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The advent of dispersion-engineered and highly nonlinear nanophotonics is expected to open up an all-optical path towards the strong-interaction regime of quantum optics by combining high transverse field confinement with ultra-short-pulse operation. Obtaining a full understanding of photon dynamics in such broadband devices, however, poses major challenges in the modeling and simulation of multimode non-Gaussian quantum physics, highlighting the need for sophisticated reduced models that facilitate efficient numerical study while providing useful physical insight. In this manuscript, we review our recent efforts in modeling broadband optical systems at varying levels of abstraction and generality, ranging from multimode extensions of quantum input-output theory for sync-pumped oscillators to the development of numerical methods based on a field-theoretic description of nonlinear waveguides. We expect our work not only to guide ongoing theoretical and experimental efforts towards next-generation quantum devices but also to uncover essential physics of broadband quantum photonics.

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A few decades ago, quantum optics stood out as a new domain of physics by exhibiting states of light with no classical equivalent. The first investigations concerned single photons, squeezed states, twin beams, and Einstein-Podolsky-Rosen states, which involve only one or two modes of the electromagnetic field. The study of the properties of quantum light then evolved in the direction of more and more complex and rich situations, involving many modes of the spatial, temporal, frequency, or polarization type. Actually, each mode of the electromagnetic field can be considered as an individual quantum degree of freedom. It is then possible, using the techniques of nonlinear optics, to couple different modes and thus build in a controlled way a quantum network [H. Jeff Kimble, Nature (London) 453, 1023 (2008)] in which the nodes are optical modes, and that is endowed with a strong multipartite entanglement. In addition, such networks can be easily reconfigurable and are subject only to weak decoherence. They indeed open many promising perspectives for optical communications and computation. Because of the linearity of Maxwell equations a linear superposition of two modes is another mode. This means that a “modal superposition principle” exists hand in hand with the regular quantum state superposition principle. The purpose of this review is to show the interest of considering these two aspects of multimode quantum light in a global way. Indeed, using different sets of modes allows one to consider the same quantum state under different perspectives: a given state can be entangled in one basis and factorized in another. It is shown that there exist some properties that are invariant over a change in the choice of the basis of modes. The method of finding the minimal set of modes that are needed to describe a given multimode quantum state is also presented. It is then shown how to produce, characterize, tailor, and use multimode quantum light while also considering the effect of loss and amplification on such light and the modal aspects of the two-photon coincidences. Switching to applications to quantum technologies, this review shows that it is possible to find not only quantum states that are likely to improve parameter estimation but also the optimal modes in which these states “live.” Finally, details on how to use such quantum modal networks for measurement-based quantum computation are presented.

We theoretically study the few- and many-body dynamics of photons in chiral waveguides. In particular, we examine pulse propagation through an ensemble of N two-level systems chirally coupled to a waveguide. We show that the system supports correlated multiphoton bound states, which have a well-defined photon number n and propagate through the system with a group delay scaling as 1/n^{2}. This has the interesting consequence that, during propagation, an incident coherent-state pulse breaks up into different bound-state components that can become spatially separated at the output in a sufficiently long system. For sufficiently many photons and sufficiently short systems, we show that linear combinations of n-body bound states recover the well-known phenomenon of mean-field solitons in self-induced transparency. Our work thus covers the entire spectrum from few-photon quantum propagation, to genuine quantum many-body (atom and photon) phenomena, and ultimately the quantum-to-classical transition. Finally, we demonstrate that the bound states can undergo elastic scattering with additional photons. Together, our results demonstrate that photon bound states are truly distinct physical objects emerging from the most elementary light-matter interaction between photons and two-level emitters. Our work opens the door to studying quantum many-body physics and soliton physics with photons in chiral waveguide QED.

Quantum evaporation may occur in a variety of systems such as superfluids, Bose-Einstein con-densates and gravitational black holes (Hawking radiation). However, to date all predictions are based on semiclassical models e.g. the Einstein equations and classical spacetime metric for a black hole and only the fluctuations are quantised. Here we use a fully quantised dynamical equation, the quantum Nonlinear Schrödinger equation, to study the evolution of quantum solitons. As a result of quantum fluctuations in the centre-of-mass position, the expectation value of the quantum soliton width increases and concomitantly evaporates through the emission of frequency-entangled photon pairs. The frequency of this emission decreases as the soliton evaporates due to the soliton spreading. In the final phase, the soliton mean field collapses irreversibly into a state with zero mean amplitude. These results may provide insight to quantum evaporation in other systems where a full quantum description in still to be developed and highlights that even classically stable systems may also be subject to quantum evaporation.

