Communicating Quantum States with Alice on a Satellite

Some authors of "Experimental Satellite Quantum Communications" [4] during a night shift: (Right to Left) Davide Bacco, Simone Gaiarin, Daniele Dequal, Giuseppe Vallone and Paolo Villoresi.

Authors: Giuseppe Vallone¹, Davide Bacco¹, Daniele Dequal¹, Simone Gaiarin¹, Vincenza Luceri², Giuseppe Bianco³, Paolo Villoresi¹

Affiliation:
¹Dipartimento di Ingegneria dell’Informazione, Università degli Studi di Padova, Italy 
²e-GEOS spa, Matera, Italy 
³Matera Laser Ranging Observatory, Agenzia Spaziale Italiana, Matera, Italy.

The exchange of quantum bits – or qubits – is a fundamental process in all Quantum Information protocols. The faithful transport of the fragile quantum content of a photon is needed inside the prototypes of photonic quantum computers as well for the teleportation of a given state.

Past 2Physics articles by this group:
August 31, 2014: "A True Randomness Generator Exploiting a Very Long and Turbulent Path" by Davide G. Marangon, Giuseppe Vallone, Paolo Villoresi.
November 24, 2013: "How to Realize Quantum Key Distribution with a Limited and Noisy Link" by Paolo Villoresi.
May 19, 2008: "The Frontier of Quantum Communication is the Space"
by Paolo Villoresi.

Image 1: The scenario where satellites uses Quantum Communications for distributing secure keys to a global communications network.

Moreover, in the relevant application of Quantum-Key-Distribution (QKD), which allows to create a private key between two terminals exploiting the laws of Quantum Physics, such exchange of qubits is expected to cover very long distances [1]. Indeed, in order to connect with secure communications two embassies, two corporate branches and so on, effective quantum communication schemes on a planetary scale are needed. The fibre channels were investigated first for the realization of QKD, such that now several commercial devices based on optical cables are already in operations. Fibers are very efficient up to about 100 km, and the present limit for QKD in fiber is 300 km as demonstrated in a recent experiment [2]; beyond that scale there is the need of quantum repeaters, which presently are in development in advanced research labs. A radically different approach is to go along a Space channels, and exploit a satellite as the sender or the receiver. From the link budget analysis and the effect of turbulence in the propagation, it is evident that the transmitter (Alice) is most conveniently located on the satellite and the receiver (Bob) on the ground [3,4,5].

The first attempt of quantum communication in Space, was made in 2008 by Villoresi et al [6,7], where photons launched from a ground station were reflected by CCRs (corner cube retroreflectors) and aimed back to the Earth. In that case it was used the Japanese satellite Ajisai for emulate an optical transmitter in space. In that work it was demonstrated the application of spectral, spatial and temporal filtering capable to point out the return photons with global losses in the up- and down-link as strong as 157 dB.

In the present experiment, reported recently in Ref. [8], we introduced novel schemes for temporal synchronization and the optical interface, realizing a significant improvement in SNR (signal to noise ratio), dark counts and total transmissivity. We proved that a generic qubit with polarization encoding preserves its characteristics in a channel starting from a source realized again by a retroreflector in orbit and measured on ground by a state analyzer connected to an astronomical telescope designed for satellite-laser ranging [9]. Moreover we were able to prove a communication protocol measuring not only one polarization degree, but a complete set of four polarization states required for protocols as QKD [5].

A very important parameter in QCs (Quantum communications) is represented by the QBER factor (Quantum bit error rate), defined as the number of wrong bits received in a slot time. In case this factor is too high (the threshold depends on the chosen protocol, and on the sending rate), the security of the generated key is not guaranteed [8]. In our experiment we showed, by using some LEO (Low Earth Orbit) satellites (Starlette, Stella, Larets, Jason-2), that our method and setup allows a secure communication in a very long distance scenario (~2000 km). The measured QBER in different runs results of the order of a few percent. It was possible to attest that even with high losses, variable attenuation, and high background a quantum key distribution system works, and an unconditionally secure key, needful for encryption, can be generated also in this case.

