Two-dimensional superexchange-mediated magnetization dynamics in an optical lattice

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Science  01 May 2015:
Vol. 348, Issue 6234, pp. 540-544
DOI: 10.1126/science.aaa1385

Simulating magnetism out of equilibrium

A major goal of quantum simulation is to help us understand problems that are difficult to describe analytically or solve with conventional computers. This goal has been very challenging to reach experimentally, requiring, for example, extremely low temperatures to determine the effects of quantum magnetism in equilibrium. Brown et al. studied a nonequilibrium system of ultracold 87Rb atoms in a two-dimensional optical lattice. They monitored the atoms' dynamics after a sudden change in the lattice parameters and were able to reach a regime where the magnetic interactions dominated the dynamics.

Science, this issue p. 540


The interplay of magnetic exchange interactions and tunneling underlies many complex quantum phenomena observed in real materials. We study nonequilibrium magnetization dynamics in an extended two-dimensional (2D) system by loading effective spin-1/2 bosons into a spin-dependent optical lattice and use the lattice to separately control the resonance conditions for tunneling and superexchange. After preparing a nonequilibrium antiferromagnetically ordered state, we observe relaxation dynamics governed by two well-separated rates, which scale with the parameters associated with superexchange and tunneling. With tunneling off-resonantly suppressed, we observe superexchange-dominated dynamics over two orders of magnitude in magnetic coupling strength. Our experiment will serve as a benchmark for future theoretical work as the detailed dynamics of this 2D, strongly correlated, and far-from-equilibrium quantum system remain out of reach of current computational techniques.

The interplay of spin and motion underlies some of the most intriguing and poorly understood behaviors in many-body quantum systems (1). A well-known example is the onset of superconductivity in cuprate compounds when mobile holes are introduced into an otherwise insulating two-dimensional (2D) quantum magnet (2). Understanding this behavior is particularly challenging because the insulating state has strong quantum spin fluctuations, and the introduction of holes impedes the use of numerically exact techniques for quantum-correlated systems in more than one dimension. Ultracold atoms in optical lattices realize tunable, idealized models of such behavior and can naturally operate in a regime where the quantum motion (tunneling) of particles and magnetic interactions (superexchange) appear simultaneously (3, 4).

For ultracold atoms in equilibrium, the extremely small energy scale associated with superexchange interactions makes the observation of magnetism challenging, and short-range antiferromagnetic correlations resulting from superexchange have only recently been observed (5, 6). Out-of-equilibrium superexchange-dominated dynamics has been demonstrated in isolated pairs of atoms (7), in 1D systems with single-atom spin impurities (8), and recently in the decay of spin-density waves (9). However, the perturbative origin of superexchange in these systems implies that it must be weak compared with tunneling, and thus the manifestation of superexchange requires extremely low motional entropy. Dipolar gases (10) and ultracold polar molecules (11) in lattices provide a promising route toward achieving large (nonperturbative) magnetic interactions (12), but technical limitations in these systems currently complicate the simultaneous observation of motional and spin-exchange effects.

Here, we study the magnetization dynamics of pseudospin-1/2 bosons in a 2D optical lattice after a global quench from an initially antiferromagnetically ordered state (13). The dynamics we observe are governed by a bosonic t-J model (14, 15) and occur in a parameter regime where quantitatively accurate calculations are not currently possible. Using a checkerboard optical lattice, we continuously tune the magnetization dynamics from a tunneling-dominated regime into a regime where superexchange is dominant, even at relatively high motional entropies. We observe decay of the antiferromagnetic order with clearly identifiable time scales that correspond to the underlying Hamiltonian parameters for tunneling and superexchange. In contrast to the oscillatory dynamics observed in isolated pairs of atoms (7), we observe exponential decay of the magnetization. Some damping of the dynamics can be attributed to the extended, many-body nature of the experiment, but the degree of the expected damping is difficult to quantify because a full theoretical description of nonequilibrium 2D systems is lacking. This experiment bridges the gap between studies of nonequilibrium behavior in systems with exclusively motional (1620) or spin (21, 22) degrees of freedom, demonstrating the requisite control to explore the intriguing intermediate territory in which they coexist. In addition, the techniques that we demonstrate lay the groundwork for adiabatic preparation of low-entropy spin states relevant for studies of equilibrium quantum magnetism (23, 24).

