Schmidt Modes Research Articles

Pre-Schmidt Mode Parameters in Quantum Entanglement

We wrote this paper to study in detail the mathematical methods used in deriving the parameters 𝑝𝑝′1, 𝑝𝑝′2, 𝑥𝑥′1, 𝑥𝑥′2, 𝑀𝑀, 𝐾𝐾 and 𝑒𝑒2𝜂𝜂 in the research papers by Han et al. (1999), Makarov (2018a), Han et al. (1993), and Han et al. (1995) to find the Schmidt modes 𝛬𝛬𝑘𝑘 in quantum entanglement. Here, we have analysed and developed a thorough calculation in exploring the rationales behind the existence of these parameters in two and three-coupled harmonic oscillators. Various mathematical approaches were applied in the study, ranging from polynomials and linear algebra to trigonometry and the Pythagorean theorem. We found the parameters 𝐾𝐾 and 𝑒𝑒2𝜂𝜂 using the matrices’ determinants and eigenvalues. With these rationales in deriving the parameters 𝐾𝐾 and 𝑒𝑒2𝜂𝜂 in the research papers for the two-coupled harmonic oscillators, we have formulated similar parameters for three-coupled harmonic oscillators as the conclusion of our study.

Space-time exchanged symplectic integrator approach to temporal-mode selective optical three-wave mixing frequency conversion

A space-time exchanged symplectic integrator (SI) method is developed to solve the three-wave resonance interaction equations commonly used in describing a number of fundamental nonlinear optical phenomena, such as second harmonic generation, optical parametric amplification, etc. Exchanging the roles of the space and time coordinates allows us to solve these nonlinearly coupled equations with hard-wired boundary conditions using a highly accurate SI scheme instead of standard finite difference methods. The developed method is then applied to the optical three-wave mixing frequency conversion processes in which an input signal light is converted into an idler light guided by a strong pump pulse. The temporal-mode selectivity in the frequency conversion processes is examined by systematically varying system parameters and the results are summarized as contour maps of selectivity on the ‘group-slownesses’, defined by the inverse of the group velocities, of the signal pulse and of the idler pulse relative to that of the pump pulse. The areas showing the highest peaks on these undergo significant and characteristic changes in their positions as well as the heights for different values of the coupling strength γ. The observed trends are rationalized on the basis of the Schmidt modes obtained by decomposing the two-time transfer function that fully characterizes the dynamics of the signal–idler wave conversion.

Single-step order shift of Schmidt modes in a linear system

In parametric down-conversion, Schmidt modes are given by special series of wave functions that have widely discussed potential applications in quantum information science. Existing technologies allow the manipulation of such modes one at a time. We report the discovery of a linear process that can simultaneously transform a wide range of Hermite-Gaussian modes, a specific type of Schmidt modes. We examine the scattering process by a single cavity under tunable coupling given by a particular coupling function, and for a wide range of Hermite-Gaussian modes, a single-step shift of mode order is shown as achievable in high fidelity. This result constitutes an effective mean of manipulating Schmidt modes with potential application in quantum information networks.

Improved Heralded Single-Photon Source with a Photon-Number-Resolving Superconducting Nanowire Detector

Deterministic generation of single photons is essential for many quantum information technologies. A bulk optical nonlinearity emitting a photon pair, where the measurement of one of the photons heralds the presence of the other, is commonly used with the caveat that the single-photon emission rate is constrained due to a trade-off between multiphoton events and pair emission rate. Using an efficient and low noise photon-number-resolving superconducting nanowire detector we herald, in real time, a single photon at telecommunication wavelength. We perform a second-order photon correlation ${g}^{2}(0)$ measurement of the signal mode conditioned on the measured photon number of the idler mode for various pump powers and demonstrate an improvement of a heralded single-photon source. We develop an analytical model using a phase-space formalism that encompasses all multiphoton effects and relevant imperfections, such as loss and multiple Schmidt modes. We perform a maximum-likelihood fit to test the agreement of the model to the data and extract the best-fit mean photon number $\ensuremath{\mu}$ of the pair source for each pump power. A maximum reduction of $0.118\ifmmode\pm\else\textpm\fi{}0.012$ in the photon ${g}^{2}(0)$ correlation function at $\ensuremath{\mu}=0.327\ifmmode\pm\else\textpm\fi{}0.007$ is obtained, indicating a strong suppression of multiphoton emissions. For a fixed ${g}^{2}(0)=7\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}$, we increase the single pair generation probability by 25%. Our experiment, built using fiber-coupled and off-the-shelf components, delineates a path to engineering ideal sources of single photons.

