Site-resolved imaging of ultracold fermions in a triangular-lattice quantum gas microscope
SSite-resolved imaging of ultracold fermionsin a triangular-lattice quantum gas microscope
Jin Yang, ∗ Liyu Liu, ∗ Jirayu Mongkolkiattichai, ∗ and Peter Schauss Department of Physics, University of Virginia, Charlottesville, Virginia 22904, USA † Quantum gas microscopes have expanded the capabilities of quantum simulation of Hubbardmodels by enabling the study of spatial spin and density correlations in square lattices. However,quantum gas microscopes have not been realized for fermionic atoms in frustrated geometries. Here,we demonstrate the single-atom resolved imaging of ultracold fermionic Li atoms in a triangu-lar optical lattice with a lattice constant of 1003 nm. The optical lattice is formed by a recyclednarrow-linewidth, high-power laser combined with a light sheet to allow for Raman sideband coolingon the D line. We optically resolve single atoms on individual lattice sites using a high-resolutionobjective to collect scattered photons while cooling them close to the two-dimensional ground vi-brational level in each lattice site. By reconstructing the lattice occupation, we measure an imagingfidelity of ∼ I. INTRODUCTION
Frustrated quantum systems pose a significant chal-lenge to condensed matter theory due to their exten-sive ground state degeneracy [1, 2] and can show frac-tional quasi-particle statistics as known from quantumHall physics [3]. There are a wide variety of interest-ing phenomena in frustrated systems. Examples includespin liquids, time-reversal symmetry breaking, and ki-netic constraints [4–6]. While small systems can besolved with tremendous computational resources, predic-tions for the low-temperature phases in the thermody-namic limit are scarce and often debated [7–9]. Existingcondensed matter realizations are complicated materialsand simpler model systems are sought after. Ultracoldatoms provide a unique way to explore quantum many-body physics through quantum simulations of frustratedquantum systems based on first principles. Prominentexamples for quantum simulation with ultracold atomsinclude the direct detection of antiferromagnetic correla-tions [10–17] and the observation of many-body localiza-tion [18]. Ultracold atoms in optical lattices implementHubbard models [19–21], where neighboring sites are cou-pled by hopping and atoms interact if they meet on thesame lattice site. Fermi-Hubbard systems were first real-ized with ultracold atoms in square lattices [22, 23]. Frus-trated lattice geometries have been studied with absorp-tion imaging of ultracold bosonic atoms [24] which led toquantum simulation of classical frustration [25]. Othergeometrically frustrated two-dimensional lattice geome-tries like kagome lattices [26] and the Lieb lattice [27]have been studied with bosonic atoms, and recently in-dividual bosonic atoms have been imaged in a triangularlattice [28]. But for the implementation antiferromag- ∗ These authors contributed equally to this work. † [email protected] netic interactions, fermions are the more natural choice[29]. For revealing intricate correlations on short lengthscales, this asks for a fermionic quantum gas microscope,where all ultracold atoms in the many-body system canbe imaged simultaneously. Existing fermionic quantumgas microscopes were used to study Hubbard models onsquare lattices [16, 30–35]. However, to obtain a geomet-rically frustrated system a non-bipartite lattice geometryis required. The triangular lattice is the paradigm exam-ple of a frustrated lattice [1], because a triangle is thesimplest structure where antiferromagnetic constraintscannot be simultaneously satisfied on all bonds. For tri-angular lattices the frustration of antiferromagnetic orderleads to a remarkable quantum phase transition for vary-ing interaction between a magnetically ordered state anda disordered state which may be a chiral spin liquid [8, 9].Here, we demonstrate the first realization of a site-resolved quantum gas microscope of ultracold fermionicatoms in a triangular lattice, thereby paving the way fora new platform to study frustrated Hubbard physics ina lattice with spacing of 1003 nm and strong tunnelingin the tight-binding limit. We load a degenerate Fermigas into the triangular lattice and obtain densities abovehalf filling.Our experiment uses fermionic Li because it possessesintriguing properties like broad Feshbach resonances anda low mass, allowing to realize Hubbard models at largertunneling and greater interaction than with other species.However, the low mass comes with difficulties localizingthe atoms during imaging. Therefore, we rely on fluo-rescence imaging during Raman sideband cooling nearthe ground state in the lattice. Raman sideband coolingof lithium is challenging due to the large lattice depthrequired to suppress tunneling and to reach the Lamb-Dicke regime. The triangular lattice adds to this diffi-culty due to extensive required optical access and con-straints on the beam geometry. We designed a sophisti-cated lattice setup to overcome these obstacles, allowingoptically resolved imaging of individual fermionic atoms a r X i v : . [ c ond - m a t . qu a n t - g a s ] F e b in the triangular optical lattice with high fidelity.In this paper, we first discuss our experimental setupto prepare Li degenerate Fermi gases and our novel ap-proach to create a triangular lattice for ultracold atoms.Then we present detailed information about the imple-mentation of single-site resolved imaging in the triangularlattice via Raman sideband cooling. We discuss recon-struction of the lattice occupation from imaging resultsand the imaging fidelity in a comparative study of single-atom imaging in three different optical lattices. We con-clude with an outlook on the study of Fermi-Hubbardphysics and frustrated quantum physics in our setup.
