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A special issue of Quantum Reports (ISSN 2624-960X).

Deadline for manuscript submissions: closed (31 December 2021) | Viewed by 5587

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Dear Colleagues,

The study and application of optics is arguably one of the longest standing fields in physics that stretches from pre-Newtonian to post-Maxwellian eras. Famous pioneers involved in the discovery of contemporary quantum mechanics during the 1920s further helped to shed new light on electromagnetism that eventually led to the birth of quantum optics. Since then, creative applications of quantum optics have been engineered to improve measurement precisions, simulate nonclassical physics, and establish secure quantum protocols, among many other new-age endeavors.

The aim of this Special Issue is to highlight the recent topics of discussion in quantum optics. We hope to attract article contributions from all researchers and experts who are engaged in cutting-edge developments in the field of quantum optics. We encourage authors to submit either research or review articles in areas in, but not limited to, the list below:

• Quantum imaging;
• Quantum sensing and metrology;
• Optical lattices and photonic quantum simulation;
• Optical angular-momentum and extended spatial-mode photonics;
• Time/frequency quantum optics;
• Light–matter interaction and quantum optomechanics;
• Quantum nonlinear optics;
• Quantum coherence;
• Cavity quantum electrodynamics;
• Photonic solid-state devices.

We believe that this Special Issue will help to trigger new and exciting frontiers and applications that will subsequently evolve this important field in quantum physics to new scientific heights.

Dr. Yong Siah Teo Guest Editor

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• Quantum imaging
• Quantum sensing and metrology
• Optical lattices and photonic quantum simulation
• Optical angular-momentum and extended spatial-mode photonics
• Time/frequency quantum optics
• Light–matter interaction and quantum optomechanics
• Quantum nonlinear optics
• Quantum coherence
• Cavity quantum electrodynamics
• Photonic solid-state devices
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Quantum Physics

Title: introduction to quantum optics.

Abstract: These are the lecture notes for a course that I am teaching at Zhiyuan College of Shanghai Jiao Tong University (available at this https URL ), though the first draft was created for a previous course I taught at the University of Erlangen-Nuremberg in Germany. It has been designed for students who have only had basic training on quantum mechanics, and hence, the course is suited for people at all levels. The notes are a work in progress, meaning that some proofs and many figures are still missing. However, I've tried my best to write everything in such a way that a reader can follow naturally all arguments and derivations even with these missing bits. Quantum optics treats the interaction between light and matter. We may think of light as the optical part of the electromagnetic spectrum, and matter as atoms. However, modern quantum optics covers a wild variety of systems, including superconducting circuits, confined electrons, excitons in semiconductors, defects in solid state, or the center-of-mass motion of micro-, meso-, and macroscopic systems. Moreover, quantum optics is at the heart of the field of quantum information. The ideas and experiments developed in quantum optics have also allowed us to take a fresh look at many-body problems and even high-energy physics. In addition, quantum optics holds the promise of testing foundational problems in quantum mechanics as well as physics beyond the standard model in table-sized experiments. Quantum optics is therefore a topic that no future researcher in quantum physics should miss.

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Quantum Optics

Quantum optics is the study of quantized light (photons) and its interaction with matter. The advancement of quantum optics theory and experiments enabled remarkably precise tests of fundamental questions in physics, as well as applications ranging from lasers to quantum computing. The University of Rochester is one of the world's leading centers of quantum optics since its birth in the 1960's, with the notable distinction of having three former students/faculty, Steven Chu in 1997 and Donna Strickland and Gérard Mourou in 2018, recognized and honored with Nobel Prizes in Physics for their pioneering contributions to quantum optics. In fact, the very term “quantum optics” was coined in Rochester, during the fruitful collaborations of Emil Wolf and Leonard Mandel.

At present there are more than a dozen different research groups at the University involved in different aspects of quantum optics, with a strong interest in foundational questions in quantum mechanics, involving counter-intuitive quantum phenomena such as superposition and entanglement. Simultaneously, our research is driven by the promise of bringing futuristic applications to fruition, including quantum computing, cryptography and teleportation. Research areas include quantum theory (Eberly, Franco, Jordan, Landi), atomic, molecular and optical (AMO) experiments with trapped atoms (Bigelow), defect centers (Vamivakas), nanophotonics (Cardenas, Lin), non-linear optics (Agrawal, Boyd), as well as quantum optics experiments in the solid state with superconducting circuits (Blok), spin qubits (Nichol), 2D materials (Wu) and optomechanics (Renninger). A broad overview of quantum research across all departments of University of Rochester can be found at the UR Quantum website .

Department Research

Departmental research in quantum optics spans a wide range of topics:

• Professor Agrawal's research interests are in the area of theoretical optics, particularly quantum electronics, nonlinear optics, and laser physics. His current research is focused on nonlinear silicon photonics, highly nonlinear fibers, and all-optical signal processing with semiconductor optical amplifiers.
• The Cooling and Trapping (CAT) Laboratory of Professor Bigelow is focusing on topological excitations of a spinor Bose-Einstein condensate for fundamental understanding and for application to quantum metrology and information. The CAT group also has a leading program on the formation and control of ultra-cold polar molecules. Experimental and theoretical work spans a range of studies of nonlinear atom (and molecular) optics.
• Professor Blok’s research focuses on the quantum mechanical properties of superconductors, including superconducting qubits and microwave resonators. Areas of interest include quantum computing with multi-level systems(qudits) and quantum simulation with superconducting circuits.
• Professor Boyd is interested in studies of the nonlinear interaction of light with matter, in the use of nonlinear optics to control the group velocity of light, in the development of nanostructured materials with exotic optical properties, in the study of quantum states of light, and in the development of applications of these techniques.
• Professor Cardenas’ research focuses on integrated photonics, nanophotonics, and nonlinear photonics. His group tackles high impact challenges using nanostructured devices on a chip. Current research is focused on four main areas: photonic packaging, 2D materials integrated photonics, nonlinear photonics, and on-chip quantum photonics.
• Professor Eberly's group is involved in theoretical studies of nonclassical states of radiation, continuous quantum entanglement, optical dark-state solitons, and electron correlation in high-field ionization.
• Professor Franco works at the interface of chemistry, physics, optics and nanoscience, using theory and simulation to develop new methods to probe and control the behavior of matter by means of external stimuli. Topics of interest include quantum dynamics, investigating basic de-coherence processes in the condensed phase, exploring frontiers of the laser-matter interaction, and advancing single-molecule spectroscopies that can be constructed in the context of nanoscale junctions.
• Professor Jordan investigates the quantum theory of dynamics and measurement in condensed matter and optical contexts. He is involved in research of electron transport and fluctuations in mesoscopic systems, many-body quantum entanglement, quantum thermodynamics, and the foundations of quantum mechanics.
• Professor Landi’s research is in the field of theoretical quantum information sciences and technologies. Areas of interest include open quantum systems, quantum thermodynamics, quantum transport and quantum metrology. His recent work focuses on reformulating the laws of thermodynamics, and concepts such as resource expenditure and irreversibility, within a quantum-coherent context.
• Professor Lin’s research focuses on understanding the fundamental physics of novel nonlinear optical, quantum optical, and optomechanical phenomena in micro-/nanoscopic photonic structures, and on finding their potential applications towards chip-scale photonic signal processing in both classical and quantum regimes.
• Professor Nichol’s group conducts research on the quantum mechanical properties of individual electrons in semiconductor quantum dots. Particular areas of interest are quantum computing with spin qubits, many-body quantum coherence, and coherent spin-phonon coupling.
• Professor Renninger’s research interest is in experimental light-matter interactions. His group focuses on ultrafast nonlinear optics and pulsed lasers for applications including imaging deep into the brain. They also investigate the coherent interactions between photons and phonons for applications such as quantum computing, high-speed networking, and dark matter detection.
• Professor Vamivakas ' research efforts center on light-matter interactions at the nanosclae, using optics to interrogate and control both artificial and naturally occurring solid state quantum emitters. Potential applications range from optical metrology to quantum information science.
• Professor Wu’s research involves using new quantum materials to create novel electronic devices beyond Moore's law computation. Topics such as spintronics, topological electronics, and multifunctional complex oxide-based transistors are explored from the perspective of materials synthesis, nano-fabrication, and low-noise device characterization.

