Bogoliubov theory {N. N. Bogoliubov, Izv. Akad. Nauk Ser. Fiz. 11, 77 (1947) [J. Phys. (USSR) 11, 23 (1947) (in English)]} provides important predictions for the low energy properties of the weakly interacting Bose gas. Recently, Bogoliubov’s predictions were justified rigorously by Boccato et al. [Acta Math. 222(2), 219–335 (2019)] for translation invariant systems in the Gross–Pitaveskii regime, where N bosons in Λ=[0;1]3⊂R3 interact through a potential whose scattering length is of size N−1. In this article, we review recent results from the work of Brennecke et al. [Ann. Henri Poincaré 23, 1583–1658 (2022)], a joint work with Schlein and Schraven, which extends the analysis for translation invariant systems to systems of bosons in R3 that are trapped by an external potential.Bogoliubov theory {N. N. Bogoliubov, Izv. Akad. Nauk Ser. Fiz. 11, 77 (1947) [J. Phys. (USSR) 11, 23 (1947) (in English)]} provides important predictions for the low energy properties of the weakly interacting Bose gas. Recently, Bogoliubov’s predictions were justified rigorously by Boccato et al. [Acta Math. 222(2), 219–335 (2019)] for translation invariant systems in the Gross–Pitaveskii regime, where N bosons in Λ=[0;1]3⊂R3 interact through a potential whose scattering length is of size N−1. In this article, we review recent results from the work of Brennecke et al. [Ann. Henri Poincaré 23, 1583–1658 (2022)], a joint work with Schlein and Schraven, which extends the analysis for translation invariant systems to systems of bosons in R3 that are trapped by an external potential.

arXiv:2202.12294Journal of Mathematical PhysicsMay 1, 20225 citations

Possibility of forming a stable Bose-Einstein condensate of $2\,^{3}\!S_1$ positronium atomsPossibility of forming a stable Bose-Einstein condensate of $2\,^{3}\!S_1$ positronium atoms

Y. Zhang, M.-S. Wu, J.-Y. Zhang +3 more

The confined variational method in conjunction with the orthogonalizing pseudo-potential method and the stabilization method is used to study the low energy elastic scattering between two spin-polarized metastable positronium Ps(2\, $^{3}\!S_1$ ) atoms. Explicitly correlated Gaussian basis functions are adopted to properly describe the complicated Coulomb interaction among the four charged particles. The calculated $s$ -wave scattering length ( $\approx8.5\,a_0$ ) is positive, indicating the possibility of forming a stable Bose-Einstein condensate of fully spin-polarized $\text{Ps}(2\,^{3}\!S_1)$ atoms. Our results will open a new way of experimental realization of Ps condensate and development of $γ$ -ray and $\text{Ps}(2\,^{3}\!S_1)$ atom lasers.The confined variational method in conjunction with the orthogonalizing pseudo-potential method and the stabilization method is used to study the low energy elastic scattering between two spin-polarized metastable positronium Ps(2\,$^{3}\!S_1$) atoms. Explicitly correlated Gaussian basis functions are adopted to properly describe the complicated Coulomb interaction among the four charged particles. The calculated $s$-wave scattering length ($\approx8.5\,a_0$) is positive, indicating the possibility of forming a stable Bose-Einstein condensate of fully spin-polarized $\text{Ps}(2\,^{3}\!S_1)$ atoms. Our results will open a new way of experimental realization of Ps condensate and development of $γ$-ray and $\text{Ps}(2\,^{3}\!S_1)$ atom lasers.

