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Bloch state tomography using Wilson lines

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Science  27 May 2016:
Vol. 352, Issue 6289, pp. 1094-1097
DOI: 10.1126/science.aad5812

Cold atoms do geometry

Electrons in solids populate energy bands, which can be simulated in cold atom systems using optical lattices. The geometry of the corresponding wave functions determines the topological properties of the system, but getting a direct look is tricky. Fläschner et al. and Li et al. measured the detailed structure of the band wave functions in hexagonal optical lattices, one resembling a boron-nitride and the other a graphene lattice. These techniques will make it possible to explore more complex situations that include the effects of interactions.

Science, this issue pp. 1091 and 1094


Topology and geometry are essential to our understanding of modern physics, underlying many foundational concepts from high-energy theories, quantum information, and condensed-matter physics. In condensed-matter systems, a wide range of phenomena stem from the geometry of the band eigenstates, which is encoded in the matrix-valued Wilson line for general multiband systems. Using an ultracold gas of rubidium atoms loaded in a honeycomb optical lattice, we realize strong-force dynamics in Bloch bands that are described by Wilson lines and observe an evolution in the band populations that directly reveals the band geometry. Our technique enables a full determination of band eigenstates, Berry curvature, and topological invariants, including single- and multiband Chern and Embedded Image numbers.

Geometric concepts play an increasingly important role in elucidating the behavior of condensed-matter systems. In band structures without degeneracies, the geometric phase acquired by a quantum state during adiabatic evolution elegantly describes a spectrum of phenomena (1). This geometric phase— known as the Berry phase— is used to formulate the Chern number (2), which is the topological invariant characterizing the integer quantum Hall effect (3). However, condensed-matter properties that are determined by multiple bands with degeneracies, such as in topological insulators (4, 5) and graphene (6), can seldom be understood with standard Berry phases. Recent work has shown that such systems can instead be described using Wilson lines (710).

Wilson lines encode the geometry of degenerate states (11), providing indispensable information for the ongoing effort to identify the topological structure of bands. For example, the eigenvalues of Wilson-Zak loops (i.e., Wilson lines closed by a reciprocal lattice vector) can be used to formulate the Embedded Image invariant of topological insulators (7) and identify topological orders protected by lattice symmetries (8, 9). Although experiments have accessed the geometry of isolated bands through various methods, including transport measurements (3, 12, 13), interferometry (14, 15), and angle-resolved photoemission spectroscopy (5, 16), Wilson lines have thus far remained a theoretical construct (710).

Using ultracold atoms in a graphene-like honeycomb lattice, we demonstrate that Wilson lines can be accessed and used as versatile probes of band structure geometry. Whereas the Berry phase merely multiplies a state by a phase factor, the Wilson line is a matrix-valued operator that can mix state populations (11) (Fig. 1A). We measure the Wilson line by detecting changes in the band populations (17) under the influence of an external force, which transports atoms through reciprocal space (18). In the presence of a force Embedded Image, atoms with initial quasimomentum Embedded Image evolve to quasimomentum Embedded Image after a time Embedded Image. If the force is sufficiently weak and the bands are nondegenerate, the system will undergo adiabatic Bloch oscillations and remain in the lowest band (18). In this case, the quantum state merely acquires a phase factor composed of the geometric Berry phase and a dynamical phase. At stronger forces, however, transitions to other bands occur, and the state evolves into a superposition over several bands.

Fig. 1 Wilson lines and effectively degenerate Bloch bands.

(A) In a nondegenerate system (left), adiabatic evolution of a state through parameter space Embedded Image results in the acquisition of a geometric phase factor, known as the Berry phase. In a degenerate system (right), the evolution is instead governed by a matrix-valued quantity called the Wilson line. If the degenerate levels can be experimentally distinguished (blue and yellow coloring), then population changes between the levels are detectable. (B) The band structure of the lowest two bands of the honeycomb lattice is given in effective energy units of Embedded Image, where Embedded Image is the applied force used to transport the atoms and Embedded Image is the distance between nearest-neighbor lattice sites. As Embedded Image is increased, the largest energy scale of the bands becomes small compared to Embedded Image. At large forces (iii), the effect of the band energies is negligible, and the system is effectively degenerate. In this regime, the evolution is governed by the Wilson line operator. We distinguish between the bands using a band-mapping technique that detects changes in the band population along the Wilson line path.

