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arXiv:1108.3004 · Physical Review A

Exact dynamics of finite Glauber-Fock photonic lattices

B. M. Rodríguez-Lara

Published Nov 23, 2011• 31 citations
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Abstract

The dynamics of Glauber-Fock lattice of size $N$ is given through exact diagonalization of the corresponding Hamiltonian; the spectra ${{\ensuremath{\lambda}}_{k}}$ is given as the roots of the $N$th Hermite polynomial, ${H}_{N}({\ensuremath{\lambda}}_{k}/\sqrt{2})=0$, and the eigenstates are given in terms of Hermite polynomials evaluated at these roots. The exact dynamics is used to study coherent phenomena in discrete lattices. Due to the symmetry and spacing of the eigenvalues ${{\ensuremath{\lambda}}_{k}}$, oscillatory behavior is predicted with highly localized spectra, that is, near complete revivals of the photon number and partial recovery of the initial state at given waveguides.

DOI
10.1103/physreva.84.053845

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The dynamics of Glauber-Fock lattice of size NNN is given through exact diagonalization of the corresponding Hamiltonian; the spectra \ensuremathλk{{\ensuremath{\lambda}}_{k}}\ensuremathλk​ is given as the roots of the NNNth Hermite polynomial, HN(\ensuremathλk/2)=0{H}_{N}({\ensuremath{\lambda}}_{k}/\sqrt{2})=0HN​(\ensuremathλk​/2​)=0, and the eigenstates are given in terms of Hermite polynomials evaluated at these roots. The exact dynamics is used to study coherent phenomena in discrete lattices. Due to the symmetry and spacing of the eigenvalues \ensuremathλk{{\ensuremath{\lambda}}_{k}}\ensuremathλk​, oscillatory behavior is predicted with highly localized spectra, that is, near complete revivals of the photon number and partial recovery of the initial state at given waveguides.
Abstract The main objective of this work is to investigate, analyze and examine in detail the behavior of quantum light in a photonic lattice having a square root law distribution for the coupling constant known as a Glauber-Fock photonic lattice. We investigate the light dynamics for both small and large size lattices for the case of spatially extended path entangled input state. We also study the squeezed state inputs, where we focus on finite size lattices. We analyze in detail the behavior of the non-classical features such as squeezing and photon correlations at the output of the system. The results and findings show the transport and revival of these features as non-classical light propagates across the waveguide array. Our findings should have applications in the areas of photonic quantum information processing and simulation and are implementable with current technology.
Coherent states and their generalizations, displaced Fock states, are of fundamental importance to quantum optics. Here we present a direct observation of a classical analogue for the emergence of these states from the eigenstates of the harmonic oscillator. To this end, the light propagation in a Glauber-Fock waveguide lattice serves as equivalent for the displacement of Fock states in phase space. Theoretical calculations and analogue classical experiments show that the square-root distribution of the coupling parameter in such lattices supports a new family of intriguing quantum correlations not encountered in uniform arrays. Because of the broken shift invariance of the lattice, these correlations strongly depend on the transverse position. Consequently, quantum random walks with this extra degree of freedom may be realized in Glauber-Fock lattices.
The fundamental position and momentum operators of quantum mechanics are unbounded, but finite rank compressions of the operators can be considered to obtain partial information on the operators and their properties. Motivated by problems in photonic quantum computing, we bring together results from quantum theory and the theory of orthogonal polynomials to show that a natural finite rank compression of the position and momentum operator representation on Fock space Hilbert space has eigenvalues given by roots of the classical Hermite polynomials. We discuss the corresponding compressed displacement operators and potential applications in approximate quantum error correction.
Here, we present the experimental analysis of light propagation in such Glauber Fock photonic lattices. The individual guides have been inscribed into fused silica employing femtosecond laser pulses and the desired coupling between the waveguides has been obtained by varying the inter-waveguide spacing accordingly. Figs. 1(a,c) show the calculated intensity evolution and experimental fluorescence images for an excitation of the first two guides, corresponding to a displacement of the Fock-states |0〉 and |1〉.
The localization of light in flat-band lattices has been recently proposed and experimentally demonstrated in several configurations, assuming a classical description of light. Here we study the problem of light localization in the quantum regime. We focus on quasi-one-dimensional and two-dimensional lattices which exhibit a perfect flat band inside their linear spectrum. Localized quantum states are constructed as eigenstates of the interaction Hamiltonian with a vanishing eigenvalue and a well defined total photon number. These are superpositions of Fock states with probability amplitudes given by positive as well as negative square roots of multinomial coefficients. The classical picture can be recovered by considering Poissonian superpositions of localized quantum states with different total photon number. We also study the separability properties of flat-band quantum states and apply them to the transmission of information via multicore fibers, where these states allow for the total passive suppression of photon crosstalk and exhibit robustness against photon losses. At the end, we propose an on-chip setup for the experimental preparation of localized quantum states of light for any number of photons.