arXiv:1203.1285

Analytical results for the Morse potential in s-wave ultracold scattering: three dimensional vs. one dimensional problem

Asaf Paris-Mandoki, Rocío Jáuregui

Published Mar 6, 2012• 0 citations

Abstract

Taking advantage of the known analytic expression of the eigenfunctions and eigenenergies of the Morse Hamiltonian, explicit expressions are found for the scattering length $a$ and the effective range $r_e$ which determine the s-wave scattering of ultracold atoms. The effects on $a$ and $r_e$ of considering the radial coordinate in the interval $[0,\infty)$ or in the extended region $(-\infty, \infty)$ are studied in detail.

DOI: 10.48550/arxiv.1203.1285

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arXiv:1203.1285

Analytical results for the Morse potential in s-wave ultracold scattering: three dimensional vs. one dimensional problem

Asaf Paris-Mandoki, Rocío Jáuregui

Published Mar 6, 2012• 0 citations

arXiv HTML PDF

Abstract

Taking advantage of the known analytic expression of the eigenfunctions and eigenenergies of the Morse Hamiltonian, explicit expressions are found for the scattering length

a

and the effective range

r_e

which determine the s-wave scattering of ultracold atoms. The effects on

a

and

r_e

of considering the radial coordinate in the interval

[0,\infty)

or in the extended region

(-\infty, \infty)

are studied in detail.

DOI: 10.48550/arxiv.1203.1285

Related papers

s -wave scattering length of a Gaussian potential

Péter Jeszenszki, Alexander Yu. Cherny, Joachim Brand

We provide accurate expressions for the $s$ -wave scattering length for a Gaussian potential well in one, two, and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold-atomic gases, where the $s$ -wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the $s$ -wave scattering length from the parameters of the Gaussian, but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the $s$ -wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.We provide accurate expressions for the $s$-wave scattering length for a Gaussian potential well in one, two, and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold-atomic gases, where the $s$-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the $s$-wave scattering length from the parameters of the Gaussian, but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the $s$-wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.

arXiv:1802.07063Physical Review AApr 23, 201829 citations

Quantum scattering approach for investigation of two interacting atoms trapped in a one-dimensional Morse potential via Lippmann-Schwinger equation

Maryam Mansoori Kermani, Ali Maghari

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.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.

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On-shell approximation for the s -wave scattering theory

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We investigate the scattering theory of two particles in a generic $D$ -dimensional space. For the $s$ -wave problem, by adopting an on-shell approximation for the $T$ -matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the $s$ -wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the $s$ -wave scattering length ${a}_{s}$ and the $s$ -wave effective range ${r}_{s}$ for $D=3$ , $D=2$ , and $D=1$ . Our results, which are strongly dependent on the spatial dimension $D$ , are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.We investigate the scattering theory of two particles in a generic $D$-dimensional space. For the $s$-wave problem, by adopting an on-shell approximation for the $T$-matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the $s$-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the $s$-wave scattering length ${a}_{s}$ and the $s$-wave effective range ${r}_{s}$ for $D=3$, $D=2$, and $D=1$. Our results, which are strongly dependent on the spatial dimension $D$, are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.

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Determination of the s -Wave Scattering Length of Chromium

