Emulating Molecular Orbitals and Electronic Dynamics with Ultracold Atoms
EEmulating Molecular Orbitals and Electronic Dynamics with Ultracold Atoms
Dirk-S¨oren L¨uhmann, Christof Weitenberg, and Klaus Sengstock Institut f¨ur Laserphysik, Universit¨at Hamburg, Luruper Chaussee 149, 22761 Hamburg, Germany
In recent years, ultracold atoms in optical lattices have proven their great value as quantum simulators forstudying strongly correlated phases and complex phenomena in solid-state systems. Here we reveal their po-tential as quantum simulators for molecular physics and propose a technique to image the three-dimensionalmolecular orbitals with high resolution. The outstanding tunability of ultracold atoms in terms of potentialand interaction offer fully adjustable model systems for gaining deep insight into the electronic structure ofmolecules. We study the orbitals of an artificial benzene molecule and discuss the effect of tunable interactionsin its conjugated π electron system with special regard to localization and spin order. The dynamical time scalesof ultracold atom simulators are on the order of milliseconds, which allows for the time-resolved monitoringof a broad range of dynamical processes. As an example, we compute the hole dynamics in the conjugated π system of the artificial benzene molecule. The structure of molecules is usually determined by x-rayor electron diffraction. Current advances with femtosecondpulses allow for the time-resolved observation of the atomicpositions [1]. In general, orbital wave functions are muchharder to image. For simple molecules, the reconstruction ofthe HOMO has been achieved recently by electron momen-tum spectroscopy [2], by using laser-induced electron diffrac-tion [3], and by higher-order harmonic generation [4, 5]. Forhydrogen atoms the nodal lines of Stark states were resolvedvia photoionization microscopy [6]. State-of-the-art scanningprobe microscopy allows resolving the HOMO as well asidentifying the chemical bonds [7] for benzene-based com-pounds attached to surfaces.Experimental access to the electronic structure ofmolecules and their dynamics is essential because even forrelatively small molecules the full many-particle problem isnot computable using classical computers. This has broughtforward the idea of quantum computation for molecules [8, 9].The quantum computation of a hydrogen molecule in a min-imal basis has already succeeded [10, 11], but estimates formore complex molecules are not promising. While the num-ber of required qubits appears manageable, the estimatednumber of quantum gate operations ( for Fe S ) is manyorders of magnitude larger than currently available [12].In the past decade, ultracold atoms in optical lattices havebeen established as quantum simulators for condensed mat-ter systems [13, 14]. In addition, ultracold atoms were pro-posed as simulators for different systems such as neutron stars[15], black holes [16, 17], quarks [18], or atoms [19]. Re-cent experimental developments include prospects such as thesingle-lattice-site imaging and addressing [20, 21] as well asthe deterministic preparation and detection of few-atom sam-ples [22, 23].Here we show that ultracold atoms can be employed as aquantum simulator for molecules using existing experimen-tal techniques. In this setting, the ultracold atoms mimic the electrons in a molecule, whereas the optical trapping poten-tial takes the role of the nuclei. We demonstrate that ultra-cold atoms can serve as a tunable model system allowing theinvestigation of open questions in molecular physics. In con-trast to the quantum computation approach aiming at the exactcalculation of molecular energies, the present quantum emu-lation approach uses model systems to investigate specific in-teraction effects and dynamical processes in moleculelike sys-tems. As a concrete example, we focus on a model system forbenzene and compute its molecular orbitals for vanishing in-teraction. For the nonsolvable interacting problem, ultracoldatoms can serve as a quantum simulator for static and dynam-ical electronic properties in molecules. We demonstrate nu-merically that the conjugated π electron system shows strong-correlation effects such as localization and spin order, evenwhen neglecting the interaction with inner particles. On thebasis of this subsystem, we also reveal that ultracold atomsimulators promise unique insight into electronic femtoseconddynamics. We show that momentum mapping in combinationwith phase retrieval allows the imaging of molecular orbitalswith 1-2 orders of magnitude better than the intramoleculardistances. We explain how to use the outstanding tunability ofinteraction and potential for studying electronic interactionsand the dynamics in artificial molecules. I. CREATING ARTIFICIAL MOLECULES