Quantum Optics

Arbitrarily structured quantum emission with a multifunctional metalens

Chi Li, Jaehyuck Jang, Trevon Badloe, Tieshan Yang, Joohoon Kim, Jaekyung Kim, Minh Nguyen, Stefan A. Maier, Junsuk Rho, Haoran Ren & Igor Aharonovich 

Letter | Published: 07 August 2023

Integrated preparation and manipulation of high-dimensional flying structured photons

Haoqi Zhao, Yichi Zhang, Zihe Gao, Jieun Yim, Shuang Wu, Natalia M. Litchinitser, Li Ge and Liang Feng

Research Article | Published: 29 June 2024

A source of entangled photons based on a cavity-enhanced and strain-tuned GaAs quantum dot

Michele B. Rota, Tobias M. Krieger, Quirin Buchinger, Mattia Beccaceci, Julia Neuwirth, Hêlio Huet, Nikola Horová, Gabriele Lovicu, Giuseppe Ronco, Saimon F. Covre da Silva, Giorgio Pettinari, Magdalena Moczała-Dusanowska, Christoph Kohlberger, Santanu Manna, Sandra Stroj, Julia Freund, Xueyong Yuan, Christian Schneider, Miroslav Ježek, Sven Höfling, Francesco Basso Basset, Tobias Huber-Loyola, Armando Rastelli and Rinaldo Trotta 

Research Article | Published: 24 July 2024

Polarization-entangled photon-pair source with van der Waals 3R-WS2 crystal

Jiangang Feng, Yun-Kun Wu, Ruihuan Duan, Jun Wang, Weijin Chen, Jiazhang Qin, Zheng Liu, Guang-Can Guo, Xi-Feng Ren and Cheng-Wei Qiu 

Research Article | Published: 23 August 2024

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Quantum optics laboratory.

17 Oxford Street Cambridge, MA 02138

Quantum Optics Theory

Topics of theoretical research in quantum optics include:

• Hybrid atom-nanophotonic systems
• Chiral and topological quantum optics
• Transformation optics and metamaterials
• Quantum acoustics
• Spin-phonon networks
• Quantum optics of transition metal dichalcogenides
• Matter-light interaction in ordered atomic arrays

• Department of Physics

Quantum Optics

Topics of study include: the properties of light; fundamentals of quantum mechanics; the interaction of light with photonic crystals, semiconductors, quantum wells and superlattices; ultrafast and nonlinear optical phenomena in condensed matter systems; Bose-Einstein condensation and neutral Fermi gases; trapped ions; plasma physics; and studies of biological systems using optical probes. Such a wide variety of interests are drawn together by a common focus on lasers as primary experimental tools, and on quantum mechanics as the primary theoretical paradigm.

Research Areas

Nonlinear optics and ultrafast lasers.

Experiment: Robin Marjoribanks ,  Dwayne Miller , Aephraim Steinberg Theory: Daniel James ,  Sajeev John ,  John Sipe

Photonic Materials and Resonant Optical Structures

Theory: Sajeev John ,  John Sipe

Precision Spectroscopy

Experiment: Boris Braverman , Amar Vutha

Quantum Information

Experiment: Boris Braverman, Aephraim Steinberg ,  Hoi-Kwong Lo

Theory: Daniel James ,  Hoi-Kwong Lo ,  John Sipe

Ultra-intense Laser-matter Interaction

Experiment: Robin Marjoribanks

Ultracold Atoms

Experiment: Boris Braverman, Aephraim Steinberg , Joseph Thywissen ,  Amar Vutha

Theory: Sajeev John ,  Daniel James , Hae-Young Kee , Yong Baek Kim ,  Arun Paramekanti ,  John Sipe

See also the research topics listed on the  condensed matter and  biological physics pages, which include QO faculty.

• Quantum Optics and AMO Physics Seminar series

The QO/AMO (quantum optics and atomic, molecular, and optical physics) seminar is held regularly during the academic year, usually on Fridays at 11:10 am.

• Toronto Quantum Information Seminars

Our Quantum Information seminar is held on Friday mornings at 11:10 am, often in the Fields Institute (222 College St), and is affiliated with the   Centre for Quantum Information and Quantum Control . The QuInf seminar ongoing work and ideas about quantum computation, cryptography, teleportation, et cetera. We hope to bring together interested parties from a variety of different backgrounds, including math, computer science, physics, chemistry, and engineering, to share ideas as well as open questions.

You also might also see the photos from our internal research  symposium , sometimes held at the end of the summer.

Affiliations

Our activities are quite interdisciplinary, overlapping with other clusters in the department —  condensed matter and  biophysics — and bridging to other departments, including  chemistry ,  mathematics , and  electrical engineering . Our members participate in two different programs of the Canadian Institute for Advanced Research: Quantum Materials and Quantum Information Processing. Many of the group are founding members of the  Centre for Quantum Information and Quantum Control .

PRX Quantum

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Five Open Problems in Quantum Information Theory

Paweł horodecki, łukasz rudnicki, and karol życzkowski, prx quantum 3 , 010101 – published 3 march 2022.

• Citing Articles (42)
• INTRODUCTION
• DISCRETE STRUCTURES IN THE HILBERT SPACE
• QUANTUM METROLOGY
• QUANTUM ENTANGLEMENT AND ITS…
• CONCLUDING REMARKS
• ACKNOWLEDGMENTS

We identify five selected open problems in the theory of quantum information, which are rather simple to formulate, are well studied in the literature, but are technically not easy. As these problems enjoy diverse mathematical connections, they offer a huge breakthrough potential. The first four concern existence of certain objects relevant for quantum information, namely a family of symmetric informationally complete generalized measurements in an infinite sequence of dimensions, mutually unbiased bases in dimension six, measurements saturating multiparameter Cramér-Rao bound and bound entangled states with negative partial transpose. The fifth problem requires checking whether a certain state of a two-ququart system is two-copy distillable.

• Received 21 December 2020
• Revised 1 December 2021

DOI: https://doi.org/10.1103/PRXQuantum.3.010101

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

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Authors & Affiliations

• 1 International Centre for Theory of Quantum Technologies (ICTQT), University of Gdańsk, Gdańsk 80-308, Poland
• 2 Faculty of Applied Physics and Mathematics, Technical University of Gdańsk, Gdańsk 80-952, Poland
• 3 National Quantum Information Centre (KCIK), University of Gdańsk, Poland
• 4 Center for Theoretical Physics, Polish Academy of Sciences, Aleja Lotników 32/46, Warsaw 02-668, Poland
• 5 Institute of Theoretical Physics, Jagiellonian University, Kraków 30-348, Poland
• * [email protected]

Popular Summary

Quantum information gradually enters the era, in which seminal theoretical and experimental research is being turned into quantum technologies. The current aim of the field mainly lays in taking well-known theoretical concepts, such as quantum cryptography, and involving them in operational devices. Such devices, while based on `standard' technologies developed so far, shall possess essential functionalities solely operating on quantum principles such as quantum entanglement.

Just looking at the case of quantum entanglement, we can therefore track the development relevant for its understanding, departing from a famous `spooky action at a distance' by Einstein, through entanglement's essential role in a theoretical proposal by Shor, providing an algorithm about how to factorize (very large) numbers, arriving at dozens of quite-well controllable entangled qubits to-be-accessible nowadays.

Being well aware of this turning moment for the community, with our contribution we aim to endorse areas of research within theoretical quantum information, which while deeply rooted in the good old quantum information, still nurture a significant potential for a wider development of the whole field. To this end, we select five well-known and very difficult open problems pertaining to existence of symmetric structures in quantum theory, quantum metrology and distillability of entanglement. We provide a thorough review of the subfields each problem belongs to, propose an extended motivation behind it, as well as sketch potential approaches to find the solution.

