Direct Measurement of the Zak phase in Topological Bloch Bands (1212.0572v1)

Abstract: Geometric phases that characterize the topological properties of Bloch bands play a fundamental role in the modern band theory of solids. Here we report on the direct measurement of the geometric phase acquired by cold atoms moving in one-dimensional optical lattices. Using a combination of Bloch oscillations and Ramsey interferometry, we extract the Zak phase - the Berry phase acquired during an adiabatic motion of a particle across the Brillouin zone - which can be viewed as an invariant characterizing the topological properties of the band. For a dimerized optical lattice, which models polyacetylene, we measure a difference of the Zak phase equal to phi_Zak=0.97(2)pi for the two possible polyacetylene phases with different dimerization. This indicates that the two dimerized phases belong to different topological classes, such that for a filled band, domain walls have fractional quantum numbers. Our work establishes a new general approach for probing the topological structure of Bloch bands in optical lattices.

• The paper introduces a novel method combining Bloch oscillations and Ramsey interferometry to directly measure the Zak phase in a dimerized optical lattice.
• The paper reports a measured Zak phase difference of approximately π between two distinct dimerization configurations, validating theoretical predictions.
• The paper demonstrates the potential of cold atom setups as versatile platforms for probing topological properties, paving the way for future research in complex systems.

Insights into the Direct Measurement of the Zak Phase in Topological Bloch Bands

The paper "Direct Measurement of the Zak phase in Topological Bloch Bands" by Atala et al. extends the paper of topological properties in periodic systems using cold atom setups. The authors focus on a specific form of geometric phase known as the Zak phase, which is a topological invariant characterizing one-dimensional (1D) systems. This paper represents a significant endeavor in experimental physics, as it provides a methodology for measuring the Zak phase directly in a 1D optical lattice simulating a dimerized chain, akin to the Su-Schrieffer-Heeger (SSH) model of polyacetylene.

Experimental Approach

The paper presents a method combining Bloch oscillations and Ramsey interferometry for the direct measurement of the Zak phase. Topological features are studied in an optical lattice formed by ultra-cold atoms. The experimental setup is designed to model a dimerized system with two possible configurations (dimerizations), and the Zak phase serves as an indicator distinguishing between these two topological classes.

Key to the methodology is the realization of spin-dependent Bloch oscillations in the lattice potential, allowing the controlled movement of atoms across the Brillouin zone. The authors detail an experimental sequence involving the coherent manipulation of a two-component spinor state, facilitating the measurement of the differential phase shift corresponding to the geometric phase in the system.

Main Findings

For a dimerized optical lattice, the paper reports a difference in the Zak phase between the two dimerization configurations D1 and D2, quantified as $\delta \varphi_{\text{Zak}} = 0.97(2) \pi$, which aligns closely with the predicted theoretical value of $\pi$. This measurement confirms that the two phases are distinct in their topological nature, a distinction encoded in the Zak phase difference. The authors underscore that where the unit cell contains two different sites, the distinction in the Zak phase correlates with the topological character of the occupying bands.

Furthermore, the paper explores the effect of an on-site energy offset $\Delta$ in the lattice, which introduces further complexity to the topological properties of the system. The variation of the Zak phase with changes in this parameter corroborates theoretical predictions and showcases the sensitivity and precision of the experimental arrangement.

Implications and Future Directions

The successful measurement of the Zak phase presented in this paper has substantial implications for the broader comprehension of topological phenomena in condensed matter physics. The results add to the understanding of phenomena like fermion number fractionalization and the quantum Hall effect by linking them to measurable phase invariants in a one-dimensional context.

The authors suggest several avenues for future research, including the investigation of edge states and fractional charges in topological insulators, and the extension of their techniques to higher dimensions where Chern numbers and other topological invariants become relevant. Additionally, they discuss potential extensions for measuring non-Abelian Berry phases in two-dimensional systems with quantum spin Hall effects, as well as applications in unconventional superconductors.

Conclusion

Atala et al. convincingly demonstrate the feasibility and utility of using cold atomic systems to probe topological properties in band structures. Their work effectively bridges the gap between theoretical predictions and experimental observation of topological invariants in simple lattice systems. The methodologies and findings reported pave the way for further explorations of more complex topological states and their associated quantum mechanical properties, underscoring the potential of cold atom systems as versatile platforms for studying fundamental physics questions.