Spontaneous parametric downconversion and quantum walk topology

Abstract

In a recent detailed research program we proposed to study the complex physics of topological phases by an all optical implementation of a discrete-time quantum walk. The main novel ingredient proposed for this study is the use of non-linear parametric amplifiers in the network which could in turn be used to emulate intra-atomic interactions and thus analyze many-body effects in topological phases even when using light as the quantum walker. In this paper, and as a first step towards the implementation of our scheme, we analize the interplay between quantum walk lattice topology and spatial correlations of bi-photons produced by spontaneous parametric down-conversion. We also describe different detection methods suitable for our proposed experimental scheme. 1.. INTRODUCTIONPhase transitions play a fundamental role in physics. From melting ice to the creation of mass in the Universe, phase transitions are at the center of most dynamical processes which involve an abrupt change in the properties of a system. Phase transitions are usually driven by some form of fluctuation. While classical phase transitions are typically driven by thermal noise, quantum phase transitions are triggered by quantum fluctuations. Quantum phase transitions have been extensively studied in a large number of fields ranging from cosmology to condensed matter and have received much attention in the field of ultra-cold atoms since the observation of Bose-Einstein condensation [1], and the subsequent experimental realization of Superfluid-Mott Insulator phase transition in optical lattices [2]. A common feature of quantum phase transitions is that they involve some form of spontaneous symmetry breaking, such that the ground state of the system has less overall symmetry than the Hamiltonian and can be described by a local order parameter.A rather distinctive class of quantum phases is present in systems characterized by a Hilbert space which is split into different topological sectors, the so called topological phases. Topological phases have received much attention after the discovery of the quantum Hall effect [3] and the interest increased following the prediction [4] and experimental realization [5] of a new class of material called topological insulators. Topological insulators are band insulators with particular symmetry properties arising from spin-orbit interactions which are predicted to exhibit surface edge states which should reflect the non-trivial topological properties of the band structure, and which should be topologically protected by time reversal symmetry. Unlike most familiar phases of matter which break different kinds of symmetries, topological phases are not characterized by a broken symmetry, they have degenerate ground states which present more symmetry than the underlying Hamiltonian, and can not be described by a local order parameter. Rather, these partially unexplored type of phases are described by topological invariants, such as the Chern number which is intimately related to the adiabatic Berry phase, and are ...

Topology in Photonic Discrete-Time Quantum Walks: A Comprehensive Review

“…We will calculate the Zak phase for two types of DTQW, the first one is the socalled split-step DTQW [19,73]. It consists of a DTQW with unitary step U given by the following expression U θ 1 , θ 2 ðÞ ¼ TR θ 1 ðÞ TR θ 2 ðÞ .…”

“…It consists of a DTQW with unitary step U given by the following expression U θ 1 , θ 2 ðÞ ¼ TR θ 1 ðÞ TR θ 2 ðÞ . Such unitary step can be readily implemented via spatial multiplexing, as described in [19,73]. For the unitary step characterizing the split-step DTQW, the components of the normal vector n i for decomposing the Hamiltonian in terms of Pauli operators can be written in the following manner:…”

Photonic Discrete-time Quantum Walks and Applications

“…We first consider the split-step quantum walk [ 3 , 62 ]. This corresponds to a quantum walk with unitary step give by , which can be readily implemented via spatial multiplexed DTQW as proposed in [ 3 , 62 ]. In this example, the normals are of the following form: …”

“…We first consider the split-step quantum walk [3,64]. This corresponds to a quantum walk with unitary step give by U (θ 1 , θ 2 ) = T R(θ 1 )T R(θ 2 ), which can be readily implemented via spatial multiplexed DTQW as proposed in [3,64].…”

“…We first consider the split-step quantum walk [3,64]. This corresponds to a quantum walk with unitary step give by U (θ 1 , θ 2 ) = T R(θ 1 )T R(θ 2 ), which can be readily implemented via spatial multiplexed DTQW as proposed in [3,64]. In this example, the normals n i are of the following form: non-trivial geometric Zak phase landscape for DTQW with non-commuting rotations, obtained by numeric integration [37].…”