Positronium formation in positron-atom collisions at intermediate energies

Bransden, B H; Joachain, Charles; McCann, J. F.

doi:10.1088/0953-4075/25/22/028

Cited by 187 publications

(270 citation statements)

References 25 publications

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“…If this Hamiltonian is time-independent and the system is initially in its ground state, then it will remain in this state. The Adiabatic Theorem (see e. g. [7]) expresses that if the Hamiltonian varies slowly enough, roughly speaking, the state of the system will stay close to the instantaneous ground state of the Hamiltonian at each time t. More specifically, let |E k ; t be the eigenstates of H(t), satisfying…”

Section: Adiabatic Theoremmentioning

confidence: 99%

Quantum search by local adiabatic evolution

2002

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The adiabatic theorem has been recently used to design quantum algorithms of a new kind, where the quantum computer evolves slowly enough so that it remains near its instantaneous ground state which tends to the solution [1]. We apply this time-dependent Hamiltonian approach to the Grover's problem, i. e., searching a marked item in an unstructured database. We find that, by adjusting the evolution rate of the Hamiltonian so as to keep the evolution adiabatic on each infinitesimal time interval, the total running time is of order √ N , where N is the number of items in the database. We thus recover the advantage of Grover's standard algorithm as compared to a classical search, scaling as N . This is in contrast with the constant-rate adiabatic approach developed in [1], where the requirement of adiabaticity is expressed only globally, resulting in a time of order N .

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Section: Adiabatic Theoremmentioning

confidence: 99%

Quantum search by local adiabatic evolution

2002

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“…Using time-independent perturbation theory [20], we can derive from Eq. (1) the equations of motion for all the small eigenvalues.…”

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confidence: 99%

Locally Optimal Control of Quantum Systems with Strong Feedback

Shabani

Jacobs

2008

Phys. Rev. Lett.

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For quantum systems with high purity, we find all observables that, when continuously monitored, maximize the instantaneous reduction in the average linear entropy. This allows us to obtain all locally optimal feedback protocols with strong feedback, and explicit expressions for the best such protocols for systems of size N ≤ 4. We also show that for a qutrit the locally optimal protocol is the optimal protocol for observables with equispaced eigenvalues, providing the first fully optimal feedback protocol for a 3-state system. PACS numbers: 03.65. Yz, 87.19.lr, 02.30.Yy, 03.65.Ta Observation and control of coherent quantum behavior has been realized in a variety of mesoscopic devices [1, 2, 3, 4]. With further refinements, such devices may well form the basis of new technologies, for example in sensing [5] and communication [6,7]. Feedback, in which a system is continuously observed and the information used to control its behavior in the presence of noise, is an important element in the quantum engineer's toolbox [6,7,8,9,10,11]. In view of this, one would like to know the limits on such control, given any relevant limitations on the measurement and/or control forces. However, except in special cases [12], the dynamics of continuously observed quantum systems is nonlinear. Further, results on the quantum-to-classical transition show that this nonlinear dynamics, described by stochastic master equations (SME's), is necessarily every bit as complex (and chaotic) as that of nonlinear classical systems [13]. Because of this, fully general and exact results regarding optimal quantum feedback are unlikely to exist; certainly no such results have been found for nonlinear classical systems [14]. Nevertheless, one would like to obtain results that give insights applicable across a range of systems.Quantum feedback control is implemented by modifying a "control" Hamiltonian, H, that is some part of the system Hamiltonian. Here we will examine feedback protocols in the regime where the controls are able to keep the system close to a pure state. This is an important regime, both because it is where many quantum control systems will need to operate, and because it allows one to simplify the problem by using a power series expansion [15]. In addition to working in the regime of good control, we make two further simplifications. The first is that the control is strong -that is, that 1) the only constraint on H is that Tr[H 2 ] ≤ µ 2 for some constant µ, and 2) that H can induce dynamics much faster than both the dynamics of the system and the rate at which the measurement extracts information. This means that H is effectively unconstrained. We thus deal strictly with a subset of the regime of good control, defined by µ ≫ k and k ≫ β. Here k is the strength of the measurement (defined precisely below), and β is the noise strength, which we define as the rate of increase of the linear entropy due to the noise. The latter inequality is essential for good control. This regime is applicable, for example, to mesoscopic supercond...

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“…In this result, the Pauli spin matrices σ 1 ,σ 3 appear (see, e.g., [63] and A). While the upper or lower placement of Cartesian indices does not matter because the Euclidean plane has trivial metric g AB = δ AB , indices in the complex basis should be raised and lowered using the metric tensor in the complex basis:…”

Section: Appendix B: Complex Basismentioning

confidence: 91%

“…In our calculations, we use a real-valued basis of Pauli spin matrices [63] as a basis for the vector space of 2 × 2 matrices:…”

Section: Appendix A: Pauli Matricesmentioning

confidence: 99%

Effective dynamics of twisted and curved scroll waves using virtual filaments

Dierckx

Verschelde

2013

Phys. Rev. E

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Scroll waves are three-dimensional excitation patterns that rotate around a central filament curve; they occur in many physical, biological, and chemical systems. We explicitly derive the equations of motion for scroll wave filaments in reaction-diffusion systems with isotropic diffusion up to third order in the filament's twist and curvature. The net drift components define at every instance of time a virtual filament which lies close to the instantaneous filament. Importantly, virtual filaments obey simpler, time-independent laws of motion which we analytically derive here and illustrate with numerical examples. Stability analysis of scroll waves is performed using virtual filaments, showing that filament curvature and twist add as quadratic terms to the nominal filament tension. Applications to oscillating chemical reactions and cardiac tissue are discussed.

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Positronium formation in positron-atom collisions at intermediate energies

Cited by 187 publications

References 25 publications

Quantum search by local adiabatic evolution

Quantum search by local adiabatic evolution

Locally Optimal Control of Quantum Systems with Strong Feedback

Effective dynamics of twisted and curved scroll waves using virtual filaments

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