Implementation of quantum and classical discrete fractional Fourier transforms

The Goldilocks principle of learning unitaries by interlacing fixed operators with programmable phase shifters on a photonic chip

Programmable photonic integrated circuits represent an emerging technology that amalgamates photonics and electronics, paving the way for light-based information processing at high speeds and low power consumption. Programmable photonics provides a flexible platform that can be reconfigured to perform multiple tasks, thereby holding great promise for revolutionizing future optical networks and quantum computing systems. Over the past decade, there has been constant progress in developing several different architectures for realizing programmable photonic circuits that allow for realizing arbitrary discrete unitary operations with light. Here, we systematically investigate a general family of photonic circuits for realizing arbitrary unitaries based on a simple architecture that interlaces a fixed intervening layer with programmable phase shifter layers. We introduce a criterion for the intervening operator that guarantees the universality of this architecture for representing arbitrary N × N unitary operators with N + 1 phase layers. We explore this criterion for different photonic components, including photonic waveguide lattices and meshes of directional couplers, which allows the identification of several families of photonic components that can serve as the intervening layers in the interlacing architecture. Our findings pave the way for efficiently designing and realizing novel families of programmable photonic integrated circuits for multipurpose analog information processing.

Auto-calibrating universal programmable photonic circuits: hardware error-correction and defect resilience

It is recently shown that discrete N × N linear unitary operators can be represented by interlacing N + 1 phase shift layers with a fixed intervening operator such as discrete fractional Fourier transform (DFrFT). Here, we show that introducing perturbations to the intervening operations does not compromise the universality of this architecture. Furthermore, we show that this architecture is resilient to defects in the phase shifters as long as no more than one faulty phase shifter is present in each layer. These properties enable post-fabrication auto-calibration of such universal photonic circuits, effectively compensating for fabrication errors and defects in phase components.

Experimental Implementation of the Optical Fractional Fourier Transform in the Time-Frequency Domain

The fractional Fourier transform (FrFT), a fundamental operation in physics that corresponds to a rotation of phase space by any angle, is also an indispensable tool employed in digital signal processing for noise reduction. Processing of optical signals in their time-frequency degree of freedom bypasses the digitization step and presents an opportunity to enhance many protocols in quantum and classical communication, sensing, and computing. In this Letter, we present the experimental realization of the fractional Fourier transform in the time-frequency domain using an atomic quantum-optical memory system with processing capabilities. Our scheme performs the operation by imposing programmable interleaved spectral and temporal phases. We have verified the FrFT by analyses of chroncyclic Wigner functions measured via a shot-noise limited homodyne detector. Our results hold prospects for achieving temporal-mode sorting, processing, and superresolved parameter estimation.

Analytical formulation of quantum interference inside coupled waveguides with unequal losses

In this paper, a theoretical framework is proposed to formulate the quantum interference inside the coupled waveguides with unequal losses. The quantum coupled mode equation is added with the Langevin noise terms to account for the impact of unequal losses, which can be solved analytically. A close form formula is derived for the correlation matrix of the Langevin noise terms, which provides full information for the density matrix of the propagation state. The theory is self-consistent and tested with a three-waveguide system, which is considered as anti-parity-time (PT) symmetric and simulated in the previous publications. An 89-waveguide system is analyzed afterwards to further demonstrate the applicability of the theory.

Waveguide-lattice-based architecture for multichannel optical transformations

We consider waveguide lattices as the architecture to implement a wide range of multiport transformations. In this architecture, a particular transfer matrix is obtained by setting step-wise profiles of propagation constants experienced by a field evolving in a lattice. To investigate the capabilities of this architecture, we numerically study the implementation of random transfer matrices as well as several notable cases, such as the discrete Fourier transform, the Hadamard, and permutation matrices. We show that waveguide lattice schemes are more compact than their traditional lumped-parameter counterparts, thus the proposed architecture may be beneficial for photonic information processing systems of the future.

Topological protection versus degree of entanglement of two-photon light in photonic topological insulators

Topological insulators combine insulating properties in the bulk with scattering-free transport along edges, supporting dissipationless unidirectional energy and information flow even in the presence of defects and disorder. The feasibility of engineering quantum Hamiltonians with photonic tools, combined with the availability of entangled photons, raises the intriguing possibility of employing topologically protected entangled states in optical quantum computing and information processing. However, while two-photon states built as a product of two topologically protected single-photon states inherit full protection from their single-photon “parents”, a high degree of non-separability may lead to rapid deterioration of the two-photon states after propagation through disorder. In this work, we identify physical mechanisms which contribute to the vulnerability of entangled states in topological photonic lattices. Further, we show that in order to maximize entanglement without sacrificing topological protection, the joint spectral correlation map of two-photon states must fit inside a well-defined topological window of protection.

Topological protection of entangled states is a promising avenue for photonic quantum technologies. Here, Tschernig et al. theoretically analyse the impact of disorder on topological protection of entangled two-photon states in periodic and aperiodic topological insulator lattices.

Two-photon interference: the Hong-Ou-Mandel effect

Nearly 30 years ago, two-photon interference was observed, marking the beginning of a new quantum era. Indeed, two-photon interference has no classical analogue, giving it a distinct advantage for a range of applications. The peculiarities of quantum physics may now be used to our advantage to outperform classical computations, securely communicate information, simulate highly complex physical systems and increase the sensitivity of precise measurements. This separation from classical to quantum physics has motivated physicists to study two-particle interference for both fermionic and bosonic quantum objects. So far, two-particle interference has been observed with massive particles, among others, such as electrons and atoms, in addition to plasmons, demonstrating the extent of this effect to larger and more complex quantum systems. A wide array of novel applications to this quantum effect is to be expected in the future. This review will thus cover the progress and applications of two-photon (two-particle) interference over the last three decades.

