ H B (1) ˆ H I = 1 N N (cid:88) i 9, 10 ).Ensembles of photons, ions, neutral atoms, molecules, and superconducting circuits are alldeveloping as quantum simulation platforms ( ). For example, a variety of quantum spin modelshave been realized with large ensembles of neutral atoms ( ) and molecules ( ), usingcontact or dipolar interactions in optical lattices and using infinite-range interactions mediatedby photons in optical cavities ( ). Trapped-ion quantum simulators can implement ˆ H T ( ) and have a number of advantages over other implementations, such as high-fidelity state2reparation and readout, long trapping and coherence times, and strong, variable-range spin-spin couplings. To date, trapped-ion simulators have been constrained to of order 20 spins( 18, 20 ), where classical numerical simulation remains tractable, but substantial engineeringefforts are underway to increase the number of ions by cryogenically cooling linear traps and2D surface-electrode traps ( 21, 22 ).Penning traps have emerged as an alternative approach to performing quantum simulationswith hundreds of ions ( ). Laser-cooled ions in a Penning trap self assemble into two-dimensional triangular lattices and are amenable to the same high-fidelity spin-state control,long trapping times, and generation of transverse-field Ising interactions as ions in linear Paultraps. Previous work in Penning traps demonstrated control of the collective spin ( ) andbenchmarked the engineered, variable-range Ising interaction in the mean-field, semi-classicallimit ( ). However, for a simulator of quantum magnetism to be trusted, quantum correla-tions generated by the Ising interaction must be observed and understood. For large, trapped-ionsimulators, this benchmarking requires a detailed accounting of many-body physics in an openquantum system. This is both a challenge and an opportunity, as existing numerical methods forcomputing strongly correlated dynamics of open quantum systems are, in general, inadequateto model large ensembles.Here, we observe and benchmark entanglement in hundreds of trapped ions generated withengineered Ising interactions in a 2D array of Be + ions in a Penning trap. To enable efficienttheoretical computation of the spin dynamics ( ), we perform experiments with a homogenousIsing interaction and without simultaneous application of the transverse field B x . We use globalspin observables, such as collective magnetization, to study quantum correlations without need-ing to perform full state tomography on the ensemble ( ). For each experimental result,we verify that a solution of the full quantum master equation gives good agreement with thedata. Specifically, we perform three measurements that characterize the quantum dynamics. We3bserve an N -dependent depolarization of the collective spin, distinguishing the destruction ofcorrelations caused by decoherence from the coherent depolarization caused by the spin-spin in-teraction. We measure the modification of the collective spin variance arising from correlationsbetween the spins, and verify that entanglement survives the decoherence using spin-squeezingas an entanglement witness ( ). For longer interaction times, when spin-squeezing is ab-sent, we observe non-Gaussian counting statistics in the collective spin state. Comparison ofthe full counting statistics with numerical calculations shows that the disappearance of spin-squeezing is due to the formation of an over-squeezed state ( 35, 36 ) and not just degradationdue to decoherence.Our experimental system consists of between 20 and 300 Be + ions confined to a single-plane Coulomb crystal in a Penning trap, described in Fig. 1 and ( ). The trap is characterizedby an axial magnetic field | (cid:126)B | = 4 . T and an axial trap frequency ω z = 2 π × . MHz. Astack of cylindrical electrodes generates a harmonic confining potential along their axis. Radialconfinement is provided by the Lorentz force from (cid:126)E × (cid:126)B -induced rotation in the axial magneticfield. Time varying potentials applied to eight azimuthally segmented electrodes generate arotating wall potential that controls the crystal rotation frequency ω r , typically between π × 172 kHz and π × 190 kHz.The spin-1/2 system is the S / ground state of the valence electron spin |↑(cid:105) ( |↓(cid:105) ) ≡| m s = +1 / (cid:105) ( | m s = − / (cid:105) ) . In the magnetic field of the Penning trap, the ground state issplit by 124 GHz. A resonant microwave source provides an effective transverse field, whichwe use to perform global rotations of the spin ensemble with a Rabi frequency of 8.3 kHz. The T spin echo coherence time is 15 ms. Optical transitions to the P / states are used for statepreparation, Doppler cooling, and projective measurement ( ).The Ising interaction is implemented by a spin-dependent optical dipole force (ODF) gen-erated from the interference of a pair of detuned lasers, shown in Fig. 1a. The ODF couples the4 microwave waveguide ‘Bottom view’ objective C oo li n g Cool Detect C Time ⌧/ ⌧/ Initialize Readout60 µ s 20 ms30 µ s Rotate µ s ⇡ | ⇡ ˆ H I ˆ H I ~B B ‘Side view’ objective N=25 N=61 N=91 N=12450 µ m z y x Figure 1: Penning trap quantum simulator. (A) A cross-section illustration of the Penningtrap (not to scale). The orange electrodes provide axial confinement and the rotating wall po-tential. The . T magnetic field is directed along the z -axis. The blue disk indicates the 2Dion crystal. Resonant Doppler cooling is performed with the beams along z and y . The spin-state dependent optical dipole force (ODF) beams enter ± degrees from the 2D ion plane.Resonant microwave radiation for coupling ground states |↑(cid:105) and |↓(cid:105) is delivered through awaveguide. State-dependent fluorescence is collected through the pair of imaging objectives,where the bright state corresponds to |↑(cid:105) . (B) Coulomb crystal images in a frame rotating at ω r with Be + ions in |↑(cid:105) . (C) The typical experiment pulse sequence, composed of cooling laserpulses (blue), microwave pulses (grey), and ODF laser pulses (green). Cooling and repump-ing initialize each ion in |↑(cid:105) , then a microwave π/ pulse prepares the spins along the x -axis.Suddenly switching on ˆ H I initiates the non-equilibrium spin dynamics. The microwave π pulseimplements a spin-echo, reducing dephasing from magnetic field fluctuations and ODF laserlight shifts. State readout consists of a final global rotation and fluorescence detection. Thefinal microwave pulse area and phase are chosen to measure the desired spin projection.5pin and motional degrees of freedom through the interaction ˆ H ODF = (cid:80) Ni =1 F cos( µt )ˆ z i ˆ σ zi ,where ˆ z i is the position operator for ion i , µ π is the ODF laser beatnote frequency, and F is theforce amplitude, typically yN. The ODF drives the axial drumhead modes of the planar ioncrystal ( 24, 25 ), generating an effective spin-spin interaction by modifying the ions’ Coulombpotential energy ( ). Detuning µ from ω z changes the effective range of the spin-spin interac-tion J i,j ∝ d − ai,j , where d i,j is the ion separation. Although a can range from 0 to 3 ( ), in thiswork we primarily drive the highest frequency, center-of-mass (COM) mode at ω z with ODFdetunings δ = µ − ω z ranging from about π × . kHz to π × kHz, such that a variesfrom . to . , respectively. The next closest axial motional mode frequency is more than π × kHz lower than ω z . Since a (cid:28) , the Ising interaction is approximately independent ofdistance, resulting in a homogeneous pairwise coupling J i,j ≈ ¯ J = F Mω z δ , where M is the ionmass.At the mean-field level, each spin precesses in an effective magnetic field determined bythe couplings to other spins, described by the Hamiltonian ˆ H MF = (cid:80) Nj =1 ¯ B j ˆ σ zj / , where ¯ B j = N (cid:80) i (cid:54) = j J i,j (cid:104) ˆ σ zi (cid:105) . We calibrate ¯ J through measurements of mean-field spin precession ( 23, 27 ),typically finding ¯ J /h ≤ ˆ σ x so that ¯ B j = 0 . This choice of initial condition ensuresthe observed physics are dominated by quantum correlations and decoherence alone.State readout is performed using fluorescence from the Doppler cooling laser on the cyclingtransition ( ). Ions in |↑(cid:105) fluoresce and ions in |↓(cid:105) are dark. Global fluorescence is collectedwith the side view objective (Fig. 1a) and counted with a photomultiplier tube. We calibratethe photon counts per ion using the bottom view image to count the number of ions (Fig. 1b).From the detected photon number, we infer the bright state population N ↑ , which is equivalentto a projective measurement of ˆ S z = ˆ N ↑ − N/ , where ˆ S z is the z component of the collectivespin vector (cid:126)S = (cid:80) Ni (ˆ σ xi , ˆ σ yi , ˆ σ zi ) . By performing a final global rotation before measuring,6e can measure the moments of any component of (cid:126)S . The directly observed variance of themeasurement (∆ S z ) is well described by the sum of two noise terms: spin noise (∆ S (cid:48) z ) andphoton shot noise (∆ S psn ) . Here ∆ X indicates the standard deviation of repeated measure-ments of ˆ X . In this paper, we use the underlying spin noise (∆ S (cid:48) z ) = (∆ S z ) − (∆ S psn ) forcomparison with theory predictions, but use the directly observed variance in the measurement (∆ S z ) for evaluating the spin-squeezing entanglement witness. The ratio (∆ S psn ) / (∆ S (cid:48) z ) istypically 0.13 (-8.8 dB), so the noise subtraction is small for all but the most squeezed statesobserved here. Other sources of technical noise in the state readout are not significant ( ).The depolarization of the collective spin length |(cid:104) (cid:126)S (cid:105)| , or contrast, due to the Ising interactionis a canonical example of non-equilibrium quantum dynamics ( ). Quantum correlationsreduce the contrast and cause the collective spin state to wrap around the Bloch sphere thatrepresents the state space (Fig. 2A). However, the contrast also decreases from decoherence,which destroys correlations, effectively shrinking the Bloch sphere. Our calculation accountsfor both effects, and for homogenous Ising interactions J i,j = ¯ J , the contrast is approximately( ) given by |(cid:104) (cid:126)S (cid:105)| = e − Γ τ N (cid:20) cos (cid:18) JN τ (cid:19)(cid:21) N − . (3)Here τ is the total ODF interaction time (Fig. 1c) and Γ is the total single-particle decoherencerate ( ) due to spontaneous emission from the ODF lasers.We show the depolarization dynamics of |(cid:104) (cid:126)S (cid:105)| in our experiment in Fig. 2B, distinguishingeffects of coherent interactions from decoherence. We determine |(cid:104) (cid:126)S (cid:105)| from measurements of (cid:104) ˆ S x (cid:105) , performing independent experiments to confirm that (cid:104) ˆ S y (cid:105) = (cid:104) ˆ S z (cid:105) = 0 after evolutionunder ˆ H I . To distinguish the depolarization due to decoherence associated with the ODF lasersalone, we perform experiments at δ = +2 π × kHz, effectively eliminating the Ising couplingwhile leaving the spontaneous emission rate unchanged. The dashed line in Fig. 2B is a fitto the observed exponential decay, measuring Γ in our system ( ). The significantly faster7 A Coherent InteractionsDecoherence Figure 2: Depolarization of the collective spin from spin-spin interactions and decoherence.(A) A diagram of the quasi-probability distribution of the collective spin state on a Bloch sphere,illustrating (top) an over-squeezed state generated by the Ising interaction with no decoherenceand (bottom) a loss of contrast only from decoherence, effectively shrinking the Bloch sphere. (B) Contrast versus interaction time for N = 21, 58, and 144 ions indicated by circles, squares,and diamonds, respectively. The error bars show one standard deviation of the mean, and thesolid lines are predictions with no free parameters. The contrast decay from decoherence dueto spontaneous emission is measured in the absence of spin-spin coupling (black squares withthe dashed line showing an exponential fit). Note that at each τ , the detuning δ is adjusted toeliminate spin-motion coupling at the end of the experiment, resulting in a different ¯ J ∝ /δ for each point. (Inset): The data collapse to a common curve with proper rescaling, indicatingthe depolarization is dominated by coherent spin-spin interactions.8ontrast decay for µ tuned near ω z is in good agreement with Eq. (3) for a range of systemsizes. For these data, δ = 4 π/τ , ensuring spin-motion decoupling of the COM mode at the endof the experiment ( ). The collapse of the data to a single curve when plotted as a functionof J τ / √ N , shown in the inset to Fig. 2B, provides strong evidence that the depolarization isprimarily the result of spin-spin interactions. However, depolarization dynamics alone are notenough to prove that entanglement exists in the ensemble.To verify entanglement, we use the Ramsey squeezing parameter ξ R , which only requiresmeasuring the variance of collective observables, instead of full state tomography. The Ramseysqueezing parameter is ξ R = N min ψ [(∆ S ψ ) ] |(cid:104) (cid:126)S (cid:105)| , (4)where ˆ S ψ = (cid:80) Ni cos( ψ )ˆ σ zi + sin( ψ )ˆ σ yi and min ψ [ ] indicates taking the minimum as a func-tion of ψ . For an unentangled spin state, polarized along the x -axis, |(cid:104) (cid:126)S (cid:105)| = N/ and thespin noise is set by Heisenberg uncertainty relations to (∆ S y ) = (∆ S z ) = N/ , so ξ R = 1 .This quantum noise limits the signal-to-noise ratio for a wide range of quantum sensors basedon ensembles of independent quantum objects ( ). Non-classical correlations can redistributequantum noise between two orthogonal quadratures of the collective spin, squeezing the noisesuch that (∆ S ψ ) < N/ and ξ R < . These squeezed states are entangled ( ), and further-more, ξ R is sufficient to quantify the usefulness of the entanglement as a resource for precisesensing. As a result, the generation of spin-squeezed states is widely studied ( 16, 32, 35, 38–43 ).At short times, the non-equilibrium spin dynamics due to the Ising interaction can producespin-squeezed states ( 16, 30, 32, 33 ). Figures 3A and 3B show the measured time evolution ofthe spin variance (∆ S (cid:48) ψ ) of 86 ions, normalized to the spin variance of the initial, unentangledstate. We compare the data to an analytic model ( ) that assumes homogenous Ising inter-actions and fully accounts for both elastic and spin-changing spontaneous emission. The dataclearly show the development of squeezed and anti-squeezed quadratures, and deviations from9 BC Figure 3: Spin variance and entanglement. (A) Spin variance as a function of tomographyangle ψ for N = ± 2. The variance is calculated from 200 trials. The solid lines are aprediction, with no free parameters, assuming homogenous Ising interactions and including de-coherence from spontaneous emission. The dashed