Temporal dynamics of tunneling: Hydrodynamic approach

Abstract

We use the hydrodynamic representation of the Gross -Pitaevskii/Nonlinear Schrödinger equation in order to analyze the dynamics of macroscopic tunneling process. We observe a tendency to a wave breaking and shock formation during the early stages of the tunneling process. A blip in the density distribution appears in the outskirts of the barrier and under proper conditions it may transform into a bright soliton. Our approach, based on the theory of shock formation in solutions of Burgers equation, allows us to find the parameters of the ejected blip (or soliton if formed) including the velocity of its propagation. The blip in the density is formed regardless of the value and sign of the nonlinearity parameter. However a soliton may be formed only if this parameter is negative (attraction) and large enough. A criterion is proposed. An ejection of a soliton is also observed numerically. We demonstrate, theoretically and numerically, controlled formation of soliton through tunneling. The mass of the ejected soliton is controlled by the initial state.

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

“…On the theoretical side, shock waves in repulsive BECs were mainly studied in the framework of mean-field theory and the GP equation for weakly-interacting Bose gases [397][398][399][400][401][402][403][404][405], but also for strongly interacting ones [406] and in the BEC-Tonks crossover [407]; additionally, the effect of temperature (see Sec. 5.7) on shock wave formation and dynamics and the effect of depleted atoms were respectively discussed in Refs.…”

Jetlike tunneling from a trapped vortex

“…Then we change the potential into V trap (x, y) allowing the wave function to tunnel through the barrier and apply the adiabatic approximation. 5,21 As a result, the Euler equation (3) takes the form m ∂v pot (r, t) ∂t…”

“…Even in the context of simple wavefunctions, without angular momentum, phase can have profound effects. Examples include the recent prediction of "blips" in the outgoing matter through a trap [5][6][7][8] and the development of dispersive shock waves 9-12 , e.g. when tunneling through a barrier.…”

“…Adiabatically slow varying density 5,21 in the trap may be described as ρ(N, r, t) = N (t)ρ 0 exp(− 2 √ 2mV 0 |r|) and Eq. (4) becomes…”

“…5), the adiabatic approximation in the Euler equation (3) yields the velocity at the exit point in the form v(x ex ) = 2V 0 /m, where V 0 is the energy difference between the top of the barrier and the chemical potential in the trap. The same definition holds for the 2d case (see Fig.…”