Our experiment uses two hyperfine states (denoted by Embedded Image and Embedded Image) of ultracold 87Rb atoms trapped in a dynamically controlled, 2D checkerboard optical lattice (25) composed of two sublattices, Embedded Image and Embedded Image (Fig. 1A). For most experimental conditions presented here, our system is well described by a Bose-Hubbard model (3) characterized by a nearest-neighbor tunneling energy Embedded Image, and an onsite interaction energy Embedded Image. In addition, we use the lattice to apply an energy offset Embedded Image between the Embedded Image and Embedded Image sublattices, consisting of a spin-independent part Embedded Image and a spin-dependent part Embedded Image acting as a staggered magnetic field, Embedded Image (26). All of these parameters can be dynamically controlled, which we exploit to prepare initial states with 2D antiferromagnetic order and to observe the resulting dynamics after a quench to different values of Embedded Image, Embedded Image, Embedded Image, and Embedded Image.

Fig. 1 Tunable exchange processes.

(A) Schematic of terms in the underlying Bose-Hubbard Hamiltonian: on-site interaction energy between two atoms Embedded Image, tunneling Embedded Image, and offset Embedded Image between sublattices A and B. The red and blue spheres represent atoms in states Embedded Image and Embedded Image, respectively. (B) Second-order magnetic coupling processes arising from exchange between occupied nearest-neighbor sites (Embedded Image) or hole-mediated exchange associated with hopping of a hole within one sublattice (Embedded Image and Embedded Image). These couplings dominate the magnetization dynamics when tunneling is suppressed by tuning Embedded Image.

At unit filling, for Embedded Image and Embedded Image, double occupation at each site is allowed only virtually, and the Bose-Hubbard model can be mapped onto a ferromagnetic Heisenberg model (4, 27) with a nearest-neighbor magnetic interaction strength Embedded Image that is second-order in the tunneling energy. In the presence of hole impurities, first-order tunneling (with the much larger energy scale Embedded Image) must be included, which substantially modifies the dynamics even at low hole concentrations (9). The offset Embedded Image provides the flexibility to tune the relative importance of first-order tunneling and second-order superexchange processes. For example, below unit filling, if Embedded Image and Embedded Image, the Bose-Hubbard model can be mapped onto a bosonic t-J model with a staggered energy offset.

Embedded Image(1)

The local spin operators are defined as Embedded Image, where a (Embedded Image) annihilates (creates) a hardcore boson of spin Embedded Image on site Embedded Image, and Embedded Image is a vector of Pauli matrices. The notation Embedded Image indicates that the sum over Embedded Image and Embedded Image is restricted to nearest neighbors, and Embedded Image indicates that the sum is restricted to sites Embedded Image, such that Embedded Image are both nearest neighbors of Embedded Image. The superexchange energy Embedded Image (Fig. 1B) can be either ferromagnetic (Embedded Image) or antiferromagnetic (Embedded Image) (7). The last term in Eq. 1 describes hole-mediated exchange, where an atom on site Embedded Image interacts via superexchange with an atom on site Embedded Image, while simultaneously hopping to site Embedded Image (Fig. 1B). Here, Embedded Image, where −(+) applies when j is in the lower (higher) energy sublattice. In writing Eq. 1, we have ignored second-order processes (28, 29) that conserve sublattice magnetization (30). When Embedded Image, first-order tunneling is resonant and dominates the magnetization dynamics except at extremely low hole densities. For Embedded Image, however, first-order tunneling is effectively suppressed, in which case superexchange dominates the magnetization dynamics. (The frequently ignored Embedded Image term should be included for a quantitative description at nonzero hole density.) Superexchange is resonant when Embedded Image but is suppressed when Embedded Image. The values of Embedded Image, Embedded Image, Embedded Image and Embedded Image are determined from an experimentally calibrated model of the lattice (30). Inhomogeneity in the system—arising, for example, from trap curvature—primarily enters the model via inhomogeneities in the parameters Embedded Image and Embedded Image.

