Arbitrary control of the temporal waveform of photons during spontaneous emission

TL;DR

This work addresses the challenge of generating single photons with arbitrary temporal waveforms from spontaneous emission, essential for high-fidelity quantum networking across heterogeneous qubits. The authors develop a general, free-space protocol that combines amplitude modulation and phase-parity control of the driving field to sculpt the emitted waveform, modeled by the envelope $g_q(t)$ and constrained by the emitter lifetime; they extend the method to a three-level $ ext{Lambda}$ system and discuss generalization to higher-dimensional qudits. They introduce numerical optimization to identify optimal excitation pulses and deploy Quantum Monte Carlo trajectory techniques to characterize multi-photon statistics, enabling post-selection strategies that improve remote entanglement fidelity. The approach is demonstrated experimentally with $^{174}$Yb$^{+}$ ions, achieving photon-shaping fidelities approaching $0.996$ in ideal conditions, while identifying micromotion and hardware limits as primary fidelity bottlenecks. Collectively, the work provides a versatile framework for in-situ photon waveform control across diverse emitters, with clear implications for improved interference, state transfer, and scalable quantum networks.

Abstract

Control of the temporal waveform and Fock state statistics of photons produced during spontaneous emission from single quantum emitters provides a crucial tool in the establishment of hybrid quantum systems, optimal state transfer and interferometric stability of network architectures based on flying qubits. We describe a method to generate photons of any temporal waveform from emitters of any lifetime. Our broadly applicable approach has only two requirements for a candidate qudit: (1) control of the phase-parity and (2) modulation of the amplitude of a field coupling a ground state to an excited manifold which produces a photon during relaxation. We detail how to find optimal excitation pulse shapes, both numerically and experimentally, by employing variational algorithms to feedback on atomic populations. Additionally, we develop Quantum Monte Carlo based tools to determine emission statistics and establish techniques for optimal post-selection to ensure maximum fidelity of photon generation protocols. We situate our work in the context of other prior research on bespoke single photon sources and networking including post-emission pulse shaping, temporal gating and cavity-based methods. In comparison, our free-space process has greater flexibility in producing any waveform, requires less infrastructure and can be readily applied across a wide domain of emitters of any frequency or lifetime. We demonstrate temporal waveform shaping in $^{174}$Yb$^+$ trapped ions. Using feedforward validation of photon waveforms, we estimate an achievable process fidelity of at least $\approx$0.996.

Figures (11)

• Figure 1: Partial level structure of $^{174}\text{Yb}^+$ relevant for photon control experiments. (a) Optical pumping to $\ket{0}$ via the application of $\sigma^-$ polarized light (not shown) initializes the system. The application of a time-dependent complex Rabi frequency, $\Omega(t)$, drives the transition from $\ket{0}$ to $\ket{1}$ to control the excited state population, $\rho_{11}(t)$. Spontaneous emission from $\ket{1}$ generates photons with a temporal waveform at rate $\Gamma \times \rho_{11}(t)$ in a superposition of $\sigma^+$ and $\pi$ polarizations with a relative amplitude of 2:1. (b) Expanded atomic level system showing relevant laser frequency for photo-ionization and repumping.
• Figure 2: The best possible photon temporal waveforms which can be produced by a single emitter as target $\langle N \rangle$ is increased. Incoherent re-excitation leads to greater weight in photon tail at later times. These effects are suppressed in long photon temporal distributions.
• Figure 3: Representative numerically predicted photon waveforms and optimized pulses. In all plots, green and blue solid lines represent in- and out-of-phase Rabi frequency relative amplitudes, dashed black lines show target photon distribution and filled red area shows predicted photon shape. Left: 'Long' photon demonstrating ability to produce waveforms with high fidelities, defined as the normalized mode overlap, and high $\langle N\rangle$ values without the use of phase modulation. Inset shows atomic populations during the excitation process. Center: An exponentially rising photon with $\tau_{\text{rise}} = 8.1 \text{ ns}$, optimal for same qubit type state transfer. Right: Balanced double gaussian for time-bin atom-photon entanglement schemes.
• Figure 4: Optimized excitation pulse shapes to produce a desired gaussian photon ($\sigma = 8 \text{ ns}$) for emitters of a variety of lifetimes. As $\tau / \sigma$ grows, the corresponding pulse shape shows earlier phase flips and greater contribution of out-of-phase contributions to $\Omega(t)$.
• Figure 5: Left: Illustration of time-of-detection based post-selection. The histogram shows simulated click-time densities for single-photon shots ($p_{1}(t)$, orange) and multi-photon shots ($p_{2}(t)$, green; $p_{\ge3}(t)$, purple), normalized to unit area over clicks. The blue line shows the expected temporal distribution for the same process based on solution of the system master equation. The vertical red line marks a threshold for shots where the probability that a photon detection came from a multi-photon emission event is greater than 0.10, $P_{N\ge 2}=0.10$. This is used to establish a cut time $t^\star$. Accepting only shots in region I (i.e. $t\le t^\star$) raises the fraction of true single-photon heralds while reducing the acceptance rate. Right: Comparison of photon distributions for one and two photon emission events for the same excitation process. (a) normalized distribution of all first photon time-of-arrivals; (b) normalized distribution of all second photon time of arrivals; (c) normalized distribution of all first photon time of arrivals conditioned on the emission of only one photon; (d) normalized distribution of all first photon time of arrivals conditioned on the emission of two photons. We observe expected behavior for single and multi-photon events including: photon anti-bunching (see by comparing top right and bottom right panes) and a tighter, earlier distribution of first photons generated during two-photon emission trajectories vis-à-vis those coming from single-photon events.