Dynamics of a Trapped Particle in a Double-Well Potential

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.