50 research outputs found
Stationary waves on nonlinear quantum graphs: General framework and canonical perturbation theory
In this paper we present a general framework for solving the stationary
nonlinear Schr\"odinger equation (NLSE) on a network of one-dimensional wires
modelled by a metric graph with suitable matching conditions at the vertices. A
formal solution is given that expresses the wave function and its derivative at
one end of an edge (wire) nonlinearly in terms of the values at the other end.
For the cubic NLSE this nonlinear transfer operation can be expressed
explicitly in terms of Jacobi elliptic functions. Its application reduces the
problem of solving the corresponding set of coupled ordinary nonlinear
differential equations to a finite set of nonlinear algebraic equations. For
sufficiently small amplitudes we use canonical perturbation theory which makes
it possible to extract the leading nonlinear corrections over large distances.Comment: 26 page
A sub-determinant approach for pseudo-orbit expansions of spectral determinants in quantum maps and quantum graphs
We study implications of unitarity for pseudo-orbit expansions of the
spectral determinants of quantum maps and quantum graphs. In particular, we
advocate to group pseudo-orbits into sub-determinants. We show explicitly that
the cancellation of long orbits is elegantly described on this level and that
unitarity can be built in using a simple sub-determinant identity which has a
non-trivial interpretation in terms of pseudo-orbits. This identity yields much
more detailed relations between pseudo orbits of different length than known
previously. We reformulate Newton identities and the spectral density in terms
of sub-determinant expansions and point out the implications of the
sub-determinant identity for these expressions. We analyse furthermore the
effect of the identity on spectral correlation functions such as the
auto-correlation and parametric cross correlation functions of the spectral
determinant and the spectral form factor.Comment: 25 pages, one figur
Stationary waves on nonlinear quantum graphs. II. Application of canonical perturbation theory in basic graph structures
We consider exact and asymptotic solutions of the stationary cubic nonlinear Schrödinger equation on metric graphs. We focus on some basic example graphs. The asymptotic solutions are obtained using the canonical perturbation formalism developed in our earlier paper [S. Gnutzmann and D. Waltner, Phys. Rev. E 93, 032204 (2016)]. For closed example graphs (interval, ring, star graph, tadpole graph), we calculate spectral curves and show how the description of spectra reduces to known characteristic functions of linear quantum graphs in the low-intensity limit. Analogously for open examples, we show how nonlinear scattering of stationary waves arises and how it reduces to known linear scattering amplitudes at low intensities. In the short-wavelength asymptotics we discuss how genuine nonlinear effects may be described using the leading order of canonical perturbation theory: bifurcation of spectral curves (and the corresponding solutions) in closed graphs and multistability in open graphs
Quantum Corrections to Fidelity Decay in Chaotic Systems
By considering correlations between classical orbits we derive semiclassical
expressions for the decay of the quantum fidelity amplitude for classically
chaotic quantum systems, as well as for its squared modulus, the fidelity or
Loschmidt echo. Our semiclassical results for the fidelity amplitude agree with
random matrix theory (RMT) and supersymmetry predictions in the universal Fermi
golden rule regime. The calculated quantum corrections can be viewed as arising
from a static random perturbation acting on nearly self-retracing interfering
paths, and hence will be suppressed for time-varying perturbations. Moreover,
using trajectory-based methods we show a relation, recently obtained in RMT,
between the fidelity amplitude and the cross-form factor for parametric level
correlations. Beyond RMT, we compute Ehrenfest-time effects on the fidelity
amplitude. Furthermore our semiclassical approach allows for a unified
treatment of the fidelity, both in the Fermi golden rule and Lyapunov regimes,
demonstrating that quantum corrections are suppressed in the latter.Comment: 14 pages, 4 figure
Collectivity and Periodic Orbits in a Chain of Interacting, Kicked Spins
The field of quantum chaos originated in the study of spectral statistics for
interacting many-body systems, but this heritage was almost forgotten when
single-particle systems moved into the focus. In recent years new interest
emerged in many-body aspects of quantum chaos. We study a chain of interacting,
kicked spins and carry out a semiclassical analysis that is capable of
identifying all kinds of genuin many-body periodic orbits. We show that the
collective many-body periodic orbits can fully dominate the spectra in certain
cases.Comment: 6 pages, 6 figures, accepted for publication in Acta Physica Polonica
A. arXiv admin note: substantial text overlap with arXiv:1611.0574
Semiclassical Prediction of Large Spectral Fluctuations in Interacting Kicked Spin Chains
While plenty of results have been obtained for single-particle quantum
systems with chaotic dynamics through a semiclassical theory, much less is
known about quantum chaos in the many-body setting. We contribute to recent
efforts to make a semiclassical analysis of many-body systems feasible. This is
nontrivial due to both the enormous density of states and the exponential
proliferation of periodic orbits with the number of particles. As a model
system we study kicked interacting spin chains employing semiclassical methods
supplemented by a newly developed duality approach. We show that for this model
the line between integrability and chaos becomes blurred. Due to the
interaction structure the system features (non-isolated) manifolds of periodic
orbits possessing highly correlated, collective dynamics. As with the invariant
tori in integrable systems, their presence lead to significantly enhanced
spectral fluctuations, which by order of magnitude lie in-between integrable
and chaotic cases.Comment: 42 pages, 19 figure