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 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.
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 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.
In this thesis, we construct an approximate series solution of the Wigner equation in terms of Airy functions, which are semiclassically concentrated on certain Lagrangian curves in two-dimensional phase space. These curves are defined by the eigenvalues and the Hamiltonian function of the associated one-dimensional Schrödinger operator, and they play a crucial role in the quantum interference mechanism in phase space. We assume that the potential of the Schrödinger operator is a single potential well, such that the spectrum is discrete. The construction starts from an eigenfunction series expansion of the solution, which is derived here for first time in a systematic way, by combining the elementary technique of separation of variables with involved spectral results for the Moyal star exponential operator. The eigenfunctions of the Wigner equation are the Wigner transforms of the Schrödinger eigenfunctions, and they are approximated in terms of Airy functions by a uniform stationary phase approximation of the Wigner transforms of the WKB expansions of the Schrödinger eigenfunctions. The approximation of the eigenfunction series is an approximated solution of the Wigner equation, which by projection onto the configuration space provides an approximate wave amplitude, free of turning point singularities. It is generally expected that, the derived wave amplitude is bounded, and correctly scaled, even on caustics, since only finite terms of the approximate terms are significant for WKB initial wave functions with finite energy. The details of the calculations are presented for the simple potential of the harmonic oscillator, in order to be able to check our approximations analytically. But, the same construction can be applied to any potential well, which behaves like the harmonic oscillator near the bottom of the well.
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.
Abstract Phase space reflection operators lie at the core of the Wigner–Weyl representation of density operators and observables. The role of the corresponding classical reflections is known in the construction of semiclassical approximations to Wigner functions of pure eigenstates and their coarsegrained microcanonical superpositions, which are not restricted to classically integrable systems. In their active role as unitary operators, they generate transitions between pairs of eigenstates specified by transition Wigner functions (or cross-Wigner functions): the square modulus of the transition Wigner function at each point in phase space is the transition probability for the reflection through that point. Coarsegraining the initial and final energies provides a transition probability density as a phase space path integral. It is here investigated in the simplest classical approximation involving microcanonical Wigner functions. A reflection operator generates a transition between a pair of energy shells with a probability density given by the integral of the inverse modulus of a Poisson bracket over the intersection of a shell with the reflection of its pair. The singularity of the pair of Wigner functions at their dominant caustics is nicely integrable over their intersection, except for a single degree of freedom. Even though this case is not directly relevant for future investigations of chaotic systems, it is shown here how the improved approximation of the spectral Wigner functions in terms of Airy functions resolves the singularity.