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arXiv:2007.15900 · Physical Review Research

Three identical bosons: Properties in noninteger dimensions and in external fields

Eric Garrido, A. S. Jensen

Published Aug 18, 2020• 13 citations
arXivHTMLPDF

Abstract

This work shows how the confinement from three to two dimensions of a three-body system by means of an external harmonic oscillator field can be equivalently described as a system moving in a non-integer dimensional space. The case of three identical bosons is considered.
DOI
10.1103/physrevresearch.2.033261

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Abstract The squeezing process of a three-dimensional quantum system by use of an external deformed one-body oscillator potential can also be described by the d-method, without external field and where the dimension can take non-integer values. In this work we first generalize both methods to N particles and any transition between dimensions below 3. Once this is done, the use of harmonic oscillator interactions between the particles allows complete analytic solutions of both methods, and a direct comparison between them is possible. Assuming that both methods describe the same process, leading to the same ground state energy and wave function, an analytic equivalence between the methods arises. The equivalence between both methods and the validity of the derived analytic relation between them is first tested for two identical bosons and for squeezing transitions from 3 to 2 and 1 dimensions, as well as from 2 to 1 dimension. We also investigate the symmetric squeezing from 3 to 1 dimensions of a system made of three identical bosons. We have in all the cases found that the derived analytic relations between the two methods work very well. This fact permits to relate both methods also for large squeezing scenarios, where the brute force numerical calculation with the external field is too much demanding from the numerical point of view, especially for systems with more than two particles.
We consider the energetics and structural properties of a many-particle system in one dimension with pairwise contact interactions confined in a parabolic external potential. To render the problem analytically solvable, we use the harmonic approximation scheme at the level of the Hamiltonian. We investigate the scaling with particle number of the ground-state energies for systems consisting of identical bosons or fermions. We then proceed to focus on bosonic systems and make a detailed comparison with known exact results in the absence of the parabolic external trap for three-body systems. We also consider the thermodynamics of the harmonic model, which turns out to be similar for bosons and fermions due to the lack of degeneracy in one dimension.
• Induced Efimov effect. • Confined three-body systems. • Non-integer dimensions. The Efimov effect can be induced by means of an external deformed one-body field that effectively reduces the allowed spatial dimensions to less than three. To understand this new mechanism, conceptually and practically, we employ a formulation using non-integer dimension, which is equivalent to the strength of an external oscillator field. The effect most clearly appears when the crucial two-body systems are unbound in three, but bound in two, dimensions. We discuss energy variation, conditions for occurrence, and number of Efimov states, as functions of the dimension. We use practical examples from cold atom physics of 133 Cs- 133 Cs- 133 Cs, 87 Rb- 87 Rb- 87 Rb, 133 Cs- 133 Cs- 6 Li, and 87 Rb- 87 Rb- 39 K. Laboratory tests of the effect can be performed with two independent parameters, i.e. the external one-body field and the Feshbach two-body tuning. The scaling and (dis)appearance of these Efimov states occur precisely as already found in three dimensions.
Abstract The quantum mechanical three-body problem is a source of continuing interest due to its complexity and not least due to the presence of fascinating solvable cases. The prime example is the Efimov effect where infinitely many bound states of identical bosons can arise at the threshold where the two-body problem has zero binding energy. An important aspect of the Efimov effect is the effect of spatial dimensionality; it has been observed in three dimensional systems, yet it is believed to be impossible in two dimensions. Using modern experimental techniques, it is possible to engineer trap geometry and thus address the intricate nature of quantum few-body physics as function of dimensionality. Here we present a framework for studying the three-body problem as one (continuously) changes the dimensionality of the system all the way from three, through two, and down to a single dimension. This is done by considering the Efimov favorable case of a mass-imbalanced system and with an external confinement provided by a typical experimental case with a (deformed) harmonic trap.