Distinguishability in quantum interference with multimode squeezed states

Distinguishability theory is developed for quantum interference of the squeezed vacuum states on unitary linear interferometers. It is found that the entanglement of photon pairs over the Schmidt modes is one of the sources of distinguishability. The distinguishability is quantified by the symmetric part of the internal state of $n$ pairs of photons, whose normalization $q_{2n}$ is the probability that $2n$ photons interfere as indistinguishable. For two pairs of photons $q_{4}=(1+ 2\mathbb{P} )/3$, where $\mathbb{P} $ is the purity of the squeezed states ($K=1/\mathbb{P} $ is the Schmidt number). For a fixed purity $\mathbb{P}$, the probability $q_{2n}$ decreases exponentially fast in $n$. For example, in the experimental Gaussian boson sampling of H.-S.~Zhong \textit{et al} [Science \textbf{370}, 1460 (2020)], the achieved purity $\mathbb{P}\approx 0.938$ for the average number of photons $2n\ge 43$ gives $q_{2n}\lesssim 0.5$, i.e., close to the middle line between $n$ indistinguishable and $n$ distinguishable pairs of photons. In derivation of all the results the first-order quantization representation based on the particle decomposition of the Hilbert space of identical bosons serves as an indispensable tool. The approach can be applied also to the generalized (non-Gaussian) squeezed states, such as those recently generated in the three-photon parametric down-conversion.

Single-Step Order Shift of Schmidt Modes in a Linear System

Single-Step Order Shift of Schmidt Modes in a Linear System

Wave Packet Control and Simulation Protocol for Entangled Two-Photon Absorption of Molecules.

Quantum light spectroscopy, providing novel molecular information nonaccessible by classical light, necessitates new computational tools when applied to complex molecular systems. We introduce two computational protocols for the molecular nuclear wave packet dynamics interacting with an entangled photon pair to produce an entangled two-photon absorption signal. The first involves summing over transition pathways in a temporal grid defined by two light-matter interaction times accompanied by the field correlation functions of quantum light. The signal is obtained by averaging over the two time distribution characteristics of the entangled photon state. The other protocol involves a Schmidt decomposition of the entangled light and requires summing over the Schmidt modes. We demonstrate how photon entanglement can be used to control and manipulate the two-photon excited nuclear wave packets in a displaced harmonic oscillator model.

Characterizing the spatial entanglement from laser modes analogous to quantum wave functions.

The Schmidt decomposition is exploited to study the spatial entanglement of laser transverse modes analogous to quantum Lissajous states. Based on the inverse Fourier transform, the stationary Lissajous state can be analytically derived as a coherent superposition of degenerate Hermite-Gaussian eigenmodes. With the derived stationary state, the Schmidt modes and the participation number N can be employed to evaluate the spatial localization and the quantum entanglement. The larger the participation number, the more localized is the stationary coherent state on the Lissajous figure. Moreover, the larger the participation number, the higher is the spatial entanglement.

Managing spectral properties and Schmidt mode content of squeezed vacuum light using sum-frequency converter

Capabilities of quantum optical sum-frequency converter seeded by squeezed light are investigated and described analytically in the case when several frequency Schmidt modes are involved in nonlinear process. Methods to manage and manipulate extensively the properties and mode content of multimode multipartite non-classical squeezed field state are developed. Possibility to control the photon numbers in different Schmidt modes of squeezed light simultaneously is demonstrated. The significant phase sensitivity of the converter is established. Tailoring the spectral distribution of the output light due to the phase effects and variable coupling between modes in the converter is demonstrated. In the case of twin-beam seeding the effect of changing the conjugated correlated frequency mode is found which allows to manipulate the correlations between photons. The developed methods to vary the efficient number of Schmidt modes of squeezed light are strongly required for important practical applications and quantum technology.