Spontaneous Parametric Down-Conversion (SPDC), also known as parametric fluorescence, parametric noise, parametric scattering and all various combinations of the abbreviation SPDC, is a non-linear optical process where a photon spontaneously splits into two other photons of lower energies. One would think that this article is about particle physics and yet it is not, as this process can occur fairly easily on a day to day basis in an optics laboratory. Nowadays, SPDC is at the heart of many quantum optics experiments for applications in quantum cryptography, quantum simulation, quantum metrology but also for testing fundamentals laws of physics in quantum mechanics. In this article, we will focus on the physics of this process and highlight a few important properties of SPDC. There will be two parts: a first theoretical one showing the particular quantum nature of SPDC, and the second part, more experimental and in particular focusing on applications of parametric down-conversion. This is clearly a non-exhaustive article about parametric down-conversion as there is a tremendous literature on the subject, but it gives the necessary first elements needed for a novice student or researcher to work on SPDC sources of light.

A powerful method to interface quantum light with matter is to propagate the light though an ensemble of atoms. Recently, a number of such interfaces have emerged, including Rydberg ensembles and atoms coupled to nanophotonic systems, in which strong nonlinear interactions between the propagating photons can be attained. An interesting and largely open problem is whether these systems can produce exotic many-body states of light, as the number of input photons is increased. To gain insight into this problem, it would be highly desirable to find approaches to numerically simulate light propagation in the many-body limit, a goal which has remained elusive thus far. Here, we describe an approach to this problem using a "spin model" that maps a quasi one-dimensional (1D) light propagation problem to the dynamics of an open 1D interacting spin system, where all of the photon correlations are obtained from those of the spins. The spin dynamics in turn can be numerically solved using the powerful toolbox of matrix product states. As a specific example, we apply this formalism to investigate vacuum induced transparency, wherein a pulse propagates with a photon number-dependent group velocity, thereby enabling separation of different photon number components at the output.

Optical solitons are waveforms that preserve their shape while travelling, relying on a balance of dispersion and nonlinearity. Data transmission schemes using solitons were heavily investigated in the 1980s promising to overcome the limitations imposed by dispersion of optical fibers. These approaches, however, were eventually abandoned in favour of WDM schemes, that are easier to implement and offer much better scalability to higher data rates. Here, we show that optical solitons may experience a comeback in optical terabit communications, this time not as a competitor, but as a key element of massively parallel WDM. Instead of encoding data on the soliton itself, we exploit continuously circulating solitons in Kerr-nonlinear microresonators to generate broadband optical frequency combs. In our experiments, we use two interleaved Kerr combs to transmit data on a total of 179 individual optical carriers that span the entire C and L bands. Using higher-order modulation formats (16QAM), net data rates exceeding 50 Tbit/s are attained, the highest value achieved with a chip-scale frequency comb source to date. Equally important, we demonstrate coherent detection of a WDM data stream by using a second Kerr soliton comb as a multi-wavelength local oscillator (LO) at the receiver. As a consequence, the microresonator soliton based sources exploit the scalability advantages for massively parallel optical communications at both the transmitter and the receiver side, contrasting commonly employed cw-lasers as optical LO for detection. Taken together the results prove the tremendous technological potential of photonic chip based microresonator soliton comb sources in high-speed communications. In combination with advanced spatial multiplexing schemes and highly integrated silicon photonic circuits, microresonator soliton combs may bring chip scale petabit/s transceiver systems into reach.

A critical question for the field of quantum computing in the near future is whether quantum devices without error correction can perform a well-defined computational task beyond the capabilities of state-of-the-art classical computers, achieving so-called quantum supremacy. We study the task of sampling from the output distributions of (pseudo-)random quantum circuits, a natural task for benchmarking quantum computers. Crucially, sampling this distribution classically requires a direct numerical simulation of the circuit, with computational cost exponential in the number of qubits. This requirement is typical of chaotic systems. We extend previous results in computational complexity to argue more formally that this sampling task must take exponential time in a classical computer. We study the convergence to the chaotic regime using extensive supercomputer simulations, modeling circuits with up to 42 qubits - the largest quantum circuits simulated to date for a computational task that approaches quantum supremacy. We argue that while chaotic states are extremely sensitive to errors, quantum supremacy can be achieved in the near-term with approximately fifty superconducting qubits. We introduce cross entropy as a useful benchmark of quantum circuits which approximates the circuit fidelity. We show that the cross entropy can be efficiently measured when circuit simulations are available. Beyond the classically tractable regime, the cross entropy can be extrapolated and compared with theoretical estimates of circuit fidelity to define a practical quantum supremacy test.