lines are theoretical predictions with thesame interaction parameters but no decoherence. (B) The explicit time dependence of the spinvariance for the ensemble in (A). The data for the squeezed (green points) and anti-squeezed(black points) quadratures are shown with theory predictions (solid lines), including decoher-ence. Since our measurement of (∆ S (cid:48) ψ ) has significant granularity, we visualize the the effectof finite sampling of ψ using the green shaded region bounded by max [(∆ S (cid:48) ψ ( ψ m ± ◦ )) ] ,where ψ m corresponds to the angle that minimizes (∆ S (cid:48) ψ ) . The ± ◦ uncertainty does not havea visible effect in the anti-squeezing component on this plot. (C) Ramsey squeezing parametermeasured for different ensemble sizes N . The black points show data for the initial unentangledspin state. The solid purple squares show the lowest directly measured ξ R with no correctionsor subtractions of any detection noise for evaluation of the entanglement witness. The opensquares show ξ R inferred by subtracting photon shot noise. The dashed line is the predictedoptimal ξ R from coherent Ising interactions with no decoherence, and the solid line shows thelimit including spontaneous emission assuming Γ / ¯ J = 0 . , which is typical for our system.The shaded purple region accounts for finite sampling of ψ as in (B). All error bars indicate onestandard error. 10erfectly coherent Ising dynamics are well described by the effects of spontaneous emissionalone. Similar data for different N are shown in ( ).Using measurements of the directly observed spin variance (∆ S ψ ) and contrast |(cid:104) (cid:126)S (cid:105)| , weobtain ξ R for a range of τ . For a given ensemble size N , we plot the minimum observed ξ R , shown in Fig. 3c, where we see that the entanglement witness ξ R < is satisfied forseven independent datasets with N ranging from 21 to 219. We also show ξ R measured forthe initial state, confirming our calibration of N . For comparison, Fig. 3c shows the absoluteminimum ξ R predicted for coherent Ising interactions. The majority of the observed discrepancyfor ensembles ranging from 60 to 150 ions is accounted for by photon shot noise, spontaneousemission, and the finite sampling of τ and ψ . For other ion numbers ( ), we still observe goodagreement in the anti-squeezed spin variance, but the minimum spin variance and ξ R deviatefurther from the prediction. We attribute the deviation to technical noise sources, describedin ( ).The Ramsey squeezing parameter is an effective entanglement witness at short times whenquantum noise is approximately Gaussian. At longer times, the growth of spin correlationscauses both the depolarization seen in Fig. 2 and the increase in min ψ [(∆ S ψ ) ] , due to theappearance of non-Gaussian quantum noise in the collective spin. Both effects cause ξ R toincrease above 1, which we call an over-squeezed state. Over-squeezed states can be entangled( ), however, ξ R can also increase simply because of decoherence.In order to observe signatures of quantum correlations at longer interaction times, in Fig.4 we show the histogram of the measurements of (cid:104) ˆ S ψ (cid:105) for an over-squeezed state of 127 ionsafter an interaction time of τ = ψ = 5 . ◦ also contains non-Gaussian characteristics in the tails away from the narrow centralfeature. We compare the data to a theoretical model of the full counting statistics and observe11 B Figure 4: Full counting statistics of a non-Gaussian spin state. Histograms showing the (A) squeezed and (B) anti-squeezed quadratures of the collective spin of N = ± ψ = ◦ and ψ = ◦ , respectively. Here τ = ), as-suming homogenous interactions and including decoherence from spontaneous emission andmagnetic field fluctuations. We account for photon shot noise by convolving the theoreticalprobability density with a Gaussian distribution with a variance (∆ S psn ) / ( N/ .12ood agreement. Even though ξ R = 26 , the theoretically predicted state is entangled, which weverify using an entanglement witness based on the Fisher information F . The Quantum Fisherinformation has been used as an entanglement witness in other trapped-ion simulators ( ).We bound the Fisher information using the approach in Ref. ( ) and find F/N > . , whichsatisfies the inequality of the entanglement witness F/N > ( ). Photon shot noise in ourmeasurement limits our capability to directly witness the entanglement experimentally ( ), butthe good agreement with theory indicates that the state of the ensemble is consistent with anentangled, over-squeezed state. The full counting statistics are only efficiently computable forhomogenous couplings, a good approximation for the small detunings δ considered here. Forfuture work with inhomogeneous Ising coupling, obtaining the full counting statistics theoreti-cally will likely be intractable for more than 20 to 30 spins.In conclusion, we have verified numerous hallmark signatures of quantum dynamics in en-sembles of hundreds of trapped ions, demonstrating spin-squeezing and showing evidence ofover-squeezed states, where the magnitude of the collective spin |(cid:104) (cid:126)S (cid:105)| is near zero and the fullcounting statistics are non-Gaussian. The techniques presented here are applicable to precisionsensors using trapped ions, where the number of ions is limited by systematic errors arisingfrom ion motion ( ), and could be useful for quantum-enhanced metrology with non-Gaussianspin states ( 42, 46–48 ). 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A.M.R. acknowl-edges support from nsf-phy 1521080, JILA-NSF-PFC-1125844, ARO, MURI-AFOSR andAFOSR. J.G.B., M.F.F., and M.L.W. acknowledge financial support from the National Re-search Council Research Associateship Award at NIST. This manuscript is the contribution17f NIST and is not subject to US copyright. Correspondence should be addressed to J.G.B.([email protected]) or J.J.B. ([email protected]).18 upplementary Material Be + Spin-1/2, Control, and Detection Reference ( ) and the supplementary material to ( ) give detailed descriptions of our spininitialization, control, and measurement capabilities with planar ion arrays in Penning traps.We briefly summarize some of that discussion, emphasizing aspects relevant for the measure-ments reported here. Figure S1 shows the relevant Be + energy levels. We use the valenceelectron spin states parallel |↑(cid:105) = (cid:12)(cid:12) m J = + (cid:11) and anti-parallel |↓(cid:105) = (cid:12)(cid:12) m J = − (cid:11) to the ap-plied magnetic field of the Penning trap as the spin- / or qubit. In the 4.46 T magnetic fieldof the trap these levels are split by approximately Ω = π × GHz. The Be + nucleus hasspin I = 3 / . We optically pump the nuclear spin to the m I = +3 / level ( ), where itremains throughout the duration of an experiment. The ions are Doppler laser-cooled to atemperature ∼ |↑(cid:105) → (cid:12)(cid:12) P / m J = +3 / (cid:11) cycling transition and directed parallel and perpendicular to themagnetic field ( ). Spins in the |↓(cid:105) state are efficiently optically pumped to the |↑(cid:105) state witha laser tuned to the |↓(cid:105) → (cid:12)(cid:12) P / m J = +1 / (cid:11) transition. A typical experimental cycle startswith ∼ ms of combined Doppler laser cooling and repumping. We estimate the fidelity of the |↑(cid:105) state preparation should be very high ( (cid:29) %).Low-phase noise microwave radiation from a 124 GHz source described