403 research outputs found
Time-convolutionless master equations for composite open quantum systems
In this work we consider the master equations for composite open quantum
systems. We provide purely algebraic formulae for terms of perturbation series
defining such equations. We also give conditions under which the Bogolubov-van
Hove limit exists and discuss some corrections to this limit. We present an
example to illustrate our results. In particular, this example shows, that
inhomogeneous terms in time-convolutionless master equations can vanish after
reservoir correlation time, but lead to renormalization of initial conditions
at such a timescale
Quantum Trajectories, State Diffusion and Time Asymmetric Eventum Mechanics
We show that the quantum stochastic unitary dynamics Langevin model for
continuous in time measurements provides an exact formulation of the Heisenberg
uncertainty error-disturbance principle. Moreover, as it was shown in the 80's,
this Markov model induces all stochastic linear and non-linear equations of the
phenomenological "quantum trajectories" such as quantum state diffusion and
spontaneous localization by a simple quantum filtering method. Here we prove
that the quantum Langevin equation is equivalent to a Dirac type boundary-value
problem for the second-quantized input "offer waves from future" in one extra
dimension, and to a reduction of the algebra of the consistent histories of
past events to an Abelian subalgebra for the "trajectories of the output
particles". This result supports the wave-particle duality in the form of the
thesis of Eventum Mechanics that everything in the future is constituted by
quantized waves, everything in the past by trajectories of the recorded
particles. We demonstrate how this time arrow can be derived from the principle
of quantum causality for nondemolition continuous in time measurements.Comment: 21 pages. See also relevant publications at
http://www.maths.nott.ac.uk/personal/vpb/publications.htm
Estimates for multiple stochastic integrals and stochastic Hamilton-Jacobi equations
We study stochastic Hamilton-Jacobi-Bellman equations and the
corresponding Hamiltonian systems driven by jump-type Lévy processes.
The main objective of the present paper is to show existence,
uniqueness and a (locally in time) diffeomorphism property of the solution:
the solution trajectory of the system is a diffeomorphism as a
function of the initial momentum. This result enables us to implement
a stochastic version of the classical method of characteristics for the
Hamilton-Jacobi equations. An –in itself interesting– auxiliary result
are pointwise a.s. estimates for iterated stochastic integrals driven by
a vector of not necessarily independent jump-type semimartingales
Tropical Kraus maps for optimal control of switched systems
Kraus maps (completely positive trace preserving maps) arise classically in
quantum information, as they describe the evolution of noncommutative
probability measures. We introduce tropical analogues of Kraus maps, obtained
by replacing the addition of positive semidefinite matrices by a multivalued
supremum with respect to the L\"owner order. We show that non-linear
eigenvectors of tropical Kraus maps determine piecewise quadratic
approximations of the value functions of switched optimal control problems.
This leads to a new approximation method, which we illustrate by two
applications: 1) approximating the joint spectral radius, 2) computing
approximate solutions of Hamilton-Jacobi PDE arising from a class of switched
linear quadratic problems studied previously by McEneaney. We report numerical
experiments, indicating a major improvement in terms of scalability by
comparison with earlier numerical schemes, owing to the "LMI-free" nature of
our method.Comment: 15 page
On time-dependent projectors and on generalization of thermodynamical approach to open quantum systems
In this paper, we develop a consistent perturbative technique for obtaining a
time-local master equation based on projective methods in the case where the
projector depends on time. We then introduce a generalization of the
Kawasaki--Gunton projector, which allows us to use this technique to derive,
generally speaking, nonlinear master equations in the case of arbitrary
ansatzes consistent with some set of observables. Most of our results are very
general, but in our discussion we focus on the application of these results to
the theory of open quantum systems
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