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Tailoring nonlinear processes for quantum optics with pulsed temporal-mode encodings

Vahid Ansari, John M. Donohue, Benjamin Brecht, Christine Silberhorn

arXiv:1803.04316v1quant-ph

TL;DR

High-dimensional time-frequency quantum information requires ways to generate diverse photon states and measure complex temporal waveforms. This review examines how group-velocity-engineered nonlinear processes tailor photon generation and frequency conversion; it surveys experimental progress and the practical challenges that remain.

  • Problem

    Generating diverse quantum states and accurately measuring complex temporal waveforms are outstanding needs for using time-frequency encodings in quantum information.

    Source: §I · p. 1, §IV · pp. 7, 10

  • Method

    The review surveys tailoring phasematching in parametric downconversion and sum-frequency generation to generate, manipulate, and measure temporal modes.

    Source: §I · p. 1, §IV · p. 7, §VII · p. 14

  • Results

    The reviewed work demonstrates temporal-mode reconstruction from quantum light with fidelity (98.8 ± 0.4)% for randomly generated seven-dimensional coherent superpositions.

    Source: §VI · p. 14

  • Takeaways & Limitations

    Dispersion-engineered waveguides provide a toolbox for generating and measuring photon temporal modes, including pure heralded photons and customizable temporal-mode structure.

    Source: §VII · p. 14

    Temporal-mode devices demonstrated with sub-picosecond or few-picosecond pulses face synchronisation challenges for long-distance communication, while longer regimes increase pulse-shaping demands.

    Source: §VII · p. 15

Abstract

from arXiv · show

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.

I. INTRODUCTION

Time-frequency encodings offer a high-dimensional photonic information space compatible with fibre networks and waveguides, but arbitrary temporal-mode control and measurement remain challenging. The review surveys nonlinear-process techniques for generating and manipulating these modes.

I. INTRODUCTION

  • Polarisation encodings are limited to two dimensions, while complex spatial modes are incompatible with spatially single-mode integrated devices and fibre networks.Source: §I · p. 1
  • Time-frequency encoding offers an in-principle unbounded Hilbert space and is intrinsically compatible with waveguides and fibre transmission.Temporal-mode bases include discrete time or frequency bins and intensity-overlapping, field-orthogonal pulses.Source: §I · p. 1, Figure 1 · p. 1
  • Arbitrary temporal-mode control and direct measurement remain significant challenges for time-frequency quantum information processing.Source: §I · p. 1
  • The review examines targeted generation and manipulation of temporal modes by controlling group-velocity relationships in nonlinear processes.It covers parametric downconversion, frequency conversion, experimental progress, applications, and remaining challenges.Source: §I · p. 1

II. TEMPORAL-MODE STRUCTURE OF PARAMETRIC DOWN-CONVERSION

Parametric downconversion produces photon pairs whose continuous time-frequency structure can be represented as discrete temporal modes using the Schmidt decomposition. This framework links mode structure to entanglement and photon purity, while ordinary configurations are often highly correlated and multimode.

II. TEMPORAL-MODE STRUCTURE OF PARAMETRIC DOWN-CONVERSION

  • PDC converts a bright classical pump photon into signal and idler daughter photons in a nonlinear medium and can generate heralded photons, squeezed states, and maximally entangled states.Source: §II · p. 2
  • The joint spectral amplitude is the product of a pump envelope reflecting energy conservation and a phasematching function expressing momentum conservation and medium dispersion.Periodic poling adds a quasi-phasematching component to the phase mismatch, allowing the phasematched centre frequencies to be tuned.Source: §II · p. 2
  • Schmidt decomposition connects the continuous time-frequency description to discrete temporal-mode pairs, with each pair occurring with probability λk.For a typical Gaussian JSA, the Schmidt modes are Hermite-Gaussian functions that overlap in spectral and temporal intensity.Source: §II · p. 3
  • The Schmidt number K quantifies the number of temporal-mode pairs: K = 1 denotes a separable single-mode state, while K ≫ 1 denotes a multimode entangled state.The Schmidt number is related to the spectral purity of individual signal photons, and PDC photon purity is experimentally accessible through unheralded signal-photon g(2)(0).Source: §II · p. 3
  • Most PDC configurations generate highly correlated states with large Schmidt number, yielding low-purity heralded photons without spectral filtering.Engineering the PDC process can overcome this limitation and facilitate direct generation of pure single photons.Source: §II · p. 3