When the force is infinite with respect to a chosen set of bands, the effect of the dispersion vanishes, and the bands appear as effectively degenerate (Fig. 1B). The system then evolves according to the formalism of Wilczek and Zee for adiabatic motion in a degenerate system (11). The unitary time-evolution operator describing the dynamics is the Wilson line matrix (19)Embedded Image(1)where the path-ordered (Embedded Image) integral runs over the path Embedded Image in reciprocal space from Embedded Image to Embedded Image and Embedded Image is the Wilczek-Zee connection, which encodes the local geometric properties of the state space.

In a lattice system with Bloch states Embedded Image in the Embedded Image band at quasimomentum Embedded Image, where Embedded Image is the position operator, the elements of the Wilczek-Zee connection are determined by the cell-periodic part Embedded Image as Embedded Image. The diagonal elements (Embedded Image) are the Berry connections of the individual Bloch bands, which yield the Berry phase when integrated along a closed path. The off-diagonal elements (Embedded Image) are the interband Berry connections, which couple the bands and induce interband transitions.

Although the evolution described by Eq. 1 must be path-ordered when the Wilczek-Zee connections at different quasimomenta do not commute, it can also be path-independent under certain circumstances (9, 20). For example, when the relevant bands span the same Hilbert space for all quasimomenta, as is the case in our system, the Wilson line operator describing transport of a Bloch state from Embedded Image to Embedded Image reduces to Embedded Image (9, 19, 21)). Consequently, the elements of the Wilson line operator simply measure the overlap between the cell-periodic Bloch functions at the initial and final quasimomenta (9, 21)

Embedded Image(2)

Hence, access to the Wilson line elements facilitates the characterization of band structure topology in both path-dependent and path-independent evolution. In both cases, the topological information is encoded in the eigenvalues of the Wilson-Zak loops. In the latter case, the simplified form of the Wilson line in Eq. 2 additionally enables a map of the cell-periodic Bloch functions over the entire Brillouin zone (BZ) in the basis of the states Embedded Image at the initial quasimomentum Embedded Image.

We create the honeycomb lattice by interfering three blue-detuned laser beams at 120(1)° angles (Fig. 2A). At a lattice depth Embedded Image, where Embedded Image is the recoil energy, Embedded Image is the laser wavelength, and Embedded Image is the mass of 87Rb, the combined width Embedded Image kHz of the lowest two bands is much smaller than the Embedded Image kHz gap to higher bands. Consequently, there exists a regime of forces where transitions to higher bands are suppressed, and the system is well approximated by a two-band model (19).

Fig. 2 Reaching the Wilson line regime in the honeycomb lattice.

(A) Schematic of the honeycomb lattice in real space with A (B) sublattice sites denoted by solid (open) circles. The lattice is formed by interfering three in-plane laser beams (blue arrows) with frequency Embedded Image. Sweeping the frequency of beam Embedded Image by Embedded Image creates a force Embedded Image in the lattice frame in the propagation direction of beam Embedded Image (19). (B) Two copies of the first BZ of the honeycomb lattice, separated by a reciprocal lattice vector Embedded Image. By changing the relative strengths of Embedded Image (red arrows), the atoms can be moved along arbitrary paths in reciprocal space. Each BZ features nonequivalent Dirac points Embedded Image and Embedded Image at the corners of the hexagonal cell. High-symmetry points Embedded Image, at the center of the BZ, and M, at the edge of the BZ, are also shown. (C) The population remaining in the first band for different forces after transport to Embedded Image (green), Embedded Image (red), and Embedded Image (blue). Inset numbers i to iii refer to band schematics in Fig. 1B, representing the diminishing effect of the dispersion for increasing force. The data agree well with a two-level, tight-binding model (dashed line) that approaches the Wilson line regime (thick shaded line) at large forces. Discrepancies at larger forces result from transfer to higher bands and match well with ab initio theory using a full band structure calculation including the first six bands (thin solid line) (19). For all subsequent data, we use Embedded Image, indicated by the dashed gray line. Error bars indicate the standard error of the mean (SEM) from 10 shots per data point.

