Normal forms for sub-Lorentzian metrics supported on Engel type distributions

We construct normal forms for Lorentzian metrics on Engel distributions under the assumption that abnormal curves are timelike future directed Hamiltonian geodesics. Then we indicate some cases in which the abnormal timelike future directed curve initiating at the origin is geometrically optimal. We also give certain estimates for reachable sets from a point.     

Stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems

The stochastic optimal control of partially observable nonlinear quasi-integrable Hamiltonian systems is investigated. First, the stochastic optimal control problem of a partially observable nonlinear quasi-integrable Hamiltonian system is converted into that of a completely observable linear system based on a theorem due to Charalambous and Elliot. Then, the converted stochastic optimal control problem is solved by applying the stochastic averaging method and the stochastic dynamical programming principle....     

Generic fractal structure of finite parts of trajectories of piecewise smooth Hamiltonian systems

Piecewise smooth Hamiltonian systems with tangent discontinuity are studied. A new phenomenon is discovered, namely, the generic chaotic behavior of finite parts of trajectories. The approach is to consider the evolution of Poisson brackets for smooth parts of the initial Hamiltonian system. It turns out that, near second-order singular points lying on a discontinuity stratum of codimension two, the system of Poisson brackets is reduced to the Hamiltonian system of the Pontryagin Maximum Principle. The corresponding optimization problem is studied and the topological structure of its optimal trajectories is constructed (optimal synthesis). The synthesis contains countably many periodic solutions on the quotient space by the scale group and a Cantor-like set of nonwandering points (NW) having fractal Hausdorff dimension. The dynamics of the system is described by a topological Markov chain. The entropy is evaluated, together with bounds for the Hausdorff and box dimension of (NW).     

Search for and elimination of dependences between vibrational coordinates in modeling spectra of large molecules

We have investigated an algorithm allowing us to reliably identify an arbitrary number of complex linear dependences between vibrational coordinates in a molecular model of very high dimensionality. These dependences are eliminated in the step for diagonalization of the kinetic part of the vibrational Hamiltonian. We have carried out computer experiments allowing us to propose optimal rules for designing appropriate computer programs for working with a vibrational Hamiltonian of very high dimensionality. __________ Translated from Zhurnal Prikladnoi Spektroskopii, Vol. 73, No. 5, pp. 561–565, September–October, 2006.     

Optimal truncation procedures in renormalization-group calculations

Renormalization or rescaling transformations generally produce more complicated interactions than are present in the initial Hamiltonian. After each rescaling it is necessary to truncate the Hamiltonian to make the next rescaling mathematically tractable. One is faced with the problem of choosing the coupling constants of the truncated Hamiltonian to obtain the best approximation. Following ideas of McMillan, we consider truncation procedures which give lower and upper bounds to the free energy. Conditions for optimal lower- and upper-bound truncations are derived. These optimal truncations are seen to yield exact results for the free energy in both the high- and low-temperature limits. Some of the problems inherent in all renormalization transformations that incorporate an optimal lower- or upper-bound truncation are discussed. Calculations for the twodimensional Ising model based on renormalization transformations which combine decimation and an optimal truncation are described. Even in the simplest approximation in which only nearest-neighbor interactions are retained the free energy is obtained to an accuracy of better than 1% for all temperatures if an optimal truncation rather than an ordinary truncation with no readjustment of the coupling constants is made. However, the simplest calculations involving optimal truncations are less successful in predicting derivatives of the free energy and critical exponents than the free energy itself.     

Hölder Continuity of the Integrated Density of States for the Fibonacci Hamiltonian

We obtain the asymptotics of the optimal global Hölder exponent of the integrated density of states of the Fibonacci Hamiltonian for large and small couplings.     

Maximum Speedup in Quantum Search: O(1) Running Time

We consider the running time of the generalized quantum search Hamiltonian. We provide the surprising result that the maximum speedup of quantum search in the generalized Hamiltonian is an O(1) running time regardless of the number of total states. This seems to violate the proof of Zalka that the quadratic speedup is optimal in quantum search. However the argument of Giovannetti et al. that a quantum speedup comes from the interaction between subsystems (or, equivalently entanglement) (and is concerned with the Margolus and Levitin theorem) supports our result.     