II. EXPERIMENTAL SETUPA. Preparation of degenerate Fermi gas
In the following, we describe the experimental setupand the path to a degenerate Fermi gas. For stabilityand fast cycle time of the experiment, we designed asingle-chamber experiment that allows for sufficient op-tical access for all required laser beams [Fig. 1]. Westart with 600,000 Li atoms in a ∼ −
30 MHz to − ∼ ◦ , where thepower of each beam is 125 W with a beam waist of90 µ m at the crossing point. The atoms evenly popu-late the states | (cid:105) ≡ | S / F = 1/2 m F = 1/2 (cid:105) and | (cid:105)≡ | S / F = 1/2 m F = − / (cid:105) after loading into theCDT, and the initial density is ∼ × cm − witha temperature of ∼ µ K. Thereafter, a three-stageevaporation (plain evaporation for 1 s; forced evapora-tion I for 0 . Li degenerate Fermi gas. During the evaporation, aFeshbach field is ramped up to 810 G, where the scat-tering length between state | (cid:105) and | (cid:105) is a s ≈ , a [36, 37], where a is the Bohr radius. The intensity ofthe dipole trap stays unchanged in plain evaporation. Inforced evaporation I, the intensity of the dipole trap is re-duced to 6% of the initial value following an exponentialdecay curve with a time constant τ = 300 ms. In forcedevaporation II, the intensity of the dipole trap continuesto reduce to 0.4% of the initial value with a time con-stant τ = 6 s. To prevent the formation of deeply boundlithium molecules and obtain degenerate Fermi gases, theFeshbach magnetic field is switched from 810 G to 300 G( a s = − a ) within 10 ms and about 500 ms beforethe end of the forced evaporation stage II, where thedensity is not yet high enough to form lithium dimersvia three-body collisions. With this experimental cycle of 12 s duration, we obtain a degenerate Fermi gas withabout 3,000 atoms and temperature below one fifth ofthe Fermi temperature, determined by a Fermi fit to anon-interacting gas. B. Triangular lattice setup
Fluorescence imaging of atoms in the triangular ge-ometry requires a strong three-dimensional confinementat each lattice site. Therefore, we need to find a tri-angular lattice configuration that provides sufficient lat-tice depth at the limited available laser power. For thispurpose, we interfere three laser beams to create a tri-angular array of one-dimensional light tubes and add astrongly oblate “light sheet” beam to complete the three-dimensional confinement. The strongly oblate light sheethas beam waists of 4 . µ m × µ m × µ m and uses powerof 24 W at 1070 nm. The trap frequency along z axis is ∼
160 kHz. In order to create a deep triangular lattice withresolvable lattice spacing we use an unusual approach.We recycle a single 1064 nm laser beam (MOPA 55W,Nd:YAG, Coherent) twice and cross all three beams atthe position of the light sheet, thereby reusing the laserpower three times [Fig. 1]. The phases of the three latticebeams do not need to be stabilized because phase driftsonly lead to translations of the triangular lattice. Tokeep these translations within tolerable bounds of aboutone lattice site per minute, the setup is very rigid andtemperature-controlled via water cooling and air condi-tioning.All three lattice beams propagate from the negative z direction (down) to the positive z direction (up) with anangle of 45 ◦ out of the x - y plane. Their projections onto x - y plane cross to each other at an angle of 120.0(6) ◦ .The power for each beam is 42 W, 40 W and 38 W, re-spectively, due to losses caused by optics during the re-cycling. All three beams have a Gaussian beam waist of ∼ µ m at the crossing. This leads us to a triangularlattice with a lattice spacing of a latt = 1003 nm. Ourconfiguration for the lattice is compatible with a stan-dard octagon vacuum chamber but requires very care-ful consideration of objective mount and magnetic fieldcoils which typically block the optical access exploitedhere, as illustrated in Fig. 1. In addition, we have acustom-designed anti-reflection coating for the vacuumwindows to