Quantum Simulation of Antiferromagnetic Spin Chains in an Optical Lattice

.1 Ising Interactions in a Tilted Optical Lattice

The quantum Ising model is a paradigmatic model of magnetism, and an Ising chain is one of the simplest many-body systems to exhibit a quantum phase transition. The Hamiltonian describing a 1D antiferromagnetic Ising chain in the presence of an applied magnetic field is given by:

Here $S_{z}^{i}$ ($S_{x}^{i}$) is the $z$ ($x$) spin-projection operator at site $i$, and $h_{z}$ ($h_{x}$) is the $z$ ($x$)-component of the magnetic field applied to site $i$. The zero temperature phase diagram of the modelNovotny and Landau (1986); Ovchinnikov et al. (2003) is shown in Fig. 1a, for a homogeneous applied field $(h_{z},h_{x})$. For small applied fields, the magnetic interactions induce staggered ordering of the spins, producing an antiferromagnet (AF). For large applied fields, the field overwhelms the interactions, and all spins align to the field, producing a paramagnet (PM).

Our approach to constructing a magnetic Hamiltonian was proposed by Sachdev et al.Sachdev et al. (2002), in the context of experiments by Greiner et al.Greiner et al. (2002), where a gradient was applied to measure the insulating properties of the Mott state. Sachdev et al. showed that under the influence of such field gradients, the dynamics of a 1D Mott insulator map onto the aforementioned Ising model (see Methods) in the neighborhood of $(h_{z},h_{x})=(1,0)$ (Fig. 1b).

In the Mott insulator regime ($U\gg t$) it is energetically forbidden for the atoms to tunnel as long as the tilt per lattice site, $E$, differs from the onsite atom-atom interaction $U$. Hence, the system remains in a state with one atom per lattice site for $E<U$ (Fig. 2a). As the tilt approaches the interaction strength ($E=U$), each atom is free to tunnel onto its neighbor, so long as its neighbor has not itself tunneled (Fig. 2b). This nearest-neighbor constraint is the source of the effective spin-spin interaction. If the tilt $E$ is increased sufficiently slowly through the transition so as to keep the system near the many-body ground state, density wave ordering results (Fig. 2c).

Spatially varying tilts in the optical lattice can give rise to site-to-site variation of $h_{z}$. Such inhomogeneity would impact the critical behaviour by breaking the translational symmetry, inducing different sites to transition at different applied tilts. Accordingly, the resulting many-body energy gaps, dynamical timescalesImry and Ma (1975); Dziarmaga (2006), and entropy of entanglementVidal et al. (2003) would differ from the homogeneous case. Controlling such inhomogeneities is thus crucial for studies of magnetism.

The mapping of the Hubbard model onto the spin-model breaks down away from $(h_{z},h_{x})\sim(1,0)$. This is because states with three atoms on a lattice site are not within the Hilbert space that maps to the spin model. As such we can study ground state dynamics and low-energy excitations of the equivalent Ising model, but not high-energy excitations associated with adjacent flipped spins. These constraints admit investigation of the Ising physics only in the neighborhood of the multicritical point $(h_{z},h_{x})\sim(1,0)$. This regime is of particular theoretical interest as the model is here not exactly solvableOvchinnikov et al. (2003). It is nonetheless in the Ising universality classSachdev et al. (2002), and so a study of its critical physics would provide insight into the behaviours of the more commonly considered transverse ($h_{z}=0$) Ising model.

