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Quantum scattering with time-decaying harmonic potentials

Masaki Kawamoto

Published Apr 12, 2017• 0 citations
OpenAlex

Abstract

By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. Deceleration makes the threshold between the short-range class of potentials and long range class of potentials $1/(1- \lambda)$ for some $0 < \lambda < 1/2$, where $\lambda$ is determined by the mass of particle and coefficients of harmonic potentials. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we prove the existence and completeness of wave operators with respect to such short-range potentials.

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By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. Deceleration makes the threshold between the short-range class of potentials and long range class of potentials 1/(1−λ)1/(1- \lambda)1/(1−λ) for some 0<λ<1/20 < \lambda < 1/20<λ<1/2, where λ\lambdaλ is determined by the mass of particle and coefficients of harmonic potentials. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we prove the existence and completeness of wave operators with respect to such short-range potentials.
By controlling coefficients and decaying order of time-decaying harmonic potentials, the velocity of a quantum particle is decelerated by the effect of harmonic potentials but the particle is non-trapping. In this paper, we consider the quantum system with controlled harmonic potentials. By defining the wave operators for this system and suitable range of these, we can prove the existence and completeness of wave operators with respect to the short-range potentials V(t,x)V(t,x)V(t,x) satisfying ∣V(t,x)∣=o((1+∣x∣)−1/(1−λ))|V(t,x)| = o ((1+|x|)^{-1/(1- λ)})∣V(t,x)∣=o((1+∣x∣)−1/(1−λ)), for some λλλ, 0<λ<1/20< λ< 1/20<λ<1/2.
We prove an almost sure existence of the modified wave operators for a class of Schrödinger operators with random long range potentials. The assumed decay of potential at infinity places it beyond the threshold of the standard class of long range potentials as described in the work of Buslaev-Matveev, Alsholm-Kato, and Hörmander. We develop an approach to quantum scattering which relies on averaging of potential over classical random trajectories. We also establish classical scattering for corresponding classical hamiltonians.
The design of a modular moving trap Zeeman decelerator capable of decelerating gas pulses produced from a supersonic source is presented here. Unlike the conventional form of Zeeman decelerator, paramagnetic particles are confined in a 3D potential throughout the deceleration process. The decelerator field is produced by flattened helical coils and currents of up to 1000 A peak. As the coils are periodic in nature, each coil produces a number of deep, quadrupole traps along the molecular beam axis. The resultant periodic field is described as a travelling wave. The application of the appropriate time dependent current allows the traps to move through the four coil modules. In order to compensate for the weaker transverse confinement, a quadrupole guide, operating at 700 A DC, is required to prevent further losses during the deceleration process. The operation of the decelerator relies on the power electronics developed specifically for the quadrupole and the decelerator coils. Due to the electromagnetic interference generated through the switching of the large currents, much of the electronics used to control the power electronics had to be developed specifically. The quadrupole power electronics have been designed to produce fast switching edges. This is necessary to minimise the interaction of the particles within the fringe field regions while maximising the interaction time within the pure quadrupole field. Even at a modest voltage applied to the circuitry, the rise time in current to 700 A has been reduced by a half. The decelerator power electronics must be capable of producing an alternating waveform with an amplitude of at least 500 A for each of the coil phases. Furthermore, the frequency of the waveforms must be tunable within a range of 10 kHz to 0 Hz. Through a combination of pulse width modulation and knowledge of the electrical properties of the coil it is possible to synthesise an alternating current waveform from a 800 V DC supply using a suitable switching circuit. %The challenge of switching such high powers is the transient voltage spikes that can be produced, however, these can be avoided through careful consideration of the layout of the circuit. Decelerators such as this do not cool the sample but instead reduces the mean velocity of a subset of particles which remained trapped. This maintains the phase space density of trapped particles. Modelling the magnetic fields generated by the decelerator coils has been necessary in order to understand the phase space acceptance of the decelerator. The helical nature of the coils required the development of a specific algorithm in order to calculate the field generated by each wire element. The resultant potential can then be interpolated using a tricubic interpolator to extract the field gradients necessary for numerical simulations of the particle trajectories. Including the effects of the pulse width modulated on the trap facilitates the characterisation of the acceptance of the decelerator and the limitations of the current iteration of the design. The numerical simulations can also be compared to experimental results gathered for metastable argon. The 3D guiding, or velocity bunching, of the gas packet over a range of velocities has been demonstrated. The ability to 3D guide and decelerate were severely hampered by the failure of key electronic components, limiting three coils to 100 A peak, moreover, these traps were sub-optimally loaded. Deceleration from 350 to 347 m/s and 342 to 310 m/s has been observed. The design of a trap capable of simultaneously loading samples of decelerated CaH and Li while allowing the cooling of Li would potentially allow for the sympathetic cooling of a molecular species with an atomic refrigerant. This particular atom-molecule system would also facilitate the examination of controlled chemistry and collisions over a range of temperatures through state selection of the reactants. The loading of the trap has been optimised in 1D for CaH with a loading efficiency of 52.2 % while only 7.3 % of Li is loaded when each of the gas packets has a mean velocity of 11 m/s. This implies that the source of the Li must be at least 130 times brighter than that of the CaH.