Phase-Controlled Ramsey Interference of XUV Photoelectrons

TL;DR

This study demonstrates CEP-controlled Ramsey-type interference in XUV photoionization by two time-delayed, linearly polarized pulses acting on Neon prepared in a 2p+ current-carrying state. The authors solve the full-dimensional TDSE within the SAE framework and show that interference fringes in both angle-resolved and energy-resolved photoelectron distributions arise from temporal phase accumulation between ionization pathways, with fringe energies obeying $E_n(φ) = (2π n + φ)/τ$ and spacing $ΔE = 2π/τ$. By varying wavelength and intensity, they show fringe spacing is governed by the interpulse delay, not by Autler–Townes splitting, and they reveal a CEP-driven bound-state dynamics, dominated by transient 2s population. A reduced two-level model captures the dynamic Stark shift of the 2s state, linking bound-state dressing to the observed continuum interference, and providing a transparent physical interpretation of the results. The findings establish CEP-enabled temporal interference as a phase-sensitive probe of ultrafast electronic dynamics in the XUV regime and offer guidance for future experiments using phase-stable XUV pulse pairs.

Abstract

We investigate Ramsey-type quantum interference in photoelectron momentum distributions generated by two time-delayed, linearly polarized extreme-ultraviolet (XUV) laser pulses. The electron dynamics are studied by solving the full-dimensional time-dependent Schrödinger equation within the single-active-electron approximation for neon initially prepared in a current-carrying $2p_+$ state. The coherent superposition of electron wave packets released by the two pulses gives rise to pronounced interference fringes in both energy-resolved spectra and angle-resolved momentum distributions. We demonstrate that the fringe positions are governed by a Ramsey phase accumulated during the interpulse delay, resulting in a linear dependence on the relative carrier-envelope phase and an inverse scaling of the fringe spacing with the delay time. By systematically varying the laser intensity, we establish that the observed modulations originate from temporal quantum interference rather than Autler--Townes splitting. Analysis of the time-resolved bound-state population dynamics reveals that carrier-envelope-phase dependent bound--bound coupling dominated by transient population transfer to the $2s$ state, which controls the interference contrast. The accumulated phase is further interpreted in terms of a dynamic Stark shift of the dressed bound states, which is quantitatively reproduced using a reduced two-level model.

Figures (7)

• Figure 1: Photoelectron momentum distribution in $x-y$ plane is shown for CEP $\phi = \pi/2$ (a), $\phi = 0$ (b) and $\phi = 3\pi/2$ (c). Total angle-integrated energy distribution for these respective cases are illustrated in (d), (e) and (f), along with the case when only single pulse is used [red curve in panel (e)]. Yield is normalized with respect to the $\phi = 0$ case for all the results in this figure. The schematic diagram of the two 5 cycle pulses, separated by the delay $\tau = 6\tau_0$ is also shown. Vertical dashed line in (d), (e) and (f) represents the $E_{\mathrm{kin}}\simeq 0.346$ a.u. [Eq. \ref{['ener_kin']}].
• Figure 2: Angle-integrated photoelectron energy yield as a function of (a) CEP $\phi$ at fixed interpulse delay $\tau=6$ optical cycles and (b) interpulse delay $\tau$ at fixed CEP $\phi=0$, for two linearly polarized XUV pulses. The bright ridges correspond to constructive interference between electron wave packets released by the two pulses. Arrows mark the energy spacing $\Delta E$ between adjacent interference fringes. The linear CEP dependence at fixed delay in (a) and the inverse scaling of the fringe spacing with delay, $\Delta E \propto 1/\tau$, in (b) are represented by dashed lines.
• Figure 3: Effect of laser wavelength and intensity on the photoelectron spectra for fixed CEP $\phi=0$ and interpulse delay $\tau=6$ optical cycles. (a) Energy-integrated angular distribution as a function of laser wavelength $\lambda$. (b) Corresponding angle-integrated energy--wavelength map showing a systematic shift of the interference fringes with photon energy. (c) Energy--intensity map with all other laser parameters fixed. Panels (d)--(f) show representative spectra extracted from panels (a)--(c), respectively. In panel (f), the intensity is expressed in units of $10^{14}$ W/cm$^2$.
• Figure 4: Time-resolved population dynamics illustrating the role of CEP in the formation of interference patterns. (a) Snapshot of the population distribution over field-free eigenstates $(n,\ell,m)$ at $t=7\tau_0$, showing population transfer from the initially prepared $2p_{+}$ state into nearby bound and continuum states. (b) Temporal evolution of the $2p_{+}$ population, demonstrating stepwise depletion during the interaction with the two time-delayed laser pulses; the schematic laser pulse envelope is shown for reference. (c) Time-resolved population summed over different angular-momentum channels, together with the population of the $2s$ state, highlighting transient population trapping and redistribution during the delay interval.
• Figure 5: CEP-dependent population dynamics for fixed interpulse delay $\tau=6\tau_0$. (a) Depletion of the initially populated $2p_{+}$ state for different CEP values. (b) Time-resolved population of the $2s$ state, showing strong sensitivity to the CEP. (c) Population summed over different principal quantum numbers and angular-momentum channels, illustrating CEP-controlled redistribution among bound and continuum states. All quantities are shown as a function of time in units of optical cycles.