Bound states in the continuum (BICs) are waves that remain localized even though they coexist with a continuous spectrum of radiating waves that can carry energy away. Their very existence defies conventional wisdom. Although BICs were first proposed in quantum mechanics, they are a general wave phenomenon and have since been identified in electromagnetic waves, acoustic waves in air, water waves and elastic waves in solids. These states have been studied in a wide range of material systems, such as piezoelectric materials, dielectric photonic crystals, optical waveguides and fibres, quantum dots, graphene and topological insulators. In this Review, we describe recent developments in this field with an emphasis on the physical mechanisms that lead to BICs across seemingly very different materials and types of waves. We also discuss experimental realizations, existing applications and directions for future work.

Cavity-enhanced frequency comb spectroscopy for molecule detection in the mid-infrared powerfully combines high resolution, high sensitivity, and broad spectral coverage. However, this technique, and essentially all spectroscopic methods, is limited in application to relatively small, simple molecules. Here we integrate comb spectroscopy with continuous, cold samples of molecules produced via buffer gas cooling, thus enabling the study of significantly more complex molecules. We report simultaneous gains in resolution, sensitivity, and bandwidth and demonstrate this combined capability with the first rotationally resolved direct absorption spectra in the CH stretch region of several complex molecules. These include nitromethane (CH$_3$NO$_2$), a model system that presents challenging questions to the understanding of large amplitude vibrational motion, as well as several large organic molecules with fundamental spectroscopic and astrochemical relevance, including naphthalene (C$_{10}$H$_8$), adamantane (C$_{10}$H$_{16}$), and hexamethylenetetramine (C$_{6}$N$_4$H$_{12}$). This general spectroscopic tool has the potential to significantly impact the field of molecular spectroscopy, simultaneously improving efficiency, spectral resolution, and specificity by orders of magnitude. This realization could open up new molecular species and new kinetics for precise investigations, including the study of complex molecules, weakly bound clusters, and cold chemistry.

For the growing demand of frequency combs in mid-infrared (mid-IR), known as
the "molecular fingerprint" region of the spectrum [1], down conversion of
near-IR frequency combs through half- harmonic generation offers numerous
benefits including high conversion efficiency and intrinsic phase and frequency
locking to the near-IR pump [2]. Hence cascaded half-harmonic generation
promises a simple path towards extending the wavelength coverage of stable
frequency combs. Here, we report a two-octave down-conversion of a frequency
comb around 1 {\mu}m through cascaded half-harmonic generation with ~64%
efficiency in the first stage, and ~18% in the second stage. We obtain
broadband intrinsically-frequency-locked frequency combs with ~50-fs pulses at
~2 {\mu}m and ~110-fs pulses at ~4 {\mu}m. These results indicate the
effectiveness of half-harmonic generation as a universal tool for efficient
phase- and frequency-locked down-conversion, which can be beneficial for
numerous applications requiring long-wavelength coherent sources.

We present a concept of non-Gaussian measurement composed of a non-Gaussian
ancillary state, linear optics and adaptive heterodyne measurement, and on the
basis of this we also propose a simple scheme of implementing a quantum cubic
gate on a traveling light beam. In analysis of the cubic gate in the Heisenberg
representation, we find that nonlinearity of the gate is independent from
nonclassicality; the nonlinearity is generated solely by a classical nonlinear
adaptive control in a measurement-and-feedforward process while the
nonclassicality is attached by the non-Gaussian ancilla that suppresses excess
noise in the output. By exploiting the noise term as a figure of merit, we
consider the optimum non-Gaussian ancilla that can be prepared within reach of
current technologies and discuss performance of the gate. It is a crucial step
towards experimental implementation of the quantum cubic gate.

Temporal modes (TMs) of photonic quantum states provide promising bases for
quantum information science (QIS), because they intrinsically span a
high-dimensional Hilbert space and lend themselves to integration into existing
single-mode fiber communication networks. We show that the three main
requirements to construct a valid framework for QIS - the controlled generation
of resource states, the targeted and highly efficient manipulation of TMs and
their efficient detection, can be fulfilled with current technology. We suggest
implementations of diverse QIS applications based on those three building
blocks.

An analysis is conducted of the multipartite entanglement for Gaussian states
generated by the parametric downconversion of a femtosecond frequency comb.
Using a recently introduced method for constructing optimal entanglement
criteria, a family of tests is formulated for mode decompositions that extend
beyond the traditional bipartition analyses. A numerical optimization over this
family is performed to achieve maximal significance of entanglement
verification. For experimentally prepared 4, 6, and 10-mode states, full
entanglement is certified for all of the 14, 202, and 115974 possible
nontrivial partitions, respectively.

