Experimental engineering of Floquet topological phases in a one-dimensional optical lattice

This paper experimentally demonstrates the engineering of a one-dimensional anomalous Floquet topological phase in an optical lattice by utilizing multi-frequency driving with a tunable relative phase to manipulate gap windings and validate the resulting topology through a band-inversion-surface-resolved Ramsey protocol.

The Gist

Imagine you have a row of identical houses (atoms) arranged in a perfect line. In the quantum world, the people living in these houses (electrons or atoms) can move between them, but they usually follow strict rules about how they can travel. Sometimes, these rules create "traffic jams" that prevent movement, and other times, they create special "highways" that only exist on the edges of the line, allowing traffic to flow smoothly without getting stuck. These special highways are called topological phases.

For a long time, scientists thought these highways only existed in static, unchanging systems. But this paper introduces a new way to build them: by shaking the system rhythmically. This is called Floquet engineering.

Here is a simple breakdown of what the researchers did, using everyday analogies:

1. The Setup: A Quantum Train Track

The scientists used a "optical lattice," which is like a train track made of light. Atoms sit in the "stations" (the dips in the light).

• The Problem: Usually, atoms in the lowest energy "station" (the $s$-orbital) are too lazy to jump to the next "station" up the hill (the $p$-orbital).
• The Trick: Instead of just shaking the track back and forth (which is like a standard drumbeat), they modulated the depth of the light wells. Imagine making the stations deeper and shallower rhythmically.
• The Magic: Because of the unique shape of the "upper" station ($p$-orbital), this depth-shaking creates a special kind of "staggered" connection. It's like if the train tracks were twisted so that every time the train moved forward, it had to switch lanes in a specific pattern. This twisting is the secret sauce that creates the topological highway.

2. The Innovation: The Two-Beat Drummer

In previous experiments, scientists used a single rhythm (one frequency) to shake the system. This opened up one type of "gap" (a space where traffic can't go) but made it hard to control the other.

In this experiment, the team acted like a drummer playing two beats at once:

• Beat 1: A slow, steady rhythm.
• Beat 2: A faster rhythm (exactly twice as fast).

The key was the relative phase between these two beats. Think of it like two people pushing a swing.

• If they push at the exact same time (in phase), the swing goes very high (strong effect).
• If one pushes while the other pulls back (out of phase), they cancel each other out (weak effect).

By adjusting the timing (phase) between the two beats, the scientists could turn the "traffic rules" on or off for different parts of the system. They could make the "highways" appear, disappear, or even reverse direction, all without changing the physical track.

3. The Measurement: The Quantum "Echo" Test

How do you know if you successfully built these invisible highways? You can't just look at them. The team used a clever trick called Ramsey Interferometry, which is like a quantum echo test.

• Step 1: They split the atoms into two groups (like splitting a wave of water).
• Step 2: They let them evolve for a moment.
• Step 3: They brought them back together.

If the "traffic rules" (topology) were normal, the waves would meet and cancel out in a predictable way. But if the topological highway existed, the waves would meet with a 180-degree phase shift (a perfect "echo" difference). By measuring this shift at specific points, they could map out exactly which "gaps" in the energy spectrum had the special highway and which didn't.

4. The Big Discovery: Canceling and Adding

The most exciting part is what happened when they played with the two beats:

• Adding: Sometimes, the two rhythms worked together to create a "super-highway" with a very high winding number (a very complex, robust highway).
• Canceling: Other times, they adjusted the phase so the two rhythms canceled each other out. The total "winding" became zero (no net highway), BUT the individual gaps still had their own hidden topological features.

This is like having two people walking in a circle. If they walk in the same direction, they cover a lot of ground. If they walk in opposite directions, they end up back where they started (net distance = 0), but they still did the work of walking! This proves that even when the "big picture" looks boring, the "small picture" (the individual gaps) can still be full of interesting quantum magic.

Why Does This Matter?

This paper is a major step forward because:

1. Precision Control: It shows we can tune quantum materials like a radio, dialing in specific topological properties just by changing the timing of the drive.
2. New Materials: It opens the door to creating "anomalous" phases—states of matter that have no static equivalent and can only exist when driven.
3. Future Tech: These robust edge states are the building blocks for future quantum computers that are resistant to errors (noise).

In a nutshell: The researchers learned how to conduct a quantum orchestra using two different drum beats. By changing the timing between the beats, they could instantly rewrite the rules of traffic for quantum particles, creating, destroying, and reshaping invisible highways that could one day power the next generation of technology.

Technical Summary

1. Problem Statement

Periodic driving (Floquet engineering) allows for the creation of topological phases that have no static counterparts. While conventional Floquet phases obey standard bulk-edge correspondence, anomalous Floquet topology is a distinct regime where topology is encoded in the full time-evolution operator rather than just static band invariants.

• The Challenge: Anomalous phases are characterized by global winding numbers $(W_0, W_\pi)$ associated with the $0$ and $\pi$ quasienergy gaps. However, experimental control over these specific gap windings, particularly in one-dimensional (1D) systems, has been limited.
• The Gap: Existing cold-atom experiments have largely relied on single-tone driving, where multiple gaps are opened via higher-order processes, making independent control of gap windings difficult. Furthermore, distinguishing between high-winding phases and vanishing-winding phases (where net winding cancels but gap indices remain non-trivial) requires precise, gap-resolved readout methods.