.2 Extracting Spin Observables

We locally detect magnetic ordering by utilizing our quantum gas microscope, capable of resolving individual lattice sites. The microscope is sensitive only to the parity of the site occupation numberBakr et al. (2009), and so PM domains (with one atom per lattice site) should appear as entirely bright regions, and AF domains (with alternating 0-2-0-2 occupation) as entirely dark regions. In the spin language, the detection parity operator $P^{i}$ measures the spin-spin correlation between adjacent spins:

Throughout this article we will characterize our spin-ordering primarily via the probability that site i has odd occupation $p_{\text{odd}}^{i}=\frac{1}{2}(1+\langle P^{i}\rangle)$. Taking the spin constraint into account, the chain average of $p_{\text{odd}}^{i}$ is equivalent to a chain-averaged measurement of $\langle S_{z}^{i}\rangle$, the mean $z$-projection of the spin: $\overline{\langle S_{z}^{i}\rangle}=\frac{1}{2}\overline{p_{\text{odd}}^{i}}$ . Here angle brackets denote realization-averages and bars denote chain-averages.

This parity measurement allows us to locally identify magnetic domains and estimate their size. This, however, is not a direct measurement of the AF order parameter described in Ref. Sachdev et al. (2002), and does not reflect the broken symmetry in the AF phase (see Methods). Accordingly, we also study the AF order parameter more directly through 1D quantum noise interferometryAltman et al. (2004).

.3 Observing the Phase Transition

Our experiments begin with a Mott insulator of 87Rb atoms in a two dimensional optical lattice with spacing $a=680\,\text{nm}$ and a depth of $35E_{r}$, in the focal plane of a high resolution imaging system which allows detection of single atoms on individual lattice sites as described in previous workBakr et al. (2009, 2010). The lattice recoil energy is given by $E_{r}=h^{2}/8ma^{2}$, where $h$ is Planck’s constant and $m$ the mass of 87Rb. We generate our effective $h_{z}$ by tilting the lattice potential by $E$ per lattice-site, which is achieved via a magnetic field gradient along the $x$-direction (defined in Fig. 3a). This gradient is applied in two steps first a fast ramp (in 8 ms) to just below the transition pointSachdev et al. (2002); Ovchinnikov et al. (2003) at $E=U+1.85t$ ($h_{z}=1-0.66h_{x}$), followed by a slow linear ramp (in 250 ms) across the transition. Before starting the slow gradient ramp, the lattice depth along the $y$-direction is increased (in 2 ms) typically to $45(7)E_{r}$, while the depth along the $x$-direction is reduced to $14(1)E_{r}$. This decouples the system into 1D chains with significant tunneling only along their lengths. Simultaneously, we compensate the tilt inhomogeneity arising from harmonic confinement, leaving only residual inhomogeneity arising from our lattice projection methodBakr et al. (2009). We then probe the system by stopping the ramp at various points and observing spin ordering, first of the entire cloud, and then specializing to the transitions of individual lattice sites in a particular chain.

We initiate the gradient ramp on the paramagnetic side of the phase transition (typically at $E/U=0.7$), as the initial Mott state has good overlap with the paramagnetic ground state (Fig. 3ai). At the end of the ramp ($E/U=1.2$), we observe an even occupation with probability 0.90(2) (Fig. 3aii), as expected for an AF phase in the magnetic model where the spin-spin interaction overwhelms the effective field $h_{z}$. In between, density-wave (AF) ordered regions begin to form, as shown in Fig. 3aiii. Fig. 3c shows $p_{\text{odd}}$ at various times during this ramp. A crucial characteristic of an adiabatic transition is that it is reversible. Fig. 3b shows $p_{\text{odd}}$ during a ramp from a PM to an AF and back. The recovery of the singly occupied sites is evidence of the reversibility of the process, and hence that the state at the end of the forward ramp is in fact an antiferromagnet.

We directly verify the existence of staggered ordering in the AF phase via a 1D quantum noise correlation measurementAltman et al. (2004). We perform this measurement by increasing the lattice depth along the chains to $35E_{r}$ within 5 ms and then rapidly switching off that lattice to realize a 1D expansion. The resulting spatial autocorrelation is plotted in Fig. 3d at both the beginning (i) and end (ii) of the ramp from the PM phase to the AF phase. In the PM phase the spectrum exhibits peaks at momentum difference $p=h/a$, characteristic of a Mott insulatorFölling et al. (2005). In the AF phase peaks at $p=h/2a$ appear, indicative of the emergence of a spatial ordering with twice the wavelength. In principle the mean domain size can be extracted from the $p=h/2a$ peak width, however our measurement is broadened by both finite expansion time and aberration arising from the fact that the 1D expansion is performed not in free space but in slightly corrugated confining tubes.

