Quantum Computations on a Topologically Encoded Qubit

From Left to Right: (top row) Daniel Nigg, Markus Müller, Esteban Martínez, Philipp Schindler, (bottom row) Markus Hennrich, Thomas Monz, Miguel Angel Martín-Delgado, Rainer Blatt.

Authors: Markus Müller¹ and Daniel Nigg²

Affiliation:
¹Departamento de Física Teórica I, Universidad Complutense, Spain.
²Institut für Experimentalphysik, Universität Innsbruck, Austria.

Email: mueller@ucm.es, daniel.nigg@uibk.ac.at

Even computers are error-prone. The slightest disturbances may alter saved information and falsify the results of calculations. To overcome these problems, computers use specific routines to continuously detect and correct errors. This also holds true for a future quantum computer, which will also require procedures for error correction. Whereas general quantum states can not be simply copied, fragile quantum information can still be protected from errors during storage and information processing by using quantum error correcting codes. Here, quantum states are encoded in entangled states that are distributed over several physical particles.

A quantum bit encoded in seven ions

In the experiment realized at the University of Innsbruck, Austria [1], we confined seven calcium ions in an ion trap, with one qubit stored in each of the ions. In our setup, we use lasers to cool the ion string to almost absolute zero temperature and to precisely control their quantum properties. We used the register of seven physical qubits to encode quantum states of one logical qubit in entangled states of these particles. The topological quantum error-correcting code employed in the experiment provided the program for this encoding process, and was proposed and developed in the theory group at the Universidad Complutense in Madrid, Spain. The code arranges the qubits on a two-dimensional lattice structure where they interact with the neighboring particles. The encoding of the logical qubit in the seven physical qubits was the experimentally most challenging step. It required a long sequence of laser pulses to effectively realize three entangling gate operations, each acting on subsets of four neighboring qubits belonging to one plaquette.

Figure 1: Schematics of the string of 7 ions stored in a linear Paul trap, with each ion hosting one physical qubit. One logical qubit is encoded in entangled states of these 7 physical qubits, by using a quantum error correcting code which arranges the qubits on a two-dimensional triangular lattice of three plaquettes.

Detection of arbitrary errors and logical quantum gate operations

After the encoding step, once the atoms are entangled in this specific way, the quantum correlations provide a resource for subsequent error correction and quantum computations on the encoded logical qubit. Using the available set of laser pulses we induced at purpose all types of single-qubit errors that can occur on any of the seven physical qubits. Our measurements demonstrate that the quantum code is indeed able to independently detect phase flip errors, bit flip errors as well as combinations of both, regardless on which of the qubits these occur.

Figure 2: Schematics of error detection by the 7-qubit code: Arbitrary errors (in the shown example a phase flip error Z on qubit 5) manifests itself as excitations on one or several plaquettes (black filled circle on the blue plaquette) and by its characteristic signature, the error syndrome. The latter allows one to deduce the type, i.e. whether a bit flip, phase flip or combined error of both has occurred, as well as the location of the error in the qubit register.

Next, we applied logical quantum gate operations onto the encoded logical qubit. The 7-qubit quantum code we used allowed us to implement individual operations and longer sequences of gate operations (the single-qubit Clifford group) on the logical qubit in a transversal way, i.e. by applying the corresponding operations bitwise on each of the 7 physical qubits.

Towards a fault-tolerant quantum computer

The 7-ion system we used for encoding one logical quantum bit can serve as a building block for larger quantum systems. Storing and processing logical quantum information in larger lattice systems with more physical qubits is predicted to further increase the robustness with respect to noise and errors. The required technology in the form of two-dimensional ion trap arrays, which would enable the storage and manipulation of larger numbers of qubits, are currently developed and tested at the University of Innsbruck as well as in other laboratories worldwide. Together with further theoretical progress and optimized quantum error correcting codes, the result of these developments might be a quantum computer that could reliably perform arbitrarily long quantum computations without being impeded by errors.

For further background information and explication, please watch this video:



Funding:
The researchers are financially supported by the Spanish Ministry of Science, the Austrian Science Fund, the U.S. Government, the European Commission and the Federation of Austrian Industries Tyrol.