We probe the lattice geometry with a nearly pure Bose-Einstein condensate of 87Rb, which is initially loaded into the lowest band at quasimomentum Embedded Image, the center of the BZ (Fig. 2B). To move the atoms in reciprocal space, we linearly sweep the frequency of the beams to uniformly accelerate the lattice, thereby generating a constant inertial force in the lattice frame. By independently controlling the frequency sweep rate of two beams (Fig. 2A), we can tune the magnitude and direction of the force and move the atoms along arbitrary paths in reciprocal space.

To verify that we can access the Wilson line regime, where the dynamics are governed entirely by geometric effects, we transport the atoms from Embedded Image to different final quasimomenta using a variable force Embedded Image and perform band-mapping (17) to measure the population remaining in the lowest band (Fig. 2C). For vanishing forces, we recover the adiabatic limit, where the population remains in the lowest band. For increasing forces (i and ii in Fig. 1B), where the gradient Embedded Image over the distance between A and B sites Embedded Image is less than the combined width Embedded Image, the population continuously decreases. However, at strong forces (iii in Fig. 1B), where Embedded Image, the population saturates at a finite value. For example, after transport by one reciprocal lattice vector (blue data in Fig. 2C), about one-quarter of the atoms remain in the first band, in stark contrast to typical Landau-Zener dynamics, where the population vanishes for strong forces (22).

Theoretically, the population in the first band after the strong-force transport directly measures the Wilson line element Embedded Image in the basis of the band eigenstates. Based on Eq. 2, the saturation value Embedded Image of the population after transport to Embedded Image is a measure of the overlap between the cell-periodic Bloch functions of the first band Embedded Image at Embedded Image and Embedded Image. Notably, for the case of transport by one reciprocal lattice vector Embedded Image, the cell-periodic parts Embedded Image are not identical, despite the unity overlap of the Bloch states Embedded Image at Embedded Image and Embedded Image. In contrast to the typical Landau-Zener case, they are also not orthogonal—hence the finite saturation value.

To corroborate that our experiment measures the Wilson line, we transport atoms initially in the ground state at Embedded Image by up to three reciprocal lattice vectors (Fig. 3). The threefold rotational symmetry of the lattice, combined with the symmetry of its s-orbitals, makes the path from Embedded Image to Embedded Image equivalent to the triangular path shown in Fig. 3A, such that the overlap between cell-periodic components of the Bloch wave functions at the two endpoints is unity (see Eq. 2). Correspondingly, we expect to recover all the population in the lowest band after transport from Embedded Image to Embedded Image. This prediction is confirmed in Fig. 3B, where we plot the population remaining in the first band after transport to final quasimomentum Embedded Image. The data are well described by a tight-binding model that takes into account the relative phase between orbitals on A and B sites of the lattice due to the Wilson line Embedded Image. Physically, this can be understood by assuming that the real-space wave function simply accumulates a position-dependent phase when the strong force Embedded Image is applied for a short time Embedded Image (Fig. 3C). Notably, the result depends crucially on the real-space embedding of the lattice and would be different in, e.g., a brick-wall incarnation (23) of the same tight-binding model. Discrepancies from the tight-binding model result from population transfer to higher bands (19).

Fig. 3 Measuring mixing angles Embedded Image at different quasimomenta.