Image 2: Picture of the SLR laser and MLRO station situated in Matera.

For the first time qubits bouncing from space were measured and analyzed in different polarization states. Moreover, all the results were obtained with existing satellites naturally used for geodetic studies and other activities usually equipped with CCRs. The optical setup used in the experiment is yet completely integrable in a lot of OGS (optical ground station) and present an easy interface between quantum and classical signals. Furthermore, the technology of SLR and classical satellite communications was exploited for synchronizing the transmitter and the receiver, even though the synchronization process is not so easy with satellite in motion.

What’s next? The possibility of sending and receiving single photons in very long distances paves the way to a lot of future experiments and brings Quantum Physics and Quantum Communication in a privilege position. Firstly, the big effort made by the Governments and by the University is surely compensated. From a scientific point of view these experimental results are very fascinating because they allow new experiments based on this technology. In particular, QKD could be realized with a small and compact device capable of changing polarization of photons, creating a base element for quantum two-way protocols. Additionally, experiments like entanglement distribution involving long distances, Bell inequality and teleportation protocol could be possible in next few years.

With this experiment, it was demonstrated that, not only free-space quantum key distribution is a ready technology, but also the quantum satellite communication is nowadays possible and realizable. The results open the way to look towards a global space quantum network, where OGS could talk with satellite and vice-versa creating a global secure network.

Acknowledgments: The work was carried out within QuantumFuture, one of ten Strategic Projects funded by the University of Padova in 2009. Coordinated by Prof. Villoresi, the project has established the Quantum Communication Laboratory and engaged four research groups in a joint activity: Quantum Communications, Quantum Control Theory, Quantum Astronomy and Quantum Optics.

References:
[1] Valerio Scarani, Helle Bechmann-Pasquinucci, Nicolas J. Cerf, Miloslav Dušek, Norbert Lütkenhaus, Momtchil Peev, "The security of practical quantum key distribution", Review of Modern Physics, 81, 1301 (2009). Abstract.
[2] Boris Korzh, Charles Ci Wen Lim, Raphael Houlmann, Nicolas Gisin, Ming Jun Li, Daniel Nolan, Bruno Sanguinetti, Rob Thew, Hugo Zbinden, “Provably Secure and Practical Quantum Key Distribution over 307 km of Optical Fibre”. Nature Photonics, 9(3), 7. doi:10.1038/nphoton.2014.327 (2014). Abstract.
[3] Cristian Bonato, Markus Aspelmeyer, Thomas Jennewein, Claudio Pernechele, Paolo Villoresi, Anton Zeilinger, “In- fluence of satellite motion on polarization qubits in a Space-Earth quantum communication link,” Optics Express, 14,  10050 (2006). Full Article.
[4] Andrea Tomaello, Cristian Bonato, Vania Da Deppo, Giampiero Naletto, Paolo Villoresi, “Link budget and background noise for satellite quantum key distribution,” Advances in Space Research, 47, 802 (2011). Abstract.
[5] C. Bonato, A. Tomaello, V. Da Deppo, G. Naletto, and P. Villoresi, “Feasibility of satellite quantum key distribution,” New Journal of Physics, 11, 45017 (2009). Full Article.
[6] P Villoresi, T Jennewein, F Tamburini, M Aspelmeyer, C Bonato, R Ursin, C Pernechele, V Luceri, G Bianco, A Zeilinger and C Barbieri, "Experimental verification of the feasibility of a quantum channel between space and Earth", New Journal of Physics, 10, 033038 (2008). Full Article. 2Physics Article.
[7] Giuseppe Vallone, Davide Bacco, Daniele Dequal, Simone Gaiarin, Vincenza Luceri, Giuseppe Bianco, and Paolo Villoresi, “Experimental Satellite Quantum Communication” Phys. Rev. Lett. 115, 040502 (2015). Abstract.
[8] Davide Bacco, Matteo Canale, Nicola Laurenti, Giuseppe Vallone, Paolo Villoresi, "Experimental quantum key distribution with finite-key security analysis for noisy channels", Nature Communications, 4:2363, doi: 10.1038/ncomms3363 (2013). Abstract. 2Physics Article.
[9] http://ilrs.gsfc.nasa.gov/