III. PDC ENGINEERING

Without dispersion engineering, pump energy conservation imposes frequency anticorrelations that produce highly multimode PDC states. Tailoring phasematching to yield separable joint spectral amplitudes enables high-quality heralded photons without additional spectral filtering.

III. PDC ENGINEERING

  • Without dispersion engineering, pump energy conservation imposes intrinsic signal-idler frequency anticorrelations, producing highly multimode PDC states.Source: §III · p. 3
  • Tailoring the phasematching function can produce separable joint spectral amplitudes and single-mode PDC.This approach allows high-quality heralded photons without additional spectral filtering.Source: §III · p. 3
  • The section introduces group-velocity matching as an approach to engineering single-mode PDC.Source: §III · p. 3

emission

Group-velocity relationships shape the phasematching angle and joint spectral amplitude in PDC. Asymmetric matching can make the JSA more separable as its phasematching bandwidth narrows, while symmetric matching supports separable or positively correlated states under different bandwidth conditions.

emission

  • Group-velocity matching shapes the phasematching function and hence the JSA; aGVM matches the pump velocity to either signal or idler, while sGVM places the pump velocity between them.The cited prior work identifies group-velocity relationships as essential to tailoring phasematching.Source: emission · p. 4
  • The analysis assumes perfect phasematching at the central frequencies and negligible group-velocity dispersion in a first-order expansion of the phase mismatch.Source: emission · p. 4
  • When the pump bandwidth exceeds the phasematching bandwidth in aGVM, the JSA becomes more separable; broader pumps or longer interactions can increase single-modedness.The resulting signal and idler photons have drastically different spectral bandwidths.Source: emission · p. 4
  • In sGVM, equal pump and phasematching bandwidths produce a separable JSA and pure single photons with identical spectral properties.A broader pump instead permits positive spectral correlations and negative temporal correlations, useful for quantum synchronisation and dispersion cancellation.Source: emission · p. 4

III.2. Experimental high-purity photon sources

Experiments progressed from bulk-crystal control of spectral correlations to bright, high-purity waveguide sources and polarization-entangled photon pairs. Group-velocity matching enabled separable sources, telecom-wavelength degeneracy, and temporal anticorrelation.

III.2. Experimental high-purity photon sources

  • A 2007 LiIO3 bulk-crystal experiment tuned photon pairs from spectrally correlated to separable by adapting the pump’s spatial mode.The control used spectral-spatial coupling and was verified by measuring the joint spectral intensity.Source: III.2. Experimental high-purity photon… · p. 5
  • In 2008, an aGVM bulk KDP source produced photon pairs around 830 nm and demonstrated 94.4% Hong-Ou-Mandel visibility between heralded photons from separate sources.Source: III.2. Experimental high-purity photon… · p. 5
  • A 2008 ppKTP sGVM design used up-conversion and relative-delay scans to measure joint temporal intensity and demonstrate temporal anticorrelation.The work showed that changing pump spectral bandwidth can control photon-pair time-frequency correlations.Source: III.2. Experimental high-purity photon… · p. 5
  • In 2011, a bulk ppKTP sGVM source produced spectrally degenerate telecom photon pairs with 95% signal-idler Hong-Ou-Mandel visibility.Source: III.2. Experimental high-purity photon… · p. 5
  • A waveguide source based on Rubidium-exchanged ppKTP achieved brightness of about 2.5 photons per pulse at 70 pJ, with measured spectral purity up to 86.7%.Signal-idler indistinguishability was around 94%, confirmed with Hong-Ou-Mandel interference.Source: III.2. Experimental high-purity photon… · p. 5
  • Later sGVM sources were incorporated into dual-pumped Sagnac schemes to construct degenerate, highly pure photon-pair sources with polarization entanglement.Source: III.2. Experimental high-purity photon… · p. 5

III.3. The problem with side lobes

Sinc-shaped phasematching side lobes limit heralded-photon purity, while filtering can improve purity at a cost in heralding rate. Apodizing the phasematching profile suppresses side lobes and can enable high purity without filtering.