Reconstructing two-dimensional spatial modes for classical and quantum light

We propose a method for finding 2D spatial modes of thermal field through a direct measurement of the field intensity and an offline analysis of its spatial fluctuations. Using this method, in a simple and efficient way we reconstruct the modes of a multimode fiber and the spatial Schmidt modes of squeezed vacuum generated via high-gain parametric down conversion. The reconstructed shapes agree with the theoretical results.

Nonlinear interference in the strongly nondegenerate regime and Schmidt mode analysis

The possibility to manage and control spectral and spatial properties of optical-terahertz nonclassical fields generated in strongly nondegenerate parametric down-conversion is demonstrated experimentally using Mach-Zehnder nonlinear interferometer and analyzed theoretically in terms of Schmidt modes. The interferometer can be used for measuring the dispersion properties of different materials in the terahertz frequency range. The key point is that the information about these properties is extracted from the measurements in the optical channel due to strong frequency-angular correlations between optical and terahertz photons. The developed theoretical approach is used to predict properties of the optical-terahertz fields and photon correlations in different regimes of nondegenerate parametric down-conversion. The considered interferometer is shown to be a promising source of terahertz fields with new features and frequency-filtered terahertz Schmidt modes.

Properties of bright squeezed vacuum at increasing brightness

This work provides a theoretical approach to describe the spatial properties of bright squeezed vacuum. The model describes these using Schmidt modes and captures the modal structure and the effects of strong pumping.

Spatial features and photon correlations of squeezed light in strongly non-degenerate parametric down conversion

Parametric down conversion in a strongly non-degenerate regime with output terahertz (THz) and optical beams is investigated. The theoretical approach based on the concept of independent Schmidt modes is developed and the shapes and weights of optical and THz Schmidt modes are analyzed both in a far-field and near-field regions. Analytical approximation for examined radial Schmidt modes is found. Strong azimuthal correlations between optical and THz photons are established. THz photons are shown to be involved simultaneously both in parametric down conversion and sum frequency generation processes. Photons in THz, Stocks and anti-Stocks channels are demonstrated to be strongly entangled. Methods to examine the properties of the THz radiation by measuring the correlations between generated Stocks and anti-Stocks components are suggested.

Bloch-Messiah reduction for twin beams of light

We study the Bloch-Messiah reduction of parametric downconversion of light in the pulsed regime with a nondegenerate phase matching providing generation of twin beams. We find that in this case every squeezing eigenvalue has multiplicity at least two. We discuss the problem of ambiguity in the definition of the squeezing eigenmodes in this case and develop two approaches to unique determination of the latter. First, we show that the modal functions of the squeezing eigenmodes can be tailored from the Schmidt modes of the signal and idler beams. Alternatively, they can be found as a solution of an eigenvalue problem for an associated Hermitian squeezing matrix. We illustrate the developed theory by an example of frequency non-degenerate collinear twin beams generated in beta barium borate crystal. On this example we demonstrate how the squeezing eigenmodes can be approximated analytically on the basis of the Mehler's formula, extended to complex kernels. We show how the multiplicity of the eigenvalues and the structure of the eigenmodes are changed when the phase matching approaches the degeneracy in frequency.

Separated Schmidt modes in the angular spectrum of biphotons

We prepared qudits based on angular multimode biphoton states by modulating the pump angular spectrum. The modes were prepared in the Schmidt basis and their intensity distributions did not overlap in space. This allows one to get rid of filtering losses while addressing single modes and to realize a single-shot qudit readout.