Article Text

Vol. 3, Iss. 1 — March - May 2022

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Four pure states | ψ i ⟩ span a regular tetrahedron inscribed in the Bloch sphere and lead to a single-qubit symmetric informationally complete measurement (SIC POVM) for N = 2 . Can you find N 2 pure states of size N such that the corresponding projectors form a simplex inscribed into the set of quantum states of a given order N ?

Three eigenbases of Pauli matrices σ x , σ y , σ z span an octahedron inscribed in the Bloch sphere and form a set of three mutually unbiased bases for N = 2 . There exist four, five, and six MUBs in dimensions N = 3 , 4 and 5 , respectively. How many MUBs exist for N = 6 ?

Pictorial comparison between quantum metrology on the single-copy level (a) and with multiple copies being subject to collective measurements visualized by a frame (b).

In the two-qubit problem, d = 2 , the set of separable states coincides with the set of PPT states and there are no bound entangled states since any entangled state is distillable. For higher dimensions, d > 2 , one asks whether the hypothetical region representing bound entangled states with negative partial transpose, depicted in yellow, is empty or not.

Sketch of the convex set of mixed quantum states for a d ⊗ d system with d > 2 , which contains the sets of separable states, a larger set of PPT states and the sets of states with various classes of distillability. The line represents the family of Werner states, Eq. ( 5 ), and the points A , B 1 , B , and C correspond to states labeled by α equal to − 1 , − 2 / d , − 1 / d , and 1 , respectively. Point A represents here the mixed state equal to the normalized projector onto the antysymmetric subspace. Problem 4 of the existence of NPT bound entanglement is equivalent to the question, whether point B ∞ , the position of which is still unknown, differs from point B . The nature of the states along the dashed line B 1 B is still unclear—to solve Problem 5 one has to decide, whether in case d = 4 the unknown point B 2 is identical with B 1 .

Example of N = 4 Greaco-Latin square prepared for bridge players. Due to works of Euler and Tarry we know that for N = 6 a similar design of 36 cards of six different suits and six different ranks (or 36 officers of different ranks and arms) does not exist. Is there a solution of the N = 6 problem if we play bridge with quantum cards, like ( | K ♣ ⟩ + | Q ♢ ⟩ ) / 2 , or allow the officers of Euler to be entangled?

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This book highlights quantum optics technologies that can revolutionize the way we encode, store, transmit, and handle information. These technologies can help us overcome bottlenecks in classical physics-based information technology in information transmission capacity, computing speed, and information security. The book provides readers with new perspectives on potential applications of the quantum theory. Besides, the book summaries the research advances in quantum optics and atom optics, including manipulation and construction of the quantum states of photons and even atoms, molecules, and matter at the quantum level, and new phenomena and technologies brought about by the interactions between photons and the quantum states of matter.

The book provides extensive and thoroughly exhaustive coverage of quantum optics. It is suitable for researchers and graduate students of optical physics and quantum optics.

Optical Quantum Computing

• Quantum Optics
• Atom Optics
• Optomechanically Induced Transparency
• Photonic Entanglement
• Quantum Storage
• Quantum Teleportation
• Phase Measurement and Quantum Noise
• Two-dimensional Magneto-optical Trap
• Heralded Single-photon Source
• Quantum Correlation in Raman Scattering
• Optical Quantum Correlation Interferometer

Table of contents (2 chapters)

Front matter.

Zeng-Bing Chen

New Progress in Quantum Optics and Atom Optics

Weiping Zhang

Back Matter

Editors and affiliations, about the editors.

Wei-Ping Zhang is currently a Zhiyuan Chair Professor at Shanghai Jiao Tong University. He received his Ph.D. degree from Shanghai Institute of Optics and Fine Mechanics, Chinese Academy of Sciences. His research interests include quantum optics and atom optics, atomic, molecular and optical physics and quantum manipulation and quantum metrology. He is Fellow of Optical Society of America (the U.S.A.) and received the Outstanding Referee Award by American Physical Society (the U.S.A.). He is also granted the honorary titles of National Distinguished Young Scholar (China) and Yangtze River Scholar (China).

Bibliographic Information

Book Title : Special Topics in Quantum Optics

Editors : Weiping Zhang, Zeng-Bing Chen

DOI : https://doi.org/10.1007/978-981-99-8454-1

Publisher : Springer Singapore

eBook Packages : Physics and Astronomy , Physics and Astronomy (R0)

Copyright Information : Shanghai Jiao Tong University Press 2024

Hardcover ISBN : 978-981-99-8453-4 Published: 01 June 2024

Softcover ISBN : 978-981-99-8456-5 Due: 15 June 2025

eBook ISBN : 978-981-99-8454-1 Published: 31 May 2024

Edition Number : 1

Number of Pages : V, 226

Number of Illustrations : 150 b/w illustrations

Topics : Quantum Optics , Quantum Computing , Quantum Physics , Atomic, Molecular, Optical and Plasma Physics

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• Open access
• Published: 13 September 2024

Unipolar quantum optoelectronics for high speed direct modulation and transmission in 8–14 µm atmospheric window

• Hamza Dely   ORCID: orcid.org/0000-0002-0457-5384 1   na1 ,
• Mahdieh Joharifar 2   na1 ,
• Laureline Durupt 3   na1 ,
• Armands Ostrovskis 4 ,
• Richard Schatz 2 ,
• Thomas Bonazzi 1 ,
• Gregory Maisons 3 ,
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• Lu Zhang 5 ,
• Sandis Spolitis 4 ,
• Yan-Ting Sun   ORCID: orcid.org/0000-0002-8545-6546 2 ,
• Vjačeslavs Bobrovs   ORCID: orcid.org/0000-0002-5156-5162 4 ,
• Xianbin Yu 5 ,
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• Xiaodan Pang   ORCID: orcid.org/0000-0003-4906-1704 2 , 4 , 7 &
• Carlo Sirtori   ORCID: orcid.org/0000-0003-1817-4554 1  

Nature Communications volume  15 , Article number:  8040 ( 2024 ) Cite this article

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• Fibre optics and optical communications
• Mid-infrared photonics

The large mid-infrared (MIR) spectral region, ranging from 2.5 µm to 25 µm, has remained under-exploited in the electromagnetic spectrum, primarily due to the absence of viable transceiver technologies. Notably, the 8–14 µm long-wave infrared (LWIR) atmospheric transmission window is particularly suitable for free-space optical (FSO) communication, owing to its combination of low atmospheric propagation loss and relatively high resilience to turbulence and other atmospheric disturbances. Here, we demonstrate a direct modulation and direct detection LWIR FSO communication system at 9.1 µm wavelength based on unipolar quantum optoelectronic devices with a unprecedented net bitrate exceeding 55 Gbit s −1 . A directly modulated distributed feedback quantum cascade laser (DFB-QCL) with high modulation efficiency and improved RF-design was used as a transmitter while two high speed detectors utilizing meta-materials to enhance their responsivity are employed as receivers; a quantum cascade detector (QCD) and a quantum-well infrared photodetector (QWIP). We investigate system tradeoffs and constraints, and indicate pathways forward for this technology beyond 100 Gbit s −1 communication.

Introduction

Driven by growing bandwidth demands, wireless communications are transitioning from microwaves to millimeter-waves (MMW) and soon to terahertz (THz). This trend points towards an all-spectra communication paradigm, utilizing any available electromagnetic (EM) resources across radio and optics for bandwidth 1 . Mid-infrared (MIR) (3–30 µm) represents a compelling segment of the EM spectrum, raising significant interest for applications in spectroscopy 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , defense 11 , 12 , 13 , astronomy 14 , 15 , and free-space optical (FSO) communications 16 , 17 . Two atmospheric transmission windows in the MIR, namely, the mid-wave infrared (MWIR, 3–5 µm) and the long-wave infrared (LWIR, 8–14 µm), hold intrinsic advantages for both terrestrial and space applications. They provide broader bandwidth and nearly 100 times lower atmospheric water absorption than MMW and THz 17 . They also experience considerably reduced Mie scattering, commonly found in meteorological phenomena such as dust, haze, and low-altitude clouds, than the 1.55-µm telecom band 18 . Therefore, MIR potentially offers high link availability through the atmosphere 19 .

