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Conditional displacement with a qubit ancilla is a critical component in continuous-variable error correction protocols. We present the generalized conditional displacement operator, conditioned on a qudit ancilla, showing how it enhances error-correction with Gottesman-Kitaev-Preskill (GKP) codes and exploring potential implementations.

The manipulation of quantum many-body systems is a frontier challenge in quantum science. Entangled quantum states that are symmetric to permutation between qubits are of growing interest. Yet, the creation and control of symmetric states has remained a challenge. Here, we find a way to universally control symmetric states, proposing a scheme that relies solely on coherent rotations and spin squeezing. We present protocols for the creation of different symmetric states including Schrodinger cat and Gottesman-Kitaev-Preskill states. The obtained symmetric states can be transferred to traveling photonic states via spontaneous emission, providing a powerful mechanism for the creation of desired quantum light states.

For decades, most research on high harmonic generation (HHG) considered matter as quantum but light as classical, leaving the quantum-optical nature of the harmonics an open question. Here we explore the quantum properties of high harmonics. We derive a formula for the quantum state of the high harmonics, when driven by arbitrary quantum light states, and then explore specific cases of experimental relevance. Specifically, for a moderately squeezed pump, HHG driven by squeezed coherent light results in squeezed high harmonics. Harmonic squeezing is optimized by syncing ionization times with the pump's squeezing phase. Beyond this regime, as pump squeezing is increased, the harmonics initially acquire squeezed thermal photon statistics, and then occupy an intricate quantum state which strongly depends on the semi-classical nonlinear response function of the interacting system. Our results pave the way for the generation of squeezed extreme-ultraviolet ultrashort pulses, and, more generally, quantum frequency conversion into previously inaccessible spectral ranges, which may enable ultrasensitive attosecond metrology.

Entangled photon pairs are a vital resource for quantum information, computation, and metrology. Although these states are routinely generated at optical frequencies, sources of quantum of light are notably lacking at extreme ultraviolet (XUV) and soft X-ray frequencies. Here, we show that strongly driven systems used for high harmonic generation (HHG) can become versatile sources of entangled photon pairs at these high frequencies. We present a general theory of photon pair emission from non-perturbatively driven systems, which we refer to as "strong field pair generation" (SFPG). We show that strongly driven noble gases can generate thousands of entangled pairs per shot over a large XUV bandwidth. The emitted pairs have distinctive properties in angle and frequency, which can be exploited to discriminate them from the background HHG signal. We connect SFPG theory to the three-step-model of HHG, showing that this pair emission originates from the impact of high frequency vacuum fluctuations on electron recombination. The light produced by SFPG exhibits attosecond Hong-Ou-Mandel correlations, and can be leveraged as a source of heralded single photon attosecond pulses. Our findings aid ongoing efforts to propel quantum optics into the XUV and beyond.

Collective spontaneous emission occurs when multiple quantum emitters decay into common radiation modes, resulting in enhanced or suppressed emission. Here, we find the quantum state of light collectively emitted from emitters exhibiting quantum correlations. We unveil under what conditions the quantum correlations are not lost during the emission but are instead transferred to the output light. Under these conditions, the inherent nonlinearity of the emitters can be tailored to create desired photonic states in the form of traveling single-mode pulses, such as Gottesman-Kitaev-Preskill and Schr\"odinger-cat states. To facilitate such predictions, our work reveals the multi-mode nature of collective spontaneous emission, capturing the role of the emitters' positions, losses, interactions, and beyond-Markov dynamics on the emitted quantum state of light. We present manifestations of these effects in different physical systems, with examples in cavity-QED, waveguide-QED, and atomic arrays. Our findings suggest new paths for creating and manipulating multi-photon quantum light for bosonic codes in continuous-variable-based quantum computation, communications, and sensing.

We show that the coherent interaction between free electrons and photons can be used for universal control of continuous-variable photonic quantum states in the form of Gottesman-Kitaev-Preskill (GKP) qubits. Specifically, we find that electron energy combs enable non-destructive measurements of the photonic state and can induce arbitrary gates. Moreover, a single electron interacting with multiple photonic modes can create highly entangled states such as Greenberger-Horne-Zeilinger states and cluster states of GKPs.