reduce the reflection at the 45 ◦ angle of inci-dence. Since the interference pattern between the threecrossing beams depends both on the wavevector direc-tion and the polarization of each beam, these parametershave to be carefully adjusted for each beam. The an-gles between the lattice beams are restricted to about1 ◦ by the optical access and we use half-wave plates tocontrol the polarizations of all lattice passes. For thefollowing experiments, we adjusted these to obtain thestrongest possible interference pattern in the triangularlattice. We found that the lattice depth is maximal forincoming linear polarization angles of about 40 ◦ , − ◦ , RPR2
R1Lattice BeamT1 T2T3 O p t i c a l den s i t y S i gna l ( a r b . u . ) z xy +T1 T2T3 FIG. 1.
Triangular-lattice quantum gas microscope. ( left ) Sketch of triangular lattice and Raman sideband imagingbeams and their alignment relative to the vacuum chamber. The stainless steel octagon chamber is equipped with an outercopper coil pair for the MOT field and inner coil pair for the Feshbach field. The triangular lattice is formed by recycling thelattice beam through the recessed top and bottom windows, leaving just enough space for the objective at the top window.The second and third focus are created by 1 : 1 imaging systems, which are not shown. Three orange arrows (T1, T2 and T3)indicate the direction of the three beams which cross at the position of the atoms where the triangular lattice is formed. Thepolarization configuration used for imaging in the lattice is illustrated in the bottom middle inset. The Raman cooling beams(R1 and R2) and the Raman repump beam (RP) are sent through the side windows. ( top right ) Kapitza-Dirac scattering of Li molecular Bose-Einstein condensate (BEC) from the triangular lattice. This image is an average of 10 absorption imagesafter a time-of-flight of 1 . µ s. For this picture,we used polarization angles of 0 ◦ for lattice beams T1, T2, and T3 to demonstrate a symmetric lattice. ( bottom right ) Rawsite-resolved fluorescence image of Li atoms in the triangular lattice. and 80 ◦ for lattice beams L1, L2, and L3, respectively,relative to the vertical polarization closest aligned to the z axis [Fig. 1]. Due to birefringence in the vacuum win-dows and coatings the polarizations may be slightly mod-ified at the atom position. The asymmetry of the con-figuration leads to anisotropic tunneling in the latticein our current configuration. We confirmed by explicitcalculation that anisotropic triangular lattice geometriescan be adiabatically transformed to a symmetric config-uration by varying the polarization of one of the threelattice beams. To implement such a scheme, we planto add the capability to dynamically switch between themaximum-lattice-depth and an isotropic-tunneling con-figuration during the experimental cycle by upgrading toa motorized wave plate mount in the future.To prepare a quasi-two-dimensional Fermi gas in thetriangular lattice, we first load the degenerate Fermi gasfrom the CDT into the light sheet and evaporate for an-other 250 ms. The intensity of the light sheet is reducedto 0.2% of its initial value following an exponential decaycurve with a time constant τ = 100 ms. This evapora- tion is necessary to remove excitations created during theloading procedure. Next, the intensity of the light sheetis increased to the initial value again, and the triangularlattice is adiabatically switched on within 100 ms. Thisconfiguration with maximal depth of lattice and lightsheet is used for imaging the atoms by collecting fluo-rescence during Raman sideband cooling.For calibration of the lattice depth, we carried outKapitza-Dirac scattering [analogous to Fig. 1] and mea-sured the atom number in the zeroth order as a functionof lattice intensity. Through fitting of the decay curveto a Bessel function, we find a maximum lattice depth of ∼ E r with E r ≡ (cid:126) π / (2 ma ) = 8 . III. RAMAN SIDEBAND COOLING