Piet O. Schmidt, S. Hensler, J. Werner +4 more

We have measured the deca-triplet $s$ -wave scattering length of the bosonic chromium isotopes $^{52}\mathrm{C}\mathrm{r}$ and $^{50}\mathrm{C}\mathrm{r}$ . From the time constants for cross-dimensional thermalization in ultracold atomic samples, we have determined the magnitudes $|a(^{52}\mathrm{C}\mathrm{r})|=(170\ifmmode\pm\else\textpm\fi{}39){a}_{0}$ and $|a(^{50}\mathrm{C}\mathrm{r})|=(40\ifmmode\pm\else\textpm\fi{}15){a}_{0}$, where ${a}_{0}=0.053\text{ }\mathrm{n}\mathrm{m}$ . By measuring the rethermalization rate of $^{52}\mathrm{C}\mathrm{r}$ over a wide temperature range and comparing the temperature dependence with the effective-range theory and numerical single-channel calculations, we have obtained strong evidence that the sign of $a(^{52}\mathrm{C}\mathrm{r})$ is positive. Rescaling our $^{52}\mathrm{C}\mathrm{r}$ model potential to $^{50}\mathrm{C}\mathrm{r}$ strongly suggests that $a(^{50}\mathrm{C}\mathrm{r})$ is positive also.We have measured the deca-triplet $s$-wave scattering length of the bosonic chromium isotopes $^{52}\mathrm{C}\mathrm{r}$ and $^{50}\mathrm{C}\mathrm{r}$. From the time constants for cross-dimensional thermalization in ultracold atomic samples, we have determined the magnitudes $|a(^{52}\mathrm{C}\mathrm{r})|=(170\ifmmode\pm\else\textpm\fi{}39){a}_{0}$ and $|a(^{50}\mathrm{C}\mathrm{r})|=(40\ifmmode\pm\else\textpm\fi{}15){a}_{0}$, where ${a}_{0}=0.053\text{ }\mathrm{n}\mathrm{m}$. By measuring the rethermalization rate of $^{52}\mathrm{C}\mathrm{r}$ over a wide temperature range and comparing the temperature dependence with the effective-range theory and numerical single-channel calculations, we have obtained strong evidence that the sign of $a(^{52}\mathrm{C}\mathrm{r})$ is positive. Rescaling our $^{52}\mathrm{C}\mathrm{r}$ model potential to $^{50}\mathrm{C}\mathrm{r}$ strongly suggests that $a(^{50}\mathrm{C}\mathrm{r})$ is positive also.

arXiv:quant-ph/0303069Physical Review LettersNov 3, 200349 citations

Scattering and effective interactions of ultracold atoms with spin-orbit coupling

Long Zhang, Youjin Deng, Peng Zhang

We derive an analytical expression for the scattering amplitude of two ultracold atoms of arbitrary spin and with general spin-orbit (SO) coupling, on the basis of our recent work [Phys. Rev. A 86, 053608 (2012)]. As an application, we demonstrate that SO coupling can induce scattering resonance in the case with finite scattering length. The same approach can be applied to calculate the two-body bound state of SO-coupled ultracold atoms. For the ultracold spin- $1/2$ Fermi gases in three- or two- dimensional systems with SO coupling, we also obtain the renormalization relation of effective contact interaction with momentum cutoff, as well as the applicability of Huang-Yang pseudopotential.We derive an analytical expression for the scattering amplitude of two ultracold atoms of arbitrary spin and with general spin-orbit (SO) coupling, on the basis of our recent work [Phys. Rev. A 86, 053608 (2012)]. As an application, we demonstrate that SO coupling can induce scattering resonance in the case with finite scattering length. The same approach can be applied to calculate the two-body bound state of SO-coupled ultracold atoms. For the ultracold spin-$1/2$ Fermi gases in three- or two- dimensional systems with SO coupling, we also obtain the renormalization relation of effective contact interaction with momentum cutoff, as well as the applicability of Huang-Yang pseudopotential.

arXiv:1211.6919Physical Review AMay 31, 201330 citations

Taking advantage of the known analytic expression of the eigenfunctions and eigenenergies of the Morse Hamiltonian, explicit expressions are found for the scattering length

a

and the effective range

r_e

which determine the s-wave scattering of ultracold atoms. The effects on

a

and

r_e

of considering the radial coordinate in the interval

[0,\infty)

or in the extended region

(-\infty, \infty)

are studied in detail.