Reference:
[1]  Daniel Nigg, Markus Müller, Esteban A. Martinez, Philipp Schindler, Markus Hennrich, Thomas Monz, Miguel Angel Martin-Delgado, Rainer Blatt, "Quantum Computations on a Topologically Encoded Qubit". Science, 345, 302 (2014). Abstract.

Few-layer Black Phosphorus Phototransistors for Fast and Broadband Photodetection

From Left to Right: (Top) Michele Buscema, Dirk J. Groenendijk, Sofya I. Blanter, (Bottom) Gary A. Steele, Herre S.J. van der Zant and Andres Castellanos-Gomez

Authors:
Michele Buscema, Dirk J. Groenendijk, Sofya I. Blanter, Gary A. Steele, Herre S.J. van der Zant, Andres Castellanos-Gomez.

Affiliation:
Kavli Institute of Nanoscience, Delft University of Technology, The Netherlands.

Introduction

The isolation of graphene has opened the door for the studying of the large family of layered two-dimensional (2D) materials, driven by the extraordinary properties that these materials show in their single and few-layer form [1-4]. Graphene, a one-atom thick layer of carbon atoms, has shown excellent electrical properties (e.g. mobility in the order of 170 000 cm²/Vs at room temperature) and large breaking strength [5,6]. However, its applicability in low-power field effect transistors (FETs) and optoelectronic devices (e.g. photodetectors) is hampered by its zero bandgap.

This absence of a bandgap has intensified the current research in other 2D materials with an intrinsic bandgap [7]. For instance, silicene, a single layer of silicon atoms, represents a semiconducting analogue to graphene but so far it has only been realized in an ultra-high-vacuum environment, a severe limitation for further studies and applications [8]. Other promising candidates for optoelectronic applications are the members of the transition metal dichalcogenides (TMDCs) material class [9-13]. The large and direct bandgap of their single-layer form provides strong light absorption – a necessary condition for large photoresponse – but operation is limited to part of the visible spectrum. A material with a direct and small bandgap is needed to extend the detection range accessible with 2D materials.

Black phosphorus

Few-layer black phosphorus is a new member of the 2D-materials family. Black phosphorus is a layered allotrope of the element phosphorus and, in bulk, it is a semiconductor with a direct bandgap of 0.35 eV [14]. In its few-layer form, the bandgap is predicted to strongly depend on the number of layers, from 0.35 eV (bulk) to 2.0 eV (single-layer). Moreover, FETs based on few-layer black-phosphorus show promising electrical properties [15-18], making them an appealing candidate for tunable photodetection from the visible to the infrared part of the spectrum.

Main Results

In our recently published work [19], we characterized the response to light excitation of FETs based on few-layer black phosphorus (thickness ranging from 3nm to 8nm). Figure 1a shows a schematic of the device and of the measurement circuit. Without illumination, the black-phosphorus FETs show ambipolar behavior, as both holes and electrons can be induced in the conducting channel by the gate electric field. The measured mobilities are in the order of 100 cm²/Vs and current on/off ratio in the order of 10³, demonstrating good electrical behavior. Under illumination, we measure a sizable photoresponse to excitation wavelengths from the visible up to 940 nm (see Figure 1b). Figure 1c shows the photocurrent measured for a single pulse of light excitation from which we estimate a rise time of 1 ms, demonstrating broadband and fast photodetection. For comparison, photodetectors based on single-layer molybdenum disulphide (MoS₂) can reach higher responsivities (~ 880 X 10³ mA/W) but their response time is limited to 0.6 sec [20].

Figure 1: (a) Device schematics (b) Source-drain current vs. gate voltage in dark (black solid line), with λ = 940 nm illumination (purple solid line), λ = 640 nm illumination (red solid line) and λ = 532 nm illumination (green solid line). The total incident optical power is 750 μW for all wavelengths. (c) Source-drain current vs. time for a single period of light modulation with a mechanical chopper (different device from panel b).

Future trends and outlook

Taking advantage of the ambipolarity, one could think of electrostatically defining a PN junction in a few-layer black phosphorus flake, as already pioneered in single-layer tungsten diselenide (WSe₂) [9-11]. A PN junction could be used to boost the photoresponse and generate electrical power via the photovoltaic effect. Given the small and direct bandgap of few-layer black-phosphorus, it would be possible to harvest photons also in the near-infrared part of the spectrum.