.4 Single-Site Study of the Transition

A high resolution study reveals that in the presence of harmonic confinement the spins undergo the transition sequentially due to the resulting spatial variation of the effective longitudinal field. Fig. 4a shows $p_{\text{odd}}$ vs. tilt for two different rows of a harmonically confined Mott insulator, separated by seven lattice sites. These two rows tune through resonance at different tilts, as can be understood from the energy level diagram Fig. 4b. To realize a homogeneous field Ising model we eliminate the harmonic confinement (see Methods) immediately before the slow ramp into the AF phase. The homogeneity is now only limited by residual lattice beam disorder, and accordingly, Fig. 4c demonstrates that different rows transition almost simultaneously, as anticipated theoretically (Fig. 4d).

After compensating the harmonic confinement, we use high resolution imaging to study the transition on the single-site level. This allows us to focus on a single six-site chain with particularly low inhomogeneity, which we will study for the remainder of this article. We identify such a chain by imaging individual lattice sites as the system is tuned across the PM-AF transition. Fig. 5 shows the average occupation of each of the six adjacent sites (black curves), versus tilt, for a 250 ms ramp across the transition. The r.m.s. variation in the fitted centers is 6 Hz, significantly less than their mean 10%–90% width of 105(30) Hz, corresponding to the effective transverse field $2^{3/2}t=28\,\text{Hz}$. By quickly jumping across the transition with tunneling inhibited, and then ramping slowly across the transition in reverse with tunneling allowed (red curves, taken under slightly different conditions), we are able to rule out large, localized potential steps that would otherwise prevent individual spins from flipping. The curves in Fig. 5 provide our best estimate of the inhomogeneity. However, exact determination of the site-to-site disorder using this technique is complicated by the many-body nature of the observed transition. New spectroscopic techniques such as single-site modulation spectroscopy would need to be developed to ensure that the inhomogeneities are small enough to study criticality in long, homogeneous Ising chains.

.5 Domain Formation in a 1D Ising Chain

Quantum fluctuations induce the formation of AF domains as the homogeneous chain is ramped through the transition. As discussed previously, these domains will appear as uninterrupted strings of dark lattice sites. Fig. 6a shows the observed mean length-weighted dark domain length extracted from 43 single-shot images per tilt, as the system is ramped from the PM phase into the AF phase. The dark domain length is here defined as the number of contiguous dark sites (see SI). On the AF side of the transition, the mean dark domain length grows to 4.9(2) sites, giving evidence that the average AF domain size approaches the system size.

While the antiferromagnetic domain formation discussed thus far occurs in spin-chains that remain in a quantum state near the many-body ground state, we can also produce antiferromagnetic domains that correspond to the highest energy state of the constrained Hilbert space. This is achieved by starting with the Mott insulator and rapidly ramping the field gradient through the transition point with tunneling inhibited, then adiabatically ramping back with tunneling permitted. This prepares a PM on the AF side of the transition, and adiabatically converts it into an AF on the PM side. The resulting data are shown in red, in Fig. 5 and Fig. 6a, demonstrating that these high energy states are sufficiently long-lived to support domain formation. Similar ideas have been proposed for preparation of difficult-to-access many-body states using the highest energy states of Hamiltonians with easily prepared ground statesSørensen et al. (2010).

.6 Conclusions and Outlook

We have experimentally realized a quantum simulation of an Ising chain in the presence of longitudinal and transverse fields. By varying the applied longitudinal field, we drive a transition between PM and AF phases, and verify the formation of spin domains via both direct in situ imaging, and noise-correlation in expansion. We study the adiabaticity requirements for transition dynamics, and observe a timescale consistent with tunneling-induced quantum fluctuations. By rapidly tuning through the transition and ramping back across it slowly, we prepare the highest energy state of a many-body spin Hamiltonian.