arXiv:1903.08353Mar 20, 20192 citations

Elementary Potential Scattering

Emilio d’Emilio, Luigi E. Picasso

UNITEXT for physicsJan 1, 20170 citations

Focusing of Carr-Purcell photon echoes, and collisional effects

C. V. Heer

Since for echo formation the off-diagonal density matrix element ${\ensuremath{\sigma}}_{\mathrm{ab}}$ evolves from ${\ensuremath{\sigma}}_{\mathrm{ba}}={\ensuremath{\sigma}}_{\mathrm{ab}}^{*}$ of the previous pulse, the eikonal $\ensuremath{\phi}\ensuremath{-}\ensuremath{\omega}t$ reverses sign with each pulse. If the radius of curvature of the first $\frac{\ensuremath{\pi}}{2}$ pulse is ${R}_{1}$ , the radius of curvature of the subsequent $\ensuremath{\pi}$ pulses ${R}_{2}$ , ${R}_{3}$ , etc., the radius of curvature of the photon echo which follows the second pulse is $\frac{1}{{R}_{E1}}=\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, the radius of curvature of the Carr-Purcell echo following the third pulse is given by $\frac{1}{{R}_{E2}}=\frac{2}{{R}_{3}}\ensuremath{-}\frac{2}{{R}_{2}}+\frac{1}{{R}_{1}}$, and for subsequent Carr-Purcell echoes by $\frac{1}{{R}_{E3}}=\frac{2}{{R}_{4}}\ensuremath{-}\frac{2}{{R}_{3}}+\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, etc. The influence of motion, adiabatic collisions, and velocity-changing collisions on the decay of echo amplitude is considered for curved wave fronts. Since the change in position from the constant-velocity position is important in the decay, a statistical theory for small displacement and its relationship to the forward scattering cross section is given. These results with ${r}^{\ensuremath{-}3}$ and ${r}^{\ensuremath{-}6}$ interactions between colliding molecules are compared with the experimental data for ${\mathrm{CH}}_{3}$ F and with an ${r}^{\ensuremath{-}6}$ potential with the data for ${\mathrm{SF}}_{6}$ and ${\mathrm{SiF}}_{4}$ .Since for echo formation the off-diagonal density matrix element ${\ensuremath{\sigma}}_{\mathrm{ab}}$ evolves from ${\ensuremath{\sigma}}_{\mathrm{ba}}={\ensuremath{\sigma}}_{\mathrm{ab}}^{*}$ of the previous pulse, the eikonal $\ensuremath{\phi}\ensuremath{-}\ensuremath{\omega}t$ reverses sign with each pulse. If the radius of curvature of the first $\frac{\ensuremath{\pi}}{2}$ pulse is ${R}_{1}$, the radius of curvature of the subsequent $\ensuremath{\pi}$ pulses ${R}_{2}$, ${R}_{3}$, etc., the radius of curvature of the photon echo which follows the second pulse is $\frac{1}{{R}_{E1}}=\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, the radius of curvature of the Carr-Purcell echo following the third pulse is given by $\frac{1}{{R}_{E2}}=\frac{2}{{R}_{3}}\ensuremath{-}\frac{2}{{R}_{2}}+\frac{1}{{R}_{1}}$, and for subsequent Carr-Purcell echoes by $\frac{1}{{R}_{E3}}=\frac{2}{{R}_{4}}\ensuremath{-}\frac{2}{{R}_{3}}+\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, etc. The influence of motion, adiabatic collisions, and velocity-changing collisions on the decay of echo amplitude is considered for curved wave fronts. Since the change in position from the constant-velocity position is important in the decay, a statistical theory for small displacement and its relationship to the forward scattering cross section is given. These results with ${r}^{\ensuremath{-}3}$ and ${r}^{\ensuremath{-}6}$ interactions between colliding molecules are compared with the experimental data for ${\mathrm{CH}}_{3}$F and with an ${r}^{\ensuremath{-}6}$ potential with the data for ${\mathrm{SF}}_{6}$ and ${\mathrm{SiF}}_{4}$.

Physical Review AMay 1, 197614 citations

Potential Scattering

WORLD SCIENTIFIC eBooksJan 1, 20110 citations

Possibility of forming a stable Bose-Einstein condensate of

2\,^{3}\!S_1

positronium atoms

^{3}\!S_1

) atoms. Explicitly correlated Gaussian basis functions are adopted to properly describe the complicated Coulomb interaction among the four charged particles. The calculated

s

-wave scattering length (

\approx8.5\,a_0

) is positive, indicating the possibility of forming a stable Bose-Einstein condensate of fully spin-polarized

\text{Ps}(2\,^{3}\!S_1)

atoms. Our results will open a new way of experimental realization of Ps condensate and development of

γ

-ray and

\text{Ps}(2\,^{3}\!S_1)

atom lasers.

{R}_{1}

, the radius of curvature of the subsequent $\ensuremath{\pi}$ pulses

{R}_{2}

{R}_{3}

, etc., the radius of curvature of the photon echo which follows the second pulse is $\frac{1}{{R}_{E1}}=\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, the radius of curvature of the Carr-Purcell echo following the third pulse is given by $\frac{1}{{R}_{E2}}=\frac{2}{{R}_{3}}\ensuremath{-}\frac{2}{{R}_{2}}+\frac{1}{{R}_{1}}$, and for subsequent Carr-Purcell echoes by $\frac{1}{{R}_{E3}}=\frac{2}{{R}_{4}}\ensuremath{-}\frac{2}{{R}_{3}}+\frac{2}{{R}_{2}}\ensuremath{-}\frac{1}{{R}_{1}}$, etc. The influence of motion, adiabatic collisions, and velocity-changing collisions on the decay of echo amplitude is considered for curved wave fronts. Since the change in position from the constant-velocity position is important in the decay, a statistical theory for small displacement and its relationship to the forward scattering cross section is given. These results with ${r}^{\ensuremath{-}3}$ and ${r}^{\ensuremath{-}6}$ interactions between colliding molecules are compared with the experimental data for

{\mathrm{CH}}_{3}

F and with an ${r}^{\ensuremath{-}6}$ potential with the data for

{\mathrm{SF}}_{6}

and

{\mathrm{SiF}}_{4}