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.
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.
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.
Potential wells are widely studied entities nowadays, because of their wide range of applications in many fields of science. Experimental and technological improvements make it possible to design and control the potential shape. For instance, one can modify a single-well potential into a double well and backwards. To investigate this kind of problems, one needs to properly take into account the time-dependency of the system Hamiltonian. We used split operator method to study the dynamics of wavepacket in general, and in time-dependent potential wells in particular. While considering time-dependent potential wells, we concentrated on the case when a single-well is transformed into a double-well. Our goal was to explore the range of validity of the adiabatic theorem. We have made observations on the dynamics of the wavepacket for different barrier heights and widths. For each case, we estimated how long the switching time should be in order to leave the system in the ground state. In doing this, we established a link between the eigenvalues of potential well and the time taken by the wavepacket to be in the ground state of potential well. The greater the difference between the first two eigenvalues of the double well, the lesser the switching time requested to go from single- to double well. Moreover, we simulated the system dynamics while going from single- to asymmetric doublewell. We noticed that the superposition of left and right states is very fragile with respect to asymmetry. By introducing any asymmetry in the double-well, the wavepacket tends to be trapped in the deeper side of the potential well. While talking in terms of eigenvalues of asymmetric double-well, we again observed that the greater the difference between eigenvalues, the lesser the time needed by the wavepacket to be in ground state of double-well. Finally, we applied the theoretically discussed approach to estimate the switching time needed by a trapped ion to be in the ground state of a double-well potential, for realistic experimental parameters.
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.
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.
We predict wideband suppression of tunneling of spin-orbit-coupled atoms (or noninteracting Bose-Einstein condensate) in a double-well potential with periodically varying depths of the potential wells. The suppression of tunneling is possible for a single state and for superposition of two states, i.e., for a qubit. By varying spin-orbit coupling one can drastically increase the range of modulation frequencies in which an atom remains localized in one of the potential wells, the effect connected with crossing of energy levels. This range of frequencies is limited because temporal modulation may also excite resonant transitions between lower and upper states in different wells. The resonant transitions enhance tunneling and are accompanied by pseudospin switching. Since the frequencies of the resonant transitions are independent of the potential modulation depth, in contrast to frequencies at which suppression of tunneling occurs, by varying this depth one can dynamically control both the spatial localization and pseudospin of the final state.
The spontaneous switching of a quantum particle between the wells of a double-well potential is a phenomenon of general interest to physics and chemistry. It was broadly believed that the switching rate decreases steadily as a function of the size of the energy barrier. This view was challenged by a recent experiment on a driven superconducting Kerr nonlinear oscillator (often called the Kerr-cat qubit or the Kerr parametric oscillator), whose energy barrier can be continuously increased by ramping up the drive. Remarkably, as the drive amplitude increases, the switching rate exhibits a steplike decrease termed the ``staircase.'' The view challenged by the experimental staircase demands a deep review of our understanding of the role of quantum effects in double wells. In this work, we use a Lindbladian model of dissipation to derive a semianalytical formula for the switching rate, resolving a continuous transition between tunneling-dominated dynamics and dissipation-dominated dynamics. These two dynamics are observed respectively in the flat part and the steep part of each step in the staircase. Our formula exposes two distinct dissipative processes that limit tunneling: the dephasing of interwell superpositions freezes tunneling via the quantum Zeno effect, and the decay from one excited state to another limits the time during which tunneling can be attempted. This physical understanding allows us to pinpoint the critical drive amplitude that separates the flat and steep part of each step using an equation involving the rates of the two dissipative processes and the tunnel splitting. In addition, analyzing the transition matrix elements in the formula shows that in the regime of a few ($\ensuremath{\lesssim}10$) states in the well and under moderate to low temperatures, highly excited states are populated predominantly via cascaded and direct thermal heating rather than quantum heating. At very low temperatures, we find that the perturbation induced by the non-Hermitian Hamiltonian part of the Lindbladian model becomes increasingly important and facilitates a form of quantum heating that has not been identified before. We numerically map the activation mechanism as a function of drive amplitude, damping rate, and temperature. Our theory deepens the understanding of switching dynamics between metastable quantum states, highlights the importance of a general interplay between tunneling and dissipation, and identifies a novel quantum regime in activated transitions.