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# Towards an Engineering Framework for Ultrafast Quantum Nonlinear Optics

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## Abstract

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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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
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.
Article
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.