As an example, we discuss how to create an artificial ben-zene molecule and compute the orbital wave functions. In realbenzene, the molecular structure is formed by a ring of sixcarbon atoms and six hydrogen atoms (Fig. 1c). Thereby, themolecular symmetry is essential for the formation of molecu-lar orbitals, where benzene belongs to the point group D . Inour case, the idea is to simulate the electrons in a molecularstructure via ultracold atoms in a tailored optical dipole poten- a r X i v : . [ c ond - m a t . qu a n t - g a s ] N ov a b y/az/bx/a c -1 (1) (2) (3) (4) (5) (6)(7) (8) (9) (10) (11) (12)(13) (14) (15) (16) (17) (18)(19) (20) (21) (22) (23) (24)(25) (26) (27) (28) (29) (30) FIG. 1. Molecular orbitals of artificial benzene. (a) Illustration of a honeycomb optical lattice superposed with a dipole trap. The hexagonalring and the adjacent sites form a trapping potential for an artificial, benzenelike molecule. (b) Calculated molecular single-particle orbitalsof the artificial benzene molecule with low-lying s orbitals (1-6) on the inner ”C” ring, hybridized sp orbitals (7-18,25-30) including theadjacent ”H” atoms, and p z orbitals (19-24) forming a conjugated π system. The orbitals are plotted for the lattice depths V = 11 E R and V z = 35 E R in units of the recoil energy E R of the lattice wavelength. The isosurfaces depict the orbital wave functions at . /a √ b (orange)and − . /a √ b (green) with the lattice constant a of the honeycomb and b of the orthogonal lattice. (c) Chemical structure of the benzenemolecule C H . tial that mimics the Coulomb interaction with the nuclei. Theoptical potential is imposed by the ac Stark shift of a laser fieldand can be created for the example of benzene by superposinga hexagonal optical lattice V V hc ( x, y ) [24] with a tight dipoletrap V dip ( x, y ) as shown in Fig. 1a (see Appendix). A similarpotential can, e.g., also be created by adding a triangular su-perlattice to the honeycomb lattice (a ”benzene lattice” [25])or by using a spatial light modulator, which renders the possi-bility of almost arbitrary two-dimensional potentials [26–28].In the orthogonal direction a one-dimensional lattice V withdepth V z or a light sheet is applied. A controlled number offermionic atoms (e.g. Li atoms with spin states ± / ) canbe loaded into this optical potential [22] (see Appendix).The molecular orbitals of the artificial molecule are com-puted using the plane-wave expansion method for vanishinginteraction (see Appendix for technical details and param-eters). The resulting single-particle orbitals are shown inFig. 1b. The six lowest s orbitals represent an energeticallyseparated shell within the inner ”C” ring, whereas the higherorbitals (7-18, 25-30) are sp -hybridized. The orbitals (7-11)and the antibonding orbital (13) form p orbitals pointing alongthe C ring. The outward-pointing p orbitals (12) and (14-18)lower their energy by maximizing the overlap to the s orbitalsof the H atoms. The hybridized orbitals (25-30) show a nodalplane between H and C ring and thus have a higher energy.Energetically in between lie the delocalized p z orbitals (19-24) with a nodal plane at z = 0 . Benzene is one of the prime examples for this type of delocalized conjugation of a π elec-tron system stabilizing the planar ring structure. This propertyis commonly referred to as aromaticity. Accounting for in to-tal 42 electrons, in benzene three of the six p z orbitals areoccupied. We can tune the energetic order of the orbitals, bytuning the parameters of the potential, e.g. strength and widthof the dipole trap as well as the lattice depths V and V z (seeFig. 3a). II. IMAGING OF MOLECULAR WAVE FUNCTIONS
