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We study the emergence of non-Gaussian quantum features in pulsed squeezed light generation with a mesoscopic number (i.e., dozens to hundreds) of pump photons. Due to the strong optical nonlinearities necessarily involved in this regime, squeezing occurs alongside significant pump depletion, compromising the predictions made by conventional semiclassical models for squeezing. Furthermore, nonlinear interactions among multiple frequency modes render the system dynamics exponentially intractable in na\"ive quantum models, requiring a more sophisticated modeling framework. To this end, we construct a nonlinear Gaussian approximation to the squeezing dynamics, defining a "Gaussian interaction frame" (GIF) in which non-Gaussian quantum dynamics can be isolated and concisely described using a few dominant (i.e., principal) supermodes. Numerical simulations of our model reveal non-Gaussian distortions of squeezing in the mesoscopic regime, largely associated with signal-pump entanglement. We argue that the state of the art in nonlinear nanophotonics is quickly approaching this regime, providing an all-optical platform for experimental studies of the semiclassical-to-quantum transition in a rich paradigm of coherent, multimode nonlinear dynamics. Mesoscopic pulsed squeezing thus provides an intriguing case study of the rapid rise in dynamic complexity associated with semiclassical-to-quantum crossover, which we view as a correlate of the emergence of new information-processing capacities in the quantum regime.

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Gaussian states have played an important role in the physics of continuous-variable quantum systems. They are appealing for the experimental ease with which they can be produced, and for their compact and elegant mathematical description. Nevertheless, many proposed quantum technologies require us to go beyond the realm of Gaussian states and introduce non-Gaussian elements. In this Tutorial, we provide a roadmap for the physics of non-Gaussian quantum states. We introduce the phase-space representations as a framework to describe the different properties of quantum states in continuous-variable systems. We then use this framework in various ways to explore the structure of the state space. We explain how non-Gaussian states can be characterized not only through the negative values of their Wigner function, but also via other properties such as quantum non-Gaussianity and the related stellar rank. For multimode systems, we are naturally confronted with the question of how non-Gaussian properties behave with respect to quantum correlations. To answer this question, we first show how non-Gaussian states can be created by performing measurements on a subset of modes in a Gaussian state. Then, we highlight that these measured modes must be correlated via specific quantum correlations to the remainder of the system to create quantum non-Gaussian or Wigner-negative states. On the other hand, non-Gaussian operations are also shown to enhance or even create quantum correlations. Finally, we demonstrate that Wigner negativity is a requirement to violate Bell inequalities and to achieve a quantum computational advantage. At the end of the Tutorial, we also provide an overview of several experimental realizations of non-Gaussian quantum states in quantum optics and beyond.

This article reviews recent progress in quasi-phasematched X⁽²⁾nonlinear nanophotonics, with a particular focus on dispersion-engineered nonlinear interactions. Throughout this article, we establish design rules for the bandwidth and interaction lengths of various nonlinear processes, and provide examples for how these processes can be engineered in nanophotonic devices. In particular, we apply these rules towards the design of sources of non-classical light and show that dispersion-engineered devices can outperform their conventional counterparts. Examples include ultra-broadband optical parametric amplification as a resource for measurement-based quantum computation, dispersion-engineered spontaneous parametric downconversion as a source of separable biphotons, and synchronously pumped nonlinear resonators as a potential route towards single-photon nonlinearities.

Growing interest in quantum computing for practical applications has led to a surge in the availability of programmable machines for executing quantum algorithms1,2. Present-day photonic quantum computers3–7 have been limited either to non-deterministic operation, low photon numbers and rates, or fixed random gate sequences. Here we introduce a full-stack hardware−software system for executing many-photon quantum circuit operations using integrated nanophotonics: a programmable chip, operating at room temperature and interfaced with a fully automated control system. The system enables remote users to execute quantum algorithms that require up to eight modes of strongly squeezed vacuum initialized as two-mode squeezed states in single temporal modes, a fully general and programmable four-mode interferometer, and photon number-resolving readout on all outputs. Detection of multi-photon events with photon numbers and rates exceeding any previous programmable quantum optical demonstration is made possible by strong squeezing and high sampling rates. We verify the non-classicality of the device output, and use the platform to carry out proof-of-principle demonstrations of three quantum algorithms: Gaussian boson sampling, molecular vibronic spectra and graph similarity⁸. These demonstrations validate the platform as a launchpad for scaling photonic technologies for quantum information processing.

