arXiv subjects

Shina Tan

Publications and source records attributed to Shina Tan.

At least 19 recordsLinked to original sources

arXiv2026

Three-body scattering area of identical bosons in two dimensions

We study the wave function $\phi^{(3)}$ of three identical bosons scattering at zero energy, zero total momentum, and zero orbital angular momentum in two dimensions, interacting via short-range potentials with a finite two-body scattering length $a$. We derive asymptotic expansions of $\phi^{(3)}$ in two regimes: the 111-expansion, where all three pairwise distances are large, and the 21-expansion, where one particle is far from the other two. In the 111-expansion, the leading term grows as $\ln^3(B/a)$ at large hyperradius $B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}$. At order $B^{-2}\ln^{-3}(B/a)$, we identify a three-body parameter $D$ with dimension of length squared, which we term the three-body scattering area. This quantity should be contrasted with the three-body scattering area previously studied for infinite or vanishing two-body scattering length. If the two-body interaction is attractive and supports bound states, $D$ acquires a negative imaginary part, and we derive its relation to the probability amplitudes for the production of two-body bound states in three-body collisions. Under weak modifications of the interaction potentials, we derive the corresponding shift of $D$ in terms of $\phi^{(3)}$ and the changes of the two-body and three-body potentials. We also study the effects of $D$ and $\phi^{(3)}$ on three-body and many-body physics, including the three-body ground-state energy in a large periodic volume, the many-body energy and the three-body correlation function of the dilute two-dimensional Bose gas, and the three-body recombination rates of two-dimensional ultracold atomic Bose gases.

cond-mat.quant-gas↗

arXiv2026

Scattering of a weakly bound dimer from a hard wall in one dimension

We consider a dimer formed by two particles with an attractive contact interaction in one dimension, colliding with a hard wall. We compute the scattering phase shifts and the reflection coefficients for various collision energies and various mass ratios of the two particles. For low-energy collisions (with dimer kinetic energies much smaller than the binding energy) our results are consistent with those of D. Lee and M. Pine, The European Physical Journal A 47, 41 (2011). For mass ratios much greater than 1 we use the Born-Oppenheimer approximation to show that the scattering length and the effective range of the dimer-wall collision both depend logarithmically on the mass ratio. For collision energies much greater than the binding energy, the dissociation probability is inversely proportional to the square of the incident momentum of the dimer and we find the constant of proportionality analytically, and we use a semiclassical analysis to approximately derive the ``angular distribution" of the dissociated pair, where the ``angle" $\theta$ depends on the ratio of the velocities of the two outgoing unbound particles.

cond-mat.quant-gas↗

arXiv2025

Three-boson scattering hypervolume for a nonzero orbital angular momentum

We analyze the zero energy collision of three identical bosons in the same internal state with total orbital angular momentum $L=2$, assuming short range interactions. By solving the Schr\"odinger equation asymptotically, we derive two expansions of the wave function when three bosons are far apart or a pair of bosons and the third boson are far apart. The scattering hypervolume $D$ is defined for this collision. Unlike the scattering hypervolume defined by one of us in 2008, whose dimension is length to the fourth power, the dimension of $D$ studied in the present paper is length to the eighth power. We then derive the expression of $D$ when the interaction potentials are weak, using the Born's expansion. We also calculate the energy shift of such three bosons with three different momenta $\hbar \mathbf{k_{1}}$, $\hbar\mathbf{k_{2}}$ and $\hbar\mathbf{k_{3}}$ in a large periodic box. The obtained energy shift depends on $D^{(0)}/\Omega^{2}$ and $D/\Omega^{2}$, where $D^{(0)}$ is the three-body scattering hypervolume defined for the three-body $L=0$ collision and $\Omega$ is the volume of the periodic box. We also calculate the contribution of $D$ to the three-body T-matrix element for low-energy collisions. We then calculate the shift of the energy and the three-body recombination rate due to $D^{(0)}$ and $D$ in the dilute homogeneous Bose gas. The contribution to the three-body recombination rate constant from $D$ is proportional to $T^2$ if the temperature $T$ is much larger than the quantum degeneracy temperature but still much lower than the temperature scale at which the thermal de Broglie wave length becomes comparable to the physical range of interaction.

cond-mat.quant-gas↗

arXiv2025

Three-body scattering hypervolume of two-component fermions in three dimensions

We study the zero-energy collision of three fermions, two of which are in the spin-down ($\downarrow$) state and one of which is in the spin-up ($\uparrow$) state. Assuming that the two-body and the three-body interactions have a finite range, we find a parameter, $D$, called the three-body scattering hypervolume. We study the three-body wave function asymptotically when three fermions are far apart or one spin-$\uparrow$ (spin-$\downarrow$) fermion and one pair, formed by the other two fermions, are far apart, and derive three asymptotic expansions of the wave function. The three-body scattering hypervolume $D$ appears in the coefficients of such expansions at the order of $B^{-5}$, where $B=\sqrt{(s_1^2+s_2^2+s_3^2)/2}$ is the hyperradius of the triangle formed by the three fermions (we assume that the three fermions have the same mass), and $s_1,s_2,s_3$ are the sides of the triangle. We compute the $T$-matrix element for three such fermions colliding at low energy in terms of $D$ in the absence of two-body interactions. When the interactions are weak, we calculate $D$ approximately using the Born expansion. We also analyze the energy shift of three two-component fermions in a large periodic cube due to $D$ and generalize this result to the many-fermion system. $D$ also determines the three-body recombination rates in two-component Fermi gases, and we calculate the three-body recombination rates in terms of $D$ and the density and temperature of the gas.

