### Quantum Walks of Correlated Photons in Integrated Waveguide Arrays

**Alberto Peruzzo**

[This is an invited article based on a recently published work by the authors -- 2Physics.com]

**Authors: Alberto Peruzzo and Jeremy L. O’Brien**

Affiliation: Centre for Quantum Photonics, H. H. Wills Physics Laboratory and Dept of Electrical & Electronic Engineering, University of Bristol, UK

Affiliation: Centre for Quantum Photonics, H. H. Wills Physics Laboratory and Dept of Electrical & Electronic Engineering, University of Bristol, UK

Since their initial development for studying the random motion of microscopic particles (such as those suspended in a fluid), random walks have been a successful model for random processes applied in many fields, from computer science to economics. Such processes are random in the sense that at a particular time the choice for a particle to make a particular step is probabilistic and decided by flipping a coin.

Past 2Physics articles based on works of this group:

Sep 20, 2009: "Shor's Quantum Factoring Algorithm Demonstrated on a Photonic Chip"

May 2, 2008: "Silicon Photonics for Optical Quantum Technologies"

In the quantum analogue – the quantum walk[1] – the walker is, at a given time, in a superposition of the possible states and different paths can interfere, exhibiting ballistic propagation with faster dynamics compared to the slow diffusion of classical random walks, prompting applications in quantum computer science and quantum communication. Indeed quantum walks have been shown to be universal for quantum computing, enable direct simulation of important physical, chemical and biological systems, and the possibility to study very large entangled states of several particles, with potential to investigate the existence of quantum- classical boundaries.

The first application of quantum walks was search algorithms on graphs (vertices connected by edges) and is more efficient than the classical search. Finding an element in a collection of N vertices using a quantum walk requires √N steps while the classical algorithm takes N steps to check all the vertices.

Quantum walks come in two types, the discrete time quantum walk (DTQW) and the continuous time quantum walk (CTQW). In a DTQW the step direction is specified by a coin and a shift operator, which are applied repeatedly, similarly to the classical random walk, but with the difference that now the coin flip is replaced by a quantum coin operation defining the superposition of the directions the step undertakes. The CTQW describes tunnelling of quantum particles through arrayed potential wells.

The theory of quantum walks have been extensively studied but only few experimental demonstration of several steps of single particle quantum walks with atoms, trapped ions, nuclear magnetic resonance and photons have been carried out so far.

Quantum walks are based on wave interference and require a stable environment to reduce the noise (decoherence) that would otherwise destroy the interference. Interferometric stability and miniaturization using phonics waveguide circuits have been shown to be a promising approach for quantum optics experiments, silica-on-silicon waveguides have been used to demonstrate high fidelity quantum information components [2, 3, 4] and a small scale quantum algorithm for prime number factorization [5].

We’ve implemented CTQW of photons designing periodic waveguide arrays in integrated photonic circuits that enable the injection of single photons and the coupling to single photon detectors at their output. The chips were fabricated in the high refractive index contrast material silicon oxynitride, enabling to quickly stop the coupling between neighbour waveguides so that high level of control over the propagation was possible.

**Integrated quantum photonic circuit used to implement a continuous time quantum walk of two correlated photons.**

In contrast to all previous demonstrations — which were restricted to single particle quantum walks that have exact classical counterparts — we have demonstrated the quantum walk of two identical photons spatially correlated within arrayed waveguide, observing uniquely quantum mechanical behaviour in the two-photon correlations at the outputs of this array [6]. Pairs of correlated photons were generated with a standard type I spontaneous parametric down-conversion process, a nonlinear process where a 402nm wavelength CW laser pumps a χ

^{2}nonlinear bismuth borate crystal generating pairs of photons at 804nm wavelength in conservation of energy and momentum. The correlated photons were coupled to the waveguide using fibre arrays and the correlations at the output were recorded by measuring two photon coincide events with a detection system of 12 avalanche single photons detectors and 3 programmable counting boards. The measured correlations fit with high similarity to our simulations.

**Artist’s impression of the two-photon quantum walk.**

Credit: Image by Proctor & Stevenson

Credit: Image by Proctor & Stevenson

We’ve shown that the results strongly depend on the input state and these correlations violate classical limits by 76 standard deviation, proving that such phenomena cannot be described using classical theory. This generalized form of quantum interference is similar to the Hong-Ou-Mandel dip effect in an optical beam splitter but in our case on a 21 mode system. Bunching of correlated photons reduces the probability of detecting two photons at the opposite sides of the array while enhancing the case of two particles at the same side.

Such two particle quantum walks have already been identified as a powerful computational tool for solving important problems such as graph isomorphism, and provide a direct route to powerful quantum simulations. Implementing new algorithms based on quantum walks will require integration of the single photon sources and detectors. These have already been showed to be compatible with integration, reducing coupling losses and considerably improving the overall performances. Reconfigurability and feedback will provide further necessary tools enabling to perform more challenging and interesting tasks.

Random walk is an extremely successful tool, employed in many scientific fields and their quantum analogues promise to be similarly powerful.

**References:**

[1]Y. Aharonov, L. Davidovich, N. Zagury, "Quantum Random Walks", Phys. Rev. A 48, 1687 (1993). doi:10.1103/PhysRevA.48.1687

[1]

**[2]**A. Politi, M. J. Cryan, J. G. Rarity, S. Yu, J. L. O'Brien, "Silica-on-silicon waveguide quantum circuits", Science 320, pp. 646-649 (2008). doi:10.1126/science.1155441

**[3]**J. C. F. Matthews, A. Politi, A. Stefanov, J. L. O'Brien, "Manipulation of multiphoton entanglement in waveguide quantum circuits", Nature Photonics, 3, pp. 346-350 (2009). doi:10.1038/nphoton.2009.93

**[4]**A. Laing, A. Peruzzo, A. Politi, M. Rodas Verde, M. Halder, T. C. Ralph, M. G. Thompson, J. L. O'Brien, "High-fidelity operation of quantum photonic circuits", Quant. Phys., e-prints, arXiv:1004.0326v2.

**[5]**A. Politi, J. C. F. Matthews, J. L. O'Brien, "Shor's quantum factoring algorithm on a photonic chip", Science 325, no. 5945, pp. 1221, 2009. doi:10.1126/science.1173731.

**[6]**A. Peruzzo, M. Lobino, J.C.F. Matthews, N. Matsuda, A. Politi, K. Poulios, X.-Q. Zhou, Y. Lahini, N. Ismail, K. Wörhoff, Y. Bromberg, Y. Silberberg, M. G. Thompson, J. L. O’Brien, "Quantum Walks of Correlated Photons", Science 329, pp. 1500-1503 (2010). doi:10.1126/science.1193515 .

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