In order to keep the atoms localized at each single siteduring the fluorescence imaging, we utilize Raman side-band cooling to collect scattered photons while keepingthe atoms near the ground-state of the harmonic poten-tial. Variations of Raman sideband cooling have beenused to detect various atomic species in optical latticeswith single-atom resolution [30–34, 38]. A two-photonRaman sideband transition transfers atoms from one hy-perfine ground state to the other hyperfine ground statewhile lowering the vibrational level in the on-site har-monic trap. The frequency difference between the twophotons needs to be calibrated to match the frequencydifference between the two hyperfine ground states plusthe lattice on-site harmonic oscillator frequency ω latt . Toclose the cooling cycle, the atoms need to be transferedback to the initial hyperfine ground state without chang-ing their vibrational levels. This is implemented throughan optical pumping process using the Raman repumplaser. In order to keep the heating in the repump processlow, a large ω latt is required to suppress recoil heating in x - y plane by operating in the Lamb-Dicke regime. Aftermany cycles, most atoms occupy the ground vibrationallevel which is a dark state in the absence of heating pro-cesses. The scattered photons in the optical pumpingprocess are then collected to image the atoms.Further specifics of our Raman sideband cooling setupare described as follows. A two-photon Raman transitionvia the D line transfers the atoms from | / F = 3/2 (cid:105) manifold to | / F = 1/2 (cid:105) while lowering the vibra-tional state by one. The incoming Raman cooling beam(R1) is locked 5 GHz red-detuned to the D line andis linearly polarized. It has a power of 2 . µ m on the atoms. After passingthrough the chamber, we use a double-pass configura-tion of an acousto-optic modulator (AOM) to gener-ate the second Raman beam (R2) with a detuning of228 . ω latt / (2 π ) and 70% efficiency. We choosethe angles between the two Raman beams and relativeto the lattice to obtain sufficient coupling in-plane aswell as in the z direction [Fig. 2 (a,b) ]. To determinethe ω latt we take sideband spectra by applying a pulse ofboth Raman beams directly after loading into the lattice,transferring a fraction of the atoms from | / F = 1/2 (cid:105) to | / F = 3/2 (cid:105) . These atoms are then detected inabsorption imaging and the sidebands show the lat-tice vibrational spacing of ω latt = 2 π × (d) ]. The Raman repump beam (RP) has a powerof 0 .
15 mW and a beam waist of 500 µ m at the focus onthe atoms. It is locked 9 . | / F = 1/2 (cid:105) to the | / F = 1/2 (cid:105) atomic transitionand is circularly polarized. The atoms excited by theRaman repump beam more likely decay down into the | / F = 3/2 (cid:105) state rather than the | / F = 1/2 (cid:105) state with a branching ratio of 8 : 1. Their vibrationalstate in the lattice remains mostly unchanged due to theLamb-Dicke factor η ≡ (cid:112) (cid:126) k R / (2 mω latt ) = 0 .
29 in ourexperiment, where k R is the wavevector of the Ramanrepump light. There may be higher vibrational statesexcited in the vertical light sheet direction. However,due to the depth of the light sheet and the absence ofnearby wells the atoms could tunnel to, the impact ofelevated temperatures in the z dimension on imaging fi- delity is low and we can rely on coupling to other dimen-sions for cooling, which is provided by the small angle α [Fig. 2 (b) ].The spatial configuration of the Raman cooling beamsand repump beam is shown in Fig. 1 and Fig. 2. To getthe best imaging result, we optimize the offset magneticfields, leading to a magnetic field of 1.1(2) G rather thanzero field. The parameters for all offset magnetic fieldsare shown in Fig. 2. The lifetime of atoms under contin-uous Raman cooling in the triangular lattice is 44(2) s,possibly limited by background gas collisions. δ =9.6(5) MHz Δ =5 GHz ω latt RP R1R2
R1 RP B R1 R2 z xy RP High-NA Objective α B (a) (b)(c) (d) C oun t s ( a r b . u . ) Frequency (MHz)
FIG. 2.
