Probing topological edge states in a molecular synthetic dimension

This paper demonstrates the use of ultracold RbCs molecules to encode a 1D synthetic lattice in their rotational states, successfully realizing the Su-Schrieffer-Heeger model with long coherence times and full site-resolved readout to probe and verify the stability of topological edge states against various perturbations.

The Gist

Imagine you are trying to build a skyscraper, but you only have a single, narrow hallway to work in. You can't build up or out, so how do you create a complex 3D structure?

This is the challenge physicists face when they want to study complex quantum phenomena that usually require extra dimensions of space. The solution? Synthetic Dimensions.

Think of a synthetic dimension not as a physical place you can walk into, but as a "menu" of internal states. In this new study, scientists used ultracold molecules (specifically Rubidium-Cesium, or RbCs) as their building blocks. Instead of using the molecule's position in space, they used its rotational states (how the molecule is spinning) as the "floors" of a building.

Here is a simple breakdown of what they did and why it matters:

1. The Elevator System (The Synthetic Lattice)

Imagine the molecule is a person standing in a hallway. The "floors" of the building aren't physical steps; they are different ways the molecule can spin.

• Floor 1: Spinning slowly.
• Floor 2: Spinning a bit faster.
• Floor 3: Even faster, and so on.

To move the molecule from one "floor" to another, the scientists used microwaves (like invisible elevators). By tuning these microwaves, they could make the molecule "tunnel" (jump) between these spinning states. This created a 1D chain or a synthetic ladder made entirely of internal energy states.

2. The SSH Model: A Bumpy Road

The scientists wanted to test a famous physics model called the Su-Schrieffer-Heeger (SSH) model.

• The Analogy: Imagine a road with stepping stones. In a normal road, the distance between every stone is the same. In the SSH model, the road is "bumpy." The distance between stone 1 and 2 is short, but between 2 and 3 is long, then short again, then long.
• The Result: This alternating pattern creates a special "topological" effect. It's like a magic trick where the ends of the road behave differently than the middle. If you put a ball on the road, it might get "stuck" at the very ends, unable to move to the middle, even if you push it. These are called Edge States.

3. The Experiment: Testing the Magic

The team built this "bumpy road" using up to 8 spinning states (8 floors). They then asked: Do the molecules get stuck at the ends like the theory predicts?

• The Spectroscopy (The Flashlight): They shined a "probe" microwave light at the molecules. If the light matched the energy of a specific state, the molecules would disappear from their starting spot. By scanning the light, they mapped out the energy levels. They saw that when the road was "bumpy" enough (a specific ratio of short/long steps), two special energy levels appeared right at the edges, just as the math predicted.
• The Time Travel (The Movie): They prepared the molecules in a specific state and watched them move over time.
  • The Result: When they started the molecule at one end, it stayed there for a very long time. It was "protected."
  • The Coherence: Usually, quantum systems are fragile; they lose their "quantumness" (decohere) very quickly due to noise. But these molecules were incredibly stable. They stayed coherent for 500 times longer than it took for the molecule to jump between floors. This is like watching a spinning top spin for 500 hours without wobbling.

4. The "Chiral" Shield

One of the coolest findings was about protection.

• The Analogy: Imagine the edge state is a VIP guest at a party.
  • Chiral Perturbation: If you change the music volume (the strength of the connection between floors) unevenly, the VIP guest stays in their seat. The "topology" protects them.
  • Non-Chiral Perturbation: If you suddenly change the temperature of the room (adding a random energy offset), the VIP guest gets up and moves around. The protection is broken.
• The Finding: The experiment confirmed that the edge states are indeed immune to certain types of "noise" (chiral perturbations) but vulnerable to others. This proves the system is truly topological.

5. Why This Matters

This isn't just about spinning molecules; it's a new way to do Quantum Simulation.

