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arXiv:1603.08718 · Physical Review A

Adiabaticity in a time-dependent trap: A universal limit for the loss by touching the continuum

D. Sokolovski, M. Ll. Pons

Published Jul 14, 2016• 0 citations
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Abstract

We consider a time-dependent trap externally manipulated in such a way that one of its bound states is brought into an instant contact with the continuum threshold and then down again. It is shown that, in the limit of slow evolution, the probability to remain in the bound state, ${P}^{\mathrm{stay}}$ tends to a universal limit and is determined only by the manner in which the adiabatic bound state approaches and leaves the threshold. The task of evaluating ${P}^{\mathrm{stay}}$ in the adiabatic limit can be reduced to studying the loss from a zero-range well and is performed numerically. Various types of trapping potentials are considered. Applications of this theory to cold atoms in traps and to propagation of transversal modes in tapered waveguides are proposed.

DOI
10.1103/physreva.94.013410

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We consider a time-dependent trap externally manipulated in such a way that one of its bound states is brought into an instant contact with the continuum threshold and then down again. It is shown that, in the limit of slow evolution, the probability to remain in the bound state, Pstay{P}^{\mathrm{stay}}Pstay tends to a universal limit and is determined only by the manner in which the adiabatic bound state approaches and leaves the threshold. The task of evaluating Pstay{P}^{\mathrm{stay}}Pstay in the adiabatic limit can be reduced to studying the loss from a zero-range well and is performed numerically. Various types of trapping potentials are considered. Applications of this theory to cold atoms in traps and to propagation of transversal modes in tapered waveguides are proposed.
We consider a time-dependent trap externally manipulated in such a way that one of its bound states is brought up towards the continuum threshold and then down again. We evaluate the probability Pstay{P}^{\mathrm{stay}}Pstay of a particle, initially in a bound state of the trap, continuing in it at the end of the passage. We use the Sturmian representation, whereby the problem is reduced to evaluating the reflecting coefficient of an absorbing potential. In the slow-passage limit, Pstay{P}^{\mathrm{stay}}Pstay goes to 1 for a state turning before reaching the continuum threshold and vanishes if the bound state crosses into the continuum. For a slowly moving state, just ``touching'' the threshold Pstay{P}^{\mathrm{stay}}Pstay tends to a universal value of about 3838%38, for a broad class of potentials. In the rapid-passage limit, Pstay{P}^{\mathrm{stay}}Pstay depends on the choice of the potential. Various types of trapping potentials are considered, with an analytical solution obtained in the special case of a zero-range well.
We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper, we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.
We expand upon the standard quantum adiabatic theorem, examining the time-dependence of quantum evolution in the near-adiabatic limit. We examine a Hamiltonian that evolves along some fixed trajectory from H^0\hat{H}_0H^0​ to H^1\hat{H}_1H^1​ in a total evolution-time τττ, and our goal is to determine how the final state of the system depends on τττ. If the system is initialized in a non-degenerate ground state, the adiabatic theorem says that in the limit of large τττ, the system will stay in the ground state. We examine the near-adiabatic limit where the system evolves slowly enough that most but not all of the final state is in the ground state, and we find that the probability of leaving the ground state oscillates in τττ with a frequency determined by the integral of the spectral gap along the trajectory of the Hamiltonian, so long as the gap is big. If the gap becomes exceedingly small, the final probability is the sum of oscillatory behavior determined by the integrals of the gap before and after the small gap. We confirm these analytic predictions with numerical evidence from barrier tunneling problems in the context of quantum adiabatic optimization.
In this paper, for the first time in the context of time orbiting potential (TOP) trap, the necessary and sufficient conditions for the adiabatic evolution of weak field seeking states have been quantitatively examined. It has been well accepted since decades that adiabaticity has to be obeyed by the atoms for successful magnetic trapping. However, we show, on the contrary, that atoms can also be confined beyond the adiabatic limit. For the demonstration, we have considered a toy model of a single weak field seeking atom in its ground state and have calculated its survival probability inside a TOP trap. Our findings open new possibilities to relax the restrictions of atom trapping in laboratories.
We present a method for accelerating adiabatic protocols for systems involving a coupling to a continuum, one that cancels both nonadiabatic errors as well as errors due to dissipation. We focus on applications to a generic quantum state transfer problem, where the goal is to transfer a state between a single level or mode, and a propagating temporal mode in a waveguide or transmission line. Our approach enables perfect adiabatic transfer protocols in this setup, despite a finite protocol speed and a finite waveguide coupling. Our approach even works in highly constrained settings, where there is only a single time-dependent control field available.