Drift Hamiltonian Research Articles

Experimental implementation of quantum gates through actuator qubits

Universal quantum computation requires the implementation of arbitrary control operations on the quantum register. In most cases, this is achieved by external control fields acting selectively on each qubit to drive single-qubit operations. In combination with a drift Hamiltonian containing interactions between the qubits, this allows the implementation of arbitrary gate operations. Here, we demonstrate an alternative scheme that does not require local control for all qubits: we implement one- and two-qubit gate operations on a set of target qubits indirectly, through a combination of gates on directly controlled actuator qubits with a drift Hamiltonian that couples actuator and target qubits. Experiments are performed on nuclear spins, using radio-frequency pulses as gate operations and magnetic-dipole couplings for the drift Hamiltonian.

Mutual influence of locality and chaotic dynamics on quantum controllability

Quantum control tasks are classified either as classical-like or as quantum requiring interference of pathways. We study the generation of interference pathways and relate them to the fidelity of the control target at a fixed time for various tasks. The model drift Hamiltonian studied is the two-dimensional Henon-Heiles (HH) potential. This system shows regular classical dynamics for low energies and chaotic dynamics for higher energies. A control operator supported by the whole momentum space and therefore connecting the entire Hilbert phase space is a random spiky potential. The other extreme is a smooth control potential. Intermediate cases are obtained by filtering the random spiky potential in momentum space. The fidelity of achieving a control task was related to the connectivity in phase space of the control operators. Typical quantum tasks such as generating a superposition of generalized coherent states rely on interfering pathways. For these cases the nonlinearity in the drift or control Hamiltonian is a necessary requirement for creating interferences. Control over rapidly diverging components of the wave function is achieved by the use of highly nonlocal control operators. Quantum control under chaotic drift was found to give a better yield than control under regular dynamics for such cases. For classical tasks we study the transformation of an initial generalized coherent state to another one. The best fidelity is obtained for regular or harmonic regions of the potential and smooth control operators. The approach to the classical limit is checked by decreasing the effective value of $\ensuremath{\hbar}$. Control under both quantum and classical tasks suffered from the decrease of $\ensuremath{\hbar}$ and the approach to classical proximity. Classical control tasks which rely heavily on maintaining a generalized coherent state throughout the evolution were found to be dysfunctional and lead to a completely uncontrolled situation once the classical chaos starts to appear.

Optimized pulses for the control of uncertain qubits

Constructing high-fidelity control fields that are robust to control, system, and/or surrounding environment uncertainties is a crucial objective for quantum information processing. Using the two-state Landau-Zener model for illustrative simulations of a controlled qubit, we generate optimal controls for \pi/2- and \pi-pulses, and investigate their inherent robustness to uncertainty in the magnitude of the drift Hamiltonian. Next, we construct a quantum-control protocol to improve system-drift robustness by combining environment-decoupling pulse criteria and optimal control theory for unitary operations. By perturbatively expanding the unitary time-evolution operator for an open quantum system, previous analysis of environment-decoupling control pulses has calculated explicit control-field criteria to suppress environment-induced errors up to (but not including) third order from \pi/2- and \pi-pulses. We systematically integrate this criteria with optimal control theory, incorporating an estimate of the uncertain parameter, to produce improvements in gate fidelity and robustness, demonstrated via a numerical example based on double quantum dot qubits. For the qubit model used in this work, post facto analysis of the resulting controls suggests that realistic control-field fluctuations and noise may contribute just as significantly to gate errors as system and environment fluctuations.

Electron Monopole Duality in Quantum Hall Effect

Starting from the duality between electric and magnetic field, we have made an attempt to discuss the quantum hall effect from the consideration of magnetic monopole in view of electron monopole duality. Starting from the dual dy-namics of electric and magnetic charges, we have reformulated a consistent theory of quantum hall effect in presence of monopole. Speculating the existence of magnetic monopoles in magnetic materials (metals), we have accordingly modi-fied the parameters; like drift velocity, current density, Hamiltonian and eigen values and eigen function for harmonic oscillator; resposible to examine the quantum Hall effect in metals.

Extensions of adiabatic invariant theory for a charged particle

The standard theory of Hamiltonian dynamics of a charged particle is extended to allow electric and magnetic fields to vary across magnetic field lines or surfaces on the Larmor radius distance scale. After the development of the general theory, the special cases of toroidally nested magnetic surfaces and of axisymmetry are considered. In a further restriction the situation with spatially slowly varying static magnetic fields but spatially rapidly varying static bounded electrostatic potentials is treated. The dynamics of the perpendicular velocity is represented by a nonlinear oscillator. The adiabatic invariant and drift Hamiltonian are constructed near an O point in the perpendicular velocity phase plane. Motion near a separatrix and X point in physical space is also briefly explored.