previously in thesupplementary material to ( ) is used to globally rotate the spins, and provides an effectivetransverse magnetic field in the rotating frame of the qubit. The length of the time intervalrequired to drive |↑(cid:105) to |↓(cid:105) ( π -pulse) was ∼ µ s. The fidelity of a π -pulse was measured tobe greater than 99.9% in a random benchmarking experiment ( ). The spin-echo coherenceduration ( T ) was measured to be ∼ ms with the magnet sitting on the floor, and greater than ms with the magnet vibrationally isolated. All measurements were done with the magnetsitting on the floor.At the end of an experimental sequence we turn on the Doppler cooling laser and make aprojective measurement of the ion spin state through state-dependent resonance fluorescence.With the Doppler cooling laser on, an ion in the |↑(cid:105) state scatters photons while an ion in |↓(cid:105) is dark. Specifically we detected, with f/5 light collection and a photomultiplier tube (PMT),the resonance fluorescence from all the ions in a direction perpendicular to the magnetic field(the side-view). The PMT counts are integrated over time periods of typically 1 ms for mea-suring averages, or 15 ms for measuring standard deviation (to reduce the impact of photonshot noise). The photon detection rate varied between 1 and 2 photons per ion per ms. Thevariation is due to day-to-day variation in the cooling beam intensities and positions, usuallychanged to optimize crystal stability. Once data collection started, the photon detection rate washeld constant by fixing the Doppler cooling beams’ positions and stabilizing their intensities.The integrated photon count is converted to state population measurement using a frequentlyrepeated calibration of the counts for all the spins in |↑(cid:105) and for all spins in |↓(cid:105) .Images of the ions in the rotating frame of the crystal were obtained by using an imaging19 s S 1/ 2 P 3/ 2 C OO L I NG R E P U M P OD F || ~ 124 GHz ~ GH z m J =+3/2m J =-3/2m J =+1/2m J =-1/2 Figure S1: Relevant energy levels of Be + at B = 4 . T (not drawn to scale). We only show m I = + levels which are prepared experimentally through optical pumping. The S / − P / transition wavelength is nm. A resonant laser beam provides Doppler laser cooling and statediscrimination, a second repumps |↓(cid:105) to the |↑(cid:105) . The ODF interaction is due to a pair of beams(derived from the same laser) with relative detuning µ . The qubit splitting Ω ∼ π × GHz.A low phase noise microwave source at Ω provides full global control over spins.20MT (maximum processing capability (cid:46) 100 kHz) to record ( x , y , t ) for each photon ( ). Theseimages were used to count the number of Be + ions in the crystal, and were recorded eitherbefore or after long sequences of measurements. The number of Be + ions slowly decreasesdue to the formation of BeH + through collisions with residual H in the room-temperaturevacuum system. This slow change in the Be + ion number was tracked by monitoring theresulting slow change in the global fluorescence. Reversing the hydride-ion formation throughphotodissociation of BeH + has been demonstrated ( ) and can be implemented in the futurewith a redesign of the vacuum envelope.We employ a recently designed and fabricated Penning trap consisting of a stack of cylin-drical electrodes (see sketch in Fig. 1 of the main text) to generate an electrostatic potential qφ trap ( ρ, z ) (cid:39) mω z ( z − ρ / near the center of the trap, where z and ρ are cylindrical co-ordinates. With potentials of up to 2 kV we obtain ω z (cid:39) π × . MHz while nulling thelowest order anharmonic ( C ) term. The direction of the magnetic field of the trap is alignedwith the symmetry axis (the z -axis) of the electrodes to better than . ◦ . The rotation fre-quency ω r of the ion array determines the strength of the radial confinement of the ion crystal,and is precisely controlled by a rotating quadrupole potential ( ). The middle electrode ofthe trap incorporates eight azimuthally segmented rotating wall electrodes at a radius of 1 cmfrom the center of the trap. This configuration enables control of smaller arrays than possiblein our previous trap, presumably because stronger rotating wall potentials are easily generated.For most of the measurements recorded here, the rotating wall potential is characterized by qφ , ( x R , y R ) = mω q ( x R − y R ) where ω q (cid:39) π × (28 kHz ) and x R , y R denote coordinates inthe rotating frame. Optical-Dipole Force and Lamb-Dicke Confinement A spin-dependent optical dipole force is obtained from a moving 1D optical lattice generatedat the intersection of two off-resonant laser beams. The set-up is identical to that describedin ( 24, 56 ) and in the supplementary material to ( ), except the beams cross with an angle θ = 20 ◦ ( ± ◦ with respect to the central z = 0 plane of the trap). The optical dipole force(ODF) beams are detuned by approximately 20 GHz from any electric dipole transitions in Fig.S1 and produce an AC Stark shift on the |↑(cid:105) and |↓(cid:105) states, ∆ ↑ ,acss = (cid:15) ↑ + U ↑ sin (cid:104) δ(cid:126)k · (cid:126) ˆ r − µt (cid:105) ∆ ↓ ,acss = (cid:15) ↓ + U ↓ sin (cid:104) δ(cid:126)k · (cid:126) ˆ r − µt (cid:105) . (S1)Here δ(cid:126)k and µ are the wave vector and frequency difference between the ODF beams. (cid:12)(cid:12)(cid:12) δ(cid:126)k (cid:12)(cid:12)(cid:12) =2 k sin ( θ/ 2) = 2 π/ (0 . µ m ) for θ = 20 o . We adjust the polarization and frequency of theODF laser beams so that (cid:15) ↑ = (cid:15) ↓ and U ↑ = − U ↓ ≡ U , producing a spin-dependent ODF21otential, ˆ H ODF = U (cid:88) i sin (cid:104) δ(cid:126)k · (cid:126) ˆ r i − µt (cid:105) ˆ σ zi . (S2)To minimize the variation of the phase of the 1D optical lattice across the ion array we align δ(cid:126)k (cid:107) ˆ z . We do this by minimizing decoherence of the spins with ˆ H ODF applied and µ tuned to aharmonic of the rotation frequency, µ = nω r . By comparing the n = 2 and n = 1 decoherencesignals, we estimate a misalignment error | ∆ θ err | (cid:46) . o . For N = 200 (largest numbers usedfor data collection), the array radius is R (cid:39) µ m, and the phase difference between thecenter and edge of the array is (cid:12)(cid:12)(cid:12) δ(cid:126)k (cid:12)(cid:12)(cid:12) R tan (∆ θ err ) (cid:46) . radians (cid:39) . ◦ .With δ(cid:126)k (cid:107) ˆ z , Eqn. S2 only involves the axial coordinates of the ions, ˆ H ODF = U (cid:80) i sin [ δk ˆ z i − µt ] ˆ σ zi , where δk = (cid:12)(cid:12)(cid:12) δ(cid:126)k (cid:12)(cid:12)(cid:12) . In the Lamb-Dicke confinement limit( δk z rms,i (cid:28) , z rms,i ≡ (cid:112) (cid:104) ˆ z i (cid:105) is the root mean square (rms) axial extent of the wave functionof ion i ), this reduces to ˆ H ODF (cid:39) F cos ( µt ) (cid:80) i ˆ z i ˆ σ zi (assuming U/µ (cid:28) where F ≡ U · δk .The spin-dependent ODF, F , is reduced outside of the Lamb-Dicke confinement limit by theDebye-Waller factor DW F i ≡ exp (cid:0) − δk z rms,i (cid:1) ( ). The Ising pair-wise coupling strengthsare reduced by the square of the Debye-Waller factor. The measured Ising coupling strengths,determined from mean-field spin precession measurements, were less than the calculated cou-pling strengths (i.e. U · δk ) by to , in rough agreement with the Debye-Waller factorsestimated below. Typical values for this work are U (cid:39) (cid:126) π × (6 . kHz ) resulting in F = 30 yN.The calculated coupling