III.3. The problem with side lobes

  • With standard phasematching, side lobes limit heralded-photon purity to around 86% for sGVM; filtering raises it, but narrower filters reduce heralding rates.In aGVM, filtering the idler as herald yields up to 97% signal purity, versus about 92% when filtering the signal; sGVM filtering leaves about 94% purity either way.Source: III.3. The problem with side lobes · p. 5, Figure 5 · p. 6
  • Apodizing the QPM grating smooths the nonlinearity profile toward a Gaussian; one early demonstration suppressed side lobes by 13 dB but reduced peak efficiency and broadened phasematching.Other approaches include changing poling periodicity, duty cycle, domain orientation, or waveguide geometry.Source: III.3. The problem with side lobes · pp. 5–6
  • Apodization methods have achieved photon purities above 99% without spectral filtering, allowing the phasematching function to shape the PDC temporal-mode structure.Source: III.3. The problem with side lobes · p. 6

III.4. Controlled generation of temporal modes

Pump shaping combined with group-velocity matching controls whether PDC generates pure shaped single photons or multimode entangled states. The review describes arbitrary-dimensional maximally entangled states as possible without changing hardware, while noting that its first-order perturbative description applies only at low gain.

III.4. Controlled generation of temporal modes

  • In aGVM PDC, shaping the pump programs the signal-photon spectrum while the idler spectrum is set mostly by phasematching, provided phasematching is narrower than the desired spectral features.KTP waveguides demonstrated high-purity shaped photons at 1411 nm.Source: III.4. Controlled generation of… · p. 6
  • In sGVM PDC, pump shaping controls the number of excited temporal modes: a first-order Hermite-Gaussian pump produces two modes, while higher orders add modes that are generally unequally occupied.Source: III.4. Controlled generation of… · pp. 6–7
  • Time-bin superpositions provide balanced Schmidt-mode distributions and can generate maximally entangled states of arbitrary dimension without hardware changes.Source: III.4. Controlled generation of… · p. 7
  • The first-order perturbative PDC description is valid only in the weakly pumped, low-gain regime; at high gain, time ordering changes Schmidt modes and their squeezing.High-gain operation can nevertheless generate bright squeezed states for studying mesoscopic quantum phenomena.Source: III.4. Controlled generation of… · p. 7
  • The review concludes that PDC engineering now offers close-to-arbitrary control over generated temporal-mode structure, bringing temporal-mode quantum-information applications within reach.Source: III.4. Controlled generation of… · p. 7

IV. MANIPULATION AND MEASUREMENT OF TEMPORAL MODES

Engineered frequency conversion can act as a quantum pulse gate (QPG), selecting a chosen temporal mode from a multimode input while leaving orthogonal modes unconverted. Its selectivity depends on the transfer function and operating regime; temporal-mode interferometry can improve it at high conversion.