A True Randomness Generator Exploiting a Very Long and Turbulent Path

From Left to Right: Paolo Villoresi,  Davide Marangon, Giuseppe Vallone

Authors:
Davide G. Marangon, Giuseppe Vallone,  Paolo Villoresi

Affiliation:
Department of Information Engineering, University of Padova, Italy.

Random numbers are the main ingredients of cryptographic protocols for both Classical and Quantum Information. However, it is well known that to rely on random numbers produced with deterministic algorithms can be very risky and it is of fundamental priority to discover physical processes to generate "pure" random numbers. Usually True Random Number Generators (TRNG) are implemented by exploiting classical or quantum microscopical processes. However it can be shown that random numbers can be extracted from macroscopic physical systems.

Past 2Physics articles by Paolo Villoresi:
November 24, 2013: "How to Realize Quantum Key Distribution with a Limited and Noisy Link".
May 19, 2008: "The Frontier of Quantum Communication is the Space".

In the 60s, the famous "Butterfly Effect" captured the idea that when one deals with the terrestrial atmosphere, very tiny perturbations such as the air moved by the tail strokes of a butterfly can lead to very huge consequences as a hurricane in some other place in the world. Terrestrial atmosphere indeed may be seen as a physical system ruled by a chaotic dynamic. Moreover, while statistical models are available for average trends, the prediction of the instantaneous motion of the air mass in a spot is out of reach.

From the textbooks we know that the propagation of light through an inhomogeneous medium is strongly influenced by the refractive index distribution. We experimentally investigated this phenomenon with the purpose of realizing if such propagation along a free-space path may induce a useful randomness. An intuition of such effect manifested during the campaigns for the experiments we carried out at the Canarias on the quantum Communications along extremely long links [1, 2]. The atmospheric turbulence in the path is very strong, preventing for example the direct application of interferometry [3]. However, the effect of turbulence is crucial there for the application of a method that exploits brief moments of high transmissivity for good communication [2]. We tried to turn it here instead into a useful resource for randomness.

FIG. 1. The experiment was set up between the islands of La Palma and Tenerife where a laser beam (with λ = 810 nm) was exchanged between the two islands. After propagating across a 143 km Free Space Optical link, the wavefront of the beam features a randomly composed speckle pattern as consequence of the distortions induced by the atmosphere.

The experiment we describe here was set between the two islands of La Palma and Tenerife: on the rooftop of the Jacobus Kaptein Telescope (JKT) building at an altitude of 2360 m; our transmitting telescope for the Quantum Communication was aimed to send a continuous laser beam towards the ESA Optical Ground Station (OGS) 143 km far away, at Izana, near the mount Teide, see Figure 1. The telescope -- that was designed and realized in Padova -- is a refractor based on a 230 mm aspheric singlet. In the path, the turbulent atmosphere is comparable to a dynamic volumetric scatterer and the electromagnetic field is subjected to phase delays and amplitude fluctuations, induced by the inhomogeneities of the refractive index of the air [4–6]. The receiver then observes a beam profile which does not feature the typical intensity Gaussian distribution, rather a collection of clear and dark spots of irregular shape, the so-called speckle pattern. The speckle pattern evolves according the unpredictable dynamic of the turbulence as consequence of the random walks the electromagnetic field suffers while propagating. Therefore at the receiving plane a continuously and randomly evolving distribution of speckles was acquired with a CMOS camera and for every frame one has a variable number of spots randomly taking different spatial configurations [7].

