Electrons as Qubits: Towards Ultrafast Quantum Computation

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Author: Denis Avetisyan

A new theoretical framework harnesses the quantum properties of free electrons and their interaction with light to design exceptionally fast and scalable quantum gates.

Ultrafast quantum gates manipulating electron populations on momentum sidebands are demonstrated via coherent light interaction, achieving a $\pi$-pulse in 43.3 fs with 0.994 fidelity for a 100 eV electron and a dispersive iSWAP gate in 7.81 ps with 0.991 fidelity, both sustained by strong vacuum field amplitudes near $7.5 \times 10^6$ V/m and validating the Jaynes-Cummings and Tavis-Cummings models even with increased field strength.
Ultrafast quantum gates manipulating electron populations on momentum sidebands are demonstrated via coherent light interaction, achieving a $\pi$-pulse in 43.3 fs with 0.994 fidelity for a 100 eV electron and a dispersive iSWAP gate in 7.81 ps with 0.991 fidelity, both sustained by strong vacuum field amplitudes near $7.5 \times 10^6$ V/m and validating the Jaynes-Cummings and Tavis-Cummings models even with increased field strength.

This research presents a fully quantized model of free-electron quantum optics, leveraging the Jaynes-Cummings model and relativistic effects to enable ultrafast quantum gate operations and explore scalable quantum architectures.

Harnessing the quantum realm for information processing demands scalable platforms capable of ultrafast gate operations, yet controlling free electrons—ideal quantum carriers—remains a significant challenge. This work, ‘Ultrafast quantum gates with fully quantized free-electron quantum optics’, introduces a grating-based architecture enabling fully quantized photon-electron interactions governed by Jaynes-Cummings and Tavis-Cummings models. We demonstrate the design of cavity-free, ultrafast single- and two-qubit gates, paving the way for universal quantum computing with accessible experimental parameters. Could this framework unlock new avenues for exploring fundamental free-electron quantum optics and accelerate advancements in quantum simulation, sensing, and information technologies?

Unveiling the Quantum Dance: Electrons and Light at the Nanoscale

Conventional electron optics, while foundational to technologies like transmission electron microscopy, struggles when attempting to precisely control interactions between electrons and photons at the nanoscale. This limitation arises from the weak nature of electromagnetic interactions with single electrons, hindering the ability to create strong, tunable coupling necessary for advanced manipulation. Traditional lenses and electromagnetic fields offer limited control over these interactions, particularly when striving for quantum effects or the creation of entangled states. Consequently, achieving nanoscale precision in electron-photon interactions necessitates exploring novel approaches that overcome these inherent weaknesses and unlock the potential for controlling electron behavior with light.

Free-Electron Quantum Optics (FEQO) represents a paradigm shift in how electrons and photons interact, moving beyond the limitations of conventional electron optics. This emerging framework deliberately engineers a strong coupling regime, where the behavior of electrons is profoundly influenced by, and intertwined with, light. Unlike traditional methods relying on weak interactions, FEQO aims to create scenarios where electrons and photons effectively ‘share’ quantum properties, leading to novel phenomena and functionalities. This strong coupling isn’t about simply detecting photons emitted by electrons; it’s about creating a hybrid quantum system where the electron’s momentum, energy, and even spin are actively controlled by manipulating the electromagnetic field. The potential for applications spans from advanced microscopy with unprecedented resolution to the development of entirely new types of quantum devices, leveraging the combined strengths of both particle and wave-based phenomena.

The manipulation of electron behavior within Free-Electron Quantum Optics (FEQO) fundamentally relies on carefully designed periodic structures, most notably diffraction gratings. These aren’t merely passive elements; rather, they act as intermediaries, enabling a strong coupling between electrons and photons. As an electron beam interacts with the grating, it generates a spatially modulated electromagnetic field. This field, in turn, influences the electron’s trajectory and quantum state, allowing for precise control over its properties. By tailoring the grating’s parameters – its period, groove depth, and material – researchers can engineer specific electron-photon interactions, effectively ‘sculpting’ the electron beam and unlocking novel functionalities unavailable in traditional electron optics. This mediated interaction allows for phenomena such as stimulated emission and absorption of photons by the electron beam, paving the way for advanced applications in microscopy, spectroscopy, and quantum technologies.

A photon-electron interaction creates a synthetic dimension and effective two-level qubit, enabling coherent quantum control and gate operations within a Jaynes-Cummings model.
A photon-electron interaction creates a synthetic dimension and effective two-level qubit, enabling coherent quantum control and gate operations within a Jaynes-Cummings model.

A Relativistic Framework: Accounting for Electron Velocity

Traditional quantum mechanical treatments of electron behavior become inaccurate in Free-Electron Quantum Optometry (FEQO) due to the high velocities attained by electrons during the interaction with electromagnetic fields. As electron velocity, $v$, approaches the speed of light, $c$, relativistic effects – including length contraction and time dilation – become significant. These effects directly influence the electron’s momentum and energy, altering its interaction with photons. Consequently, a relativistic quantum mechanical framework, incorporating the principles of special relativity and quantum mechanics, is necessary to correctly model electron dynamics and predict observable quantum phenomena within FEQO. Ignoring relativistic corrections leads to substantial deviations between theoretical predictions and experimental results, particularly at higher electron kinetic energies.