We introduce and implement a novel route to studying low entropy magnetism in optical lattices. Complexities associated with cooling of spin mixturesWeld et al. (2009); Medley et al. (2010); McKay and DeMarco (2010) are circumvented by employing a low entropy, gapped Mott insulator as an initially spin-polarized state, and then adiabatically opening a spin degree of freedomGarcía-Ripoll et al. (2004). This recipe is directly applicable to more traditional approaches to quantum magnetism, including those employing super-exchange interactionsDuan et al. (2003).

The spin-mapping demonstrated here opens a new avenue for future work on quantum magnetism and control. Strong effective magnetic interactions make possible the creation of states with coherently generated long-range order. Combined with a lower-disorder lattice, in-depth studies of criticality become possible. Tilting by the band excitation energy will enable studies of the transverse Ising modelPlẗz et al. (2010). It will be interesting to investigate the impact of various types of controlled disorder on criticality and transition dynamics, as well as the possible existence of non-thermalizing statesBañuls et al. (2011). A particularly intriguing direction is the extension of the tilted Mott insulator physics to higher dimensions. The simple square lattice geometry will provide access to phases with longitudinal density wave ordering and transverse superfluiditySachdev et al. (2002). More sophisticated geometries will produce frustrated systems with novel quantum liquid and dimer covered ground statesPielawa et al. (2011).

References

Appendix A Methods

We follow Sachdev et al. in formally mapping a 1D Mott insulator of spinless bosons in a tilted lattice onto a chain of interacting dipoles (doublon-hole pairs, in a singly occupied Mott shell), and then onto a chain of spin-1/2 particles with AF Ising interactions in longitudinal and transverse fields. In a homogeneously tilted lattice, the 1D Bose-Hubbard Hamiltonian reads:

Here $t$ is the nearest-neighbor tunneling rate, $U$ is the onsite interaction, $E$ is the tilt per lattice site, $a_{j}^{\dagger}$ ($a_{j}$) is the creation (annihilation) operator for a particle on site $j$, and $n_{j}=a^{\dagger}_{j}a_{j}$ is the occupation number operator on site $j$.

For a tilt near $E=U$, the onsite interaction energy cost for an atom to tunnel onto its neighbor is almost precisely cancelled by the tilt energy. If one starts in a Mott insulator with $M$ atoms per site, an atom can then resonantly tunnel onto the neighboring site to produce a dipole excitation with a pair of sites with $M+1$ and $M-1$ atoms. The resonance condition is only met if adjacent sites contain equal numbers of atoms, so only one dipole can be created per link and neighboring links cannot both support dipoles. We define a (properly normalized) dipole creation operator $d_{j}^{\dagger}=\frac{a_{j}a_{j+1}^{\dagger}}{\sqrt{M(M+1)}}$.

The Bose-Hubbard Hamiltonian above can hence be mapped onto the dipole Hamiltonian:

subject to the constraints $d_{j}^{\dagger}d_{j}\leq 1$, $d_{j+1}^{\dagger}d_{j+1}d_{j}^{\dagger}d_{j}=0$.

The factor of $\sqrt{M(M+1)}$ arises due to bosonic enhancement.

To map from the dipole Hamiltonian to the spin-1/2 Hamiltonian, we define a link without (with) a dipole excitation to be an up (down) spin along $\hat{z}$. Then the creation/annihilation of dipoles are related to the flipping of spins, and we can write:

The constraint forbidding adjacent dipoles can be implemented by introducing a positive energy term $Jd_{j+1}^{\dagger}d_{j+1}d_{j}^{\dagger}d_{j}$ to the Hamiltonian, where $J$ is of order $U$. This term gives rise to nearest-neighbor interactions and an effective longitudinal field for the spins.

