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arXiv:1201.2962 · New Journal of Physics

Effective renormalized multi-body interactions of harmonically confined ultracold neutral bosons

P R Johnson, D. Blume, Xiangyu Yin, William F. Flynn, Eite Tiesinga

Published May 25, 2012• 29 citations
arXivHTMLPDF

Abstract

We calculate the renormalized effective two-, three- and four-body interactions for N neutral ultracold bosons in the ground state of an isotropic harmonic trap, assuming two-body interactions modeled with the combination of a zero-range and energy-dependent pseudopotential. We work to third-order in the scattering length a t (0) defined at zero collision energy, which is necessary to obtain both the leading-order effective four-body interaction and consistently include corrections for realistic two-body interactions. The leading-order, effective three- and four-body interaction energies are and , where ω and σ ( ω ) are the harmonic oscillator frequency and length, respectively, and energies are in units of ℏ ω . The one-standard deviation error ±0.0001 for the third-order coefficient in U 3 ( ω ) is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. The effective three- and four-body interactions can play an important role in the dynamics of tightly confined and strongly correlated systems. We also performed numerical simulations for a finite-range (FR) boson–boson potential, and it was comparison to the zero-range predictions which revealed that finite-range effects must be taken into account for a realistic third-order treatment. In particular, we show that the energy-dependent pseudopotential accurately captures, through third order, the finite-range physics, and in combination with the multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.