Two main MIR FSO communication approaches are wavelength conversion and direct-emitting sources 20 . The former, by employing nonlinear parametric conversions, leverages mature fiber-optic components, supports very high data rates, offers compatibility with fibre-optic systems, and facilitates multi-dimensional multiplexing 21 , 22 , 23 , 24 . The latter focuses more on footprint and energy consumption to reinvent compact MIR FSO transceivers. An early study utilized a direct-emitting PbCdS diode laser at 3.5 μm for 100 Mb/s data transmission, with speed limited by carrier lifetime 25 . Since the 1990s, there has been an advent of a new assemblage of optoelectronic devices based on intersubband transitions, such as quantum cascade laser (QCL) 26 , quantum-well infrared detector (QWIP) 27 and quantum cascade detector (QCD) 28 , 29 , 30 . Using III-V semiconductor heterostructures, these unipolar devices can target any wavelength from MIR to THz 31 . They bring opportunities to build novel integrated systems for sensing and communication 32 , 33 . In particular, data transmission benefits from the intrinsic high-speed properties of intersubband devices due to their fast electron relaxation time under a picosecond 34 . Previous characterizations at the component-level have demonstrated promising results regarding QCL modulation response 35 , 36 , 37 , 38 , 39 , detectors’ bandwidth, as well as receiver responsivity 40 , 41 , 42 , 43 , 44 . These results encourage further explorations at the system level, which is far more complicated to characterize than the single components and devices. The implementation of an operational system present unique challenges arising from the holistic optimizations and evaluations of the overall trade-offs of each component, in terms of bandwidth, power, and linearity under real operating conditions. Furthermore, the characteristics of the optoelectronic components must be synthesized with information and coding theory, along with advanced communication technologies such as modulation and signal processing techniques, to maximize transmission performance in such systems 20 . Even though preliminary data transmission efforts were initiated in the early 2000s, most devices at that time needed to be operated at cryogenic temperature 45 , 46 . The subsequent two decades have witnessed remarkable advancements in both unipolar lasers and detectors, paving the way for room-temperature operation. To date, several MIR FSO transmissions are carried out using directly modulated QCL 47 , 48 , 49 , 50 , 51 , reaching speed over 10 Gbit s −1 . However, these devices’ potential remains underutilized, mainly due to QCL’s suboptimal RF mounting and detectors’ low signal-to-noise ratio (SNR) at room temperature. Here, we’ve optimized the bandwidth of a QCL chip through refined RF design and bonding, and enhanced room-temperature QCD and QWIP’s responsivity and speed by combining metamaterials with III-V heterostructures 43 . Utilizing these refined devices, we achieved unprecedented >55 Gbit s −1 net bitrate LWIR FSO transmission at room temperature.

Characteristics of unipolar quantum optoelectronic devices

Two distributed feedback (DFB) QCLs were designed, fabricated and compared for FSO transmission (see details in Methods). Both lasers originating from the same process flow, emit at around 9.1 µm wavelength and operate in continuous-wave (CW) mode at room temperature. However, the two QCL chips differ in width, wiring, and submount soldering configurations, addressing the tradeoff between power and bandwidth. The first QCL chip, referred to as Standard-QCL, is optimized for high output power through a dedicated thermal dissipation design. It is 2-µm wide, soldered epi-down, and bonded onto a standard submount tailored for DC operation. It is then placed on a copper block with a cooling system, achieving an output power of more than 30 mW in CW mode at 15 °C. In contrast, the second QCL chip, referred to as RF-QCL here, is optimized for RF characteristics. It has a width of 4 µm, and is soldered epi-up onto a cleaved submount of the exact dimensions as the QCL chip. This cleavage enables laser bonding using short wires for high-speed operation. Then, the QCL chip is soldered onto a copper base plate, followed by bonding onto a printed circuit board (PCB). Additional wirebonds connect the bottom of the laser to the PCB on each side of the central line, as shown in Fig.  1a . The PCB was designed to accommodate an SMA connector for RF injection, utilizing the high-speed capabilities of the QCL. The RF-QCL emits a relatively lower power of 13 mW at 15 °C, mainly due to the thermal dissipation challenge from the suboptimal indium soldering at the interface between the submount and the copper base plate. In summary, the Standard-QCL prioritized output power with optimized heat dissipation, whereas the RF-QCL emphasized bandwidth with refined RF design.

a A microscopic photo showing the the wirebonding between the cleavered RF-QCL submount and the PCB to enhance the frequency response. b The measured L-I-V curves of both the Standard-QCL and the RF-QCL at 15 °C in CW mode. Note that these measurements are performed in the system-level setup, and characterized power values are lower than chip-level measurements due to the beam divergence. While the actual output power of the QCL chips are higher, these measured values correspond to the actual power levels the detectors receive in the free-space transmission setup. The measured optical spectra with normalized intensity with respect to the bias current of c the Standard-QCL, and d the RF-QCL. e The characterized modulation bandwidth of the RF-QCL with electrical rectification at different bias current points.

The light-current-voltage (L-I-V) curves for both QCLs are shown in Fig.  1b , measured at 15 °C with mounted FSO setup. The Standard-QCL has a threshold current of around 400 mA, while the RF-QCL’s is about 350 mA. The RF-QCL also experiences slight roll-off beyond 460 mA. The normalized spectra of both lasers at different bias points are depicted in Fig.  1c, d , measured at 15 °C with a spectrometer (Nicolet iS50 FTIR). For both lasers, the emission peak shifts to longer wavelengths with higher bias current, as theory predicts. The current-tunning coefficient is calculated to be (264 ± 4) MHz/mA for the Standard-QCL, and (193 ± 10) MHz/mA for the RF-QCL. To verify the enhanced modulation bandwidth of the RF-QCL, additional characterizations were performed using electrical rectification 52 (see Methods and Supplementary Information Note  2 ). Modulation bandwidth of around 10 GHz can be obtained at different bias points with a flat frequency response, with neither relaxation oscillations nor electrical free spectral range (FSR) resonances, as shown in Fig.  1e . This result represents a substantial improvement compared to a previous design 51 .

Figure  2a, b present SEM images of the QCD, which layout is analogous to the QWIP. The device fits in a 60 × 60 µm 2 area. A 50 Ω coplanar waveguide and an air bridge are processed on the sample for easier electrical connection and enhanced frequency performance. These home-made detectors use a one-dimensional stripe array metamaterial design where the active region (AR) is embedded in stripe-shaped metal-semiconductor-metal resonators, with width S matching half the AR’s resonant radiation wavelength (Fig.  2c ). Conceptually, metamaterial-based intersubband detectors function like two coupled oscillators, the cavity and the intersubband transition, exchanging energy 53 , 54 . S is chosen so that the cavity resonant frequency matches the AR peak absorption wavelength for each device. The resonator height H can be adjusted to modify both the cavity mode’s coupling efficiency with incident radiation and its quality factor. Both QCD and QWIP operate in the weak light-matter coupling regime.

a SEM picture of the stripe-based QCD and the view with its coplanar waveguide. The QWIP used in this experiment has the same structure and appearance. b The zoomed-in image on the device. Each stripe is 1.5 μm wide and the space between each stripe is 7 μm. Both the QCD and the QWIP are metamaterial-patterned. c Lateral view of a single detector stripe cavity resonator with its active region (AR). d Sketch of the stripe-array detector with illustrations of the electric field \(\overrightarrow{{{{{\bf{E}}}}}}_{{{{\bf{cav}}}}}\) induced by the incident electromagnetic signal \({S}_{{in}}\) polarized along the resonant dimension of the stripe. \(\overrightarrow{{{{{\bf{E}}}}}}_{{{{\bf{cav}}}}}\) turns vertical to both meet the polarization selection requirement. e The normalized photocurrent response spectra for both the QCD (orange), and the QWIP (green). f The characterized detection bandwidth of the QCD (orange) and the QWIP (green) with electrical rectification.

