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arXiv:quant-ph/0410129 · Journal of Physics A Mathematical and General

Pure state correlations: chords in phase space

Alfredo M. Ozorio de Almeida, Raúl O. Vallejos, Marcos Saraceno

Published Feb 3, 2005• 17 citations
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Abstract

The intensity of the overlap of a quantum state with all its phase space translations defines its quantum correlations. In the case of pure states, these are invariant with respect to Fourier transformation. The overlaps themselves are here studied in terms of the Wigner function and its Fourier transform, i.e., the characteristic function or chord function. Unlike the Wigner function, the chord function need not be real, but eventual symmetry with respect to reflections about a phase space point may relate these representations. Semiclassical approximations for the "classical-like" region of small chords and for large chords are derived. These lead to an interpretation of the Fourier invariance in terms of conjugate chords. The interrelation of large and small (sub-Planck) phase space structures previously noted in the literature are thus reinterpreted.

DOI
10.1088/0305-4470/38/7/005

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The intensity of the overlap of a quantum state with all its phase space translations defines its quantum correlations. In the case of pure states, these are invariant with respect to Fourier transformation. The overlaps themselves are here studied in terms of the Wigner function and its Fourier transform, i.e., the characteristic function or chord function. Unlike the Wigner function, the chord function need not be real, but eventual symmetry with respect to reflections about a phase space point may relate these representations. Semiclassical approximations for the "classical-like" region of small chords and for large chords are derived. These lead to an interpretation of the Fourier invariance in terms of conjugate chords. The interrelation of large and small (sub-Planck) phase space structures previously noted in the literature are thus reinterpreted.
Characteristic functions contain complete information about all the moments of a classical distribution and the same holds for the Fourier transform of the Wigner function: a quantum characteristic function , or the chord function. However, knowledge of a finite number of moments does not allow for accurate determination of the chord function. This provides the overlap of the state with all its possible rigid translations (or displacements). We here present a semiclassical approximation of the chord function for large Bohr-quantized states, which is accurate right up to a caustic, beyond which the chord function becomes evanescent. It is verified to pick out blind spots, which are displacements for zero overlaps. These occur even for translations within a Planck area of the origin. We derive a simple approximation for the closest blind spots, depending on the Schrödinger covariance matrix, which is verified for Bohr-quantized states.
Abstract A finite Hilbert space can be associated to a periodic phase space, that is, a torus. Then a subgroup of operators corresponding to reflections and translations on the torus form respectively the basis for the discrete Weyl representation, including the Wigner function, and for its Fourier conjugate, the chord representation. They are invariant under Clifford transformations and obey analogous product rules to the continuous representations, so allowing for the calculation of expectations and correlations for observables. We here import new identities from the continuum for products of pure state Wigner and chord functions, involving, for instance the inverse phase space participation ratio and correlations of a state with its translate. New identities are derived involving transition Wigner or chord functions of transition operators . Extension of the reflection and translation operators to a doubled torus phase space leads to the representation of superoperators and so to the construction of the propagator of Wigner functions from the Weyl representation of the evolution operator.
The semiclassical Wigner function for a Bohr-quantized energy eigenstate is known to have a caustic along the corresponding classical closed phase space curve in the case of a single degree of freedom. Its Fourier transform, the semiclassical chord function, also has a caustic along the conjugate curve defined as the locus of diameters, i.e. the maximal chords of the original curve. If the latter is convex, so is its conjugate, resulting in a simple fold caustic. Indentations in the quantized curve generate self-intersections of the closed fold line and isolated cusp points. The uniform approximation through the fold caustic, that is derived here, describes the transition undergone by the overlap of the state with its translation, from an oscillatory regime for small chords, to evanescent overlaps, rising to a maximum near the caustic. The diameter caustic for the Wigner function is also treated.
The Lindblad master equation governs the general Markovian evolution of a density operator for an open quantum system. Semiclassical Wigner functions represent density operators in phase space in terms of chords on a classical manifold, so that the amplitude and phase of each chord contribution is classically defined. Inserting such a Wigner function into a phase space version of the master equation, its explicit evolution is derived in the absence of dissipation. There results a simple extension of the unitary evolution of the semiclassical Wigner function, which does not affect the phase of each chord contribution, while dampening its amplitude exponentially. Projecting the Wigner function on to an orthogonal position or momentum basis, the dampening of long chords emerges as the exponential decay of off-diagonal elements of the density matrix.
The exact solution of the Lindblad equation with a quadratic Hamiltonian and linear coupling operators was derived within the chord representation, that is, for the Fourier transform of the Wigner function, also known as the characteristic function. It is here generalized for several degrees of freedom, so as to provide an explicit expression for the reduced density operator of any subsystem, as well as moments expressed as derivatives of this evolving chord function. The Wigner function is then the convolution of its straightforward classical evolution with a widening multidimensional Gaussian window, eventually ensuring its positivity. Futher on, positivity also holds for the Glauber-Sundarshan P function, which guarantees separability of the components. In the context of several degrees of freedom, a full dissipation matrix is defined, whose trace is equal to twice the previously derived dissipation coefficient. This governs the rate at which the phase space volume of the argument of the Wigner function contracts, while that of the chord function expands. Examples of Markovian evolution of a triatomic molecule and of an array of harmonic oscillators are discussed.
Abstract Operators in quantum mechanics—either observables, density or evolution operators, unitary or not—can be represented by c-numbers in operator bases. The position and momentum bases are in one-to-one correspondence with lagrangian planes in double phase space, but this is also true for the well known Wigner–Weyl correspondence based on translation and reflection operators. These phase space methods are here extended to the representation of superoperators. We show that the Choi–Jamiolkowsky isomorphism between the dynamical matrix and the linear action of the superoperator constitutes a ‘double’ Wigner or chord transform when represented in double phase space. As a byproduct several previously unknown integral relationships between products of Wigner and chord distributions for pure states are derived.