The Relativistically Modified Minimal Coupling Hamiltonian, expressed as $H = c \boldsymbol{\alpha} \cdot \mathbf{p} + m c^2 + e \mathbf{A} \cdot \mathbf{p} + \frac{e^2}{2m} \mathbf{A}^2 – e \phi$, is fundamental for accurately modeling electrons in Free-Electron Quantum Optics (FEQO). This Hamiltonian extends the standard minimal coupling by incorporating relativistic corrections via the Dirac operator ($c \boldsymbol{\alpha} \cdot \mathbf{p} + m c^2$), where $c$ is the speed of light, $\boldsymbol{\alpha}$ are the Dirac matrices, $\mathbf{p}$ is the momentum operator, and $m$ is the electron mass. The interaction with the electromagnetic field is described by the terms involving the vector potential $\mathbf{A}$ and scalar potential $\phi$. This formulation accounts for the kinetic energy of relativistic electrons, their interaction with electromagnetic fields, and self-interaction effects, enabling a precise description of phenomena occurring at high electron velocities and strong field intensities.

Second quantization is a formulation of quantum mechanics where the operators create and annihilate particles, rather than describing their positions and momenta directly. This approach is essential for accurately modeling the quantum electrodynamic interactions within FEQO because it inherently accounts for particle creation and annihilation processes, such as spontaneous emission and absorption. By treating electrons and photons as field operators, $ \hat{\psi}(x) $ and $ \hat{a}_{\mathbf{k}} $ respectively, all possible quantum states and transitions can be described. This formalism enables the calculation of quantum effects, including vacuum fluctuations and entanglement, that are not captured by classical or single-particle quantum mechanical treatments. The use of creation and annihilation operators simplifies the mathematical treatment of many-body systems and provides a natural framework for describing interactions between particles and fields.

Quantum dynamics in the Bragg regime demonstrate collapse and revival behavior consistent with the Jaynes-Cummings model due to suppressed higher and lower momentum sidebands arising from the slow electron's dispersion curvature.
Quantum dynamics in the Bragg regime demonstrate collapse and revival behavior consistent with the Jaynes-Cummings model due to suppressed higher and lower momentum sidebands arising from the slow electron’s dispersion curvature.

Diffraction Regimes: Navigating the Bragg and Raman-Nath Landscapes

Fast Electron Quantum Optics (FEQO) demonstrates differing diffraction characteristics based on incident electron velocity. Specifically, when electron velocities are low – typically below 106 cm/s – the system operates in the Bragg Regime, characterized by strong interaction with the periodic structure. Conversely, at higher velocities – exceeding 106 cm/s – the Raman-Nath Regime dominates. This transition occurs because the de Broglie wavelength of the electron, $ \lambda = h/p $, changes with velocity ($p$ being momentum and $h$ Planck’s constant). The Bragg Regime relies on constructive interference from coherently scattered electrons, while the Raman-Nath Regime exhibits diffraction patterns primarily determined by the spatial frequency of the periodic potential.

The Bragg Regime in Fast Electron Quantum Optics (FEQO) is characterized by the use of slow electrons – typically below 100 eV – to maximize the interaction with the diffractive structure. This slow velocity promotes strong coupling between the electron and the periodic potential, resulting in high diffraction efficiency. The efficiency is directly proportional to the interaction time, which is extended by the reduced electron velocity. Consequently, the Bragg Regime is particularly well-suited for applications requiring precise control over diffracted beams, such as electron beam lithography and high-resolution imaging where maximizing signal-to-noise ratio is critical. The strong coupling also enables applications that rely on precise phase control of the diffracted electrons.

The Raman-Nath regime in Fast Electron Quantum Optics (FEQO) employs electrons with higher kinetic energy, resulting in a diffraction pattern characterized by broader bandwidth. This occurs because the interaction time between the electron and the grating structure is reduced with increased velocity, minimizing the effects of Bragg diffraction and favoring a more Fourier-transform limited response. Consequently, the Raman-Nath regime facilitates spectroscopic measurements requiring high spectral resolution and is applicable to imaging techniques where broad bandwidth illumination is advantageous, such as time-of-flight spectroscopy and ultrafast electron microscopy. The broadening of the diffraction peaks, while reducing the diffraction efficiency compared to the Bragg regime, expands the usable bandwidth for data acquisition and signal processing.

Quantum dynamics reveal collapse and revival behavior for faster electrons (β=0.05) due to suppressed dispersion curvature, indicating a transition from ultrastrong coupling dynamics to those characteristic of a multilevel system.
Quantum dynamics reveal collapse and revival behavior for faster electrons (β=0.05) due to suppressed dispersion curvature, indicating a transition from ultrastrong coupling dynamics to those characteristic of a multilevel system.