We review the concepts of temporal modes (TMs) in quantum optics, highlighting Roy Glauber’s crucial and historic contributions to their development, and their growing importance in quantum information science. TMs are orthogonal sets of wave packets that can be used to represent a multimode light field. They are temporal counterparts to transverse spatial modes of light and play analogous roles—decomposing multimode light into the most natural basis for isolating statistically independent degrees of freedom. We discuss how TMs were developed to describe compactly various processes: superfluorescence, stimulated Raman scattering, spontaneous parametric down conversion, and spontaneous four-wave mixing. TMs can be manipulated, converted, demultiplexed, and detected using nonlinear optical processes such as three-wave mixing and quantum optical memories. As such, they play an increasingly important role in constructing quantum information networks.

We present a new method for the spectral characterization of pulsed twin-beam sources in the high-gain regime, using cascaded stimulated emission. We show an implementation of this method for a periodically poled potassium titanyl phosphate spontaneous parametric down-conversion source generating up to 60 photon pairs per pulse, and we demonstrate excellent agreement between our experiments and our theory. This work enables the complete and accurate experimental characterization of high-gain effects in parametric down-conversion, including self- and cross-phase modulation. Moreover, our theory allows the exploration of designs with the goal of improving the specifications of twin-beam sources for application in quantum information, computation, sampling, and metrology.

We consider pulsed-pump spontaneous parametric downconversion (SPDC) as well as pulsed single- and dual-pump spontaneous four-wave mixing processes in waveguides within a unified Hamiltonian theoretical framework. Working with linear operator equations in k -space, our approach allows inclusion of linear losses, self- and cross-phase modulation, and dispersion to any order. We describe state evolution in terms of second-order moments, for which we develop explicit expressions. We use our approach to calculate the joint spectral amplitude of degenerate squeezing using SPDC analytically in the perturbative limit, benchmark our theory against well-known results in the limit of negligible group velocity dispersion, and study the suitability of recently proposed sources for quantum sampling experiments.

We introduce a general method for building neural networks on quantum computers. The quantum neural network is a variational quantum circuit built in the continuous-variable (CV) architecture, which encodes quantum information in continuous degrees of freedom such as the amplitudes of the electromagnetic field. This circuit contains a layered structure of continuously parameterized gates which is universal for CV quantum computation. Affine transformations and nonlinear activation functions, two key elements in neural networks, are enacted in the quantum network using Gaussian and non-Gaussian gates, respectively. The non-Gaussian gates provide both the nonlinearity and the universality of the model. Due to the structure of the CV model, the CV quantum neural network can encode highly nonlinear transformations while remaining completely unitary. We show how a classical network can be embedded into the quantum formalism and propose quantum versions of various specialized models such as convolutional, recurrent, and residual networks. Finally, we present numerous modeling experiments built with the strawberry fields software library. These experiments, including a classifier for fraud detection, a network which generates tetris images, and a hybrid classical-quantum autoencoder, demonstrate the capability and adaptability of CV quantum neural networks.

We develop a resource theory for continuous-variable systems grounded on operations routinely available within current quantum technologies. In particular, the set of free operations is convex and includes quadratic transformations and conditional coarse-grained measurements. The present theory lends itself to quantify both quantum non-Gaussianity and Wigner negativity as resources, depending on the choice of the free-state set --- i.e., the convex hull of Gaussian states or the states with positive Wigner function, respectively. After showing that the theory admits no maximally resourceful state, we define a computable resource monotone --- the CV-mana. We use the latter to assess the resource content of experimentally relevant states --- e.g., photon-added, photon-subtracted, cubic-phase, and cat states --- and to find optimal working points of some resource concentration protocols. We envisage applications of this framework to sub-universal and universal quantum information processing over continuous variables.