References:
[1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, "Two-dimensional gas of massless Dirac fermions in graphene". Nature, 438, 197-200 (2005). Abstract.
[2] K.S. Novoselov, D. Jiang, F. Schedin, T.J. Booth, V.V. Khotkevich, S.V. Morozov, A.K. Geim, "Two-dimensional atomic crystals". Proceedings of the National Academy of Sciences of the United States of America, 102, 10451(2005). Full Article.
[3] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A. A. Firsov, "Electric Field Effect in Atomically Thin Carbon Films". Science, 306, 666 (2004). Abstract.
[4] Qing Hua Wang, Kourosh Kalantar-Zadeh, Andras Kis, Jonathan N. Coleman, Michael S. Strano, "Electronics and optoelectronics of two-dimensional transition metal dichalcogenides". Nature nanotechnology, 7, 699–712 (2012). Abstract.
[5] L. Wang, I. Meric, P.Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe, L. M. Campos, D.A. Muller, J. Guo, P. Kim, J. Hone, K. L. Shepard, C. R. Dean, "One-Dimensional Electrical Contact to a Two-Dimensional Material". Science, 342, 614-617 (2013). Abstract.
[6] Changgu Lee, Xiaoding Wei, Jeffrey W. Kysar, James Hone, "Measurement of the Elastic Properties and Intrinsic Strength of Monolayer Graphene". Science, 321, 385-388 (2008). Abstract.
[7] B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, A. Kis, "Single-layer MoS₂ transistors". Nature Nanotechnology, 6, 147-150 (2011). Abstract.
[8] Patrick Vogt, Paola De Padova, Claudio Quaresima, Jose Avila, Emmanouil Frantzeskakis, Maria Carmen Asensio, Andrea Resta, Bénédicte Ealet, Guy Le Lay, "Silicene: Compelling Experimental Evidence for Graphenelike Two-Dimensional Silicon". Physical Review Letters, 108, 155501 (2012). Abstract.
[9] Britton W. H. Baugher, Hugh O. H. Churchill, Yafang Yang, Pablo Jarillo-Herrero, "Optoelectronic devices based on electrically tunable p–n diodes in a monolayer dichalcogenide". Nature Nanotechnology, 9, 262-267 (2014). Abstract.
[10] Andreas Pospischil, Marco M. Furchi, Thomas Mueller, "Solar-energy conversion and light emission in an atomic monolayer p–n diode". Nature Nanotechnology, 9, 257-261 (2014). Abstract.
[11] Jason S. Ross, Philip Klement, Aaron M. Jones, Nirmal J. Ghimire, Jiaqiang Yan, D. G. Mandrus, Takashi Taniguchi, Kenji Watanabe, Kenji Kitamura, Wang Yao, David H. Cobden, Xiaodong Xu, "Electrically tunable excitonic light-emitting diodes based on monolayer WSe₂ p–n junctions". Nature Nanotechnology, 9, 268-272 (2014). Abstract.
[12] Zongyou Yin, Hai Li, Hong Li, Lin Jiang, Yumeng Shi, Yinghui Sun, Gang Lu, Qing Zhang, Xiaodong Chen, Hua Zhang, "Single-Layer MoS₂ Phototransistors". ACS Nano, 6, 74-80 (2012). Abstract.
[13] Néstor Perea-López, Ana Laura Elías, Ayse Berkdemir, Andres Castro-Beltran, Humberto R. Gutiérrez, Simin Feng, Ruitao Lv, Takuya Hayashi, Florentino López-Urías, Sujoy Ghosh, Baleeswaraiah Muchharla, Saikat Talapatra, Humberto Terrones, Mauricio Terrones, "Photosensor Device Based on Few-Layered WS₂ Films". Advanced Functional Materials, 23, 5511-5517 (2013). Abstract.
[14] Yuichi Akahama, Shoichi Endo, Shin-ichiro Narita, "Electrical Properties of Black Phosphorus Single Crystals". Journal of the Physical Society of Japan, 52, 2148-2155 (1983). Abstract.
[15] Han Liu, Adam T. Neal, Zhen Zhu, David Tomanek, Peide D. Ye, "Phosphorene: A New 2D Material with High Carrier Mobility". arXiv:1401.4133 [cond-mat.mes-hall] (2014).
[16] Likai Li, Yijun Yu, Guo Jun Ye, Qingqin Ge, Xuedong Ou, Hua Wu, Donglai Feng, Xian Hui Chen, Yuanbo Zhang, "Black phosphorus field-effect transistors". Nature Nanotechnology, 9, 372–377 (2014). Abstract.
[17] Steven P. Koenig, Rostislav A. Doganov, Hennrik Schmidt, A. H. Castro Neto, Barbaros Özyilmaz, "Electric field effect in ultrathin black phosphorus". Applied Physics Letters, 104, 103106 (2014). Abstract.
[18] Han Liu, Adam T. Neal, Zhen Zhu, Zhe Luo, Xianfan Xu, David Tománek, Peide D. Ye, "Phosphorene: An Unexplored 2D Semiconductor with a High Hole Mobility". ACS Nano, 8, 4033-4041 (2014). Abstract.
[19] Michele Buscema, Dirk J. Groenendijk, Sofya I. Blanter, Gary A. Steele, Herre S. J. van der Zant, Andres Castellanos-Gomez, "Fast and Broadband Photoresponse of Few-Layer Black Phosphorus Field-Effect Transistors". Nano letters, 14, 3347-3352 (2014). Abstract.
[20] Oriol Lopez-Sanchez, Dominik Lembke, Metin Kayci, Aleksandra Radenovic, Andras Kis, "Ultrasensitive photodetectors based on monolayer MoS₂" Nature Nanotechnology, 8, 497-501 (2013). Abstract.