Randomness is then extracted by using the geometrical complexity of the frames evaluating the centers of mass, the so-called centroids, of those speckle areas with the same intensity. For the implementation of the method the relevant pixels in CCD are labelled sequentially with an index s, s ∈ {1, . . . , N}, the n_f  speckle centroids of the frame f are elaborated, an ordered sequence S_f = {s₁ , s₂ , . . . , s_nf } with s₁ < s₂ < · · · < s_nf  is formed, by considering then the pixels where a centroid falls in. The pixel grid can be regarded as the classical collection of urns where the turbulence randomly throws balls (the centroids) in, see Figure 2. Because of the random nature of the process, the centroids visit every part of the grid with the same probability. A given frame f  “freezes” one S_f  out of the

possible and equally likely sequences of n_f centroids. Among all of combinations, a given S_f can be univocally identified with its lexicographic index I (S_f )

with 0 ≤ I (S_f ) ≤ T_f  - 1. Basically, (2) enumerates all the possible arrangements which succeed a given centroids configuration. As an uniform RNG is supposed to yield numbers identically and independently distributed (i.i.d.) in a range [X,Y ], as this method generates a random integer in the range [0, T_f - 1]. In order then to optimize the conversion from integer to random bits without introducing any bias, an efficient algorithmic procedure was applied to the bits [8].

FIG. 2. (click on the figure to view higher resolution) The figure represents a scheme of the mechanism employed to extract randomness from the frames of the captured video. Every frame features a different spatial disposition of centroids (the yellow crosses). To every centroid configuration, a univocal lexicographic index is associated. The lexicographic index then is converted in random bits.

In this proof of principle, a generation rate of 400 kbit/s was achieved but it can be easily enhanced by using cameras with higher resolutions. Another point, worth to be stressed, is that this method does not rely on sensitive and hardly detectable processes which require extremely tuned hardware: indeed unavoidable hardware non-idealities can induce bits dependencies and bias. In addition, from the theoretical point of view, the strength of the method lies in the fact the dynamic of turbulent atmosphere on such a long link represents a physical process which is practically impossible to be predicted, both analytically (at the present time only statical models are given) and numerically (it would require an unbounded computational power).

In addition to a sound knowledge of the physical process employed, it is necessary to apply statistical tests in order to exclude the presence of defects caused by a faulty hardware. This has been done by applying the most stringent test batteries for randomness such as the Alphabit and Rabbit batteries belonging to the TESTU01, the NIST SP-800-22 suite and the AIS31 suite. All the tests were successfully passed.

The presented procedure then could be an efficient method to generate random numbers to be employed in long range QC setups. More in detail, bits generated in this way could be used in connection with other protocols involving Quantum Random Number Generator: for example in the first well known experiment of randomness expansion by means of non-locality [9] the initial seed was obtained by mixing numbers obtained with several generators including atmospheric radio electromagnetic noise. Finally, the extraction algorithm can be easily adapted to other paradigms involving spatial random complex patterns.