Defining $\Delta=E-U$ the Hamiltonian for the spins now reads:

A.2 Experimental Details

Our experiments start with a single layer 2D Mott insulator of 87Rb atoms in a $35E_{r}$ lattice with 680 nm spacing as described in previous work. The atoms are in the $|F=1,m_{f}=-1\rangle$ state and the initial fidelity of the Mott insulator is 0.95(2), with local fidelities as high as 0.98. A magnetic field gradient along the $x$-direction is ramped up within 8 ms to tilt the lattice potential by $0.7U$ per lattice site. At this point, the depth of the lattice along the chains is ramped down to $14E_{r}$, while the potential transverse to the chains is ramped up to $45E_{r}$, within 2 ms. At the same time, the optical potential providing harmonic confinement is ramped down. Tunneling between chains is negligible over the experimental timescale (see Supplementary notes for further discussion). The gradient is then ramped adiabatically through the transition point using a linear ramp that ends at a tilt of $1.2U$ per lattice site, typically within 250 ms.

We can then perform either an in situ measurement or a 1D expansion of the chains to achieve noise correlation interferometry. In both cases, we use fluorescence imaging after pinning the atoms in a deep lattice to obtain the density distribution with single atom/single lattice-site resolution. Images far on the Mott side of the transition are used to select chains of atoms within the first shell of the insulator. The phase transition is then studied only within these chains, with quantitative curves employing data only from the single chain with lowest disorder.

For noise correlation measurements, the magnetic field gradient and the lattice along the chain are switched off, while the interchain lattice and the potential confining the atoms in the third direction remain on. After a 1D expansion for 8 ms, the atoms are pinned for imaging. To extract information about density wave ordering in the chains, several hundred images (250 for paramagnetic, 500 for antiferromagnetic phase) each containing 15 chains, are fitted to extract the atom positions, and then spatially autocorrelated and averaged as described in Ref. Fölling (2008).

Lattice depths are calibrated to 15% using Kapitza-Dirac scattering, however the width of single-site transition regions was found to be a more sensitive probe of the longitudinal tunneling rate and hence the longitudinal lattice depth (see Supplementary Fig. S1), and accordingly was employed throughout this article.

The magnetic field gradient is calibrated using lattice modulation spectroscopy. In the presence of a potential gradient $E$ per lattice site, modulation of the lattice depth along the chains causes resonant excitation at two frequencies, $U+E$ and $U-E$ corresponding to an atom in the Mott insulator moving up or down gradient. We detect these excitations as a reduction in the value of $p_{\text{odd}}$ using in-situ imaging (see Supplementary Fig. S2). Using the mean of the two resonances, we obtain the interaction energy $U=430(20)\,\text{Hz}$ at $16E_{r}$ longitudinal lattice, $45E_{r}$ transverse lattice (corresponding to $U=413(19)\,\text{Hz}$ at $14E_{r}$ longitudinal lattice, where the experiment operates, which agrees with a band-structure calculation of 401(25) Hz). The separation between the resonances as a function of applied gradient is used to calibrate $E$. At zero applied magnetic field gradient, we find the stray gradients to be less than $0.02U$.

A.3 Local and Long-range Observables

In-situ detection gives the atom number modulo 2 due to light assisted collisions of atoms on each lattice site during imaging. By averaging the occupation of a site over multiple images, we obtain the probability of an odd occupation on the $j$th lattice site ($p^{j}_{\text{odd}}$), which corresponds to the probability of having a single atom on a site within the subspace of our model. This is related to the spin observables in the effective model by $\langle S_{z}^{j-1}S_{z}^{j}\rangle=\frac{1}{2}\left(p_{\text{odd}}^{j}-\frac{1}{2}\right)$. We average over all the atoms in the chain to obtain $p_{\text{odd}}=\overline{p^{j}_{\text{odd}}}$, which in combination with the constraint that neighboring down spins are not allowed permits us to relate the chain averaged mean $z$-projection of spin to $p_{\text{odd}}$ according to: $\overline{\langle S_{z}^{j}\rangle}=\frac{p_{\text{odd}}}{2}$. This quantity varies across the transition and depends only weakly on the chain length. On the other hand, the order parameter for the transition O=\biggr{\langle}\biggr{(}\frac{1}{N}\sum_{j}(-1)^{j}S_{z}^{j}\biggr{)}^{2}\biggr{\rangle} depends on the chain length (Supplementary Fig. S3) and is non-analytic across the transition for a thermodynamic system.

