34 research outputs found
Complexified coherent states and quantum evolution with non-Hermitian Hamiltonians
The complex geometry underlying the Schr\"odinger dynamics of coherent states
for non-Hermitian Hamiltonians is investigated. In particular two seemingly
contradictory approaches are compared: (i) a complex WKB formalism, for which
the centres of coherent states naturally evolve along complex trajectories,
which leads to a class of complexified coherent states; (ii) the investigation
of the dynamical equations for the real expectation values of position and
momentum, for which an Ehrenfest theorem has been derived in a previous paper,
yielding real but non-Hamiltonian classical dynamics on phase space for the
real centres of coherent states. Both approaches become exact for quadratic
Hamiltonians. The apparent contradiction is resolved building on an observation
by Huber, Heller and Littlejohn, that complexified coherent states are
equivalent if their centres lie on a specific complex Lagrangian manifold. A
rich underlying complex symplectic geometry is unravelled. In particular a
natural complex structure is identified that defines a projection from complex
to real phase space, mapping complexified coherent states to their real
equivalents.Comment: 18 pages, small improvements made, similar to published versio
Semiclassical quantisation for a bosonic atom-molecule conversion system
We consider a simple quantum model of atom-molecule conversion where bosonic
atoms can combine into diatomic molecules and vice versa. The many-particle
system can be expressed in terms of the generators a deformed algebra,
and the mean-field dynamics takes place on a deformed version of the Bloch
sphere, a teardrop shaped surface with a cusp singularity. We analyse the
mean-field and many-particle correspondence, which shows typical features of
quantum-classical correspondence. We demonstrate that semiclassical methods can
be employed to recover full many-particle features from the mean-field
description in cold atom systems with atom-molecule conversion, and derive an
analytic expression for the many-particle density of states in the limit of
large particle numbers.Comment: 10 pages, 10 figures, corrected typos, further small changes, similar
to published versio
Propagation of Gaussian beams in the presence of gain and loss
We consider the propagation of Gaussian beams in a waveguide with gain and
loss in the paraxial approximation governed by the Schr\"odinger equation. We
derive equations of motion for the beam in the semiclassical limit that are
valid when the waveguide profile is locally well approximated by quadratic
functions. For Hermitian systems, without any loss or gain, these dynamics are
given by Hamilton's equations for the center of the beam and its conjugate
momentum. Adding gain and/or loss to the waveguide introduces a non-Hermitian
component, causing the width of the Gaussian beam to play an important role in
its propagation. Here we show how the width affects the motion of the beam and
how this may be used to filter Gaussian beams located at the same initial
position based on their width
Mixed-state evolution in the presence of gain and loss
A model is proposed that describes the evolution of a mixed state of a
quantum system for which gain and loss of energy or amplitude are present.
Properties of the model are worked out in detail. In particular, invariant
subspaces of the space of density matrices corresponding to the fixed points of
the dynamics are identified, and the existence of a transition between the
phase in which gain and loss are balanced and the phase in which this balance
is lost is illustrated in terms of the time average of observables. The model
is extended to include a noise term that results from a uniform random
perturbation generated by white noise. Numerical studies of example systems
show the emergence of equilibrium states that suppress the phase transition.Comment: 5 pages, 2 figures (published version
Random matrix ensembles for -symmetric systems
Recently much effort has been made towards the introduction of non-Hermitian
random matrix models respecting -symmetry. Here we show that there is a
one-to-one correspondence between complex -symmetric matrices and
split-complex and split-quaternionic versions of Hermitian matrices. We
introduce two new random matrix ensembles of (a) Gaussian split-complex
Hermitian, and (b) Gaussian split-quaternionic Hermitian matrices, of arbitrary
sizes. They are related to the split signature versions of the complex and the
quaternionic numbers, respectively. We conjecture that these ensembles
represent universality classes for -symmetric matrices. For the case of
matrices we derive analytic expressions for the joint probability
distributions of the eigenvalues, the one-level densities and the level
spacings in the case of real eigenvalues.Comment: 9 pages, 3 figures, typos corrected, small changes, accepted for
publication in Journal of Physics