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Multiphoton interference is crucial to many photonic quantum technologies. In particular, interference forms the basis of optical quantum information processing platforms and can lead to significant computational advantages. It is therefore interesting to study the interference arising from various states of light in large interferometric networks. Here, we implement a quantum walk in a highly stable, low-loss, multiport interferometer with up to 24 ultrafast time bins. This time-bin interferometer comprises a sequence of birefringent crystals which produce pulses separated by 4.3\,ps, all along a single optical axis. Ultrafast Kerr gating in an optical fiber is employed to time-demultiplex the output from the quantum walk. We measure one-, two-, and three-photon interference arising from various input state combinations, including a heralded single-photon state, a thermal state, and an attenuated coherent state at one or more input ports. Our results demonstrate that ultrafast time bins are a promising platform to observe large-scale multiphoton interference.

Atmospheric turbulence is the main barrier to large-scale free-space quantum communication networks. Aberrations distort optical information carriers, thus limiting or preventing the possibility of establishing a secure link between two parties. For this reason, forecasting the turbulence strength within an optical channel is highly desirable, as it allows for knowing the optimal timing to establish a secure link in advance. Here, we train a Recurrent Neural Network, TAROCCO, to predict the turbulence strength within a free-space channel. The training is based on weather and turbulence data collected over 9 months for a 5.4 km intra-city free-space link across the City of Ottawa. The implications of accurate predictions from our network are demonstrated in a simulated high-dimensional Quantum Key Distribution protocol based on orbital angular momentum states of light across different turbulence regimes. TAROCCO will be crucial in validating a free-space channel to optimally route the key exchange for secure communications in real experimental scenarios.

Strongly-driven nonlinear optical processes such as spontaneous parametric down-conversion and spontaneous four-wave mixing can produce multiphoton nonclassical beams of light which have applications in quantum information processing and sensing. In contrast to the low-gain regime, new physical effects arise in a high-gain regime due to the interactions between the nonclassical light and the strong pump driving the nonlinear process. Here, we describe and experimentally observe a gain-induced group delay between the multiphoton pulses generated in a high-gain type-II spontaneous parametric down-conversion source. Since the group delay introduces distinguishability between the generated photons, it will be important to compensate for it when designing quantum interference devices in which strong optical nonlinearities are required., Comment: 18 pages, 9 figures (including supplemental material)

We propose a quantum information processing platform that utilizes the ultrafast time-bin encoding of photons. This approach offers a pathway to scalability by leveraging the inherent phase stability of collinear temporal interferometric networks at the femtosecond-to-picosecond timescale. The proposed architecture encodes information in ultrafast temporal bins processed using optically induced nonlinearities and birefringent materials while keeping photons in a single spatial mode. We demonstrate the potential for scalable photonic quantum information processing through two independent experiments that showcase the platform's programmability and scalability, respectively. The scheme's programmability is demonstrated in the first experiment, where we successfully program 362 different unitary transformations in up to 8 dimensions in a temporal circuit. In the second experiment, we show the scalability of ultrafast time-bin encoding by building a passive optical network, with increasing circuit depth, of up to 36 optical modes. In each experiment, fidelities exceed 97\%, while the interferometric phase remains passively stable for several days., Comment: 7 pages, 3 figures

The quantum walk (QW) has proven to be a valuable testbed for fundamental inquiries in quantum technology applications such as quantum simulation and quantum search algorithms. Many benefits have been found by exploring implementations of QWs in various physical systems, including photonic platforms. Here, we propose a novel platform to perform quantum walks using an ultrafast time-bin encoding (UTBE) scheme. This platform supports the scalability of quantum walks to a large number of steps while retaining a significant degree of programmability. More importantly, ultrafast time bins are encoded at the picosecond time scale, far away from mechanical fluctuations. This enables the scalability of our platform to many modes while preserving excellent interferometric phase stability over extremely long periods of time without requiring active phase stabilization. Our 18-step QW is shown to preserve interferometric phase stability over a period of 50 hours, with an overall walk fidelity maintained above $95\%$, Comment: 13 pages, 8 figures

