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Journal of Physics A Mathematical and General

Properties of a discretized coherent state representation and the relation to Gabor analysis

L. Mauritz Andersson, Johan  berg, Hans Karlsson, Osvaldo Goscinski

Published Aug 27, 2002• 5 citations
Publisher

Abstract

Properties of a discretized coherent state representation (DCSR) and its connection to Gabor frame analysis are discussed. The DCSR approach was recently shown (Andersson L M 2001 J. Chem. Phys. 115 1158) to yield a practical computational scheme for quantum dynamics, and an iterative scheme for finding the identity operator was proposed. In the present work, we suggest a proof of fast convergence of the iterative scheme for computing the canonical dual to any given countable frame in a Hilbert space. The method of frames is concerned with the use of a non-orthogonal, over-complete set of functions for expansion of an arbitrary function. We also introduce the concept of 'representations of the identity operator' and show how to expand arbitrary vectors using the frame elements, without explicit diagonalization to an orthonormal basis. Numerical examples that illustrate the method are shown.

DOI
10.1088/0305-4470/35/36/307

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Properties of a discretized coherent state representation (DCSR) and its connection to Gabor frame analysis are discussed. The DCSR approach was recently shown (Andersson L M 2001 J. Chem. Phys. 115 1158) to yield a practical computational scheme for quantum dynamics, and an iterative scheme for finding the identity operator was proposed. In the present work, we suggest a proof of fast convergence of the iterative scheme for computing the canonical dual to any given countable frame in a Hilbert space. The method of frames is concerned with the use of a non-orthogonal, over-complete set of functions for expansion of an arbitrary function. We also introduce the concept of 'representations of the identity operator' and show how to expand arbitrary vectors using the frame elements, without explicit diagonalization to an orthonormal basis. Numerical examples that illustrate the method are shown.
We introduce a discretized coherent state representation (DCSR) for quantum dynamics. Expansion of a wave function in the nonorthogonal slightly overcomplete set is made with an identity operator computed using an iterative refinement method. Calculating the inverse of the overlap matrix is not necessary. The result is an accurate and efficient representation, where you only put basis functions in the region of phase space where the wave function is nonvanishing. Compared to traditional spatial grid methods, fewer grid points are needed. The DCSR can be viewed as an application of the Weyl–Heisenberg frame and extends it into a useful computational method. A scheme for fully quantum mechanical propagation is constructed and applied to the realistic problem of highly excited vibration in the heavy diatomic molecule Rb2. Compared to split-operator propagation in a conventional spatial grid, an order of magnitude longer time steps can be taken and fewer grid points are needed. The computational effort scales linearly with the number of basis functions. Nonreflecting boundary conditions are a natural property of the representation and is illustrated in a model of predissociation.
A Hilbert space is usually described using an orthonormal basis, but a more general description can be defined using a tight frame. In the case of continuous-variable Gaussian states, the coherence is usually described using the Hermite-Gauss basis. We present an alternative description using the tight frame of all canonical coherent states. The description of Gaussian states in terms of coherent states is simpler than in terms of Hermite-Gauss basis, and this allows us to obtain an explicit formula describing the frame-dependent coherence of a mixed Gaussian state as a function of the determinant and the trace of covariance matrix. For each mixed Gaussian state, we construct a sequence of discrete-variable quantum states rapidly convergent to it (as concern the purity and coherence) when the dimension of Hilbert space becomes large enough (15 or more). In the discrete case, the coherence is measured with respect to the tight frame of all discrete coherent states.
The resolution of the identity is typically utilized as a passive representation tool in standard quantum mechanics textbooks. In this paper, we reinterpret this traditional perspective by introducing the resolution of the identity associated with Glauber coherent states as a direct generator of exact complex Gaussian integral identities. By systematically projecting the continuous coherent-state completeness relation onto the discrete Fock basis, highly generalized Gaussian integral relations emerge naturally as a direct consequence of state preservation, rather than being introduced as independent mathematical postulates. We demonstrate that this overarching formal structure simultaneously dictates both discrete (Kronecker delta) and continuous (Dirac delta) localization kernels under appropriate basis projections, providing a sharp conceptual contrast between the structural features of overcomplete and strictly orthogonal representations. Crucially, we highlight an intriguing non-commuting behavior in the parameter limits of the master identity. Requiring only the elementary properties of coherent states and the Dirac formalism, this approach offers a transparent and visually intuitive illustration of how Hilbert-space completeness inherently encodes vast libraries of exact, solvable mathematical relations -- providing a unifying perspective suitable for advanced undergraduate and graduate instruction.
In a Hilbert space ℋ, discrete families of vectors {hj} with the property that f=∑j〈hj‖ f〉hj for every f in ℋ are considered. This expansion formula is obviously true if the family is an orthonormal basis of ℋ, but also can hold in situations where the hj are not mutually orthogonal and are ‘‘overcomplete.’’ The two classes of examples studied here are (i) appropriate sets of Weyl–Heisenberg coherent states, based on certain (non-Gaussian) fiducial vectors, and (ii) analogous families of affine coherent states. It is believed, that such ‘‘quasiorthogonal expansions’’ will be a useful tool in many areas of theoretical physics and applied mathematics.
The quantum observables used in the case of quantum systems with finite-dimensional Hilbert space are defined either algebraically in terms of an orthonormal basis and discrete Fourier transformation or by using a continuous system of coherent states. We present an alternative approach to these important quantum systems based on the finite frame quantization. Finite systems of coherent states, usually called finite tight frames, can be defined in a natural way in the case of finite quantum systems. Novel examples of such tight frames are presented. The quantum observables used in our approach are obtained by starting from certain classical observables described by functions defined on the discrete phase space corresponding to the system. They are obtained by using a finite frame and a Klauder–Berezin–Toeplitz-type quantization. Semi-classical aspects of tight frames are studied through lower symbols of basic classical observables.