2. Methodology

The authors propose and experimentally realize a 1D Floquet topological phase using ultracold $^{87}$Rb atoms in an optical lattice. Their approach combines orbital physics with multi-frequency driving.

A. Physical Platform: Orbital Lattice Depth Modulation (LDM)

• System: Atoms are loaded into the $s$ and $p$ orbitals of a 1D optical lattice.
• Mechanism: Instead of lattice position shaking (LPS), the authors use Lattice Depth Modulation (LDM).
• Key Insight: Due to the odd parity of the $p$-orbital Wannier function, LDM naturally induces a staggered nearest-neighbor $s$-$p$ coupling at the first order. This provides a direct, strong route to engineering an effective orbital-ladder Hamiltonian, superior to the weaker second-order processes typical of LPS.

B. Multi-Frequency Driving Scheme

• Single-Tone: A single frequency $\omega_0$ opens a gap and induces a specific winding.
• Two-Tone Control: The authors introduce a second tone at $2\omega_0$. The modulation is defined as:
  $$\delta V(t) = \delta V_0^{(1)} \cos(\omega_0 t + \phi_0^{(1)}) + \delta V_0^{(2)} \cos(2\omega_0 t + \phi_0^{(2)})$$
• Control Knob: The relative phase $\phi_0^{(2)}$ between the two tones acts as a tunable parameter. It controls the relative signs of the effective coupling terms in the $0$ and $\pi$ quasienergy gaps. By adjusting this phase, the windings of the two gaps can be made to add (constructive interference) or cancel (destructive interference).

C. Readout Protocol: BIS-Resolved Ramsey Interferometry

To measure the topology gap-by-gap, the authors developed a specific protocol:

1. Band Inversion Surfaces (BIS): They identify specific quasimomenta ($k_L, k_R$) where the energy gap closes (resonance conditions).
2. Ramsey Sequence:
   • Preparation: A lattice position shaking (LPS) pulse creates a superposition of $s$ and $p$ states (topologically trivial).
   • Evolution: Atoms evolve under the static Hamiltonian for a time $\tau_d$.
   • Readout: A LDM pulse projects the state.
3. Measurement: By measuring the spin imbalance (population difference between $s$ and $p$ orbitals) at the left ($k_L$) and right ($k_R$) BIS momenta, they detect a $\pi$ phase shift in the Ramsey fringes. This phase contrast is the dynamical hallmark of a non-trivial winding in that specific gap.

3. Key Contributions

1. First Experimental Realization of 1D Anomalous Floquet Phases: The team successfully engineered and detected anomalous Floquet topology in a 1D optical lattice, a regime previously difficult to access in cold atoms.
2. Phase-Tunable Gap Windings: They demonstrated that multi-frequency LDM allows for quantitative control over the winding numbers $(W_0, W_\pi)$. By tuning the relative phase of the drive, they can switch between:
   • High-Winding Phase: $(W_0, W_\pi) = (1, 1) \rightarrow W_{total} = 2$.
   • Vanishing-Winding Phase: $(W_0, W_\pi) = (1, -1) \rightarrow W_{total} = 0$, while retaining non-trivial edge modes in individual gaps.
3. Novel Readout Technique: The implementation of a BIS-resolved Ramsey protocol allows for the direct, gap-resolved measurement of topological invariants, overcoming the limitations of time-averaged spin texture measurements used in previous works.
4. Coexistence of Drive Modalities: The experiment proves that LPS (for preparation) and LDM (for readout/driving) can coexist coherently within a single sequence, enabling complex mixed-drive protocols.

4. Key Results

• Single-Tone Verification: Using single-tone driving, they observed the expected $\pi$ phase shift between $k_L$ and $k_R$ at the BIS, confirming the presence of a non-trivial winding in the opened gap.
• Two-Tone Control:
  • With $\phi_0^{(2)} = 0$, the windings of the $0$ and $\pi$ gaps aligned, resulting in a total winding number $W=2$.
  • With $\phi_0^{(2)} = \pi$, the windings opposed each other, resulting in $W=0$.
• Topological Charge Observation: In the $W=2$ case, a "topological charge" (a node where the effective transverse coupling $h_{F,y}(k)$ vanishes) appeared between the BIS momenta. This suppressed dynamics at that specific momentum, a feature absent in the $W=0$ case.
• Robustness: The phase contrast between $k_L$ and $k_R$ remained robust even when the global fringe phase was shifted, confirming the topological nature of the measurement.

5. Significance

• Engineering Tool: This work establishes multi-frequency phase control as a quantitative tool for engineering complex Floquet topologies. It moves beyond simple band inversion to the precise manipulation of gap-resolved invariants.
• Beyond Non-Interacting Limits: The use of orbital degrees of freedom ($s$-$p$ hybridization) opens pathways for studying topological phases with tunable interactions, extending beyond the non-interacting limit.
• Fundamental Physics: It provides a clear experimental demonstration of anomalous Floquet topology, where edge transport is tied to quasienergy gaps rather than individual bands, and where global invariants can vanish while local gap topology remains non-trivial.
• Future Applications: The ability to create high-winding phases and control mixed-drive protocols paves the way for realizing exotic quantum states, such as those with higher-order topology or specific edge-mode configurations, in ultracold atom systems.

In summary, this paper represents a significant leap in Floquet engineering, demonstrating that by leveraging orbital physics and multi-frequency control, researchers can now quantitatively design and verify complex topological phases that were previously theoretical constructs.