149 research outputs found
Deterministic submanifolds and analytic solution of the stochastic differential master equation describing a qubit
This paper studies the stochastic differential equation (SDE) associated to a
two-level quantum system (qubit) subject to Hamiltonian evolution as well as
unmonitored and monitored decoherence channels. The latter imply a stochastic
evolution of the quantum state (density operator), whose associated probability
distribution we characterize. We first show that for two sets of typical
experimental settings, corresponding either to weak quantum non demolition
measurements or to weak fluorescence measurements, the three Bloch coordinates
of the qubit remain confined to a deterministically evolving surface or curve
inside the Bloch sphere. We explicitly solve the deterministic evolution, and
we provide a closed-form expression for the probability distribution on this
surface or curve. Then we relate the existence in general of such
deterministically evolving submanifolds to an accessibility question of control
theory, which can be answered with an explicit algebraic criterion on the SDE.
This allows us to show that, for a qubit, the above two sets of weak
measurements are essentially the only ones featuring deterministic surfaces or
curves
Accelerating Consensus by Spectral Clustering and Polynomial Filters
It is known that polynomial filtering can accelerate the convergence towards
average consensus on an undirected network. In this paper the gain of a
second-order filtering is investigated. A set of graphs is determined for which
consensus can be attained in finite time, and a preconditioner is proposed to
adapt the undirected weights of any given graph to achieve fastest convergence
with the polynomial filter. The corresponding cost function differs from the
traditional spectral gap, as it favors grouping the eigenvalues in two
clusters. A possible loss of robustness of the polynomial filter is also
highlighted
Control limitations from distributed sensing: theory and Extremely Large Telescope application
We investigate performance bounds for feedback control of distributed plants
where the controller can be centralized (i.e. it has access to measurements
from the whole plant), but sensors only measure differences between neighboring
subsystem outputs. Such "distributed sensing" can be a technological necessity
in applications where system size exceeds accuracy requirements by many orders
of magnitude. We formulate how distributed sensing generally limits feedback
performance robust to measurement noise and to model uncertainty, without
assuming any controller restrictions (among others, no "distributed control"
restriction). A major practical consequence is the necessity to cut down
integral action on some modes. We particularize the results to spatially
invariant systems and finally illustrate implications of our developments for
stabilizing the segmented primary mirror of the European Extremely Large
Telescope.Comment: submitted to Automatic
Synchronization with partial state coupling on SO(n)
This paper studies autonomous synchronization of k agents whose states evolve
on SO(n), but which are only coupled through the action of their states on one
"reference vector" in Rn for each link. Thus each link conveys only partial
state information at each time, and to reach synchronization agents must
combine this information over time or throughout the network. A natural
gradient coupling law for synchronization is proposed. Extensive convergence
analysis of the coupled agents is provided, both for fixed and time-varying
reference vectors. The case of SO(3) with fixed reference vectors is discussed
in more detail. For comparison, we also treat the equivalent setting in Rn,
i.e. with states in Rn and connected agents comparing scalar product of their
states with a reference vector.Comment: to be submitted to SIAM Journal on Control and Optimizatio
Robust open-loop stabilization of Fock states by time-varying quantum interactions
A quantum harmonic oscillator (spring subsystem) is stabilized towards a
target Fock state by reservoir engineering. This passive and open-loop
stabilization works by consecutive and identical Hamiltonian interactions with
auxiliary systems, here three-level atoms (the auxiliary ladder subsystem),
followed by a partial trace over these auxiliary atoms. A scalar control input
governs the interaction, defining which atomic transition in the ladder
subsystem is in resonance with the spring subsystem. We use it to build a
time-varying interaction with individual atoms, that combines three
non-commuting steps. We show that the resulting reservoir robustly stabilizes
any initial spring state distributed between 0 and 4n+3 quanta of vibrations
towards a pure target Fock state of vibration number n. The convergence proof
relies on the construction of a strict Lyapunov function for the Kraus map
induced by this reservoir setting on the spring subsystem. Simulations with
realistic parameters corresponding to the quantum electrodynamics setup at
Ecole Normale Superieure further illustrate the robustness of the method
Contraction and stability analysis of steady-states for open quantum systems described by Lindblad differential equations
For discrete-time systems, governed by Kraus maps, the work of D. Petz has
characterized the set of universal contraction metrics. In the present paper,
we use this characterization to derive a set of quadratic Lyapunov functions
for continuous-time systems, governed by Lindblad differential equations, that
have a steady-state with full rank. An extremity of this set is given by the
Bures metric, for which the quadratic Lyapunov function is obtained by
inverting a Sylvester equation. We illustrate the method by providing a strict
Lyapunov function for a Lindblad equation designed to stabilize a quantum
electrodynamic "cat" state by reservoir engineering. In fact we prove that any
Lindblad equation on the Hilbert space of the (truncated) harmonic oscillator,
which has a full-rank equilibrium and which has, among its decoherence
channels, a channel corresponding to the photon loss operator, globally
converges to that equilibrium.Comment: Submitted (10 pages, 1 figure
String Stability towards Leader thanks to Asymmetric Bidirectional Controller
This paper deals with the problem of string stability of interconnected
systems with double-integrator open loop dynamics (e.g.~acceleration-controlled
vehicles). We analyze an asymmetric bidirectional linear controller, where each
vehicle is coupled solely to its immediate predecessor and to its immediate
follower with different gains in these two directions. We show that in this
setting, unlike with unidirectional or symmetric bidirectional controllers,
string stability can be recovered when disturbances act only on a small
(-independent) set of leading vehicles. This improves existing results from
the literature with this assumption. We also indicate that string stability
with respect to arbitrarily distributed disturbances cannot be achieved with
this controller.Comment: Version 2 corrects a typo in the proof, and adds the proof of
stability before string stability. Slightly longer than published versio
Convergence and adiabatic elimination for a driven dissipative quantum harmonic oscillator
We prove that a harmonic oscillator driven by Lindblad dynamics where the
typical drive and loss channels are two-photon processes instead of
single-photon ones, converges to a protected subspace spanned by two coherent
states of opposite amplitude. We then characterize the slow dynamics induced by
a perturbative single-photon loss on this protected subspace, by performing
adiabatic elimination in the Lindbladian dynamics.Comment: submitted to IEEE-CDC 201
Adding a single state memory optimally accelerates symmetric linear maps
International audiencePrevious papers have proposed to add memory registers to the dynamics of discrete-time linear systems in order to accelerate their convergence. In particular, it has been proved that adding one memory slot per agent allows faster convergence towards average consensus. We here prove that this situation cannot be improved by adding more memory slots, when the knowledge about the self-adjoint linear map to be accelerated reduces to bounds on its extreme eigenvalues
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