IV. MANIPULATION AND MEASUREMENT OF TEMPORAL MODES

  • Tailored asymmetric group-velocity-matched SFG has K ≈1.01, compared with K ≈3.7 for non-engineered SFG.The tailored cases use Gaussian and first-order Hermite-Gauss pump functions.Source: §IV · p. 8
  • A separable transfer function makes the QPG select one input temporal mode for conversion while orthogonal modes pass through undisturbed.The pump pulse shape sets the selected input mode, and the phasematching envelope sets the output mode; shaping the pump adapts the selected mode.Source: §IV · pp. 8–9
  • A single QPG reaches maximum selectivity of about 83%, as time-ordering effects make its operation multimode at high conversion.The transfer function broadens and develops additional correlations as pump power and conversion efficiency increase.Source: Figure 7 · p. 9, §IV · p. 10
  • Two 50%-efficient QPG stages in a Mach-Zehnder-like interferometer can achieve more than 98% selectivity through interference.The phases are adjusted so that interference produces complete conversion of the targeted input mode.Source: §IV · p. 10
  • Separability measures how well a QPG isolates a target mode from a mixture, independent of incomplete conversion.The suppression or extinction ratio instead quantifies suppression of signals from modes orthogonal to the target.Source: §IV · p. 10

V. EXPERIMENTAL PROGRESS ON TM SELECTION

Experimental work has realised temporal-mode selection using group-velocity-engineered frequency conversion and shaped pumps. Demonstrations span mode-selective measurement, high-efficiency conversion, and interferometric improvements, while material choice involves trade-offs between noise suppression and nonlinear strength.

V. EXPERIMENTAL PROGRESS ON TM SELECTION

  • A type-II lithium-niobate configuration provides group-velocity matching while avoiding phase-matched pump second-harmonic background, but its weaker nonlinearity requires stronger pump fields.The example uses a 1550 nm input, 875 nm pump, and green upconverted signal.Source: §V · p. 10, Figure 8 · p. 10
  • Group-velocity-mismatched SHG enabled ultrashort-pulse code-division access, with mismatched codes showing over 27 dB extinction after central-frequency filtering.The demonstration used a 20-mm PPLN sample and a telecommunications-wavelength pulse split into 16 channels.Source: §V · p. 11
  • A type-II PPLN quantum pulse gate converted a Gaussian mode with nearly 88% efficiency, while extinction was about 7 dB and limited by pulse-shaper resolution.Follow-up measurements reached 12.8 dB extinction for PDC photons and over 20 dB for shaped coherent light, at approximately 20% conversion efficiency.Source: §V · p. 11
  • A 17-element frequency comb in type-II PPKTP demonstrated efficiencies near 80% and 8 dB extinction ratios.A later near-group-velocity-matched type-0 PPLN setup demonstrated above 75% efficiency for a four-dimensional Hermite-Gaussian alphabet, with separabilities above 65%.Source: §V · p. 11
  • A type-0 PPLN gate reached 85% efficiency with 3.5 mW pump power using 500-fs pulses.In the high-efficiency regime, pump shapes differing from the low-efficiency analytical modes can yield greater conversion efficiency and mode selectivity.Source: §V · p. 12
  • Temporal-mode interferometry in a double-pass waveguide showed enhanced mode-selective Ramsey interference relative to single-stage expectations.The two 50%-efficient stages and phase reshaping suppress higher-order corrective terms.Source: §V · p. 12

V.1. Mode selection in quantum memories

Raman ensemble memories manipulate temporal modes by mapping an addressed optical mode to a spinwave and back. Their bandwidth varies widely by physical system, while single-mode operation can degrade with efficiency; a cavity is one proposed route to combine mode selectivity and high efficiency.

V.1. Mode selection in quantum memories

  • A strong control pulse maps an addressed input temporal mode to a spinwave, and another control pulse retrieves it as an optical field.The interaction can be described as a broadband beam splitter whose selected modes are determined by the control-pulse shape.Source: §V.1 · p. 12
  • Raman memories span bandwidths from a few MHz to THz depending on the physical system, and have been explored for multimode frequency-comb storage and photon frequency conversion.Process tomography demonstrated single-temporal-mode operation.Source: §V.1 · p. 12
  • As in a single-stage QPG, Raman-memory single-modedness degrades as efficiency increases; placing the memory inside a cavity is proposed to enable both high efficiency and mode selectivity.Source: §V.1 · p. 12

V.2. Multimode manipulations with sum-frequency generation

Multimode sum-frequency generation (SFG) can reshape temporal-mode fields for applications such as memory interfaces and temporal-mode rotations. Dispersion and SFG also provide a route to temporal imaging, with group-velocity matching and crystal length constraining the design.