• The Problem: Building real 3D or 4D quantum computers or simulators is incredibly hard because it requires complex lasers and traps.
• The Solution: By using the internal "spinning" states of molecules, you can simulate complex, high-dimensional physics using a simple 1D trap.
• The Future: Because molecules have strong interactions (they can "talk" to each other via electric forces), this platform could eventually simulate exotic materials, like superconductors or magnetic strings, that we can't build in the real world.

In a Nutshell

The scientists turned ultracold molecules into a programmable quantum playground. They used microwaves to create a "bumpy road" of energy states and proved that the molecules get stuck at the ends of this road, protected by the laws of topology. They did this with incredible precision and stability, paving the way for using molecules to simulate the most complex physics problems in the universe.

Technical Summary

1. Problem and Motivation

The paper addresses the challenge of simulating complex quantum phenomena that are inaccessible in standard real-space lattices. While synthetic dimensions (encoding spatial dimensions into internal quantum states) have been realized in photonic systems and cold atoms (using hyperfine, Rydberg, or momentum states), ultracold polar molecules offer a unique platform with distinct advantages:

• Rich Internal Structure: Access to large Hilbert spaces via rotational and hyperfine degrees of freedom.
• Strong Coupling: Rotational states are connected via strong electric-dipole transitions in the microwave domain.
• Dipolar Interactions: Molecules possess long-range, anisotropic dipolar interactions, enabling the study of interacting many-body systems.
• Coherence: Rotational states are immune to radiative decay, offering potentially long coherence times.

The authors aim to benchmark these advantages by realizing a Su-Schrieffer-Heeger (SSH) model—a minimal 1D topological model—in a synthetic dimension encoded within the rotational states of ultracold RbCs molecules. The goal is to probe topological phase transitions, measure edge state protection, and extract topological invariants with high precision.

2. Methodology

System Encoding

• Platform: Ultracold $^{87}\text{Rb}^{133}\text{Cs}$ molecules prepared in the absolute ground state ($X^1\Sigma$, $v=0$).
• Synthetic Lattice: The 1D chain is encoded in the rotational states $|N\rangle$ (where $N$ is the rotational quantum number).
  • The ground state $|0\rangle$ serves as an auxiliary probe state.
  • Sites $|1\rangle$ through $|N\rangle$ form the synthetic lattice.
  • The study utilized chains of 4 sites and 8 sites.
• State Selection: "Spin-stretched" states were used ($m_N=N, m_{\text{Rb}}=3/2, m_{\text{Cs}}=7/2$) to maximize coherence and simplify coupling.

Hamiltonian Engineering (Floquet Engineering)

• Stroboscopic Driving: The SSH Hamiltonian, characterized by alternating tunneling rates $J_1$ and $J_2$, is implemented using stroboscopic microwave pulses.
  • A single microwave source rapidly alternates between driving specific transitions ($|1\rangle \leftrightarrow |2\rangle$, $|2\rangle \leftrightarrow |3\rangle$, etc.).
  • The pulse duration ($\sim 1-200 \, \mu\text{s}$) is much shorter than the tunneling timescales, allowing the system to approximate the time-averaged target Hamiltonian (Trotterization).
  • Rabi frequencies ($\Omega$) control the effective tunneling strengths ($J \propto \Omega$).
• Trap Optimization: Molecules are confined in a near-magic-wavelength optical dipole trap. The wavelength is tuned to minimize differential light shifts between rotational states, maximizing coherence times (up to $\sim 500$ times the tunneling period).

Probing Techniques

1. Spectroscopy: An auxiliary weak probe field addresses the $|0\rangle \to |1\rangle$ transition. By scanning the probe frequency, the authors measure the loss of molecules from $|0\rangle$ to map the eigenenergy spectrum of the dressed SSH Hamiltonian.
2. Time-Domain Dynamics: Molecules are initialized in specific states (e.g., edge states or superpositions) and allowed to evolve under the SSH Hamiltonian. Site-resolved populations are measured at various evolution times to observe tunneling and oscillations.
3. Perturbation Tests: The system is subjected to:
   • Chiral perturbations: Imbalanced Rabi frequencies (preserving chiral symmetry).
   • Non-chiral perturbations: Local detunings (breaking chiral symmetry).
4. Winding Number Extraction: The mean chiral displacement is calculated from the time-evolved populations to extract the topological winding number ($W$).