Realizing Two-Dimensional Optics with Metal Antennas and Graphene Plasmons

(From Left to Right) Pablo Alonso-González, Alexey Nikitin and Rainer Hillenbrand.

Authors: 
Pablo Alonso-González¹, Alexey Nikitin^1,2, Federico Golmar^1,3, Alba Centeno⁴, Amaia Pesquera⁴, Saül Vélez¹, Jianing Chen¹, Gabriele Navickaite⁵, Frank Koppens⁵, Amaia Zurutuza⁴, Félix Casanova^1,2, Luis E. Hueso^1,2, Rainer Hillenbrand^1,2.

Affiliation:
¹CIC nanoGUNE, 20018 Donostia-San Sebastián, Spain.
²IKERBASQUE, Basque Foundation for Science, 48011 Bilbao, Spain.
³I.N.T.I-CONICET and ECyT-UNSAM, San Martín, Bs. As., Argentina.
⁴Graphenea SA, 20018 Donostia-San Sebastián, Spain.
⁵ICFO-Institut de Ciéncies Fotoniques, Mediterranean Technology Park, Barcelona, Spain.

Optical circuits and devices could make signal processing and computing much faster. However, although light is very fast, it needs too much space. In fact, propagating light needs at least the space of half its wavelength, which is much larger than state-of-the-art electronic building blocks in our computers. For that reason, a quest for squeezing light to propagate it through nanoscale materials arises.

Graphene, a single layer of carbon atoms with extraordinary properties, has been proposed as one solution. The wavelength of light captured by a graphene layer can be strongly shortened by a factor of 10 to 100 compared to light propagating in free space [1, 2]. As a consequence, this light propagating along the graphene layer - called graphene plasmon - requires much less space and promises ultra-compact photonic devices [3,4].

Past 2Physics articles by this Group:
July 22, 2012: "Capturing, Tuning and Controlling Light with a Single Sheet of Carbon Atoms"
by Jianing Chen, Michela Badioli, Pablo Alonso-González, Susokin Thongrattanasiri, Florian Huth, Johann Osmond, Marko Spasenović, Alba Centeno, Amaia Pesquera, Philippe Godignon, Amaia Zurutuza, Nicolas Camara, Javier García de Abajo, Rainer Hillenbrand, Frank Koppens

Converting light efficiently into graphene plasmons, however, has been a major challenge. In our recent work [5], we demonstrate that the antenna concept of radio wave technology could be a promising solution. We show that a nanoscale metal rod on graphene (acting as an antenna for light) can capture infrared light and convert it into graphene plasmons, analogous to a radio antenna converting radio waves into electromagnetic waves in a metal cable. The excitation of graphene plasmons is purely optical, the device is compact and the phase and wavefronts of the graphene plasmons can be directly controlled by geometrically tailoring the antennas. The latter is essential for the development of applications that require focusing and guiding of graphene plasmons.