Quantum gas microscopes have demonstrated optical imag-ing with single-site resolution in an optical lattice [20, 21].However, the imaging of molecular wave functions of artifi-cially created molecules requires an optical resolution severaltimes better than the optical lattice spacing. The idea is toovercome the limited spatial resolution by imaging the par-ticles in momentum space. In ultracold atom experiments,this can be achieved by turning off the optical potential, let-ting the particles freely expand and imaging the particle den-sity ρ TOF ( r ) after a certain time of flight. In the language ofmolecular physics, this would be equivalent to a sudden re-moval of all nuclei. During the expansion, the interactionsquickly become negligible, because there are only few par-ticles with high momenta. This allows for a free expansionin two dimensions with an unaltered lattice in the z directionavoiding problems with a limited depth of focus of the imag-ing system. In addition, the interaction among the particlescan be tuned to zero via a Feshbach resonance during the ex-pansion. For suitably long expansion times t (see Supplemen-tal Material [25]), the momentum density can be expressed as ρ p ( q ) ∝ ρ TOF ( p t/m ) with momenta p = h q /a , a the latticeconstant of the honeycomb lattice, and h the Planck constant.In the example shown in Fig. 2b, we assume that we canimage the momentum density after the time of flight with P = 49 × discrete momenta using a so-called pinninglattice for imaging (see Appendix). Initially preparing oneparticle (or two with opposite spin without interaction) onlythe lowest orbital n = 1 is occupied and per experimentalsequence only - momenta are recorded. To show the fea-sibility of the proposed imaging method, we use a randomnumber generator with an accumulated number of mo-menta. Note that in principle satisfying results with fewer ob-served momenta are possible (see Supplemental Material [25],Fig. S4). Figure 2b (first row) shows that the statics is clearlysufficient to map out the momentum distribution of the or-bitals. The phase of the distribution can be retrieved by iden-tifying the nodal lines of the momentum density ( ρ p ( q ) = 0 )and mapping it piecewise to the momentum wave functionwith p ( q ) = ± (cid:112) ρ p ( q ) (second row). Since the potentialis symmetric, p ( q ) can be assumed to be real. The compo-nents of p ( q ) are the Fourier coefficients of the wave functionand therefore allow the direct transformation to orbital wavefunctions in real space via ψ ( x, y ) = (cid:88) q p ( q ) | q (cid:105) (1)with plane waves | q (cid:105) = 1 S e π i( q x x + q y y ) /a , (2)lattice constant a and the number of grid points S per recip-rocal lattice vector. The resulting density is plotted in thethird row of Fig. 2b. The figure demonstrates that the effec-tive resolution of the imaging method is very high. In prin-ciple, the resolution is given by the diameter in momentumspace. However, since the momentum wave function dropsexponentially to zero for higher values of q , we can assume p ( q ) ≈ for | q | (cid:38) . . Since the diameter in momentumspace is not limiting, the spatial resolution in the experimentis mainly influenced by the statistics of the momentum den-sity (see Supplemental Material [25], Fig. S4). In contrary,the maximal diameter of the artificial molecule to be imagedis given by the resolution in momentum space and estimatedas P/ (2 q max x − q min x ) using the Nyquist theorem ( . a for ourexample).In analogy, higher single-particle orbital wave functions canbe imaged as depicted in Fig. 2c. The preparation of higherorbitals is analogous to the preparation of higher bands in op-tical lattices via Bragg transfer [29, 30] or via amplitude mod- Theory Wave reconstruction a b c D en s i t y q x q x q x q x q x -2 q x q x x/a x/a x/a n = 1 n = 1 n = 12 (cid:43) M o m en t u m w .f. M o m en t u m den s i t y ay z/b x/ay/a x/a x/a q y q y n = 19 201-1 d x/a q z q z z/b q y (cid:43)(cid:45)(cid:43)(cid:45)(cid:43)(cid:43) (cid:43)(cid:43)(cid:43)(cid:43) -2 0 10(arb. units) FIG. 2. Imaging of molecular wave functions via momentum-spacemapping. The first row shows the momentum density in two dimen-sions, the second row the momentum wave function after phase