The quantum dynamics of an ultracold diatomic molecule tunneling and diffusing in a one-dimensional optical lattice exhibits unusual features. While it is known that the process of quantum tunneling through potential barriers can break up a bound-state molecule into a pair of dissociated atoms, interference and reassociation produce intricate patterns in the time-evolving site-dependent probability distribution for finding atoms and bound-state molecules. We find that the bound-state molecule is unusually resilient against break up at ultralow binding energy Eb (Eb much smaller than the barrier height of the lattice potential). After an initial transient, the bound-state molecule spreads with a width that grows as the square root of time. Surprisingly, the width of the probability of finding dissociated atoms does not increase with time as a power law.
We investigate, in one spatial dimension, the quantum mechanical tunnelling of a diatomic, homogeneous molecule with a single bound state incident upon an external barrier. Both time-independent and time-dependent tunnelling are investigated, using analytical and numerical methods. In the time-independent case, we first derive a formal solution for the molecule's wave function. Then, using the method of variable reflection and transmission amplitudes, we find that the probabilities of reflection and transmission in the bound state decrease with decreased binding strength, while the probabilities of refection and transmission in an unbound state increase with decreased binding energy. In the time-dependent case, we consider a molecule with discrete unbound states. The molecular wave function is modeled as a Gaussian wave packet, and its propagation is calculated numerically using Crank-Nicholson integration. It is found that, in addition to reflecting and transmitting, the molecule may also straddle the potential barrier in an unbound state.
In this article, we present the results of studies on the quantum mechanical tunneling and reflection of a diatomic, homonuclear molecule with a single bound state incident upon a potential barrier. In the first study, we investigate the tunneling of a molecule using a time-dependent formulation. The molecular wave function is modeled as a Gaussian wave packet, and its propagation is calculated numerically using Crank-Nicholson integration. It is found that a molecule may transition between the bound state and an unbound state numerous times during the process of reflection from or transmission past the barrier. It is also found that, in addition to reflecting and transmitting, the molecule may also temporarily straddle the potential barrier in an unbound state. In the second study, we consider the case of a molecule incident in the bound state upon a step potential with energy less than the step. We show that in the limit where the binding energy
e0 approaches zero and the step potential
V0 goes to infinity, the molecule cannot remain in a bound state if the center of mass gets closer to the step than an arbitrarily large distance
x0 which increases as the magnitude of
e0 decreases, as
V0 increases, or both. We also show that, for ${e}_{0}\ensuremath{\rightarrow}{0}^{\ensuremath{-}}$ and ${V}_{0}\ensuremath{\rightarrow}\ensuremath{\infty}$, if the molecule is incident in the bound state, it is reflected in the bound state with probability equal to unity, when the center of mass reaches the reflection distance
x0. We verify that the unbound states exhibit the expected physical behavior. We discuss some surprising results. Connections between our results and investigations done in cold atoms, excitons, Cooper pairs, and Rydberg atoms are discussed.
We investigate tunneling properties of a bound pair of Fermi atoms in an optical lattice. Including a realistic cosine-shape optical lattice potential, we calculate the molecular wavefunction ψ. Starting from the weak-coupling regime, we show how ψ shrinks as one increases the strength of a pairing interaction between atoms. When the molecular size becomes comparable to the lattice constant in the intermediate coupling regime, we find from the spatial structure of ψ that the molecular tunneling between lattice sites is accompanied by virtual dissociation. This phenomenon is shown to be suppressed in the strong-coupling regime due to very strong binding energy. We point out that, while the molecular tunneling accompanied by virtual dissociation can be described by the Hubbard model, it cannot describe the molecular tunneling with no virtual dissociation in the strong-coupling regime.
Quantum tunneling is a quantum phenomenon in which a microscopic object crosses through a potential barrier even if its energy cannot overcome the barrier. A general belief is that tunneling occurs only when the barrier width is comparable to, or smaller than the de Broglie’s wavelength of the object. Here, we study the tunneling of an ultracold atom among N far-separated trapping potentials in a state-selective optical lattice and present a mechanism to realize long-range tunneling. We find that, mediated by the propagating matter wave emitted from the atom, coherent tunneling of the atom to the remote lattices occurs as long as bound states are present in the energy spectrum of the system formed by the atom and its matter-wave. Going beyond the Markovian approximation, and breaking through the conventional distance constraint, our result opens another avenue to realizing tunneling and gives a guideline for developing tunneling devices. It is generally believed that quantum tunneling cannot occur when the distance is larger than the particle’s wavelength. This paper reveals that a long-range quantum tunneling of the atom to the remote trapping potentials mediated by its emitted matter wave occurs as long as bound states are present in the energy spectrum of the total system formed by the atom and its matter-wave.
We summarize our study of a diatomic homonuclear molecule incident upon a potential barrier in one dimension [G.L. Goodvin and M.R.A. Shegelski. Phys. Rev. A, 71, 032719 (2005).]. We study various initial states for the molecule and allow transitions between vibrational states of the molecule during the process of tunneling. By applying an elegant method to solve this problem, we show that transmission resonances occur in several cases and that adding more binding energy levels for the molecule can tend to decrease the probability of tunneling. We also discuss the extension of our results to more realistic three-dimensional and experimental systems.PACS Nos.: 34.20.Cf, 03.65.Xp, 34.20.Gj, 34.50.Pi