Role of Statistics in Two-Particle Anderson Localization
Authors: Roberto Osellame1 and Fabio Sciarrino2
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 >>
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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  and, on the other hand, it can turn into an advantageous property for developing completely new devices . 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 , is due to coherent scattering of the electron wavefunction in the disordered crystal and is general to any wave propagating in a disordered media . 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 . 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 . 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). 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 . 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  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 . 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 .
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  (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.
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