Long-range coherence and correlations between electrons in solids are the cornerstones for developing future quantum materials and devices. In 1954, Dicke described correlated spontaneous emission from closely packed quantum emitters, forming the theoretical basis of superradiance and superfluorescence. Since then, it has remained an open challenge to observe such phenomena with nanometer spatial resolution, precisely the important scale at which the collective correlations occur. Here, we report the first instance of free-electron-driven superfluorescence - superfluorescent cathodoluminescence - enabling us to excite and observe correlations at nanometer spatial scales. To exemplify this concept in our experiments, superlattices of lead halide perovskite quantum dots are excited by focused pulses of multiple free electrons. The electrons trigger superfluorescence: collective ultrafast emission observed at rates faster than both the lifetime and decoherence time. By controlling the area illuminated by the electron beam, we create a transition from a non-correlated spontaneous emission to a correlated superfluorescent emission. The observed signatures of superfluorescence are a reduction of the intrinsic emitter lifetime, a narrower linewidth, and a distinct redshift. We develop the theory of superfluorescent cathodoluminescence, which matches the results and highlights the unique features of electron-driven versus light-driven superfluorescence. Our observation and theory introduce a novel way to characterize coherence and correlations in quantum materials with nanometer spatial resolution, a key for future engineering of quantum devices., Comment: 34 pages, 22 figures

High harmonic generation (HHG) is an extreme nonlinear process where intense pulses of light drive matter to emit high harmonics of the driving frequency, reaching the extreme ultraviolet (XUV) and x-ray spectral ranges. So far, the HHG process was always generated by intense laser pulses that are well described as a classical electromagnetic field. Advances in the generation of intense squeezed light motivate us to revisit the fundamentals of HHG and ask how the photon statistics of light may alter this process, and more generally alter the field of extreme nonlinear optics. The role of photon statistics in non-perturbative interactions of intense light with matter has remained unexplored in both experiments and theory. Here we show that the defining spectral characteristics of HHG, such as the plateau and cutoff, are sensitive to the photon statistics of the driving light. While coherent (classical) and Fock light states induce the established HHG cutoff law, thermal and squeezed states substantially surpass it, extending the cutoff compared to classical light of the same intensity. Hence, shaping the photon statistics of light enables producing far higher harmonics in HHG. We develop the theory of extreme nonlinear optics driven by squeezed light, and more generally by arbitrary quantum states of light. Our work introduces quantum optical concepts to strong-field physics as new degrees of freedom in the creation and control of HHG, and finally shows that experiments in this field are feasible. Looking forward, HHG driven by quantum light creates quantum states of XUV and X-rays, enabling applications of quantum optics in new spectral regimes.

In strong-field physics and attosecond science, intense light induces ultrafast electron dynamics. Such ultrafast dynamics of electrons in matter is at the core of phenomena such as high harmonic generation (HHG), where these dynamics lead to emission of extreme UV bursts with attosecond duration. So far, all ultrafast dynamics of matter were understood to originate purely from the classical vector potential of the driving light, disregarding the influence of the quantum nature of light. Here we show that dynamics of matter driven by bright (intense) light significantly depend on the quantum state of the driving light, which induces an effective photon-statistics force. To provide a unified framework for the analysis & control over such a force, we extend the strong-field approximation (SFA) theory to account for non-classical driving light. Our quantum SFA (qSFA) theory shows that in HHG, experimentally feasible squeezing of the driving light can shift & shape electronic trajectories and attosecond pulses at the scale of hundreds of attoseconds. Our work presents a new degree-of-freedom for attosecond spectroscopy, by relying on nonclassical electromagnetic fields, and more generally, introduces a direct connection between attosecond science and quantum optics.

Strongly driven systems of emitters offer an attractive source of light over broad spectral ranges up to the X-ray region. A key limitation of these systems is that the light they emit is for the most part classical. We challenge this paradigm by building a quantum-optical theory of strongly driven many-body systems, showing that the presence of correlations among the emitters creates emission of nonclassical many-photon states of light. We consider the example of high-harmonic generation (HHG), by which a strongly driven system emits photons at integer multiples of the drive frequency. In the conventional case of uncorrelated emitters, the harmonics are in an almost perfectly multi-mode coherent state lacking any correlation between harmonics. By contrast, a correlation of the emitters prior to the strong drive is converted onto nonclassical features of the output light, including doubly-peaked photon statistics, ring-shaped Wigner functions, and quantum correlations between harmonics. We propose schemes for implementing these concepts, creating the correlations between emitters via an interaction between them or their joint interaction with the background electromagnetic field (as in superradiance). By tuning the time at which these processes are interrupted by the strong drive, one can control the amount of correlations between the emitters, and correspondingly the deviation of the emitted light from a classical state. Our work paves the way towards the engineering of novel many-photon states of light over a broadband spectrum of frequencies, and suggests HHG as a diagnostic tool for characterizing correlations in many-body systems with attosecond temporal resolution., Comment: 26 pages main (5 figures), 23 pages Supplementary Information