The robust generation of quantum states in the presence of decoherence is a primary challenge for explorations of quantum
mechanics at larger scales. Using the mechanical motion of a single trapped ion, we utilize reservoir engineering to generate
squeezed, coherent, and displaced-squeezed states as steady states in the presence of noise. We verify the created state by
generating two-state correlated spin-motion Rabi oscillations, resulting in high-contrast measurements. For both cooling and
measurement, we use spin-oscillator couplings that provide transitions between oscillator states in an engineered Fock state
basis. Our approach should facilitate studies of entanglement, quantum computation, and open-system quantum simulations in
a wide range of physical systems.

We derive frequency correlation and exit probability expressions for photons generated via spontaneous parametric downconversion (SPDC) in nonlinear waveguides that exhibit linear scattering loss. Such loss is included within a general Hamiltonian formalism by connecting waveguide modes to reservoir modes with a phenomenological coupling Hamiltonian, the parameters of which are later related to the usual loss coefficients. In the limit of a low probability of SPDC pair production, the presence of loss requires that we write the usual lossless generated pair state as a reduced density operator, and we find that this density operator is naturally composed of two photon, one photon, and zero photon contributions. The biphoton probability density, or joint spectral intensity (JSI), associated with the two-photon contribution is determined not only by a phase matching term, but also by a loss matching term. The relative sizes of the loss coefficients within this term lead to three qualitatively different regimes of SPDC JSIs. If either the pump or generated photon loss is much higher than the other, the side lobes of the phase matching squared sinc function are washed out. On the other hand, if pump and generated photon loss are appropriately balanced, the lossy JSI is identical to the lossless JSI. Finally, if the generated photon loss is frequency dependent, the shape of the JSI can be altered more severely, potentially leading to generated photons that are less frequency correlated though also produced less efficiently when compared to photons generated in low-loss waveguides.

We have developed a full multimode theory of a synchronously pumped
type-I optical parametric oscillator. We calculate the output quantum
fluctuations of the device and find that, in the degenerate case
(coincident signal and idler set of frequencies), significant squeezing
is obtained when one approaches threshold from below for a set of
well-defined “supermodes,” or frequency combs, consisting of
a coherent linear superposition of signal modes of different frequencies
which are resonant in the cavity.

Highly entangled quantum networks cluster states lie at the heart of recent
approaches to quantum computing \cite{Nielsen2006,Lloyd2012}. Yet, the current
approach for constructing optical quantum networks does so one node at a time
\cite{Furusawa2008,Furusawa2009,Peng2012}, which lacks scalability. Here we
demonstrate the \emph{single-step} fabrication of a multimode quantum network
from the parametric downconversion of femtosecond frequency combs. Ultrafast
pulse shaping \cite{weiner2000} is employed to characterize the comb's spectral
entanglement \cite{vanLoock2003}. Each of the 511 possible bipartitions among
ten spectral regions is shown to be entangled; furthermore, an eigenmode
decomposition reveals that eight independent quantum channels
\cite{Braunstein2005} (qumodes) are subsumed within the comb. This multicolor
entanglement imports the classical concept of wavelength-division multiplexing
(WDM) to the quantum domain by playing upon frequency entanglement as a means
to elevate quantum channel capacity. The quantum frequency comb is easily
addressable, robust with respect to decoherence, and scalable, which renders it
a unique tool for quantum information.

Frequency conversion (FC) and type-II parametric down-conversion (PDC)
processes serve as basic building blocks for the implementation of quantum
optical experiments: type-II PDC enables the efficient creation of quantum
states such as photon-number states and Einstein-Podolsky-Rosen-states
(EPR-states). FC gives rise to technologies enabling efficient atom-photon
coupling, ultrafast pulse gates and enhanced detection schemes. However,
despite their widespread deployment, their theoretical treatment remains
challenging. Especially the multi-photon components in the high-gain regime as
well as the explicit time-dependence of the involved Hamiltonians hamper an
efficient theoretical description of these nonlinear optical processes.
In this paper, we investigate these effects and put forward two models that
enable a full description of FC and type-II PDC in the high-gain regime. We
present a rigorous numerical model relying on the solution of coupled
integro-differential equations that covers the complete dynamics of the
process. As an alternative, we develop a simplified model that, at the expense
of neglecting time-ordering effects, enables an analytical solution.
While the simplified model approximates the correct solution with high
fidelity in a broad parameter range, sufficient for many experimental
situations, such as FC with low efficiency, entangled photon-pair generation
and the heralding of single photons from type-II PDC, our investigations reveal
that the rigorous model predicts a decreased performance for FC processes in
quantum pulse gate applications and an enhanced EPR-state generation rate
during type-II PDC, when EPR squeezing values above 12 dB are considered.