Discrete normalized Bargmann transform through the gyrator transform

Linear canonical transforms (LCTs) are important in several areas of signal processing; in particular, they were extended to complex-valued parameters to describe optical systems. A special case of these complex LCTs is the Bargmann transform. Recently, Pei and Huang [J. Opt. Soc. Am. A34, 18 (2017)JOAOD60740-323210.1364/JOSAA.34.000018] presented a normalization of the Bargmann transform so that it becomes possible to delimit it near infinity. In this paper, we follow the Pei-Huang algorithm to introduce the discrete normalized Bargmann transform by the relationship between Bargmann and gyrator transforms in the SU(2) finite harmonic oscillator model, and we compare it with the discrete Bargmann transform based on coherent states of the SU(2) oscillator model. This transformation is invertible and unitary. We show that, as in the continuous analog, the discrete normalized Bargmann transform converts the Hermite-Kravchuk functions into Laguerre-Kravchuk functions. In addition, we demonstrate that the discrete su(1,1) repulsive oscillator functions self-reproduce under this discrete transform with little error. Finally, in the space spanned by the wave functions of the SU(2) harmonic oscillator, we find that the discrete normalized Bargmann transform commutes with the fractional Fourier-Kravchuk transform.

Quantum interference enables constant-time quantum information processing

It is an open question how fast information processing can be performed and whether quantum effects can speed up the best existing solutions. Signal extraction, analysis, and compression in diagnostics, astronomy, chemistry, and broadcasting build on the discrete Fourier transform. It is implemented with the fast Fourier transform (FFT) algorithm that assumes a periodic input of specific lengths, which rarely holds true. A lesser-known transform, the Kravchuk-Fourier (KT), allows one to operate on finite strings of arbitrary length. It is of high demand in digital image processing and computer vision but features a prohibitive runtime. Here, we report a one-step computation of a fractional quantum KT. The quantum d-nary (qudit) architecture we use comprises only one gate and offers processing time independent of the input size. The gate may use a multiphoton Hong-Ou-Mandel effect. Existing quantum technologies may scale it up toward diverse applications.

Photonic quantum information processing: a review

Photonic quantum technologies represent a promising platform for several applications, ranging from long-distance communications to the simulation of complex phenomena. Indeed, the advantages offered by single photons do make them the candidate of choice for carrying quantum information in a broad variety of areas with a versatile approach. Furthermore, recent technological advances are now enabling first concrete applications of photonic quantum information processing. The goal of this manuscript is to provide the reader with a comprehensive review of the state of the art in this active field, with a due balance between theoretical, experimental and technological results. When more convenient, we will present significant achievements in tables or in schematic figures, in order to convey a global perspective of the several horizons that fall under the name of photonic quantum information.

Totally Destructive Many-Particle Interference

In a general, multimode scattering setup, we show how the permutation symmetry of a many-particle input state determines those scattering unitaries that exhibit strictly suppressed many-particle transition events. We formulate purely algebraic suppression laws that identify these events and show that the many-particle interference at their origin is robust under weak disorder and imperfect indistinguishability of the interfering particles. Finally, we demonstrate that all suppression laws so far described in the literature are embedded in the general framework that we here introduce.

Interferometric spatial mode analyzer with a bucket detector

A spatial mode analyzer based on a Michelson interferometer with a bucket detector is experimentally implemented. The delay line in the interferometer is an optical implementation of the fractional Fourier transform (fFT) which enables the spatial mode analysis of a given input field in the Hermite-Gaussian (HG) mode basis. Modal weights for both 1D and 2D input fields are experimentally measured. Results for input fields comprising of multiple HG modes are also presented.

Tapering of femtosecond laser-written waveguides

The vast development of integrated quantum photonic technology enables the implementation of compact and stable interferometric networks. In particular, laser-written waveguide structures allow for complex 3D circuits and polarization-encoded qubit manipulation. However, the main limitation in the scaling up of integrated quantum devices is the single-photon loss due to mode-profile mismatch when coupling to standard fibers or other optical platforms. Here we demonstrate tapered waveguide structures realized by an adapted femtosecond laser writing technique. We show that coupling to standard single-mode fibers can be enhanced up to 77% while keeping the fabrication effort negligible. This improvement provides an important step for processing multiphoton states on chip.

Non-Hermitian engineering of synthetic saturable absorbers for applications in photonics

We introduce a new type of synthetic saturable absorber based on quantum-inspired Jx photonic arrays. We demonstrate that the interplay between optical Kerr nonlinearity, interference effects and non-Hermiticity through radiation loss leads to a nonlinear optical filtering response with two distinct regimes of small and large optical transmissions. More interestingly, we show that the boundary between these two regimes can be very sharp. The threshold optical intensity that marks this abrupt "phase transition" and its steepness can be engineered by varying the number of the guiding elements. The practical feasibility of these structures as well as their potential applications in laser systems and optical signal processing are also discussed.

General optical discrete z transform: design and application

This paper presents a generalization of the discrete z transform algorithm. It is shown that the GOD-ZT algorithm is a generalization of several important conventional discrete transforms. Based on the GOD-ZT algorithm, a tunable general optical discrete z transform (GOD-ZT) processor is synthesized using the silica-based finite impulse response transversal filter. To demonstrate the effectiveness of the method, the design and simulation of a tunable optical discrete Fourier transform (ODFT) processor as a special case of the synthesized GOD-ZT processor is presented. It is also shown that the ODFT processor can function as a real-time optical spectrum analyzer. The tunable ODFT has an important potential application as a tunable optical demultiplexer at the receiver end of an optical orthogonal frequency-division multiplexing transmission system.