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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Phase-space tomography of the truncated harmonic oscillator states

Marta C. González, Celso L. Ladera

Published Jul 19, 1999• 0 citations
Publisher

Abstract

Phase-Space Tomography has been recently applied, in complete analogy to classical optics tomography, for the reconstruction of quantum states. The quantum systems thus far considered move in potentials for which only the coordinates of phase space are modified by simple affine transformations--shearing and rotation--, the state Wigner quasi-probability distribution remaining invariant. In this work we consider the case of the truncated harmonic oscillator, a more general, and potentially useful, potential, and show that state reconstruction is still feasible.

DOI
10.1117/12.354780

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Phase-Space Tomography has been recently applied, in complete analogy to classical optics tomography, for the reconstruction of quantum states. The quantum systems thus far considered move in potentials for which only the coordinates of phase space are modified by simple affine transformations--shearing and rotation--, the state Wigner quasi-probability distribution remaining invariant. In this work we consider the case of the truncated harmonic oscillator, a more general, and potentially useful, potential, and show that state reconstruction is still feasible.
In a previous work the concept of quantum potential is generalized into extended phase space (EPS) for a particle in linear and harmonic potentials.It was shown there that in contrast to the Schrödinger quantum mechanics by an appropriate extended canonical transformation one can obtain the Wigner representation of phase space quantum mechanics in which the quantum potential is removed from dynamical equation.In other words, one still has the form invariance of the ordinary Hamilton-Jacobi equation in this representation.The situation, mathematically, is similar to the disappearance of the centrifugal potential in going from the spherical to the Cartesian coordinates.Here we show that the Husimi representation is another possible representation where the quantum potential for the harmonic potential disappears and the modified Hamilton-Jacobi equation reduces to the familiar classical form.This happens when the parameter in the Husimi transformation assumes a specific value corresponding to Q-function.
The quasiprobability distribution of the discrete Wigner function provides a complete description of a quantum state and is, therefore, a useful alternative to the usual density matrix description. Moreover, the experimental quantum state tomography in discrete Wigner phase space can also be implemented. We observe that for a certain class of states, such as harmonic states, the Wigner matrix is far more sparse compared to the density matrix in the computational basis. Additionally, reading only a small part of the Wigner matrix may suffice to infer certain behavior of quantum dynamics. In such cases, selective Wigner phase space tomography (SWPST) can be more efficient than the usual density matrix tomography (DMT). Employing nuclear magnetic resonance methods on a three-qubit nuclear spin register, we experimentally estimate Wigner matrices of various two-qubit quantum states. As a specific example application of SWPST, we study the evolution of spin coherent states under the quantum chaotic kicked top model and extract signatures of quantum-classical correspondence in the Wigner phase space.
Most quantum tomographic methods for massive particles using destructive measurement can only be used for motion in one dimension. We show how to infer the quantum state of a nonrelativistic, NNN-dimensional harmonic oscillator system by simple inverse Radon transforms. We assume a time-independent Hamiltonian for which the dynamics along each coordinate always occurs simultaneously with the others and measurements on all coordinates are performed simultaneously. This is unlike light systems where different modes can be delayed and phase shifted with respect to each other. A requirement of the procedure is that the angular frequencies of the NNN harmonic potentials be incommensurable. We discuss the information available if the requirement of incommensurability is not fulfilled and also under what conditions the state can be reconstructed from finite-time measurements. As further examples, we consider free particles in NNN dimensions with periodic boundary conditions and particles in an NNN-dimensional box, where we find a similar condition of incommensurability and finite recurrence time.
We review the problem of state reconstruction in classical and in quantum physics, which is rarely considered at the textbook level. We review a method for retrieving a classical state in phase space, similar to that used in medical imaging known as Computer Aided Tomography. We explain how this method can be taken over to quantum mechanics, where it leads to a description of the quantum state in terms of the Wigner function which, although may take on negative values, plays the role of the probability density in phase space in classical physics. We explain another approach to quantum state reconstruction based on the notion of Mutually Unbiased Bases, and indicate the relation between these two approaches. Both are for a continuous, infinite-dimensional Hilbert space. We then study the finite-dimensional case and show how the second method, based on Mutually Unbiased Bases, can be used for state reconstruction.
We show how to reconstruct the quantum state of completely unpolarized light of type I [J. Lehner, U. Leonhardt, and H. Paul, Phys. Rev. A 53, 2727 (1996)] via optical homodyne tomography. Our analysis can be extended to any two-mode situation with complete two-mode randomization and, moreover, to any two-dimensional harmonic oscillator with SU(2) symmetric states. Interestingly, we have shown that the Laplace operator applied to the quadrature distribution of unpolarized photons is orthogonal on the set of these distributions.