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The generalized Talbot effect and its potential for processing quantum information

Samuel Mauricio Hernández Caballero

Published Apr 8, 2026• 0 citations
OpenAlex

Abstract

This thesis develops a phase-space generalization of the Talbot effect using the modular-variable formalism of quantum mechanics. Starting from the standard Fresnel-diffraction description of self-imaging behind periodic gratings, the work reframes the effect as the action of symplectic transformations on a discrete phase-space lattice, visualized through Wigner functions. The central result is that free propagation, responsible for the ordinary integer Talbot effect, is just one instance of a much larger symmetry: every element of the group SL(2,Z) maps modular eigenstates onto modular eigenstates, generating its own Talbot-like recurrence. Closed-form expressions for the associated quantum phases, written in terms of quadratic Gauss sums, are derived and shown to reduce to the classical Fresnel results in the appropriate limit. The extension to rational transformations, SL(2,Q), reproduces the fractional Talbot effect and yields consistent analytical formulas for rational shears, though a discrete path-integral construction reveals a structural obstruction that remains unresolved and is proposed as a direction for future work. The thesis also explores a first, preliminary application to quantum information processing: an encoding of qubits and qutrits in modular eigenstates based on torsion points of the modular torus, together with generalized Pauli gates and a phase-readout scheme based on SL(2,Z) interference. This application is left incomplete, since the displacement-operator formalism used cannot by itself generate the superposition required for universal quantum computation.

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Summary form only given.Recent developments in atom optics have led to different approaches to lithographic techniques with neutral atoms. One of these is the use of diffraction from free-standing masks, which is sometimes called atom holography. There, narrow structures arise due to matter-wave interference in the Fresnel regime. In the presented experiment, we demonstrate that the wave nature of atoms can be used to produce periodic structures with small periodicities. We used the Talbot effect in the Fresnel diffraction regime behind a microfabricated transmission grating. The Talbot effect is the self-imaging of a grating of periodicity d illuminated with plane coherent waves.

