The theoretical description of broadband, multimode quantum pulses undergoing a second-order $χ^{(2)}$-nonlinear interaction can be quite intricate, due to the large dimensionality of the underlying phase space. However, in many cases only a few broadband (temporal) modes are relevant before and after the nonlinear interaction. Here we present an efficient framework to calculate the relation between the quantum states at the input and output of a nonlinear element in their respective relevant modes. Since the number of relevant input and output modes may differ, resulting in an open quantum system, we introduce the generalized Bloch-Messiah decomposition (GBMD), reducing the description to an equal number of input and output modes. The GBMD enables us to calculate the multimode Wigner function of the output state by convolving the rescaled Wigner function of the reduced input quantum pulse with a multivariate Gaussian phase-space function. We expand on this result by considering two examples input states: A Fock state in a single broadband mode and a two-mode squeezed vacuum, both in the THz-frequency regime, up-converted to a single output broadband mode of optical frequencies. We investigate the effect, the convolution and thermalization due to entanglement breakage have on the output Wigner function by calculating the von Neumann entropy of the output Wigner function. The methods presented here can be used to optimize the amplification or frequency conversion of broadband quantum states, opening an avenue to the generation and characterization of optical quantum states on ultrafast time scales.
The theoretical description of broadband, multimode quantum pulses undergoing a second-order χ(2)-nonlinear interaction can be quite intricate, due to the large dimensionality of the underlying phase space. However, in many cases only a few broadband (temporal) modes are relevant before and after the nonlinear interaction. Here we present an efficient framework to calculate the relation between the quantum states at the input and output of a nonlinear element in their respective relevant modes. Since the number of relevant input and output modes may differ, resulting in an open quantum system, we introduce the generalized Bloch-Messiah decomposition (GBMD), reducing the description to an equal number of input and output modes. The GBMD enables us to calculate the multimode Wigner function of the output state by convolving the rescaled Wigner function of the reduced input quantum pulse with a multivariate Gaussian phase-space function. We expand on this result by considering two examples input states: A Fock state in a single broadband mode and a two-mode squeezed vacuum, both in the THz-frequency regime, up-converted to a single output broadband mode of optical frequencies. We investigate the effect, the convolution and thermalization due to entanglement breakage have on the output Wigner function by calculating the von Neumann entropy of the output Wigner function. The methods presented here can be used to optimize the amplification or frequency conversion of broadband quantum states, opening an avenue to the generation and characterization of optical quantum states on ultrafast time scales.
The quantum features of ultrashort-pulse optical parametric oscillators (OPOs) are investigated theoretically in the nonlinear regime near and above threshold. Viewing the pulsed OPO as a multimode open quantum system, we rigorously derive a general input-output model that features nonlinear coupling among many cavity (i.e., system) signal modes and a broadband single-pass (i.e., reservoir) pump field. Under appropriate assumptions, our model produces a Lindblad master equation with multimode nonlinear Lindblad operators describing two-photon dissipation and a multimode four-wave-mixing Hamiltonian describing a broadband, dispersive optical cascade, which we show is required to preserve causality. To simplify the multimode complexity of the model, we employ a supermode decomposition to perform numerical simulations in the regime where the pulsed supermodes experience strong single-photon nonlinearity. We find that the quantum nonlinear dynamics induces pump depletion as well as corrections to the below-threshold squeezing spectrum predicted by linearized models. We also observe the formation of non-Gaussian states with Wigner-function negativity and show that the multimode interactions with the pump, both dissipative and dispersive, can act as effective decoherence channels. Finally, we briefly discuss some experimental considerations for potentially observing such quantum nonlinear phenomena with ultrashort-pulse OPOs on nonlinear nanophotonic platforms.
Ideally, strong nonlinearities could be used to implement quantum gates for photonic qubits by well-controlled two-photon interactions. However, the dependence of the nonlinear interaction on frequency and time makes it difficult to preserve a coherent pulse shape that could justify a single-mode model for the time-frequency degree of freedom of the photons. In this paper, we analyze the problem of temporal multimode effects by considering the pulse shape of the average output field obtained from a coherent input pulse. It is shown that a significant part of the two-photon state transformation can be derived from this semiclassical description of the optical nonlinearity. The effect of a nonlinear system on a two-photon state can then be determined from the density-matrix dynamics of the coherently driven system using input-output theory. As an example, the resonant nonlinearity of a single two-level atom is characterized. The results indicate that the most efficient nonlinear effect may not be the widely studied single-mode phase shift, but the transfer of one of the photons to an orthogonal mode distinguished by its temporal and spectral properties.
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.