References:
[1] Ivan Capraro, Andrea Tomaello, Alberto Dall’Arche, Francesca Gerlin, Ruper Ursin, Giuseppe Vallone, Paolo Villoresi, "Impact of turbulence in long range quantum and classical communication". Physical Review Letters, 109, 200502, (2012). Abstract.
[2] Giuseppe Vallone, Davide Marangon, Matteo Canale, Ilaria Savorgnan, Davide Bacco, Mauro Barbieri, Simon Calimani, Cesare Barbieri, Nicola Laurenti, Paolo Villoresi, "Turbulence as a Resource for Quantum Key Distribution in Long Distance Free-Space Links". arXiv:1404.1272 [quant-ph] (2014).
[3] Cristian Bonato, Alexander V. Sergienko, Bahaa E. A. Saleh, Stefano Bonora, Paolo Villoresi, "Even-Order Aberration Cancellation in Quantum Interferometry". Physical Review Letters, 101, 233603 (2008). Abstract.
[4] Larry C. Andrews and Ronald L. Phillips, "Laser beam propagation through random media", volume 152 (SPIE press, 2005). 
[5] R. L. Fante, "Electromagnetic beam propagation in turbulent media". Proceedings of the IEEE, 63, 1669,(1975). Abstract.
[6] R. L. Fante, "Electromagnetic beam propagation in turbulent media - An update". Proceedings of the IEEE, 68, 1424 (1980). Abstract.
[7] Davide G. Marangon, Giuseppe Vallone, Paolo Villoresi, "Random bits, true and unbiased, from atmospheric turbulence". Scientific Reports, 4 : 5490 (2014). Full Article.
[8] Peter Elias. "The efficient construction of an unbiased random sequence". Annals of Mathematical Statistics, 43, 865 (1972). Full Article.
[9] S. Pironio, A. Acín, S. Massar, A. Boyer de la Giroday, D.N. Matsukevich, P. Maunz, S. Olmschenk, D. Hayes, L. Luo, T.A. Manning,  C. Monroe. "Random numbers certified by Bell’s theorem". Nature, 464, 1021 (2010). Abstract.

How to Realize Quantum Key Distribution with a Limited and Noisy Link

The authors of the paper "Experimental quantum key distribution with finite-key security analysis for noisy channels" [5]: (Left to Right) Paolo Villoresi, Nicola Laurenti, Giuseppe Vallone, Davide Bacco, Matteo Canale.

Author: Paolo Villoresi

Affiliation: Dept of Information Engineering, University of Padua, Italy.

A huge amount of sensitive data travels every day on the internet: credit card numbers, emails, medical reports, social network contents, etc. Such traffic is constantly increasing on a global scale, and data protection is becoming not only a privacy issue for an individual, but also an economical and national security asset. This call for an intense use of cryptography, as we see in everyday transactions and queries over the internet. In particular, depending on the application context, security services may be required such as data integrity, confidentiality and authenticity. The corresponding security mechanisms are implemented by means of algorithms that require cryptographic keys, that is, secret bit sequences shared only by the sender and the receiver, and totally hidden from unintended users or devices in the network [1].

Past 2Physics article by Paolo Villoresi:
May 19, 2008: "The Frontier of Quantum Communication is the Space".

By leveraging the laws of quantum physics, two distant parties are able to share cryptographic keys with unconditional security, i.e., with provably negligible information leaked to the eavesdropper. When photons are used as bit carriers, in fact, observations of the attacker on the quantum transmission statistically produce a perturbation on the system itself, thus allowing the legitimate users to estimate the information that (s)he may possess and to take the appropriate countermeasures (e.g., discarding the current key). On the contrary, the security of classical cryptography – which is currently used on the internet – is based on mathematical problems for which no efficient solution exists nowadays. Such a solution, however, may appear in the near future, thanks to mathematical and computational breakthroughs as well as for the development of quantum computers.

Therefore, these keys need to be securely exchanged. As opposed to classical methods, by using quantum cryptography this key exchange (the quantum-key-distribution or QKD) can be brought to the level of being unconditionally secure, envisaging the distant parties with quanta of lights – photons – as the information carrier. The ultimate security may be obtained using the One-Time-Pad or OTP technique, which requires for each encryption and decryption a fresh and truly random key, which shall never be used again. Therefore, besides being the most secure, OTP is also the most key-demanding cipher.

Against this growing need for cryptographic keys, the optimization of the key exchange is a major issue, except for partners that may share a box of hard drives with multi-terabytes of random bits when needed. In the case of distant terminal that are in the need to implement QKD, simple aspects such as the duration of the communication and the background level are critical aspects and useful to assess the viability of a QKD implementation.

In particular, in protected environment like in research laboratories QKD systems are deployed in isolated environment and -- assuming that an arbitrarily high number of photons can be exchanged -- in real world applications these conditions are no longer verified. In realistic scenarios, in fact, the ratio of the number of final secret bits to the number of sent photons decreases on one hand with the number of sent photons, and on the other hand with increasing noise in the transmission channel and in the receiver apparatus.