We study the dynamics of a one-dimensional Bose gas after a sudden change of
the interaction strength from zero to a finite value using the numerical
time-evolving block decimation (TEBD) algorithm. It is shown that despite the
integrability of the system, local quantities such as the two-particle
correlation $g^{(2)}(x,x)$ attain steady state values in a short characteristic
time inversely proportional to the Tonks parameter $\gamma$ and the square of
the density. The asymptotic values are very close to those of a finite
temperature grand canonical ensemble with a local temperature corresponding to
initial energy and density. Non-local density-density correlations on the other
hand approach a steady state on a much larger time scale determined by the
finite propagation velocity of oscillatory correlation waves.

The generation of pulsed squeezed light using an optical parametric oscillator (OPO) is discussed. This mode-locked optical parametric oscillator consists of a nonlinear crystal in a cavity which is resonant for both signal and idler waves and which is synchronously pumped by the second-harmonic of an acousto-optically mode-locked cw Nd: YAG laser. The fundamental wavelength of the pump laser provides local oscillator pulses for balanced homodyne detection of squeezed vacuum pulses emitted by the oscillator when operated below oscillation threshold. Photocurrent noise reduction to 30% below the classical shot-noise limit is observed, corresponding to squeezing of the field to a level approximately a factor of two below the mean square vacuum noise.

Quantum models for synchronously pumped type I optical parametric
oscillators (SPOPO) are presented. The study of the dynamics of SPOPOs,
which typically involves millions of coupled signal longitudinal modes, is
significantly simplified when one considers the “supermodes", which are
independent linear superpositions of all the signal modes diagonalizing the
parametric interaction. In terms of these supermodes the SPOPO dynamics
becomes that of about a hundred of independent, single mode degenerate OPOs,
each of them being a squeezer. One derives a general expression for the
squeezing spectrum measured in a balanced homodyne detection experiment,
valid for any temporal shape of the local oscillator. Realistic cases are
then studied using both analytical and numerical methods: the oscillation
threshold is derived, and the spectral and temporal shapes of the squeezed
supermodes are characterized.

Multimode nonclassical states of light are an essential resource in quantum computation with continuous variables, for example, in cluster state computation. We report in this Letter the first experimental evidence of a multimode nonclassical frequency comb in a femtosecond synchronously pumped optical parametric oscillator. In addition to a global reduction of its quantum intensity fluctuations, the system features quantum correlations between different parts of its frequency spectrum. This allows us to show that the frequency comb is composed of several uncorrelated eigenmodes having specific spectral shapes, two of them at least being squeezed, and to characterize their spectral shapes.

Ultra-short pulses propagating in nonlinear nanophotonic waveguides can simultaneously leverage both temporal and spatial field confinement, promising a route towards single-photon nonlinearities in an all-photonic platform. In this multimode quantum regime, however, faithful numerical simulations of pulse dynamics naively require a representation of the state in an exponentially large Hilbert space. Here, we employ a time-domain, matrix product state (MPS) representation to enable efficient simulations by exploiting the entanglement structure of the system. In order to extract physical insight from these simulations, we develop an algorithm to unravel the MPS quantum state into constituent temporal supermodes, enabling, e.g., access to the phase-space portraits of arbitrary pulse waveforms. As a demonstration, we perform exact numerical simulations of a Kerr soliton in the quantum regime. We observe the development of non-classical Wigner-function negativity in the solitonic mode as well as quantum corrections to the semiclassical dynamics of the pulse. A similar analysis of chi2 simultons reveals a unique entanglement structure between the fundamental and second harmonic. Our approach is also readily compatible with quantum trajectory theory, allowing full quantum treatment of propagation loss and decoherence. We expect this work to establish the MPS technique as part of a unified engineering framework for the emerging field of broadband quantum photonics.

We propose a deterministic, measurement-free implementation of a cubic phase gate for continuous- variable quantum information processing. In our scheme, the applications of displacement and squeezing operations allow us to engineer the effective evolution of the quantum state propagating through an optical Kerr nonlinearity. Under appropriate conditions, we show that the input state evolves according to a cubic phase Hamiltonian, and we find that the cubic phase gate error decreases inverse quartically with the amount of quadrature squeezing, even in the presence of linear loss. We also show how our scheme can be adapted to deterministically generate a nonclassical approximate cubic phase state with high fidelity using a ratio of native nonlinearity to linear loss of only 1e−4, indicating that our approach may be experimentally viable in the near term even on all-optical platforms, e.g., using quantum solitons in pulsed nonlinear nanophotonics.