V.2. Multimode manipulations with sum-frequency generation

  • Multimode SFG can reshape fields beyond add/drop mode selection, including matching PDC photons to memory acceptance and implementing temporal-mode rotations.The passage identifies mode-selective interfaces as unable to reshape multimode fields except through resource-intensive add/drop devices.Source: §V.2 · p. 12
  • Frequency conversion and bandwidth manipulation have preserved time-bin and polarization entanglement in experiments.Source: §V.2 · p. 13
  • Dispersion plus SFG manipulates PDC-photon bandwidth and timescale and reshapes entangled-pair time-frequency structure; broad phasematching requires group-velocity matching of all fields.Source: §V.2 · p. 13
  • SFG time lenses often need short crystals, yet can in principle achieve high efficiency without mode-selective measurement's time-ordering roadblocks; FWM, electro-optic modulation, and cross-phase modulation are alternatives.Source: §V.2 · p. 13

VI. TOWARDS APPLICATIONS OF TEMPORAL MODES IN QUANTUM INFORMATION SCIENCE

Temporal-mode applications include quantum-state characterization and projective measurements, with calibrated QPG reconstructions reaching (98.8 ± 0.4)% fidelity for seven-dimensional superpositions. The review also describes experimental progress and practical challenges in synchronization, pulse shaping, and scaling.

VI. TOWARDS APPLICATIONS OF TEMPORAL MODES IN QUANTUM INFORMATION SCIENCE

  • Complete d-dimensional tomography needs projections onto at least d^2 states; a complete set of d + 1 mutually unbiased bases is sufficient, as illustrated for Hermite-Gauss modes.Source: §VI · p. 13, Figure 11 · p. 13
  • Time-bin measurements can use interferometers or SFG time-to-frequency conversion, while frequency-bin schemes use SFG recombination or electro-optic sidebands and rotations.SFG time-to-frequency projection resolves superpositions below detector time resolution but is limited to a maximum efficiency of 1/d; recombination-based frequency-bin schemes are difficult to extend to quantum networks.Source: §VI · pp. 13–14
  • Dispersion-engineered SFG can project onto arbitrary temporal-mode bases when its transfer function remains separable, but high-dimensional protocols require low-noise, coherent operation on superpositions.The review describes this application as an active research field.Source: §VI · pp. 13–14
  • (98.8 ± 0.4)% fidelity was achieved reconstructing randomly generated seven-dimensional coherent temporal-mode superpositions with a detector-tomography-calibrated QPG.The review states these were the only dispersion-engineered temporal-mode measurements performed with a quantum light source rather than attenuated coherent light.Source: §VI · p. 14
  • A QPG enabled mode-selective photon subtraction from multimode squeezed light, with 96% subtraction-matrix purity for a seven-dimensional Hermite-Gauss superposition at both bright and single-photon levels.The photon-subtraction QPG used 0.1% efficiency to minimize disturbance of the quantum state.Source: §VI · p. 14

VII. OUTLOOK AND CHALLENGES

  • Dispersion-engineered waveguides provide tools to generate and measure temporal modes, including pure heralded photons for high-visibility multiphoton interference and customizable temporal-mode entanglement.The review also describes considerable experimental progress toward realizing this toolbox.Source: §VII · pp. 14–15
  • Temporal-mode devices have been demonstrated at sub- or few-picosecond scales, creating synchronization challenges for long-distance communication; longer memory or cavity regimes ease jitter concerns but increase pulse-shaping demands.Source: §VII · p. 15
  • High-dimensional scaling remains difficult because single-mode isolation needs multiple shaped pulses and physical media; temporal-mode demultiplexers are identified as essential for scaling networks.Multi-peak phasematching structures are described as a promising route to such demultiplexers.Source: §VII · p. 15

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