Raman sideband cooling. (a), (b) Raman side-band cooling beam configuration. Blue dots mark the trian-gular lattice sites. The Raman repump beam propagates inthe lattice plane ( x - y plane). The first Raman beam (R1)has horizontal linear polarization and propagates in negative z direction with a shallow angle of α = 7 . ◦ relative to thelattice plane. The second Raman beam (R2) is perpendicularto R1 and consists of a mix of horizontally and vertically lin-ear polarizations in a ratio of 4 : 1. Two green arrows show theprojection of the magnetic field on x - y plane and x - z planewith angles of − ◦ and − ◦ relative to x axis, respectively.(c) Raman sideband cooling transition scheme showing thelevels connected by the Raman repump RP and the Ramanbeams R1 and R2 and the respective detunings δ and ∆. (d)Raman sideband spectrum in the triangular lattice. The cen-ter peak is the carrier corresponding to hyperfine splitting inthe ground state while the sidebands show the lattice vibra-tional spacing of ω latt = 2 π × +3 − in x and y direction. The dots representexperimental data and the solid line is a Gaussian fit. Errorbars are the standard deviation of four repetitions. IV. HIGH-RESOLUTION IMAGING
To achieve imaging of Li atoms in the triangular lat-tice with single-site resolved sensitivity, a high-resolutionimaging system is used to collect the fluorescence dur-ing Raman sideband cooling. The imaging system con-sists of a custom objective (54-25-25@671nm, Navitar)with a focal length of 25 mm and a numerical aperture(NA) of 0.5, and an achromatic doublet (AC508-750-B)with a focal length of 750 mm, leading to a theoreticalmagnification of 30. The measured magnification is 33.Scattered photons are detected with an exposure time of500 ms by a low-noise scientific CMOS camera (AndorZyla 4.2 plus) with quantum efficiency of 77% and pixelsize 6 . × . µ m . The total transmission of imaging op-tics and narrow-band filters is ∼ ∼ (a) ]. This revealsa full width at half maximum (FWHM) of 720(18) nm,consistent with our expectation of 711 nm. From the firstminimum of an Airy fit, we extract the resolution ac-cording to the Rayleigh criterion as 818(8) nm (4.3 pix-els). This is smaller than our triangular lattice spacing of1003 nm (5.1 pixels), and we therefore resolve individualatoms in the triangular lattice without post-processing. V. IMAGE RECONSTRUCTION ANDANALYSIS
We apply a reconstruction algorithm to extract a dig-itized occupation matrix of the lattice [39]. In order toobtain the geometric parameters of the triangular lattice,we determine lattice angles and lattice constants. Thisrelies on identifying individual isolated atoms and deter-mining their center via Gaussian fits. Then, we projectthe coordinates of isolated atoms onto an axis with vary-ing rotational angle in the lattice plane. By introducingequidistant bins on this axis we generate a histogram ofatom projections [Fig. 3 (c) ]. If the rotation angle is veryclose to the lattice angle, the histogram shows multiplepeaks with minimal width and the separation betweenthe peaks is related to the lattice constant. The angleswith respect to x axis for both lattice vectors are deter-mined with high precision to − . ◦ and 13 . ◦ ,leading to 59.36(3) ◦ between the lattice vectors. Theselattice angles allow us to extract the precision of the rel- (a) (b)(c) C oun t s ( a r b . u . ) Distance from center ( µ m) N u m be r o f l a tt i c e s i t e s Detected photons per site A1 A2 Counts C oun t s b1b2 xy z FIG. 3.