Bifurcation structures and dominant modes near relative equilibria in the one-dimensional discrete nonlinear Schrödinger equation

We investigate the bifurcation structure of a family of relative equilibria of a ring of seven oscillators described by the discrete nonlinear Schrödinger equation (DNLSE) when the period of these orbits and a suitable defect act as bifurcation parameters. We find a reduced Hamiltonian that gives substantial insight into the dynamics of this system. The convexity of this Hamiltonian at given nonresonant equilibria supports the stability of nearby quasiperiodic solutions. We show that the local loss of convexity in the reduced Hamiltonian is determined by the Hessian of its integrable part in the family of relative equilibria under study. Stable quasiperiodic solutions are studied by considering the power spectral densities of a set of suitable fast and slow actions, whose origin is suggested by the averaging principle. We also show that the return times form an optimal embedding to characterize the system dynamics. We show that the power spectral density of a suitable interference signal, arising from a ring of Bose-Einstein condensates and described by the DNLSE, has a single prominent peak at the breather-like relative equilibria.     

Optimal control for unitary preparation of many-body states: application to Luttinger liquids

Many-body ground states can be prepared via unitary evolution in cold atomic systems. Given the initial state and a fixed time for the evolution, how close can we get to a desired ground state if we can tune the Hamiltonian in time? Here we study this optimal control problem focusing on Luttinger liquids with tunable interactions. We show that the optimal protocol can be obtained by simulated annealing. We find that the optimal interaction strength of the Luttinger liquid can have a nonmonotonic time dependence. Moreover, the system exhibits a marked transition when the ratio τ/L of the preparation time to the system size exceeds a critical value. In this regime, the optimal protocols can prepare the states with almost perfect accuracy. The optimal protocols are robust against dynamical noise.     

Thermal Quantum Discord in Pure Dzyaloshinskii-Moriya Model with Magnetic Field

We investigate the effects of the directions of Dzyaloshinskii-Moriya (DM) interaction vector and magnetic field on the quantum discord in the pure DM model. For different directions of DM vector, we find that there are different optimal parameter components of magnetic field. Moreover, we find that the optimal parameter components rules are the same for the Hamiltonian H₁ and H₂. According to the rules, for a certain axial DM vector, we can get the maximal quantum discord by adjusting the direction of the external magnetic field, which is feasible under the current experimental technology.     

Optimal block-tridiagonalization of matrices for coherent charge transport

Numerical quantum transport calculations are commonly based on a tight-binding formulation. A wide class of quantum transport algorithms require the tight-binding Hamiltonian to be in the form of a block-tridiagonal matrix. Here, we develop a matrix reordering algorithm based on graph partitioning techniques that yields the optimal block-tridiagonal form for quantum transport. The reordered Hamiltonian can lead to significant performance gains in transport calculations, and allows to apply conventional two-terminal algorithms to arbitrarily complex geometries, including multi-terminal structures. The block-tridiagonalization algorithm can thus be the foundation for a generic quantum transport code, applicable to arbitrary tight-binding systems. We demonstrate the power of this approach by applying the block-tridiagonalization algorithm together with the recursive Green’s function algorithm to various examples of mesoscopic transport in two-dimensional electron gases in semiconductors and graphene.     

A New Scheme of Integrability for (bi)Hamiltonian PDE

We develop a new method for constructing integrable Hamiltonian hierarchies of Lax type equations, which combines the fractional powers technique of Gelfand and Dickey, and the classical Hamiltonian reduction technique of Drinfeld and Sokolov. The method is based on the notion of an Adler type matrix pseudodifferential operator and the notion of a generalized quasideterminant. We also introduce the notion of a dispersionless Adler type series, which is applied to the study of dispersionless Hamiltonian equations. Non-commutative Hamiltonian equations are discussed in this framework as well.     

Optimal Lyapunov quantum control of two-level systems: Convergence and extended techniques

Taking a two-level system as an example, we show that a strong control field may enhance the efficiency of optimal Lyapunov quantum control but could decrease its control fidelity. A relationship between the strength of the control field and the control fidelity is established. An extended technique, which combines free evolution and external control, is proposed to improve the control fidelity. We analytically demonstrate that the extended technique can be used to design a control law for steering a two-level system exactly to one predetermined eigenstate of the free Hamiltonian. In such a way, the convergence of the extended optimal Lyapunov quantum control can be guaranteed.     