In recent years, neural networks have been used to solve phase retrieval problems in imaging with superior accuracy and speed than traditional techniques, especially in the presence of noise. However, in the context of interferometric imaging, phase noise has been largely unaddressed by existing neural network architectures. Such noise arises naturally in an interferometer due to mechanical instabilities or atmospheric turbulence, limiting measurement acquisition times and posing a challenge in scenarios with limited light intensity, such as remote sensing. Here, we introduce a 3D-2D Phase Retrieval U-Net (PRUNe) that takes noisy and randomly phase-shifted interferograms as inputs, and outputs a single 2D phase image. A 3D downsampling convolutional encoder captures correlations within and between frames to produce a 2D latent space, which is upsampled by a 2D decoder into a phase image. We test our model against a state-of-the-art singular value decomposition algorithm and find PRUNe reconstructions consistently show more accurate and smooth reconstructions, with a x2.5 - 4 lower mean squared error at multiple signal-to-noise ratios for interferograms with low (< 1 photon/pixel) and high (~100 photons/pixel) signal intensity. Our model presents a faster and more accurate approach to perform phase retrieval in extremely low light intensity interferometry in presence of phase noise, and will find application in other multi-frame noisy imaging techniques.

In quantum key distribution (QKD), protocols are tailored to adopt desirable experimental attributes, including high key rates, operation in high noise levels, and practical security considerations. The round-robin differential phase shift protocol (RRDPS), falling in the family of differential phase shift protocols, was introduced to remove restrictions on the security analysis, such as the requirement to monitor signal disturbances, improving its practicality in implementations. While the RRDPS protocol requires the encoding of single photons in high-dimensional quantum states, at most, only one bit of secret key is distributed per sifted photon. However, another family of protocols, namely high-dimensional (HD) QKD, enlarges the encoding alphabet, allowing single photons to carry more than one bit of secret key each. The high-dimensional BB84 protocol exemplifies the potential benefits of such an encoding scheme, such as larger key rates and higher noise tolerance. Here, we devise an approach to extend the RRDPS QKD to an arbitrarily large encoding alphabet and explore the security consequences. We demonstrate our new framework with a proof-of-concept experiment and show that it can adapt to various experimental conditions by optimizing the protocol parameters. Our approach offers insight into bridging the gap between seemingly incompatible quantum communication schemes by leveraging the unique approaches to information encoding of both HD and DPS QKD., Comment: 14 pages, 6 figures

Time-bin qudits have emerged as a promising encoding platform in many quantum photonic applications. However, the requirement for efficient single-shot measurement of time-bin qudits instead of reconstructive detection has restricted their widespread use in experiments. Here, we propose an efficient method to measure arbitrary superposition states of time-bin qudits and confirm it up to dimension 4. This method is based on encoding time bins at the picosecond time scale, also known as ultrafast time bins. By doing so, we enable the use of robust and phase-stable single spatial mode temporal interferometers to measure time-bin qudit in different measurement bases., Comment: 9 pages, 8 figures

The increased phase sensitivity of N00N states has been used in many experiments, often involving photon paths or polarization. Here we experimentally combine the phase sensitivity of N00N states with the orbital angular momentum (OAM) of photons up to 100$\,\hslash$, to resolve rotations of a light field around its optical axis. The results show that both a higher photon number and larger OAM increase the resolution and achievable sensitivity. The presented method opens a viable path to unconditional angular super-sensitivity and accessible generation of N00N states between any transverse light fields., Comment: 12 pages, 8 figures

The photonic temporal degree of freedom is one of the most promising platforms for quantum communication over fiber networks and free-space channels. In particular, time-bin states of photons are robust to environmental disturbances, support high-rate communication, and can be used in high-dimensional schemes. However, the detection of photonic time-bin states remains a challenging task, particularly for the case of photons that are in a superposition of different time-bins. Here, we experimentally demonstrate the feasibility of picosecond time-bin states of light, known as ultrafast time-bins, for applications in quantum communications. With the ability to measure time-bin superpositions with excellent phase stability, we enable the use of temporal states in efficient quantum key distribution protocols such as the BB84 protocol., Comment: 8 pages, 6 figures

Structured light harnessing multiple degrees of freedom has become a powerful approach to use complex states of light in fundamental studies and applications. Here, we investigate the light field of an ultrafast laser beam with a wavelength-depended polarization state, a beam we term spectral vector beam. We demonstrate a simple technique to generate and tune such structured beams and demonstrate their spectroscopic capabilities. By only measuring the polarization state using fast photodetectors, it is possible to track pulse-to-pulse changes in the frequency spectrum caused by, e.g. narrowband transmission or absorption. In our experiments, we reach read-out rates of around 6 MHz, which is limited by our technical ability to modulate the spectrum and can in principle reach GHz read-out rates. In simulations we extend the spectral range to more than 1000 nm by using a supercontinuum light source, thereby paving the way to various applications requiring high-speed spectroscopic measurements., Comment: 11 pages, 12 figures