Non-Gaussian states and operations are crucial for various continuous-variable quantum information processing tasks. To quantitatively understand non-Gaussianity beyond states, we establish a resource theory for non-Gaussian operations. In our framework, we consider Gaussian operations as free operations, and non-Gaussian operations as resources. We define entanglement-assisted non-Gaussianity generating power and show that it is a monotone that is non-increasing under the set of free super-operations, i.e., concatenation and tensoring with Gaussian channels. For conditional unitary maps, this monotone can be analytically calculated. As examples, we show that the non-Gaussianity of ideal photon-number subtraction and photon-number addition equal the non-Gaussianity of the single-photon Fock state. Based on our non-Gaussianity monotone, we divide non-Gaussian operations into two classes: (1) the finite non-Gaussianity class, e.g., photon-number subtraction, photon-number addition and all Gaussian-dilatable non-Gaussian channels; and (2) the diverging non-Gaussianity class, e.g., the binary phase-shift channel and the Kerr nonlinearity. This classification also implies that not all non-Gaussian channels are exactly Gaussian-dilatable, disproving a recent conjecture in [Phys. Rev. A 95, 062309 (2017)]. Our resource theory enables a quantitative characterization and a first classification of non-Gaussian operations, paving the way towards the full understanding of non-Gaussianity.

The time-frequency degree of freedom is a powerful resource for implementing high-dimensional quantum information processing. In particular, field-orthogonal pulsed temporal modes offer a flexible framework compatible with both long-distance fibre networks and integrated waveguide devices. In order for this architecture to be fully utilised, techniques to reliably generate diverse quantum states of light and accurately measure complex temporal waveforms must be developed. To this end, nonlinear processes mediated by spectrally shaped pump pulses in group-velocity engineered waveguides and crystals provide a capable toolbox. In this review, we examine how tailoring the phasematching conditions of parametric downconversion and sum-frequency generation allows for highly pure single-photon generation, flexible temporal-mode entanglement, and accurate measurement of time-frequency photon states. We provide an overview of experimental progress towards these goals, and summarise challenges that remain in the field.

We demonstrate an ultralow loss monolithic integrated lithium niobate photonic platform consisting of dry-etched subwavelength waveguides. We show microring resonators with a quality factor of 10$^7$ and waveguides with propagation loss as low as 2.7 dB/m.

We propose an experimentally-feasible method for enhancing the atom-field coupling as well as the ratio between this coupling and dissipation (i.e., cooperativity) of two atoms in an optical cavity. Our method also enables the generation of steady-state nearly-maximal quantum entanglement. Our approach exploits optical parametric amplification to exponentially enhance the atom-cavity interaction and, hence, the cooperativity of the system, with the squeezing-induced fluctuation noise being completely eliminated. Thus, an effective cooperativity much larger than $100$ can be achieved even for modest values of a squeezing parameter. We demonstrate that the entanglement infidelity (which quantifies the deviation of the generated state from a maximally-entangled state) is exponentially smaller than the lower bound on the infidelities obtained in other dissipative entanglement preparations without applying squeezing. Thus, this infidelity can be arbitrarily small. Our generic method for enhancing atom-cavity cooperativities can be implemented in a wide range of physical systems, and provides diverse applications for quantum information processing based on entanglement.

We present an open source computational framework geared towards the efficient numerical investigation of open quantum systems written in the Julia programming language. Built exclusively in Julia and based on standard quantum optics notation, the toolbox offers great speed, comparable to low-level statically typed languages, without compromising on the accessibility and code readability found in dynamic languages. After introducing the framework, we highlight its novel features and showcase implementations of generic quantum models. Finally, we compare its usability and performance to two well-established and widely used numerical quantum libraries.

Weakly nonlinear degrees of freedom in dissipative quantum systems tend to localize near manifolds of quasi-classical states. We present a family of analytical and computational methods for deriving optimal unitary model transformations based on representations of finite dimensional Lie groups. The transformations are optimal in that they minimize the quantum relative entropy distance between a given state and the quasi-classical manifold. This naturally splits the description of quantum states into quasi-classical coordinates that specify the nearest quasi-classical state and a transformed quantum state that can be represented in fewer basis levels. We derive coupled equations of motion for the coordinates and the transformed state and demonstrate how this can be exploited for efficient numerical simulation. Our optimization objective naturally quantifies the non-classicality of states occurring in some given open system dynamics. This allows us to compare the intrinsic complexity of different open quantum systems.