strengths are based on well known atomic physics parameters for Be + ,and calibration of the ODF laser intensity through AC Stark shift measurements for differentpolarizations of the ODF beams.Measurements indicate the temperature of the axial drumhead modes is close to the Dopplercooling limit of ∼ . mK ( 24, 56 ). Neglecting the Coulomb interaction between the ions, weestimate z rms,i (cid:39) (cid:113) (cid:126) mω z (2¯ n + 1) (cid:39) nm, δk · z rms,i = 0 . , and DW F i (cid:39) . . Animproved estimate of z rms,i is obtained by summing the contributions from all of the transversemodes m z rms,i = (cid:32)(cid:88) m ( b i,m ) (cid:126) mω m (2¯ n m + 1) (cid:33) / (S3)where ¯ n m (cid:39) k B T / (cid:126) ω m and b i,m is the amplitude of the m th normal mode at site i . With N = 127 , ω r = 2 π × kHz, ω z = 2 π × . MHz, and T = 0 . mK (typical parametersused in this work), z rms,i (cid:39) nm, δk · z rms,i = 0 . , DW F i (cid:39) . in the center of the array,changing to z rms,i (cid:39) , δk · z rms,i = 0 . , DW F i (cid:39) . at the radial edge of the array.In addition to reducing the average strength of the spin-dependent ODF, a non-zero Lamb-Dicke confinement parameter gives rise to fluctuations in the spin-dependent ODF from onerealization of the experiment to the next ( ). These fluctuations can produce fluctuations in theinduced spin-spin interactions. This appears to be a challenging problem to accurately modelfor a many-ion array, but large numbers of ions will tend to average out the effects of thermal22otional fluctuations. For our work where µ − ω z is small compared to µ − ω m for any non-COM mode m , thermal motional fluctuations give rise to fluctuations in the single-axis twistingstrength produced by the spin-dependent coupling to the COM mode. We estimate fractionalfluctuations in the single-axis twisting strength to be less than 3% for N = 100 and T = 0 . mK. Sub-Doppler cooling the axial drumhead modes can reduce z rms,i . Spin Variance Measurements with Different Ion Numbers Figure S2 shows spin variance measurements, analogous to Fig 3A of the main text, with differ-ent numbers of ions. The uncertainty, one standard error σ S , in the measured variance (∆ S ψ ) is calculated as σ S = (∆ S ψ ) (cid:112) /N trials where N trials is the number of experimental trials.Then in Fig. S2 and Fig. 3 of the main text, one standard error on the normalized variance isdetermined by following standard error propagation. Sources of Noise and Decoherence For the analysis in the main text, we account for decoherence due to spontaneous light scatteringand photon shot noise. In particular we model the measured variance in the transverse spin (∆ S ψ ) as (∆ S ψ ) = (∆ S ψ | Γ ) + mK . (S4)Here (∆ S ψ | Γ ) denotes the prediction for the transverse spin variance obtained with the en-gineered Ising interaction in the presence of spontaneous light scattering from the ODF laserbeams ( ). The contribution of photon shot noise to the variance is (∆ S psn ) = m/K . Here m is the mean number of photons collected in a global fluorescence measurement and K is thenumber of photons collected per ion in the bright state |↑(cid:105) . The angle ψ , defined in the maintext, denotes the angle along which the transverse spin variance is measured. In the main text, (cid:0) ∆ S (cid:48) ψ (cid:1) ≡ (∆ S ψ ) − (∆ S psn ) . We separately discuss decoherence due to spontaneous emis-sion and photon shot noise. We also discuss a few potential sources of decoherence that maybe contributing to the increase in the variance of the squeezed spin quadrature observed withincreasing ion number. Spontaneous emission The primary source of decoherence in the simulator arises from spontaneous emission from theoff-resonant ODF laser beams. Decoherence due to spontaneous light scattering from an off-resonant laser beam has been carefully studied in this system ( ). The off-diagonal elementsof the density matrix for an individual spin decay exponentially with rate Γ ≡ (Γ el + Γ Ram ) / where Γ el and Γ Ram are the decoherence rates for elastic and Raman scattering, respectively. Γ Ram = Γ ud + Γ du where Γ ud and Γ du are the rates for spontaneous transitions from | ↑(cid:105) to | ↓(cid:105) and from | ↓(cid:105) to | ↑(cid:105) , respectively. Reference ( ) provides expressions for Γ el and Γ Ram in23 =21 N=33N=66 N=100N=149 N=219 A BC DE F Figure S2: Spin variance as a function of tomography angle ψ for different ion numbers N ,calculated from N trials = 200 trials. The error bars are one standard error on the variance.The solid lines are a prediction, with no free parameters, assuming homogenous Ising interac-tions and including decoherence from spontaneous emission. The dashed lines are a theoreticalprediction with the same interaction parameters but no decoherence.24erms of atomic matrix elements, and laser beam polarizations and intensities. For our set-up, Γ el ∼ Ram .We use the ions to measure the individual laser beam intensities (typically ∼ . W / cm )through measurements of the AC Stark shift with the polarization rotated parallel to ˆ z (themagnetic field axis). We directly measure Γ by measuring the exponential decrease in |(cid:104) (cid:126)S (cid:105)| as afunction of the time interval the ODF beam(s) is (are) turned on. We use a spin-echo sequencesimilar to that described in ( ) and illustrated in Fig. 1 of the main text. We observe goodagreement between the calculated and measured decoherence rates with the application of asingle ODF laser beam. A typical single beam decoherence rate is Γ (cid:39) . s − . With the 1Doptical lattice and an ODF beat note µ tuned ∼ to ∼ kHz above the axial COM mode, weobserve exponential decay (dashed line in Fig. 2 of the main text) of the system Bloch vector ata rate ∼ higher than the sum of the rates from each beam. This excess decoherence rate isobserved to be relatively independent of the ODF beat note µ , and is presently not understood.For each data set, measurements of the decoherence rate with ( µ − ω z ) ∼ π × kHz wereused, along with the measured Ising interaction strength and the theory of ( ), to generate (∆ S ψ | Γ ) , displayed by the solid lines in Fig. S2 and Fig. 3 of the main text. We assumethe measured excess decoherence is due to an increase in Γ el . More details on the theoreticalmodeling is given in a subsequent section. The impact of decoherence due to spontaneous lightscattering can be decreased by increasing the angle θ with which the ODF beams cross. Photon shot noise, classical detection noise Photon shot noise contributes to the measured transverse spin variance. For a global fluores-cence measurement where m photons are collected, the variance in the number of collectedphotons is m . We assume the same contribution of photon shot noise to the variance of a seriesof global fluorescence measurements where the mean number of photons collected per measure-ment is m . For measuring the variance of a spin component, typical detection times were 15 msresulting in at least K = 15 photons collected for an ion in the bright state. For a spin state inthe equatorial plane of the Bloch sphere, the mean number of photons collected is m = N K , sophoton shot noise will contribute to (∆ S ψ ) at the level mK = N K . Relative to projection noise N , photon shot noise contributes at the level of K , or less than 13% ( − . dB ) for K > . Forthe variance measurements in Fig. 3 of the main text, the photon shot