At the fundamental level, quantum communication is ultimately limited by noise. For instance, quantum signals cannot be amplified without the introduction of noise in the amplified states. Furthermore, photon loss reduces the signal-to-noise ratio, accentuating the effect of noise. Thus, most of the efforts in quantum communications have been directed towards overcoming noise to achieve longer communication distances, larger secret key rates, or to operate in noisier environmental conditions. Here, we propose and experimentally demonstrate a platform for quantum communication based on ultrafast optical techniques. In particular, our scheme enables the experimental realization of high-rates and quantum signal filtering approaching a single spectro-temporal mode, resulting in a dramatic reduction in channel noise. By experimentally realizing a 1-ps optically induced temporal gate, we show that ultrafast time filtering can result in an improvement in noise tolerance by a factor of up to 1200 compared to a 2-ns electronic filter enabling daytime quantum key distribution or quantum communication in bright fibers., Comment: 11 pages, 9 figures

Nearly 30 years ago, two-photon interference was observed, marking the beginning of a new quantum era. Indeed, two-photon interference has no classical analogue, giving it a distinct advantage for a range of applications. The peculiarities of quantum physics may now be used to our advantage to outperform classical computations, securely communicate information, simulate highly complex physical systems and increase the sensitivity of precise measurements. This separation from classical to quantum physics has motivated physicists to study two-particle interference for both fermionic and bosonic quantum objects. So far, two-particle interference has been observed with massive particles, among others, such as electrons and atoms, in addition to plasmons, demonstrating the extent of this effect to larger and more complex quantum systems. A wide array of novel applications to this quantum effect is to be expected in the future. This review will thus cover the progress and applications of two-photon (two-particle) interference over the last three decades., Comment: 21 pages, 10 figures

Underwater quantum communication has recently been explored using polarization and orbital angular momentum. Here, we show that spatially structured modes, e.g., a coherent superposition of beams carrying both polarization and orbital angular momentum, can also be used for underwater quantum cryptography. We also use the polarization degree of freedom for quantum communication in an underwater channel having various lengths, up to $30$ meters. The underwater channel proves to be a difficult environment for establishing quantum communication as underwater optical turbulence results in significant beam wandering and distortions. However, the errors associated to the turbulence do not result in error rates above the threshold for establishing a positive key in a quantum communication link with both the polarization and spatially structured photons. The impact of the underwater channel on the spatially structured modes is also investigated at different distances using polarization tomography., Comment: 7 pages, 3 figures, 3 tables

Whenever a quantum system undergoes a cycle governed by a slow change of parameters, it acquires a phase factor: the geometric phase. Its most common formulations are known as the Aharonov-Bohm, Pancharatnam and Berry phases, but both prior and later manifestations exist. Though traditionally attributed to the foundations of quantum mechanics, the geometric phase has been generalized and became increasingly influential in many areas from condensed-matter physics and optics to high energy and particle physics and from fluid mechanics to gravity and cosmology. Interestingly, the geometric phase also offers unique opportunities for quantum information and computation. In this Review we first introduce the Aharonov-Bohm effect as an important realization of the geometric phase. Then we discuss in detail the broader meaning, consequences and realizations of the geometric phase emphasizing the most important mathematical methods and experimental techniques used in the study of geometric phase, in particular those related to recent works in optics and condensed-matter physics., Comment: This is the author's accepted manuscript. The finalized version is available on: https://www.nature.com/articles/s42254-019-0071-1

We explore the complex design space of behaviour planning for autonomous driving. Design choices that successfully address one aspect of behaviour planning can critically constrain others. To aid the design process, in this work we decompose the design space with respect to important choices arising from the current state of the art approaches, and describe the resulting trade-offs. In doing this, we also identify interesting directions of future work., Comment: 8 pages, 6 figures