cond-mat.quant-gas↗

arXiv2024

Three-body scattering area for particles with infinite or zero scattering length in two dimensions

We derive the asymptotic expansions of the wave function of three particles having equal mass with finite-range interactions and infinite or zero two-dimensional scattering length colliding at zero energy and zero orbital angular momentum, from which a three-body parameter $D$ is defined. The dimension of $D$ is length squared, and we call $D$ three-body scattering area. We find that the ground state energy per particle of a zero-temperature dilute Bose gas with these interactions is approximately $\frac{\hbar^2 D }{6m}\rho^2$, where $\rho$ is the number density of the bosons, $m$ is the mass of each boson, and $\hbar$ is Planck's constant over $2\pi$. Such a Bose gas is stable at $D\geq 0$ in the thermodynamic limit, and metastable at $D<0$ in the harmonic trap if the number of bosons is less than $N_{cr}\approx 3.6413 \sqrt{\frac{\hbar}{m\omega |D|}}$, where $\omega$ is the angular frequency of the harmonic trap. If the two-body interaction supports bound states, $D$ typically acquires a negative imaginary part, and we find the relation between this imaginary part and the amplitudes of the pair-boson production processes. We derive a formula for the three-body recombination rate constant of the many-boson system in terms of the imaginary part of $D$.

cond-mat.quant-gas↗

arXiv2023

Universal Energy Functionals for Trapped Fermi Gases in Low Dimensions

We study the system of trapped two-component Fermi gases with zero-range interaction in two dimensions (2D) or one dimension (1D). We calculate the one-particle density matrices of these systems at small displacements, from which we show that the $N$-body energies are linear functionals of the occupation probabilities of single-particle energy eigenstates. A universal energy functional was first derived in 2011 for trapped zero-range interacting two-component Fermi gases in three dimensions (3D). We also calculate the asymptotic behaviors of the occupation probabilities of single-particle energy eigenstates at high energies.

cond-mat.quant-gas↗

arXiv2023

The three-body scattering hypervolume of identical fermions in one dimension

We study the zero-energy collision of three identical spin-polarized fermions with short-range interactions in one dimension. We derive the asymptotic expansions of the three-body wave function when the three fermions are far apart or one pair and the third fermion are far apart, and the three-body scattering hypervolume $D_F$ appears in the coefficients of such expansions. If the two-body interaction is attractive and supports two-body bound states, $D_F$ acquires a negative imaginary part related to the amplitudes of the outgoing waves describing the departure of the resultant bound pair and the remaining free fermion. For weak interaction potentials, we derive an approximate formula of the hypervolume by using the Born expansion. For the square-barrier and the square-well potentials and the Gaussian potential, we solve the three-body Schr\"{o}dinger equation to compute $D_F$ numerically. We also calculate the shifts of energy and of pressure of spin-polarized one-dimensional Fermi gases due to a nonzero $D_F$ and the three-body recombination rate in one dimension.

cond-mat.quant-gas↗

arXiv2022

Scattering Hypervolume of Fermions in Two Dimensions

We define the three-body scattering hypervolume $D_F$ for identical spin-polarized fermions in two dimensions, by considering the wave function of three such fermions colliding at zero energy and zero orbital angular momentum. We derive the asymptotic expansions of such a wave function when three fermions are far apart or one pair and the third fermion are far apart, and $D_F$ appears in the coefficients of such expansions. For weak interaction potentials, we derive an approximate formula of $D_F$ by using the Born expansion. We then study the shift of energy of three such fermions in a large periodic area due to $D_F$. This shift is proportional to $D_F$ times the square of the area of the triangle formed by the momenta of the fermions. We also calculate the shifts of energy and of pressure of spin-polarized two-dimensional Fermi gases due to a nonzero $D_F$ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases in two dimensions.

cond-mat.quant-gas↗

arXiv2021

Scattering Hypervolume of Spin-Polarized Fermions

We analyze the collision of three identical spin-polarized fermions at zero collision energy, assuming arbitrary finite-range potentials, and define the corresponding three-body scattering hypervolume $D_F$. The scattering hypervolume $D$ was first defined for identical bosons in 2008 by one of us. It is the three-body analog of the two-body scattering length. We solve the three-body Schrödinger equation asymptotically when the three fermions are far apart or one pair and the third fermion are far apart, deriving two asymptotic expansions of the wave function. Unlike the case of bosons for which $D$ has the dimension of length to the fourth power, here the $D_F$ we define has the dimension of length to the eighth power. We then analyze the interaction energy of three such fermions with momenta $\hbar\mathbf{k}_1$, $\hbar\mathbf{k}_2$ and $\hbar\mathbf{k}_3$ in a large periodic cubic box. The energy shift due to $D_F$ is proportional to $D_F/Ω^2$, where $Ω$ is the volume of the box. We also calculate the shifts of energy and pressure of spin-polarized Fermi gases due to a nonzero $D_F$ and the three-body recombination rate of spin-polarized ultracold atomic Fermi gases at finite temperatures.

