Robust Bilayer Charge-Pumping for Spin- and Density-Resolved Quantum Gas Microscopy
Joannis Koepsell, Sarah Hirthe, Dominik Bourgund, Pimonpan Sompet, Jayadev Vijayan, Guillaume Salomon, Christian Gross, Immanuel Bloch
RRobust Bilayer Charge-Pumping forSpin- and Density-Resolved Quantum Gas Microscopy
Joannis Koepsell,
1, 2, ∗ Sarah Hirthe,
1, 2
Dominik Bourgund,
1, 2
Pimonpan Sompet,
1, 2
Jayadev Vijayan,
1, 2
Guillaume Salomon,
1, 2
Christian Gross,
1, 2, 3 and Immanuel Bloch
1, 2, 4 Max-Planck-Institut f¨ur Quantenoptik, 85748 Garching, Germany Munich Center for Quantum Science and Technology (MCQST), 80799 M¨unchen, Germany Physikalisches Institut, Eberhard Karls Universit¨at T¨ubingen, 72076 T¨ubingen, Germany Fakult¨at f¨ur Physik, Ludwig-Maximilians-Universit¨at, 80799 M¨unchen, Germany (Dated: February 19, 2020)Quantum gas microscopy has emerged as a powerful new way to probe quantum many-bodysystems at the microscopic level. However, layered or efficient spin-resolved readout methods haveremained scarce as they impose strong demands on the specific atomic species and constrain thesimulated lattice geometry and size. Here we present a novel high-fidelity bilayer readout, whichcan be used for full spin- and density-resolved quantum gas microscopy of two-dimensional systemswith arbitrary geometry. Our technique makes use of an initial Stern-Gerlach splitting into adjacentlayers of a highly-stable vertical superlattice and subsequent charge pumping to separate the layersby 21 µ m. This separation enables independent high-resolution images of each layer. We benchmarkour method by spin- and density-resolving two-dimensional Fermi-Hubbard systems. Our techniquefurthermore enables the access to advanced entropy engineering schemes, spectroscopic methods orthe realization of tunable bilayer systems. Quantum simulation has opened a new and uniquewindow to explore static and dynamical properties ofquantum matter [1–5], difficult to access with classicalnumerical computations. Strongly-correlated materialsare typically simulated with Fermi-Hubbard systems, inwhich the intricate interplay between density (charge)and spin degrees of freedom is believed to contain es-sential ingredients for the physics of high-temperaturesuperconductivity [6, 7]. Despite its simple form, thetwo-dimensional Fermi-Hubbard model still poses majorchallenges to the exploration of its phase diagram. In ad-dition, it is an open question to what extent the layeredstructure in real materials like the cuprates affects theresulting physical properties [8, 9].Quantum gas microscopy of two-dimensional Fermi-Hubbard systems promises to shed new light on the inter-play between antiferromagnetic order and mobile density(charge) dopants. Several aspects were recently explored,such as long-range antiferromagnetic correlations [10],spin and density transport [11, 12] or the effect of dopingon two-point spin-correlations [13]. However, most exper-iments are strongly constrained in their accessible observ-ables. It is usually possible to measure either density ob-servables, without being able to distinguish between dou-blons or holes (parity projection) [14, 15], or single spin-component observables alone [16, 17], thereby severelyrestricting the potential of quantum gas microscopy, es-pecially for doped systems. Recent studies at full density-and spin-resolution have shown the capability of simul-taneous detection of occupation and spin, including thestatic [18, 19] or dynamic [20] aspects of spin-charge sep-aration in one dimension or magnetic polarons in twodimensions [21]. However the applied technique [22] typ-ically strongly constrains available lattice geometries, re- duces system size and leads to weaker couplings in thesystem.Alternative methods for spin-resolved readout arebased on vector light shifts or electronic dark states andcan be partly used to overcome such deficiencies [23, 24],however, they were demonstrated for atomic species notavailable for Fermi-Hubbard experiments. In addition, itwould be desirable to extend quantum gas microscopytechniques beyond two dimensions, thus enabling thecontrol and study of multi-layered systems. This is highlychallenging, because the high density of the analyzed sys-tems, on scales of the optical resolution, prevents employ-ment of multi-layer readout schemes that are suitable forsystems with large lattice spacing and vanishing tunnelcoupling between sites [25–27]. For bosons, a first exam-ple of such a scheme has been demonstrated [28].Here we demonstrate a novel approach that overcomesall these challenges. It allows to realize and image bilayersystems and to obtain full spin- and