Unitary transformations are the fundamental building blocks of gates and operations in quantum information processing allowing the complete manipulation of quantum systems in a coherent manner. In the case of photons, optical elements that can perform unitary transformations are readily available only for some degrees of freedom, e.g. wave plates for polarisation. However for high-dimensional states encoded in the transverse spatial modes of light, performing arbitrary unitary transformations remains a challenging task for both theoretical proposals and actual implementations. Following the idea of multi-plane light conversion, we show that it is possible to perform a broad variety of unitary operations when the number of phase modulation planes is comparable to the number of modes. More importantly, we experimentally implement several high-dimensional quantum gates for up to 5-dimensional states encoded in the full-field mode structure of photons. In particular, we realise cyclic and quantum Fourier transformations, known as Pauli $\hat{X}$-gates and Hadamard $\hat{H}$-gates, respectively, with an average visibility of more than 90%. In addition, we demonstrate near-perfect "unitarity" by means of quantum process tomography unveiling a process purity of 99%. Lastly, we demonstrate the benefit of the two independent spatial degrees of freedom, i.e. azimuthal and radial, and implement a two-qubit controlled-NOT quantum operation on a single photon. Thus, our demonstrations open up new paths to implement high-dimensional quantum operations, which can be applied to various tasks in quantum communication, computation and sensing schemes.

We examine the propagation of optical beams possessing different polarization states and spatial modes through the Ottawa River in Canada. A Shack-Hartmann wavefront sensor is used to record the distorted beam's wavefront. The turbulence in the underwater channel is analysed, and associated Zernike coefficients are obtained in real-time. Finally, we explore the feasibility of transmitting polarization states as well as spatial modes through the underwater channel for applications in quantum cryptography., Comment: 5 pages, 4 figures. Comments welcome

High-dimensional encoding schemes have emerged as a novel way to perform quantum information tasks. For high dimensionality, temporal and transverse spatial modes of photons are the two paradigmatic degrees of freedom commonly used in such experiments. Nevertheless, general devices for multi-outcome measurements are still needed to take full advantage of the high-dimensional nature of encoding schemes. We propose a general full-field mode sorting scheme consisting only of up to two optimized phase elements based on evolutionary algorithms that allows for joint sorting of azimuthal and radial modes in a wide range of bases. We further study the performance of our scheme through simulations in the context of high-dimensional quantum cryptography, where high-fidelity measurement schemes are crucial.

Noise can be considered the natural enemy of quantum information. An often implied benefit of high-dimensional entanglement is its increased resilience to noise. However, manifesting this potential in an experimentally meaningful fashion is challenging and has never been done before. In infinite dimensional spaces, discretisation is inevitable and renders the effective dimension of quantum states a tunable parameter. Owing to advances in experimental techniques and theoretical tools, we demonstrate an increased resistance to noise by identifying two pathways to exploit high-dimensional entangled states. Our study is based on two separate experiments utilising canonical spatio-temporal properties of entangled photon pairs. Following these different pathways to noise resilience, we are able to certify entanglement in the photonic orbital-angular-momentum and energy-time degrees of freedom up to noise conditions corresponding to a noise fraction of 72 % and 92 % respectively. Our work paves the way towards practical quantum communication systems that are able to surpass current noise and distance limitations, while not compromising on potential device-independence., Comment: 12 pages main text, 7 pages supplementary information, 6 figures

Electromagnetic plane waves, solutions to Maxwell's equations, are said to be `transverse' in vacuum. Namely, the waves' oscillatory electric and magnetic fields are confined within a plane transverse to the waves' propagation direction. Under tight-focusing conditions however, the field can exhibit longitudinal electric or magnetic components, transverse spin angular momentum, or non-trivial topologies such as M\"obius strips. Here, we show that when a suitably spatially structured beam is tightly focused, a 3-dimensional polarization topology in the form of a ribbon with two full twists appears in the focal volume. We study experimentally the stability and dynamics of the observed polarization ribbon by exploring its topological structure for various radii upon focusing and for different propagation planes., Comment: 10 pages, and 4 figures