(A) Because of the threefold-rotational symmetry of the honeycomb lattice, a path from Embedded Image to Embedded Image is equivalent to a triangle-shaped path with each leg of length Embedded Image, beginning and ending at Embedded Image. Colored dots correspond to colored quasimomentum labels in (B). (B) The population remaining in the first band after transport to final quasimomentum Embedded Image. Theory lines are a single-particle solution to the dynamics using a full lattice potential and including the first six bands (solid) and a two-band, tight-binding model (dashed) (19). The inset Bloch sphere depicts the transported state at Embedded Image (red), Embedded Image (blue), and Embedded Image (green) in the basis of the cell-periodic Bloch functions at Embedded Image. Error bars represent the SEM from averaging 9 to 11 shots, with the exception of Embedded Image and Embedded Image, which show the average of 20 shots. (C) Transport of a Bloch state by one reciprocal lattice vector corresponds to a Embedded Image phase shift in the real-space wave functions of each sublattice site. Projecting the combined lattice and gradient potential V(x) along the path shown in red onto the x axis, which is the direction of the applied force in the measurements of Figs. 2B and 3B, highlights the effect of the real-space embedding of the honeycomb lattice. Since the distance between A (solid circles) and B sites (open circles) is 1/3 the distance between sites of the same type, there is a phase difference of Embedded Image between the real-space wave functions of A and B sites, which gives rise to the band mixing.

As the Wilson line enables a comparison of the cell-periodic Bloch functions at any two quasimomenta (Eq. 2), it can in principle be applied to fully reconstruct these states throughout reciprocal space. To this end, it is convenient to represent the state Embedded Image at quasimomentum Embedded Image in the basis of cell-periodic Bloch functions Embedded Image and Embedded Image at a fixed reference quasimomentum Embedded Image as

Embedded Image(3)

Mapping out the geometric structure of the lowest band therefore amounts to obtaining Embedded Image and Embedded Image, which parametrize the amplitude and phase of the superposition between the reference Bloch states, for each quasimomentum Embedded Image (24, 25). Whereas the total phase of Embedded Image is gauge dependent—i.e., it can be chosen for each Embedded Image—the relative phase Embedded Image is fixed for all Embedded Image once the basis states Embedded Image and Embedded Image are fixed. Throughout this work, we choose the basis states at reference point Embedded Image.

In this framework, the population measurements in Fig. 3B constitute a reconstruction of the mixing angle Embedded Image. This can be visualized as the rotation of a pseudospin on a Bloch sphere, where the north (south) pole represents Embedded Image Embedded Image. As a function of quasimomentum Embedded Image, the angle Embedded Image winds by Embedded Image per reciprocal lattice vector (see inset of Fig. 3B).

To obtain the relative phase Embedded Image, which is directly connected to the Wilson line via Embedded Image (19), we perform a procedure analogous to Ramsey or Stückelberg interferometry (26, 27). We initialize atoms in the lowest band at Embedded Image and rapidly transport them by one reciprocal lattice vector to prepare a superposition of band eigenstates at the reference point Embedded Image (i in Fig. 4A). We then hold the atoms at Embedded Image for a variable time (ii), during which the phase of the superposition state precesses at a frequency set by the energy difference between the bands at Embedded Image. Following this preparation sequence, we rapidly transport the superposition state to a final quasimomentum Embedded Image, lying at angular coordinate Embedded Image on a circle of radius Embedded Image centered at Embedded Image. Measuring the population of the first band as a function of hold time yields an interference fringe that reveals the relative phase Embedded Image (19).

Fig. 4 Measuring relative phases Embedded Image at different quasimomenta.

(A) Schematic of the interferometric sequence in the extended BZ scheme (left) and the corresponding rotation on the Bloch sphere (right). To create a superposition state, atoms initially in the lowest eigenstate at Embedded Image are rapidly transported to Embedded Image (i). The phase of the superposition state is controlled by varying the hold time at Embedded Image (ii). After the state preparation, the atoms are transported to a final quasimomentum Embedded Image, which is parametrized by the angle Embedded Image and lies on a circle of radius Embedded Image centered at Embedded Image (iii). (B) Phases Embedded Image referenced to Embedded Image = 180° for the lattice with AB-site degeneracy (blue) and AB-site offset (red). Data in blue have been offset by +120° for visual clarity. Dashed lines are a two-band, tight-binding calculation with Embedded Image (blue) and Embedded Image (red), where Embedded Image Hz. Error bars indicate fit errors.