density-resolved im-ages of two-dimensional quantum gases with arbitrary ge-ometries, including two-dimensional Fermi-Hubbard sys-tems studied here. By using a vertical bichromatic super-lattice, we gain full control over coupled layers to imple-ment a charge pump [29–34], which makes our schemeespecially robust and efficient. We use this quantumpump to separate two initially coupled layers over largedistances. The large separation between the layers, farbeyond the depth of focus of our imaging system, enablessingle-site fluorescence imaging of a single layer withoutsignificant background contribution from the other layer.By using a vertical magnetic field gradient to split differ-ent hyperfine states of atoms into different layers in aninitial step, we can use the scheme to implement high-fidelity spin- and density-resolution of two-dimensional a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b xyz Bilayer Charge Pump a Δ z Image 2Image 1 b c T i m e z XZ Bilayer1st2nd3rd
21 µm O . D . φ SL t z t z EG FIG. 1.
Schematics of setup and charge pump. a , Weuse a bichromatic optical superlattice (red, green) in the verti-cal direction of our quantum gas microscope to control bilayersystems (gray) and their mutual tunnel couplings t z or energyoffsets ∆. The two layers of a bilayer system are pumped toa separation of 21 µ m, which enables subsequent site-resolvedmicroscopy of both layers by shifting the focal plane by ∆ z between the images. b , Charge pumping is achieved by time-dependent modulation of superlattice parameters. Atoms ini-tialized in opposite wells ( G and E ) experience transport inopposite directions. For technical reasons we reset the su-perlattice phase after each adiabatic passage (see Supplemen-tary Material). c , Absorption images (side view) of an ini-tially coupled bilayer system, whose planes are subsequentlypumped in opposite directions for 1,2 and 3 pumping steps,leading to a final separation of 21 µ m between the two layers. Fermi-Hubbard systems. We first benchmark the pump-ing and bilayer readout technique by imaging the indi-vidual site occupations of a bilayer system, consisting oftwo coupled two-dimensional Mott insulators. Next, wedemonstrate full spin- and density-resolution of a singleFermi-Hubbard layer and reveal strong antiferromagneticspin correlations in the Mott insulating regime.Our quantum gas microscope realizes spin-1 / Li in a square optical latticeof spacing a xy = 1 . µ m and tunnel coupling t in the xy -direction [35]. In the vertical z -direction the atomicsystem is usually confined to a single layer of a highlystable bichromatic optical superlattice (see Fig. 1a andSupplementary Material). This vertical lattice exhibitsshort (long) lattice spacings of a sz = 3 µ m ( a lz = 6 µ m)and is created by interfering two laser beams of wave-length λ s = 532 nm ( λ l = 1064 nm) at an angle of 5 . ◦ .The phase difference φ SL between the two 532 nm lat-tice beams is controlled by shifting the frequency of the532 nm light, thereby enabling full and dynamical controlof the resulting superlattice potential (see SupplementaryMaterial). Strongly coupled bilayer systems with tunnelcouplings of up to t z /h = 571(1) Hz between two layerscan be realized with typical lattice depths of V s = 11 E sR and V l = 100 E lR , where E iR denotes the respective recoilenergy.An important feature of time-modulated superlatticesis the existence of two distinct bands G and E with op-posite Chern numbers [29, 33, 36]. These bands causetransport in opposite directions upon the same adiabaticpassage of the double-well tilt ∆. This is referred toas geometric pumping and lies at the heart of topologi-cal charge pumping in time-modulated superlattices [30–34]. As depicted in Fig. 1b, an atom initialized in G canbe transferred to a neighbouring well by adiabaticallychanging the energy offset ∆ between the wells at a con-stant interwell tunnel coupling t z . For the same ramp,an atom in E will be transported to a neighbouring wellin the opposite direction. Performing n such adiabaticpumping steps can be used to separate two layers, ini-tially displaced by only a sz , over macroscopic distances of a sz + na lz . In Fig. 1c we show absorption images of atomsinitialized in neighbouring layers of the vertical super-lattice. Initially, the distance between the two layers isunresolved, however, after three adiabatic pumping cy-cles we are able to clearly resolve the wide separationof 21 µ m. Optimized pump parameters for high-fidelitytransport are described in the Supplementary Material.A large separation between planes enables the inde-pendent microscopy of each layer. Our imaging system(NA=0 .
