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arXiv:1108.0803 · Physical Review Letters

Classical Analogue of Displaced Fock States and Quantum Correlations in Glauber-Fock Photonic Lattices

Robert Keil, Armando Pérez-Leija, Felix Dreisow, Matthias Heinrich, H. M. Moya-Cessa, Stefan Nolte, Demetrios N. Christodoulides, Alexander Szameit

Published Aug 31, 2011• 95 citations
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Abstract

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.

DOI
10.1103/physrevlett.107.103601

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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.
We demonstrate that superpositions of coherent states and displaced Fock states, also referred to as generalized Schr\"odinger cat states, can be generated by application of a deformed Glauber displacement operator on number states, coherent states, and displaced Fock states. Based on such a deformed displacement operator we introduce a deformed version of the so-called Glauber photonic lattices. These novel lattices are endowed with alternating positive and negative coupling coefficients and give rise to classical analogs of Schr\"odinger cat states. Finally, we show that the analytic propagator of these new Glauber-Fock arrays explicitly contains the Wigner operator opening the possibility to emulate Wigner functions of the quantum harmonic oscillator in the classical domain.
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〉.
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.
Random processes are ubiquitous in the natural world as well as in man-made environments. While classical random walks may behave highly complex or even chaotic, there outcome can, in principle, always be predicted from the parameters of the system and the initial conditions. In the realm of quantum mechanics, however, this is not possible as the underlying wave mechanics leads to intrinsically indeterministic outcomes. Moreover, if multiple indistinguishable particles are subjected to such a quantum random walk, their exchange symmetry causes quantum interference, thereby enriching the dynamics of the system even further. \nIn this work, quantum random walks of pairs of indistinguishable photons, the quanta of light, are investigated. Networks of coupled optical waveguides are chosen as the experimental platform of choice, offering a high degree of coherence and versatility. In these photonic lattices, light propagates along one spatial dimension, whereas the individual waveguides are connected by evanescent coupling in the transverse dimensions. \nIn particular, it is investigated how the various degrees of freedom, which are available in such photonic lattices, affect the trajectories in the quantum walks and their complexity. It is shown how the facilitation of both transverse dimensions allows for much richer quantum walks with properties unencountered in planar arrangements. But even if just a single transverse dimension is available, the coupling properties of the lattice along this dimension are a potent degree of freedom, which can be used to manipulate the quantum walk. Finally, an experimental technique is developed which enables a convenient characterisation of the expected quantum walk in an arbitrary waveguide lattice by classical light. \nThe thesis concludes with a summary of the results and an outlook onto further developments in the field.
The dynamics of Glauber-Fock lattice of size NNN is given through exact diagonalization of the corresponding Hamiltonian; the spectra ${{\ensuremath{\lambda}}_{k}}$ is given as the roots of the NNNth 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.