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.
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.
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 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.
Abstract The strictly classical propagation of an initial Wigner function, referred to as TWA or LSC-IVR, is considered to provide approximate averages, despite not being a true Wigner function: it does not represent a positive operator. We here show that its symplectic Fourier transform, the truncated chord approximation (TCA), coincides with the full semiclassical approximation to the evolved quantum characteristic function (or chord function) in a narrow neighbourhood of the origin of the dual chord phase space. Surprisingly, this small region accounts for purely quantum features, such as blind spots and local wave function correlations, as well as the expectation of observables with a close classical correspondence. Direct numerical comparison of the TCA with exact quantum results verifies the semiclassical predictions for an initial coherent state evolving under the Kerr Hamiltonian. The resulting clear criterion for any further features, which may be estimated by classical propagation, is that, within the chord representation, they are concentrated near the origin.
The overlap of a large quantum state with its image, under tiny translations, oscillates swiftly. We here show that complete orthogonality occurs generically at isolated points. Decoherence, in the Markovian approximation, lifts the correlation minima from zero much more quickly than the Wigner function is smoothed into a positive phase space distribution. In the case of a superposition of coherent states, blind spots depend strongly on positions and amplitudes of the components, but they are only weakly affected by relative phases and the various degrees and directions of squeezing. The blind spots for coherent state triplets are special in that they lie close to a hexagonal lattice: further superpositions of translated triplets, specified by nodes of one of the sublattices, are quasi-orthogonal to the original triplet and to any state likewise constructed on the other sublattice.
A semiclassical approximation for an evolving density operator, driven by a “closed” hamiltonian operator and “open ” markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra “open ” term is added to the double Hamiltonian by the non-hermitian part of the Lindblad operators in the general case of dissipative markovian evolution. The particular case of generic hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighborhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further “small-chord” approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.
A semiclassical approximation for an evolving density operator, driven by a 'closed' Hamiltonian operator and 'open' Markovian Lindblad operators, is obtained. The theory is based on the chord function, i.e. the Fourier transform of the Wigner function. It reduces to an exact solution of the Lindblad master equation if the Hamiltonian operator is a quadratic function and the Lindblad operators are linear functions of positions and momenta. Initially, the semiclassical formulae for the case of Hermitian Lindblad operators are reinterpreted in terms of a (real) double phase space, generated by an appropriate classical double Hamiltonian. An extra 'open' term is added to the double Hamiltonian by the non-Hermitian part of the Lindblad operators in the general case of dissipative Markovian evolution. The particular case of generic Hamiltonian operators, but linear dissipative Lindblad operators, is studied in more detail. A Liouville-type equivariance still holds for the corresponding classical evolution in double phase space, but the centre subspace, which supports the Wigner function, is compressed, along with expansion of its conjugate subspace, which supports the chord function. Decoherence narrows the relevant region of double phase space to the neighbourhood of a caustic for both the Wigner function and the chord function. This difficulty is avoided by a propagator in a mixed representation, so that a further 'small-chord' approximation leads to a simple generalization of the quadratic theory for evolving Wigner functions.