Semiclassical trace formulae using coherent states

Path integration over closed loops and Gutzwiller's trace formula

We discuss a basic mathematical approach to Feynman path integrals as infinite-dimensional oscillatory integrals. We present new results on asymptotics of such integrals which exploit recently developed approximation techniques via finite dimensional oscillatory integrals. Applications are also given, namely to the study of the trace of the time evolution operator in quantum mechanics and to the interpretation of “Gutzwiller's trace formula” as a leading term in an asymptotic expansion around classical periodic orbits.

We investigate the resonance spectrum of the Hénon-Heiles potential up to twice the barrier energy. The quantum spectrum is obtained by the method of complex coordinate rotation. We use periodic orbit theory to approximate the oscillating part of the resonance spectrum semiclassically and Strutinsky smoothing to obtain its smooth part. Although the system in this energy range is almost chaotic, it still contains stable periodic orbits. Using Gutzwiller's trace formula, complemented by a uniform approximation for a co-dimension-two bifurcation scenario, we are able to reproduce the coarse-grained quantum-mechanical density of states very accurately, including only a few stable and unstable orbits.

Bogomolny's transfer operator method plays a significant role in the study of quantum chaos, along with other well known methods like Gutzwiller's trace formula and the dynamical zeta function, which generalize the Einstein-Brillouin-Keller quantization rule from integrable systems to chaotic systems. According to the theory, the Fredholm determinant of the transfer operator, defined on a Poincaré section of a classical physical system, provides a quantization condition to the energy spectrum of the corresponding quantum system. This study presents two factorization formulas, which relate different quantization conditions defined on different classical trajectory segments. These explicit relations answer the question of why all these classical quantization conditions determine exactly the same energy spectrum of the corresponding quantum systems. As an example, these formulas are illustrated in the equilateral triangular billiard.

We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional billiard systems with corners. This is achieved by using the exact Sommerfeld solution for the Green function of a wedge. We obtain a uniformly valid formula for single-diffractive orbits which interpolates between formerly separate approaches (the geometrical theory of diffraction and Gutzwiller's trace formula). It yields excellent numerical agreement with exact quantum results, also in cases where other methods fail.

The scattering of a point particle from three hard discs in a plane is studied in the semiclassical approximation, using the Gutzwiller trace formula. Using a previously introduced coding of the classical dynamics, the needed summation over the classical periodic orbits is performed. The trace function is then given in terms of Ruelle zeta functions. A semiclassical limit upper bound is obtained on the lifetimes of the scattering resonances. This bound is larger than the classical lifetime when the classical repellor is chaotic but coincides with it when the repellor is periodic. We conclude that classical chaos dramatically influences the lifetimes of the scattering resonances. Our upper bound for the resonance lifetime is compared with the results of numerical calculation of the full quantum dynamics. The distribution of the imaginary parts of the complex wave numbers of the resonances is also calculated.

(1) What is semiclassical Analysis? (2) What is a Trace Formula? (3) Physical Motivations : “Quantum Chaos” (4) Gutzwiller Trace Formula. (Rigorous Approach)

Trace formulas are derived for the coefficients of a matrix differential equation representing wave propagation in a medium which supports N types of waves. In general these waves are coupled to each other but are assumed to become uncoupled as lzl -> ?o. The two coefficients of the ordinary differential equation, N x N matrices, are assumed known for z 0. A point source is located in the region z 0. The unique features of this work are a derivation of the trace formula in which the Jost functions are not needed, an alternate measurement method based on an impedance concept and a trace equation based on the impedance concept. Introduction. Motivated by the work of Trubowitz (1), trace formula methods were introduced by Deift and Trubowitz (2) to study one-dimensional inverse quantum mechanical scattering problems on the line. The trace formula expresses the potential as a functional of the reflection coefficient. Greene (3) used the same techniques to study the one-dimensional acoustical wave equation with the angular frequency playing the role of the quantum mechanical energy. Stickler and Deift (4) examined the acoustic problem of a point source in an inhomogeneous half space bounded by a pressure release surface. The sound speed was recovered from a measurement of the normal component of the acoustic velocity at the pressure release surface. The role of the quantum mechanical energy was played by a vertical wave number, which does not correspond to either a spatial or temporal derivative. Using again the analogy between vertical wave number (the wave number in the direction of the coefficients' variation) and the quantum mechanical energy, Stickler (5) showed how the trace formula methods could be used to recover the sound speed when it approaches different values at plus and minus infinity. In a quantum mechanical context this means that the potential approaches zero, say at minus infinity, but some positive constant, not necessarily known, at plus infinity. (In the Deift-Trubowitz analysis the potential approached zero in both limits.) Stickler (6) also discussed this same problem in an electromagnetic context. Tomei (7) has used the trace methods to recover the coefficients of a linear, third order equation associated with the Boussinesq equation. In the present paper the trace method is extended to a one-dimensional vector Helmholtz equation which supports N types of waves and for which the waves are coupled to each other except at infinity. The trace method yields a functional relation- ship between the coefficients of the Helmholtz equation, a reflection coefficient described below and assumed known and a specific solution of the vector Helmholtz equation with known and prescribed initial conditions. These relationships are called the trace formulas and are derived here. The coefficients of the vector Helmholtz differential equation can be replaced by these trace formulas and this results in a system of coupled, nonlinear ordinary differential equations with known initial condi- tions. Once this solution is determined, the coefficients can be recovered. The existence of a solution of this coupled system is not given.