With the emergence of the field of quantum communications, the appropriate choice of photonic degrees of freedom used for encoding information is of paramount importance. Highly precise techniques for measuring the polarisation, frequency, and arrival time of a photon have been developed. However, the transverse spatial degree of freedom still lacks a measurement scheme that allows the reconstruction of its full transverse structure with a simple implementation and a high level of accuracy. Here we show a method to measure the azimuthal and radial modes of Laguerre-Gaussian beams with a greater than 99% accuracy, using a single phase screen. We compare our technique with previous commonly used methods and demonstrate the significant improvements it presents for quantum key distribution and state tomography of high-dimensional quantum states of light. Moreover, our technique can be readily extended to any arbitrary family of spatial modes, such as mutually unbiased bases, Hermite-Gauss, and Ince-Gauss. Our scheme will significantly enhance existing quantum and classical communication protocols that use the spatial structure of light, as well as enable fundamental experiments on spatial-mode entanglement to reach their full potential., Comment: 7 pages, 4 figures, 1 table

The characterization of quantum processes, e.g. communication channels, is an essential ingredient for establishing quantum information systems. For quantum key distribution protocols, the amount of overall noise in the channel determines the rate at which secret bits are distributed between authorized partners. In particular, tomographic protocols allow for the full reconstruction, and thus characterization, of the channel. Here, we perform quantum process tomography of high-dimensional quantum communication channels with dimensions ranging from 2 to 5. We can thus explicitly demonstrate the effect of an eavesdropper performing an optimal cloning attack or an intercept-resend attack during a quantum cryptographic protocol. Moreover, our study shows that quantum process tomography enables a more detailed understanding of the channel conditions compared to a coarse-grained measure, such as quantum bit error rates. This full characterization technique allows us to optimize the performance of quantum key distribution under asymmetric experimental conditions, which is particularly useful when considering high-dimensional encoding schemes., Comment: 13 pages, 6 figures

Quantum key distribution (QKD) offers the possibility for two individuals to communicate a securely encrypted message. From the time of its inception in 1984 by Bennett and Brassard, QKD has been the result of intense research. One technical challenge is the monitoring of signal disturbance in a QKD system to bound the information leakage towards an unwanted eavesdropper. Recently, the round-robin differential phase-shift (RRDPS) protocol, which encodes bits of information in a high-dimensional state space, was proposed to solve this exact problem. Since its introduction, many realizations of the RRDPS protocol were demonstrated using trains of coherent pulses. Here, we propose and experimentally demonstrate an implementation of the RRDPS protocol using the photonic orbital angular momentum degree of freedom. In particular, we show that Alice's generation stage and Bob's detection stage can each be reduced to a single phase element, greatly simplifying its implementation. Our scheme offers a practical demonstration of the RRDPS protocol which will suppress the need for monitoring signal disturbance in free-space channels., Comment: 5 pages, 3 figures

Quantum key distribution is on the verge of real world applications, where perfectly secure information can be distributed among multiple parties. Several quantum cryptographic protocols have been theoretically proposed and independently realized in different experimental conditions. Here, we develop an experimental platform based on high-dimensional orbital angular momentum states of single photons that enables implementation of multiple quantum key distribution protocols with a single experimental apparatus. Our versatile approach allows us to experimentally survey different classes of quantum key distribution techniques, such as the 1984 Bennett \& Brassard (BB84), tomographic protocols including the six-state and the Singapore protocol, and to investigate, for the first time, a recently introduced differential phase shift (Chau15) protocol using twisted photons. This enables us to experimentally compare the performance of these techniques and discuss their benefits and deficiencies in terms of noise tolerance in different dimensions., Comment: 13 pages, 4 figures, 1 table

Quantum communication has been successfully implemented in optical fibres and through free-space [1-3]. Fibre systems, though capable of fast key rates and low quantum bit error rates (QBERs), are impractical in communicating with destinations without an established fibre link [4]. Free-space quantum channels can overcome such limitations and reach long distances with the advent of satellite-to-ground links [5-8]. Shorter line-of-sight free-space links have also been realized for intra-city conditions [2, 9]. However, turbulence, resulting from local fluctuations in refractive index, becomes a major challenge by adding errors and losses [10]. Recently, an interest in investigating the possibility of underwater quantum channels has arisen, which could provide global secure communication channels among submersibles and boats [11-13]. Here, we investigate the effect of turbulence on an underwater quantum channel using twisted photons in outdoor conditions. We study the effect of turbulence on transmitted QBERs, and compare different QKD protocols in an underwater quantum channel showing the feasibility of high-dimensional encoding schemes. Our work may open the way for secure high-dimensional quantum communication between submersibles, and provides important input for potential submersibles-to-satellite quantum communication.