Image 1: Quantum Communications using satellite are the best example of QKD with finite – and often short - duration and noisy channels.

For real-world scenarios of crucial data exchange and of remote location, as for the satellite communication, the key may be exchanged during satellite passages – which may last as short as a few minutes per hour, and are affected by background illumination which induces spurious light in the detectors [2]. Only recently the theoretical security analysis was made for the case of finite communication time [3], providing a theory for the maximum rate and the measurement strategy in the case of QKD along a modified BB84 protocol (For details of BB84 protocol, visit the Wikipedia page [4]).

In the work recently appeared on 'Nature Communications' by our group [5], including Davide Bacco, Matteo Canale, Nicola Laurenti, Giuseppe Vallone and Paolo Villoresi, all with the Department of Information Engineering at the University of Padova, we have experimentally shown the upper limit to quantum key distribution, in the presence of environmental noise, with the transmission of a limited number of photons and by considering different attack models.

Image 2: Details of the QKD receiver used in the experiment.

In the 'Nature Communications' paper [5] we described a fully fledged realization of a QKD system in the finite key regime with all details, which could represent a practical guide for the experimental realizations of the modified BB84 protocol. Moreover, in our work we extend the analysis on the key rate: the most general security analysis includes all the possible attack from an eavesdropper, that may store the sniffed photons in a quantum memory (q-memory) and analyze them at leisure. In addition, we decided to introduce the notion of pragmatic security, which is relevant for today attack possibilities since large q­memories are not yet available. This limitation allows to extract more secure bits with respect to the ones obtainable given a requirement of general security, that is, assuming that Eve is limited only by the law of physics.

However, pragmatic secrecy offers forward security: if a key is produced today with pragmatic secrecy (without a q­memory available for Eve), the key or a message encrypted with it will be secure for any future task, even when a q­memory will be present. This is opposed to computationally secure classical cryptography or key agreement where the available information can be stored by the Eavesdropper and decrypted in the future with higher computational power (either technological or algorithmic).

This idea is supported by considering that in a long-­term perspective (more than 50 years), a general security is the goal. In the near future (5-­10 years), we know that an ideal intercept-­resend attack is the best option that an eavesdropper can choose because the quantum memory needed for a general or coherent attack is not yet available. This analysis is crucial for an actual implementation of QKD beyond fiber, such as satellite quantum communication, a situation characterized by a short key in general due to a low rate and an high background noise.

This result opens perspectives for scenarios where the transmission window is limited by physical constraints, as for satellite communications, where the passage of one terminal over the other is restricted to a few minutes. The Padua team has been active on Satellite Quantum Communications for years, and obtained the first experimental demonstration of single photon exchange with an orbiting terminal in 2008 [2].

Acknowledgments: The work was carried out within QuantumFuture, one of ten Strategic Projects funded by the University of Padova in 2009. Coordinated by Prof. Villoresi, the project has established the Quantum Communication Laboratory and engaged four research groups in a joint activity: Quantum Communications, Quantum Control Theory, Quantum Astronomy and Quantum Optics.

References:
[1] Valerio Scarani, Helle Bechmann-Pasquinucci, Nicolas J. Cerf, Miloslav Dušek, Norbert Lütkenhaus, Momtchil Peev, "The security of practical quantum key distribution", Review of Modern Physics, 81, 1301 (2009). Abstract.
[2] P Villoresi, T Jennewein, F Tamburini, M Aspelmeyer, C Bonato, R Ursin, C Pernechele, V Luceri, G Bianco, A Zeilinger and C Barbieri, "Experimental verification of the feasibility of a quantum channel between space and Earth", New Journal of Physics, 10, 033038 (2008) [IOP select paper]. Abstract. 2Physics Article.
[3] Marco Tomamichel, Charles Ci Wen Lim, Nicolas Gisin, Renato Renner, "Tight finite-key analysis for quantum cryptography", Nature Communications, 3:634, doi:10.1038/ncomms1631 (2012). Abstract.
[4] Wikipedia page on BB84 protocol.
[5] Davide Bacco, Matteo Canale, Nicola Laurenti, Giuseppe Vallone, Paolo Villoresi, "Experimental quantum key distribution with finite-key security analysis for noisy channels", Nature Communications, 4:2363, doi: 10.1038/ncomms3363 (2013). Abstract.