Nonlinear frequency conversion plays a crucial role in advancing the functionality of next-generation optical systems. Portable metrology references and quantum networks will demand highly efficient second-order nonlinear devices, and the intense nonlinear interactions of nanophotonic waveguides can be leveraged to meet these requirements. Here we demonstrate second harmonic generation (SHG) in GaAs-on-insulator waveguides with unprecedented efficiency of 40 W-1 for a single-pass device. This result is achieved by minimizing the propagation loss and optimizing phase-matching. We investigate surface-state absorption and design the waveguide geometry for modal phase-matching with tolerance to fabrication variation. A 2.0 µm pump is converted to a 1.0 µm signal in a length of 2.9 mm with a wide signal bandwidth of 148 GHz. Tunable and efficient operation is demonstrated over a temperature range of 45 °C with a slope of 0.24 nm/°C. Wafer-bonding between GaAs and SiO2 is optimized to minimize waveguide loss, and the devices are fabricated on 76 mm wafers with high uniformity. We expect this device to enable fully integrated self-referenced frequency combs and high-rate entangled photon pair generation.

Quasi-phase-matched interactions in waveguides with quadratic nonlinearities enable highly efficient nonlinear frequency conversion. In this paper, we demonstrate the first generation of devices that combine the dispersion engineering available in nanophotonic waveguides with quasi-phase-matched nonlinear interactions available in periodically poled lithium niobate (PPLN). This combination enables quasi-static interactions of femtosecond pulses, reducing the pulse energy requirements by several orders of magnitude compared to conventional devices, from picojoules to femtojoules. We experimentally demonstrate two effects associated with second harmonic generation (SHG). First, we observe efficient quasi-phase-matched SHG with $ {\lt} {100}\;{\rm fJ}$<100fJ of pulse energy. Second, in the limit of strong phase-mismatch, we observe spectral broadening of both harmonics with as little as 2 pJ of pulse energy. These results lay a foundation for a new class of nonlinear devices, in which coengineering of dispersion with quasi-phase-matching enables efficient nonlinear optics at the femtojoule level.

Entanglement is the key resource for measurement-based quantum computing. It is stored in quantum states known as cluster states, which are prepared offline and enable quantum computing by means of purely local measurements. Universal quantum computing requires cluster states that are both large and possess (at least) a two-dimensional topology. Continuous-variable cluster states—based on bosonic modes rather than qubits—have previously been generated on a scale exceeding one million modes, but only in one dimension. Here, we report generation of a large-scale two-dimensional continuous-variable cluster state. Its structure consists of a 5- by 1240-site square lattice that was tailored to our highly scalable time-multiplexed experimental platform. It is compatible with Bosonic error-correcting codes that, with higher squeezing, enable fault-tolerant quantum computation.

Periodically poled lithium niobate (PPLN) waveguides are a powerful platform for efficient wavelength conversion. Conventional PPLN converters, however, typically require long device lengths and high pump powers due to the limited nonlinear interaction strength. Here we use a nanostructured PPLN waveguide to demonstrate an ultrahigh normalized efficiency of 2600%/W−cm2 for second-harmonic generation of 1.5 μm radiation, more than 20 times higher than that in state-of-the-art diffused waveguides. This is achieved by a combination of sub-wavelength optical confinement and high-fidelity periodic poling at a first-order poling period of 4 μm. Our highly integrated PPLN waveguides are promising for future chip-scale integration of classical and quantum photonic systems.

We show how to use the input-output formalism to compute the propagator for a low-dimensional quantum system coupled to an optical field under the rotating wave and Markovian approximation. The propagator for this coupled system is expressed in terms of the Green's functions of the low-dimensional system, and it is shown that these Green's functions can be computed entirely from the evolution of the low-dimensional system under an effective non-Hermitian Hamiltonian. Our formalism generalizes the previous works that have focused on time-independent Hamiltonians to systems with time-dependent Hamiltonians (e.g., coherently driven systems), making it a suitable computational tool for the analysis of a number of experimentally interesting quantum systems. We illustrate our formalism by applying it to analyze photon emission and scattering from the driven and undriven two-level system and the three-level lambda system.

In recent years, notions drawn from non-Hermitian physics and parity-time (PT) symmetry have attracted considerable attention. In particular, the realization that the interplay between gain and loss can lead to entirely new and unexpected features has initiated an intense research effort to explore non-Hermitian systems both theoretically and experimentally. Here we review recent progress in this emerging field, and provide an outlook to future directions and developments. © 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

The current shift in the quantum optics community towards large-size experiments -- with many modes and photons -- necessitates new classical simulation techniques that go beyond the usual phase space formulation of quantum mechanics. To address this pressing demand we formulate linear quantum optics in the language of tensor network states. As a toy model, we extensively analyze the quantum and classical correlations of time-bin interference in a single fiber loop. We then generalize our results to more complex time-bin quantum setups and identify different classes of architectures for high-complexity and low-overhead boson sampling experiments.