This chapter is dedicated to the derivation of semiclassical trace formulae for particles with spin. We will focus on Pauli Hamiltonians with arbitrary spin and only briefly describe the corresponding calculations for the Dirac equation. We will also comment on a different limit in which the spectral asymptotics of the Pauli Hamiltonian can be considered.KeywordsPeriodic OrbitDirac EquationJacobi EquationTrace FormulaOrbit CouplingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The connection between the wavefunctions and the classical periodic orbits in a 2D harmonic oscillator is analytically constructed by using the representation of SU(2) coherent states. It is found that the constructed wavefunction generally corresponds to an ensemble of classical trajectories and its localization is extremely efficient. With the constructed wavefunction, we also analyse the property of the probability current density associated with the classical periodic orbit. The appearance of vortex structure in the quantum flow is clearly found to arise from the wave interference.

The relationship between quantum degeneracies and the emergence of classical periodic orbits in the Fock-Darwin (FD) system is systematically explored by using the ladder operators in the Cartesian coordinates. The quantum-classical connection is analytically developed with the time-dependent coherent state that is theoretically verified to correspond to the Gaussian wave packet state with unitary transformation. The time-dependent coherent state is further used to derive the stationary coherent state that can be expressed as a superposition of degenerate eigenstates. More importantly, the stationary coherent state can alternatively be expressed as an integral of the Gaussian wave packet over a periodic orbit. With the integral representation, the quantum vortex structures of the stationary coherent states can be precisely characterized up to extremely high order. The vortex arrays are generally observed when the phase structures are near the crossing points and cusps of the classical trajectories. Finally, the quantum Fourier transform is employed to verify that the circularly symmetric eigenstate physically corresponds to an ensemble of classical hypotrochoid orbits. This work can provide a pedagogical insight into the formation of Laguerre-Gaussian eigenmodes and hypotrochoid geometric modes in degenerate spherical cavities.

Quantum chaotic transport in double barrier tunnel structures

We use the SYK family of models with N Majorana fermions to study the complexity of time evolution, formulated as the shortest geodesic length on the unitary group manifold between the identity and the time evolution operator, in free, integrable, and chaotic systems. Initially, the shortest geodesic follows the time evolution trajectory, and hence complexity grows linearly in time. We study how this linear growth is eventually truncated by the appearance and accumulation of conjugate points, which signal the presence of shorter geodesics intersecting the time evolution trajectory. By explicitly locating such “shortcuts” through analytical and numerical methods, we demonstrate that: (a) in the free theory, time evolution encounters conjugate points at a polynomial time; consequently complexity growth truncates at O( sqrt{N} ), and we find an explicit operator which “fast-forwards” the free N-fermion time evolution with this complexity, (b) in a class of interacting integrable theories, the complexity is upper bounded by O(poly(N)), and (c) in chaotic theories, we argue that conjugate points do not occur until exponential times O(eN), after which it becomes possible to find infinitesimally nearby geodesics which approximate the time evolution operator. Finally, we explore the notion of eigenstate complexity in free, integrable, and chaotic models.

Using quantum maps, we study the accuracy of semiclassical trace formulas. The role of chaos in improving the semiclassical accuracy in some systems is demonstrated quantitatively. However, our study of the standard map cautions that this may not be most general. While studying a sawtooth map we demonstrate the rather remarkable fact that at the level of the time one trace even in the presence of fixed points on singularities the trace formula may be exact, and in any case has no logarithmic divergences observed for the quantum bakers map. As a byproduct we introduce fantastic periodic curves akin to curlicues.

The localization of eigenfunctions around classical periodic orbits is studied numerically for the H-atom in a strong magnetic field by calculating their Husimi distribution in phase space. In contrast to the configuration space representation, the phase space distributions are simply structured: about 90% of eigenstates may be unambigously related to fixed points and invariant manifolds of periodic orbits, indicating that scars are the rule rather than the exception. In order to measure the influence of one particular orbit, we calculate the integrals of the energetically lowest 500 Husimi distributions along the orbit. Their incoherent superposition defines the scar strength distribution for the particular periodic orbit which is analyzed by Fourier transformation. The Husimi distribution at (q, p) in phase space may be represented as a scalar product of the wavefunction with a coherent state of the unperturbed system, i.e., a radial Gaussian wave packet located at (q, p) in the (regularized) Coulomb system. This simplifies the actual calculation of the Husimi distribution and allows to treat their incoherent superposition within Gutzwillers theory extended to matrix elements of an operator A, if we choose A to be the projector on a coherent state.