5) exhibits a depth of focus below 3 µ m. Whentaking a fluorescence image with one layer in focus, theother layer will be out-of-focus and only contribute aweak and homogeneous background. After capturing thefirst image, we shift the focal plane of our imaging systemto take an image of the other layer. The parasitic back-ground of the opposite layer is extremely weak, such thatno further image processing is necessary and occupationscan be reconstructed with high fidelity using our usualdeconvolution algorithm [22]. To demonstrate the inde-pendent readout of two layers, we initialize and imagea weakly coupled ( t z /t = 1 .
3, ∆ = 0) Mott-insulatingbilayer system. After freezing the atomic occupations,three pumping steps are applied to separate the atomicplanes. Then we take fluorescence images of both lay-ers in a dedicated pinning lattice [35] and reconstruct : empty: single : double: triple3 μ m 21 μ m(i) (ii) ad be xyz c C o u p l e d B il a y e r C h a r g e P u m p t z A t o m nu m b e r ( a . u . ) xy xy FIG. 2.
Bilayer readout and charge pump fidelity. a ,An initially coupled Mott insulating bilayer system is sep-arated over a distance of 21 µ m using three charge pumpingsteps. b,c , Single-site resolved fluorescence images and recon-struction (inset) of the respective site occupations of the twoseparated layers. The images were obtained by shifting thehigh-resolution objective to sequentially image the two layers. d , Summed occupation of vertically combined sites (super-sites), relevant for bilayer systems. Here, the two verticallyoverlapping Mott insulators show a large region of double oc-cupation on a super-site. e , Averaged and normalized num-ber of atoms in a monolayer system as a function of pumpingsteps. A fit to the data (black line) yields a transfer-fidelityfor each pumping step of η = 0 . the single-site resolved occupations (see Fig. 2). Further-more, the combined occupation between vertically neigh-bouring sites (forming a super-site) is highly relevant forbilayer systems, as a suppression of density fluctuationson such a super-site is expected for band-insulating [37]or dimer phases [38]. This observable is now readily ac-cessible (see Fig. 2d) as well as three-dimensional chargecorrelators to study bilayer physics in the future.In order to determine the probability of adiabatictransfer per pumping step η , we track the number ofatoms initialized in a single layer for a variable amountof pumping steps n ∈ [0 , , , μ m 21 μ m(ii) (iii)(i) a : hole: doublon : spin : spin d be A t o m nu m b e r xyz M o n o l a y e r c S p i n S p li t C h a r g e P u m p xy xy FIG. 3.