3. Key Results

Spectral Characterization

• Phase Transition: The authors successfully mapped the energy spectrum as a function of the tunneling ratio $J_2/J_1$.
  • Trivial Phase ($J_2 < J_1$): No edge states; eigenstates are delocalized.
  • Topological Phase ($J_2 > J_1$): Two edge states emerge, localized at the ends of the chain ($|1\rangle$ and $|N\rangle$).
• 8-Site Chain: In the 8-site chain, the energy splitting between the two edge states becomes extremely small ($\sim 3.1 \, \text{Hz}$ for $J_2/J_1 = 5.4$), demonstrating the exponential suppression of splitting with chain length ($\Delta E \propto e^{-L/\xi}$).

Dynamics and Coherence

• Long Coherence: The system exhibited coherence times of $\sim 100$ ms, which is approximately 500 times the lattice tunneling period ($\tau_1$). This significantly exceeds previous synthetic dimension experiments using Rydberg atoms ($\sim 10 \tau_1$).
• Edge State Stability:
  • When initialized in an edge state superposition ($|\Phi_{e+}\rangle = \frac{1}{\sqrt{2}}(|1\rangle + |4\rangle)$), the population remained static, confirming the state is an eigenstate.
  • When initialized in a single edge site ($|1\rangle$), the population oscillated between $|1\rangle$ and $|4\rangle$ with a frequency corresponding to the energy splitting of the edge states.
• Energy Splitting Measurement: Using interferometric techniques, the energy splitting was measured with Hertz-level precision (e.g., $379.0(7) \, \text{Hz}$ for the 4-site chain).

Topological Protection

• Chiral Perturbations: When Rabi frequencies were imbalanced (breaking symmetry but preserving chiral symmetry), the edge states remained robust. The state fidelity remained high ($1-F \approx 2 \times 10^{-4}$).
• Non-Chiral Perturbations: When a local detuning was applied (breaking chiral symmetry), the edge states were destroyed. The initialized edge state showed significant population oscillations, and fidelity dropped significantly ($1-F \approx 0.02$).

Winding Number

• The mean chiral displacement $\langle C(\tau) \rangle$ was measured over time.
• In the topological regime ($J_2/J_1 = 5.0$), the mean displacement converged to a value corresponding to a winding number $W \approx 1$.
• In the trivial regime ($J_2/J_1 = 0.3$), it converged to $W \approx 0$.
• The experimental data matched theoretical predictions (exact diagonalization) with high accuracy, confirming the topological nature of the synthetic lattice.

4. Significance and Outlook

• Benchmarking Molecular Platforms: This work demonstrates that ultracold polar molecules are a superior platform for synthetic dimensions due to their long coherence times and precise control over tunneling rates, enabling the observation of dynamics over hundreds of tunneling periods.
• Topological Physics: It provides a robust experimental realization of the SSH model in a synthetic dimension, allowing for the direct observation of topological phase transitions, edge state localization, and the measurement of topological invariants (winding number).
• Future Applications:
  • Interacting Systems: The combination of synthetic dimensions with dipolar interactions opens the door to simulating exotic many-body phases, such as dipolar string phases and 2D synthetic lattices.
  • Adiabatic Preparation: The long coherence times enable the adiabatic preparation of complex many-body ground states.
  • Complex Geometries: The approach can be extended to simulate higher-dimensional topological models (e.g., 4D quantum Hall effect) and non-trivial geometries (loops, 2D grids) using the rich internal structure of molecules.

In conclusion, the paper establishes a new paradigm for quantum simulation using ultracold molecules, proving that their internal degrees of freedom can be engineered to create highly coherent, tunable synthetic lattices capable of exploring complex topological and many-body physics.