Discrete Coherent States Research Articles

Discrete coherent states for higher Landau levels

We consider the quantum dynamics of a charged particle evolving under the action of a constant homogeneous magnetic field, with emphasis on the discrete subgroups of the Heisenberg group (in the Euclidean case) and of the SL(2,R) group (in the Hyperbolic case). We investigate completeness properties of discrete coherent states associated with higher order Euclidean and hyperbolic Landau levels, partially extending classic results of Perelomov and of Bargmann, Butera, Girardello and Klauder. In the Euclidean case, our results follow from identifying the completeness problem with known results from the theory of Gabor frames. The results for the hyperbolic setting follow by using a combination of methods from coherent states, time-scale analysis and the theory of Fuchsian groups and their associated automorphic forms.

Symmetric discrete coherent states for n-qubits

We put forward a method of constructing discrete coherent states for n qubits. After establishing appropriate displacement operators, the coherent states appear as displaced versions of a fiducial vector that is fixed by imposing a number of natural symmetry requirements on its Q-function. Using these coherent states, we establish a partial order in the discrete phase space, which allows us to picture some n-qubit states as apparent distributions. We also analyze correlations in terms of sums of squared Q-functions.This article is part of a special issue of Journal of Physics A: Mathematical and Theoretical devoted to ‘Coherent states: mathematical and physical aspects’.

Isotropic and squeezed fluctuations in an n-qubit system

We show that 2n discrete coherent states of an n-qubit system, generated by application of the discrete displacement operators to a symmetric fiducial state, have isotropic fluctuations, with n/2 ⩽ 〈ΔS2〉 ⩽ n, in a specific tangent plane, which in general is not orthogonal to the mean spin direction. This allows us to use them as reference states to define a discrete squeezing for non-symmetric n-qubit states. Examples of states with reduced fluctuations, obtained after application of XOR gates to correlate (partially entangle) qubits, are analyzed.

Quasidistributions and coherent states for finite-dimensional quantum systems

In recent years, an approach to discrete quantum phase spaces which comprehends all the main quasiprobability distributions known has been developed. It is the research that started with the pioneering work of Galetti and Piza, where the idea of operator bases constructed of discrete Fourier transforms of unitary displacement operators was first introduced. Subsequently, the discrete coherent states were introduced, and finally, the s-parametrized distributions, that include the Wigner, Husimi, and Glauber–Sudarshan distribution functions as particular cases. In the present work, we adapt its formulation to encompass some additional discrete symmetries, achieving an elegant yet physically sound formalism.

DISCRETE COHERENT STATES FOR n QUBITS

Discrete coherent states for a system of n qubits are introduced in terms of eigenstates of the finite Fourier transform. The properties of these states are pictured in phase space by resorting to the discrete Wigner function.

Geometry and dynamics in the fractional discrete Fourier transform

The N x N Fourier matrix is one distinguished element within the group U(N) of all N x N unitary matrices. It has the geometric property of being a fourth root of unity and is close to the dynamics of harmonic oscillators. The dynamical correspondence is exact only in the N-->infinity contraction limit for the integral Fourier transform and its fractional powers. In the finite-N case, several options have been considered in the literature. We compare their fidelity in reproducing the classical harmonic motion of discrete coherent states.

Holomorphic representation of coherent states of a singular oscillator and their Darboux representation

We consider the coherent states of a singular oscillator; these states are defined to be the characteristic states of an operator that reduces the number of basis functions for a discrete spectrum to one. We use the Darboux transform to study the coherent states of a transformed Hamiltonian. We obtain an expression for measures that can be used to decompose unity. We construct a holomorphic representation for the state vectors in the space of functions holomorphic everywhere in the complex plane, including vectors for discrete spectra and coherent states. We obtain a holomorphic representation of the Darboux transformation operators.

Discrete Coherent States and Probability Distributions in Finite-Dimensional Spaces

Operator bases are discussed in connection with the construction of phase space representatives of operators in finite-dimensional spaces, and their properties are presented. It is also shown how these operator bases allow for the construction of a finite harmonic oscillator-like coherent state. Creation and annihilation operators for the Fock finite-dimensional space are discussed and their expressions in terms of the operator bases are explicitly written. The relevant finite-dimensional probability distributions are obtained and their limiting behavior for an infinite-dimensional space are calculated which agree with the well known results

Wigner functions and quantum interference in optics

A new, approximate generalized Wigner function based on discrete coherent state superpositions is introduced. It is shown that in contrast to the exact generalized Wigner function that may not exist in some region of its parameter the proposed approximate function exists everywhere.

Monte Carlo evaluation of real time coherent state path integrals

An alternative formulation of the discrete real time coherent state path integral is examined. Following the development of a previous work [T. L. Marchioro II, J. Math Phys. 31, 2935 (1990)] the discrete coherent state action is transformed to second order form. The resulting path integral expression contains an integrand which factors into a Gaussian probability density and an oscillatory term, making the integral directly amenable to Monte Carlo techniques. Several simple examples in one and two spatial dimensions are presented and the possibilities of extending such an approach to general multidimensional systems are discussed.

Corrections to the coherent state path integral: Comments upon a speculation of L. S. Schulman

Intuitively, the Feynman path integral corresponds to a weighted sum over classical paths, an interpretation that fails for the phase space path integral. To address the question of whether there exists a path integral expression conforming to a sum over paths in phase space, an examination of the discrete coherent state path integral (CSPI) is undertaken. Via an alternative formulation of the CSPI, it is shown that the coherent state action for a broad class Hamiltonians can be transformed from the variables (q,p) to (q,q̇). For these Hamiltonians, such a transformation along with the inclusion of all terms O(ε(zi−zi−1)) yields an expression which, for finite ε, can be interpreted as a sum over classical paths with Gaussian weight. The numerical evaluation of this expression through importance sampling (Monte Carlo) is demonstrated.

Coherent states and lattice sums

The expansion of harmonic oscillator states in discrete coherent states on a von Neumann lattice leads to relationships between lattice sums and expansion coefficients of the Weierstrass σ function. It is shown that these relationships can be generalized to arbitrary lattices. Some interesting identities are obtained between infinite sums of different convergence rates.