Spin resolution of two-dimensional systems.a,
The two different spin states of a monolayer spin mixture(here a two-dimensional Fermi-Hubbard system) are split intodifferent layers (red/blue) of a bilayer system with a verticalmagnetic gradient. Subsequently, charge pumping and bilayerreadout is applied with single-site resolution. b,c , Snapshot ofthe two spin components of a two-dimensional Fermi-Hubbardsystem after separation and the full spin and density recon-struction of the original single layer in d . The correct re-combination of the spin-layers is ensured by the unique min-imization of reconstructed doublons (see Supplementary Ma-terial). e , Number of atoms detected in the upper (red) orlower (blue) layer as a function of the superlattice splittingphase. The magnetic field gradient during the initial splittingprocess creates a broad plateau of superlattice phases φ SL ,for which a balanced spin-mixture is successfully split into itsconstituents. η n to this curve yields a very high pumping efficiency of η = 0 . (cid:39)
300 Hz, caused by the xy -lattices. Itdetunes distant double-well structures appreciably com-pared to the superlattice energy scales t z and reduces thefidelity of pumping in both directions.The bilayer readout scheme presented above canbe readily extended to obtain full spin- and density-resolution of a single 2D Fermi-Hubbard plane. Afterrealizing such a single two-dimensional system, we firstfreeze the motion of the atoms in a deep xy -lattice and a bc d q x (1/ a ) q y ( / a ) C( d ) x10 -2 π π π π m z O cc u r e n ce S p i n c o rr e l a t i o n Distance (sites) S z ( q )- S z ( ) x- distance (sites) y - d i s t a n ce ( s i t e s ) ^ FIG. 4.
Two-dimensional spin correlations , a Averagedspin correlations versus distance of undoped two-dimensionalFermi-Hubbard systems at
U/t = 9 . n < .
84 areexcluded from the analysis. b , Sign-corrected spin correla-tions versus radial distance. Error bars denote one s.e.m. c ,Spin structure-factor, exhibiting a strong peak at the antifer-romagnetic momentum ( π, π ). d , Full counting statistics ofthe staggered magnetization ˆ m z . The figure is based on 150experimental realizations. confine them in a single layer of the large-scale verti-cal lattice. A strong vertical magnetic field gradient of45 G/cm is then applied to pull the two spins in oppositedirections and the short spaced vertical lattice is turnedon adiabatically with a phase φ SL set to obtain sym-metric double wells (∆ = 0) along the vertical direction.We thereby realize a controlled Stern-Gerlach separation[22] of the two spin components into two adjacent planesof the bilayer optical lattice (see Fig. 3). For our exper-iment, we employ a magnetic offset field below 13 G tomaximize the differential magnetic moments between thelowest hyperfine states of Li during the splitting pro-cess. Once the spin states are split into different layers,we separate the two planes over larger distances using thebilayer readout sequence introduced above. We measurethe occupation in each layer as a function of the super-lattice phase φ SL during the splitting to demonstrate theaccuracy of the spin splitting (see Fig. 3e). Working witha spin-balanced system, successful spin-splitting mani-fests itself as a broad range of superlattice phases φ SL ,for which the occupation in both layers is constant. Thewidth of this plateau is determined by the strength of themagnetic field gradient and the lattice parameters along the z -direction. Combining the spin-resolved occupationsof each layer, we obtain the full density and spin re-construction of the two-dimensional system (see Fig. 3d).The excellent agreement between densities and densityfluctuations of unpumped and spin-resolved pumped lay-ers excludes the presence of any significant transversemotion caused by the vertical splitting and pumping (seeSupplementary Material).The most stringent benchmark, however, for ourspin-resolution technique is the measurement of spin-correlations in a two-dimensional system. For undopedMott insulators, strong antiferromagnetic correlationsare expected to arise in the system in the regime oflarge on-site interaction versus tunneling U/t and forlow enough temperatures, due to the antiferromagneticHeisenberg spin couplings J . We take spin- and density-resolved measurements of an undoped system at an in-plane lattice depth of 6 . E xyR and U/t = 9 . C ( d ) = (cid:104) ˆ S zi ˆ S zi + d (cid:105) as well as re-veal a strong peak in the corresponding spin structurefactor S ( q ) at the antiferromagnetic quasimomentum q = ( π, π ). We find a mean staggered magnetizationof m z = (cid:112) (cid:104) ( ˆ m z ) (cid:105) = 0 . U/t = 7 .