The amplitude of the noise correlation signal at separation $d$ for a chain consisting of a large number of atoms is C(d)=1+\frac{1}{N^{2}}\biggr{|}\sum_{j}e^{-i\frac{madj}{\hbar t}}n_{j}\biggr{|}^{2} where $N$ is the number of lattice sites, $j$ is the lattice site index, $a$ is the lattice spacing, $m$ is the mass of the atom, $t$ is the expansion time, $n_{j}$ is the occupation of the $j$th site. It can be shown that the correlation signal at $d=\frac{\pi\hbar t}{ma}$ is related to the order parameter $O=C\left(\frac{\pi\hbar t}{ma}\right)-1$.

Appendix B Supplementary Information

The magnetic interactions should produce even-length domains of dark sites, corresponding to AF spin domains. To quantify the length of these AF domains we study the length of the measurable dark domains, defining a “dark domain” as a contiguous string of dark sites that is bounded either by a site with an atom or an edge of the region of interest. We then calculate the mean length-weighted dark chain length from this data.

Defects in the initial Mott insulator (MI) reduce the effective system size. Their appearance can produce an overestimate of the dark chain length by either connecting two dark chains, or appearing on the end of a dark chain. The initial MI defect probability is typically 4% per site over an entire $N=1$ shell, after correcting for losses during imaging.

Losses and higher order tunneling processes during the ramp can have similar consequences for the observed dark domain length, and can also suppress the observed dark domain length by perturbing atoms near the end of the ramp once the AF has already formed. The rate of such processes can be estimated from the MI $1/e$ squeezing lifetime in the tilted lattice, measured to be 3.3 seconds. To perform this measurement we first ramp to a tilt of 300 Hz/lattice site and tune the lattice depths to $45E_{r}$ and $14E_{r}$ for transverse and longitudinal lattices, respectively. We then hold for a variable time, and measure the observed in situ atom number. The lifetime is dominated by higher-order tunneling processes. Parametric heating and inelastic scattering become the dominant loss channel in deeper lattices once tunneling is inhibited.

A worst-case estimate for the impact of missing atoms can be reached from the fraction of the time that the system is missing no atoms at the end of the ramp. The six-site chain analyzed in the main text is initially fully occupied 79% of the time. During the time it takes the dark-domain length to grow to 4 lattice sites (60 ms), the aforementioned effects only reduce this number to 73%.

B.2 Supplementary Discussion of Entropy and Thermalization

Supplementary Table 1 shows the entropy per particle ($S/Nk_{B}$) for several different Mott insulator fidelities ($p_{\text{odd}}$), assuming a chemical potential $\mu=U/2$, as well as the mean length-weighted AF domain size $D$ in an infinite 1D magnetic system with the same entropy per particle. Here $D=2(2-\epsilon)/\epsilon$, where the spin-dislocation probability in the AF $\epsilon$ is defined by $S/Nk_{B}\approx(\epsilon/2)\left[1+\log(1/\epsilon)\right]$. Spin defects are ignored as they are both dynamically and thermodynamically unlikely. The entropy per particle can be related to the Mott insulator fidelity byJördens et al. (2008): $S/Nk_{B}=\log\left[\frac{2}{1-p_{\text{odd}}}\right]-p_{\text{odd}}\log\left[\frac{2p_{\text{odd}}}{1-p_{\text{odd}}}\right]$. If such thermalization took place in our finite length chain of six sites (with initial fidelity 97.5%), the mean domain size would be limited by the system size to 5.3 sites.

