9,047 research outputs found
Approximate stabilization of an infinite dimensional quantum stochastic system
We propose a feedback scheme for preparation of photon number states in a
microwave cavity. Quantum Non-Demolition (QND) measurements of the cavity field
and a control signal consisting of a microwave pulse injected into the cavity
are used to drive the system towards a desired target photon number state.
Unlike previous work, we do not use the Galerkin approximation of truncating
the infinite-dimensional system Hilbert space into a finite-dimensional
subspace. We use an (unbounded) strict Lyapunov function and prove that a
feedback scheme that minimizes the expectation value of the Lyapunov function
at each time step stabilizes the system at the desired photon number state with
(a pre-specified) arbitrarily high probability. Simulations of this scheme
demonstrate that we improve the performance of the controller by reducing
"leakage" to high photon numbers.Comment: Submitted to CDC 201
Explicit approximate controllability of the Schr\"odinger equation with a polarizability term
We consider a controlled Schr\"odinger equation with a dipolar and a
polarizability term, used when the dipolar approximation is not valid. The
control is the amplitude of the external electric field, it acts non linearly
on the state. We extend in this infinite dimensional framework previous
techniques used by Coron, Grigoriu, Lefter and Turinici for stabilization in
finite dimension. We consider a highly oscillating control and prove the
semi-global weak stabilization of the averaged system using a Lyapunov
function introduced by Nersesyan. Then it is proved that the solutions of the
Schr\"odinger equation and of the averaged equation stay close on every finite
time horizon provided that the control is oscillating enough. Combining these
two results, we get approximate controllability to the ground state for the
polarizability system
Quantum walks in higher dimensions
We analyze the quantum walk in higher spatial dimensions and compare
classical and quantum spreading as a function of time. Tensor products of
Hadamard transformations and the discrete Fourier transform arise as natural
extensions of the quantum coin toss in the one-dimensional walk simulation, and
other illustrative transformations are also investigated. We find that
entanglement between the dimensions serves to reduce the rate of spread of the
quantum walk. The classical limit is obtained by introducing a random phase
variable.Comment: 6 pages, 6 figures, published versio
Nonlocally-induced (quasirelativistic) bound states: Harmonic confinement and the finite well
Nonlocal Hamiltonian-type operators, like e.g. fractional and
quasirelativistic, seem to be instrumental for a conceptual broadening of
current quantum paradigms. However physically relevant properties of related
quantum systems have not yet received due (and scientifically undisputable)
coverage in the literature. In the present paper we address
Schr\"{o}dinger-type eigenvalue problems for , where a kinetic term
is a quasirelativistic energy operator of mass particle. A potential we assume
to refer to the harmonic confinement or finite well of an arbitrary depth. We
analyze spectral solutions of the pertinent nonlocal quantum systems with a
focus on their -dependence. Extremal mass regimes for eigenvalues and
eigenfunctions of are investigated: (i) spectral affinity
("closeness") with the Cauchy-eigenvalue problem () and (ii) spectral affinity with the nonrelativistic eigenvalue
problem (). To this end we generalize to
nonlocal operators an efficient computer-assisted method to solve
Schr\"{o}dinger eigenvalue problems, widely used in quantum physics and quantum
chemistry. A resultant spectrum-generating algorithm allows to carry out all
computations directly in the configuration space of the nonlocal quantum
system. This allows for a proper assessment of the spatial nonlocality impact
on simulation outcomes. Although the nonlocality of might seem to stay in
conflict with various numerics-enforced cutoffs, this potentially serious
obstacle is kept under control and effectively tamed.Comment: 23 pages, 16 figure
Amplification of non-Markovian decay due to bound state absorption into continuum
It is known that quantum systems yield non-exponential (power law) decay on
long time scales, associated with continuum threshold effects contributing to
the survival probability for a prepared initial state. For an open quantum
system consisting of a discrete state coupled to continuum, we study the case
in which a discrete bound state of the full Hamiltonian approaches the energy
continuum as the system parameters are varied. We find in this case that at
least two regions exist yielding qualitatively different power law decay
behaviors; we term these the long time `near zone' and long time `far zone.' In
the near zone the survival probability falls off according to a power
law, and in the far zone it falls off as . We show that the timescale
separating these two regions is inversely related to the gap between the
discrete bound state energy and the continuum threshold. In the case that the
bound state is absorbed into the continuum and vanishes, then the time scale
diverges and the survival probability follows the power law even
on asymptotic scales. Conversely, one could study the case of an anti-bound
state approaching the threshold before being ejected from the continuum to form
a bound state. Again the power law dominates precisely at the point of
ejection.Comment: 15 pages, 9 figure
Time-dependent stabilization in AdS/CFT
We consider theories with time-dependent Hamiltonians which alternate between
being bounded and unbounded from below. For appropriate frequencies dynamical
stabilization can occur rendering the effective potential of the system stable.
We first study a free field theory on a torus with a time-dependent mass term,
finding that the stability regions are described in terms of the phase diagram
of the Mathieu equation. Using number theory we have found a compactification
scheme such as to avoid resonances for all momentum modes in the theory. We
further consider the gravity dual of a conformal field theory on a sphere in
three spacetime dimensions, deformed by a doubletrace operator. The gravity
dual of the theory with a constant unbounded potential develops big crunch
singularities; we study when such singularities can be cured by dynamical
stabilization. We numerically solve the Einstein-scalar equations of motion in
the case of a time-dependent doubletrace deformation and find that for
sufficiently high frequencies the theory is dynamically stabilized and big
crunches get screened by black hole horizons.Comment: LaTeX, 38 pages, 13 figures. V2: appendix C added, references added
and typos correcte
Periodic excitations of bilinear quantum systems
A well-known method of transferring the population of a quantum system from
an eigenspace of the free Hamiltonian to another is to use a periodic control
law with an angular frequency equal to the difference of the eigenvalues. For
finite dimensional quantum systems, the classical theory of averaging provides
a rigorous explanation of this experimentally validated result. This paper
extends this finite dimensional result, known as the Rotating Wave
Approximation, to infinite dimensional systems and provides explicit
convergence estimates.Comment: Available online
http://www.sciencedirect.com/science/article/pii/S000510981200286
Simultaneous local exact controllability of 1D bilinear Schr\"odinger equations
We consider N independent quantum particles, in an infinite square potential
well coupled to an external laser field. These particles are modelled by a
system of linear Schr\"odinger equations on a bounded interval. This is a
bilinear control system in which the state is the N-tuple of wave functions.
The control is the real amplitude of the laser field. For N=1, Beauchard and
Laurent proved local exact controllability around the ground state in arbitrary
time. We prove, under an extra generic assumption, that their result does not
hold in small time if N is greater or equal than 2. Still, for N=2, we prove
using Coron's return method that local controllability holds either in
arbitrary time up to a global phase or exactly up to a global delay. We also
prove that for N greater or equal than 3, local controllability does not hold
in small time even up to a global phase. Finally, for N=3, we prove that local
controllability holds up to a global phase and a global delay
- …