arXiv:1103.5781 · Journal of Physics B Atomic Molecular and Optical Physics

Acceleration of trapped particles and beams

Er'el Granot, Boris A. Malomed

Published Aug 24, 2011• 5 citations

Abstract

The dynamics of a quantum particle bound by an accelerating delta-functional potential is investigated. Three cases are considered, using the reference frame moving along with the δ-function, in which the acceleration is converted into the additional linear potential. (i) A stationary regime, which corresponds to a resonance state, with a minimum degree of delocalization, supported by the accelerating potential trap. (ii) A pulling scenario : an initially bound particle follows the accelerating delta-functional trap, within a finite time. (iii) The pushing scenario : the particle, which was initially localized to the right of the repulsive delta-function, is shoved to the right by the accelerating potential. For the two latter scenarios, the lifetime of the trapped particle and the largest velocity to which it can be accelerated while staying trapped are found. Analytical approximations are developed for the cases of small and large accelerations in the pulling regime and also for a small acceleration in the stationary situation, and in the regime of pushing. The same regimes may be realized by Airy-like planar optical beams guided by a narrow bending potential channel or crest. Physical estimates are given for an atom steered by a stylus of a scanning tunnelling microscope, and for the optical beam guided by a bending stripe.

DOI: 10.1088/0953-4075/44/17/175005

Related papers

Optimization of conveyance of quantum particles by moving potential well

Satoshi Morita, Yoshiaki Teranishi, Seiji Miyashita

Quantum mechanical control of the position of a particle by using a trapping potential well is an important problem for the manipulation of a quantum particle. We study the probability of successful conveyance of the particle trapped in a potential well for a given length within a given fixed time, i.e., survival probability after the motion. For the actual motion of conveyance, we need to accelerate the particle to move and then decelerate it to stop at the destination. Acceleration and deceleration cause a dropoff of a particle from the trapping potential well. The relaxation of the survival probability in a process with a constant acceleration rate is studied in detail. First, the process is studied as the relaxation of the survival probability of the trapped particle by direct numerical calculations. The survival probability obeys an exponential decay in a long time, which is analyzed from a viewpoint of eigenvalue problem. The value of survival probability is also estimated by the Wentzel-Kramers-Brillouin method with connection formulas using the Airy function and the Weber function. The value is further estimated by a method of the resonance states. We emphasize the fact that an important source of dropoff comes from a nonanalytic change of velocity at the starting point. When the rested particle begins to move, the ground state of the rest frame is redistributed to eigenstates of the moving frame, and then each eigenstate of the moving frame evolves in time. The dephasing of wave functions of the distributed populations reduces the probability of successful conveyance. In general, a smooth start gives a small initial disturbance but it requires a large acceleration during the process to reach the destination in the fixed time which causes a larger dropoff in the process. Considering these conflicting facts, we study the survival probability in concrete conveyance schemes, i.e., protocols with (1) a sudden change of velocity to a constant velocity, (2) a sudden change of acceleration rate to a constant acceleration, and (3) a smooth change of acceleration by studying the real-time change of populations of the adiabatic (instantaneous) eigenstates. We observe the time evolution of the trapped probability and the population distribution during the conveyance process. In cases that the potential well has several bound states, we propose a method to select the particle trapped at the ground state by making use of the difference of survival probabilities of bound states. Published by the American Physical Society 2024

arXiv:2408.04480Physical Review ResearchDec 30, 20240 citations

Accelerating Quantum Decay by Multiple Tunneling Barriers

Stefano Longhi, Ermanno Pinotti

A quantum particle constrained between two high potential barriers provides a paradigmatic example of a system sustaining quasi-bound (or resonance) states. When the system is prepared in one of such quasi-bound states, the wave function approximately maintains its shape but decays in time in a nearly exponential manner radiating into the surrounding space, the lifetime being of the order of the reciprocal of the width of the resonance peak in the transmission spectrum. Naively, one could think that adding more lateral barriers would preferentially slow down or prevent the quantum decay since tunneling is expected to become less probable and due to quantum backflow induced by multiple scattering processes. However, this is not always the case and in the early stage of the dynamics quantum decay can be accelerated (rather than decelerated) by additional lateral barriers, even when the barrier heights are arbitrarily large. The decay acceleration originates from resonant tunneling effects and is associated to large deviations from an exponential decay law. We discuss such a counterintuitive phenomenon by considering the hopping dynamics of a quantum particle on a tight-binding lattice with on-site potential barriers.