There exist two families of superradiance phenomena: relying on correlated emission by an ensemble of atoms or by free electrons. Here we investigate emission from an ensemble of atoms driven by coherently shaped electrons. This interaction creates superradiance emerging from both the atoms' and the electrons' coherence - with emission intensity that scales quadratically in both the number of atoms and number of electrons. This phenomenon enables electrons to become novel probes of quantum correlations in matter, with high temporal and spatial resolution., Comment: 4 figures

We analyze the interaction between a free electron and an ensemble of identical optical emitters. The mutual coherence and correlations between the emitters can enhance the interaction with each electron and become imprinted on its energy spectrum. We present schemes by which such collective interactions can be realized. As a possible application, we investigate free-electron interactions with superradiant systems, showing how electrons can probe the ultrafast population dynamics of superradiance., Comment: 20 pages, 4 figures

Cat states and Gottesman-Kitaev-Preskill (GKP) states play a key role in quantum computation and communication with continuous variables. The creation of such states relies on strong nonlinear light-matter interactions, which are widely available in microwave frequencies as in circuit quantum electrodynamics platforms. However, strong nonlinearities are hard to come by in optical frequencies, severely limiting the use of continuous variable quantum information in the optical range. Here we propose using the strong interactions of free electrons with light as a source for optical cat and GKP states. The strong interactions can be realized by phase-matching of free electrons with photonic structures such as optical waveguides and photonic crystals. Our approach enables the generation of optical GKP states with above 10 dB squeezing and fidelities above 90% at post-selection probability of 10%, even reaching >30% using an initially squeezed vacuum state. Furthermore, the free-electron interaction allows for conditional rotations on the photonic state, enabling to entangle a pair of GKP states into a GKP Bell state. Since electrons can interact resonantly with light across the electromagnetic spectrum, our approach may be used for the generation of cat and GKP states over the entire electromagnetic spectrum, from radio-waves to X-rays., Comment: 26 pages, 4 figures, 1 table

The ultrafast dynamics of charge carriers in solids plays a pivotal role in emerging optoelectronics, photonics, energy harvesting, and quantum technology applications. However, the investigation and direct visualization of such non-equilibrium transport phenomena remains as a long-standing challenge, owing to the nanometer-femtosecond spatio-temporal scales at which the charge carriers evolve. Here, we propose and demonstrate a novel interaction mechanism enabling nanoscale imaging of the femtosecond dynamics of charge carriers in solids. This imaging modality, which we name charge dynamics electron microscopy (CDEM), exploits the strong interaction between terahertz (THz) electromagnetic near fields produced by the moving charges and synchronized free-electron pulses in an ultrafast scanning transmission electron microscope. The measured free-electron energy at different spatio-temporal coordinates allows us to directly retrieve the THz near-field amplitude and phase, from which we reconstruct movies of the generated charges by comparison with microscopic theory. The introduced CDEM technique thus allows us to investigate previously inaccessible spatio-temporal regimes of charge dynamics in solids, for example revealing new insight into the photo-Dember effect, showing oscillations of photo-generated electron-hole distributions inside a semiconductor. Our work lays the foundation for exploring a wide range of previously inaccessible charge-transport phenomena in condensed matter using ultrafast electron microscopy.

For over 80 years of research, the conventional description of free-electron radiation phenomena, such as Cherenkov radiation, has remained unchanged: classical three-dimensional electromagnetic waves. Interestingly, in reduced dimensionality, the properties of free-electron radiation are predicted to fundamentally change. Here, we present the first observation of Cherenkov surface waves, wherein free electrons emit narrow-bandwidth photonic quasiparticles propagating in two-dimensions. The low dimensionality and narrow bandwidth of the effect enable to identify quantized emission events through electron energy loss spectroscopy. Our results support the recent theoretical prediction that free electrons do not always emit classical light and can instead become entangled with the photons they emit. The two-dimensional Cherenkov interaction achieves quantum coupling strengths over two orders of magnitude larger than ever reported, reaching the single-electron-single-photon interaction regime for the first time with free electrons. Our findings pave the way to previously unexplored phenomena in free-electron quantum optics, facilitating bright, free-electron-based quantum emitters of heralded Fock states.

