55 research outputs found
Monotonically convergent optimal control theory of quantum systems under a nonlinear interaction with the control field
We consider the optimal control of quantum systems interacting non-linearly
with an electromagnetic field. We propose new monotonically convergent
algorithms to solve the optimal equations. The monotonic behavior of the
algorithm is ensured by a non-standard choice of the cost which is not
quadratic in the field. These algorithms can be constructed for pure and
mixed-state quantum systems. The efficiency of the method is shown numerically
on molecular orientation with a non-linearity of order 3 in the field.
Discretizing the amplitude and the phase of the Fourier transform of the
optimal field, we show that the optimal solution can be well-approximated by
pulses that could be implemented experimentally.Comment: 24 pages, 11 figure
Control of trapped-ion quantum states with optical pulses
We present new results on the quantum control of systems with infinitely
large Hilbert spaces. A control-theoretic analysis of the control of trapped
ion quantum states via optical pulses is performed. We demonstrate how resonant
bichromatic fields can be applied in two contrasting ways -- one that makes the
system completely uncontrollable, and the other that makes the system
controllable. In some interesting cases, the Hilbert space of the
qubit-harmonic oscillator can be made finite, and the Schr\"{o}dinger equation
controllable via bichromatic resonant pulses. Extending this analysis to the
quantum states of two ions, a new scheme for producing entangled qubits is
discovered.Comment: Submitted to Physical Review Letter
Monotonically convergent optimal control theory of quantum systems with spectral constraints on the control field
We propose a new monotonically convergent algorithm which can enforce
spectral constraints on the control field (and extends to arbitrary filters).
The procedure differs from standard algorithms in that at each iteration the
control field is taken as a linear combination of the control field (computed
by the standard algorithm) and the filtered field. The parameter of the linear
combination is chosen to respect the monotonic behavior of the algorithm and to
be as close to the filtered field as possible. We test the efficiency of this
method on molecular alignment. Using band-pass filters, we show how to select
particular rotational transitions to reach high alignment efficiency. We also
consider spectral constraints corresponding to experimental conditions using
pulse shaping techniques. We determine an optimal solution that could be
implemented experimentally with this technique.Comment: 16 pages, 4 figures. To appear in Physical Review
Reduced-basis Output Bound Methods for Parametrised Partial Differential Equations
An efficient and reliable method for the prediction of outputs of interest of partial differential equations with affine parameter dependence is presented. To achieve efficiency we employ the reduced-basis method: a weighted residual Galerkin-type method, where the solution is projected onto low-dimensional spaces with certain problem-specific approximation properties. Reliability is obtained by a posteriori error estimation methods - relaxations of the standard error-residual equation that provide inexpensive but sharp and rigorous bounds for the error in outputs of interest. Special affine parameter dependence of the differential operator is exploited to develop a two-stage off-line/on-line blackbox computational procedure. In the on-line stage, for every new parameter value, we calculate the output of interest and an associated error bound. The computational complexity of the on-line stage of the procedure scales only with the dimension of the reduced-basis space and the parametric complexity of the partial differential operator; the method is thus ideally suited for the repeated and rapid evaluations required in the context of parameter estimation, design, optimization, and real-time control. The theory and corroborating numerical results are presented for: symmetric coercive problems (e.g. problems in conduction heat transfer), parabolic problems (e.g. unsteady heat transfer), noncoercive problems (e.g. the reduced-wave, or Helmholtz, equation), the Stokes problem (e.g flow of highly viscous fluids), and certain nonlinear equations (e.g. eigenvalue problems)
Enriching some recent coincidence theorems for nonlinear contractions in ordered metric spaces
Sliding mode control of quantum systems
This paper proposes a new robust control method for quantum systems with
uncertainties involving sliding mode control (SMC). Sliding mode control is a
widely used approach in classical control theory and industrial applications.
We show that SMC is also a useful method for robust control of quantum systems.
In this paper, we define two specific classes of sliding modes (i.e.,
eigenstates and state subspaces) and propose two novel methods combining
unitary control and periodic projective measurements for the design of quantum
sliding mode control systems. Two examples including a two-level system and a
three-level system are presented to demonstrate the proposed SMC method. One of
main features of the proposed method is that the designed control laws can
guarantee desired control performance in the presence of uncertainties in the
system Hamiltonian. This sliding mode control approach provides a useful
control theoretic tool for robust quantum information processing with
uncertainties.Comment: 18 pages, 4 figure
Normal Cones and Thompson Metric
The aim of this paper is to study the basic properties of the Thompson metric
in the general case of a real linear space ordered by a cone . We
show that has monotonicity properties which make it compatible with the
linear structure. We also prove several convexity properties of and some
results concerning the topology of , including a brief study of the
-convergence of monotone sequences. It is shown most of the results are
true without any assumption of an Archimedean-type property for . One
considers various completeness properties and one studies the relations between
them. Since is defined in the context of a generic ordered linear space,
with no need of an underlying topological structure, one expects to express its
completeness in terms of properties of the ordering, with respect to the linear
structure. This is done in this paper and, to the best of our knowledge, this
has not been done yet. The Thompson metric and order-unit (semi)norms
are strongly related and share important properties, as both are
defined in terms of the ordered linear structure. Although and
are only topological (and not metrical) equivalent on , we
prove that the completeness is a common feature. One proves the completeness of
the Thompson metric on a sequentially complete normal cone in a locally convex
space. At the end of the paper, it is shown that, in the case of a Banach
space, the normality of the cone is also necessary for the completeness of the
Thompson metric.Comment: 36 page
Optimal control theory for unitary transformations
The dynamics of a quantum system driven by an external field is well
described by a unitary transformation generated by a time dependent
Hamiltonian. The inverse problem of finding the field that generates a specific
unitary transformation is the subject of study. The unitary transformation
which can represent an algorithm in a quantum computation is imposed on a
subset of quantum states embedded in a larger Hilbert space. Optimal control
theory (OCT) is used to solve the inversion problem irrespective of the initial
input state. A unified formalism, based on the Krotov method is developed
leading to a new scheme. The schemes are compared for the inversion of a
two-qubit Fourier transform using as registers the vibrational levels of the
electronic state of Na. Raman-like transitions through the
electronic state induce the transitions. Light fields are found
that are able to implement the Fourier transform within a picosecond time
scale. Such fields can be obtained by pulse-shaping techniques of a femtosecond
pulse. Out of the schemes studied the square modulus scheme converges fastest.
A study of the implementation of the qubit Fourier transform in the Na
molecule was carried out for up to 5 qubits. The classical computation effort
required to obtain the algorithm with a given fidelity is estimated to scale
exponentially with the number of levels. The observed moderate scaling of the
pulse intensity with the number of qubits in the transformation is
rationalized.Comment: 32 pages, 6 figure
Growth of Sobolev norms and controllability of Schr\"odinger equation
In this paper we obtain a stabilization result for the Schr\"odinger equation
under generic assumptions on the potential. Then we consider the Schr\"odinger
equation with a potential which has a random time-dependent amplitude. We show
that if the distribution of the amplitude is sufficiently non-degenerate, then
any trajectory of system is almost surely non-bounded in Sobolev spaces
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