Conference on Lasers and Electro-Optics•Nov 27, 2002•0 citations
This thesis develops a phase-space generalization of the Talbot effect using the modular-variable formalism of quantum mechanics. Starting from the standard Fresnel-diffraction description of self-imaging behind periodic gratings, the work reframes the effect as the action of symplectic transformations on a discrete phase-space lattice, visualized through Wigner functions. The central result is that free propagation, responsible for the ordinary integer Talbot effect, is just one instance of a much larger symmetry: every element of the group SL(2,Z) maps modular eigenstates onto modular eigenstates, generating its own Talbot-like recurrence. Closed-form expressions for the associated quantum phases, written in terms of quadratic Gauss sums, are derived and shown to reduce to the classical Fresnel results in the appropriate limit. The extension to rational transformations, SL(2,Q), reproduces the fractional Talbot effect and yields consistent analytical formulas for rational shears, though a discrete path-integral construction reveals a structural obstruction that remains unresolved and is proposed as a direction for future work. The thesis also explores a first, preliminary application to quantum information processing: an encoding of qubits and qutrits in modular eigenstates based on torsion points of the modular torus, together with generalized Pauli gates and a phase-readout scheme based on SL(2,Z) interference. This application is left incomplete, since the displacement-operator formalism used cannot by itself generate the superposition required for universal quantum computation.
Quantum information can be processed in two fundamentally different ways, using either discrete or continuous variable implementations. Each one of them provides different practical advantages and drawbacks. In this thesis we study theoretical means allowing to implement quantum information protocols originally formulated for discrete quantum systems in physical objects characterized by continuous variables. At the heart of our considerations is the use of modular variables as helpful technique to reveal discrete structure in continuous-variable states, operations and observables. The present work is strongly guided by the experimental applicability of our ideas in quantum optics experiments, with a particular focus on the transverse degrees of freedom of single photons.One of the main themes of this thesis is the formulation of a framework for quantum information processing in phase-space based on the use of modular variables. The term modular variables refers to a specific class of observables that are periodic with respect to some pair of conjugate variables. In our framework we use these periodic observables in order to encode discrete quantum information in Hilbert spaces of infinite dimension. In particular, we consider protocols that involve measurements of judiciously chosen logical observables enabling the readout of the encoded discrete quantum information from the corresponding logical states. Using this framework we show how to perform tests of fundamental properties of quantum mechanics, such as entanglement, Bell nonlocality and contextuality, in Hilbert spaces of various dimensions. Particularly, we generalize known tests of each of these properties, that were originally formulated for measurements with discrete outcomes, to more general measurement contexts comprising the case of bounded continuous outcomes. Concerning experimental implementations of the presented theoretical elaborations we discuss the transverse degrees of freedom of single photons as a natural platform to manipulate and measure modular variables. In particular, we demonstrate how to process discrete quantum information encoded in the spatial distribution of single photons via the optical Talbot effect - a near-field interference effect. Finally, we show how to produce d-dimensional entangled Talbot photon pairs without post-selection, using spontaneous parametric down-conversion and linear optical elements only. As last topic we explore the nonlocal properties of a specific class of hybrid entanglement between particle-like and wave-like optical qubits. Using a hybrid measurement scheme of Pauli and displaced parity measurements we show that, even after including realistic experimental losses, a violation of local-realism is theoretically possible.
The Fresnel diffraction of periodic objects at rational fractions of the Talbot distance is described in terms of the Wigner distribution function (WDF). The analysis provides a heuristic model for understanding the formation of the diffraction patterns as well as for evaluating the complex amplitude at any fractional Talbot plane. Furthermore, certain symmetry properties of the Fresnel-diffracted wave field can be derived directly from the WDF. Additionally, a discussion is given on how periodic signals and information about the phase are encoded in the WDF.
Single-photon interference due to passage through a periodic grating is considered in a novel proposal for processing DDD-dimensional quantum systems (quDits) encoded in the spatial degrees of freedom of light. We show that free-space propagation naturally implements basic single-quDit gates by means of the Talbot effect: an intricate time-space carpet of light in the near-field diffraction regime. By adding a diagonal phase gate, we show that a complete set of single-quDit gates can be implemented. We then introduce a spatially dependent beam splitter that allows for projective measurements in the computational basis and can be used for the implementation of controlled operations between two quDits. Universal quantum information processing can then be implemented with linear optics and ancilla photons via postselection and feed-forward following the original proposal of Knill-Laflamme and Milburn. Although we consider photons, our scheme should be directly applicable to a number of other physical systems. Interpretation of the Talbot effect as a quantum logic operation provides a beautiful and interesting way to visualize quantum computation through wave propagation and interference.
We develop a path integral representation for the dynamics of quantum systems with a finite-dimensional Hilbert space, formulated entirely within a discrete phase space. Starting from the discrete Wigner function on Zd×Zd{\mathbb{Z}_d} \times {\mathbb{Z}_d}Zd​×Zd​ (ddd an odd prime) and the associated Weyl transform built from generalized displacement operators, we derive an exact kernel that propagates the discrete Wigner function in time and, by iterating its composition law through a short-time approximation, obtain a sum-over-paths expression weighted by a discrete phase-space action -- the natural finite-dimensional counterpart of Marinov's functional. For Hamiltonians linear in the phase-space coordinates and at times strictly commensurate with the lattice, the fluctuation sum collapses to a deterministic shift, realizing the discrete analog of classical Hamiltonian flow. Applying the formalism to one and to two interacting qutrits (d=3d=3d=3), we show that the full entanglement dynamics -- captured by a closed-form linear entropy valid for all times -- requires the coherent contribution of all fluctuation sectors; the boundary-term (mean-field) sector alone fails to reproduce it. For a non-stabilizer Hamiltonian, where the short-time kernel is only approximate, the time-sliced path integral converges to the exact dynamics, including the dynamical generation of Wigner negativity. We discuss implications for the semiclassical simulation of many-body spin systems and for the characterization of non-classicality through Wigner negativity.
Summary form only given.Recent developments in atom optics have led to different approaches to lithographic techniques with neutral atoms. One of these is the use of diffraction from free-standing masks, which is sometimes called atom holography. There, narrow structures arise due to matter-wave interference in the Fresnel regime. In the presented experiment, we demonstrate that the wave nature of atoms can be used to produce periodic structures with small periodicities. We used the Talbot effect in the Fresnel diffraction regime behind a microfabricated transmission grating. The Talbot effect is the self-imaging of a grating of periodicity d illuminated with plane coherent waves.