8 research outputs found
Optimal Control of Quantum Dissipative Dynamics: Analytic solution for cooling the three level system
We study the problem of optimal control of dissipative quantum dynamics.
Although under most circumstances dissipation leads to an increase in entropy
(or a decrease in purity) of the system, there is an important class of
problems for which dissipation with external control can decrease the entropy
(or increase the purity) of the system. An important example is laser cooling.
In such systems, there is an interplay of the Hamiltonian part of the dynamics,
which is controllable and the dissipative part of the dynamics, which is
uncontrollable. The strategy is to control the Hamiltonian portion of the
evolution in such a way that the dissipation causes the purity of the system to
increase rather than decrease. The goal of this paper is to find the strategy
that leads to maximal purity at the final time. Under the assumption that
Hamiltonian control is complete and arbitrarily fast, we provide a general
framework by which to calculate optimal cooling strategies. These assumptions
lead to a great simplification, in which the control problem can be
reformulated in terms of the spectrum of eigenvalues of , rather than
itself. By combining this formulation with the Hamilton-Jacobi-Bellman
theorem we are able to obtain an equation for the globaly optimal cooling
strategy in terms of the spectrum of the density matrix. For the three-level
system, we provide a complete analytic solution for the optimal
cooling strategy. For this system it is found that the optimal strategy does
not exploit system coherences and is a 'greedy' strategy, in which the purity
is increased maximally at each instant.Comment: 9 pages, 3 fig
Encoding a qubit into multilevel subspaces
We present a formalism for encoding the logical basis of a qubit into
subspaces of multiple physical levels. The need for this multilevel encoding
arises naturally in situations where the speed of quantum operations exceeds
the limits imposed by the addressability of individual energy levels of the
qubit physical system. A basic feature of the multilevel encoding formalism is
the logical equivalence of different physical states and correspondingly, of
different physical transformations. This logical equivalence is a source of a
significant flexibility in designing logical operations, while the multilevel
structure inherently accommodates fast and intense broadband controls thereby
facilitating faster quantum operations. Another important practical advantage
of multilevel encoding is the ability to maintain full quantum-computational
fidelity in the presence of mixing and decoherence within encoding subspaces.
The formalism is developed in detail for single-qubit operations and
generalized for multiple qubits. As an illustrative example, we perform a
simulation of closed-loop optimal control of single-qubit operations for a
model multilevel system, and subsequently apply these operations at finite
temperatures to investigate the effect of decoherence on operational fidelity.Comment: IOPart LaTeX, 2 figures, 31 pages; addition of a numerical simulatio