arXiv subjects
Publications and source records attributed to Julio Guerrero.
We introduce Gilmore-Perelomov coherent states for non-unitary representations of non-compact groups, and discuss the main similarities and differences with respect to ordinary unitary Gilmore-Perelomov coherent states. The example of coherent states for the non-unitary finite dimensional representations of $SU(1,1)$ is considered and they are used to describe the propagation of light in coupled PT-symmetric optical devices.
Group theory (GT) provides a rigorous framework for studying symmetries in various disciplines in physics ranging from quantum field theories and the standard model to fluid mechanics and chaos theory. To date, the application of such a powerful tool in optical physics remains limited. Over the past few years however, several quantum-inspired symmetry principles (such as parity-time invariance and supersymmetry) have been introduced in optics and photonics for the first time. Despite the intense activities in these new research directions, only few works utilized the power of group theory. Motivated by this status quo, here we present a brief overview of the application of GT in optics, deliberately choosing examples that illustrate the power of this tool in both continuous and discrete setups. We hope that this review will stimulate further research that exploits the full potential of GT for investigating various symmetry paradigms in optics, eventually leading to new photonic devices.
In this paper we achieve the quantization of a particle moving on the $SU(2)$ group manifold, that is, the three-dimensional sphere $S^{3}$, by using group-theoretical methods. For this purpose, a fundamental role is played by contact, non-point symmetries, i.e., symmetries that leave the Poincaré-Cartan form semi-invariant at the classical level, although not necessarily the Lagrangian. Special attention is paid to the role played by the basic quantum commutators, which depart from the canonical, Heisenberg-Weyl ones, as well as the relationship between the integration measure in the Hilbert space of the system and the non-trivial topology of the configuration space. Also, the quantization on momentum space is briefly outlined.
In this paper we exploit the use of symmetries of a physical system so as to characterize algebraically the corresponding solution manifold by means of Noether invariants. This constitutes a necessary preliminary step towards the correct quantization in non-linear cases, where the success of Canonical Quantization is not guaranteed in general. To achieve this task "point symmetries" of the Lagrangian are generally not enough, and the notion of contact transformations is in order: the solution manifold can not be in general parametrized by means of Noether invariants associated with basic point symmetries. The use of the contact structure given by the Poincaré-Cartan form permits the definition of the symplectic form on the solution manifold, through some sort of Hamilton-Jacobi transformation. It also provides the required basic symmetries, realized as Hamiltonian vector fields associated with global functions on the solution manifold (thus constituting an inverse of the Noether Theorem), lifted back to the evolution space through the inverse of this Hamilton-Jacobi mapping. In this framework, solutions and symmetries, as a whole, are somehow identified and this correspondence is also kept at a perturbative level. We present non-trivial examples of this interplay between symmetries and solutions pointing out the usefulness of this mechanism in approaching the corresponding quantization. In particular, we achieve the proper generalization of the Heisenberg-Weyl algebra for the non-linear particle sigma model in $S^3$ within this framework, and notice that a subset of the classical symmetries corresponding with this quantizing algebra (those generalizing boosts) are necessarily of non-point character.
We address the problem of efficient sparse fixed-rank (S-FR) matrix decomposition, i.e., splitting a corrupted matrix $M$ into an uncorrupted matrix $L$ of rank $r$ and a sparse matrix of outliers $S$. Fixed-rank constraints are usually imposed by the physical restrictions of the system under study. Here we propose a method to perform accurate and very efficient S-FR decomposition that is more suitable for large-scale problems than existing approaches. Our method is a grateful combination of geometrical and algebraical techniques, which avoids the bottleneck caused by the Truncated SVD (TSVD). Instead, a polar factorization is used to exploit the manifold structure of fixed-rank problems as the product of two Stiefel and an SPD manifold, leading to a better convergence and stability. Then, closed-form projectors help to speed up each iteration of the method. We introduce a novel and fast projector for the $\text{SPD}$ manifold and a proof of its validity. Further acceleration is achieved using a Nystrom scheme. Extensive experiments with synthetic and real data in the context of robust photometric stereo and spectral clustering show that our proposals outperform the state of the art.
We revise the Lewis-Riesenfeld invariant method for solving the quantum time-dependent harmonic oscillator in light of the Quantum Arnold Transformation previously introduced and its recent generalization to the Quantum Arnold-Ermakov-Pinney Transformation. We prove that both methods are equivalent and show the advantages of the Quantum Arnold-Ermakov-Pinney transformation over the Lewis-Riesenfeld invariant method. We show that, in the quantum time-dependent and damped harmonic oscillator, the invariant proposed by Dodonov & Man'ko is more suitable and provide some examples to illustrate it, focusing on the damped case.
In this work, we address the problem of outlier detection for robust motion estimation by using modern sparse-low-rank decompositions, i.e., Robust PCA-like methods, to impose global rank constraints. Robust decompositions have shown to be good at splitting a corrupted matrix into an uncorrupted low-rank matrix and a sparse matrix, containing outliers. However, this process only works when matrices have relatively low rank with respect to their ambient space, a property not met in motion estimation problems. As a solution, we propose to exploit the partial information present in the decomposition to decide which matches are outliers. We provide evidences showing that even when it is not possible to recover an uncorrupted low-rank matrix, the resulting information can be exploited for outlier detection. To this end we propose the Robust Decomposition with Constrained Rank (RD-CR), a proximal gradient based method that enforces the rank constraints inherent to motion estimation. We also present a general framework to perform robust estimation for stereo Visual Odometry, based on our RD-CR and a simple but effective compressed optimization method that achieves high performance. Our evaluation on synthetic data and on the KITTI dataset demonstrates the applicability of our approach in complex scenarios and it yields state-of-the-art performance.