Experimentally, we find most Mott defects to be unbound doublons and holes, which do not directly map to excitations in the spin model. The large energy gap present in our tilted lattice, combined with conservation of particle number, make it difficult for these Mott defects to thermalize with spin degrees of freedom. Such thermalization would require, for example, migration of a doublon to a hole, or decay via a very high order processCapogrosso-Sansone et al. (2009) into several spin defects- quite unlikely within the experimental timescale. Consequently, these nearly static defects act as fixed boundary conditions that limit the effective length of the simulated spin chains. Supplementary Table one also provides the expected uninterrupted chain length, computed as $L_{\text{sys}}=(1+p_{\text{odd}})/(1-p_{\text{odd}})$.

B.3 Supplementary Discussion of Higher Order Effects

Tunneling between chains is excluded from the spin-mapping described in the main text, though under certain conditions it produces exotic transverse superfluidity, as described in Ref. Sachdev et al. (2002). For our purposes, these tunneling processes serve only to take the system out of the Hilbert space described by the spin model. Most of our experiments were performed at a transverse lattice depth of $45E_{r}$, corresponding to an interchain tunneling rate of $t_{\text{transverse}}=2\pi\times 0.07\,\text{Hz}$. This tunneling rate is basically negligible on our experiment timescale of 250 ms. The noise correlation data, as well as the shell pictures and reversibility curve in Fig. 2 of the main text, were taken at $35E_{r}$ transverse lattice depth. At this depth the transverse tunneling rate is $t_{\text{transverse}}=2\pi\times 0.27\,\text{Hz}$, which is small compared to our lattice inhomogeneities, and so results in highly-suppressed, off-resonant Rabi-flopping. In practice, increasing the transverse lattice from $35E_{r}$ to $45E_{r}$ results in a modest $\sim 5\%$ improvement in the quality of the Mott insulator after transitioning to the antiferromagnetic state and back.

B.3.2 Second Order Tunneling

In addition to nearest-neighbor tunneling which creates doublon-hole pairs, and proceeds at a rate $\sqrt{2}t$ when the tilt $E=U$, there remain second-order tunneling processes which create triplons at a rate $t_{\text{S.O.}}\sim\frac{\sqrt{3}t^{2}}{U}$. For our longitudinal lattice depth of $14E_{r}$, and interaction energy $U=2\pi\times 416\,\text{Hz}$, we find $t_{\text{S.O.}}\sim 2\pi\times 0.4\,\text{Hz}$.

Because our system is continuously tilted, all such transitions will be tuned through resonance. For our typical experiment, $R_{\text{ramp}}\approx\frac{\frac{1}{2}U}{250\,\text{ms}}\approx 2\pi\times 840\,\text{Hz}^{2}$, so the Landau-Zener adiabatic transition probability to the triplon state $P_{\text{triplon}}=1-\exp\left[-2\pi\times\frac{t_{\text{S.O.}}^{2}}{R_{\text{ramp}}}\right]\sim 1\%$. In future experiments with slower ramps, both this effect and the closely related second-order Stark-shift will become more of a concern. These can be further suppressed relative to the desired dynamics by increasing the longitudinal lattice depth, at the expense of slower many-body dynamics. It bears mentioning that for our experimental parameters the triplon state should experience an additional energy shift calculated to be 22 Hz, due to multi-orbital interactionsWill et al. (2010); Johnson et al. (2009)

B.3.3 Impact of physics beyond the Hubbard Model

For a $14E_{r}$ lattice, the next-nearest neighbor tunneling rate is suppressed relative to that of the nearest neighborJaksch et al. (1998) by a factor of $\sim 300$, making the total rate $t_{\text{NextNeighbor}}=2\pi\times 0.04\,\text{Hz}$, which is negligible on present experiment timescales. The longitudinal nearest-neighbor interaction shift for one atom per lattice site is $\sim 10^{-3}\,\text{Hz}$, and interaction driven tunnelingFölling et al. (2007) occurs with a rate of $2\pi\times 0.3\,\text{Hz}$.