Interband Berry connection measurement in the optical honeycomb lattice

This paper demonstrates that the interband Berry connection in an optical honeycomb lattice can be directly mapped by measuring the resonant excitation strength of ultracold fermionic atoms under periodic lattice modulation, revealing key geometric features such as transparency lines and irreducible Dirac strings between specific energy bands.

The Gist

Imagine a crystal not as a rigid block of stone, but as a vast, invisible dance floor made of light. In this dance floor, tiny particles (like electrons in a real metal, or atoms in this experiment) are forced to move in specific patterns. These patterns are called "Bloch bands."

Usually, scientists can only guess at the shape of these dance floors by looking at how the particles behave from far away. But in this paper, the researchers at UC Berkeley built a special "quantum simulator" to peek directly at the geometry of these dance floors. They didn't use real electrons; they used ultracold potassium atoms trapped in a grid of laser beams that looks like a honeycomb (think of a beehive pattern).

Here is how they did it, explained simply:

1. The Setup: A Shaking Honeycomb

The researchers created a honeycomb-shaped trap for their atoms using three laser beams. Once the atoms were settled into the lowest energy level (the "ground floor" of the dance), they started shaking the entire laser grid.

Think of it like holding a tray of Jell-O and shaking it back and forth. If you shake it in just the right rhythm, the Jell-O starts to wobble and jump to a higher level. In their experiment, the "Jell-O" is the cloud of atoms, and the "shaking" is a precise vibration of the laser grid.

2. The Discovery: The "Invisible Compass"

The paper focuses on a concept called the Interband Berry Connection. That's a fancy physics term for a hidden "compass" that exists between two different energy levels (bands).

• The Analogy: Imagine you are trying to push a swing. If you push it in the right direction (matching the swing's natural motion), it goes high. If you push it in the wrong direction (perpendicular to the motion), nothing happens.
• The Experiment: The researchers shook their honeycomb grid in different directions (up-down, left-right, diagonal). They found that for certain specific locations on the grid, shaking in a specific direction did nothing. The atoms refused to jump to the higher energy level.
• The Result: These "do nothing" spots formed invisible lines across the grid, which the authors call "transparency lines." By mapping where these lines were, they could draw a complete map of the hidden "compass" (the Berry connection) that dictates how the atoms move between energy levels.

3. The "String" Mystery

The most exciting part of their discovery involves a strange feature they found between the ground level and the third excited level.

They found a line connecting two special points in the honeycomb grid (called K and K' points). Along this line, the "compass" direction flips abruptly, like a sudden 180-degree turn.

• The Metaphor: Imagine a field of wind socks. Most of the time, they point smoothly in a flowing direction. But along this specific line, the wind socks suddenly snap to point the opposite way.
• The "Dirac String": The researchers call this a Dirac string. It's a "knot" in the geometry of the system. They proved that no matter how they tried to smooth out the map or change their perspective (a concept called "gauge"), this string could not be erased. It is a fundamental, unchangeable feature of the honeycomb lattice's geometry.

4. Why This Matters

The paper claims that by simply shaking the atoms and watching where they jump (or don't jump), they can directly measure the complex geometric shapes of energy bands.

• Before: Scientists had to use complicated math or indirect measurements to guess these shapes.
• Now: They have a direct tool. They can "see" the geometry by observing the optical response (the atoms' reaction to the shake).

In summary: The team used a shaking honeycomb of light to reveal a hidden map of directions between energy levels. They found that this map has "blind spots" (transparency lines) and a permanent, un-erasable "knot" (a Dirac string) connecting two key points, proving that the geometry of these quantum systems is as real and measurable as the physical world around us.

Technical Summary

Technical Summary: Interband Berry Connection Measurement in the Optical Honeycomb Lattice

Problem and Context
The geometry of Bloch bands fundamentally influences the physical properties of crystalline solids and spatially periodic systems, ranging from the anomalous Hall effect to electric polarization and wavepacket motion. While topological invariants derived from band geometry are well-established, the direct experimental determination of the geometric properties of Bloch-band manifolds remains an active area of research. Specifically, the interband Berry connection, $\vec{A}_{nn'}(\vec{q}) = i \langle u_{n\vec{q}} | \nabla_{\vec{q}} | u_{n'\vec{q}} \rangle$, characterizes the relative geometry between two Bloch bands ($n \neq n'$). In solids, optical transitions between bands are driven by the off-diagonal matrix elements of the position operator, which are directly proportional to this interband Berry connection. However, bulk measurements in solids (e.g., photoconductivity) typically sum over all quasimomenta, obscuring the detailed vector field structure of the connection across the Brillouin zone.