High harmonic generation (HHG) is a highly nonlinear emission process in which systems driven by intense laser pulses emit integer multiples (harmonics) of the driving field. This feature is considered universal to all occurrences of HHG. Here we show that a strong nonlinear response of certain systems can split the HHG spectrum into non-harmonic Mollow-type triplets, imprinting the internal electronic characteristics and confining potential on the HHG spectrum. The spectral lines become tunable by the driver field. These phenomena are universal to any material system with isolated bound states. We identify the conditions under which our predictions become accessible and propose experimental systems to observe them., Comment: 13 pages, 4 figures

Entanglement of photons is a fundamental feature of quantum mechanics, which stands at the core of quantum technologies such as photonic quantum computing, communication, and sensing. An ongoing challenge in all these is finding an efficient and controllable mechanism to entangle photons. Recent experimental developments in electron microscopy enable to control the quantum interaction between free electrons and light. Here, we show that free electrons can create entanglement and bunching of light. Free electrons can control the second-order coherence of initially independent photonic states, even in spatially separated cavities that cannot directly interact. Free electrons thus provide a type of optical nonlinearity that acts in a nonlocal manner, offering a new way of heralding the creation of entanglement. Intriguingly, pre-shaping the electron's wavefunction provides the knob for tuning the photonic quantum correlations. The concept can be generalized to entangle not only photons but also photonic quasiparticles such as plasmon-polaritons and phonons.

The fundamental interaction between free electrons and light stands at the base of both classical and quantum physics, with applications in free-electron acceleration, radiation sources, and electron microscopy. Yet, to this day, all experiments involving free-electron light interactions are fully explained by describing the light as a classical wave, disregarding its quantum nature. Here, we observe quantum statistics effects of photons on free-electron-light interactions. We demonstrate interactions passing continuously from Poissonian to super-Poissonian and up to thermal statistics, unveiling a surprising manifestation of Bohr's Correspondence Principle: the transition from quantum walk to classical random walk on the free-electron energy ladder. The electron walker serves as the probe in non-destructive quantum detection, measuring the photon-correlation ${g^{(2)} (0)}$ and higher-orders ${g^{(n)} (0)}$. Unlike conventional quantum-optical detectors, the electron can perform both quantum weak measurements and projective measurements by evolving into an entangled joint-state with the photons. Our findings suggest free-electron-based non-destructive quantum tomography of light, and constitute an important step towards combined attosecond-temporal and sub-A-spatial resolution microscopy.

Angular momentum plays a central role in a multitude of phenomena in quantum mechanics, recurring in every length scale from the microscopic interactions of light and matter to the macroscopic behavior of superfluids. Vortex beams, carrying intrinsic orbital angular momentum (OAM), are now regularly generated with elementary particles such as photons and electrons, and harnessed for numerous applications including microscopy and communication. Untapped possibilities remain hidden in vortices of non-elementary particles, as their composite structure can lead to coupling of OAM with internal degrees of freedom. However, thus far, the creation of a vortex beam of a non-elementary particle has never been demonstrated experimentally. We present the first vortex beams of atoms and molecules, formed by diffracting supersonic beams of helium atoms and dimers, respectively, off binary masks made from transmission gratings. By achieving large particle coherence lengths and nanometric grating features, we observe a series of vortex rings corresponding to different OAM states in the accumulated images of particles impacting a detector. This method is general and can be applied to most atomic and molecular gases. Our results may open new frontiers in atomic physics, utilizing the additional degree of freedom of OAM to probe collisions and alter fundamental interactions.

