🔬 mesoscale physics

Resolving the phase of a Dirac topological state via interferometric photoemission

This paper presents a quantum-path electron interferometer based on time- and angle-resolved photoemission spectroscopy that successfully reconstructs the previously inaccessible phase of electronic wavefunctions, demonstrated by resolving the phase jumps and helicity of Dirac states in a topological insulator.

Original authors: Shiri Gvishi, Ittai Sidilkover, Yun Yen, Shaked Rosenstein, Nir Hen Levin, Adi Perelmuter, Omer Pasternak, Costel R. Rotundu, Ido Biran, Semën Gorfman, Naaman Amer, Michael Sentef, Hadas Soifer

Published 2026-07-07

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Shiri Gvishi, Ittai Sidilkover, Yun Yen, Shaked Rosenstein, Nir Hen Levin, Adi Perelmuter, Omer Pasternak, Costel R. Rotundu, Ido Biran, Semën Gorfman, Naaman Amer, Michael Sentef, Hadas Soifer

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand a complex song. Usually, when scientists study electrons in materials, they can hear the "volume" (how many electrons are there) and the "pitch" (what energy they have), but they are completely deaf to the "rhythm" or the "phase" of the wave. In the quantum world, this missing rhythm is crucial because it holds the secret to how electrons are connected, how they spin, and why certain materials act like magic (topological insulators).

For decades, measuring this "phase" in matter has been like trying to catch a ghost: you know it's there, but you can't grab it. This paper introduces a new way to catch that ghost using a clever trick called a Quantum-Path Interferometer.

Here is how they did it, broken down into simple concepts:

1. The Problem: The Invisible Rhythm

Think of an electron's wavefunction like a ripple in a pond. You can easily measure how high the ripple is (amplitude) and how fast it moves. But the timing of the ripple—whether it's at the peak or the trough at a specific moment—is the "phase." In crystals, this phase tells us about the material's hidden topology (its shape in a mathematical sense), but standard tools can't see it.

2. The Solution: A Two-Path Race

The researchers built a "race track" for electrons inside a crystal of Bismuth Selenide ( $Bi_2Se_3$ ), a material known for its special "Dirac" electrons (electrons that act like massless particles).

They shot two laser pulses at the material:

A Low-Energy Pulse (LE): Like a gentle tap.
A High-Energy Pulse (HE): Like a strong kick.

When these pulses hit the electrons, the electrons have two different ways to escape the material and be detected. Think of these as two runners taking different routes to the finish line:

Route A (The Resonant Path): The electron gets tapped by the low-energy laser, lands perfectly on a "stepping stone" (an intermediate energy state), and then gets kicked out by the high-energy laser. This path is sensitive to the material's internal rhythm.
Route B (The Reference Path): The electron gets kicked first by the high-energy laser to a high spot, then tapped by the low-energy laser to get out. This path is a "boring" reference; it doesn't pick up much of the material's special rhythm.

3. The Magic Switch: Polarization

Here is the genius part. Usually, you can't easily turn one of these routes on or off. But in this specific material, the "stepping stone" for Route A is picky about the angle of the light.

If the high-energy laser is vertically polarized, it ignores the stepping stone. Only Route A happens.
If the high-energy laser is horizontally polarized, it hits the stepping stone hard. Now, both Route A and Route B happen at the same time.

When both routes happen, the electron waves from Route A and Route B crash into each other at the finish line. Just like two ocean waves crashing, they either amplify each other (constructive interference) or cancel each other out (destructive interference). By comparing the "volume" of the signal when the switch is OFF (only Route A) versus when it is ON (both routes), the scientists can mathematically calculate the hidden rhythm (phase) of Route A.

4. What They Found: The Helical Dance

By using this "interferometer," they mapped out the phase of the electrons in the material's Dirac cone (a specific energy shape where the electrons live). They discovered two major things:

The Phase Jump: As the electrons moved through a specific resonance (a sweet spot in energy), their phase suddenly flipped by 180 degrees (like a wave turning upside down). This is a known phenomenon in physics, but seeing it clearly in a solid material with such detail is a first.
The Helical Twist: The most exciting discovery was that the phase wasn't just random; it was twisted. On one side of the electron's path, the phase was positive; on the other side, it was negative. This "handedness" (helicity) is the fingerprint of the material's topological nature. It proves that the electrons are locked in a specific spin-momentum dance that is protected by the laws of quantum mechanics.

5. Why It Matters (According to the Paper)

The paper claims this is a breakthrough because it turns a standard photoemission experiment (which usually just sees "where" electrons are) into a phase-sensitive tool.

It acts like a quantum camera that can now see the "shape" of the electron's wave, not just its brightness.
It works by using conditions that are already possible in labs (specific laser energies and polarizations), meaning this isn't just a theory; it's a practical tool that can be applied to many other materials, not just the one they tested.

In a nutshell: The researchers built a microscopic "wave-catcher" using lasers and a special crystal. By switching a laser's polarization, they forced electrons to take two different paths that interfered with each other. By listening to how these paths canceled or boosted each other, they successfully "heard" the invisible rhythm of the electrons, revealing the hidden, twisted nature of the material's quantum state.