DOI
10.1088/1367-2630/14/5/053037

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We calculate the renormalized effective two-, three- and four-body interactions for N neutral ultracold bosons in the ground state of an isotropic harmonic trap, assuming two-body interactions modeled with the combination of a zero-range and energy-dependent pseudopotential. We work to third-order in the scattering length a t (0) defined at zero collision energy, which is necessary to obtain both the leading-order effective four-body interaction and consistently include corrections for realistic two-body interactions. The leading-order, effective three- and four-body interaction energies are and , where ω and σ ( ω ) are the harmonic oscillator frequency and length, respectively, and energies are in units of ℏ ω . The one-standard deviation error ±0.0001 for the third-order coefficient in U 3 ( ω ) is due to numerical uncertainty in estimating a slowly converging sum; the other two coefficients are either analytically or numerically exact. The effective three- and four-body interactions can play an important role in the dynamics of tightly confined and strongly correlated systems. We also performed numerical simulations for a finite-range (FR) boson–boson potential, and it was comparison to the zero-range predictions which revealed that finite-range effects must be taken into account for a realistic third-order treatment. In particular, we show that the energy-dependent pseudopotential accurately captures, through third order, the finite-range physics, and in combination with the multi-body effective interactions gives excellent agreement with the numerical simulations, validating our theoretical analysis and predictions.
We study systems of few two-component fermions interacting via short-range interactions within a harmonic-oscillator trap. The dominant interactions, which are two-body interactions, are organized according to the number of derivatives and defined in a two-body truncated model space made from a bound-state basis. Leading-order (LO) interactions are solved for exactly using the formalism of the no-core shell model, whereas corrections are treated as many-body perturbations. We show explicitly that next-to-LO and next-to-next-to-LO interactions improve convergence as the model space increases. We present results at unitarity for three- and four-fermion systems, which show excellent agreement with the exact solution (for the three-body problem) and results obtained by other methods (in the four-body case). We also present results for finite scattering lengths and nonzero range of the interaction, including (at positive scattering length) observation of a change in the structure of the three-body ground state and extraction of the atom-dimer scattering length.
Ultracold atomic gases have been of great theoretical and experimental interest in the last two decades. In these systems, the de Broglie wavelength of the particles is much greater than the two body van der Waals length. As a result, the details of the two body interaction potential are irrelevant for a large number of applications and the realistic two body interaction potential can be replaced by a simple finite range or zero range model potential that reproduces the scattering quantities of the full interaction potential. This thesis develops zero range pseudopotentials and applies them to trapped two-particle systems. Ultracold gases loaded into optical lattices can be used to realize two particle systems under approximately harmonic confinement. We use pseudopotentials to obtain the eigenspectrum of two particles under external harmonic confinement semi-analytically. Advancements in trapping technology have resulted in the realization of low dimensional systems. We develop pseudopotentials to treat two body interactions in one and two dimensions. We also elaborate on the physics that is unique to one and two dimensional systems. Feshbach resonances allow for the tunability of the effective two body interaction strength in the presence of a magnetic field. To model Feshbach resonances in two and three dimensions we develop coupled two channel zero range potentials. Dipole-dipole interactions in Chromium and polar molecules have been the subject of a lot of recent research. Unlike the interactions between two alkali atoms, these interactions are long range and anisotropic. We explore the scattering properties of two aligned dipoles using a simple shape dependent model potential. To understand a system two aligned dipoles under confinement, we develop a pseudopotential treatment for cylindrically symmetric interaction potentials under cylindrically symmetric harmonic confinement. This pseudopotential can be used to model any cylindrically symmetric interaction potential. Our zero range treatment describes many features of two aligned dipoles under spherically symmetric harmonic confinement accurately but breaks down when applied to two aligned dipoles under cylindrically symmetric harmonic confinement. The breakdown is unique to the long range nature of the dipole-dipole interaction and expected to be absent for short range interactions.
Ultracold atomic systems have been of great theoretical and experimental interest in the past few decades.They provide ideal paradigms to study quantum phenomena, simulate condensed matter systems, and could potentially become a platform for quantum computations.This thesis focuses on the theoretical investigation of universal and non-universal properties of ultracold few-atom systems.The goal of this thesis is to answer or provide insights into several interesting open questions.First, what is the role of the non-universal three-body parameter in weakly-interacting harmonically trapped Bose gases and at which order does it kick in?Second, how large is the ground state energy of small two-component Fermi gases with infinitely strong interactions compared to their non-interacting counterparts?Lastly, what is the interplay between the spin-orbit coupling and the s-wave two-body interaction and how does it affect the energy spectrum of the two-atom system?In order to answer these questions, we develop and employ different analytical and numerical techniques.A perturbative effective field theory treatment to study the weaklyinteracting trapped Bose gas is employed.The treatment is extended to up to fourth-order perturbation theory and techniques to regularize the diverging sums