Quantum - or classically correlated - light can be employed in various ways to improve resolution and measurement sensitivity. In an "interaction-free" measurement, a single photon can be used to reveal the presence of an object placed within one arm of an interferometer without being absorbed by it. This method has previously been applied to imaging. With a technique known as "ghost imaging", entangled photon pairs are used for detecting an opaque object with significantly improved signal-to-noise ratio while preventing over-illumination. Here, we integrate these two methods to obtain a new imaging technique which we term "interaction-free ghost-imaging" that possesses the benefits of both techniques. While maintaining the image quality of conventional ghost-imaging, this new technique is also sensitive to phase and polarisation changes in the photons introduced by a structured object. Furthermore, thanks to the "interaction-free" nature of this new technique, it is possible to reduce the number of photons required to produce a clear image of the object (which could be otherwise damaged by the photons) making this technique superior for probing light-sensitive materials and biological tissues., Comment: 5 pages, 4 figures

The speed of light in vacuum, $c$, is a fundamental constant of nature. Photons belonging to a structured beam of finite transverse size, generated by a spatial light modulator, have been observed to travel with a group velocity, $v_g$, which is smaller than $c$ also when propagating in vacuum [1-3]. This is an effect that depends on the geometry of the beam. From quantum mechanical considerations, these photons must in any case propagate at the speed of light. This paradox can be resolved by taking into account a projection effect. What was measured in these experiments as group velocity was in fact its projection onto the beam propagation axis [4]. This depends on the divergence of the beams used in these experiments. We have found that for hypergeometric beams carrying orbital angular momentum (OAM), generated by sources with equal aperture [5-8], $v_g$ obeys an OAM/velocity relationship similar to that proposed by Majorana between spin and mass for bosonic and fermionic relativistic particles. This relationship, depending on the geometrical properties of the beam, can pave the way for an alternative estimation of OAM or to implement a time buffer in optical communications., Comment: 6 pages, 1 figure

Free electrons with a helical phase front, referred to as "twisted" electrons, possess an orbital angular momentum (OAM) and, hence, a quantized magnetic dipole moment along their propagation direction. This intrinsic magnetic moment can be used to probe material properties. Twisted electrons thus have numerous potential applications in materials science. Measuring this quantity often relies on a series of projective measurements that subsequently change the OAM carried by the electrons. In this Letter, we propose a nondestructive way of measuring an electron beam's OAM through the interaction of this associated magnetic dipole with a conductive loop. Such an interaction results in the generation of induced currents within the loop, which are found to be directly proportional to the electron's OAM value. Moreover, the electron experiences no OAM variations and only minimal energy losses upon the measurement, and, hence, the nondestructive nature of the proposed technique., Comment: 5 pages, 3 figures, and supplemental material that is comprised of text and 4 figures

Quantum key distribution (QKD) promises information-theoretically secure communication, and is already on the verge of commercialization. Thus far, different QKD protocols have been proposed theoretically and implemented experimentally [1, 2]. The next step will be to implement high-dimensional protocols in order to improve noise resistance and increase the data rate [3-7]. Hitherto, no experimental verification of high-dimensional QKD in the single-photon regime has been conducted outside of the laboratory. Here, we report the realization of such a single-photon QKD system in a turbulent free-space link of 0.3 km over the city of Ottawa, taking advantage of both the spin and orbital angular momentum photonic degrees of freedom. This combination of optical angular momenta allows us to create a 4-dimensional state [8]; wherein, using a high-dimensional BB84 protocol [3, 4], a quantum bit error rate of 11\% was attained with a corresponding secret key rate of 0.65 bits per sifted photon. While an error rate of 5\% with a secret key rate of 0.43 bits per sifted photon is achieved for the case of 2-dimensional structured photons. Even through moderate turbulence without active wavefront correction, it is possible to securely transmit information carried by structured photons, opening the way for intra-city high-dimensional quantum communications under realistic conditions., Comment: 5 pages, 3 figures, and Supplementary Information comprising of 4 sections and 2 figures

Recently, we showed experimentally that light carrying orbital angular momentum experiences a slight subluminality under free-space propagation [1]. We thank Saari [2] for pointing out an apparent discrepancy between our theoretical results and the well-known results for the simple case of Laguerre-Gauss modes. In this reply, we note that the resolution of this apparent discrepancy is the distinction between Laguerre-Gauss modes and Hypergeometric-Gauss modes, which were used in our experiment and in our theoretical analysis, which gives rise to different subluminal effects., Comment: 2 pages, and one figure

