Abstract
Three-body systems that are continuously squeezed from a three-dimensional (3D) space into a two-dimensional (2D) space are investigated. Such a squeezing can be obtained by means of an external confining potential acting along a single axis. However, this procedure can be numerically demanding, or even undoable, especially for large squeezed scenarios. An alternative is provided by use of the dimension $d$ as a parameter that changes continuously within the range $2\leq d \leq 3$. The simplicity of the $d$-calculations is exploited to investigate the evolution of three-body states after progressive confinement. The case of three identical spinless bosons with relative $s$-waves in 3D, and a harmonic oscillator squeezing potential is considered. We compare results from the two methods and provide a translation between them, relating dimension, squeezing length, and wave functions from both methods. All calculations are then possible entirely within the simpler $d$-method, but simultaneously providing the equivalent geometry with the external potential.
Highlights
Specific cold atomic or molecular gases can be controlled by external fields to previously unprecedented accuracy [1,2]
The method presented in the previous section appears as an alternative to the natural way of confining an N-particle system, which is to put it under the effect of an external potential that forces the particles to move in a limited region of space
The energy obtained in this way, E3b, is the total energy, system plus external field, in such a way that the energy of the confined system will be obtained after subtraction of the harmonic oscillator energy, i.e., Eext = E3b − Eho, which, in our case of squeezing two coordinates along one direction [see Eq (13)], means Eext = E3b − hω
Summary
Specific cold atomic or molecular gases can be controlled by external fields to previously unprecedented accuracy [1,2]. The required numerical effort is similar to a standard calculation for an integer dimension, but where the external potential has disappeared In these works, [9,10], the equivalence between the d-dependent method and the more direct procedure working in three dimensions, including explicitly the external squeezing potential, was investigated. III we describe how the adiabatic expansion can be used as well to treat the same problem in a direct way, i.e., describing the system in three dimensions, but introducing explicitly the external squeezing potential This procedure, formally not very complicated, often leads to calculations that, especially for large squeezing, are out of numerical reach.
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