Nonsingular canonical coordinates for the drift Hamiltonian in a magnetic field with a separatrix

Magnetic coordinates are singular on a separatrix. A method is given for transforming to nonsingular coordinates that are closely related to magnetic coordinates and can be used in the canonical coordinates of the Hamiltonian for particle drifts. The case of a magnetic separatrix with a single X point is emphasized and approximate expressions are given for the functions that appear in the drift Hamiltonian.

The relativistic drift Hamiltonian

The relativistic Hamiltonian for the guiding center motion of particles in electric and magnetic fields is given in its canonical form. This drift Hamiltonian is valid if the particle gyroradius is sufficiently small compared to the spatial scale, and the cyclotron frequency is sufficiently large compared to the temporal scale, of the magnetic and electric fields. A method for determining the required magnetic coordinates for an earth-like magnetic field is also given.

The exact and drift Hamiltonian in a toroidal magnetic field

The exact Hamiltonian and the first-order corrections to the standard drift Hamiltonian are given in magnetic coordinates for a full static electromagnetic field with magnetic surfaces. The exact Hamiltonian depends on both the field strength and the shape of the magnetic surfaces (the metric of the magnetic coordinates) while the standard drift Hamiltonian depends on the field strength only. The first-order correction to the standard drift Hamiltonian, in gyroradius to system size, depends in a generic way on the metric of the magnetic coordinates, as does the exact Hamiltonian. The finite-Larmor-radius effect and the polarization drift are in the first-order corrections.

The exact and drift Hamiltonian

A canonical Hamiltonian treatment for the motion of the guiding center of a charged particle in a magnetic field of a toroidal configuration is given. The Hamiltonian of the exact trajectory, first- and second-order corrections to the standard drift Hamiltonian are given in the Boozer coordinates for a vacuum magnetic field with magnetic surfaces. The exact Hamiltonian depends on both the field strength and the shape of the magnetic surfaces (the metric of the magnetic coordinates) while the standard drift Hamiltonian depends only on the field strength. The first-order correction, in gyroradius to system size, to the standard drift Hamiltonian depends in a generic way on the metric of the magnetic coordinates, as does the exact Hamiltonian.

The drift Hamiltonian in a magnetic field with a separatrix

In a plasma equilibrium, the simplest coordinates to use for the calculation of particle drift trajectories are magnetic coordinates. Unfortunately, these coordinates are singular on a magnetic separatrix, which makes their use impractical in this region. Here a coordinate system that is nonsingular on a magnetic separatrix while retaining most of the attractive features of magnetic coordinates is developed. In particular, a simple expression for the Hamiltonian for particle drift trajectories is obtained and a simple prescription for making quantities, such as the electric potential, constant on the magnetic surfaces is given.

Monte Carlo calculations for transport due to MHD modes

The three basic mechanisms that produce either classical or anomalous transport are spatial variation of magnetic field strength, spatial variation of electrostatic potential in magnetic surfaces, and loss of magnetic surfaces. A Monte Carlo code is written to study transport due to these three mechanisms interacting with collisional effects. The equations of motion are obtained from the canonical drift Hamiltonian, but non-canonical co-ordinates are used to simplify the integrations. The code is applied to the reversed-field-pinch ZT-40 and the Tokapole II. For ZT-40 the Bessel-function model is used to represent the magnetic field geometry. The effects of pitch-angle scattering, loop voltage and the break-up of magnetic surfaces resulting from resistive MHD perturbations on the drift particle trajectories are illustrated. The particle diffusion coefficients are obtained for varying amplitudes of resistive MHD perturbations. For Tokapole II the spectrum of both the ideal and resistive MHD perturbations is constructed from the experimental data. The drift trajectories for trapped and passing electrons in the presence of such perturbations are obtained. The particle diffusion coefficients for the neo-classical regime in Tokapole II are obtained for varying collision frequency. By comparing the transport coefficients for various groups of particles with the experimental data, we hope to obtain far more information on the transport mechanisms than can be obtained by the standard confinement time measurements. The various groups of particles that can be studied using the code include runaway electrons, thermal electrons, and both passing and trapped diagnostic beam ions.