Quantum complementarity states that particles, e.g. electrons, can exhibit wave-like properties such as diffraction and interference upon propagation. \textit{Electron waves} defined by a helical wavefront are referred to as twisted electrons~\cite{uchida:10,verbeeck:10,mcmorran:11}. These electrons are also characterised by a quantized and unbounded magnetic dipole moment parallel to their propagation direction, as they possess a net charge of $-|e|$~\cite{bliokh:07}. When interacting with magnetic materials, the wavefunctions of twisted electrons are inherently modified~\cite{lloyd:12b,schattschneider:14a,asenjo:14}. Such variations therefore motivate the need to analyze electron wavefunctions, especially their wavefronts, in order to obtain information regarding the material's structure~\cite{harris:15}. Here, we propose, design, and demonstrate the performance of a device for measuring an electron's azimuthal wavefunction, i.e. its orbital angular momentum (OAM) content. Our device consists of nanoscale holograms designed to introduce astigmatism onto the electron wavefunctions and spatially separate its phase components. We sort pure and superposition OAM states of electrons ranging within OAM values of $-10$ and $10$. We employ the device to analyze the OAM spectrum of electrons having been affected by a micron-scale magnetic dipole, thus establishing that, with a midfield optical configuration, our sorter can be an instrument for nano-scale magnetic spectroscopy., Comment: 5 pages, 3 figures, 3 pages of supplementary materials

Converting spin angular momentum to orbital angular momentum has been shown to be a practical and efficient method for generating optical beams carrying orbital angular momentum and possessing a space-varying polarized field. Here, we present novel liquid crystal devices for tailoring the wavefront of optical beams through the Pancharatnam-Berry phase concept. We demonstrate the versatility of these devices by generating an extensive range of optical beams such as beams carrying $\pm200$ units of orbital angular momentum along with Bessel, Airy and Ince-Gauss beams. We characterize both the phase and the polarization properties of the generated beams, confirming our devices' performance., Comment: 9 pages, and 7 figures

Attempts at cloning a quantum system result in the introduction of imperfections in the state of the copies. This is a consequence of the no-cloning theorem, which is a fundamental law of quantum physics and the backbone of security for quantum communications. Although such perfect copies are prohibited, a quantum state may be copied with maximal accuracy via various optimal cloning schemes. Optimal quantum cloning, which lies at the border of the physical limit imposed by the no-signalling theorem and the Heisenberg uncertainty principle, has been experimentally realized for low dimensional photonic states. However, an increase in the dimensionality of quantum systems is greatly beneficial to quantum computation and communication protocols. Nonetheless, no experimental demonstration of optimal cloning machines has hitherto been shown for high-dimensional quantum systems. Here, we perform optimal cloning of high-dimensional photonic states by means of the symmetrization method. We show the universality of our technique by conducting cloning of numerous arbitrary input states, and fully characterize our cloning machine by performing quantum state tomography on \emph{cloned} photons. In addition, a cloning attack on a Bennett and Brassard (BB84) quantum key distribution protocol is experimentally demonstrated in order to reveal the robustness of high-dimensional states in quantum cryptography., Comment: 4 pages, 4 figures and a supplementary material

We study the nonlinear optical propagation of two different classes of space-varying polarized light beams -- radially symmetric vector beams and Poincar\'e beams with lemon and star topologies -- in a rubidium vapour cell. Unlike Laguerre-Gauss and other types of beams that experience modulational instabilities, we observe that their propagation is not marked by beam breakup while still exhibiting traits such as nonlinear confinement and self-focusing. Our results suggest that by tailoring the spatial structure of the polarization, the effects of nonlinear propagation can be effectively controlled. These findings provide a novel approach to transport high-power light beams in nonlinear media with controllable distortions to their spatial structure and polarization properties., Comment: 5 pages, and 4 figures

