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.

Numerical simulations of the NLSE (or GPE) are presented demonstrating emission of short pulses of the matter (light) density formed in the course of tunneling in wave-guided light and/or trapped BEC. The phenomenon is observed under various conditions, for nonlinearities of different signs, zero nonlinearity included. We study, both numerically and analytically, pulsations of matter (light) remaining within the trap and use the results in order to induce emission of sequential pulses by properly narrowing the trap. This allows us to propose a mechanism for a realization of Atom Pulse Laser.

Tunneling of a quasibound state is a non-smooth process in the entangled many-body case. Using time-evolving block decimation, we show that repulsive (attractive) interactions speed up (slow down) tunneling, which occurs in bursts. While the escape time scales exponentially with small interactions, the maximization time of the von Neumann entanglement entropy between the remaining quasibound and escaped atoms scales quadratically. Stronger interactions require higher order corrections. Entanglement entropy is maximized when about half the atoms have escaped. Tunneling is one of the most pervasive concepts in quantum mechanics and is essential to contexts as diverse as α-decay of nuclei [1], vacuum states in quantum cosmology [2] and chromodynamics [3], and photo-synthesis [4]. Macroscopic quantum tunneling (MQT), the aggregate tunneling behavior of a quantum many-body wavefunction, has been demonstrated in many condensed matter systems [5, 6] and is one of the remarkable features of Bose-Einstein Condensates (BECs), ranging from Landau-Zener tunneling in tilted optical lattices [7] to the AC and DC Josephson effects in double wells [8, 9], as well as their quantum entangled generalizations [10]. The original vision of quantum tunneling was in fact the quantum escape or quasibound problem by Gurney and Condon in 1929 [1], and recently the first mean-field or semiclassical observation of quantum escape has been made in Toronto [11]. However, with the rise of entanglement as a key perspective on quantum many-body physics, the advent of powerful entangled dynamics matrix-product-state (MPS) methods [12, 13], and the possibility of observing the moment-to-moment time evolution of quasibound tunneling dynamics directly in the laboratory [11, 14-17] it is the right time to revisit quantum escape. In this Letter, we take advantage of the powerful new toolset for quantum many-body simulations [13, 18] to show that the many-body quantum tunneling problem differs in key respects from our expectations from semiclassical and other well-established approaches to tunneling. Specifically, we use time-evolving block decimation (TEBD) to follow lowly entangled matrix product states [12, 19] for the quantum escape of a quasibound ultracold Bose gas initially confined behind a potential barrier. Our use of a Bose-Hubbard Hamiltonian [20] can be viewed either as a discretization scheme or as an explicitly enforced optical lattice used to control the tunneling dynamics. Unlike instanton and semiclassical approaches, we are able to follow the von Neumann en-tanglement entropy, number fluctuations, quantum depletion , and other quantum many-body aspects of time evolution of the many-body wavefunction. Such measures clarify when semiclassical approaches are and are not applicable. They also show that hiding in the semi-classical averaged picture are other many-body features with radically different scalings: the escape time t esc , i.e., the time at which the average number of remaining qua-sibound atoms falls to 1/e of its initial valu...