Squeezed light generation has come of age. Significant advances on squeezed
light generation has been made over the last 30 years - from the initial,
conceptual experiment in 1985 till todays top-tuned, application-oriented
setups. Here we review the main experimental platforms for generating
quadrature squeezed light that has been investigated the last 30 years.

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.

The control of strongly interacting many-body systems enables the creation of tailored quantum matter with complex properties. Atomic ensembles that are optically driven to a Rydberg state provide many examples for this atom-atom entanglement(1,2), many-body Rabi oscillations(3), strong photon-photon interaction(4) and spatial pair correlations(5). In its most basic form Rydberg quantum matter consists of an isolated ensemble of strongly interacting atoms spatially confined to the blockade volume-a superatom. Here we demonstrate the controlled creation and characterization of an isolated mesoscopic superatom by means of accurate density engineering and excitation to Rydberg p-states. Its variable size allows the investigation of the transition from effective two-level physics to many-body phenomena. By monitoring continuous laser-induced ionization we observe a strongly anti-bunched ion emission under blockade conditions and extremely bunched ion emission under off-resonant excitation. Our measurements provide insights into both excitation statistics and dynamics. We anticipate applications in quantum optics and quantum information as well as many-body physics experiments.

In this work, we develop several new simulation algorithms for one-dimensional (1D) many-body quantum mechanical systems combining the Matrix Product State variational ansatz (vMPS) with Taylor, Padé and Arnoldi approximations to the evolution operator. By comparing with previous techniques based on vMPS and Density Matrix Renormalization Group (DMRG), we demonstrate that the Arnoldi method is the best one, reaching extremely good accuracy with moderate resources. Finally, we apply this algorithm to studying how correlations are transferred from the atomic to the molecular cloud when crossing a Feschbach resonance with two-species hard-core bosons in a 1D optical lattice.

Optical $\chi^{(2)}$ non-linearity can be used for parametric amplification
and producing down-converted entangled photon pairs that have broad
applications. It is known that weak non-linear media exhibit dispersion and
produce a frequency response. It is therefore of interest to know how spectral
effects of a strong $\chi^{(2)}$ crystal affect the performance. Here we model
the spectral effects of the dispersion of a strong $\chi^{(2)}$ crystal and
illustrate how this affects its ability to perform Bell measurements and
influence the performance of a quantum gates that employ such a Bell
measurement. We show that a Dyson series expansion of the unitary operator is
necessary in general, leading to unwanted spectral entanglement. We identify a
limiting situation employing periodic poling, in which a Taylor series
expansion is a good approximation and this entanglement can be removed.

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.

Photonic integrated circuits are a key component of future telecommunication networks, where demands for greater bandwidth, network flexibility, and low energy consumption and cost must all be met. The quest for all-optical components has naturally targeted materials with extremely large nonlinearity, including chalcogenide glasses and semiconductors, such as silicon and AlGaAs (ref. 4). However, issues such as immature fabrication technology for chalcogenide glass and high linear and nonlinear losses for semiconductors motivate the search for other materials. Here we present the first demonstration of nonlinear optics in integrated silica-based glass waveguides using continuous-wave light. We demonstrate four-wave mixing, with low (5 mW) continuous-wave pump power at lambda = 1,550 nm, in high-index, doped silica glass ring resonators. The low loss, design flexibility and manufacturability of our device are important attributes for low-cost, high-performance, nonlinear all-optical photonic integrated circuits.

A measure of non-classicality of quantum states based on the volume of the negative part of the Wigner function is proposed. We analyse this quantity for Fock states, squeezed displaced Fock states and cat-like states defined as coherent superposition of two Gaussian wavepackets.

Wigner functions and Weyl transforms of operators offer a formulation of quantum mechanics that is equivalent to the standard approach given by the Schrodinger equation. We give a short introduction and emphasize features that give insight into the nature of quantum mechanics and its relation to classical physics. A careful discussion of the classical limit and its difficulties is also given. The discussion is self-contained and includes complete derivations of the results presented. (c) 2008 American Association of Physics Teachers.

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.