The possibility of focusing light to an ever tighter spot has important implications for many applications and fields of optics research, such as nano-optics and plasmonics, laser-scanning microscopy, optical data storage and many more. The size of lateral features of the field at the focus depends on several parameters, including the numerical aperture of the focusing system, but also the wavelength and polarization, phase and intensity distribution of the input beam. Here, we study the smallest achievable focal feature sizes of coherent superpositions of two co-propagating beams carrying opposite orbital angular momentum. We investigate the feature sizes for this class of beams not only in the scalar limit, but also use a fully vectorial treatment to discuss the case of tight focusing. Both our numerical simulations and our experimental results confirm that lateral feature sizes considerably smaller than those of a tightly focused Gaussian light beam can be observed. These findings may pave the way for improving the resolution of imaging systems or may find applications in nano-optics experiments., Comment: 6 pages, 3 figures

The Born rule, a foundational axiom used to deduce probabilities of events from wavefunctions, is indispensable in the everyday practice of quantum physics. It is also key in the quest to reconcile the ostensibly inconsistent laws of the quantum and classical realms, as it confers physical significance to reduced density matrices, the essential tools of decoherence theory. Following Bohr's Copenhagen interpretation, textbooks postulate the Born rule outright. However, recent attempts to derive it from other quantum principles have been successful, holding promise for simplifying and clarifying the quantum foundational bedrock. A major family of derivations is based on envariance, a recently discovered symmetry of entangled quantum states. Here, we identify and experimentally test three premises central to these envariance-based derivations, thus demonstrating, in the microworld, the symmetries from which the Born rule is derived. Further, we demonstrate envariance in a purely local quantum system, showing its independence from relativistic causality., Comment: 21 pages, 6 figures, 6 appendices - Submitted!

Einstein's theory of relativity establishes the speed of light in vacuum, c, as a fundamental constant. However, the speed of light pulses can be altered significantly in dispersive materials. While significant control can be exerted over the speed of light in such media, no experimental demonstration of altered light speeds has hitherto been achieved in vacuum for ``twisted'' optical beams. We show that ``twisted'' light pulses exhibit subluminal velocities in vacuum, being slowed by 0.1\% relative to c. This work does not challenge relativity theory, but experimentally supports a body of theoretical work on the counterintuitive vacuum group velocities of twisted pulses. These results are particularly important given recent interest in applications of twisted light to quantum information, communication and quantum key distribution., Comment: 16 pages, and 4 figures

Orbital angular momentum associated with the helical phase-front of optical beams provides an unbounded \qo{space} for both classical and quantum communications. Among the different approaches to generate and manipulate orbital angular momentum states of light, coupling between spin and orbital angular momentum allows a faster manipulation of orbital angular momentum states because it depends on manipulating the polarisation state of light, which is simpler and generally faster than manipulating conventional orbital angular momentum generators. In this work, we design and fabricate an ultra-thin spin-to-orbital angular momentum converter, based on plasmonic nano-antennas and operating in the visible wavelength range that is capable of converting spin to an arbitrary value of OAM $\ell$. The nano-antennas are arranged in an array with a well-defined geometry in the transverse plane of the beam, possessing a specific integer or half-integer topological charge $q$. When a circularly polarised light beam traverses this metasurface, the output beam polarisation switches handedness and the OAM changes in value by $\ell = \pm2q\hbar$ per photon. We experimentally demonstrate $\ell$ values ranging from $\pm 1$ to $\pm 25$ with conversion efficiencies of $8.6\pm0.4~\%$. Our ultra-thin devices are integratable and thus suitable for applications in quantum communications, quantum computations and nano-scale sensing., Comment: 4 pages, 3 figures - submitted!

We describe a novel approach for generating light beams that carry orbital angular momentum (OAM) by means of total internal reflection in an isotropic medium. A continuous space-varying cylindrically symmetric reflector, in the form of \textit{two glued hollow axicons}, is used to introduce a nonuniform rotation of polarisation into a linearly polarised input beam. This device acts as a full spin-to-orbital angular momentum convertor. It functions by switching the helicity of the incoming beam's polarisation, and by conservation of total angular momentum thereby generates a well-defined value of OAM. Our device is broadband, since the phase shift due to total internal reflection is nearly independent of wavelength. We verify the broad-band behaviour by measuring the conversion efficiency of the device for three different wavelengths corresponding to the RGB colours, red, green and blue. An average conversion efficiency of $95\%$ for these three different wavelengths is observed. %, which confirms its wavelength independence. This device may be used to generate a white vector vortex beam, as we show, and may find applications in imaging from micro- to astronomical systems where a white vortex beam is needed., Comment: 15 pages, 6 figures and 1 table