Resolving the phase of a Dirac topological state via interferometric photoemission

Abstract

The electronic wavefunction is at the heart of physical phenomena, defining the frontiers of quantum materials research. While the amplitude of the electron wavefunction in crystals can be measured with state-of-the-art probes in unprecedented resolution, its phase has remained largely inaccessible, obscuring rich electronic information. Here we develop a quantum-path electron interferometer based on time- and angle-resolved photoemission spectroscopy, that enables the reconstruction of the phase of electronic states in quantum materials - with energy and momentum resolution. We demonstrate the scheme by resolving the phase along the Dirac electronic band of a prototypical topological insulator and observe a resonance-associated phase jump as well as a momentum and phase synchronized inversion revealing the helicity of the Dirac cone. We show the interferometer can be optically controlled by the polarization of the absorbed light, allowing a differential measurement of the phase - a crucial component for extracting phase information from an interferogram. This photo-electron-interferometer is a purely experimental scheme and does not rely on any specific theoretical model. It can be extended to a variety of materials, opening up the phase dimension in quantum materials research.

Figures (6)

• Figure 1: Quantum-path interferometer in trARPES. LE and HE photons are depicted by the blue and purple arrows, respectively, throughout the figure. (a) Schematic diagram of the two QPs available in two-color 2PPE with the same initial and final states, where QP1 goes through a resonant intermediate state, while QP2 goes through a virtual one. (b) Experimental geometry: the plane of incidence is along the $\Gamma-M$ direction (red plane) in the hexagonal Brillouin-zone of $\mathrm{Bi}_2\mathrm{Se}_3$, and the measured momentum cut is along $\Gamma-K$ (yellow line). (c) Conceptual QP-interferometer: The electronic wavefunction originating from the initial states (gray Lorentzian on the left) is split into the two QPs (depicted by the two gray wavepackets in QP1 and QP2). In QP1, the electron wavepacket acquires energy- and momentum-dependent phase due to the resonant transition into the intermediate state, represented by the red-blue shading, while in QP2, the non-resonant transition does not modify the phase structure. Upon absorption of the second photon, both paths reach the same final state (beam-combiner) and the wavepackets interfere. The intensity of the combined wavepacket in energy and momentum is the trARPES spectrum measured on the detector. (d) Schematic visualization of the optical transitions of both QPs in $\mathrm{Bi}_2\mathrm{Se}_3$. The initial and intermediate states involved in the process (bottom panel) are extracted from the experimental data (see Methods). The top panel presents the measured spectrum of the final states originating from both QPs. The intermediate states of QP1 (QP2), upshifted by $\hbar\omega_\mathrm{HE}$ ($\hbar\omega_\mathrm{LE}$) are marked in dashed red (green). The right axis on the top panel is the energy of QP1 intermediate states relative to the Fermi level.
• Figure 2: Identifying interference regions. (a-b) Normalized $I_\mathrm{CD}$ spectra for S- and P-polarized HE, respectively. (c) k-symmetrized difference between the CD spectra in (a) and (b), showing the interference signatures. Regions displaying significant interference are highlighted by the numbered rectangles. (d) Schematic 2PPE pathway pairs (QP1 - left, QP2 - right), corresponding to the spectral regions #1-3 in (c). (e) 2PPE intensity ($I^-+I^+$) of region #2 for S-pol HE (top) and P-pol HE (bottom). Magenta arrows highlight the region where destructive interference is observed in (e). For all spectra in the figure, the intermediate states for QP1 (QP2) are marked by solid black (green) lines. The upshifted initial states (by LE = 3.08 eV) are marked by black dashed lines.
• Figure 3: Interference signature in CD spectrum. (a)-(b) 2PPE intensity $I^- + I^+$ and (c)-(d) LE-CD spectra $I^- - I^+$, measured with LE = 3.08 eV, for S-polarized HE (a,c) and P-polarized HE (b,d). (e-f) EDCs of CD spectra integrated over the momentum range marked by gray lines in (c-d), respectively. (g) Schematic visualization of the optical transition from initial states to intermediate state of QP1 (black lines). The resonant transition for LE=3.08 eV (3.53 eV) is marked by a blue (green) arrow, with the corresponding upshifted initial state marked by a dashed line. (h-i) LE-CD spectra for LE = 3.53 eV, for S-polarized HE and P-polarized HE, respectively. In all spectra, the intermediate states are marked by solid lines, and the upshifted initial states by dashed lines.
• Figure 4: Phase reconstruction. (a)-(b) Experimental $\cos(\Delta\phi_-)$ and $\cos(\Delta\phi_+)$ respectively. (c)-(d) Simulated $\cos(\Delta\phi_-)$ and $\cos(\Delta\phi_+)$ from the phase in (i)-(j). Dispersion of the Dirac cone (intermediate state of QP1) is shown in solid lines in (a-d,i-j). (e)-(f) Momentum Distribution Curves (MDCs) of experimental (gray) and simulated (orange) $\cos(\Delta\phi_\pm)$. Integration region is marked on the corresponding spectra (a-d). The simulated phase $\Delta\phi$(k,E), calculated according to the equations in (g), is presented in (i)-(j) for $\sigma_-$ and $\sigma_+$ LE polarizations respectively. (h) EDCs and (k) MDCs of the simulated $\Delta\phi$ for $\sigma_-$ ($\sigma_+$) polarized LE in solid (dashed) lines. The corresponding energy or momentum integration regions are marked in solid (dashed) gray lines on panels j (i). The EDCs for $k<0$ ( $k>0$ ) are on the left (right) of (h).
• Figure 5: Image Potential State Final states spectrum of $I^- + I^+$, at $\Delta t = -75$ fs, for (a) S-polarized (b) P-polarized HE. The QP1 intermediate states are marked by a solid black line, the IPS (QP2 intermediate state) is marked by a solid red line. (c) EDC fit (solid black line) of the center of the IPS band, red dots are the data. (d) The estimated intensity from QP2, $|\psi_2|^2$, in logarithmic scale. The purple rectangle marks the region used for $\cos(\phi)$ reconstruction (Fig. \ref{['fig:4_phase']}).