We present dynamical calculations for the quantum parametric oscillator using both number-state and coherent-state bases. The coherent-state methods use the positive-P representation, which has a nonclassical phase space-an essential requirement in obtaining an exact stochastic representation of this nonlinear problem. This also provides a way to directly simulate quantum tunneling between the two above-threshold stable states of the oscillator. The coherent-state methods provide both analytic results at large photon numbers, and numerical results for any photon number, while our number-state calculations are restricted to numerical results in the low-photon-number regime. The number-state and coherent-state methods give precise agreement within the accuracy of the numerical calculations. We also compare our results with methods based on a truncated Wigner representation equivalent to stochastic electrodynamics, and find that these are unable to correctly predict the tunneling rate given by the other methods. An interesting feature of the results is the much faster tunneling predicted by the exact quantum-theory methods compared with earlier semiclassical calculations using an approximate potential barrier. This is similar to the faster tunneling found when comparing quantum penetration of a barrier to classical thermal activation. The quantum parametric oscillator, which has an exact steady-state solution, therefore provides a useful and accessible system in which nonlinear quantum effects can be studied far from thermal equilibrium.

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.

The science of quantum information has arisen over the last two decades
centered on the manipulation of individual quanta of information, known as
quantum bits or qubits. Quantum computers, quantum cryptography and quantum
teleportation are among the most celebrated ideas that have emerged from this
new field. It was realized later on that using continuous-variable quantum
information carriers, instead of qubits, constitutes an extremely powerful
alternative approach to quantum information processing. This review focuses on
continuous-variable quantum information processes that rely on any combination
of Gaussian states, Gaussian operations, and Gaussian measurements.
Interestingly, such a restriction to the Gaussian realm comes with various
benefits, since on the theoretical side, simple analytical tools are available
and, on the experimental side, optical components effecting Gaussian processes
are readily available in the laboratory. Yet, Gaussian quantum information
processing opens the way to a wide variety of tasks and applications, including
quantum communication, quantum cryptography, quantum computation, quantum
teleportation, and quantum state and channel discrimination. This review
reports on the state of the art in this field, ranging from the basic
theoretical tools and landmark experimental realizations to the most recent
successful developments.

We report the realization of a bright ultrafast type II parametric down-conversion source of twin beams free of any spatiotemporal correlations in a periodically poled KTiOPO4 (PP-KTP) waveguide. From a robust, single-pass setup it emits pulsed two-mode squeezed vacuum states: photon-number entangled pairs of single-mode pulses or, in terms of continuous variables quantum optics, pulsed Einstein-Podolsky-Rosen states in the telecom wavelength regime. We verify the single-mode character of our source by measuring Glauber correlation functions g(2) and demonstrate with a pump energy as low as 75 pJ per pump pulse a mean photon number of 2.5.

In classical estimation theory, the central limit theorem implies that the
statistical error in a measurement outcome can be reduced by an amount
proportional to n^(-1/2) by repeating the measures n times and then averaging.
Using quantum effects, such as entanglement, it is often possible to do better,
decreasing the error by an amount proportional to 1/n. Quantum metrology is the
study of those quantum techniques that allow one to gain advantages over purely
classical approaches. In this review, we analyze some of the most promising
recent developments in this research field. Specifically, we deal with the
developments of the theory and point out some of the new experiments. Then we
look at one of the main new trends of the field, the analysis of how the theory
must take into account the presence of noise and experimental imperfections.

We present a theoretical technique for solving the quantum transport problem
of a few photons through a one-dimensional, strongly nonlinear waveguide. We
specifically consider the situation where the evolution of the optical field is
governed by the quantum nonlinear Schr\"odinger equation (NLSE). Although this
kind of nonlinearity is quite general, we focus on a realistic implementation
involving cold atoms loaded in a hollow-core optical fiber, where the atomic
system provides a tunable nonlinearity that can be large even at a
single-photon level. In particular, we show that when the interaction between
photons is effectively repulsive, the transmission of multi-photon components
of the field is suppressed. This leads to anti-bunching of the transmitted
light and indicates that the system acts as a single-photon switch. On the
other hand, in the case of attractive interaction, the system can exhibit
either anti-bunching or bunching, which is in stark contrast to semiclassical
calculations. We show that the bunching behavior is related to the resonant
excitation of bound states of photons inside the system.