Technical Summary: Resolving the Phase of a Dirac Topological State via Interferometric Photoemission

Problem Statement
The electronic wavefunction is central to understanding quantum materials, yet while the amplitude of electron wavefunctions in crystals can be measured with high resolution using state-of-the-art probes like Scanning Tunneling Microscopy (STM) and Angle-Resolved Photoemission Spectroscopy (ARPES), the phase remains largely inaccessible. This missing phase information obscures critical details regarding internal coherence, correlations, and topology. Conventional time- and angle-resolved photoemission spectroscopy (trARPES) provides energy, momentum, and time resolution but typically resolves only incoherent population dynamics, leaving the phase of the electronic wavefunction experimentally unresolved. Retrieving this phase is essential for capturing the full complexity of quantum matter, particularly near singularities like the Dirac point where quantum geometry dictates nontrivial topology.

Methodology
The authors introduce a quantum-path (QP) electron interferometer implemented within a trARPES setup to reconstruct phase information encoded in electronic transition amplitudes. The scheme utilizes a two-color, two-photon photoemission (2PPE) process driven by two pulses: a low-photon-energy (LE) pulse and a high-photon-energy (HE) pulse.

Quantum Path Interference: The interferometer exploits two indistinguishable excitation pathways that lead to the same final electron state (same energy and momentum):
- QP1 (Signal Arm): The LE pulse excites an electron from an initial state to a resonant intermediate state (the Dirac surface state), followed by photoemission via the HE pulse.
- QP2 (Reference Arm): The HE pulse excites an electron to a high-energy intermediate state (specifically an Image Potential State, IPS), followed by photoemission via the LE pulse.
Controllability via Polarization: A critical innovation is the use of polarization selection rules to control the interference. In the material studied ( $Bi_2Se_3$ $B i_{2} S e_{3}$ ), the coupling to the IPS is strongly polarization-dependent.
- With S-polarized HE light, coupling to the IPS is negligible, effectively turning off QP2. This provides a "one-path" measurement (reference intensity $I_S$ ).
- With P-polarized HE light, coupling to the IPS is strong, activating QP2. This creates a coherent superposition of QP1 and QP2, resulting in an interference pattern in the measured intensity ( $I_P$ ).
Phase Extraction: By comparing the intensities $I_S$ and $I_P$ , the relative phase difference ( $\Delta\phi$ ) between the two paths can be extracted using the relation $I_P = | \psi_1 + \psi_2 |^2$ , where $\psi_1$ and $\psi_2$ are the wavefunctions of the two paths. The circular dichroism (CD) of the LE pulse is used to identify regions where interference occurs, distinguishing true phase effects from matrix-element variations.

Key Contributions and Results
The scheme was demonstrated on the second Dirac cone (an unoccupied topological surface state) of the prototypical topological insulator $Bi_2Se_3$ .

Interference Identification: The authors identified specific energy-momentum regions where the difference between S- and P-polarized HE measurements could not be explained by matrix elements alone, confirming the presence of coherent QP interference.
Resonance-Associated Phase Jump: The reconstructed phase reveals a sharp $\pi$ -phase jump across the resonant energy of the Dirac band. This manifests as a sign change in the circular dichroism signal and a transition from constructive to destructive interference, analogous to Fano-type resonances.
Momentum-Dependent Phase Inversion (Helicity): The study resolved a momentum-odd phase dichroism between left- and right-circularly polarized LE excitations. The phase difference between these polarizations ( $\Delta\phi_{CD}$ ) exhibits a $\pi$ shift across the Dirac cone, directly reflecting the helical spin- and orbital-angular-momentum texture of the Dirac spinor.
Phase Reconstruction: The authors successfully reconstructed the phase distribution $\Delta\phi(E, k_y)$ , showing excellent agreement between experimental data and a theoretical model based on an arctan phase step across the resonance. The reconstruction captures the symmetry-to-antisymmetry switch in the interference pattern when changing the LE polarization from $\sigma^-$ to $\sigma^+$ .

Significance and Claims
The paper claims to provide the first direct experimental access to the phase of electronic transition amplitudes in quantum materials with simultaneous energy and momentum resolution.

Direct Phase Access: Unlike previous methods that infer phase indirectly or rely on specific optical field phases, this method extracts the intrinsic electronic phase accumulated during the transition.
Experimental Feasibility: The approach relies on experimentally accessible conditions: the presence of a suitable intermediate state (like an Image Potential State) and polarization-selective coupling. It does not require cryogenic or ultra-high vacuum conditions beyond standard trARPES requirements, nor does it rely on a detailed theoretical model of the underlying electronic bands for the reconstruction itself.
Broad Applicability: The authors argue that because Image Potential States are generic surface features and the scheme operates in the perturbative regime with visible-to-UV photons, this interferometric approach can be extended to a wide class of materials beyond topological insulators.
New Observables: The work demonstrates that phase-sensitive phenomena, previously restricted to photonics and ultrafast optics, can be measured directly in quantum materials. This opens a route to studying topological states, symmetry-breaking phases, and coherence mechanisms where the phase structure is the defining characteristic.

The authors maintain a modest scope, noting that while the strong spin-orbit coupling in $Bi_2Se_3$ aided in isolating the interference, it is not a fundamental requirement for the scheme. The primary achievement is the establishment of a controllable, momentum-resolved electron interferometer capable of revealing the hidden phase structure of quantum wavefunctions.