Image analysis. (a) Point spread function. Az-imuthal average of the point spread function (red), with Gaus-sian fit (blue), and Airy fit (green). The measured FWHMof 720(18) nm of the PSF is consistent with the FWHM of711 nm expected from the numerical aperture of the objec-tive. The inset shows the PSF obtained by averaging isolatedatoms. (b) Single atom count histogram. The left peak cor-responds to empty sites and the right peak indicates sitesoccupied by single atoms. The threshold value between noatom and a single atom (vertical orange line) is determinedas the intersection point of two Gaussian fits to backgroundand atom signal distribution, respectively. The reconstruc-tion error caused by the overlap is negligible compared to theobserved hopping and loss. (c) Determining lattice angles andlattice constant. Single atom positions (exemplary shown inblue) are orthogonally projected onto a line of varying angle,here exemplified by A1 and A2. For each angle, a histogramof the projected positions is depicted. At the lattice angle, theexperimental histogram has perfect contrast (right graph) cor-responding to lattice vector b
1. At other angles, for exampleA2, there is almost no structure in the histograms. ative angles between the lattice beams to 120.0(6) ◦ . Thelattice constants in pixels are 5 . . (b) shows a well-separatedpeak of single atom signal. Due to light-induced colli-sions, doubly occupied sites are detected as empty sites[40–42]. From the histogram, we obtain an optimizedthreshold between the signal of no atom and single atomsto decide if a lattice site is occupied. As a result of thereconstruction, a matrix with entries zero (empty) or one(occupied) is generated. To handle the triangular latticestructure, we interpret it as a square lattice with diagonaltunneling, sheared by 30 ◦ . A. Imaging fidelity
We evaluate the imaging fidelity by taking a series offive images with 500 ms exposure time each and 50 msseparation in between. Hopping and loss rate are esti-mated by comparing two adjacent images [Fig. 4]. Thehopping rate is defined by the fraction of sites detectedas occupied in the second image only while the loss rateis given by the fraction of atoms lost from picture topicture. We define the imaging fidelity by the fraction ofatoms that remained in the same lattice sites. Our single-site imaging has a field of view of 90 × µ m , with goodhopping rates in a region of 25 × µ m in the centerof the atom distribution, which includes 625 lattice sites.We obtain a hopping rate of 2.0(2)% and a loss rate of0.4(2)% at a detected occupation of up to 50% by aver-aging over twelve pairs of pictures. This demonstratesan imaging fidelity of 97.6(3)%. The detected densityis reduced by light-induced pair-wise losses at the be-ginning of the first image which lowers the density of theloaded Fermi gas from an initial density of approximately1.3 atoms per lattice site, determined independently byhigh-field absorption imaging. B. Comparison to square lattices
In addition to the triangular lattice, we implementeda versatile square lattice at the same experimental setupwhich can be superimposed with the triangular lattice.The square lattice setup can be used at 532 nm or 752 nmlattice spacing. We create the square lattices usingthe recycled lattice setup as described in refs. [16, 43][Fig. 5 (a) ]. For vertical polarization, four-beam interfer-ence leads to a 752 nm spacing lattice, while an in-planepolarization creates a 532 nm spacing lattice. The powerof the four passes is 41 W, 39 W, 37 W and 36 W, respec-tively, with a Gaussian beam waist of 70 µ m. The trapdepths are 1900 E r and 7500 E r and trap fre-quencies are 1 . . S i gna l ( a r b . u . ) (a) (b)(c) (d) FIG. 4.
Imaging fidelity. (a), (b) Two adjacent images ofindividual Li atoms (white dots) in a triangular lattice (blackdots) imaged with 500 ms exposure and separation of 50 ms.(c) Reconstructed occupation of picture (a) convolved withthe PSF. (d) Hopping and loss during imaging, stationaryatoms (blue), hopped atoms (green) and lost atoms (red). and 752 nm spacing lattices, respectively [Fig. 5 (b) ].The square lattices have smaller lattice spacing thanthe triangular lattice, however, our reconstruction algo-rithm is able to determine the lattice occupation withan error only limited by the observed hopping and loss[Fig. 5 (c,d) ]. We confirmed this by comparing differentfitting subroutines which lead to differences much smallerthan the imaging infidelity. The 532 nm spacing lattice isimaged using the same Raman cooling configuration asthe triangular lattice, while for the 752 nm square latticethe Raman beam R2 is the retroreflection of the incom-ing Raman beam R1, instead of the orthogonal config-uration described above. For the triangular and 532 nmspacing square lattices with smaller trap frequencies, weobserved that the orthogonal Raman beam configurationis necessary, but for trap frequencies beyond 1 . z confinement [34]. However, the imaging fi-delity in the 752 nm spacing lattice is comparable withprevious results [16]. Due to the large sideband fre-quency in our 752 nm spacing lattice, it would be pos-sible to double the system size while maintaining suf- C oun t s ( a r b . u . ) Frequency (MHz)00.51 -2 -1 0 1 2
L1L2L3L4 y xz S i gna l ( a r b . u . ) (a) (b)(c) (d) FIG. 5.