re-trieval, the third row the density (the sign of the wave function is in-dicated), and the fourth row the isosurface of the three-dimensionalwave function for ± . /a √ b . (a) Theoretical calculation for thelowest orbital ( n = 1 ) of the artificial benzene molecule. (b) Simu-lated time-of-flight measurements with observed momenta inthe x - y plane (randomly distributed) for the orbitals n = 1 and (c) n = 12 . (d) The momentum density in y - z direction with randommomenta for the lowest p z orbital ( n = 19 ), integrated momentumwave function and density in z direction. In combination with thereconstructed wave function in x - y coordinates (see (b)), this allowsone to retrieve the three-dimensional wave function for the lowest p z orbital (analogous for (b) and (c) using the lowest Wannier orbital in z direction). ulation [31]. Alternatively, increasing the number of parti-cles in the artificial molecule automatically occupies higherorbitals. In this case, the recorded images correspond to thesum of the respective momentum densities (for vanishing in-teractions). Imaging the wave functions in three dimensionsrequires measuring the momentum distribution also in the z direction, where Fig. 2d depicts the reconstruction of the p z orbital. Here, a statistics of particles is sufficient to re-solve the orbital structure. In our case, the separable den-sities in the x - y and z direction can be simply combined tothe three-dimensional orbital wave functions shown in Fig. 2(fourth row). III. QUANTUM SIMULATION OFTHE MANY-BODY PROBLEM
The single-particle energies of the orbitals are plotted inFigs. 3a and b, where the six p z orbitals are indicated in red.Because of the separability of the potential in the z direction,these orbitals are analogous to the six lowest s orbitals butwith a nodal plane for z = 0 . The lowest and highest orbitalare symmetric (A ) and antisymmetric (B ∗ ) under rotation(compare Fig. 1b), whereas the second-third and fourth-fifthorbitals are doubly degenerate and labeled as E and E ∗ . Thelower three orbitals have a bonding character and the higherthree orbitals an antibonding character. While molecular po-tential curves are usually plotted against the nuclei distance R , the distances are fixed in an optical lattice potential. Forthe same reason the repulsion of the nuclei, dominating thepotential curves at R → , is not present in optical potentials.However, even though the positions are fixed in our case, wecan effectively change the intramolecular distances by tuningthe depth V of the lattice potential. A deeper lattice potentialdecreases the density between the sites and is analogous to alarger spacing R between the nuclei coordinates.A quantum simulator for molecular problems is needed fornon-vanishing interactions, since solving the many-particleproblem using the full configuration-interaction method isnot feasible for complex molecules even accounting for themolecular symmetries. Unlike electrons, neutral atoms do notinteract via Coulomb force, but by either a short-range contactpotential or a long-range dipolar potential [32]. For ultracoldatoms, the interaction strength between ultracold atoms canbe tuned in a wide range via an external magnetic field due tothe existence of Feshbach resonances. As a concrete example,the s -wave scattering length a s for Li atoms can be tuned be-tween − . b and . b , where b = λ/ is the latticespacing in the z direction. Experimentally, we can probe theenergies of the interacting problem (benzene has 42 electrons)by exciting a particle to a reference orbital using a microwavetransition, which also changes the internal state. This avoidsproblems with postinteraction, assuming that the particles inthe two different internal atomic states do not interact. Fur-thermore, the population in this internal atomic state servesas an observable measuring the transfer rate as a function ofexcitation energy.To demonstrate how interactions change the electronic con-figuration, let us turn to the conjugated π system, i.e. the p z orbitals with delocalized particles, and neglect all interac-tions with the lower-lying orbitals. When restricting ourselvesto this subsystem, the problem can be solved using the fullconfiguration-interaction