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Jens Bolte

Publications and source records attributed to Jens Bolte.

36 records · Page 2Linked to original sources
arXiv2005

The spectral form factor for quantum graphs

We consider quantum graphs with spin-orbit couplings at the vertices. Time-reversal invariance implies that the bond S-matrix is in the orthogonal or symplectic symmetry class, depending on spin quantum number s being integer or half-integer, respectively. The periodic-orbit expansion of the spectral form factor is shown to acquire additional weights from spin rotations along orbits. We determine the spin contribution to the coefficients in an expansion of the form factor from properties of the representation of the group of spin transformations on the graph. Consistency with the Circular Orthogonal and Circular Symplectic Ensemble, respectively, of random matrices is obtained.

nlin.CD↗
arXiv2005

Stability of wave packet dynamics under perturbations

We introduce a novel method to investigate the stability of wave packet dynamics under perturbations of the Hamiltonian. Our approach relies on semiclassical approximations, but is non-perturbative. Two separate contributions to the quantum fidelity are identified: one factor derives from the dispersion of the wave packets, whereas the other factor is determined by the separation of a trajectory of the perturbed classical system away from a corresponding unperturbed trajectory. We furthermore estimate both contributions in terms of classical Lyapunov exponents and find a decay of fidelity that is, generically, at least exponential, but may also be doubly exponential. The latter case is shown to be realised for inverted harmonic oscillators.

nlin.CD↗
arXiv2004

Semiclassical propagation of coherent states with spin-orbit interaction

We study semiclassical approximations to the time evolution of coherent states for general spin-orbit coupling problems in two different semiclassical scenarios: The limit \hbar to zero is first taken with fixed spin quantum number s and then with \hbar*s held constant. In these two cases different classical spin-orbit dynamics emerge. We prove that a coherent state propagated with a suitable classical dynamics approximates the quantum time evolution up to an error of size \sqrt{\hbar} and identify an Ehrenfest time scale. Subsequently an improvement of the semiclassical error to an arbitray order \hbar^{N/2} is achieved by a suitable deformation of the state that is propagated classically.

math-ph↗
arXiv2004

Zitterbewegung and semiclassical observables for the Dirac equation

In a semiclassical context we investigate the Zitterbewegung of relativistic particles with spin 1/2 moving in external fields. It is shown that the analogue of Zitterbewegung for general observables can be removed to arbitrary order in \hbar by projecting to dynamically almost invariant subspaces of the quantum mechanical Hilbert space which are associated with particles and anti-particles. This not only allows to identify observables with a semiclassical meaning, but also to recover combined classical dynamics for the translational and spin degrees of freedom. Finally, we discuss properties of eigenspinors of a Dirac-Hamiltonian when these are projected to the almost invariant subspaces, including the phenomenon of quantum ergodicity.

quant-ph↗
arXiv2003

The spin contribution to the form factor of quantum graphs

Following the quantisation of a graph with the Dirac operator (spin-1/2) we explain how additional weights in the spectral form factor K(τ) due to spin propagation around orbits produce higher order terms in the small-τasymptotics in agreement with symplectic random matrix ensembles. We determine conditions on the group of spin rotations sufficient to generate CSE statistics.

nlin.CD↗
arXiv2002

Semiclassical time evolution and quantum ergodicity for Dirac-Hamiltonians

Within the framework of Weyl calculus we establish a quantum-classical correspondence for the time evolution of observables generated by a Dirac-Hamiltonian. This includes a semiclassical separation of particles and antiparticles. We then prove quantum ergodicity for Dirac-Hamiltonians under the condition that a skew product of the classical relativistic translational motion and relativistic spin precession is ergodic.

math-ph↗
arXiv2002

Spectral Statistics for the Dirac Operator on Graphs

We determine conditions for the quantisation of graphs using the Dirac operator for both two and four component spinors. According to the Bohigas-Giannoni-Schmit conjecture for such systems with time-reversal symmetry the energy level statistics are expected, in the semiclassical limit, to correspond to those of random matrices from the Gaussian symplectic ensemble. This is confirmed by numerical investigation. The scattering matrix used to formulate the quantisation condition is found to be independent of the type of spinor. We derive an exact trace formula for the spectrum and use this to investigate the form factor in the diagonal approximation.

nlin.CD↗
arXiv2002

A semiclassical Egorov theorem and quantum ergodicity for matrix valued operators

We study the semiclassical time evolution of observables given by matrix valued pseudodifferential operators and construct a decomposition of the Hilbert space $L^2(\rz^d)\otimes\kz^n$ into a finite number of almost invariant subspaces. For a certain class of observables, that is preserved by the time evolution, we prove an Egorov theorem. We then associate with each almost invariant subspace of $L^2(\rz^d)\otimes\kz^n$ a classical system on a product phase space $\TRd\times\cO$, where $\cO$ is a compact symplectic manifold on which the classical counterpart of the matrix degrees of freedom is represented. For the projections of eigenvectors of the quantum Hamiltonian to the almost invariant subspaces we finally prove quantum ergodicity to hold, if the associated classical systems are ergodic.