2, these correla-tions correspond to a temperature of k B T (cid:39) . J , where k B is the Boltzmann constant. Such temperatures areamong the lowest that have been reported for ultracoldfermionic systems and underline that the detection pro-cess employed preserves the intricate correlations in thesystem.Our work underlines the unique detection possibilitiesafforded by versatile and highly controlled superlatticesetups for quantum gas microscopy. Next to realizingtunable bilayer systems with independent density read-out, we showed how our scheme can be used as a univer-sal technique to obtain full spin- and density-resolutionof single 2D Fermi-Hubbard layers for arbitrary 2D lat-tice geometries. The methods demonstrated here com-bine robust charge pumping and efficient Stern-Gerlachseparation to provide an entirely new degree of controland readout for quantum gas microscopes. Our workextends the capabilities of such systems to coupled lay-ered systems, which are highly relevant in the context ofhigh- T c superconductivity [37, 39–42]. An extension ofthe scheme to spin detection of a bilayer system requiresfour images and can be implemented in the future. Fur-thermore, advanced cooling schemes for two-dimensionalsystems based on dynamical disentangling of layers arenow within reach [43] and schemes for non-trivial observ-ables like entanglement entropy [44, 45] or angle-resolved-photoemission-spectroscopy [46, 47] can be realized onthe repulsive side ( U >
0) and for higher dimensionalsystems, based on the technique reported here.
Acknowledgments:
This work was supported by theMax Planck Society (MPG), the European Union (FET-Flag 817482, PASQUANS), the Max Planck Harvard Re-search Center for Quantum Optics (MPHQ) and underGermany’s Excellence Strategy – EXC-2111 – 39081486.J.K. gratefully acknowledges funding from Hector FellowAcademy. We would like to thank Michael H¨ose for earlycontributions to the superlattice setup.
SUPPLEMENTARY MATERIAL:Superlattice Setup
As a laser source for the 1064 nm light of our super-lattice we use around 8 W from a Mephisto MOPA. 1 Wof that light is used as seed light for second-harmonic-generation of the 532 nm light. The seed light is sentin a double-pass configuration through two consecutiveAOMs (AODF 4225-2, G&H), which have a tailored ex-tremely large bandwidth of about 130 MHz. By con-trolling the frequency drive of the AOMs, we tune theabsolute frequency by 410 MHz. The remaining 30 mWafter AOMs is amplified to 40 W with a fiber amplifier(Azurlight Systems) and used to generate 12 W of 532 nmlight in a single-pass second-harmonic generation with aperiodically-poled crystal (PPMgSLT, OXIDE). The twobeams, which are interfered for the vertical lattice, origi-nate from a single overlapped bichromatic beam. For op-timal phase stability, the single overlapped beam is splitinto two bichromatic beams and aligned onto the atoms,all in a dedicated evacuated and temperature-stabilizedchamber. A path length difference of ∆ L = 56 cm be-tween the two bichromatic beams in the evacuated cham-ber allows control of the relative superlattice phase φ SL without moving physical parts, by shifting the frequencyof the 532 nm light. Elliptical beam shaping is used toguarantee a low in-plane harmonic confinement at theposition of the atoms. Superlattice Control
The optical superlattice potential is given by V = V s cos ( k s z + φ SL )+ V l cos ( k l z + π/ V s,l , wave vectors k s,l = π/a s,l and superlattice phase φ SL . Due to the arm length difference of the two inter-fering beams of the short lattice, a change in frequencychanges not only k s , but also φ SL . As the frequencychange is small compared to the absolute frequency ofthe light, the wavevector change is irrelevant. The su-perlattice phase, on the other hand, is tunable by anamplitude of 1 . π with ∆ ν = 820 MHz.We verify the tunnel coupling and coherence between twolayers by measuring Rabi oscillations. A Mott insulatingstate was prepared at 26 E xyR in an isolated single layerand quenched to a strong coupling with its neighbour-ing layer. After quenching off the coupling, we applybilayer readout and measure the single-site resolved oc-cupation per layer as a function of the coupling duration.In Fig. S1, the oscillation of the occupation is shown fora quench to V s = 11 E sR and V l = 100 E lR . We extracta tunnel coupling of 571(1) Hz for those parameters byan exponentially decaying sinusoidal fit, where the errordenotes the uncertainty of the fit. Pump Sequence
Our range of available superlattice phases contains twodistinguishable symmetric double-well configurations at − π/ π/
2. We realize a coupled bilayer system orthe splitting of a monolayer into its spin components ata symmetric superlattice phase of π/
2. During the appli-cation of splitting and threefold pumping, the atoms arefrozen in the xy -plane with 43 E xyR and vertically with49 E sR and 100 E lR for a duration of 400 ms. To preparethe first pump, φ SL is ramped within 5 ms from π/ − π/ δφ/
2, where δφ = 0 . π . Then the double-welltunnel coupling t z is turned on by ramping the shortlattice down to 11 E sR in 20 ms. The first pump is per-formed by ramping the superlattice phase within 3 ms by δφ across the symmetric configuration to a final value of φ SL = − π/ − δφ/
2. During this phase sweep the trans-port of atoms to the neighbouring layer happens. Even-tually, the short lattice is ramped back to 49 E sR and thenext pump is performed at φ SL = ( − χ +1 π/
2, where χ is the number of already applied pumps. These parame-ters were optimized for highest overall pump fidelity. I m b a l a n ce n n FIG. S1.