method. Experimentally, this sub-system is also accessible by populating only the six lowest s orbitals that are equivalent to the p z orbitals in the x and y direction. In the case of p z orbitals, the experimental systemwould incorporate the interaction with the energetically lower-lying particles constituting a much more complex situation. a e E ( E R ) V ( E R ) b c V ( E R ) E ( E R ) E E E A E a s / b E ( E R ) U x/a ay d B *E *A E B *E * FIG. 3. Interactions in the conjugated π system. (a) Single-particleenergy spectrum showing the lowest orbitals ( p z orbitals in red) asa function of the lattice depth V (taking the role of the effectivenuclei distance R ). See Appendix for parameters. The energy ofthe p z orbitals can be tuned independently by lattice depth V z (here E R ). (b) Close-up showing only the p z orbitals with three bond-ing and three antibonding orbitals, where the orbitals 2-3 and 4-5are degenerate. (c) Energy spectrum of the conjugated system incor-porating the on-site interaction U for V = 6 E R (see Appendix).The colored arrows indicate the lowest excitations (labeled in (e))for vanishing interaction. (d) Logarithmic density plot of the two-dimensional Wannier function for the p z (or s ) band. Each contourindicates a density drop by a factor / . (e) Level scheme depictingorbitals and spin configuration. For the description of strongly correlated electron systems, itis often desirable to switch to Wannier orbitals (see Appendix)that are localized at individual sites (potential minima). TheWannier orbital shown in Fig. 3d allows the formulation ofa tight-binding Hubbard Hamiltonian taking into account thetunneling element t and the on-site interaction U , where U scales linearly with the scattering length a s . The next-nearest-neighbor tunneling t is considerably smaller than t but largeenough to cause an asymmetry in the p z band ( E (cid:54) = E inFig. 3e). The fermionic many-particle Hamiltonian reads ˆ H = − (cid:88) i,σ,d t d ˆ c † i,σ ˆ c [ i + d ] ,σ + c.c. + U (cid:88) i ˆ n i, ↑ ˆ n i, ↓ , (3)where the operator ˆ c i,σ annihilates a particle on site i in spinstate σ ∈ {↑ , ↓} and ˆ n i,σ = ˆ c † i,σ ˆ c i,σ corresponds to the re-spective particle number. Here, the tunneling matrix elements t ( d = 1 ) and t ( d = 2 ) are included and [ i ] = i mod 6 in-corporates the periodic boundary conditions.For vanishing interaction ( a s = 0 ), the energy spectrum(Fig. 3c) represents all possible single-particle excitations ofthe half filled band with delocalized particles (Fig. 3e). In-creasing the interaction causes two drastic changes on the”electronic” structure. First, the particles undergo the tran-sition from delocalized orbitals to the Mott insulator state,where the particles occupy the localized Wannier orbitals. Atthe same time, an interaction gap is formed separating themany-particle bands by the on-site interaction energy U . Sec-ond, the spins of the particles align in an alternating order onthe Wannier orbitals (antiferromagnetic alignment), therebygaining the second-order energy t /U per particle. The low-est Mott band contains possible spin excitations from thisantiferromagnetic ground state. For the interaction strength a s = 0 . b , the ratio U/t ≈ is in the vicinity of the pro-posed values for aromatic hydrocarbons which are still underdebate [33, 34]. The tunability of interaction strength and po-tential depth allows an independent control of the ratios U/t and t /t (or the next-neighbor interaction for long-rage in-teractions [32]). IV. CORRELATED ELECTRON DYNAMICS
One of the outstanding advantages of ultracold atomic sys-tems is that the dynamical time scale is on the order of milli-or microseconds. Therefore, artificial molecules would allowmonitoring of dynamical processes taking place on femto- orattoseconds time scales in real molecules. As an example,we compute the hole dynamics after the removal of a particle(see Fig. 4). Again, we restrict ourselves to the conjugated π system, for which the dynamics can be computed. The site-selective manipulation has been demonstrated for an opticallattice using a tightly focused addressing beam [26, 35]. Afteran evolution time t , suddenly ramping up the optical latticeallows freezing the particle number distribution