math-ph↗
arXiv2001

The Gutzwiller Trace Formula for Quantum Systems with Spin

The Gutzwiller trace formula provides a semiclassical approximation for the density of states of a quantum system in terms of classical periodic orbits. In its original form Gutzwiller derived the trace formula for quantum systems without spin. We will discuss the modifications that arise for quantum systems with both translational and spin degrees of freedom and which are either described by Pauli- or Dirac-Hamiltonians. In addition, spectral densities weighted by expectation values of observables will be considered. It turns out that in all cases the semiclassical approximation yields sums over periodic orbits of the translational motion. Spin contributes via weight factors that take a spin precession along the translational orbits into account.

nlin.CD↗
arXiv2001

Quantum and classical ergodicity of spinning particles

We give a formulation of quantum ergodicity for Pauli Hamiltonians with arbitrary spin in terms of a Wigner-Weyl calculus. The corresponding classical phase space is the direct product of the phase space of the translational degrees of freedom and the two-sphere. On this product space we introduce a combination of the translational motion and classical spin precession. We prove quantum ergodicity under the condition that this product flow is ergodic.

nlin.CD↗
arXiv2000

Semiclassical expectation values for relativistic particles with spin 1/2

For relativistic particles with spin 1/2, which are described by the Dirac equation, a semiclassical trace formula is introduced that incorporates expectation values of observables in eigenstates of the Dirac-Hamiltonian. Furthermore, the semiclassical limit of an average of expectation values is expressed in terms of a classical average of the corresponding classical observable.

nlin.CD↗
arXiv2000

Quantum ergodicity for Pauli Hamiltonians with spin 1/2

Quantum ergodicity, which expresses the semiclassical convergence of almost all expectation values of observables in eigenstates of the quantum Hamiltonian to the corresponding classical microcanonical average, is proven for non-relativistic quantum particles with spin 1/2. It is shown that quantum ergodicity holds, if a suitable combination of the classical translational dynamics and the spin dynamics along the trajectories of the translational motion is ergodic.

math-ph↗
arXiv1999

Semiclassical form factor for chaotic systems with spin 1/2

We study the properties of the two-point spectral form factor for classically chaotic systems with spin 1/2 in the semiclassical limit, with a suitable semiclassical trace formula as our principal tool. To this end we introduce a regularized form factor and discuss the limit in which the so-called diagonal approximation can be recovered. The incorporation of the spin contribution to the trace formula requires an appropriate variant of the equidistribution principle of long periodic orbits as well as the notion of a skew product of the classical translational and spin dynamics. Provided this skew product is mixing, we show that generically the diagonal approximation of the form factor coincides with the respective predictions from random matrix theory.

chao-dyn↗
arXiv1998

A semiclassical approach to the Dirac equation

We derive a semiclassical time evolution kernel and a trace formula for the Dirac equation. The classical trajectories that enter the expressions are determined by the dynamics of relativistic point particles. We carefully investigate the transport of the spin degrees of freedom along the trajectories which can be understood geometrically as parallel transport in a vector bundle with SU(2) holonomy. Furthermore, we give an interpretation in terms of a classical spin vector that is transported along the trajectories and whose dynamics, dictated by the equation of Thomas precession, gives rise to dynamical and geometric phases every orbit is weighted by. We also present an analogous approach to the Pauli equation which we analyse in two different limits.

quant-ph↗
arXiv1998

Semiclassical Time Evolution and Trace Formula for Relativistic Spin-1/2 Particles

We investigate the Dirac equation in the semiclassical limit \hbar --> 0. A semiclassical propagator and a trace formula are derived and are shown to be determined by the classical orbits of a relativistic point particle. In addition, two phase factors enter, one of which can be calculated from the Thomas precession of a classical spin transported along the particle orbits. For the second factor we provide an interpretation in terms of dynamical and geometric phases.

quant-ph↗
arXiv1998

A spectral correspondence for Maass waveforms

Let O^1 be a (cocompact) Fuchsian group, given as the group of units of norm one in a maximal order O in an indefinite quaternion division algebra over Q. Using the (classical) Selberg trace formula, we show that the eigenvalues of the automorphic Laplacian for O^1 and their multiplicities coincide with the eigenvalues and multiplicities of the Laplacian defined on the Maass newforms for the Hecke congruence group Gamma_0(d), when d is the discriminant of the maximal order O. We also show the equality of the traces of certain Hecke operators defined on the Laplace eigenspaces for O^1 and the newforms of level d, respectively.

math.SP↗
arXiv1997

Semiclassical Trace Formulae and Eigenvalue Statistics in Quantum Chaos

A detailed discussion of semiclassical trace formulae is presented and it is demonstrated how a regularized trace formula can be derived while dealing only with finite and convergent expressions. Furthermore, several applications of trace formula techniques to quantum chaos are reviewed. Then local spectral statistics, measuring correlations among finitely many eigenvalues, are reviewed and a detailed semiclassical analysis of the number variance is given. Thereafter the transition to global spectral statistics, taking correlations among infinitely many quantum energies into account, is discussed. It is emphasized that the resulting limit distributions depend on the way one passes to the global scale. A conjecture on the distribution of the fluctuations of the spectral staircase is explained in this general context and evidence supporting the conjecture is discussed.

chao-dyn↗