The experiments begin with Embedded Image 87Rb atoms loaded into a square 3D optical lattice with one atom per site (30), initially spin-polarized in the state Embedded Image. We use the hyperfine states Embedded Image and Embedded Image to represent the pseudospin-1/2 system. (Embedded Image and Embedded Image are the quantum numbers labeling the total angular momentum and its projection along the quantization axis, respectively.) The 3D lattice is composed of a vertical lattice along Embedded Image, which confines the atoms to an array of independent 2D planes, along with the dynamic 2D checkerboard lattice in the x-y plane. The vertical lattice depth is typically Embedded Image, held constant throughout the experiment, and the 2D lattice depth is initially Embedded Image with no staggered offset, Embedded Image (the recoil energy Embedded Image, Embedded Image kHz, where Embedded Image is the mass of 87RbRb and Embedded Image nm). The atoms occupy roughly 13 to 15 2D planes, with the central plane containing 800 to 1100 atoms. The ratio of surface lattice sites to total lattice sites of the trapped cloud is Embedded Image15% and sets a zero-temperature lower bound for the number of sites with neighboring holes. Based on spectroscopic measurements and assuming a thermal distribution (30), we estimate the hole density at the center of the cloud to be about 8%.

To measure the spin population independently on each sublattice, we map the four spin/sublattice states Embedded Image, Embedded Image, Embedded Image and Embedded Image onto four distinct Zeeman states and determine their populations with absorption imaging (Fig. 2A). To perform the experiment, we start with a spin-polarized configuration (Fig. 2B) and construct an initial state with staggered magnetization by applying the addressing offset Embedded Image and transferring the Embedded Image-site atoms to Embedded Image (Fig. 2C). After returning Embedded Image to zero, we initiate dynamics by quenching to a given configuration with lattice depth Embedded Image and offsets Embedded Image and Embedded Image (Fig. 2D). The ramp time for the quench of 200 μs was chosen to be fast with respect to subsequent dynamics but slow enough to avoid band excitation. After a variable hold time Embedded Image, we freeze the dynamics by raising Embedded Image to Embedded Image and read out the populations Embedded Image, from which we determine the staggered magnetization Embedded Image and the sublattice population difference Embedded Image.

Fig. 2 Experimental sequence.

(A) Spin/sublattice mapping. Atoms in Embedded Image (red) or Embedded Image (blue) occupy either the Embedded Image or Embedded Image sublattice (shown on the left). Applying a spin-dependent addressing offset to the Embedded Image sublattice spectroscopically resolves the Embedded Image and Embedded Image sublattices (colored lines correspond to the potentials and energy levels seen by different hyperfine states). The Embedded Image and Embedded Image populations are microwave transferred to two different hyperfine states (yellow and green respectively), and the four mapped populations are measured by absorption imaging after Stern-Gerlach separation (shown on the right). (B) Initial lattice loading: a spin-polarized Embedded Image unit filled Mott insulator. (C) Microwave state preparation. Embedded Image sites are microwave-transferred from Embedded Image to Embedded Image using techniques similar to those employed for the state readout shown in (A). (D) Time evolution. After the lattice is quenched to a specific configuration, the spin/sublattice populations are measured as a function of time (including the nonparticipating Embedded Image hyperfine state shown in gray).

Embedded Image (2)

The exchange terms in Eq. 1 conserve Embedded Image, whereas the first-order tunneling does not, allowing for population transport between sublattices. We also monitor the total spin imbalance Embedded Image and the Embedded Image population to quantify unwanted spin-changing processes that drive the atoms out of the pseudospin-1/2 manifold containing Embedded Image and Embedded Image. The measured time for depopulation of the pseudospin-1/2 states is greater than 6 s, and the atom number lifetime in the lattice is greater than 3 s.

For the lattice parameters studied in this paper, the magnetization dynamics is well described by exponential decay, with decay time scales ranging between 0.5 ms and 500 ms. Example decay curves are shown in Fig. 3A for a lattice depth Embedded Image and different offsets Embedded Image. For some Embedded Image and Embedded Image, the exponential decay clearly occurs on two well-separated time scales (Fig. 3A, inset): a fast time scale, Embedded Image, which dominates the behavior in shallow lattices when Embedded Image or Embedded Image, and a slow time scale, Embedded Image, which dominates the behavior in deep lattices with larger offset, Embedded Image.

Fig. 3 Identification and control of tunneling.