Rabi oscillation between two layers.
Imbalanceof the local densities n and n of two layers after quenchingon inter-layer tunneling, averaged over a region of three sites.By fitting the observed Rabi frequency (solid black), we ex-tract the interlayer tunneling strength of t z = h × E sR and 100 E lR . Error barsdenote one s.e.m. Bilayer Readout
The bilayer system was prepared from a harmonicallyconfined atom cloud, held in a single layer of the large-spaced vertical lattice. At a fixed balanced superlatticephase (∆ = 0), the in-plane lattice potential and short-spaced vertical lattice were ramped simultaneously to adepth of V xy = 11 E xyR , V s = 19 E sR with a final on-site in-teraction U versus in-plane tunneling t of U/t (cid:39)
50. Themotion of atoms was then quenched rapidly, by rampingto a lattice depth of V xy (cid:39) E xyR , V s (cid:39) E sR . Thedeconvolution of the fluorescence images obtained from each layer was done with a Lucy-Richardson algorithmimplemented in our previous work [22]. The resulting his-tograms of counts per lattice site are shown in Fig. S2.By fitting a gaussian to the zero- and single-occupationpeaks, we determine a fidelity to distinguish holes fromsinglons by summing the probability for false negativesand false positives. For the reconstruction of a bilayersystem of two Mott-insulator, this fidelity is 99 . . O cc u r e n ce s ( a r b )
50 100 150 200Counts (arb)0.00.51.0 O cc u r e n ce s ( a r b ) ab FIG. S2.
Bilayer reconstruction.
Histogram of counts perlattice site in layer one ( a ) and two ( b ). Solid and dashedblack lines are gaussian fits to the peaks of single and zerooccupation. Vertical dashed grey lines denote the thresholdfor single and double occupation. Spin Reconstruction
To combine the reconstruction of the two spin-layersto obtain the reconstructed spin and density informationof the two-dimensional parent system, the right latticesites along the vertical direction had to be paired. Thevertical movement of our objective by 21 µ m betweentwo images sometimes caused small movements of theobjective in the transverse direction. This resulted inshot to shot fluctuation of the displacement of the twoimages on the order of up to one lattice site. To pairthe sites correctly, we compared 25 pairing possibilities,given by shifting one layer in x, y direction by ± . Total doublons a b C o n f i g u r a t i o n s p e r s h o t correct FIG. S3.
Centering of two layers for spin reconstruc-tion.
To accurately center the two spin layers, one layer isshifted versus the other by up to ± x - and y - direction and the reconstructed total doublon number iscompared. a , Total number of reconstructed doublons for the5 × b , Average num-ber of configurations grouped by the total number of doublonsin excess of the minimum number found. There always ex-ists exactly one correct shift configuration, which contains aminimum number of doublons. Density Fidelity of Spin Readout
In order to show no particle motion occurs during spin-splitting and pumping, we compared the two-dimensionaldensity and its fluctuations measured with our spin-readout to an identical un-splitted and un-pumped sys-tem. As shown in Fig. S4, the density and its normalizedfluctuations are in good agreement for the two methods.