Fig. 1: Launching graphene plasmons with a gold antenna. The oscillations of the calculated electromagnetic field around the antenna reveal the graphene plasmons.

Based on calculations (Fig. 1), we fabricated gold nanoantennas on graphene provided by Graphenea. We then used the Neaspec near-field microscope to image how infrared graphene plasmons are launched and propagate along the graphene layer. In the experimental near-field images, we observed that indeed electromagnetic waves on the graphene propagate away from the antenna, with a wavelength that is about 30 times smaller than that of the incident light (Fig. 2).

Fig. 2: Top: Topography of a gold nanoantenna on graphene. Bottom: Near-field image showing the fields of the antenna and the graphene plasmons around the antenna. The image was taken at an illumination wavelength of 11.06 μm and shows the real part of the imaged field. The distance between fringes of the same color reveals the graphene plasmon wavelength.

In order to test whether the two-dimensional propagation of light waves along a one-atom-thick carbon layer follow the laws of conventional optics, we tried to focus and refract the waves. For the focusing experiment, we curved the antenna. The images then showed that the graphene plasmons focus away from the antenna, similar to the light beam that is concentrated with a lens or concave mirror.

We also observed that graphene plasmons refract (bend) when they pass through a prism-shaped graphene bilayer (Fig. 3), analogous to the bending of a light beam passing through a glass prism. The big difference is that the graphene prism is only two atoms thick. By measuring the graphene plasmon wavelengths in the bi- and monolayer, λ₁ and λ₂, as well as the propagation angles α₁ and α₂, we could demonstrate that the refraction of graphene plasmons qualitatively follows the fundamental law of refraction (Snell´s law): sin(α₁)/sin(α₂) = λ₁/λ₁.

Fig. 3: (a) Illustration of a graphene bilayer prism next to a gold antenna. (b) Near-field image (taken at an illumination wavelength of 10.20 μm) of graphene plasmons refracting at a graphene bilayer prism. The yellow lines and arrows illustrate the plasmon wavefronts and their refraction.

Intriguingly, the graphene plasmons are refracted because the conductivity in the two-atom-thick prism is larger than in the surrounding one-atom-thick layer. In the future, local conductivity changes in graphene could be generated by simple electronic means, such as gating, allowing for highly efficient electrical control of refraction, among others for steering applications.

Altogether, the experiments show that the fundamental and most important principles of conventional optics also apply for graphene plasmons, in other words, squeezed light propagating along a one-atom-thick layer of carbon atoms. Future developments based on these results could lead to extremely miniaturized optical circuits and devices that could be useful for sensing and computing, among other applications.

References:
[1] Jianing Chen, Michela Badioli, Pablo Alonso-González, Sukosin Thongrattanasiri, Florian Huth, Johann Osmond, Marko Spasenović, Alba Centeno, Amaia Pesquera, Philippe Godignon, Amaia Zurutuza Elorza, Nicolas Camara, F. Javier García de Abajo, Rainer Hillenbrand, Frank H. L. Koppens, "Optical nano-imaging of gate-tunable graphene plasmons". Nature, 487, 77-81 (2012). Abstract. 2Physics Article.
[2] Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. Castro Neto, C. N. Lau, F. Keilmann, D. N. Basov, "Gate-tuning of graphene plasmons revealed by infrared nano-imaging". Nature, 487, 82-85 (2012). Abstract. 2Physics Article.
[3] Ashkan Vakil, Nader Engheta, “Transformation optics using graphene”. Science, 332, 1291-1294 (2011). Abstract.
[4] A.N. Grigorenko, M. Polini, K.S. Novoselov, “Graphene plasmonics”, Nature Photonics, 6, 749-758 (2012). Abstract.
[5] P. Alonso-González, A.Y. Nikitin, F. Golmar, A. Centeno, A. Pesquera, S. Vélez, J. Chen, G. Navickaite, F. Koppens, A. Zurutuza, F. Casanova, L.E. Hueso and R. Hillenbrand. “Controlling grapheme plasmons with resonant metal antennas and spatial conductivity patterns”. Science,  344, 1369-1373 (2014). Abstract.