(A) Decay of magnetization at a lattice depth of Embedded Image for different offsets Embedded Image of 1000 Hz (green), 300 Hz (blue), and –50 Hz (red). (Inset) Short time evolution, with two time scales (Embedded Image 2 ms and Embedded Image 50 ms), both visible in the Embedded Image= 300 Hz (blue) trace. The solid lines are double exponential fits. The vertical gray line indicates the fixed decay time at which the data in (B) were taken. (B) Magnetization Embedded Image (filled circles) and sublattice population Embedded Image (open circles) as a function of Embedded Image at a fixed wait time of Embedded Image after the quench. The fast magnetization decay is resonant at Embedded Image and Embedded Image, whereas sublattice transport occurs only near Embedded Image. The vertical gray band represents the calculated Embedded Image with an uncertainty due to parameter extraction from the two-band model (30). (C) The fast time scale, Embedded Image, versus calculated tunneling time scale Embedded Image for different lattice depths and Embedded Image. The solid line is Embedded Image, and the gray band represents the uncertainty in the location of the 2D superfluid-insulator transition reported in (31). Error bars, ±1 SD statistical uncertainties from fitting.

To investigate the faster time scale, we measure Embedded Image and Embedded Image at a fixed decay time for different Embedded Image, as shown in Fig. 3B for Embedded Image15(0.5) Embedded Image. The 5-ms decay time (vertical line in Fig. 3A, inset) was chosen so that nearly all of the fast decay but little of the slow decay occurred. The fast magnetization decay reveals resonant features at Embedded Image and Embedded Image, with the decay rate at Embedded Image twice as fast as at Embedded Image. (Near the resonance at Embedded Image, the condition Embedded Image is not satisfied and the system must be described with the full Bose-Hubbard Hamiltonian, rather than Eq. 1.) In addition, Embedded Image shows sublattice transport from Embedded Image to Embedded Image sites at Embedded Image, indicative of resonant first-order tunneling. At Embedded Image the demagnetizing sublattice transport Embedded Image and Embedded Image are balanced. We theoretically estimate the expected width of the Embedded Image resonance in Embedded Image to be Embedded Image Hz (30), which is narrower than the 530(60) Hz width that we observe experimentally, suggesting inhomogeneous broadening. We note, however, that the observed broadening is beyond what is expected from the measured trap curvature and is inconsistent with estimates of light-shift inhomogeneity from spectroscopic measurements (30). A residual Embedded Image could account for the width.

The measured decay times Embedded Image for a range of lattice depths are plotted against the calculated tunneling time Embedded Image in Fig. 3C, showing a decay rate linear in Embedded Image, with Embedded Image. This slope is comparable to a simple theoretical estimate taking only resonant tunneling into account, which predicts Embedded Image, with Embedded Image the lattice coordination number. Given the relatively large hole density near the surface of the cloud, the agreement with a noninteracting estimate is not surprising, although we would expect interactions to reduce the decay rate.

To investigate the slow dynamics, we measure the magnetization decay time Embedded Image for Embedded Image, where first-order tunneling is negligible and superexchange should dominate the dynamics. To determine the dependence of Embedded Image on the spin-dependent staggered offset Embedded Image, we measure the remaining staggered magnetization Embedded Image and population difference Embedded Image after a fixed wait time Embedded Image, as shown in Fig. 4A for a lattice depth 9.4(3) Embedded Image and offset Embedded Image kHz. As expected for superexchange dynamics, the magnetization decay is resonant in Embedded Image. The full width at half maximum of the Lorentzian fit to the resonance is 126(14) Hz, considerably narrower than the observed tunneling resonances shown in Fig. 3B, and is most likely dominated by inhomogeneous broadening (second-order exchange processes are sensitive to inhomogeneity in both Embedded Image and Embedded Image). Figure 4A also shows that there is negligible sublattice transport associated with the demagnetization resonance. We note that at these values of Embedded Image, the ground state of the system would have all atoms on the lower sublattice, and the conservation of Embedded Image indicates that the spin dynamics occurs within a metastable manifold with respect to population.

Fig. 4 Resonant superexchange.

