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arXiv:2203.02895 · Quantum

The Synthetic Hilbert Space of Laser-Driven Free-Electrons

Guy Braiman, Ori Reinhardt, Chen Mechel, Omer Levi, Ido E. Kaminer

Published Jan 3, 2023• 4 citations
arXivHTMLPDF

Abstract

Recent advances in laser interactions with coherent free electrons have enabled to shape the electron's quantum state. Each electron becomes a superposition of energy levels on an infinite quantized ladder, shown to contain up to thousands of energy levels. We propose to utilize the quantum nature of such laser-driven free electrons as a "synthetic Hilbert space" in which we construct and control qudits (quantum digits). The question that motivates our work is what qudit states can be accessed using electron-laser interactions, and whether it is possible to implement any arbitrary quantum gate. We find how to encode and manipulate free-electron qudit states, focusing on dimensions which are powers of 2, where the qudit represents multiple qubits implemented on the same single electron – algebraically separated, but physically joined. As an example, we prove the possibility to fully control a 4-dimenisonal qudit, and reveal the steps required for full control over any arbitrary dimension. Our work enriches the range of applications of free electrons in microscopy and spectroscopy, offering a new platform for continuous-variable quantum information.

DOI
10.22331/q-2023-01-03-888

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Recent advances in laser interactions with coherent free electrons have enabled to shape the electron's quantum state. Each electron becomes a superposition of energy levels on an infinite quantized ladder, shown to contain up to thousands of energy levels. We propose to utilize the quantum nature of such laser-driven free electrons as a "synthetic Hilbert space" in which we construct and control qudits (quantum digits). The question that motivates our work is what qudit states can be accessed using electron-laser interactions, and whether it is possible to implement any arbitrary quantum gate. We find how to encode and manipulate free-electron qudit states, focusing on dimensions which are powers of 2, where the qudit represents multiple qubits implemented on the same single electron – algebraically separated, but physically joined. As an example, we prove the possibility to fully control a 4-dimenisonal qudit, and reveal the steps required for full control over any arbitrary dimension. Our work enriches the range of applications of free electrons in microscopy and spectroscopy, offering a new platform for continuous-variable quantum information.
Free-electron interactions with laser-driven nanophotonic nearfields can quantize the electrons' energy spectrum and provide control over this quantized degree of freedom. We propose to use such interactions to promote free electrons as carriers of quantum information and find how to create a qubit on a free electron. We find how to implement the qubit's non-commutative spin-algebra, then control and measure the qubit state with a universal set of 1-qubit gates. These gates are within the current capabilities of femtosecond-pulsed laser-driven transmission electron microscopy. Pulsed laser driving promise configurability by the laser intensity, polarizability, pulse duration, and arrival times.
Abstract Free‐electron interactions with laser‐driven nearfields can quantize the electrons’ energy spectrum and provide control over this quantized degree of freedom. The study proposes to use such interactions to promote free electrons as carriers of quantum information and show how to create a qubit on a free electron, which holds promise for applications in electron microscopy and spectroscopy. A method to implement the qubit's noncommutative spin algebra, and to control and measure the qubit state with a universal set of 1‐qubit gates is shown. These gates are within the current capabilities of ultrafast transmission electron microscopes. Laser‐driven free‐electron qubits promise configurability by the laser intensity, frequency, and polarizability, simultaneously with high‐resolution temporal control on femtosecond timescales.
Free electrons interacting coherently with optical fields provide a powerful platform for quantum simulation and quantum control. For kiloelectron-volt electron energies, even optical photon emission and absorption produce appreciable quantum recoils, endowing the electron with a discrete and controllable energy ladder. Starting from relativistic quantum electrodynamics, we derive an exact recoil-resolved interaction Hamiltonian in a traveling wave picture. The resulting recoil ladder forms a high-dimensional qudit with programmable couplings and sufficient controllability for universal quantum computation. We demonstrate applications to quantum simulation, including one-dimensional analogue black-hole models including Hawking radiation physics, and to quantum information processing, where multiple logical qubits and high-fidelity gates can be realized with a single electron. In parallel, the same recoil-enabled dynamics enable the controlled creation of complex hybrid electron--photon states, in which engineered ladder transitions imprint nonclassical correlations and structure onto the emitted light. Together, these results establish recoiled free electrons as a versatile platform bridging quantum optics, Hamiltonian engineering, and quantum simulation.
Free electrons provide a powerful tool for probing material properties at atomic resolution. Recent advances in ultrafast electron microscopy enable the manipulation of free-electron wave functions using laser pulses. It would be of great importance if one could combine the spatial resolution of electron microscopes with the ability of laser pulses to probe coherent phenomena in quantum systems. To this end, we propose a novel concept that leverages free electrons that are coherently shaped by laser pulses to measure quantum coherence in materials. We develop the quantum theory of interactions between shaped electrons and arbitrary qubit states in materials, and show how the postinteraction electron energy spectrum enables measuring the qubit state (on the Bloch sphere) and the decoherence or relaxation times (T_{2}/T_{1}). Finally, we describe how such electrons can detect and quantify superradiance from multiple qubits. Our scheme can be implemented in ultrafast transmission electron microscopes (UTEM), opening the way toward the full characterization of the state of quantum systems at atomic resolution.
Determining the quantum state of a given quantum-mechanical system is a fundamental task in physics. Quantum-state tomography has been pivotal for establishing quantum optics [1-4] and for revealing the properties of bound charges in materials [5-7]. An emerging other object for studying and utilizing quantum effects are free electrons, elementary particles that are central to high-resolution microscopy [8,9], electron-based quantum optics [10-17], ul-trafast electron microscopy [18-24] and particle accelerators [25-27]. However, free electrons are intrinsically incoherent, and we lack a broadly applicable method to measure and control their quantum state beyond special cases with discrete energy sidebands [28,29]. Here, we report a universal approach to measure arbitrary free-electron quantum states in continuous variables. Two monochromatic but spectrally shifted laser waves produce interfering quan-tum paths that directly reveal the density matrix and thus all essential properties of the pure wavepackets, the ensemble, and their interlinks. As a first application, we show how the quantum state of a single electron is modified by many-body Coulomb interactions of a sur-rounding electron gas. The reported concepts and results provide insight into otherwise hid-den correlations in electron beams and enable the controlled optimization of exceptional quantum states for free-electron quantum optics or quantum electron microscopy.
We propose free-electron Rabi oscillation by creating an isolated two-level system in a synthetic energy space induced by laser. The π/2-pulse and π-pulse in synthetic Rabi dynamics can function as 'beam splitters' and 'mirrors' for free-electron interferometry, allowing us to detect local electromagnetic fields and plasmonic excitations. When the coupling field is quantized, we can observe quantum and vacuum Rabi oscillations of the two-level electron, which can be used to investigate the quantum statistics of optical excitations and electron-photon entanglement. Recent advances in laser control of electron microscopes and spectroscopes makes the experimental detection of synthetic Rabi oscillations possible. However, observing the quantum Rabi oscillation of electrons remains challenging. Our work has the potential to advance various fundamentals and applications of resonant light-matter interactions between low-energy electrons and quatum light.