noise is subtracted fromthe measured spin variance. Shot noise was accounted for in the counting statistics of Fig. 4in the main text by convolving the theoretical probability distribution with the distribution ofshot noise, as discussed in more detail in a later section. The relative contribution of photonshot noise can be reduced with longer detection times. We estimate the detection time intervalcan be increased by an order of magnitude before optical pumping between |↓(cid:105) and |↑(cid:105) is aconsideration.Classical fluctuations in the detection laser power, frequency, and position can contributeto the measured spin variance in our global fluorescence detection. We measure this classicaldetection noise σ t by initializing all the ions in |↑(cid:105) (bright state) and measuring the variance in25 Free Precession Time τ (ms) D e p h a s i n g ( ∆ φ r m s ) ( r a d i a n s ) Figure S3: Dephasing due to magnetic field fluctuations measured with N = 124(3) ions in theabsence of the ODF beams. Plotted is the variance of the dephasing angle ∆ φ ( τ ) determinedfrom 300 trials of a spin echo experiment measuring the transverse spin noise along ψ = 90 ◦ .Photon shot noise was subtracted; τ is the sum of the two free precession intervals. The red lineis a 2-parameter fit ∆ φ rms ( τ ) = (2 . × − / ms ) τ + (1 . × − / ms ) τ where τ is in ms.the total photon count σ total , and then infer σ t = σ total − m . We measured σ t to be less than of photon shot noise (-14 dB below projection noise). We neglect this small contributionof classical detection noise relative to photon shot noise in our analysis. Specifically we do notsubtract any classical detection noise. Magnetic field fluctuations We measure a small amount of dephasing when running the experiments described in the textin the absence of the ODF lasers. This is due to fluctuations in the homogeneous magnetic fieldproduced by vibrations of the superconducting magnet ( ). Without the ODF beams, homo-geneous fluctuations produce a dephasing proportional to the square of the Bloch vector length, N / . We write its contribution to the transverse spin variance as ( N / 4) ∆ φ rms ( τ ) sin ( ψ ) where ∆ φ rms ( τ ) only depends on the magnetic field noise spectrum and the length τ of the ex-perimental sequence ( 60, 61 ). For spin-echo sequences and magnetic field noise dominated bylow frequencies ( ), we anticipate ∆ φ rms ( τ ) ∝ τ . Figure S3 shows dephasing measurementsobtained without the ODF beams. Both τ and τ dependences are observed. By taking careto minimize sources of vibration in the lab, the measured ∆ φ rms ( τ ) did not significantly varyfrom day to day.Dephasing is described by the Hamiltonian B ( t ) (cid:80) i ˆ S zi where B ( t ) is a stochastic process.26his Hamiltonian commutes with the Ising interaction (and also with the elastic Rayleigh scat-tering decoherence in the master equation). The impact of magnetic-field-induced dephasingcan therefore be accurately modeled, and we find its contribution to be small compared to pho-ton shot noise for both the variance measurements (Fig. 3, main text) and the histograms (Fig.4, main text). A more complete discussion of the impact of magnetic field fluctuations is givenin a later section. We note that relative to projection noise ( N/ , the contribution of homo-geneous dephasing scales as the length of the Bloch vector ( ∝ N ) , becoming more importantfor larger numbers of ions. The measured magnetic field noise can be reduced by more than afactor of 5 by vibrationally isolating the magnet. Other potential sources of noise We briefly discuss a few other potential sources of dephasing that do not appear to significantlycontribute to the work discussed here, but could become factors, in particular if photon shotnoise is reduced.Heating of the axial COM mode during application of the spin-dependent force is a sourceof dephasing. Following the discussion in Ref. ( ), we calculate the dephasing ∆ φ rms (∆ n ) due to a stochastic increase ∆ n in the COM mode occupation number during the applicationof a spin-dependent force F for a time τ s = 2 π/δ where δ = µ − ω z . A spin echo sequenceconsists of two such applications, and results in twice the dephasing (in variance), ∆ φ rms (∆ n )∆ φ proj (cid:39) ∆ n · F z (cid:126) δ . (S5)Here ∆ φ proj = 1 /N is the angle determined by the projection noise limit and z = (cid:112) (cid:126) / (2 mω z ) .For close detunings δ = 2 π × kHz, F z (cid:126) δ ∼ , so ∆ n ∼ / ms can cause dephasing on theorder of projection noise. For trapped ions, the COM mode is typically heated by noisy electricfields. In this case, the heating rate scales linearly with N ( ). Measurements place an up-per limit on the COM mode heating rate of (quanta/s)/ion. For τ s = 1 ms and N =100 ions, ∆ n ≤ . . Because the COM mode heating rate may scale linearly with N , this source ofdephasing will likely become more important as the ion number increases. We note that pho-ton recoil from spontaneous light scattering with the ODF beams will produce dephasing bythe mechanism described above. We estimate its contribution to be small compared with the ∆ n = 0 . estimate.For small detunings δ from the COM mode, fluctuations and drifts in the axial COM modefrequency ω z can produce spin-motion entanglement because the decoupling condition τ s =2 π/δ may no longer be satisfied. Here τ s is the duration of a single arm period of the spin-echosequence. Spin-motion entanglement produces dephasing, and we calculate this dephasing withEqs. (30) and (32) of Ref. ( ). Let δτ s = 2 π + (cid:15) where (cid:15) is a measure of the incomplete fullcircle due to error in measuring ω z . We note that a spin echo sequence suppresses the error due27igure S4: Spontaneous emission from the Raman beams creating the spin-dependent forcecauses three types of decoherence: Raman decoherence processes with rates Γ ud and Γ du , whichproject spins to be down or up, respectively, and elastic decoherence processes, which causedephasing of a spin superposition state. Figure from Ref. ( ).to a non-zero (cid:15) (relative to a Ramsey sequence ( )), and calculate a dephasing, ∆ φ rms ( (cid:15) )∆ φ proj (cid:39) F z (cid:126) δ (cid:15) ( (cid:15) + δt π ) (2¯ n + 1) . (S6)Here t π (cid:39) µ s is the duration of the π -pulse in the spin echo sequence. For δ = 2 π × (1 kHz), F (cid:39) yN, ω z = 2 π × (1 . MHz), T COM = 1 . mK (¯ n COM (cid:39) , we estimate ∆ φ rms ( (cid:15) ) / ∆ φ proj < requires (cid:15) < . . This places an upper limit on the uncertainty of theaxial COM mode frequency of ∆ ω z < π × Hz. During data collection we checked for a shiftin the COM mode frequency every 2 s, and used this information to update µ to fix δ = 2 π/τ s . Theoretical Modeling of Spin-spin Interactions with Spontaneous Emission As discussed in the earlier section on sources of noise and decoherence, spontaneous emissionmust be accounted for during the interaction time. Following Ref. ( ), there are three differ-ent types of spontaneous emission processes, shown in Fig. S4. The processes with rates Γ ud and Γ du , which induce spontaneous transitions from | ↑(cid:105) to | ↓(cid:105) and from | ↓(cid:105) to | ↑(cid:105) , respec-tively, arise from Raman scattering. On the other hand, Rayleigh scattering, with associateddecoherence rate Γ el , produces dephasing of a spin superposition state. In typical experimentalrealizations, Γ ud , Γ du ∼ s − , and Γ el ∼ − s − including