There are many approaches for the description of dissipative systems coupled to some kind of environment. This environment can be described in different ways; only effective models will be considered here. In the Bateman model, the environment is represented by one additional degree of freedom and the corresponding momentum. In two other canonical approaches, no environmental degree of freedom appears explicitly but the canonical variables are connected with the physical ones via non-canonical transformations. The link between the Bateman approach and those without additional variables is achieved via comparison with a canonical approach using expanding coordinates since, in this case, both Hamiltonians are constants of motion. This leads to constraints that allow for the elimination of the additional degree of freedom in the Bateman approach. These constraints are not unique. Several choices are studied explicitly and the consequences for the physical interpretation of the additional variable in the Bateman model are discussed.
We construct a Continuous Wavelet Transform (CWT) on the torus $\mathbb T^2$ following a group-theoretical approach based on the conformal group $SO(2,2)$. The Euclidean limit reproduces wavelets on the plane $\mathbb R^2$ with two dilations, which can be defined through the natural tensor product representation of usual wavelets on $\mathbb R$. Restricting ourselves to a single dilation imposes severe conditions for the mother wavelet that can be overcome by adding extra modular group $SL(2,\mathbb Z)$ transformations, thus leading to the concept of \emph{modular wavelets}. We define modular-admissible functions and prove frame conditions.
The previously introduced Quantum Arnold Transformation, a unitary operator mapping the solutions of the Schrödinger equation for time dependent quadratic Hamiltonians into the solutions for the free particle, is revised and some interesting extensions are introduced, providing in particular a generalization of the Ermakov-Pinney equation.
We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number $N$ of phase samples $\{θ_k=2πk/N\}_{k=0}^{N-1}$ for a given mean number $p$ of particles. The resulting Coherent State (CS) subsystem ${\cal S}=\{|z_k=p^{1/2}e^{iθ_k}>\}$ is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, ${\cal S}$ is "almost complete" (a \textit{pseudo-frame}) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when $p<N$ and/or $N\to\infty$.
Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of $N$ samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over- and under-sampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to $J$, is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform.
Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk $\mathbb D_1$), seen as a homogeneous space of the pseudo-unitary group SU(1,1). We provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from $N$ samples properly chosen. We also study the case of undersampling of band-unlimited functions and the conditions under which a partial reconstruction from $N$ samples is still possible and the accuracy of the approximation, which tends to be exact in the limit $N\to\infty$.
We put forward the concatenation of Quantum Arnold Transformations as a tool to obtain the wave function of a particle subjected to a harmonic potential which is switched on and off successively. This simulates the capture and release process of an ion in a trap and provides a mathematical picture of this physical process.
Different families of states, which are solutions of the time-dependent free Schrödinger equation, are imported from the harmonic oscillator using the Quantum Arnold Transformation introduced in a previous paper. Among them, infinite series of states are given that are normalizable, expand the whole space of solutions, are spatially multi-localized and are eigenstates of a suitably defined number operator. Associated with these states new sets of coherent and squeezed states for the free particle are defined representing traveling, squeezed, multi-localized wave packets. These states are also constructed in higher dimensions, leading to the quantum mechanical version of the Hermite-Gauss and Laguerre-Gauss states of paraxial wave optics. Some applications of these new families of states and procedures to experimentally realize and manipulate them are outlined.
By a quantum version of the Arnold transformation of classical mechanics, all quantum dynamical systems whose classical equations of motion are non-homogeneous linear second-order ordinary differential equations, including systems with friction linear in velocity, can be related to the quantum free-particle dynamical system. This transformation provides a basic (Heisenberg-Weyl) algebra of quantum operators, along with well-defined Hermitian operators which can be chosen as evolution-like observables and complete the entire Schrödinger algebra. It also proves to be very helpful in performing certain computations quickly, to obtain, for example, wave functions and closed analytic expressions for time-evolution operators.
We present a unified group-theoretical derivation of the Continuous Wavelet Transform (CWT) on the circle $\mathbb S^1$ and the real line $\mathbb{R}$, following the general formalism of Coherent States (CS) associated to unitary square integrable (modulo a subgroup, possibly) representations of the group $SL(2,\mathbb{R})$. A general procedure for obtaining unitary representations of a group $G$ of affine transformations on a space of signals $L^2(X,dx)$ is described, relating carrier spaces $X$ to (first or higher-order) ``polarization subalgebras'' ${\cal P}_X$. We also provide explicit admissibility and continuous frame conditions for wavelets on $\mathbb S^1$ and discuss the Euclidean limit in terms of group contraction.
The linearized Kepler problem is considered, as obtained from the Kustaanheimo-Stiefel (K-S)transformation, both for negative and positive energies. The symmetry group for the Kepler problem turns out to be SU(2,2). For negative energies, the Hamiltonian of Kepler problem can be realized as the sum of the energies of four harmonic oscillator with the same frequency, with a certain constrain. For positive energies, it can be realized as the sum of the energies of four repulsive oscillator with the same (imaginary) frequency, with the same constrain. The quantization for the two cases, negative and positive energies is considered, using group theoretical techniques and constrains. The case of zero energy is also discussed.