We observe quantized jumps of Embedded Image in the phase of the interference fringe each time Embedded Image is swept through a Dirac point, i.e., every 60° (blue circles in Fig. 4B) (28, 29). The binary nature of the phases is a consequence of the degeneracy between A and B sites, which dictates that the band eigenstates at each quasimomentum be an equal superposition of states Embedded Image and Embedded Image on the A and B sublattices (19). Therefore, on the Bloch sphere, the pseudospin is constrained to rotate on a meridian about an axis whose poles represent the corresponding cell-periodic functions Embedded Image and Embedded Image (inset of Fig. 3B). When we remove this constraint by introducing an energy offset between A and B sites (19, 30)), we observe smoothly varying phases that are always less than Embedded Image (red circles in Fig. 4B). The dependence of the phase on angle Embedded Image indicates both the breaking of inversion symmetry and the preservation of the threefold rotational symmetry of the lattice.

Apart from reconstructing the cell-periodic Bloch functions, our method also provides access to eigenvalues of Wilson-Zak loops, Embedded Image, which is essential for determining various topological invariants (79). To this end, we split the Wilson-Zak-loop matrix into a global phase factor, which can be measured by extending previous methods (1315, 31), and an Embedded Image matrix with eigenvalues Embedded Image. Using the data from Figs. 3B and 4B, we reconstruct the eigenvalues for a loop transporting from Embedded Image to Embedded Image, up to multiples of Embedded Image (19). We find the eigenvalue phases to be Embedded Image, in good agreement with the value of Embedded Image predicted from the two-band model. Notably, we measure the same eigenvalues even when the band eigenstates are modified by an energy offset between A and B sites (19). This invariance is a direct consequence of the real-space representation of the Wilson-Zak loop, Embedded Image [see Eq. 2 and (19)]. Because the Wilson-Zak loop depends only on the position operator Embedded Image, the eigenvalues are determined solely by the physical locations of the lattice sites, which are unchanged by the energy offset.

Our versatile approach only employs standard techniques that are broadly applicable in ultracold-atom experiments and can be extended to higher numbers of bands by adopting ideas from quantum process tomography (32). Provided that the relevant local Hilbert space is identical for all quasimomenta, our method provides a complete map of the eigenstates over the BZ, giving access to the Berry curvature and Chern number. When this is not the case, the same techniques enable the reconstruction of Wilson-Zak loop eigenvalues, which directly probe the geometry of the Wannier functions (9) and, therefore, the polarization of the system (21, 33). Consequently, these eigenvalues can reveal the topology of bands with path-dependent and non-Abelian Wilson lines (9, 11). Such systems can be realized in cold-atom experiments by periodically modulating the lattice (12, 30, 3436) to create a quasimomentum-dependent admixture of additional bands (37) or coupling between internal states (3841). Moreover, the addition of spin-orbit coupling (42) would enable the investigation of the Embedded Image invariant characterizing time-reversal invariant topological insulators (33, 41, 4346).

Supplementary Materials

Supplementary Text

Figs. S1 to S7

References (4752)

References and Notes

  1. See supplementary materials on Science Online
Acknowledgments: We acknowledge illuminating discussions with A. Alexandradinata, J.-N. Fuchs, N. Goldman, D. Greif, L.-K. Lim, G. Montambaux, A. Polkovnikov, and G. Refael. This work was supported by the Alfred P. Sloan Foundation, the European Commision (UQUAM, AQuS), Nanosystems Initiative Munich, the Harvard Quantum Optics Center, the Harvard–Massachusetts Institute of Technology Center for Ultracold Atoms, NSF grant DMR–1308435, the Defense Advanced Research Projects Agency Optical Lattice Emulator program, the Air Force Office of Scientific Research, Quantum Simulation Multidisciplinary University Research Initiative (MURI), the Army Research Office (ARO)–MURI on Atomtronics, and the ARO-MURI Qubit Enabled Imaging, Sensing, and Metrology program.
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