Methodology
This work utilizes an ultracold-atom quantum simulator to resolve the resonant response of atoms at individual quasimomenta, thereby mapping the interband Berry connection directly. The experiment employs a degenerate Fermi gas of approximately $8 \times 10^4$ $^{40}$K atoms trapped in a static optical honeycomb lattice with a depth of $V_0 = 8.95 E_R$.

To simulate the optical excitation of electrons in solids, the researchers periodically modulate the relative phases of the lattice beams, effectively "shaking" the lattice position. In the frame co-moving with the lattice, this introduces a perturbation $\hat{H}' = 2F \text{Re}[e^{-i\omega_{PM}t}\vec{\epsilon}] \cdot \hat{\vec{r}}$, where $F$ is the shaking force amplitude and $\vec{\epsilon}$ is the polarization vector.

The excitation probability from the ground band ($n=1$) to excited bands ($n'$) at a specific quasimomentum $\vec{q}$ is governed by the relation:
$$P_{nn'}(\vec{q}) \propto |\vec{\epsilon} \cdot \vec{A}_{nn'}(\vec{q})|^2$$
By applying linear and circular shaking with various polarizations and frequencies, and measuring the fractional depletion of atomic population ($D_{nn'}$) via time-of-flight imaging, the authors extract the magnitude and orientation of $\vec{A}_{nn'}(\vec{q})$. The experiment resolves excitations between the ground band ($n=1$) and the first three excited bands ($n' = 2, 3, 4$).

Key Results

1. Polarization-Dependent Selection Rules: The study demonstrates that the direction of the interband Berry connection imposes optical selection rules. For the transition between the lowest two bands ($n=1$ to $n'=2$), the excitation vanishes when the shaking polarization is perpendicular to $\vec{A}_{12}(\vec{q})$. The experiment observes "transparency lines" in quasimomentum space where the response to specific polarizations is zero. The angular positions of these lines shift predictably with the shaking direction, confirming the double-winding nature of the vector field $\vec{A}_{12}$ around the $\Gamma$ point, even in the absence of band singularities at that point.

2. Mapping the Vector Field: The authors successfully map the interband Berry connection $\vec{A}_{14}$ (and $\vec{A}_{13}$) across the entire first Brillouin zone. The measurements reveal distinct transparency lines for both $x$ and $y$ polarizations, anchored at high-symmetry points ($\Gamma$ and $K$), indicating where the connection vanishes. The data also captures the gauge-invariant relative phase between the $x$ and $y$ components of the connection.

3. Dirac Strings and Gauge Dependence: A significant finding concerns the transition between the ground band and the second excited band ($n=1$ to $n'=3$). The orientation of $\vec{A}_{13}$ exhibits "irreducible Dirac strings"—lines along which the vector orientation abruptly changes sign. These strings connect the $K$ and $K'$ points in the Brillouin zone. While the specific location of these strings depends on the gauge choice, the parity of the number of strings crossed by closed loops around high-symmetry points is conserved. For $\vec{A}_{13}$, the parity indicates that these Dirac strings cannot be gauged away, signifying a non-trivial topological feature in the relative geometry of these bands.

4. Quantitative Comparison: The measured vector fields agree qualitatively with numerical simulations based on non-interacting particle dynamics. However, the measured amplitudes are systematically smaller (by a factor of approximately two) than theoretical predictions. The authors attribute this underestimation to the dynamic evolution of holes created by excitations (smearing the response) and limitations in distinguishing atomic populations in neighboring Brillouin zones during imaging. Despite this, the technique accurately extracts the direction and topological features of the connection.

Significance and Claims
The paper establishes optical response (via lattice shaking) as a powerful tool for characterizing the geometrical and topological properties of band structures. By resolving the response at each quasimomentum, the method provides a direct measurement of the interband Berry connection, a quantity previously difficult to access with such spatial resolution in solid-state systems.

The work highlights the direct physical connection between the optical selection rules of a material and the relative geometry of its bands. The observation of irreducible Dirac strings in the $\vec{A}_{13}$ connection demonstrates the capability to detect non-trivial multi-band topology. The authors suggest this methodology paves the way for studying complex topological phases in geometrically frustrated flat band systems, such as the kagome lattice, where the ground band manifold is predicted to possess non-zero Euler classes and fragile topological phases. The technique offers a pathway to experimentally probe the "Euler connection" and test multi-band topological braiding schemes.