at the indi-vidual sites of the ring. Using fluorescence imaging, we cancount the occupation number n i, ↑ + n i, ↓ on each site of theartificial molecule [20, 21]. The spin state can be identifiedby selectively removing one of the spin states before imagingvia resonant light [35].In Fig. 4 the site-resolved dynamics is plotted after theremoval of a spin-up particle at site i = 3 . Without inter-actions (Fig. 4a), the spin-down particles are not influencedby the particle removal, whereas the initial hole in the spinup component oscillates between sites i = 3 and . Thisquantum revival occurs at the revival time T = h/E (with E = 0 . E R ). This corresponds to the energy difference be-tween A and E orbitals. For Li, the revival time corre-sponds to . for a recoil energy E R = h × . at alattice wavelength λ = 1064 nm. Switching on the (repul-sive) interaction among the particles (Fig. 4b), the dynamicalbehavior becomes very complex. For t = 0 , the antiferromag-netic order is apparent in the spin-down component causingthe almost vanishing total density on site i = 3 . The neigh-boring sites are highly occupied with the down component,which fills the empty site on short time scales. This inter- v34 Site Site0 5 10 15 0 1 n ( t ) i (cid:114) ( r , t ) (units of 1/a )1 23465 a s / b = 0 a s / b = 0.02 ba t ( h / E R ) spin downspin up FIG. 4. Correlated dynamics after particle removal. (a) Quantumrevival for the noninteracting system and (b) dynamics in the antifer-romagnetically correlated state for a s /b = 0 . and V = 6 E R . Theplots display the occupation number of the Wannier state at site i ofthe ring for spin up (left) and spin down (right) as a function of time t . The images on the left depict the summed density of both spinstates. The three-dimensional isosurface is plotted for /a b , wherethe coloring refers to the two-dimensional density. feres strongly with the dynamics of the up component leadingto complex density modulations. A priori , it is not clear ifthe spin order persists at the typical temperatures of ultracoldexperiment. Therefore, the dynamics shown in Fig. 4 incor-porates a finite temperature of . E R /k B with k B the Boltz-mann constant, which corresponds to
70 nK for Li, demon-strating that the quantum simulation of electronic dynamics isfeasible with state-of-the-art temperatures.
V. CONCLUSIONS
In conclusion, ultracold atoms have great potential as aquantum simulator for molecular physics, where solving thefull many-particle problem is still out of reach using eitherclassical or quantum computers [12]. This includes the for-mation of molecular orbitals, electronic interactions as wellas the time-resolved monitoring of a broad range of dynam-ical processes. With the proposed quantum emulation, wehope to shed light on open questions regarding the influence ofelectronic correlations and localization in molecular systems.In particular, the ability of directly observing spatial corre-lations goes far beyond the available techniques in molecu-lar physics, where correlations are probed indirectly by pho-toionization processes. This also touches on the fundamentalquestions of how interactions change the picture of effectivemolecular single-particle orbitals. The electronic correlationscan fundamentally affect the electron dynamics in a molec-ular system. While we discuss the creation of an artificialbenzene molecule using state-of-the-art experimental tech-niques, the results can be easily generalized covering morecomplex molecules. In particular, spatial light modulators canbe used to create arbitrary two-dimensional molecular struc-tures [28]. To emulate nonplanar molecules, one could utilizethree-dimensional nonseparable lattice potentials formed byseveral interfering laser beams [36] or by holographic projec-tion techniques. The nuclei positions, kept fixed in the presentstudy, can be controlled using slowly varying time-dependentoptical potentials (e.g. using a spatial light modulator or an ac-cordion lattice [37]) or coupled to phonons using other atomicspecies such as ions [38]. This allows simulation of the elec-tronic response to vibrational modes. In principle, one couldalso imagine accessing regimes where the Born-Oppenheimerapproximation breaks down.