Comparison to square lattices. (a) Squarelattice setup. Orange and blue arrows denote polarizationsof 532 nm and 752 nm spacing square lattices, respectively.(b) Raman sideband spectra in 532 nm spacing (orange) and752 nm spacing (blue) square lattice. The dots denote exper-imental data and solid lines are Gaussian fits. The sidebandsare at 1 . . . . +0 . − . in the 752 nm lattice.(c), (d) Single-site-resolved images of Li atoms with latticestructure overlay in the 532 nm spacing and 752 nm spacinglattice, respectively. The gray circles indicate occupied latticesites. For the 532 nm lattice, the Raman configuration is thesame as for the triangular, but for the 752 nm lattice we usecounter-propagating Raman beams. ficient lattice depth for high-fidelity imaging. Super-imposing the triangular lattice with the square latticecan form a two-dimensional quasi-crystalline lattice [44],which could be used to study many-body localization ina non-separable two-dimensional quasi-periodic lattice.Our setup is ready to superimpose both lattices by split-ting the laser power between both simultaneously real-ized optical paths and will be capable to study such sys-tems on a single-atom level.
VI. CONCLUSION AND OUTLOOK
We have presented the first single-site imaging of ul-tracold fermionic atoms in a triangular lattice, demon-strating a state-of-the-art imaging fidelity of 97.6(3)%.Our triangular lattice with spacing 1003 nm enables fasttunneling rates of ∼
700 Hz in the strongly interactingHubbard-regime. The interactions are tunable via theFeshbach resonance in lithium and are only limited bymulti-band effects. In our current configuration, we es-timate that about 20 sites are in the Hubbard regimewhen loading the lattice at maximum light sheet depth,making it very challenging to observe interaction effects.Through the addition of a vertical lattice we will increasethe vertical confinement to suppress multi-band effects toobtain Hubbard systems of several hundred atoms. Ourplatform will enable studies of the Fermi-Hubbard modelin the triangular lattice and, in the limit of strong interac-tions, the triangular Heisenberg spin model. By varyingthe polarizations of the three lattice beams, we can adia-batically change the triangular lattice between symmet-ric and asymmetric tunneling configurations, enablingthe study of the complete tunneling-imbalance param-eter space. Furthermore, the platform is ideally suited todirectly measure emergent quantum correlations, studysignatures of frustration and possibly even detect signa-tures of quantum spin liquids, depending on the lowestentropy states that can be prepared. It will become pos-sible to study spin-spin correlations in analogy with re-sults for square lattices in the Mott-insulating regime[13–16], possibly detecting the cross-over from three-sublattice order to a non-magnetic state [8, 9]. Even attemperatures previously reached in ultracold Hubbardsimulations, remnants of chiral correlations could be de-tected which would directly show time-reversal symmetrybreaking [3, 8]. Our new quantum gas microscope plat-form provides the basis for measuring these three-pointcorrelations. Moreover, the triangular lattice Hubbardmodel exhibits kinetic frustration, which could be probedusing a transient grating approach [45, 46] or by detect-ing magnon-hole bound states. The bound states in thetriangular lattice have binding energies that scale withthe tunneling energy and are therefore at experimentallyaccessible temperatures [47].
ACKNOWLEDGMENTS
This work was supported by the University of Virginia.We thank W. S. Bakr, S. S. Kondov and C. A. Sackettfor comments on the manuscript and acknowledge dis-cussions with D. Mitra, P. T. Brown, and E. Guardado-Sanchez. We thank J. W. Kim for early contributions tothe experiment and S. Kuhr for sharing the initial codebase for experiment control and reconstruction softwarewhich we extended for generalized lattice geometries. [1] G. H. Wannier, Antiferromagnetism. The TriangularIsing Net, Phys. Rev. , 357 (1950).[2] P. W. Anderson, The Resonating Valence Bond Statein La CuO and Superconductivity, Science , 1196(1987).[3] X. G. Wen, F. Wilczek, and A. Zee, Chiral spin statesand superconductivity, Phys. Rev. B , 11413 (1989).[4] L. Balents, Spin liquids in frustrated magnets, Nature , 199 (2010).[5] C. D. Batista, S.-Z. Lin, S. Hayami, and Y. Kamiya, Frus-tration and chiral orderings in correlated electron sys-tems, Rep. Prog. Phys. , 084504 (2016).[6] Y. Zhou, K. Kanoda, and T.-K. 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