Rapid progress in photonics and nanotechnology brings many examples of resonant optical phenomena associated with the physics of Fano resonances, with applications in optical switching and sensing. For successful design of photonic devices, it is important to gain deep insight into different resonant phenomena and understand their connection. Here, we review a broad range of resonant electromagnetic effects by using two effective coupled oscillators, including the Fano resonance, electromagnetically induced transparency, Kerker and Borrmann effects, and parity–time symmetry breaking. We discuss how to introduce the Fano parameter for describing a transition between two seemingly different spectroscopic signatures associated with asymmetric Fano and symmetric Lorentzian shapes. We also review the recent results on Fano resonances in dielectric nanostructures and metasurfaces.

An experimental scheme is introduced to measure multiple parameters that are encoded in the phase quadrature of a light beam. Using a modal description and a spectrally-resolved homodyne detection, it is shown that all of the information is collected simultaneously, such that a single measurement allows extracting the value of multiple parameters \emph{post-facto}. With a femtosecond laser source, we apply this scheme to a measurement of the delay between two pulses with a shot-noise limited sensitivity as well as extracting the dispersion value of a dispersive medium.

Dual-comb spectroscopy is an emerging new spectroscopic tool that exploits the frequency resolution, frequency accuracy, broad bandwidth, and brightness of frequency combs for ultrahigh-resolution, high-sensitivity broadband spectroscopy. By using two coherent frequency combs, dual-comb spectroscopy allows a sample’s spectral response to be measured on a comb tooth-by-tooth basis rapidly and without the size constraints or instrument response limitations of conventional spectrometers. This review describes dual-comb spectroscopy and summarizes the current state of the art. As frequency comb technology progresses, dual-comb spectroscopy will continue to mature and could surpass conventional broadband spectroscopy for a wide range of laboratory and field applications.

Quantum superpositions of distinct coherent states in a single-mode harmonic
oscillator, known as "cat states", have been an elegant demonstration of
Schrodinger's famous cat paradox. Here, we realize a two-mode cat state of
electromagnetic fields in two microwave cavities bridged by a superconducting
artificial atom, which can also be viewed as an entangled pair of single-cavity
cat states. We present full quantum state tomography of this complex cat state
over a Hilbert space exceeding 100 dimensions via quantum non-demolition
measurements of the joint photon number parity. The ability to manipulate such
multi-cavity quantum states paves the way for logical operations between
redundantly encoded qubits for fault-tolerant quantum computation and
communication.

The realization of strong interactions between individual photons is a long-standing goal of both fundamental and technological significance. Scientists have known for over half a century that light fields can interact inside nonlinear optical media, but the nonlinearity of conventional materials is negligible at the light powers associated with individual photons. Nevertheless, remarkable advances in quantum optics have recently culminated in the demonstration of several methods for generating optical nonlinearities at the level of individual photons. Systems exhibiting strong photon-photon interactions enable a number of unique applications, including quantum-by-quantum control of light fields, single-photon switches and transistors, all-optical deterministic quantum logic, and the realization of strongly correlated states of light and matter.

We provide a systematic treatment of $N$-photon transport in a waveguide
coupled to a local system, using the input-output formalism. The main result of
the paper is a general connection between the $N$-photon S matrix and the Green
functions of the local system. We also show that the computation can be
significantly simplified, by exploiting the connectedness structure of both the
S matrix and the Green function, and by computing the Green function using an
effective Hamiltonian that involves only the degrees of freedom of the local
system. We illustrate our formalism by computing $N$-photon transport through a
cavity containing a medium with Kerr nonlinearity, with $N$ up to 3.

In this tutorial, we introduce the basic concepts and mathematical tools needed for phase-space description of a very common class of states, whose phase properties are described by Gaussian Wigner functions: the Gaussian states. In particular, we address their manipulation, evolution and characterization in view of their application to quantum information.

We demonstrate a planar, tunable superconducting qubit with energy relaxation times up to 44 μs. This is achieved by using a geometry designed to both minimize radiative loss and reduce coupling to materials-related defects. At these levels of coherence, we find a fine structure in the qubit energy lifetime as a function of frequency, indicating the presence of a sparse population of incoherent, weakly coupled two-level defects. We elucidate this defect physics by experimentally varying the geometry and by a model analysis. Our "Xmon" qubit combines facile fabrication, straightforward connectivity, fast control, and long coherence, opening a viable route to constructing a chip-based quantum computer.