Methods to decompose nonlinear optical transformation vary from setting to setting, leading to apparent differences in the treatments used to model photon pair sources, compared to those used to model degenerate down-conversion processes. The Bloch-Messiah reduction of Gaussian processes to single-mode squeezers and passive (linear) unitaries appears juxtaposed against the practicalities of the Schmidt-decomposition for photon pair sources into two-mode squeezers and passive unitaries. Here, we present a general framework which unifies these forms as well as elucidating more general structure in multimode Gaussian transformations. The decomposition is achieved by introducing additional constraints into the Bloch-Messiah reduction used to diagonalise Gaussian processes, these constraints motivated by physical constraints following from the inequivalence of different physical degrees of freedom in a system, ie. the temporal-spectral degrees of freedom vs different spatial modes in a transformation. The result is the emergence of the two-mode squeezing picture from the reduction, as well as the potential to generalise these constraints to accommodate spectral imperfections in a source generating 3-mode continuous variable GHZ-like states. Furthermore, we consider the practical scenario in which a transformation aims to generate a multiphoton entangled state, whereby spatial modes provide desirable degrees of freedom, whilst undesired spectral mode structure contributes noise, and show that this spectral impurity can be efficiently modeled by finding an optimal low dimensional bases for its simulation.
Photonics offers the potential for large-scale, room-temperature, and ultrafast quantum operations. Multiplexing pulses of light allows for high throughput speeds and dense encoding, with tens of thousands of optical modes being present in a time as short as a microsecond. However, the measurement speed of electronic devices such as photodetectors is currently one bottleneck in such time-multiplexed scalability, because many quantum optical protocols rely on measurement, including state preparation, feed-forward, and feedback. Furthermore, characterization of quantum states and modes is essential for their effective use. Using nanophotonic optical parametric amplifiers (OPAs) presents one way to circumvent the measurement limitations of current quantum photonic schemes. Their large amplification bandwidth, enabled by dispersion engineering on nanophotonic integrated platforms, allows information encoded in femtosecond-scale, THz-bandwidth, multimode quantum optical states to be accessed with nonlinear optical interactions. In this work, we show how ultra-broadband integrated nanophotonic OPAs can be used to measure ultrafast and multimode quantum states of light. First, we explore the single-photon detection capabilities of OPAs, showing that current OPAs operating in the Gaussian regime are capable of 250 MHz photon count rates with 26% efficiency and a 2% dark count probability. We also show how non-Gaussian operation through pump depletion, with performance that approaches state-of-the-art photon detectors in terms of efficiency and dark count rate while retaining ultrafast operation, can become experimentally possible with a higher nonlinear coupling rate and lower loss. Next, we use nanophotonic OPAs to both generate and characterize multimode ultrafast squeezed vacuum. We use the photocurrent distribution of amplified squeezed vacuum to recover 2.41 dB of squeezing in one mode of a 154-fs multimode squeezed pulse and reconstruct its Wigner distribution. Finally, we investigate the capabilities of broadband nanophotonic OPAs to determine the temporal mode structure and quadrature variances of ultra-broadband temporally multimode quantum states by adapting frequency-resolved optical gating to the quantum regime. We numerically show the successful full characterization of a multimode squeezed state, even in the presence of noise. Together, these results establish OPAs as a valuable measurement device for measuring ultrafast quantum pulses and learning the structure of multimode quantum states, and they provide one building block towards a framework for scalable continuous-variable quantum photonics in which state generation, manipulation, and characterization occur within the same nanophotonic platform.
Photonic integrated circuits with second-order (χ(2)) nonlinearities are rapidly scaling to remarkably low powers. At this time, state-of-the-art devices achieve saturated nonlinear interactions with thousands of photons when driven by continuous-wave lasers, and further reductions in these energy requirements enabled by the use of ultrafast pulses may soon push nonlinear optics into the realm of single-photon nonlinearities. This tutorial reviews these recent developments in ultrafast nonlinear photonics, discusses design strategies for realizing few-photon nonlinear interactions, and presents a unified treatment of ultrafast quantum nonlinear optics using a framework that smoothly interpolates from classical behaviors to the few-photon scale. These emerging platforms for quantum optics fundamentally differ from typical realizations in cavity quantum electrodynamics due to the large number of coupled optical modes. Classically, multimode behaviors have been well studied in nonlinear optics, with famous examples including soliton formation and supercontinuum generation. In contrast, multimode quantum systems exhibit a far greater variety of behaviors, and yet closed-form solutions are even sparser than their classical counterparts. In developing a framework for ultrafast quantum optics, we will identify what behaviors carry over from classical to quantum devices, what intuition must be abandoned, and what new opportunities exist at the intersection of ultrafast and quantum nonlinear optics. While this article focuses on establishing connections between the classical and quantum behaviors of devices with χ(2) nonlinearities, the frameworks developed here are general and are readily extended to the description of dynamical processes based on third-order (χ(3)) nonlinearities.