Role of Statistics in Two-Particle Anderson Localization

Roberto Osellame (left)  and Fabio Sciarrino (right)

Authors: Roberto Osellame¹ and Fabio Sciarrino²

Affiliations:
¹Istituto di Fotonica e Nanotecnologie (IFN) – CNR, Milan, Italy.
Link to the Femtosecond Laser Micromachining group >>
²Dipartimento di Fisica – Sapienza Università di Roma, Rome, Italy.
Link to the Quantum Optics group >>

3D QUEST >>

Disorder in our daily life typically has a negative connotation. Also in science it has been normally considered as a source of noise or imperfection. However, disorder is ubiquitous in nature: indeed it plays, on the one hand, a crucial role in understanding the behavior of complex physical phenomena [1] and, on the other hand, it can turn into an advantageous property for developing completely new devices [2]. One of the most striking effects of disorder is the suppression of transport of electrons in a disordered crystal. This phenomenon, known as Anderson localization after the 1958 paper by P.W. Anderson [3], is due to coherent scattering of the electron wavefunction in the disordered crystal and is general to any wave propagating in a disordered media [4]. Being a coherent effect, Anderson localization can be directly observed with photons, due to their little interaction with the environment and long coherence time. In addition, photonic structures, e.g. waveguide lattices, can be manufactured with a very high level of control of the structure parameters and are therefore prone to implementing and investigating different kinds of disorder.

Fig. 1 Femtosecond laser writing of optical waveguides. The glass sample is translated with respect to the writing laser beam. The bright spot in the glass comes from electron plasma generated by the focused laser; the energy transferred to the glass matrix after plasma relaxation is responsible for the local increase of refractive index.

A very recent technique (Fig. 1) that allows the accurate fabrication of photonic lattices is femtosecond laser waveguide writing [5]. With respect to standard microfabrication techniques it enables rapid prototyping of devices, being a direct write method, and extreme flexibility in the layout reconfiguration, being a maskless process [6]. In addition, it has the unique capability of exploiting the third dimension in the fabrication of the photonic circuits, which opens the possibility for completely new architectures [7,8].

The simplest way of observing Anderson localization is the study of a single particle in a 1D periodic crystal with static disorder (i.e. disorder that is spatially uncorrelated, but does not vary with time)[9]. This is analogous to observing the quantum walk of a single photon in a disordered waveguide array. If the observation in time is not continuous, but periodical, we are studying a discrete quantum walk. Femtosecond laser waveguide writing can be effectively exploited to produce matrices of integrated optical interferometers constituting a discrete quantum walk for photons [10]. The same technology enables a straightforward implementation of arbitrary phases in the different optical paths, thus introducing disorder in the structure. In our recent paper, published on Nature Photonics [11] in collaboration with Scuola Normale di Pisa, we implemented random phase maps representing static disorder, as in the chip depicted in Fig. 2a.

Anderson localization is essentially a single particle process, however in this work we experimentally investigated for the first time the role of particle statistics in the localization of two non-interacting photons. In order to mimick bosonic and fermionic statistics we exploited the symmetric and antisymmetric wavefunction of polarization entangled photons [10]. We observed Anderson localization for the two particles obeying both statistics, however when two bosonic particles were propagating they tended to localize on the same site, while the fermionic ones localized on adjacent sites but not on the same one, as expected from the Pauli exclusion principle (Fig. 2b). We also observed that the mean position between the two particles has a stronger localization for fermions than for bosons, while the relative distance has a smaller expectation value for bosons than for fermions [11].