Since the birth of quantum optics, the measurement of quantum states of nonclassical light has been of tremendous importance for advancement in the field. To date, conventional detectors such as photomultipliers, avalanche photodiodes, and superconducting nanowires, all rely at their core on linear excitation of bound electrons with light, posing fundamental restrictions on the detection. In contrast, the interaction of free electrons with light in the context of quantum optics is highly nonlinear and offers exciting possibilities. The first experiments that promoted this direction appeared over the past decade as part of photon-induced nearfield electron microscopy (PINEM), wherein free electrons are capable of high-order multi-photon absorption and emission. Here we propose using free electrons for quantum-optical detection of the complete quantum state of light. We show how the precise control of the electron before and after its interaction with quantum light enables to extract the photon statistics and implement full quantum state tomography using PINEM. This technique can reach sub-attosecond time resolutions, measure temporal coherence of any degree (e.g., g(1), g(2)), and simultaneously detect large numbers of photons with each electron. Importantly, the interaction of the electron with light is non-destructive, thereby leaving the photonic state (modified by the interaction) intact, which is conceptually different from conventional detectors. By using a pulse of multiple electrons, we envision how PINEM quantum detectors could achieve a single-shot measurement of the complete state of quantum light, even for non-reproducible emission events. Altogether, our work paves the way to novel kinds of photodetectors that utilize the ultrafast duration, high nonlinearity, and non-destructive nature of electron-light interactions., Comment: 5 figs, 20 pages + Supplementary Material

Free electrons provide a powerful tool to probe material properties at atomic-scale spatial resolution. Recent advances in ultrafast electron microscopy enable the manipulation of free electron wavefunctions using laser pulses. It would be of great importance if one could combine the spatial resolution of electron probes with the ability of laser pulses to probe coherent phenomena in quantum systems. To this end, we propose a novel technique that leverages free electrons that are coherently-shaped by laser pulses to measure quantum coherence in materials. Developing a quantum theory of electron-qubit interactions in materials, we show how the energy spectrum of laser-shaped electrons enables measuring the qubit Block-sphere state and decoherence time (T2). Finally, we present how such shaped electrons can detect and quantify superradiance from multiple qubits. Our scheme could be implemented in an ultrafast transmission electron microscope (UTEM), opening the way towards the full characterization of the state of quantum systems at atomic-scale resolution., Comment: 6 main text sections and 8 SM sections, 4 main text figures and 2 SM figures

High harmonic generation (HHG) is an extremely nonlinear effect, where a medium is driven by a strong laser field, generating coherent broadband radiation with photon energies ranging up to the X-ray and pulse durations reaching attosecond timescales. Conventional models of HHG treat the medium quantum mechanically, while the driving and emitted fields are treated classically. Such models are usually very successful, but inherently cannot capture the quantum-optical nature of the process. Despite prior works considering quantum HHG, it is still not known in what circumstances the spectral and statistical properties of the radiation considerably depart from the known phenomenology of HHG. Finding such regimes in HHG could lead to novel sources of attosecond light with intrinsically quantum statistics such as squeezing and entanglement. In this work, we present a fully quantum electrodynamical theory of extreme nonlinear optics with which we predict new quantum effects that alter both spectral and statistical properties of HHG. We predict the emission of shifted frequency combs resulting from transitions between perturbed states of the driven atom, and identify new spectral features that arise from the breakdown of the dipole approximation for the emission. Moreover, we find that each HHG emitted photon is a superposition of all frequencies in the spectrum - e.g., HHG creates single photon combs. We also describe how the HHG process changes in the single atom regime, and discuss how our various predictions can be tested experimentally. Our approach is also applicable to a wide range of nonlinear optical processes, paving the way toward novel quantum information techniques in the EUV and X-ray, and in ultrafast quantum optics., Comment: 19 pages, 4 figures

Electric and magnetic resonances of dielectric particles have recently uncovered a range of exciting applications in steering of light at the nanoscale. Breaking of particle inversion symmetry further modifies its electromagnetic response giving rise to bianisotropy known also as magneto-electric coupling. Recent studies suggest the crucial role of magneto-electric coupling in realization of photonic topological metamaterials. To further unmask this fundamental link, we design and test experimentally one-dimensional array composed of dielectric particles with overlapping electric and magnetic resonances and broken mirror symmetry. Flipping over half of the meta-atoms in the array, we observe the emergence of interface states providing photonic realization of the celebrated Jackiw-Rebbi model. We trace the origin of these states to the fact that local modification of particle bianisotropic response affects its effective coupling with the neighboring meta-atoms which provides a promising avenue to engineer topological states of light., Comment: 5 pages, 5 figures