the excess decoherencediscussed in the above section on spontaneous emission. The spin dynamics in this case canbe modeled by a master equation in Lindblad form, and this master equation admits an exactsolution, as discussed in Ref. ( ). Using this exact solution, we compute the contrast and spin28ariance expectations from the correlation functions (cid:104) ˆ σ + j (cid:105) = e − Γ t (cid:89) k (cid:54) = j Φ ( J jk , t ) , (S7) (cid:104) ˆ σ aj ˆ σ bk (cid:105) = e − t (cid:89) l / ∈{ j,k } Φ ( aJ jl + bJ kl , t ) , (S8) (cid:104) ˆ σ aj ˆ σ zk (cid:105) = e − Γ t aJ jk , t ) (cid:89) l / ∈{ j,k } Φ ( aJ jl , t ) , (S9)where a, b ∈ { + , −} and Φ ( J, t ) = e − ( Γud+Γdu ) t (cid:104) cos (cid:18) t (cid:113) (2 iγ + 2 J/N ) − Γ ud Γ du (cid:19) + Γ ud + Γ du t sinc (cid:18) t (cid:113) (2 iγ + 2 J/N ) − Γ ud Γ du (cid:19) (cid:105) , (S10) Ψ ( J, t ) = e − ( Γud+Γdu ) t [ i (2 iγ + 2 J/N ) − γ ] t × sinc (cid:18) t (cid:113) (2 iγ + 2 J/N ) − Γ ud Γ du (cid:19) . (S11)In these expressions, the initial state is the product state of all spins pointing along the x direc-tion, γ = (Γ ud − Γ du ) / , and Γ = (Γ ud + Γ du + Γ el ) / . In the case that the couplings betweenspins are uniform, J j,k = ¯ J for all j and k , these results simplify to become (cid:104) ˆ σ + (cid:105) = e − Γ t N − (cid:0) ¯ J , t (cid:1) , (S12) (cid:104) ˆ σ a ˆ σ b (cid:105) = e − t N − (cid:0) ( a + b ) ¯ J , t (cid:1) , (S13) (cid:104) ˆ σ a ˆ σ z (cid:105) = e − Γ t (cid:0) a ¯ J , t (cid:1) Φ N − (cid:0) a ¯ J , t (cid:1) . (S14) Computation of the spin-spin coupling constants We compute the ion crystal equilibrium structure and normal modes numerically followingRef. ( ), where it is shown that the spin-spin coupling constants are given by J ij = F N (cid:126) M N (cid:88) m =1 b i,m b j,m µ − ω m . (S15)29 J j , k ( k H z ) | r j r k | ( µ m) Figure S5: Spin-spin couplings for the experimental parameters given in the main text (redpoints), together with their best power-law fit α ∼ . (blue solid line) and the uniform cou-pling approximation (black dashed line).Here, ω m and b m are the frequency and amplitude of normal mode m , respectively. We find thatfor the experimental parameters used in this work the coupling constants are well-representedby the form J j,k ∝ / | r j − r k | α with r j the position of ion j and α ∼ . − . . As aparticular example, the spin-spin coupling constants for N = 127 ions, ω z = 2 π × . MHz,a rotating wall rotation frequency of π × kHz and potential chosen to match experimentalmode spectra, and a force and detuning chosen such that ¯ J /h = 3 . kHz are shown in Fig. S5.A best fit to the computed spin-spin couplings gives a power law α ∼ . . Validation of the uniform coupling approximation Here, we show that the uniform coupling approximation Eqs. (S12)-(S14) used in the maintext is a good approximation to the solutions computed using Eqs. (S7)-(S9), with the spin-spin couplings directly determined from the phonon modes Eqn. (S15). We compare the twofor the parameters of Fig. S5. The decoherence rates are taken to match measured rates of Γ el = 171 . s − , Γ ud = 9 . s − , Γ du = 6 . s − . The normalized contrast and spin noise varianceare compared in Fig. S6(a)-(b) and very little difference is observed. These comparisons validatethe use of the uniform coupling approximation.30 -2 0 2 4 6 8 10 12 14 16 0 20 40 60 80 100 120 140 160 180 time (ms) C o n t r a s t h ~ S i / N Tomography angle (deg) Sp i n v a r i a n ce ( S ) / N / ( d B ) | % D i ↵ e r e n ce | Tomography angle (deg) (a)(b) . time (ms) | % D i ↵ e r e n ce | Figure S6: Normalized contrast (panel (a)) and spin noise variance (panel (b)) computed us-ing Eqs. (S7)-(S9) (solid red lines) and the uniform coupling approximation Eqs. (S12)-(S14)(dashed blue lines) for the spin-spin couplings of Fig. S5. The insets show the percent differ-ence between the results. The parameters used are N = 127 , ¯ J /h = 3 . kHz, Γ el = 171 . s − , Γ ud = 9 . s − , Γ du = 6 . s − , and an interaction time of ms in panel (b).31 P r o b a b ili t y d e n s i t y Spin Projection 2 S /N = 174 . P r o b a b ili t y d e n s i t y Spin Projection 2 S /N = 88 Figure S7: Full counting statistics with (red solid) and without (blue dashed) decoherence fromspontaneous emission for the squeezed (upper panel, ψ = 174 . ◦ ) and anti-squeezed (lowerpanel, ψ = 88 ◦ ) quadratures. Decoherence more significantly affects the anti-squeezed quadra-ture. 32 omputation of the Full Counting Statistics While a computation of the full counting statistics for general spin-spin couplings J jk is ex-ponentially difficult in the number of ions, the computation can be performed efficiently withthe uniform coupling approximation, J jk = ¯ J . Rather than directly computing the probabilitydistribution to measure n spins along ˆ S ψ = cos ψ ˆ S z + sin ψ ˆ S y , P ψ ( n ) , it is advantageous tocompute the characteristic function C ψ ( q ) = (cid:104) e iq (cid:80) Nj =1 ˆ σ ψj (cid:105) . (S16)From the characteristic function, the probability distribution is obtained by Fourier transforma-tion as P ψ ( n ) = 1 N + 1 N (cid:88) k =0 e − i πkN +1 n C ψ (cid:18) πkN + 1 (cid:19) . (S17)Because the spin operators ˆ σ ψj in Eqn. (S16) mutually commute, we can write the characteristicfunction as C ψ ( q ) = (cid:104) N (cid:89) j =1 (cid:104) cos ( q ) ˆ I j + i sin ( q ) ˆ σ ψj (cid:105) (cid:105) , (S18)where ˆ I j is the identity operator for spin j . In the uniform coupling approximation, this becomes C ψ ( q ) = (cid:104) (cid:16) cos ( q ) ˆ I + i sin ( q ) ˆ σ ψ (cid:17) N (cid:105) , (S19)where it is understood that multiple instances of ˆ σ ψ are interpreted to correspond to operatorson different spins. This result can be expanded using the binomial theorem as C ψ ( q ) = N (cid:88) n =0 (cid:18) Nn (cid:19) cos N − n ( q ) ( i sin ( q )) n (cid:104) ˆ σ ψ ( n ) (cid:105) , (S20)where (cid:104) ˆ σ ψ ( n ) (cid:105) denotes the expectation with n ˆ σ ψ operators on different spins, e.g., (cid:104) ˆ σ ψ (2) (cid:105) = (cid:104) ˆ σ ψ ˆ σ ψ (cid:105) . In the uniform coupling approximation, validated in Fig. S6, the system is permuta-tionally symmetric and so (cid:104) ˆ σ ψi ˆ σ ψj (cid:105) = (cid:104) ˆ σ ψ (2) (cid:105) for all i (cid:54) = j . Expanding a product of n ˆ σ ψ in termsof n + ˆ σ + s, n − ˆ σ − s, and ( n − n + − n − ) ˆ σ z s, the final result for the characteristic function in the33niform coupling approximation is (cid:104) C ψ ( q ) (cid:105) = N (cid:88) n =0 (cid:18) Nn (cid:19) cos N − n ( q ) ( i sin ( q )) n × n (cid:88) n + =0 n − n + (cid:88) n − =0 n ! n + ! n − ! ( n − n + − n − )! ( − i sin ψ ) n + ( i sin ψ ) n − × (cos ψ ) n − n + − n − (cid:104) ˆ σ +( n + ) ˆ σ − ( n − ) ˆ σ z ( n − n + − n − ) (cid:105) . (S21)Using the methods of Ref. ( ), we can write the correlation function as (cid:104) ˆ σ +( n + ) ˆ σ − ( n − ) ˆ σ z ( n z ) (cid:105) = e − ( n + + n − )Γ t n + + n − Ψ n z (( n + − n − ) J, t ) × Φ N − ( n + + n − + n z ) (( n + − n − ) J, t ) , (S22)which leads to a fully analytic representation of the counting statistics. A comparison of thecounting statistics with and without the effects of decoherence from spontaneous emission forthe parameters of Fig. S6 is given in Fig. S7. The quadrature for ˆ S ψ at ψ = 88 ◦ , which ischaracteristic of antisqueezing, is more strongly affected by decoherence than the quadraturealong the squeezed direction, ψ = 174 . ◦ . The fast oscillations