Fig. 2 (a) Scheme of the device implementing a discrete quantum walk with static disorder. The m ports represents the sites of the 1D crystal, the n steps represents the discrete observation times. The colors of the phase shifters represent different implemented phases, which are constant along n to implement a static disorder. (b) Experimental correlation maps representing the joint probability of finding one photon in output port i and the other in output port j; with respect to the case without disorder (where ballistic propagation is observed), a clear localization is observed when static disorder is introduced [11] (Click on the image to view a version of better resolution).

These results demonstrate that even without interaction, particle statistics is capable of influencing the way two particles localize in a disordered media. In addition they show the potential of femtosecond laser waveguide writing for implementing arbitrary quantum walks with controlled disorder. The capability of our technology to implement arbitrary phase maps in quantum walks will enable the experimental quantum simulation of the quantum dynamics of multiparticle correlated systems and its ramifications towards the implementation of realistic universal quantum computation with quantum walks.

References
[1] Liad Levi, Yevgeny Krivolapov, Shmuel Fishman & Mordechai Segev, “Hyper-transport of light and stochastic acceleration by evolving disorder”, Nature Physics, 8, 912-917 (2012). Abstract.
[2] Diederik S. Wiersma, “Disordered photonics”, Nature Photonics, 7, 188-196 (2013). Abstract.
[3] P.W. Anderson, “Absence of diffusion in certain random lattices”, Physical Review, 109, 1492-1505 (1958). Abstract.
[4] Mordechai Segev, Yaron Silberberg & Demetrios N. Christodoulides, “Anderson localization of light”, Nature Photonics, 7, 197–204 (2013). Abstract.
[5] Rafael R. Gattass, Eric Mazur, “Femtosecond laser micromachining in transparent materials”, Nature Photonics, 2, 219 - 225 (2008). Abstract.
[6] G. Della Valle, R. Osellame, P. Laporta, “Micromachining of photonic devices by femtosecond laser pulses”, J. Opt. A 11, 013001(2009). Abstract.
[7] Nicolò Spagnolo, Chiara Vitelli, Lorenzo Aparo, Paolo Mataloni, Fabio Sciarrino, Andrea Crespi, Roberta Ramponi & Roberto Osellame, “Three-photon bosonic coalescence in an integrated tritter”, Nature Communications, doi:10.1038/ncomms2616 (Published March 19, 2013). Abstract.
[8] Mikael C. Rechtsman, Julia M. Zeuner, Andreas Tünnermann, Stefan Nolte, Mordechai Segev & Alexander Szameit, “Strain-induced pseudomagnetic field and photonic Landau levels in dielectric structures”, Nature Photonics, 7, 153-158 (2013). Abstract.
[9] Yoav Lahini, Assaf Avidan, Francesca Pozzi, Marc Sorel, Roberto Morandotti, Demetrios N. Christodoulides and Yaron Silberberg, “Anderson Localization and Nonlinearity in One-Dimensional Disordered Photonic Lattices”, Physical Review Letters, 100, 013906 (2008). Abstract.
[10] Linda Sansoni, Fabio Sciarrino, Giuseppe Vallone, Paolo Mataloni, Andrea Crespi, Roberta Ramponi and Roberto Osellame, “Two-particle bosonic–fermionic quantum walk via integrated photonics”, Physical Review Letters, 108, 010502 (2012). Abstract.
[11] Andrea Crespi, Roberto Osellame, Roberta Ramponi, Vittorio Giovannetti, Rosario Fazio, Linda Sansoni, Francesco De Nicola, Fabio Sciarrino & Paolo Mataloni, “Anderson localization of entangled photons in an integrated quantum walk”, Nature Photonics, doi:10.1038/nphoton.2013.26 (Published online March 3, 2013). Abstract.