Stripe-array detectors, whose layout is depicted in Fig.  2d , have three advantanges. First, they confine the incoming EM energy S in in a TM 01 mode within the subwavelength cavity, intensifying the electric field in the AR. Second, the direction of the electric field \(\overrightarrow{{{{{\bf{E}}}}}}_{{{{\bf{cav}}}}}\) becomes vertical, satisfying polarization selection rules for intersubband transitions. Lastly, they reduce the device’s electrical surface, decreasing electric noise and capacitance. The outgoing EM energy S out and the detector absorbed energy can be calculated with the coupled-mode theory 54 . Figure  2e shows responsivity spectra at room temperature of the QCD and the QWIP, with notably more localized peak responses than conventional designs. Specifically, the absorption peak of the QCD (no bias) is found to be around 10 µm (124 meV) with a peak responsivity of 26 mA/W, and for the QWIP (at 1.1 V bias) it is at 9.3 µm (134 meV) with 320 mA/W peak responsivity. The full-width half-maximum of the QCD spectrum is slightly narrower than the QWIP, both around 20 meV. For the transmission experiment, the peak of the QWIP response almost aligns with the emission energy of the two QCLs (see Fig.  1c, d ), whereas the QCD is slightly off-resonance. The bandwidth of both detectors are characterized with electrical rectification, and the measurement results are shown in Fig.  2f . The QCD has a 3-dB bandwidth of about 12 GHz and a smoother frequency rolloff afterwards, while the QWIP has around 9 GHz bandwidth, beyond which a sharper rolloff is observed. To summarize the detectors’ tradeoffs: the QCD has higher bandwidth but lower responsivity, whereas the QWIP provides slightly lower bandwidth but greater responsivity. In addition, the QWIP signal is noisier due to thermally excited electrons being accelerated by the applied voltage, while the unbiased QCD has better noise performances. These tradeoffs in both lasers and detectors lead to the necessity of a thorough transmission performance evaluation.

FSO transmission setup

The FSO transmission setup is shown in Fig.  3a . Different signal formats, i.e., non-return to zero (NRZ) and multilevel pulse amplitude modulation (PAM), were generated with an arbitrary waveform generator (AWG). These signals were amplified and then added to the DC bias current through a bias-tee to drive the QCLs. The output LWIR beam was collimated, transmitted through an FSO link, and focused onto the detector. The detector output was amplified and sampled by a real-time digital storage oscilloscope (DSO). Transmission performance was evaluated by bit error rate (BER) after offline digital signal processing (DSP). We determined performance based on the highest achievable symbol rates across various modulation formats and laser-detector combinations. Setup details are described in Methods, and the DSP specifics are described in the Supplementary Information. The end-to-end system response calibrations for all test cases are shown in Fig.  3b , with calibration process details described in Methods and Supplementary Information Note  5 .

a The modulated signal is generated offline at a lab computer with MATLAB and loaded to the arbitrary waveform generator (AWG). The AWG output is firstly amplified and then combine with a DC bias current at a high-current bias-tee (2 A, 40 GHz), before driving the QCLs. The QCLs are mounted on a Peltier element with thermoelectric cooling (TEC) to configure and stabilize the operational temperature. Two types of detectors, i.e., QCD and QWIP, are mounted 20 cm away from the QCL mount. A pair of f/1” ZnSe collimation lenses are placed between the laser mount and the detector mount to collect the emitted energy focusing on the detectors. For the QWIP, a second bias-tee with a voltage source is connected to provide the bias voltage, whereas for the QCD no bias voltage is required. It’s worth noting that no cooling or temperature control is needed for both types of detectors. The received signal is amplified and captured by a real-time storage oscilloscope (DSO), and the converted digital samples are sent back to the lab computer for demodulation. b The system’s characterized end-to-end S21 amplitude response, including the cascaded frequency response of all the electrical and optoelectronic devices in the setup.

FSO transmission performance with the Standard-QCL

First, we evaluated the Standard-QCL transmission performance and compared the QCD and QWIP. The BER results using the QCD are shown in Fig.  4a . The Standard-QCL’s high output power produced a satisfactory SNR, given the QCD’s limited responsivity. The system supports up to 33 Gbaud NRZ, 18 Gbaud PAM4, and 13 Gbaud PAM6, meeting the 6.25% overhead hard-decision forward error correction code (HD-FEC) threshold 55 of 4.5 × 10 −3 . This translates to net bitrates of 31.05 Gbit s −1 for NRZ, 33.8 Gbit s −1 for PAM4 and 30.5 Gbit s −1 for PAM6. As shown in Fig.  4b , clear eye diagrams and distinct separations in the symbol distribution are observed. When weighing the system tradeoffs between bandwidth, SNR, and linearity, PAM4 stands out as the most optimal of the three tested modulation formats, as it supports the highest achievable bitrate within this experimental configuration.

a BER results with the QCD at the receiver. The BER are measured as a function of the QCL bias current for different modulation formats, namely, NRZ, PAM4, and PAM6. Two FEC code thresholds, i.e., 6.25% overhead HD-FEC of 4.5 × 10 −3 BER, and 20% overhead SD-FEC of 2 × 10 −2 BER, are shown to benchmark the performance. b Selected eye diagrams of the received signal detected with the QCD, measured at the highest bias current after receiver equalization. The distribution of recovered symbols at the decision points are shown in the histograms. c BER results with the QWIP at the receiver. d Selected eye diagrams and symbol distribution histograms of the signals detected with the QWIP, measured at the highest bias current after receiver equalization.

We subsequently assessed the performance of the QWIP. Figure  4c, d show the BER results, and received signal eye diagrams with distribution histograms. Notably, the QWIP enabled higher symbol rates across all three modulation formats due to its higher responsivity. Specifically, transmissions reached 38 Gbaud for NRZ, 21 Gbaud for PAM4, and 15 Gbaud for PAM6, all with BER performances below the HD-FEC threshold. This translated to net bitrates of 35.7 Gbit s −1 (NRZ), 39.5 Gbit s −1 (PAM4), and 35.2 Gbit s −1 (PAM6).

FSO transmission performance with the RF-QCL

We then switched to the RF-QCL and performed the same system evaluation. We firstly placed the QCD at the receiver and the results are shown in Fig.  5 , which reflect the tradeoff between the RF-QCL’s enhanced bandwidth and its lower output power. Consequently, as one can observe from Fig.  5a , the system achieves up to 42 Gbaud NRZ with BER below the HD-FEC threshold. This translates to a net bitrate of 39.5 Gbit s −1 , a result of the enhanced RF performance of the laser. The eye diagram and the symbol distribution histogram are shown in Fig.  5b . In contrast, when we apply PAM4 signaling, due to the limited SNR of the system, we only achieved 5 Gbaud below the HD-FEC limit, i.e., 9.4 Gbit s −1 net rate. We’d like to note the potential for a higher symbol rate with this configuration. We pinpointed this rate by sweeping over various symbol rates when biased at the maximum current of 480 mA (refer to Supplementary Fig.  12 ). However, it was later detected that when the bias current exceeded 460 mA, the performance degraded due to modulation nonlinearity. Examining the eye diagram and symbol distribution in Fig.  5c , there’s noticeable compression between the top amplitude levels compared to the lower ones, leading to increased bit errors. Another contributing factor is the peak-to-peak voltage of the modulated signal. The QCD’s low responsivity requires a higher modulation signal amplitude to combat SNR limitations. This, however, caused the QCL entering modulation non-linearities, i.e., the L-I roll-off region (see Fig.  1b ), more quickly with increased bias current. As the system is strictly SNR limited, we didn’t test higher modulation levels than PAM4 with the RF-QCL.

a BER versus the laser bias current for NRZ and PAM4 at the highest symbol rates achievable to meet the HD-FEC threshold. b, c Selected eye diagrams and the symbol distribution histograms for both modulation formats, measured at the highest bias current point after receiver equalization.