Institutes:

CIC nanoGUNE
The nanoGUNE Cooperative Research Center, located in Donostia-San Sebastian, Basque Country, is a research centre set up with the mission to conduct excellence research into nanoscience and nanotechnology with the aim of increasing the Basque Country’s business competitiveness and economic and social development.

GRAPHENEA S.A.
Graphenea is a pioneer graphene production start-up company founded in 2010 by private investors and CIC nanoGUNE. The company produces and commercializes graphene films by Chemical Vapor Deposition technology and graphene powders by Chemical Exfoliation techniques.

ICFO
ICFO is a young research institution located in Barcelona that aims to advance the very limits of knowledge in Photonics, namely the science and technology of harnessing Light. Its research programs target the global forefront of photonics, and aim to tackle important challenges faced by society at large. ICFO is focused on current and future problems in Health, Energy, Information, Safety, Security and caring for the Environment.

Micropillar Laser Mimics 'Excitability' of Neurons

Left to Right: S. Barbay and F. Selmi

Authors: S. Barbay, F. Selmi

Affiliation:
Laboratoire de Photonique et de Nanostructures, Marcoussis, France.

It is well known in neurophysiology textbooks (see e.g. [12]) that neurons are biological excitable systems possessing an absolute and a relative refractory period. These properties are fundamental for the propagation of nerve impulse and for information processing in the brain. Excitability is a generic property that is also found in biological [8], chemical [11], and optical systems [5]. An excitable system possess a rest state. If perturbed above a certain threshold - the excitable threshold - with a single perturbation, it emits a pulse with a characteristic shape (light pulse in optics, electrical pulse in neurons). If perturbed with two successive pulses above the excitable threshold, it can respond by emitting two identical pulses if the perturbation pulses are well separated temporally. It can emit a unique pulse if the second perturbation occurs too early after the first one : we are then in the absolute refractory period. However, in an intermediate regime, the relative refractory period, it is possible but harder to trigger a second response and this second response is inhibited and has smaller amplitude.

Figure 1: Schematic representation and SEM image of the micropillar laser with embedded saturable absorber. The pillar microcavity is coated with a thick, 2 microns, layer of SiN. The active zone is sandwiched between two aperiodic multilayer mirrors and is composed of a gain and an absorber section made by respectively 2 and 1 quantum wells. The laser emits around 980nm and is 4μm in diameter.

Our system is a special kind of micropillar laser embedding in its center a saturable absorber, and optimized for optical pumping [4]. Semiconductor lasers are interesting systems for the study of excitability [2, 3, 6, 7, 14-19] since they can have a small footprint and have short timescales. They can thus lead to short response times, necessary in view of their utilization in a neuromorphic information processing context. It was also recognized recently that microlasers with saturable absorber can work as leaky integrate-and-fire laser neurons [10] paving the way to fast cognitive computing.

We have shown in [13] that a micropillar laser with saturable absorber is a fast excitable unit with an absolute, and for the first time, a relative refractory period. The response time is of the order of 200ps, similar to the absolute refractory period, and the relative refractory period is about 350ps. This demonstrates that this system behaves analogously to a neuron but with much faster timescales (sub-nanosecond vs millisecond). The existence of a relative refractory period proves that the system keeps memory of its past state. Contrarily to a common belief, the excitable threshold is not a constant of the system and may be controlled either externally, via the bias pumping of the system, or dynamically via the past state memory. These properties can be utilized for neuro-mimetic optical processing of information, for instance by coupling several micropillar lasers together for building logic gates [1] or for neuromorphic processing [9,10].