Using ultrashort laser pulses, we have observed quadrature squeezing and parametric gain in quasi-phase-matched KTiOPO(4) waveguides. Using a local oscillator pulse that is 2.5 times shorter than the squeezed pulse, we observed noise reduction of 12 +/- 1% below the shot-noise level. Parametric amplification and deamplification of a coherent seed pulse have also been observed, with no indication of gain-induced diffraction.

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 provide an efficient method for the calculation of high-gain, twin-beam generation in waveguides derived from a canonical treatment of Maxwell's equations. Equations of motion are derived that naturally accommodate photon generation via spontaneous parametric down-conversion (SPDC) or spontaneous four-wave mixing and, also, include the effects both of self-phase modulation of the pump and of cross-phase modulation of the twin beams by the pump. The equations we solve involve fields that evolve in space and are labeled by a frequency. We provide a proof that these fields satisfy bona fide commutation relations and that in the distant past and future they reduce to standard time-evolving Heisenberg operators. Having solved for the input-output relations of these Heisenberg operators we also show how to construct the ket describing the quantum state of the twin beams. Finally, we consider the example of high-gain SPDC in a waveguide with a flat nonlinearity profile, for which our approach provides an explicit solution that requires only a single matrix exponentiation.

In this Letter, we present a universal approach enabling the full characterization of the quantum properties of a multimode optical system in terms of squeezing and morphing supermodes. These are modes undergoing a continuous evolution that allow uncoupling the system dynamics in terms of statistically independent physical observables. This dynamical feature, never considered so far, enables the description and investigation of an extremely broad variety of key resources for experimental quantum optics, ranging from optical parametric oscillators to silicon-based microring resonators, as well as optomechanical systems.

We report the efficient generation of high-gain parametric down-conversion, including pump depletion, with pump powers as low as 100 µW (energies 0.1 µJ/pulse) and conversion efficiencies up to 33%. In our simple configuration, the pump beam is tightly focused into a bulk periodically poled lithium niobate crystal placed in free space. We also observe a change in the photon number statistics for both the pump and down-converted beams as the pump power increases to reach the depleted pump regime. The experimental results are a clear signature of the interplay between the pump and the down-converted beams in highly efficient parametric down-conversion sources.

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.

We show that relatively simple integrated photonic circuits have the potential to realize a high fidelity deterministic controlled-phase gate between photonic qubits using bulk optical nonlinearities. The gate is enabled by converting travelling continuous-mode photons into stationary cavity modes using strong classical control fields that dynamically change the effective cavity-waveguide coupling rate. This architecture succeeds because it reduces the wave packet distortions that otherwise accompany the action of optical nonlinearities [J. Shapiro, Phys. Rev. A 73, 062305 (2006); J. Gea-Banacloche, Phys. Rev. A 81, 043823 (2010)]. We show that high-fidelity gates can be achieved with self-phase modulation in χ(3) materials as well as second-harmonic generation in χ(2) materials. The gate fidelity asymptotically approaches unity with increasing storage time for an incident photon wave packet with fixed duration. We also show that dynamically coupled cavities enable a trade-off between errors due to loss and wave packet distortion. Our proposed architecture represents a new approach to practical implementation of quantum gates that is room-temperature compatible and only relies on components that have been individually demonstrated.

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.

We demonstrate that nondegenerate four-wave mixing in a Si3N4 microring resonator can result in a nonlinear coupling rate between two optical fields exceeding their energy dissipation rate in the resonator, corresponding to strong nonlinear coupling. We demonstrate that this leads to a Rabi-like splitting, for which we provide a theoretical description in agreement with our experimental results. This yields new insight into the dynamics of nonlinear optical interactions in microresonators and access to novel phenomena.

The well-known metrological linear squeezing parameters (such as quadrature or spin squeezing) efficiently quantify the sensitivity of Gaussian states. Yet, these parameters are insufficient to characterize the much wider class of highly sensitive non-Gaussian states. Here, we introduce a class of metrological nonlinear squeezing parameters obtained by analytical optimization of measurement observables among a given set of accessible (possibly nonlinear) operators. This allows for the metrological characterization of non-Gaussian quantum states of discrete and continuous variables. Our results lead to optimized and experimentally feasible recipes for a high-precision moment-based estimation of a phase parameter and can be used to systematically construct multipartite entanglement and nonclassicality witnesses for complex quantum states.