Spin Correlations
Undoped two-dimensional Fermi-Hubbard systemswere prepared from a harmonically confined cloud byramping the in-plane lattice depth to a final value of6 . E xyR within 100 ms. The final scattering length wasset to 810 a B , where a B denotes the Bohr radius and thesystem was vertically confined in a strongly tilted super-lattice ( φ SL = 0) of depths V s = 50 E sR and V l = 100 E lR .Spin correlations (cid:104) ˆ S zi ˆ S zj (cid:105) are evaluated on sites with meandensity above 0 .
84, which comprises N = 47 lattice sites. D e n s i t y n δ n / n a b FIG. S4.
Comparison of methods.
Averaged radial den-sity ( a ) and fluctuation profiles ( b ) of an identical Mott-insulating system measured without (black) and with (red)our spin-resolved technique. Error bars denote one s.e.m. andare smaller than the point size. For the spin structure factor, we pad the N sites of eachsnapshot with zeros to a 9 × S z ( q ) = (cid:104) (cid:80) i ˆ S zi e − i qR i × (cid:80) j ˆ S zj e i qR j (cid:105) / ( (cid:80) k (cid:104) ˆ S zk ˆ S zk (cid:105) ).The staggered magnetization is computed according toˆ m z = (cid:80) i ( − i ˆ S zi / ( SN ) with S = 1 /
2. To evaluate themean and its standard error of m z , we performed a boot-strap by resampling our data 200 times with replacement. ∗ [email protected][1] C. Gross and I. Bloch, Science , 995 (2017).[2] I. M. Georgescu, S. Ashhab, and F. Nori, Rev. Mod.Phys. , 153 (2014).[3] R. Blatt and C. F. Roos, Nat. Phys. , 277 (2012).[4] H. Weimer, M. M¨uller, I. Lesanovsky, P. Zoller, and H. P.B¨uchler, Nat. Phys. , 382 (2010).[5] M. H. Devoret and R. J. Schoelkopf, Science , 1169(2013).[6] P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. , 17 (2006).[7] B. Keimer, S. A. Kivelson, M. R. Norman, S. Uchida,and J. Zaanen, Nature , 179 (2015).[8] A. Damascelli, Z. Hussain, and Z. X. Shen, Rev. Mod.Phys. , 473 (2003).[9] O. Yuli, I. Asulin, O. Millo, D. Orgad, L. Iomin, andG. Koren, Phys. Rev. Lett. , 057005 (2008).[10] A. Mazurenko, C. S. Chiu, G. Ji, m. F. Parsons,M. Kan´asz-Nagy, R. Schmidt, F. Grusdt, E. Demler,D. Greif, and M. Greiner, Nature , 462 (2017).[11] M. A. Nichols, L. W. Cheuk, M. Okan, T. R. Hartke,E. Mendez, T. Senthil, E. Khatami, H. Zhang, andM. W. Zwierlein, Science , 383 (2019).[12] P. T. Brown, D. Mitra, E. Guardado-Sanchez,R. Nourafkan, A. Reymbaut, C. D. H´ebert, S. Bergeron,A. M. Tremblay, J. Kokalj, D. A. Huse, P. Schauß, andW. S. Bakr, Science , 379 (2019).[13] C. S. Chiu, G. Ji, A. Bohrdt, M. Xu, M. Knap, E. Demler,F. Grusdt, M. Greiner, and D. Greif, Science , 251(2019).[14] W. S. Bakr, J. I. Gillen, A. Peng, S. F¨olling, andM. Greiner, Nature , 74 (2009).[15] J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau,I. Bloch, and S. Kuhr, Nature , 68 (2010). [16] M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif,and M. Greiner, Science , 1253 (2016).