Quantum phase-sensitive diffraction and imaging using entangled photons

We propose a quantum diffraction imaging technique whereby one photon of an entangled pair is diffracted off a sample and detected in coincidence with its twin. The image is obtained by scanning the photon that did not interact with matter. We show that when a dynamical quantum system interacts with an external field, the phase information is imprinted in the state of the field in a detectable way. The contribution to the signal from photons that interact with the sample scales as [Formula: see text], where [Formula: see text] is the source intensity, compared with [Formula: see text] of classical diffraction. This makes imaging with weak fields possible, providing high signal-to-noise ratio, avoiding damage to delicate samples. A Schmidt decomposition of the state of the field can be used for image enhancement by reweighting the Schmidt modes contributions.

Broadband pseudothermal states with tunable spectral coherence generated via nonlinear optics

It is well known that the reduced state of a two-mode squeezed vacuum state is a thermal state---i.e. a state whose photon-number statistics obey a geometric distribution. More exotic \emph{broadband} states can be realized as the reduced state of two spectrally-entangled beams generated using nonlinear optics. We show that these broadband "pseudothermal" states are tensor products of states in spectral Schmidt modes, whose photon-number statistics obey a geometric distribution. We study the spectral and temporal coherence properties of these states and show that their spectral coherence can be tuned---from perfect coherence to complete incoherence---by adjusting the pump spectral width. In the limit of a cw pump, these states are tensor products of true thermal states, but with different temperatures at each frequency. This could be an interesting state of light for investigating the interplay between spectral, temporal, and photon-number coherences.

Certification of high-dimensional entanglement and Einstein-Podolsky-Rosen steering with cold atomic quantum memory

The advent of complex, structured, and high-dimensional entangled states brings both new possibilities for experimental and theoretical scenarios and new challenges for the generation and characterization of such states. In particular, spatially structured photonic states offer applications in quantum imaging, information processing, and quantum key distribution. Here we experimentally generate a spatially entangled high-dimensional state composed of at least 10 Schmidt modes in a quantum memory setup and perform characterization using the entropic Einstein-Podolsky-Rosen (EPR)-steering inequality, yielding a genuine violation of $1.06\ifmmode\pm\else\textpm\fi{}0.15$ bits. The entanglement of formation of at least $0.70\ifmmode\pm\else\textpm\fi{}0.15$ ebit for the measured noisy state is certified using the entropic witness method. We point out and solve the difficulties in estimating the entropy, achieving characterization of the high-dimensional entangled state with highly undersampled data. Finally, the practical supremacy of the entropic EPR-steering witness over a variance-based witness is demonstrated for a wide class of states typical in an experimental scenario, giving prospects for EPR-steering demonstrations and applications in noisy systems or with lossy quantum channels.

Intense spectrally broad-band twin beams from poled nonlinear crystals

Properties of intense twin beams generated in parametric down-conversion in periodically poled LiNbO3 crystals and their chirped variants by intense pump fields are analyzed along the model of intense parametric down-conversion that allows for pump depletion and uses the dual spatio-spectral Schmidt modes. Spectral and spatial intensity auto- and cross-correlation functions are determined as they depend on the pump power. Temporal correlations in intense twin beams at the fs time scale are investigated using the Hong-Ou-Mandel interferometer and the process of sum-frequency generation.

Quantum entanglement of a harmonic oscillator with an electromagnetic field

At present, there are many methods for obtaining quantum entanglement of particles with an electromagnetic field. Most methods have a low probability of quantum entanglement and not an exact theoretical apparatus based on an approximate solution of the Schrodinger equation. There is a need for new methods for obtaining quantum-entangled particles and mathematically accurate studies of such methods. In this paper, a quantum harmonic oscillator (for example, an electron in a magnetic field) interacting with a quantized electromagnetic field is considered. Based on the exact solution of the Schrodinger equation for this system, it is shown that for certain parameters there can be a large quantum entanglement between the electron and the electromagnetic field. Quantum entanglement is analyzed on the basis of a mathematically exact expression for the Schmidt modes and the Von Neumann entropy.