References:
[1] Adrian Jacobo, Damià Gomila, Manuel A Matías, Pere Colet, "Logical operations with localized structures". New Journal of Physics, 14, 013040 (2012). Abstract.
[2] Stéphane Barland, Oreste Piro, Massimo Giudici, Jorge R. Tredicce, Salvador Balle, "Experimental evidence of van der Pol—Fitzhugh—Nagumo dynamics in semiconductor optical amplifiers". Physical Review E, 68, 036209 (2003). Abstract.
[3] Maia Brunstein, Alejandro M. Yacomotti, Isabel Sagnes, Fabrice Raineri, Laurent Bigot, Ariel Levenson, "Excitability and self-pulsing in a photonic crystal nanocavity". Physical Review A, 85:031803, 2012. Abstract.
[4] T. Elsass, K. Gauthron, G. Beaudoin, I. Sagnes, R. Kuszelewicz, S. Barbay, "Control of cavity solitons and dynamical states in a monolithic vertical cavity laser with saturable absorber". Euro Physics Journal D, 59, 91 (2010). Abstract.
[5] F. Plaza, M. G. Velarde, F. T. Arecchi, S. Boccaletti, M. Ciofini, R. Meucci. "Excitability following an avalanche-collapse process". Europhysics Letters, 38, 85 (1997). Abstract.
[6] M. Giudici, C. Green, G. Giacomelli, U. Nespolo, J. R. Tredicce. "Andronov bifurcation and excitability in semiconductor lasers with optical feedback". Physical Review E, 55, 6414 (1997). Abstract.
[7] D. Goulding, S. P. Hegarty, O. Rasskazov, S. Melnik, M. Hartnett, G. Greene, J. G. McInerney, D. Rachinskii, G. Huyet, "Excitability in a Quantum Dot Semiconductor Laser with Optical Injection". Physical Review Letters, 98, 153903 (2007). Abstract.
[8] J. D. Murray. Mathematical biology. Springer, New York, 1990. 
[9] Wolfgang Maass, Thomas Natschläger, Markram Henry, "Real-Time Computing Without Stable States: A New Framework for Neural Computation Based on Perturbations". Neural Computation, 14, 2531-2560 (2002). Abstract.
[10] M.A. Nahmias, B.J. Shastri, A.N. Tait, P.R. Prucnal, "A Leaky Integrate-and-Fire Laser Neuron for Ultrafast Cognitive Computing". IEEE Journal of Selected Topics in Quantum Electronics, 19(5), 1-12 (2013).
[11] Adolphe Pacault, Patrick Hanusse, Patrick De Kepper, Christian Vidal, Jacques Boissonade. Phenomena in homogeneous chemical systems far from equilibrium. Accounts of Chemical Research, 9(12), 438-445 (1976). Abstract.
[12] D. Randall, W. Burggren, K. French, R. Eckert. Eckert Animal Physiology. W. H. Freeman, 2002. Google Book.
[13] F. Selmi, R. Braive, G. Beaudoin, I. Sagnes, R. Kuszelewicz, S. Barbay, "Relative Refractory Period in an Excitable Semiconductor Laser". Physics Review Letters, 112, 183902 (2014). Abstract.
[14] Stefano Beri, Lilia Mashall, Lendert Gelens, Guy Van der Sande, Gabor Mezosi, Marc Sorel, Jan Danckaert, Guy Verschaffelt, "Excitability in optical systems close to Z₂-symmetry". Physics Letters A, 374, 739 (2010). Abstract.
[15] Sylvain Barbay, Robert Kuszelewicz, Alejandro M. Yacomotti. Excitability in a semiconductor laser with saturable absorber. Opt. Lett., 36(23):4476—4478, 2011. Abstract.
[16] Thomas Van Vaerenbergh, Martin Fiers, Pauline Mechet, Thijs Spuesens, Rajesh Kumar, Geert Morthier, Benjamin Schrauwen, Joni Dambre, Peter Bienstman, "Cascadable excitability in microrings". Optics Express, 20, 20292 (2012). Abstract.
[17] Sebastian Wieczorek, Bernd Krauskopf, Daan Lenstra, "Multipulse Excitability in a Semiconductor Laser with Optical Injection". Physical  Review Letters, 88, 063901 (2002). Abstract.
[18] H. J. Wünsche, O. Brox, M. Radziunas, F. Henneberger, "Excitability of a Semiconductor Laser by a Two-Mode Homoclinic Bifurcation". Physical  Review Letters, 88, 023901 (2001). Abstract.
[19] A. M. Yacomotti, P. Monnier, F. Raineri, B. Ben Bakir, C. Seassal, R. Raj, J. A. Levenson, "Fast Thermo-Optical Excitability in a Two-Dimensional Photonic Crystal". Physical  Review Letters, 97, 143904 (2006). Abstract.