Shape dynamics in 2 + 1 dimensions

Shape Dynamics is a formulation of General Relativity where refoliation invariance is traded for local spatial conformal invariance. In this paper we explicitly construct Shape Dynamics for a torus universe in 2 + 1 dimensions through a linking gauge theory that ensures dynamical equivalence with General Relativity. The Hamiltonian we obtain is formally a reduced phase space Hamiltonian. The construction of the Shape Dynamics Hamiltonian on higher genus surfaces is not explicitly possible, but we give an explicit expansion of the Shape Dynamics Hamiltonian for large CMC volume. The fact that all local constraints are linear in momenta allows us to quantize these explicitly under a certain assumption on the kinematic Hilbert space, and the quantization problem for Shape Dynamics turns out to be equivalent to reduced phase space quantization. We consider the large CMC-volume asymptotics of conformal transformations of the wave function. We then discuss the similarity of Shape Dynamics on the 2-torus with the explicitly constructible strong gravity Shape Dynamics Hamiltonian in higher dimensions.     

Phase transitions in social networks

We study a model of network with clustering and desired node degree. The original purpose of the model was to describe optimal structures of scientific collaboration in the European Union. The model belongs to the family of exponential random graphs. We show by numerical simulations and analytical considerations how a very simple Hamiltonian can lead to surprisingly complicated and eventful phase diagram.     

Time optimal quantum control of two-qubit systems

We study the optimal quantum control of heteronuclear two-qubit systems described by a Hamiltonian containing both nonlocal internal drift and local control terms.We derive an explicit formula to compute the minimum time required to steer the system from an initial state to a specified final state.As applications the minimal time to implement Controlled-NOT gate,SWAP gate and Controlled-U gate is calculated in detail.The experimental realizations of these quantum gates are explicitly presented.     

Dynamics of Confined Crowd Modelled Using Fermionic Operators

An operatorial method based on fermionic operators is used to describe the dynamics of a crowd made of different kind of populations mutually interacting and moving in a two–dimensional bounded closed region. The densities of the populations are recovered through the Heisenberg equation and the diffusion process is driven by the Hamiltonian operator defined by requiring that the populations move along optimal paths. We apply the model obtained in a concrete situation and we discuss the effect of the interaction between the populations during their motion.     

An exactly solvable model for the large polaron

We present an exactly diagonalizable model Hamiltonian for the large polaron derived by analyzing the variational ansatz by Haga-Larsen (HL) for the Fröhlich Hamiltonian. The lowest energy eigenvalue of the model Hamiltonian for fixed wave numbers reproduces the energy of the variational ansatz by Haga-Larsen and is, therefore, an upper bound with respect to the corresponding energy eigenvalue of the Fröhlich Hamiltonian. This is valid for any momentum which is proven by extending the Haga-Larsen approach. Furthermore, since all integrations can be performed analytically, the model Hamiltonian is easily tractable. The energy eigenvalue spectrum of the model Hamiltonian is studied below and above the phonon-emission threshold. The quality of the model Hamiltonian is determined by the variational ansatz of Haga and Larsen. Incorporating an improved energy-momentum relation, a generalized model Hamiltonian is derived possessing a larger validity range with respect to the coupling strength. Furthermore, a second exactly diagonalizable model Hamiltonian based on improved Wigner-Brillouin perturbation theory due to Warmenbol, Peeters, and Devreese (WPD) is presented. It is briefly demonstrated that one is able to construct all mentioned model Hamiltonians also in the 2D polaron problem. In contrast to the 3D case, where the HL-type model Hamiltonian possesses the higher quality for any momentum, in the 2D case, it works well only for small momenta. For large momenta, only the WPD-type model Hamiltonian describes the energy-momentum relation correctly. We demonstrate the usefulness of the model Hamiltonian concept by exactly calculating the one-electron Green’s function for all mentioned model Hamiltonians and comment why significant advantages of the model Hamilton concept for the treating of low-dimensional systems (planar semiconducting quantum-well structures) can be expected.     

Covariant Hamiltonian Field Theory: Path Integral Quantization

The Hamiltonian counterpart of classical Lagrangian field theory is covariant Hamiltonian field theory where momenta correspond to derivatives of fields with respect to all world coordinates. In particular, classical Lagrangian and covariant Hamiltonian field theories are equivalent in the case of a hyperregular Lagrangian, and they are quasi-equivalent if a Lagrangian is almost-regular. In order to quantize covariant Hamiltonian field theory, one usually attempts to construct and quantize a multisymplectic generalization of the Poisson bracket. In the present work, the path integral quantization of covariant Hamiltonian field theory is suggested. We use the fact that a covariant Hamiltonian field system is equivalent to a certain Lagrangian system on a phase space which is quantized in the framework of perturbative quantum field theory. We show that, in the case of almost-regular quadratic Lagrangians, path integral quantizations of associated Lagrangian and Hamiltonian field systems are equivalent.