Quantum tunneling of particles plays an important role in many chemical reactions. Studying quantum tunneling in time-dependent potential wells is tricky since most of the available solutions of time-dependent potential wells are coupled to specific properties of the Hamiltonian. Here, we investigate the tunneling probability of a quantum wavepacket in time-dependent potential well by using the split operator method. This numerical method can give us an overview of the tunneling probability of a quantum wavepacket for any temporal change in the shape of the potential well. We study a time-dependent potential well model evolving from symmetric to asymmetric quartic double well since quartic potential wells resemble certain practically available potential well models. Quanta 2024; 13: 11–19.
Quantum tunneling of particles plays an important role in many chemical reactions. Studying quantum tunneling in time-dependent potential wells is tricky since most of the available solutions of time-dependent potential wells are coupled to specific properties of the Hamiltonian. Here, we investigate the tunneling probability of a quantum wavepacket in time-dependent potential well by using the split operator method. This numerical method can give us an overview of the tunneling probability of a quantum wavepacket for any temporal change in the shape of the potential well. We study a time-dependent potential well model evolving from symmetric to asymmetric quartic double well since quartic potential wells resemble certain practically available potential well models. Quanta 2024; 13: 11–19.
The roles of spatial symmetry and strength of external time-dependent perturbation on the dynamics of a quantum particle, initially localized in one of the wells of an asymmetric double-well potential are studied using the recently developed techniques incorporating quantum theory of motion and time-dependent Fourier grid Hamiltonian methods. The model used here includes a mimic of the related experimental situations which is considered as a perturbation to the static double-well potential. Analysis of localized and delocalized phase space structures and corresponding time-profile of tunneling probability reveal the recipe toward controlling the tunneling oscillations by modulating the parameters of applied perturbation. A study on a stochastic pulsating potential also reveals the root to the quantum localization, even in moderate field strength.
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.
Solutions to explicit time-dependent problems in quantum mechanics are rare. In fact, all known solutions are coupled to specific properties of the Hamiltonian and may be divided into two categories: one class consists of time-dependent Hamiltonians which are not higher than quadratic in the position operator, like e.g. the driven harmonic oscillator with time-dependent frequency. The second class is related to the existence of additional invariants in the Hamiltonian, which can be used to map the solution of the time-dependent problem to that of a related time-independent one. In this paper we discuss and develop analytic methods for solving time-dependent tunnelling problems, which cannot be addressed by using quadratic Hamiltonians. Specifically, we give an analytic solution to the problem of tunnelling from an attractive time-dependent potential which is embedded in a long-range repulsive potential. Recent progress in atomic physics makes it possible to observe experimentally time-dependent phenomena and record the probability distribution over a long range of time. Of special interest is the observation of macroscopical quantum-tunnelling phenomena in Bose–Einstein condensates with time-dependent trapping potentials. We apply our model to such a case in the last section.
An asymmetric double-well potential is considered, assuming that the minima of the wells are quadratic with a frequency ω and the difference of the minima is close to a multiple of �ω. A WKB wave function is constructed on both sides of the local maximum between the wells, by matching the WKB function to the exact wave functions near the classical turning points. The continuities of the wave function and its first derivative at the local maximum then give the energy-level splitting formula, which not only reproduces the instanton result for a symmetric potential, but also elucidates the appearance of resonances of tunneling in the asymmetric potential. Key words: quantum tunneling, energy splitting, double-well potential PACS: 03.65.Sq, 03.65.Ge, 82.20.Xr 1
Quantum wave dynamics in systems with potential wells, bathers and outer impenetrable walls has been studied using solutions of time-dependent Schrodinger equation. Numerical integration of Schrodinger equation, calculation of dynamical variables and its average values, Fourier-spectra were carried out for different initial conditions and different relations between wave packet energy and bather height. In non-symmetric wells the oscillations of wave packets are accompanied by multistep discrete tunneling of probability fluid from well through barrier. In double well potential (symmetric wells) the temporal scales of oscillations and tunneling were investigated in details. Influence of additional time-dependent potential on quantum wave packet dynamics in above listed systems is also discussed.
The time it takes a particle to tunnel through the asymmetric Eckart barrier potential is investigated using Gaussian wavepackets, where the barrier serves as a model for the potential along a chemical reaction coordinate. We have previously shown that the, in principle experimentally measurable, tunneling flight time, which determines the time taken by the transmitted particle to traverse the barrier, vanishes for symmetric potentials like the Eckart and square barrier [ Petersen , J. ; Pollak , E. J. Phys. Chem. Lett. 2017 , 9 , 4017 ]. Here we show that the same result is obtained for the asymmetric Eckart barrier potential, and therefore, the zero tunneling flight time seems to be a general result for one-dimensional time-independent potentials. The wavepacket dynamics is simulated using both an exact quantum mechanical method and a classical Wigner prescription. The excellent agreement between the two methods shows that quantum coherences are not important in pure one-dimensional tunneling and reinforces the conclusion that the tunneling flight time vanishes.