4,379 research outputs found
Introduction to PT-Symmetric Quantum Theory
In most introductory courses on quantum mechanics one is taught that the
Hamiltonian operator must be Hermitian in order that the energy levels be real
and that the theory be unitary (probability conserving). To express the
Hermiticity of a Hamiltonian, one writes , where the symbol
denotes the usual Dirac Hermitian conjugation; that is, transpose and
complex conjugate. In the past few years it has been recognized that the
requirement of Hermiticity, which is often stated as an axiom of quantum
mechanics, may be replaced by the less mathematical and more physical
requirement of space-time reflection symmetry (PT symmetry) without losing any
of the essential physical features of quantum mechanics. Theories defined by
non-Hermitian PT-symmetric Hamiltonians exhibit strange and unexpected
properties at the classical as well as at the quantum level. This paper
explains how the requirement of Hermiticity can be evaded and discusses the
properties of some non-Hermitian PT-symmetric quantum theories
Collective stability of networks of winner-take-all circuits
The neocortex has a remarkably uniform neuronal organization, suggesting that
common principles of processing are employed throughout its extent. In
particular, the patterns of connectivity observed in the superficial layers of
the visual cortex are consistent with the recurrent excitation and inhibitory
feedback required for cooperative-competitive circuits such as the soft
winner-take-all (WTA). WTA circuits offer interesting computational properties
such as selective amplification, signal restoration, and decision making. But,
these properties depend on the signal gain derived from positive feedback, and
so there is a critical trade-off between providing feedback strong enough to
support the sophisticated computations, while maintaining overall circuit
stability. We consider the question of how to reason about stability in very
large distributed networks of such circuits. We approach this problem by
approximating the regular cortical architecture as many interconnected
cooperative-competitive modules. We demonstrate that by properly understanding
the behavior of this small computational module, one can reason over the
stability and convergence of very large networks composed of these modules. We
obtain parameter ranges in which the WTA circuit operates in a high-gain
regime, is stable, and can be aggregated arbitrarily to form large stable
networks. We use nonlinear Contraction Theory to establish conditions for
stability in the fully nonlinear case, and verify these solutions using
numerical simulations. The derived bounds allow modes of operation in which the
WTA network is multi-stable and exhibits state-dependent persistent activities.
Our approach is sufficiently general to reason systematically about the
stability of any network, biological or technological, composed of networks of
small modules that express competition through shared inhibition.Comment: 7 Figure
A Coherent Ising Machine Based On Degenerate Optical Parametric Oscillators
A degenerate optical parametric oscillator network is proposed to solve the
NP-hard problem of finding a ground state of the Ising model. The underlying
operating mechanism originates from the bistable output phase of each
oscillator and the inherent preference of the network in selecting oscillation
modes with the minimum photon decay rate. Computational experiments are
performed on all instances reducible to the NP-hard MAX-CUT problems on cubic
graphs of order up to 20. The numerical results reasonably suggest the
effectiveness of the proposed network.Comment: 18 pages, 6 figure
Pseudo-Hermiticity of an Exactly Solvable Two-Dimensional Model
We study a two-dimensional exactly solvable non-Hermitian non-symmetric
quantum model with real spectrum, which is not amenable to separation of
variables, by supersymmetrical methods. Here we focus attention on the property
of pseudo-Hermiticity, biorthogonal expansion and pseudo-metric operator. To
our knowledge this is the first time that pseudo-Hermiticity is realized
explicitly for a nontrivial two-dimensional case. It is shown that the
Hamiltonian of the model is not diagonalizable.Comment: 14 page
The algebraic criteria for the stability of control systems
This paper critically examines the standard algebraic criteria for the stability of linear control systems and their proofs, reveals important previously unnoticed connections, and presents new representations. Algebraic stability criteria have also acquired significance for stability studies of non-linear differential equation systems by the Krylov-Bogoljubov-Magnus Method, and allow realization conditions to be determined for classes of broken rational functions as frequency characteristics of electrical network
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
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