ACKNOWLEDGMENTS
We acknowledge funding by the Deutsche Forschungsge-meinschaft (Grants SFB 925 and the The Hamburg Centre forUltrafast Imaging (CUI)).
Appendix A: Proposed experimental sequence
The optical potential for the artificial benzene molecule isgenerated by three laser beams in the x - y plane, which cre-ate a honeycomb hexagonal optical lattice V V hc ( x, y ) , and anorthogonal one-dimensional lattice V z cos ( πz/b ) (see Sup-plemental Material [25]). Superposing this lattice potentialwith a tight dipole trap V dip ( x, y ) = − (cid:15) dip V e − x + y ) /w dip centered on a plaquette of the lattice as sketched in Fig. 1acauses a finite-size system with a hexagonal inner and outerring, where V is the height of the potential (in units of therecoil energy E R = h / λ m with wavelength λ and atomicmass m ). For concreteness, we choose if not stated other-wise the lattice depths V = 11 E R and V z = 35 E R , the rel-ative strength (cid:15) dip = 10 of the dipole potential and the width w dip = 4 a of the dipole trap. Here, a = 2 λ/ is the width ofthe unit cell in the honeycomb lattice, whereas b = λ/ is thelattice spacing in the z direction. The orbital order can betuned by adjusting the lattice depths V and V z (see Fig. 3a).A controlled number of ultracold fermionic atoms can beprepared in the tight two-dimensional potential formed by adipole trap and a single antinode of the orthogonal lattice.Atoms above a certain energy are expelled from the trap via astrong magnetic field gradient [22]. When adiabatically ramp- ing up the honeycomb lattice, the particles fill up the lowestorbitals of the artificial molecule. To image the momentumdistribution a time-of-flight measurement is performed, wherethe atoms expand for a certain time. Subsequently, a deepso-called pinning lattice is ramped up that holds the atomsin place while they are illuminated by an optical molasses[20, 21]. The pinning lattice can be a square lattice of thesame wavelength λ containing P central lattice sites. Usinga high-resolution imaging system the individual sites of thepinning lattice can be resolved [20, 21]. In addition to theobserved momenta, the number of particles is also retrieved.For observing the momentum density in the z direction, athree-dimensional expansion can be performed. In this case,the imaging (from the x direction) needs a large depth of fieldand is therefore restricted to a larger optical resolution. How-ever, for dilute filling of the optical lattice, the atomic posi-tions can still be identified even if the optical resolution isseveral times the lattice spacing [39] (see Supplemental Ma-terial [25]). Appendix B: Computation of orbitals
We compute the molecular orbitals by applying the plane-wave expansion method for nonperiodic systems. By decom-posing the two-dimensional potential in Fourier components V ( x, y ) = (cid:80) q v q | q (cid:105) , the Hamiltonian matrix can be calcu-lated in the plane-wave basis (2). Here, S is the number ofunit cells per dimension, q = ( q x , q y ) and q i = − / ν/S + d are the wave vectors with d = {− D, − D +1 , ..., D } and ν = 0 , , ..., S − ( ν = 1 / , ..., S − / for S odd). The eigen-vectors of the Hamiltonian matrix of dimension (2 D + 1) S represent the solution for the single-particle orbitals in theplane-wave basis | q (cid:105) . The orthogonal optical lattice in the z direction separates. Assuming that only the central latticeplane is filled in the experiment, we can restrict ourselves tothe Wannier function w ( n ) z ( z ) of band n in the z direction. Forthe orbitals in Figs. 1 and 3 we use S = 5 , D = 7 and S = 15 , D = 5 for Fig. 2a. Appendix C: Molecular Wannier functions
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