QuantumN-boson states and quantized motion of solitonic droplets: Universal scaling properties in low dimensions

Abstract

In this article, we illustrate the scaling properties of a family of solutions for A attractive bosonic atoms in the limit of large A. These solutions represent the quantized dynamics of solitonic degrees of freedom in atomic droplets. In dimensions lower than two, or d = 2 -e, we demonstrate that the number of isotropic droplet states scales as A 3/2/ e 1/2, and for e = 0, or d = 2, scales as N 2. The ground-state energies scale as A 2/e+1 in d -2 -e, and when d = 2, scale as an exponential function of A. We obtain the universal energy spectra and the generalized Tjon relation; their scaling properties are uniquely determined by the asymptotic freedom of quantum bosonic fields at short distances, a distinct feature in low dimensions. We also investigate the effect of quantum loop corrections that arise from various virtual processes and show that the resultant lifetime for a wide range of excited states scales as N e/2E '~e/2.

Discrete scale invariant quantum dynamics and universal quantum beats in Bose gases

“…where δH λ is a small correction due to the anisotropic many body fluctuations [25], the effect of which will be discussed towards the end. In this effective description, |λ is an eigenstate of the operator λ: λ|λ = λ|λ , representing a condensate with wave function φ 0 ( x), while Pλ is the momentum conjugate to λ.…”

“…The phase of φ 0 ( x), is chosen to satisfy the conservation law [25] and is of little importance for the rest of the discussion. The quantity λ parametrizes the slowly evolving field and f (x) is a smooth normalizable function that is regular at the origin.…”

“…In this effective description, |λ is an eigenstate of the operator λ: λ|λ = λ|λ , representing a condensate with wave function φ 0 ( x), while Pλ is the momentum conjugate to λ. The constants m and V are given by C 1 N and C 3 gN 2 − C 2 N respectively, where C 1 , C 2 , and C 3 have been calculated previously [25]. Finally, it is important to note that Eq.…”

“…For this reason the theory of dynamics is often restricted to semiclassical approaches [7][8][9]19,22], or the Heisenberg equations of motion [23,27]. However, one can make two further simplifications for low dimensions and dense condensates [28]. The first is to note that {φ( x )}|{φ ( x )} ≈ δ({φ ( x )} − {φ( x )}), which requires that the two field configurations are identical.…”

Universal Scaling Properties of Cold Atom Scattering Dynamics in Confined Low Dimensional Geometries

“…These scattering experiments provide detailed information about the interaction between individual atoms, and can be used as a starting point for many-body theories. The scattering properties of cold atoms are generally examined at low energy scales, where the interaction can be parametrized by the d-dimensional effective scattering length [25]. In the case of harmonically confined geometries, it is possible to integrate out the transverse degrees of freedom and write down an effective low-dimensional model with scattering lengths defined in terms of the harmonic trap length, a ⊥ , and the three dimensional scattering length, a [10,19].…”