Finally, we placed the QWIP at the receiver, which allowed us to achieve the highest bitrate among all tested laser-detector combinations. The BER results of the NRZ and PAM4 transmissions are shown in Fig.  6a, b . For NRZ, the highest achievable symbol rate fulfilling HD-FEC was 55 Gbaud, yielding a net bitrate of 51.7 Gbit s −1 . We also benchmarked the performance against two other FEC thresholds. The first one has a lower coding gain yet with lower complexity and latency, namely, the Reed–Solomon (RS) (528,514) code 56 , here referred to as the KR-FEC limit, which has an overhead of 2.7% and a pre-FEC BER threshold of 2.2 × 10 −5 . This FEC scheme is ideal for short-reach, latency-sensitive scenarios, e.g., terrestrial FSO supporting radio access networks (RAN). The second one, i.e., 20% overhead soft-decision FEC code 57 , referred to as the SD-FEC limit in this paper. It has a higher coding gain and a pre-FEC BER limit of 2 × 10 −2 . Such an FEC typically adds complexity and latency to optical transceivers, thus mostly used in long-distance scenarios like ground-to-satellite FSO to avoid re-transmission. As shown in Fig.  6a , 40 Gbaud NRZ can be transmitted and received with BER below the KR-FEC limit, resulting in a net bitrate of 39.1 Gbit s −1 . Subsequently, up to 65 Gbaud NRZ can achieve below the SD-FEC limit, achieving a net bitrate of 54.1 Gbit s −1 . Selected NRZ eye diagrams and symbol distribution histograms for the highest achievable symbol rate for each FEC threshold are displayed in Fig.  6c .

a BER results for NRZ signals at different symbol rates, measured against different laser bias current points. The KR-FEC limit of 5.2 × 10 −5 BER is also shown as a benchmark. b BER results for PAM4 signals at different symbol rates, measured against different laser bias current points. c , d Selected eye diagrams and the symbol distribution histograms measured at highest bias current point after receiver equalization for NRZ and PAM4 at different symbol rates, respectively.

For PAM4, as illustrated in Fig.  6b , we reached a BER below the HD-FEC threshold at 30 Gbaud, corresponding to a net bitrate of 56.4 Gbit s −1 . When benchmarked against the higher SD-FEC threshold, up to 35 Gbaud PAM4, i.e., 58.3 Gbit s −1 net bitrate, can be achieved. This is the highest demonstrated bitrate across all test cases. Further increasing the symbol rate to 40 Gbaud results in an above SD-FEC performance. Selected eye diagrams and symbol distribution histograms for PAM4 are shown in Fig.  6d . Different from the PAM4 results with the QCD as the receiver, negligible nonlinear compression is observed with this configuration at high bias currents. This change is attributed to the superior responsivity of the QWIP, which permits a reduced peak-to-peak voltage for the modulated PAM4 signal, thus preventing the RF-QCL from operating in a nonlinear regime.

Our high-speed LWIR FSO setup relies on two key enabling factors. First, the RF-QCL with optimized high-frequency characteristics enhances the system bandwidth, thus the transmission rate, despite a cost of approximately 3-dB output power compared to the Standard-QCL. Second, metamaterial-enhanced unipolar detectors, i.e., QCD and QWIP, outperform our prior conventional design 51 , offering both enhanced SNR and bandwidth owing to their higher responsivity and reduced electrical surface.

There are still potential improvements to be made in the current system. The first is the limited output power of the RF-QCL. The consistent downward trend in the BER curves shown in Fig.  6a, b suggests that the system is still noise limited, suggesting that better results can be expected with higher power. For this experiment, one potential way to increase RF-QCL output power was to lower its operational temperature, which, however, would consume higher TEC power and pose a risk of condensation to damage the QCL chips. We can indeed improve the sealing of the QCL mounting to prevent moisture accumulation, potentially leading to higher output power, and subsequently higher data rates. A more sustainable approach is to enhance the heat dissipation design. The most straightforward approach would be to optimize the indium soldering between the submount and the copper base plate, to allow higher output power without lowering the TEC temperature. However, the performances would be greatly improved with direct soldering of the laser chip on a specifically RF-designed AlN submount which comprises RF connections. This would allow to move away from indium soldering on a copper plate and benefit from industry standard AuSn soldering on the AlN submount.

There are also limitations on the detector side. For the QCD, its high bandwidth merit has been severely hindered by its low responsivity, partly due to the offset between the QCL emitting wavelength and the detector’s responsivity peak. And for the QWIP, its relatively lower bandwidth and higher thermal noise level could be further improved to enhance the system performance. One way forward to improve the response of both detectors is to replace the stripes in the current design with a patch array layout, so as to benefit from a reduced electrical area for high speed operation, and also avoid polarization issues 42 , 43 . Further enhancement of the detectors’ bandwidth can also be made by reducing the laser submount thickness to shorten the wirebonds.

Integrating these potential enhancements from both laser and detector should support higher symbol rates and modulation levels, which would potentially elevate the speed of room-temperature LWIR FSO links to near or even beyond 100 Gbit s −1 on a single channel in the near term. In the longer term, this atmospheric transmission window targets long-distance applications. A preliminary link budget analysis is performed (see Supplementary Information Note  4 ). We foresee that extensive engineering efforts will be required to enhance the transmission distance to meet practical requirements, building upon the foundation laid by this work. Furthermore, facilitating non-line-of-sight (NLOS) transmission would benefit many terrestrial applications. Inspired by the reconfigurable metasurface concept proposed for THz communication 58 , intersubband polaritonic metasurfaces operating in the MIR could potentially enable NLOS FSO communication 59 .

Finally, we acknowledge other technological alternatives for MIR FSO semiconductor transceivers. First, promising results have been demonstrated with interband cascade lasers (ICL) 49 , 60 , 61 , primarily targeting the the MWIR window. Compared to QCL, ICL requires lower bias current, conceivably leading to reduced power consumption. Recently, up to 14 Gbit s −1 PAM4 transmission has been demonstrated with a directly modulated Fabry–Perot ICL at 4.18 μm 61 . Lately, high-temperature ( > 200 K) ICLs operating in the LWIR window have been demonstrated 62 , indicating their potential for LWIR FSO. Another approach is external modulation, which we believe offers more long-term potential for high-speed coherent LWIR FSO communications 33 , 63 , 64 . With an external Stark-effect modulator, over 20 Gbit s −1 FSO transmission at 9 μm has been demonstrated using a room-temperature QCD and over 30 Gbit s −1 using a nitrogen-cooled QWIP at 77 K 64 . The main challenges of this coherent technology roadmap include precise modulator light coupling and its associated power loss, independent phase modulation 65 , and linear coherent reception. While MIR semiconductor transceiver technologies are still emerging, they’ve advanced significantly in the past decade. As they mature, they should be benchmarked using standard metrology for thorough performance comparisons.

Fabrication of the quantum cascade lasers

The design and operational principle of unipolar quantum optoelectonic devices, i.e., QCL, QCD and QWIP have been thoroughly studied (see details in Supplementary Information Note  1 ). For the two QCLs used in the experiment, both are distributed feedback lasers with length of 4 mm and they are fabricated by mirSense. Their active region consists of successive GaInAs wells and AlInAs barriers with a strained composition, grown using Molecular Beam Epitaxy (MBE) on an InP substrate. A buried geometry was chosen to meet the continuous wave (CW) operation requirement. In this configuration, InP:Fe regrowth was performed using hydride vapor phase epitaxy (HVPE) by KTH after ridge etching, reducing waveguide losses and thermal heating of the active region.