(A) Magnetization (filled circles) and sublattice population difference (open circles) at a fixed wait time of 70 ms versus spin-dependent energy offset Embedded Image for Embedded Image and Embedded Image kHz. The initial and final states of the second-order processes are resonant when Embedded Image= 0, resulting in increased magnetization decay. Lattice potentials (blue and red solid lines) show the sign change of Embedded Image across resonance for a fixed offset Embedded Image. (B) Measured slow decay time Embedded Image versus calculated superexchange time Embedded Image. The filled purple, red, blue, yellow, and green markers represent lattice depths of 14.7, 13.2, 11.3, 9.4, and 7.5 Embedded Image, respectively. (Inset) Measured slow decay rate versus applied staggered offset Embedded Image. The decay time scale, Embedded Image, collapses with Embedded Image over roughly two orders of magnitude in Embedded Image. The black line is a perturbative estimate of the scaling, which was checked in small systems by comparing to exact diagonalization averaged over hole-induced disorder (30). The gray line is a fit to a saturated linear dependence of Embedded Image on Embedded Image.

Figure 4B shows the measured resonant decay times Embedded Image versus calculated Embedded Image for different Embedded Image and Embedded Image, with Embedded Image and Embedded Image chosen to be larger than Embedded Image but considerably less than the next excited band. The decay time Embedded Image collapses to a single curve over two orders of magnitude in Embedded Image. The solid gray line through the data is a fit to Embedded Image, where Embedded Image is needed to capture the apparent saturation of Embedded Image at large Embedded Image; the fitted values are Embedded Image and Embedded Images. A quantitative calculation of the decay rate in 2D, including the effects of holes, is extremely challenging. However, a short-time perturbative calculation supports the experimental observation that Embedded Image scales with the single energy scale Embedded Image, which is not a priori expected given the existence of other (Embedded Image) energy scales in the Hamiltonian (30). Surprisingly, the perturbative calculations suggest that the decay rate is nearly independent of hole density, which can be attributed to the approximate cancellation of two competing effects of holes: they decrease the rate of superexchange dynamics but simultaneously open new demagnetization channels through the hole-mediated spin-exchange term in Eq. 1. We therefore compare the data to an analytical estimate, strictly valid at unit filling, of Embedded Image (black line in Fig. 4B). The empirically determined time scale Embedded Image (beyond which deviations from linear scaling become apparent in Fig. 4B) is shorter than the measured times for depopulation of the pseudospin manifold or loss of atoms. For times substantially larger than Embedded Image, interplane tunneling may also play a role in the short-time magnetization dynamics. This background decay rate may also be related to the nonzero relaxation processes observed outside the Embedded Image resonance in Fig. 4A. The mechanism for this off-resonant decay is not clear, but because the initial and final states differ in energy by substantially more than Embedded Image, it must arise from energy-nonconserving processes such as noise-assisted relaxation or doublon production (20). Corrections to Embedded Image caused by excited-band virtual processes (8), which we estimate to be on the order of 10 to 20% at the largest Embedded Image and smallest Embedded Image shown in Fig. 4B, may partially explain the observed saturation.

The scaling and resonant behavior of the fast and slow relaxation processes clearly reveal their origin as first-order tunneling and superexchange, respectively. For Embedded Image, our experiment realizes an unusual situation where tunneling, which is only active within a given sublattice, is comparable in strength to the superexchange coupling. This feature—which is crucial to our ability to observe superexchange-dominated dynamics—may have interesting implications for the equilibration of a doped antiferromagnetic state, because it determines the extent to which entropy (initially introduced in the motional degrees of freedom) is shared by the spin degrees of freedom. For smaller but nonzero Embedded Image, the ability to observe both tunneling and superexchange, often simultaneously and at experimentally accessible entropies, opens exciting opportunities to explore the nonequilibrium interplay of spin exchange and motion. Understanding the detailed dynamics of this strongly correlated, 2D quantum system is a formidable challenge, which may require the development of new theoretical techniques.

Supplementary Materials

Materials and Methods

Figs. S1 to S3

References (3235)

References and Notes

  1. Strictly speaking, the validity of Eq. 1 also requires Embedded Image and Embedded Image to avoid narrower second-order resonances that are associated with double and triple occupancy, respectively.
  2. Materials and methods are available as supplementary materials on Science Online.
  3. Acknowledgments: This work was partially supported by the Army Research Office’s atomtronics Multidisciplinary University Research Initiative and NIST. M.F.F. and E.A.G. acknowledge support from the National Research Council Research Associateship program. We thank B. Grinkemeyer for his contributions to the data taking effort, E. Tiesinga and S. Paul for discussions about tight-binding models, and A. V. Gorshkov and S. Sugawa for a critical reading of the manuscript.
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