exhibited by the Hamiltonianevolution are washed out by decoherence. Theoretical modeling of magnetic field fluctuations and photon shot noise As mentioned in an earlier section, homogenous fluctuations in the magnetic field caused byvibrations of the magnet contribute to dephasing. In the absence of decoherence, the effect ofa time-fluctuating, homogeneous magnetic field with Hamiltonian ˆ H B = B ( t ) (cid:80) i ˆ S zi on anarbitrary permutation-symmetric correlation function is (cid:104) ˆ σ +( n + ) ˆ σ − ( n − ) ˆ σ z ( n z ) (cid:105) B = e iϕ ( τ )( n + − n − ) (cid:104) ˆ σ +( n + ) ˆ σ − ( n − ) ˆ σ z ( n z ) (cid:105) B → , (S23)where ϕ ( τ ) = (cid:82) τ/ dtB ( t ) − (cid:82) ττ/ dtB ( t ) for the spin-echo sequence used in the experiment.While this expression is no longer exact in the presence of Raman decoherence, it is an excellentapproximation in the experimentally relevant case that B ( t ) is small and elastic decoherencedominates over Raman decoherence. Averaging over realizations of the fluctuating field ϕ ( τ ) ,we find e iϕ ( τ )( n + − n − ) = 1 − ( n + − n − ) φ rms ( τ ) + . . . = e − ( n + − n − ) ∆ φ rms ( τ ) , (S24)34 pin Projection 2 S /N Spin Projection 2 S /N P r o b a b ili t y d e n s i t y P r o b a b ili t y d e n s i t y (a)(b) = 174 . = 88 Figure S8: Experimentally measured histogram of counting statistics in the absence of ODFbeams (bars) compared with the theoretical prediction (solid line) including magnetic field andphoton shot noise. The upper panel is for the squeezed quadrature ψ = 174 . ◦ , while the lowerpanel is for the anti-squeezed quadrature ψ = 88 ◦ .35 P r o b a b ili t y d e n s i t y Spin Projection 2 S /N = 174 . P r o b a b ili t y d e n s i t y Spin Projection 2 S /N = 88 Figure S9: Full counting statistics without (red solid) and with (blue dashed) homogeneousmagnetic field noise for the squeezed (upper panel, ψ = 174 . ◦ ) and anti-squeezed (lowerpanel, ψ = 88 ◦ ) quadratures. The effects of magnetic field noise are strongly suppressed bydecoherence. The asymmetry in the peaks at positive and negative S ψ are due to unequal Ramandecoherence rates Γ du (cid:54) = Γ ud . 36here the overbar denotes averaging of the stochastic variable, ∆ φ rms ( τ ) = ϕ ( τ ) , and wehave used the fact that ϕ ( τ ) = 0 . The variance ∆ φ rms ( τ ) is determined experimentally, asshown in Fig. S3. With this, we find that the correlation functions in the presence of a fluctuatingmagnetic field are obtained as (cid:104) ˆ σ +( n + ) ˆ σ − ( n − ) ˆ σ z ( n z ) (cid:105) B (S25) ≈ e − ( n + − n − ) φ rms ( τ )2 (cid:104) ˆ σ +( n + ) ˆ σ − ( n − ) ˆ σ z ( n z ) (cid:105) B → . A comparison of the experimentally measured counting statistics with no ODF beams ( ¯ J and alldecoherence rates set to zero) with the theory that models magnetic field noise with Eqn. (S25)is shown in Fig. S8, demonstrating that the noise is accounted for consistently. In particular, wesee that the dependence on tomography angle ψ of the variance due to magnetic field noise isaccurately captured. In Fig. S9, we show the difference between the theoretical predictions forthe counting statistics with and without homogeneous magnetic field noise, using the parametersof Fig. S6, a total interaction time of τ = 3 ms, and ∆ φ rms ( τ ) = 0 . . We see that themagnetic field noise has a relatively slight effect on the full counting statistics. This can beunderstood by noting that the correlations which are significantly affected by the magnetic fieldnoise (those with ( n + − n − ) (cid:54) = 0 ) are already suppressed by the factor e − ( n + + n − )Γ t due todecoherence from spontaneous emission. The final source of noise we include in our theoreticalpredictions is photon shot noise, which is accounted for by convolving the theoretical countingstatistics P ψ with the distribution of photon shot noise P psn . For the present case, the distributionof shot noise is taken to be Gaussian with standard deviation . in units of S ψ / ( N/ (11% ofspin projection noise). A comparison of the results with and without this convolution is givenin Fig. S10. Also, we note that the dashed lines in Fig. S10, which include all the sources ofnoise described above, are plotted in Fig. 4 of the main text with a different normalization. Inorder to compare the discrete probability distribution which does not include shot noise withthe continuous distribution resulting from the convolution, the latter is evaluated on the set ofpoints where the former has support and normalized so that its sum is unity. Extraction of the Fisher information from the Hellinger distance The Fisher information F , which measures the distinguishability of quantum states with re-spect to small phase rotations, is a many-particle entanglement witness, with a measurement of F/N > n implying that the state is n -particle entangled. Importantly, this characterization ofentanglement holds even for non-Gaussian states, where spin squeezing is no longer an effectivewitness. Further, the bound F/N > is both a necessary and sufficient condition for the abilityto perform sub-shot-noise phase estimation with a quantum state.We characterize the Fisher information following the method of Ref. ( ), which utilizesthe Euclidean distance in the space of probability amplitudes known as the (squared) Hellinger37 P r o b a b ili t y d e n s i t y Spin Projection 2 S /N = 174 . P r o b a b ili t y d e n s i t y Spin Projection 2 S /N = 88 Figure S10: Full counting statistics without (red solid) and with (blue dashed) convolution withphoton shot noise. Both curves also include homogeneous magnetic field noise. The upperpanel is for the squeezed quadrature ψ = 174 . ◦ , while the lower panel is for the anti-squeezedquadrature ψ = 88 ◦ . In order to compare the discrete distribution without shot noise to thecontinuous distribution with shot noise, the latter was evaluated on the support of the formerand normalized to sum to 1. Note that the normalization of the data is different than in Fig. 4of the main text, but the blue lines are otherwise identical.38istance, d H ( θ ) = 12 (cid:88) n (cid:16)(cid:112) P θ ( n ) − (cid:112) P ( n ) (cid:17) . (S26)Here, P ( n ) denotes the counting statistics along the optimal tomography angle ψ , P θ ( n ) de-notes the counting statistics after rotation by θ around y , and (cid:80) n denotes the metric such that (cid:80) n P θ ( n ) = 1 . As shown in Ref. ( ), for small angles θ , the squared Hellinger distancesatisfies d H ( θ ) = F θ + O (cid:0) θ (cid:1) . (S27)We use a quartic fit to the squared Hellinger distance for small rotation angles to extract thequadratic coefficient, and from this the Fisher information per particle. In the absence of photonshot noise, but including the decoherence from spontaneous emission and magnetic field noise,we find F/N = 2 . , while including photon shot noise drops this value to F/N = 0 . ,as shown in Fig. S11. This comparison demonstrates that the experiment measures an over-squeezed state consistent with entanglement, but that this entanglement cannot be verified withthe present magnitude of photon shot noise. For comparison, we also show the result in theabsence of noise or decoherence of any kind, in which case the Fisher information per particleis F/N = 34 . . 39 q u a r e d H e lli n g e r d i s t a n ce d H ✓ (rad) N o s h o t n o i s e , F / N = . W i t h s h o t n o i s e , F / N = . N o d ec o h e r e n ce , F / N = .4