Alternative quantisation condition for wavepacket dynamics in a hyperbolic double well

Abstract

We propose an analytical approach for computing the eigenspectrum and corresponding eigenstates of a hyperbolic double well potential of arbitrary height or width, which goes beyond the usual techniques applied to quasi-exactly solvable models. We map the time-independent Schrödinger equation onto the Heun confluent differential equation, which is solved by using an infinite power series. The coefficients of this series are polynomials in the quantisation parameter, whose roots correspond to the system’s eigenenergies. This leads to a quantisation condition that allows us to determine a whole spectrum, instead of individual eigenenergies. This method is then employed to perform an in depth analysis of electronic wave-packet dynamics, with emphasis on intra-well tunneling and the interference-induced quantum bridges reported in a previous publication Chomet et al (2019 New J. Phys. 21 123004). Considering initial wave packets of different widths and peak locations, we compute autocorrelation functions and Wigner quasiprobability distributions. Our results exhibit an excellent agreement with numerical computations, and allow us to disentangle the different eigenfrequencies that govern the phase-space dynamics.

Attoscience in phase space

“…The quantum bridges could be controlled with the appropriate coherent superposition of states or different driving fields, which opens a wide range of possibilities for studying quantum effects in enhanced ionisation. We have verified that, for the model potential in [139], the frequencies computed analytically are quite robust upon inclusion of an external static field. This is illustrated in Fig.…”

“…Further work, however, showed that the momentum gates are intrinsic to the molecular sys-tem and exist even for static fields, or no fields at all. Thereby, quantum interference plays an important role by providing a bridge for quasiprobability to flow from one molecular well to the other [78], and the frequencies can be estimated for double-well potential models [139]. Further studies of nonadiabatic effects and bifurcation in strong-field ionisation were conducted in [77].…”

“…These are used to classify regions with qualitatively different behaviours and understand how processes such tunnelling and rescattering unfold. For details, we refer to the original publications [76,78,82,139]. We also intend to go beyond these articles, by bringing together aspects that have not been emphasised so far.…”

“…This definition of ionisation rate was used in the seminal paper [138] in the context of enhanced ionisation of molecules, and in our previous publications [78,139].…”

“…As seen in [78], the quantum bridge and its frequency are inherent to the molecular system, and are present even in the absence of an external field. In [139] an analytical method based upon those employed in quasisolvable models [207][208][209] is used to determine the origin and the value of the frequency of the quantum bridge in a field-free system. Using the autocorrelation function of different initial wave packets, it is clear that this frequency is due to the coupling of different eigenstates.…”