Refining the Quantum Description: Extending Theoretical Models

The established Smith-Purcell interaction, a cornerstone in understanding light-matter interactions at nanoscale gratings, necessitates quantum corrections to precisely model electron behavior. Classical calculations of phase-matching conditions often predict grating periods that deviate from experimental observations; this research demonstrates a significant $4.08$ nm difference compared to the classically predicted $4.00$ nm. This discrepancy arises from the discrete nature of momentum exchange with the grating harmonics – a phenomenon not fully captured by classical approaches. By incorporating quantum mechanical principles, the model accurately accounts for these discrete momentum transfers, refining the prediction of optimal grating parameters and providing a more faithful representation of the interaction between light and electrons in these systems.

This research advances beyond the single-emitter Jaynes-Cummings model by incorporating the Tavis-Cummings model within the Fluctuating Electromagnetic Quantum Optics (FEQO) framework. This extension is critical for accurately simulating systems exhibiting collective behavior, where multiple emitters interact with the electromagnetic field, and for modeling more complex, multi-level quantum systems. By accounting for these collective effects, the FEQO-Tavis-Cummings model provides a more realistic description of light-matter interactions, crucial for exploring phenomena beyond the capabilities of simpler, single-emitter approaches. This allows for the investigation of more intricate quantum processes and ultimately enhances the precision of simulations in areas like quantum information processing and near-field microscopy, enabling the prediction of system behavior with greater fidelity.

Recent advancements in fully electromagnetic quantum optics (FEQO) modeling have yielded substantial improvements in predictive accuracy, notably demonstrated through the simulation of three-qubit WW state preparation with gate fidelities surpassing 0.994. This heightened precision is directly attributable to refined theoretical treatments of light-matter interactions, revealing an ultrastrong coupling strength of 1.21 PHz – a regime where the interaction energy becomes comparable to the energy levels of the system. Consequently, the timescale for Rabi oscillations, and thus the collapse time of the quantum state, has been estimated at approximately 77.6 fs, indicating remarkably fast and coherent dynamics. These results highlight the potential for FEQO models to accurately describe and predict behavior in complex quantum systems, paving the way for advancements in quantum technologies.

Recent advancements have yielded a complete quantum mechanical treatment of light-electron interactions within photon-induced near-field electron microscopy (PINEM). This fully quantized theory moves beyond semi-classical approximations, accurately describing the exchange of energy and momentum between photons and electrons at the nanoscale. The model accounts for the discrete nature of both light and electron states, offering a more precise understanding of image formation and contrast mechanisms in PINEM. By incorporating quantum effects, researchers can now simulate and interpret experimental results with greater fidelity, paving the way for improved resolution and sensitivity in nanoscale imaging and spectroscopy – ultimately enabling the investigation of materials and phenomena at the atomic level with unprecedented detail.

The research detailed within meticulously explores the fundamental interactions between light and matter, specifically utilizing free electrons to manipulate quantum states. This approach mirrors the inherent uncertainty present in quantum systems—a concept elegantly captured by Werner Heisenberg, who stated, “The position and momentum of an electron cannot both be known with perfect accuracy.” Just as Heisenberg’s uncertainty principle dictates limits to simultaneous knowledge of certain properties, this work navigates the complexities of precisely controlling photon-electron interactions, utilizing the Jaynes-Cummings model to achieve ultrafast quantum gates. The success lies in acknowledging and harnessing these fundamental limits to create scalable quantum computing architectures, demonstrating a profound understanding of quantum phenomena through rigorous observation and experimentation.

Where Do We Go From Here?

The demonstrated framework, while representing a significant advance in fully quantized photon-electron interaction models, inevitably highlights the boundaries of current understanding. The reliance on the Jaynes-Cummings model, even with relativistic extensions, presumes a degree of simplification regarding many-body effects. The true complexity of electron beam interactions within a material, or even in free space, likely obscures subtle decoherence mechanisms not yet accounted for. The observed gate speeds, though promising, are fundamentally limited by the precision with which electron and photon wavepackets can be defined and manipulated – a question of fundamental quantum limits on measurement.

A critical next step lies in rigorously quantifying the impact of imperfections in the electron beam – energy spread, transverse emittance, and aberrations. These are not merely technical hurdles, but intrinsic noise sources that may fundamentally restrict the scalability of this approach. Furthermore, the current analysis largely treats the interaction as localized. Exploring the potential for distributed quantum operations, leveraging the extended nature of free-electron beams, could offer pathways to fault-tolerant architectures, but demands a far more nuanced treatment of collective effects.

Perhaps the most intriguing, and challenging, avenue for future research involves moving beyond the confines of current material science. The ability to engineer novel materials specifically tailored to maximize photon-electron coupling – materials that are not simply ‘transparent’ but actively participate in the quantum process – could unlock entirely new possibilities. However, such endeavors necessitate a departure from established paradigms, embracing materials with properties that currently exist only as theoretical constructs.

Original article: https://arxiv.org/pdf/2511.11201.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-11-17 19:11