[17] L. W. Cheuk, M. A. Nichols, K. R. Lawrence, M. Okan,H. Zhang, E. Khatami, N. Trivedi, T. Paiva, M. Rigol,and M. W. Zwierlein, Science , 1260 (2016).[18] T. A. Hilker, G. Salomon, F. Grusdt, A. Omran, M. Boll,E. Demler, I. Bloch, and C. Gross, Science , 484(2017).[19] G. Salomon, J. Koepsell, J. Vijayan, T. A. Hilker, J. Ne-spolo, L. Pollet, I. Bloch, and C. Gross, Nature , 56(2019).[20] J. Vijayan, P. Sompet, G. Salomon, J. Koepsell,S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and C. Gross,Science , 186 (2019).[21] J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A.Hilker, E. Demler, G. Salomon, I. Bloch, and C. Gross,Nature , 358 (2019).[22] M. Boll, T. A. Hilker, G. Salomon, A. Omran, J. Nespolo,L. Pollet, I. Bloch, and C. Gross, Science , 1257(2016).[23] T. Y. Wu, A. Kumar, F. Giraldo, and D. S. Weiss, Nat.Phys. , 538 (2019).[24] M. Kwon, M. F. Ebert, T. G. Walker, and M. Saffman,Phys. Rev. Lett. , 180504 (2017).[25] K. D. Nelson, X. Li, and D. S. Weiss, Nat. Phys. , 556(2007).[26] D. Barredo, V. Lienhard, S. de L´es´eleuc, T. Lahaye, andA. Browaeys, Nature , 79 (2018).[27] O. El´ıasson, R. Heck, J. S. Laustsen, R. M¨uller, C. A.Weidner, J. J. Arlt, and J. F. Sherson, (2019),arXiv:1912.03079.[28] P. M. Preiss, R. Ma, M. E. Tai, J. Simon, andM. Greiner, Phys. Rev. A , 041602 (2015).[29] D. J. Thouless, Phys. Rev. B , 6083 (1983).[30] O. Romero-Isart and J. J. Garc´ıa-Ripoll, Phys. Rev. A (2007).[31] Y. Qian, M. Gong, and C. Zhang, Phys. Rev. A ,13608 (2011). [32] L. Wang, M. Troyer, and X. Dai, Phys. Rev. Lett. ,026802 (2013).[33] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger,and I. Bloch, Nat. Phys. , 350 (2016).[34] S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa,L. Wang, M. Troyer, and Y. Takahashi, Nat. Phys. ,296 (2016).[35] A. Omran, M. Boll, T. A. Hilker, K. Kleinlein, G. Sa-lomon, I. Bloch, and C. Gross, Phys. Rev. Lett. , 1(2015).[36] M. Atala, M. Aidelsburger, J. T. Barreiro, D. Abanin,T. Kitagawa, E. Demler, and I. Bloch, Nat. Phys. , 795(2013).[37] M. Golor, T. Reckling, L. Classen, M. M. Scherer, andS. Wessel, Phys. Rev. B , 195131 (2014).[38] R. Chen, S. Lee, and L. Balents, Phys. Rev. B , 161119(2013).[39] T. A. Maier and D. J. Scalapino, Phys. Rev. B , 180513(2011).[40] A. I. Liechtenstein, I. I. Mazin, and O. K. Andersen,Phys. Rev. Lett. , 2303 (1995).[41] N. Bulut, D. J. Scalapino, and R. T. Scalettar, Phys.Rev. B , 5577 (1992).[42] S. Okamoto and T. A. Maier, Phys. Rev. Lett. ,156401 (2008).[43] A. Kantian, S. Langer, and A. J. Daley, Phys. Rev. Lett. , 060401 (2018).[44] H. Pichler, L. Bonnes, A. J. Daley, A. M. L¨auchli, andP. Zoller, New J. Phys. , 63003 (2013).[45] R. Islam, R. Ma, P. M. Preiss, M. Eric Tai, A. Lukin,M. Rispoli, and M. Greiner, Nature , 77 (2015).[46] A. Bohrdt, D. Greif, E. Demler, M. Knap, and F. Grusdt,Phys. Rev. B , 125117 (2018).[47] P. T. Brown, E. Guardado-Sanchez, B. M. Spar, E. W.Huang, T. P. Devereaux, and W. S. Bakr, Nat. Phys.16