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
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
In this paper, we study the quantum dynamics of a particle conveyed by a moving potential well, with a focus on its survival probability. In physical systems, this process is inevitably subjected to external disturbances, such as environmental coupling, which reduce survival probability. Even when external noise is suppressed, however, intrinsic disturbances remain, leading to unwanted leakage from the trapping potential. Specifically, dynamic parameter variations induce nonadiabatic transitions. Various types of nonadiabatic transitions arise from the time-dependence of the potential parameters. When the potential well is smoothly accelerated, the trapped particle may escape due to inertia, a phenomenon known as adiabatic tunneling. Beyond this, the switching procedures (starting and stopping the motion) exert a significant impact that depends heavily on the abruptness of the operation. Consequently, precise dynamical control of the system Hamiltonian is essential, yet it remains a highly nontrivial task. We analytically investigate the mechanism underlying these switching effects and derive a closed-form formula to quantify them. By combining this switching contribution with the adiabatic tunneling rate, we accurately approximate the total nonadiabatic effect. Our framework reproduces survival probabilities across various acceleration protocols with high precision. Since the switching effect is derived analytically and the adiabatic tunneling rate is determined by constant-acceleration analysis, the model is independent of specific, complex protocols. Once the fundamental parameters determined from a few numerical simulations of typical cases are established, the survival probabilities for arbitrary acceleration protocols can be predicted. This provides a practical framework for real-time quantum control without the need for full dynamical simulations.
The properties of the conveyance of quantum particles by a moving potential well are studied. We sweep a bell-shape potential well with constant velocity, and study how the potential well traps the particle. In particular, we study the case where we suddenly change the velocity up to a constant velocity. We investigate the number of particles conveyed as a function of the sweep velocity. The effect of the discretization of space in the numerical method is also investigated. We also study how the potential well carries the particle up from a region where the potential energy is low, to another region of high potential energy. Here, we find that the tunneling effect reduces the number of particles carried up. This process is well described by the Landau–Zener mechanism. The dynamics of the density of particles under the sweeping potential well is also studied in noninteracting Fermi particle systems, where the position of the Fermi level plays an important role.
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.
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.
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.
Abstract We study the behavior of a quantum particle, trapped in localized potential, when the trapping potential starts suddenly to move with constant velocity. In one dimension we have reproduced the results obtained by (Granot and Marchewka 2009 EPL 86 20007), for an attractive delta function, using an approach based on a spectral decomposition, rather than on the propagator. We have also considered the cases of Pöschl-Teller and simple harmonic oscillator potentials (in one dimension) and to the hydrogen atom (in three dimensions). In this last case we have calculated explicitly the leading contribution to the ionization probability for the hydrogen atom due to a sudden movement.