Experimental configuration of the free-space transmission

The experimental setup is shown in Fig.  3a . The digital waveforms of signals with different modulation formats and symbol rates were generated offline with typical transmitter-side digital signal processing (DSP) routine that is widely used in fibre-optic datacom systems, detailed in the next section. The generated digital samples are converted to the analog domain by an arbitrary waveform generator (AWG, Keysight M8195A) with a digital-to-analog converter (DAC) of 65 GSample s −1 sampling rate, 8-bit resolution and a memory length of 16 GSamples. The output of the AWG is configured to be between 120 mV peak-to-peak (mV pp ) and 250 mV pp , depending on the modulation formats and symbol rates. An electrical amplifier (SHF 804B) of 66 GHz bandwidth and 22 dB gain is used to amplify the signal to the range between 2 V pp and 3.1 V pp . A high-current broadband bias-tee (Maki Microwave BT2-0040), which can handle up to 2 A DC current with an RF bandwidth of 40 GHz, is used for combining the modulation signal with a DC current before sending it to the QCL mount. The QCLs are mounted on Peltier element thermoelectric cooling (TEC) to stabilize the operational temperature to 15 °C for both lasers during all test cases. A pair of ZnSe Aspheric lenses (Thorlabs AL72512-E3) with a focal length of 12.7 mm are used to collimate the QCL output beam and focus the beam on the detectors at the receiver. In this experiment, due to the high precision requirement on focusing the beam with small spot size to the detectors (60 × 60 µm 2 ) to maximize the incident optical power, we limit the distance to 20 cm between the transmitter and the receiver. It is noted that this distance was chosen as an ease of implementation rather than an upper limit, as the beam collimation and focusing performance within the lab space ( < 10 m) is expected to be virtually the same since nearly all the emitting power can be collected and sent to the detector by careful collimation and focusing. When transmitting over longer distances, one challenge is the broadening of the beam waist for such long-wavelength signals. This broadening can lead to a reduction of received signal power due to the limitations of the receiver’s aperture size. More analysis of beam broadending is described in Supplementary Information Note  4 .

The receiver consists of a photodetector, a second electrical amplifier (SHF 804B) and a real-time digital storage oscilloscope (DSO, Keysight DSAZ334A) of 80 GSample s −1 . The photodetector used is either the QCD or the QWIP. The QCD operates entirely passively, requiring no bias voltage during operation. In contrast, the QWIP requires a second bias-tee to supply a 1.1 V bias voltage. After photodetection, the generated photocurrent is amplified and sampled before sending back to the lab computer for receiver DSP and performance evaluation.

Bandwidth characterization on the device level and on the system level

Bandwidth characterisations of the unipolar optoelectronics were performed using the electrical rectification method 52 at the device-level. For the QCD and the QWIP, a DC voltage and a RF signal are combined at a bias-tee and delivered to the detectors over a 50 Ω impedance transmission line. The rectified DC current is generated when applying specific bias voltage so the devices’ I-V characteristics are in the nonlinear region, which is propotional to the magnitude of the device transfer function at the AC frequency. The rectified DC current, which is in the order of a few µA, can be directly measured with the DC source (Keithley 2450 SourceMeter). Subsequently, sweeping the RF frequency enables the acquisition of the complete amplitude frequency response. For the QCL, which is an active device requiring a large DC current of hundreds of mA, an additional low-frequency signal (much lower than the RF frequency) is modulated on the RF signal. This produces a rectified current displaced from DC, detectable and analyzable with a lock-in amplifier. Similarly, by continuously measuring this rectified current while sweeping the RF frequency, we can obtain the overall amplitude modulation response of the QCL. Detailed insight into the operational mechanics of this electrical rectification method is available in Supplementary Information Note  2 .

System-level bandwidth characteristisation is performed by generating flat frequency combs with the AWG, transmitting them through the whole system and capturing them with the DSO. In this way the end-to-end system frequency response can be obtained by comparing both the amplitude and the phase of the frequency comb lines. The characterization results of all test cases are shown in Supplementary Fig.  7 .

Digital signal processing (DSP) at the transmitter and the receiver for data transmission

The DSP algorithms used in this experiment are standard routines developed for short-reach fibre-optic communications. The transmitter DSP consists of pseudo-random binary sequence (PRBS) generation, root-raise-cosine (RRC) pulse shaping, resampling to match the AWG sampling rate, and a static 2-tap pre-emphasis filter. The receiver DSP routine consists of a matched RRC filter, up-sampling and timing recovery, a data-aided decision feedback equalizer (DFE) with 99 feedforward taps and 99 feedback taps, symbol demodulation, and bit error rate (BER) counting. The number of DFE taps were fixed during all test cases for simplicity and maximizing the performance, which can be potentially reduced for some of the test cases. A comprehensive analysis of the effect of DFE taps on the BER performance is presented in Supplementary Figs.  8 – 11 . These figures illustrate example test cases using RF-QCL in conjunction with QCD and QWIP, respectively. A block diagram and more details of the DSP configuration can be found in Supplementary Information Note  3 .

Calculation of net bitrates for different modulation formats

Calculating the net bitrates, expressed in Gbit s −1 , from the symbol rates, measured in Gbaud, consists of two steps. The first step involves converting the symbol rate to the gross bitrate. In the second step, we calculate the net bitrate from the gross bitrate by deducting the FEC overhead (OH).

In the first step, when utilizing single carrier signal formats such as those employed in this study, the formula for calculating the gross bitrate (Gbit s −1 ) is: Gbit s −1 = bits/symbol × Gbaud. Specifically, for a binary NRZ signal, the gross bitrate directly matches the baud rate; for instance, a 55 Gbaud NRZ signal corresponds to a gross bitrate of 55 Gbit s −1 . For M-level PAM signals, the calculation of bits/symbol is given by: bits/symbol = log 2 (M). Consequently, in the case of PAM4, each symbol equates to 2 bits, resulting in a bitrate of Gbit s −1  = 2 × Gbaud. For PAM6, where bits per symbol are approximately 2.585, we often round it to 2.5 for practical configurations to prevent the requirement for a long bit-to-symbol mapping memory.

In the subsequent step, the net bitrate is calculated by dividing the gross bitrate by (1 + FEC OH). The overhead for the hard-decision FEC threshold, which we evaluated in our study, is 6.25%. For the soft-decision FEC, which tolerates a higher BER, the overhead is 20%.

Summarizing both steps, the relation between symbol rate and net bitrate for M-level PAM signal is expressed as:

Data availability

The measurement data generated in this study have been deposited in https://doi.org/10.5281/zenodo.12515593 .

Code availability

The algorithms used for the digital signal processing at the transmitter and the receiver are standard and are outlined in detail in the Methods and Supplementary Information. All codes of the DSP algorithms used in this study are embedded in a larger framework, which, together with specific user instructions can be available from the corresponding authors upon request.

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Acknowledgements

This work was supported in part by the EU H2020 cFLOW Project (828893), in part by the Swedish Research Council (VR) project 2019-05197 and project ‘BRAIN’ 2022-04798, in part by the COST Action CA19111 NEWFOCUS, VINNOVA-funded project ‘A-FRONTHAUL’ 2023-00659, and in part by the LZP FLPP project ‘MIR-FAST’ (lzp-2023/1-0503). The authors from ENS acknowledge the financial support of the ENS-Thales Chair, Direction Générale de l’Armement (DGA), PEPR Electronique, ANR project LIGNEDEMIR (ANR- 18CE09-0035), FET Open projects cFLOW (Grant No. 828893) and CNRS Renatech network.

Open access funding provided by Royal Institute of Technology.

Author information

These authors contributed equally: Hamza Dely, Mahdieh Joharifar, Laureline Durupt.