We provide accurate expressions for the

s

-wave scattering length for a Gaussian potential well in one, two, and three spatial dimensions. The Gaussian potential is widely used as a pseudopotential in the theoretical description of ultracold-atomic gases, where the

s

-wave scattering length is a physically relevant parameter. We first describe a numerical procedure to compute the value of the

s

-wave scattering length from the parameters of the Gaussian, but find that its accuracy is limited in the vicinity of singularities that result from the formation of new bound states. We then derive simple analytical expressions that capture the correct asymptotic behavior of the

s

-wave scattering length near the bound states. Expressions that are increasingly accurate in wide parameter regimes are found by a hierarchy of approximations that capture an increasing number of bound states. The small number of numerical coefficients that enter these expressions is determined from accurate numerical calculations. The approximate formulas combine the advantages of the numerical and approximate expressions, yielding an accurate and simple description from the weakly to the strongly interacting limit.

At long distances, interactions between neutral ground-state atoms can be described by the van der Waals potential. In the ultracold regime, atom-atom scattering is dominated by

s

-waves phase shifts given by an effective range expansion in terms of the scattering length ${\ensuremath{\alpha}}_{0}$ and the effective range

{r}_{0}

. We show that while the scattering length cannot be predicted for these potentials, the effective range is given by the universal low-energy theorem ${r}_{0}=A+B/{\ensuremath{\alpha}}_{0}+C/{\ensuremath{\alpha}}_{0}^{2}$, where

A

B

, and

C

depend on the dispersion coefficients

{C}_{n}

and the reduced diatom mass. We confront this formula to about 100 determinations of

{r}_{0}

and ${\ensuremath{\alpha}}_{0}$ and show why the result is dominated by the leading dispersion coefficient

{C}_{6}

. Universality and scaling extend much beyond naive dimensional analysis estimates.

We investigate the scattering theory of two particles in a generic

D

-dimensional space. For the

s

-wave problem, by adopting an on-shell approximation for the

T

-matrix equation, we derive analytical formulas which connect the Fourier transform $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)$ of the interaction potential to the

s

-wave phase shift. In this way we obtain explicit expressions of the low-momentum parameters ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}$ and ${\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}$ of $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{V}(k)={\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{0}+{\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{g}}_{2}{k}^{2}+\ensuremath{\cdots}$ in terms of the

s

-wave scattering length

{a}_{s}

and the

s

-wave effective range

{r}_{s}

for

D=3

D=2

, and

D=1

. Our results, which are strongly dependent on the spatial dimension

D

, are a useful benchmark for few-body and many-body calculations. As a specific application, we derive the zero-temperature pressure of a two-dimensional uniform interacting Bose gas with a beyond-mean-field correction which includes both scattering length and effective range.

We have measured the deca-triplet

s

-wave scattering length of the bosonic chromium isotopes

^{52}\mathrm{C}\mathrm{r}

and

^{50}\mathrm{C}\mathrm{r}

. From the time constants for cross-dimensional thermalization in ultracold atomic samples, we have determined the magnitudes $|a(^{52}\mathrm{C}\mathrm{r})|=(170\ifmmode\pm\else\textpm\fi{}39){a}_{0}$ and $|a(^{50}\mathrm{C}\mathrm{r})|=(40\ifmmode\pm\else\textpm\fi{}15){a}_{0}$, where

{a}_{0}=0.053\text{ }\mathrm{n}\mathrm{m}

. By measuring the rethermalization rate of

^{52}\mathrm{C}\mathrm{r}

over a wide temperature range and comparing the temperature dependence with the effective-range theory and numerical single-channel calculations, we have obtained strong evidence that the sign of

a(^{52}\mathrm{C}\mathrm{r})

is positive. Rescaling our

^{52}\mathrm{C}\mathrm{r}

model potential to

^{50}\mathrm{C}\mathrm{r}

strongly suggests that

a(^{50}\mathrm{C}\mathrm{r})

is positive also.

1/2

Fermi gases in three- or two- dimensional systems with SO coupling, we also obtain the renormalization relation of effective contact interaction with momentum cutoff, as well as the applicability of Huang-Yang pseudopotential.