The problem of expanding a density operator rho in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function P(alpha) of the P representation, the Wigner distribution W(alpha), and the function , where |alpha> is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function P(alpha) of the P representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function P(alpha) as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators a and a†. The Wigner distribution W(alpha) is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function , which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the P representation. A parametrized integral expansion of the density operator is introduced in which the weight function W(alpha,s) may be identified with the weight function P(alpha) of the P representation, with the Wigner distribution W(alpha), and with the function when the order parameter s assumes the values s=+1, 0, -1, respectively. The function W(alpha,s) is shown to be the expectation value of the ordered operator analog of the delta function defined in the preceding paper. This operator is in the trace class for Res

A linearized quantum theory of soliton squeezing and detection is presented. The linearization reduces the quantum problem to a classical one. The classical formulation provides physical insight. It is shown that a quantized soliton exhibits uncertainties in photon number and phase, position (time), and momentum (frequency). Detectors for the measurement of all four operators are discussed. The squeezing of the soliton in the fiber is analyzed. An optimal homodyne detector for detection of the squeezing is presented that suppresses the noise associated with the continuum and the uncertainties in position and momentum.

The Q-function of the quantum soliton in a fibre is derived in a new form suitable for the assessment of the possibilities for experimental observations of specific quantum-soliton effects. The characteristic effects of soliton evolution are associated with a cubic term in the nonlinear phase angle and appear at distances well beyond the attenuation length (long range). At the accessible distances, the nonlinear dynamics of a fundamental soliton follows essentially the single-mode dynamics (short and middle range). However, for the higher-order solitons, the form of the evolution parameter suggests strong deviations of soliton dynamics from the single-mode with a square of soliton number. The use of enhanced-nonlinearity fibres and higher-order solitons might make the experimental studies of the specific quantum-soliton features more viable.

The performance of superconducting qubits has improved by several orders of magnitude in the past decade. These circuits benefit from the robustness of superconductivity and the Josephson effect, and at present they have not encountered any hard physical limits. However, building an error-corrected information processor with many such qubits will require solving specific architecture problems that constitute a new field of research. For the first time, physicists will have to master quantum error correction to design and operate complex active systems that are dissipative in nature, yet remain coherent indefinitely. We offer a view on some directions for the field and speculate on its future.

A model is presented to describe spontaneous type-II parametric down-conversion pumped by a broadband source. This process differs from the familiar cw-pumped down-conversion in that a broader range of pump energies is available for down-conversion. The properties of the nonlinear crystal determine how these energies are distributed into the down-converted photons. Because the two photons are polarized along different crystal axes, they have different spectral characteristics and are no longer exactly anticorrelated. As the pump bandwidth is increased, this effect becomes more pronounced. A fourth-order interference experiment is proposed, illustrating some of the features of broadband pumped down-conversion.

Recently it was realized that linear optics and photodetectors with feedback can be used for theoretically efficient quantum information processing. The first of three steps toward efficient linear optics quantum computation is to design a simple postselected gate that implements a nonlinear phase shift on one mode. Here a computational strategy is given for finding postselected gates for bosonic qubits with helper photons. A more efficient conditional sign flip gate is obtained. What is the maximum efficiency for such gates? This question is posed and it is shown that the probability of success cannot be 1.

We show that in the k-photon down-conversion a fundamental limit on the energy transfer does exist providing the signal mode is initially in the vacuum state and the pump mode is excited. In particular, in the two-photon down-conversion less than 2/3 of the pump energy can be transferred from the pump to the signal mode. With the increase of the order of the nonlinear process under consideration the efficiency is even smaller. On the contrary, we show that no restriction on the efficiency of the energy transfer in the kth harmonic generation does exist, i.e., in this process the total energy from the initially excited mode can be transferred into the mode which initially was in the vacuum state. We study restrictions implied by the fundamental limit on the energy transfer in the k-photon down-conversion on nonclassical effects which can be observed in the signal mode in the process.

Using the truncated Wigner function approach and the
Bloembergen solutions for nondegenerate down-conversion we
calculate the conversion efficiency of spontaneous parametric
down-conversion. In addition we determine higher moments of the
pump and signal photon number. We find the upper bound for the
efficiency of energy transfer and give a physically intuitive
explanation for its existence. The depletion of the pump mode is
immediately characterized by a strong destruction of its
coherence.

We develop a quantum theory of propagation in dispersive nonlinear media from the foundations of a correctly quantized field theory. Quantum fluctuations are handled by coherent-state expansions of localized field states. A stochastic nonlinear Schrödinger equation in the field variables is obtained for media with an intensity-dependent refractive index. This predicts squeezing for a continuous-wave input, over a wide bandwidth with anomalous dispersion, and over a gradually reducing bandwidth with normal dispersion. The equation is easily modified to include thermal-noise sources as well. For solitons, fluctuations are reduced over the soliton bandwidth. This leads to quantum solitons that have quadrature fluctuations less than the level of vacuum fluctuations. The complementary quadrature has a correspondingly increased fluctuation level.