Optimal control in a quantum cooling problem

The optimal control for cooling a quantum harmonic oscillator by controlling its frequency is considered. It is shown that this singular problem may be transformed with the proper choice of coordinates to an equivalent problem which is no longer singular. The coordinates used are sufficiently simple that a graphical solution is possible and eliminates the need to use a Weierstrass-like approach to show optimality. The optimal control of this problem is of significance in connection with cooling physical systems to low temperatures. It is also mathematically significant in showing the power and limitations of coordinate transformations for attacking apparently singular problems.     

Lyapunov control of a quantum particle in a decaying potential

A Lyapunov-based approach for the trajectory generation of an N  -dimensional Schrödinger equation in whole RN${\mathbb{R}}^N$ is proposed. For the case of a quantum particle in an N-dimensional decaying potential the convergence is precisely analyzed. The free system admitting a mixed spectrum, the dispersion through the absolutely continuous part is the main obstacle to ensure such a stabilization result. Whenever, the system is completely initialized in the discrete part of the spectrum, a Lyapunov strategy encoding both the distance with respect to the target state and the penalization of the passage through the continuous part of the spectrum, ensures the approximate stabilization.     

Pairwise correlations via quantum discord and its geometric measure in a four-qubit spin chain

The dynamic of pairwise correlations, including quantum entanglement (QE) and discord (QD) with geometric measure of quantum discord (GMQD), are shown in the four-qubit Heisenberg XX spin chain. The results show that the effect of the entanglement degree of the initial state on the pairwise correlations is stronger for alternate qubits than it is for nearest-neighbor qubits. This parameter results in sudden death for QE, but it cannot do so for QD and GMQD. With different values for this entanglement parameter of the initial state, QD and GMQD differ and are sensitive for any change in this parameter. It is found that GMQD is more robust than both QD and QE to describe correlations with nonzero values, which offers a valuable resource for quantum computation.     

Monotonic time-discretized schemes in quantum control

Most of the numerical simulations in quantum (bilinear) control have used monotonically convergent algorithms of Krotov (introduced by Tannor et al. [15]), of Zhu & Rabitz [16] or their unified formulation in [17]. However, the properties of the discrete version of these procedures have not been yet tackled with. We present in this paper a stable time and space discretization which preserves the monotonic properties of the monotonic algorithms. Numerical results show that the newly derived algorithms are stable and enable various experimentations.     

Modelling of quantum mechanical control systems

A program of studies of quantum mechanical control systems is initiated. Following the historical development of quantum mechanics, the quantum control model is obtained from a corresponding classical structure. Second order linear and bilinear control systems as well as first order linear control systems are investigated; it is further demonstrated that the analysis may be extended to some nonlinear control problems. The results derived for these systems form an interesting example of the general theory of quantization.     

Tunneling through a quantum dot in a quantum waveguide

The problem is considered of scattering in a system consisting of a quantum waveguide and a quantum dot weakly coupled to the waveguide. It is assumed that the quantum waveguide is described by the Pauli equations, and the Rashba spin-orbit interaction is taken into account. The possibility of tunneling through the quantum dot is proved.     

Selected topics in dynamics and control of open quantum systems

This work reviews several topics in the non-equilibrium dynamics and control of open quantum systems, including the dynamics of the system and the environment in the weak coupling and low density regimes, incoherent control, production of universal Kraus maps and engineering of arbitrary pure and mixed quantum states.     

Methods of geometric control theory for quantum computations

The applications of geometric control theory methods on Lie groups and homogeneous spaces to the theory of quantum computations are investigated. It is shown that these methods are very useful for the problem of constructing a universal set of gates for quantum computations. The well-known result that the set of all one-bit gates, together with no more than one two-bit gate, is universal is considered from the control theory viewpoint. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 27, Algebra and Geometry, 2005.     

Application of quantum stochastic calculus to optimal control

Given a plant whose output is described by a Hudson-Parthasarathy quantum stochastic differential equation [1–3] driven by standard quantum Brownian motion, we compute explicitly the control process that rapidly makes the size of the plant's output small, and keeps the energy used at a minimum. The solution to the quantum stochastic analogue of the linear regulator problem of classical stochastic control theory ([4], [5]) follows as a special case.Published in Matematicheskie Zametki, Vol. 53, No. 5, pp. 48–56, May, 1993.     

Embedding a quantum rank three quadric in a quantum P3

We continue the classification, begun in [11], [14] and [12], of quadratic Artin-Schelter regular algebras of global dimension 4 which map onto a twisted homogeneous coordinate ring of a quadric hypersurfcice in P³. In this paper, we consider those cases where the quadric has rank 3. We also give sufficient conditions for the point scheme of any quadratic regular algebra of global dimension 4 to be the graph of an automorphism.     

Approximation of quantum tori by finite quantum tori for the quantum Gromov-Hausdorff distance

We establish that, given a compact Abelian group G endowed with a continuous length function l and a sequence (H_n)_n∈N of closed subgroups of G converging to G for the Hausdorff distance induced by l, then is the quantum Gromov-Hausdorff limit of any sequence for the natural quantum metric structures and when the lifts of σ_n to converge pointwise to σ. This allows us in particular to approximate the quantum tori by finite-dimensional C^*-algebras for the quantum Gromov-Hausdorff distance. Moreover, we also establish that if the length function l is allowed to vary, we can collapse quantum metric spaces to various quotient quantum metric spaces.     

Pair correlations in a multicomponent fluid placed in a porous medium

We consider a multicomponent fluid placed in a porous medium. The Ornstein—Zernike approximation is used to calculate the pair correlation functions for density fluctuations in the mixture components. We show that light scattering in the neighborhood of the critical state of the system is determined (in the single-scattering approximation) by the commonly known Ornstein—Zernike formula. We investigate the shift in the critical parameters due to the porous medium. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 146, No. 3, pp. 525–528, March, 2006.     

Formulation and numerical solution of finite-level quantum optimal control problems

Optimal control of finite-level quantum systems is investigated, and iterative solution schemes for the optimization of a control representing laser pulses are developed. The purpose of this external field is to channel the system's wavefunction between given states in its most efficient way. Physically motivated constraints, such as limited laser resources or population suppression of certain states, are accounted for through an appropriately chosen cost functional. First-order necessary optimality conditions and second-order sufficient optimality conditions are investigated. For solving the optimal control problems, a cascadic non-linear conjugate gradient scheme and a monotonic scheme are discussed. Results of numerical experiments with a representative finite-level quantum system demonstrate the effectiveness of the optimal control formulation and efficiency and robustness of the proposed approaches.     

Approximation of quantum Lévy processes by quantum random walks

Every quantum Lévy process with a bounded stochastic generator is shown to arise as a strong limit of a family of suitably scaled quantum random walks.