Authors and Affiliations

Laboratoire de Physique de l’ENS, Département de Physique, École Normale Supérieure, Université PSL, Sorbonne Université, Université Paris Cité, CNRS, 75005, Paris, France

Hamza Dely, Thomas Bonazzi, Djamal Gacemi, Angela Vasanelli & Carlo Sirtori

Department of Applied Physics, KTH Royal Institute of Technology, 106 91, Stockholm, Sweden

Mahdieh Joharifar, Richard Schatz, Yan-Ting Sun, Oskars Ozolins & Xiaodan Pang

mirSense, 2 Bd Thomas Gobert, 91120, Palaiseau, France

Laureline Durupt & Gregory Maisons

Institute of Telecommunications, Riga Technical University, 1048, Riga, Latvia

Armands Ostrovskis, Toms Salgals, Sandis Spolitis, Vjačeslavs Bobrovs, Oskars Ozolins & Xiaodan Pang

College of Information Science and Electrical Engineering, Zhejiang University, Hangzhou, 310027, China

Lu Zhang & Xianbin Yu

Université Paris-Saclay, CNRS, Centre de Nanosciences et de Nanotechnologies, 91120, Palaiseau, France

Isabelle Sagnes & Konstantinos Pantzas

RISE Research Institutes of Sweden, 164 40, Kista, Sweden

Oskars Ozolins & Xiaodan Pang

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Contributions

H.D. and X.P. proposed the long-wave IR FSO transmission study with directly modulated QCL and QCD/QWIP detectors. H.D., M.J., L.D., R.S., and X.P. designed the overall experiment. I.S. and K.P. provided the epitaxial growth for the detectors. L.D., G.M. and Y.-T. S. developed the QCL chips and submounts, and characterised the QCLs. H.D., T.B., D.G., A.V., and C.S. developed the QCD and QWIP chips, performed RF bounding and device characterisation. L.Z., X.Y., O.O., and X.P. developed the PAM modulation and DSP routine. T.S., S.S., and V.B. calibrated and performed test instrumentation, H.D., M.J., L.D., A.O., R.S., and X.P. carried out the FSO transmission experiment. M.J. and X.P. processed and analyzed the experimental data. R.S., G.M., D.G., Y.-T. S., O.O., and C.S. assisted in discussing and interpreting the results. X.P. and C.S. coordinated and supervised the experiment. H. D., M.J., L.D., and X.P. wrote the draft of the manuscript. All the authors reviewed and edited the manuscript.

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Correspondence to Hamza Dely , Xiaodan Pang or Carlo Sirtori .

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Dely, H., Joharifar, M., Durupt, L. et al. Unipolar quantum optoelectronics for high speed direct modulation and transmission in 8–14 µm atmospheric window. Nat Commun 15 , 8040 (2024). https://doi.org/10.1038/s41467-024-52053-7

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Two-way mathematical 'dictionary' could connect quantum physics with number theory

by Institute of Science and Technology Austria

Several fields of mathematics have developed in total isolation, using their own "undecipherable" coded languages. In a new study published in Proceedings of the National Academy of Sciences , Tamás Hausel, professor of mathematics at the Institute of Science and Technology Austria (ISTA), presents "big algebras," a two-way mathematical 'dictionary' between symmetry, algebra, and geometry, that could strengthen the connection between the distant worlds of quantum physics and number theory.

Mathematics, the most exact among scientific disciplines, could be viewed as the ultimate quest for absolute truth. However, the mathematical roads to truth often need to overcome tremendous obstacles, much like conquering unimaginably high mountain peaks or building giant bridges between isolated continents.

The mathematical world abounds with mysteries and several mathematical disciplines have developed along convoluted paths—in complete isolation from one another. Thus, establishing an irrefutable truth around complex phenomena in the physical world draws on intuition and a good deal of abstraction.

Even fundamental aspects of physics push mathematics to new heights of complexity. This is especially true for symmetries, with the help of which physicists have theorized and discovered an entire zoo of subatomic particles that make up our universe.

In an exceptionally ambitious endeavor, Hausel, professor at the Institute of Science and Technology Austria (ISTA), not only conjectured but also proved a new mathematical tool called "big algebras."

This new theorem is comparable to a "dictionary" that deciphers the most abstract aspects of mathematical symmetry using algebraic geometry. By operating at the intersection of symmetry, abstract algebra, and geometry, big algebras use more tangible geometric information to recapitulate sophisticated mathematical information about symmetries.

"With big algebras, information from the ' tip of the mathematical iceberg ' can give us unprecedented insights into the hidden depths of the mysterious world of symmetry groups," says Hausel.

With this mathematical breakthrough, Hausel seeks to consolidate the connection between two distant fields of mathematics.

"Imagine, on the one hand, a world of mathematical representations of quantum physics, and on the other hand, very, very far away, the purely mathematical world of number theory . With the present work, I hope to have come one step closer to establishing a stable connection between these two worlds."

No longer lost in translation

The 17th-century philosopher and mathematician René Descartes showed us that we could understand the geometry of objects by using algebraic equations. Thus, he was the first to "translate" mathematical information between these previously separate fields.

"I like to view the relations between different mathematical fields as dictionaries that translate information between often non-mutually intelligible mathematical languages," says Hausel.

So far, several such mathematical "dictionaries" have been developed, but some only translate the information in one direction, leaving the information about the way back entirely encrypted. Furthermore, the term "algebra" nowadays encompasses both classical algebra, as in Descartes' time, and abstract algebra, i.e. the study of mathematical structures that cannot necessarily be expressed with numerical values. This adds another layer of complexity. Now, Hausel uses abstract algebra and algebraic geometry as a two-way "dictionary."

A skeleton and nerves

In mathematics, symmetry is defined as a form of "invariance." The group of transformations that keep a mathematical object unchanged is called a "symmetry group." These are classified as "continuous" (e.g., the rotation of a circle or sphere) or "discrete" (e.g., the mirroring of an object). Continuous symmetry groups are represented mathematically by matrices—rectangular arrays of numbers.

Starting from a matrix representation of a continuous symmetry group, Hausel can compute the big algebra and represent its essential properties geometrically by drawing its "skeleton" and "nerves" on a mathematical surface.

The big algebra's skeleton and nerves give rise to interesting, 3D-printable shapes that recapitulate sophisticated aspects of the original mathematical information, thus closing the translation circle.

"I am particularly excited about this work, as it provides us with a completely novel approach to studying representations of continuous symmetry groups. With big algebras, the mathematical 'translation' does not only work in one direction but in both."

Bridging isolated continents in a vast world of mathematics

How could big algebras strengthen the link between quantum physics and number theory, two fields of mathematics seemingly worlds apart? Firstly, the math behind quantum physics makes extensive use of matrices—rectangular arrays of numbers.

However, these matrices are typically "non-commutative," meaning that multiplying the first matrix by the second does not yield the same result as multiplying the second one by the first. This poses a problem in algebra and algebraic geometry as non-commutative algebra is not yet well understood.

Big algebras now solve this problem: when computed, a big algebra is a commutative "mathematical translation" of a non-commutative matrix algebra. Thus, the information initially enclosed within non-commutative matrices can be decoded and represented geometrically to reveal their hidden properties.

Secondly, Hausel shows that big algebras not only reveal relationships between related symmetry groups, but also when their so-called "Langlands duals" are related. These duals are a central concept in the purely mathematical world of number theory. In the Langlands Program , a highly intricate, large-scale dictionary that seeks to bridge isolated mathematical "continents," the Langlands duality is a concept or tool that allows 'mapping' mathematical information between different categories.

"In my work, big algebras seem to relate different symmetry groups precisely when their Langlands duals are related, a quite surprising outcome with possible applications in number theory," says Hausel.

"Ideally, big algebras would allow me to relate the Langlands duality in number theory with quantum physics," says Hausel.

For now, he was able to demonstrate that big algebras solve problems on both of these continents. The fog has started to dissipate, and the continents of quantum physics and number theory have caught a glimpse of each others' mountains and shores on the horizon. Soon, rather than only connecting the continents by boat, a bridge of big algebras might allow an easier crossing of the mathematical strait separating them.

Journal information: Proceedings of the National Academy of Sciences

Provided by Institute of Science and Technology Austria

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