are developed.A logarithmic divergence in the three-body sector is identified and the need for a non-universal three-body parameter is shown.v The explicitly correlated Gaussian method is used to calculate the ground state energy of small trapped two-component Fermi gases with up to ten particles.A code is developed from scratch and parallelized.An improved optimization scheme is developed.A novel technique to remove the finite-range dependence is proposed and tested.Our calculations serve as benchmarks for other numerical techniques.The explicitly correlated Gaussian method is extended to treat systems with periodic boundary conditions for the first time.Standard perturbation theory is used to treat the spin-orbit coupling term in trapped two-atom systems.Our treatment is based on the exact two-body zero-range solution in the trap.The interplay between the spin-orbit coupling and the two-body s-wave interaction is investigated and shown to have a great impact on the energy spectrum for certain parameter combinations.
The two-body scattering amplitude and energy spectrum of confined ultracold atoms are of fundamental importance for both theoretical and experimental studies of ultracold-atom physics. For many systems, one can efficiently calculate these quantities via the zero-range Huang-Yang pseudopotential (HYP), in which the interatomic interaction is characterized by the scattering length aaa. Furthermore, when the scattering length is dependent on the kinetic energy ${\ensuremath{\varepsilon}}_{\mathrm{r}}$ of two-atom relative motion, i.e., $a=a({\ensuremath{\varepsilon}}_{\mathrm{r}})$, the results are applicable for a broad energy region. However, when the free Hamiltonian of atomic internal state (e.g., the Zeeman Hamiltonian) does not commute with the interatomic interaction, or the center-of-mass (c.m.) motion is coupled to the relative motion, the generalization of this technique is still lacking. In this work we solve this problem and construct a reasonable energy-dependent multichannel HYP, which is characterized by a ``scattering length operator'' ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$, for the above complicated cases. Here, ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$ is an operator for atomic internal states and c.m. motion, and depends on both the total two-atom energy and the external field as well as the trapping parameter. The effects from the internal-state or c.m.-relative motion coupling can be self-consistently taken into account by ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$. We further show a method based on the quantum defect theory, with which ${\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{a}}_{\mathrm{eff}}$ can be analytically derived for systems with van der Waals interatomic interaction. To demonstrate our method, we calculate the spectrum of two ultracold fermionic alkaline-earth-like atoms [in electronic 1S0^{1}S_{0}1S0​ $(|g\ensuremath{\rangle})$ and 3P0^{3}P_{0}3P0​ $(|e\ensuremath{\rangle})$ states, respectively] confined in an optical lattice. By comparing our results with the recent experimental measurements for two 173Yb^{173}\mathrm{Yb}173Yb atoms and two 171Yb^{171}\mathrm{Yb}171Yb atoms, we calibrate the scattering lengths ${a}_{\ifmmode\pm\else\textpm\fi{}}$ with respect to antisymmetric and symmetric nuclear-spin states to be a+=2012(19)a0{a}_{+}=2012(19){a}_{0}a+​=2012(19)a0​ and ${a}_{\ensuremath{-}}=193(4){a}_{0}$ for 173Yb^{173}\mathrm{Yb}173Yb, and a+=232(3)a0{a}_{+}=232(3){a}_{0}a+​=232(3)a0​ and ${a}_{\ensuremath{-}}=372(1){a}_{0}$ for 171Yb^{171}\mathrm{Yb}171Yb.
Abstract We study the effects of the effective range of interaction on the eigenvalues and eigenstates of two particles confined in a three-dimensional (3D) isotropic as well as one- or quasi-one dimensional harmonic (1D) traps. For this we employ model potentials which mimic finite-range s -wave interactions over a wide range of s -wave scattering length a s including the unitarity limits a s → ± ∞ . Our results show that when the range is larger than the 3D or 1D harmonic oscillator length scale, the eigenvalues and eigenstates are nearly similar to those of noninteracting two particles in the 3D or 1D trap, respectively. In case of 3D, we find that when the range goes to zero, the results of contact potential as derived by Busch et al (1998 Foundations of Physics 28 549 ) are reproduced. However, in the case of 1D, such reproducibility does not occur as the range goes to zero. We have calculated the eigenvalues and eigenstates in a 1D harmonic trap taking one dimensional finite-range model potential. We have also calculated the bound state properties of two particles confined in a highly anisotropic quasi-1D trap taking three-dimensional finite-range model potential, and examined whether these quasi-1D results approach towards 1D ones as the aspect ratio η of the radial to axial frequency of the trap increases. We find that if the range is very small compared to the axial size of the trap, then one can reach 1D regime for η ≥ 10000 . However, for a large range, one can almost get 1D results for smaller values of η . This study will be important for the exploration of two-body or many body physics of trapped ultracold atoms interacting with narrow Feshbach resonance for which the effective range can be large.
We consider a harmonically trapped dilute NNN-boson system described by a low-energy Hamiltonian with pairwise interactions. We determine the condensate fraction, defined in terms of the largest occupation number, of the weakly interacting NNN-boson system ($N\ensuremath{\geqslant}2$) by employing a perturbative treatment within the framework of second quantization. The one-body density matrix and the corresponding occupation numbers are compared with those obtained by solving the two-body problem with zero-range interactions exactly. Our expressions are also compared with high precision ab initio calculations for Bose gases with N=2−−4N=2--4N=2−−4 that interact through finite-range two-body model potentials. Nonuniversal corrections are identified to enter at subleading order, confirming that different low-energy Hamiltonians, constructed to yield the same energy, may yield different occupation numbers. Lastly, we consider the strongly interacting three-boson system under spherically symmetric harmonic confinement and determine its occupation numbers as a function of the three-body ``Efimov parameter.''