Quantum illumination takes advantage of quantum entanglement to achieve low error probability for detecting a low reflective object embedded in a noisy thermal bath. The two-mode squeezed state (TMSS), which is a Gaussian state, has been applied to quantum illumination as the detecting states in experiment. The photon-subtracted TMSS has also been proposed to achieve even lower error probability. Here we study quantum illumination with non-Gaussian states generated by photon subtraction and photon addition. Helstrom limit and quantum Chernoff bound are evaluated for comparison between performance of states with the same squeezing strength and with the same signal strength, respectively. Particularly, states generated by asymmetrical coherent superposition of photon subtraction and addition are studied, which are shown by us to have better performance than symmetrical ones. We show that non-Gaussian operations enhance the quantum illumination by introducing both stronger entanglement and signal strength. We then give a strategy on how to choose the optimal states for the best performance in different scenarios.

We present and analyze a method where parametric (two-photon) driving of a cavity is used to exponentially enhance the light-matter coupling in a generic cavity QED setup, with time-dependent control. Our method allows one to enhance weak-coupling systems, such that they enter the strong coupling regime (where the coupling exceeds dissipative rates) and even the ultrastrong coupling regime (where the coupling is comparable to the cavity frequency). As an example, we show how the scheme allows one to use a weak-coupling system to adiabatically prepare the highly entangled ground state of the ultrastrong coupling system. The resulting state could be used for remote entanglement applications.

We generate pulsed, two-mode squeezed states in a single spatiotemporal mode with mean photon numbers up to 20. We directly measure photon-number correlations between the two modes with transition edge sensors up to 80 photons per mode. This corresponds roughly to a state dimensionality of 6400. We achieve detection efficiencies of 64% in the technologically crucial telecom regime and demonstrate the high quality of our measurements by heralded nonclassical distributions up to 50 photons per pulse and calculated correlation functions up to 40th order.

We study time ordering corrections to the description of spontaneous
parametric down-conversion (SPDC), four wave mixing (SFWM) and frequency
conversion (FC) using the Magnus expansion. Analytic approximations to the
evolution operator that are unitary are obtained. They are Gaussian preserving,
and allow us to understand order-by-order the effects of time ordering. We show
that the corrections due to time ordering vanish exactly if the phase matching
function is sufficiently broad. The calculation of the effects of time ordering
on the joint spectral amplitude of the photons generated in spontaneous SPDC
and SFWM are reduced to quadrature.

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.

By embedding an atom capable of electromagnetically induced transparency inside an appropriate photonic-crystal microcavity it may become possible to realize an optical nonlinearity that can impart a π-rad-peak phase shift in response to a single-photon excitation. Such a device, if it operated at high fidelity, would then complete a universal gate set for all-optical quantum computation. It is shown here that the causal, noninstantaneous behavior of any χ(3) nonlinearity is enough to preclude such a high-fidelity operation. In particular, when a single-photon-sensitive χ(3) nonlinearity has a response time that is much shorter than the duration of the quantum computer’s single-photon pulses, essentially no overall phase shift is imparted to these pulses by cross-phase modulation. Conversely, when this nonlinearity has a response time that is much longer than this pulse duration a single-photon pulse can induce a π-rad overall phase shift through cross-phase modulation, but the phase noise injected by the causal, noninstantaneous response function precludes this from being a high-fidelity operation.

The properties of a unique set of quantum states of the electromagnetic field are reviewed. These 'squeezed states' have less uncertainty in one quadrature than a coherent state. Proposed schemes for the generation and detection of squeezed states as well as potential applications are discussed.

The Einstein-Podolsky-Rosen paradox is demonstrated experimentally for dynamical variables having a continuous spectrum. As opposed to previous work with discrete spin or polarization variables, the continuous optical amplitudes of a signal beam are inferred in turn from those of a spatially separated but strongly correlated idler beam generated by nondegenerate parametric amplification. The uncertainty product for the variances of these inferences is observed to be 0.70±0.01, which is below the limit of unity required for the demonstration of the paradox.