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arXiv:2012.00531 · Journal of Physics Communications

A chain of three dimensional scatterers under quasi-one dimensional quantum confinement: a Green’s function approach to energy band engineering

J. I. Kim

Published May 1, 2021• 0 citations
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Abstract

Abstract A mechanism to modify the energy band structure is proposed by considering a chain of periodic scatterers forming a linear lattice (akin to an optical lattice for ultracold atoms) around which an external cylindrical trapping potential is applied along the chain axis. It is analytically shown that, when this trapping (confining) potential is tight enough so as to reduce the dimensionality of the problem to an effective one dimension, it may modify the bound and scattering states of the local lattice potential, whose three-dimensional nature around each site is fully taken into account and going beyond the zero-range contact-potential approximation. Since these states contribute to the formation of the energy bands, the latter could thereby be continuously tuned by manipulating the confinement without the need to change the lattice potential. Such dimensionality reduction by quantum confinement is analyzed by using a Green’s functions method that can describe the scattering off the lattice sites in the presence of strong confinement and that can collect the contribuitions of several scattering partial waves.

DOI
10.1088/2399-6528/abfff9

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Abstract A mechanism to modify the energy band structure is proposed by considering a chain of periodic scatterers forming a linear lattice (akin to an optical lattice for ultracold atoms) around which an external cylindrical trapping potential is applied along the chain axis. It is analytically shown that, when this trapping (confining) potential is tight enough so as to reduce the dimensionality of the problem to an effective one dimension, it may modify the bound and scattering states of the local lattice potential, whose three-dimensional nature around each site is fully taken into account and going beyond the zero-range contact-potential approximation. Since these states contribute to the formation of the energy bands, the latter could thereby be continuously tuned by manipulating the confinement without the need to change the lattice potential. Such dimensionality reduction by quantum confinement is analyzed by using a Green’s functions method that can describe the scattering off the lattice sites in the presence of strong confinement and that can collect the contribuitions of several scattering partial waves.
Finite-size effects not only alter the energy levels of small systems, but can also lead to additional effective interactions within these systems. Here the problem of low-energy quantum scattering by a spherically symmetric short-range potential in the presence of a general cylindrical confinement is investigated. A Green's function formalism is developed which accounts for the full three-dimensional (3D) nature of the scattering potential by incorporating all phase shifts and their couplings. This quasi-1D geometry gives rise to scattering resonances and weakly localized states, whose binding energies and wave functions can be systematically calculated. Possible applications include, e.g., impurity scattering in ballistic quasi-1D quantum wires in mesoscopic systems and in atomic matter-wave guides. In the particular case of parabolic confinement, the present formalism can also be applied to pair collision processes such as two-body interactions. Weakly bound pairs and quasimolecules induced by the confinement and having zero or higher orbital angular momentum can be predicted, such as ppp- and ddd-wave pairings.
We generalize the textbook Kronig-Penney model to realistic conditions for a quantum-particle moving in the quasi-one-dimensional (quasi-1D) waveguide, where motion in the transverse direction is confined by a harmonic trapping potential. Along the waveguide, the particle scatters on an infinite array of regularized delta potentials. Our starting point is the Lippmann-Schwinger equation, which for quasi-1D geometry can be solved exactly, based on the analytical formula for the quasi-1D Green's function. We study the properties of eigenenergies as a function of particle quasimomentum, which form band structure, as in standard Kronig-Penney model. We test our model by comparing it to the numerical calculations for an atom scattering on an infinite chain of ions in quasi-1D geometry. The agreement is fairly good and can be further improved by introducing energy-dependent scattering length in the regularized delta potential. The energy spectrum exhibits the presence of multiple overlapping bands resulting from excitations in the transverse direction. At large lattice constants, our model reduces to standard Kronig-Penney result with 1D coupling constant for quasi-1D scattering, exhibiting confinement-induced resonances. In the opposite limit, when lattice constant becomes comparable to harmonic oscillator length of the transverse potential, we calculate the correction to the quasi-1D coupling constant due to the quantum interference between scatterers. Finally, we calculate the effective mass for the lowest band and show that it becomes negative for large and positive scattering lengths.
In experiments with ultra-cold gases, two alkali atoms, that interact with repulsive or attractive potentials and are confined to an optical lattice, can form bound states. In order to compute the energy of such states formed by atoms in the lowest Bloch band, one needs to take into account the intra-band corrections arising from contributions by higher Bloch bands. As it is hard to implement, known calculations tend to neglect them altogether thus setting up a limit for the precision of such computations. To address the problem we apply an approach that uses renormalization-group equations for an effective potential we introduce. It allows for the expression of the bound state energy in terms of the free-space interaction scattering length and parameters of confining potentials. Expressions for bound state energies in 1D, 2D and 3D optical lattices are reported. We show that the method we use can be easily tailored to various cases of atoms confined by external fields of other geometries. A known result for atoms confined to a quasi-2D system is reproduced as an example. Universality of the approach makes it a useful tool for such class of problems.
A dual confinement induced resonance effect first observed numerically in a paper [J. I. Kim, V. S. Melezhik and P. Schmelcher, Phys. Rev. Letters 97 (2006), 193203] predicts the possibility of suppressing the effective two-body interaction in a quasi-one dimensional (1D) cylindrical geometry. Here this low dimensionality effect is analytically modeled by applying a previous formalism to treat low energy quantum scattering processes under tight confinement [J. I. Kim, J. Schmiedmayer and P. Schmelcher, Phys. Rev. A 72 (2005), 042711]. This analytic formalism is improved in order to account non-perturbatively for the three dimensional (3D) scattering phase shift l = 1 for a given spherically symmetric interaction potential between distinguishable colliding particles. The role of the confinement on the scattering properties can then be systematically calculated up to the p-wave.
In this work, a system including two neutral atoms confined to an external one-dimensional Morse potential was modelled. The problem can be relevant to cold atom physics, where neutral atoms may be effectively confined in radially tight tubes formed by optical lattices. The atom-atom interaction was considered as a nonlocal separable potential. Analytical expressions for wave-function as well as transition matrix were derived. The contributions of bound states and resonances in the complex energy plane were calculated. For numerical computations, the bound states in a system of argon gas confined in graphite were considered. Since the most important quantity in the low energy quantum scattering problems is “scattering length,” considering various values of Morse parameters, the behavior of this parameter was described versus the reduced energy.
A Green's function formalism to analyze the scattering properties in confined geometries is developed. This includes scattering from a central field inside the guide created e.g. by impurities. For atomic collisions our approach applies to the case of parabolic confinement and, with certain restrictions, also to an anharmonic one. The coupling between the angular momentum phase shifts δlδ_lδl​ of a spherically symmetric scattering potential V(r)V(r)V(r) due to the cylindrical confinement is analysed. Under these general conditions, a broad range of scattering energies covering many transversal excitations is considered and changes to the bound states of V(r)V(r)V(r) are derived. For collisions between identical atoms, the boson-fermion and fermion-boson mappings are demonstrated.