EntropySep 16, 20232 citations

Quantum particle displacement by a moving localized potential trap

E. Granot, A. Marchewka

We describe the dynamics of a bound state of an attractive δ-well under displacement of the potential. Exact analytical results are presented for the suddenly moved potential. Since this is a quantum system, only a fraction of the initially confined wave function remains confined to the moving potential. However, it is shown that besides the probability to remain confined to the moving barrier and the probability to remain in the initial position, there is also a certain probability for the particle to move at double speed. A quasi-classical interpretation for this effect is suggested. The temporal and spectral dynamics of each one of the scenarios is investigated.

arXiv:0812.1632Europhysics Letters (EPL)Apr 1, 200913 citations

Trapping of quantum particles and light beams by switchable potential wells

Eduard Sonkin, Boris A. Malomed, Er’el Granot +1 more

We consider basic dynamical effects in settings based on a pair of local potential traps that may be effectively switched on and off, or suddenly displaced, by means of appropriate control mechanisms, such as scanning tunneling microscopy or photo-switchable quantum dots. The same models, based on the linear Schr\"odinger equation with time-dependent trapping potentials, apply to the description of optical planar systems designed for the switching of trapped light beams. The analysis is carried out in the analytical form, using exact solutions of the Schr\"odinger equation. The first dynamical problem considered in this work is the retention of a particle released from a trap which was suddenly turned off, while another local trap was switched on at a distance---immediately or with a delay. In this case, we demonstrate that the maximum of the retention rate is achieved at a specific finite value of the strength of the new trap, and at a finite value of the temporal delay, depending on the distance between the two traps. Another problem is retrapping of the bound particle when the addition of the second trap transforms the single-well setting into a double-well potential (DWP). In that case, we find probabilities for the retrapping into the ground or first excited state of the DWP. We also analyze effects entailed by the application of a kick to a bound particle, the most interesting one being a kick-induced transition between the DWP's ground and excited states. In the latter case, the largest transition probability is achieved at a particular strength of the kick.

arXiv:1007.2424Physical Review ASep 17, 20103 citations

Quantum particles in a suddenly accelerating potential

Paolo Amore, Francisco Marcelo Fernández, José Luis Valdéz

We study the behavior of a quantum particle trapped in a confining potential in one dimension under multiple sudden changes of velocity and/or acceleration. We develop the appropriate formalism to deal with such situation and we use it to calculate the probability of transition for simple problems such as the particle in an infinite box and the simple harmonic oscillator. For the infinite box of length L under two and three sudden changes of velocity, where the initial and final velocity vanish, we find that the system undergoes quantum revivals for Δt=τ0≡4mL2πℏ, regardless of other parameters (Δt is the time elapsed between the first and last change of velocity). For the simple harmonic oscillator we find that the states obtained by suddenly changing (one change) the velocity and/or the acceleration of the potential, for a particle initially in an eigenstate of the static potential, are coherent states. For multiple changes of acceleration or velocity we find that the quantum expectation value of the Hamiltonian is remarkably close (possibly identical) to the corresponding classical expectation values. Finally, the probability of transition for a particle in an accelerating harmonic oscillator (no sudden changes) calculated with our formalism agrees with the formula derived long time ago by Ludwig [Z. Phys. 130(4), 468–475 (1951)], and recently modified by Dodonov [J. Russ. Laser Res. 42(3), 243–249 (2021)], but with a different expression for the dimensionless parameter γ. Our probability agrees with the one of Dodonov for γ ≪ 1 but is not periodic in time (it decays monotonously), contrary to the result derived by Dodonov.

arXiv:2206.01354Journal of Mathematical PhysicsOct 1, 20230 citations

Capture of a quantum particle by a moving trapping potential

Mark R. A. Shegelski, Tyler Poole, Cole Thompson

We investigate the capture of a quantum particle in free space by a moving trapping potential. The capture is investigated for various initial conditions. We examine the dependence of the probability of capture on the shape and depth of the trapping potential, initial speed and mass of the particle, and other parameters. We take the trapping potential to move with an initial speed v 0 and to decelerate with constant acceleration a c until the well stops moving. We study the motion during the time the well is moving and after the well has stopped. We compare the probability of capture to the probability that the particle is in a stationary state (while the well is moving) and the probability the particle is in a bound state (after the well's motion has stopped). Our work could be of interest to instructors and students in upper-year undergraduate quantum mechanics courses.

European Journal of PhysicsMar 12, 20136 citations