cond-mat.quant-gas↗

arXiv2021

Three-Body Scattering Hypervolume of Particles with Unequal Masses

We analyze the collision of three particles with arbitrary mass ratio at zero collision energy, assuming arbitrary short-range potentials, and generalize the three-body scattering hypervolume $D$ first defined for identical bosons in 2008. We solve the three-body Schrödinger equation asymptotically when the three particles are far apart or one pair and a third particle are far apart, deriving two asymptotic expansions of the wave function, and the parameter $D$ appears at the order $1/B^4$, where $B$ is the overall size of the triangle formed by the particles. We then analyze the ground state energy of three such particles with vanishing or negligible two-body scattering lengths in a large periodic volume of side length $L$, where the three-body parameter contributes a term of the order $D/L^6$. From this result we derive some properties of a two-component Bose gas with negligible two-body scattering lengths: its energy density at zero temperature, the corresponding generalized Gross-Pitaevskii equation, the conditions for the stability of the two-component mixture against collapse or phase separation, and the decay rates of particle densities due to three-body recombination.

physics.atom-ph↗

arXiv2019

$d$-dimensional Lüscher's formula and the near-threshold three-body states in a finite volume

We study two particles colliding in a $d$-dimensional finite volume and generalize Lüscher's formula to arbitrary $d$ spatial dimensions. We obtain the $s$- and $p$-wave approximations of the generalized Lüscher's formula. For resonant $s$- or $p$-wave interactions, we analytically determine the energies of the low-lying states at large box size $L$. At $s$-wave resonance, we discover two low-lying states with nearly opposite energies, which are proportional to $\pm 1/L^{d/2}$ for $d\ge 5$, or $\pm 1/L^{2}\sqrt{\ln L}$ for $d=4$. This provides important insights into the near-threshold states of three bosons at a three-body resonance in a 2- or higher-dimensional finite volume.

nucl-th↗

arXiv2018

Comparative study of one-dimensional Bose and Fermi gases with contact interactions from the viewpoint of universal relations for correlation functions

One-dimensional spinless Bose and Fermi gases with contact interactions have the close interrelation via Girardeau's Bose-Fermi mapping, leading to the correspondences in their energy spectra and thermodynamics. However, correlation functions are in general not identical between these systems. We derive in both systems the exact universal relations for correlation functions, which hold for any energy eigenstate and any statistical ensemble of the eigenstates with or without a trapping potential. These relations include the large-momentum behaviors of static structure factors and of momentum distributions as well as energy relations, which connect the sums of kinetic and interaction energies to the momentum distributions. The relations involve two- and three-body contacts, which are the integrals of local pair and triad correlations, respectively. We clarify how the relations for bosons and fermions differ and are connected with each other. In particular, we find that the three-body contact makes no contribution to the bosonic energy relation, but it plays a crucial role in the fermionic one. In addition, we compute the exact momentum distribution for any number of fermions in the unitary limit.

cond-mat.quant-gas↗

arXiv2017

Contact theory for spin-orbit-coupled Fermi gases

We develop the contact theory for spin-orbit-coupled Fermi gases. By using a perturbation method, we derive analytically the universal two-body behavior at short distance, which does not depend on the short-range details of interatomic potentials. We find that two new scattering parameters need to be introduced because of spin-orbit coupling, besides the traditional $s$- and $p$-wave scattering length (volume) and effective ranges. This is a general and unique feature for spin-orbit-coupled systems. Consequently, two new adiabatic energy relations with respect to the new scattering parameters are obtained, in which a new contact is involved because of spin-orbit coupling. In addition, we derive the asymptotic behavior of the large-momentum distribution, and find that the subleading tail is corrected by the new contact. This work paves the way for exploring the profound properties of spin-orbit-coupled many-body systems, according to two-body solutions.

cond-mat.quant-gas↗

arXiv2017

Three-body scattering hypervolumes of particles with short-range interactions

The low-energy scattering of three bosons or distinguishable particles with short-range interactions is characterized by a fundamental parameter, the three-body scattering hypervolume. Its imaginary part is directly related to the three-body recombination rate in a quantum gas consisting of such particles. We derive an analytical formula of it for weak interactions, and perform its first numerical calculations for bosons with a variable nonzero-range potential. For attractive interactions, we identify several three-body resonances at which the three-body scattering hypervolume becomes divergent or anomalously large.

cond-mat.quant-gas↗

arXiv2016

Higher-order local and non-local correlations for 1D strongly interacting Bose gas

The correlation function is an important quantity in the physics of ultracold quantum gases because it provides information about the quantum many-body wave function beyond the simple density profile. In this paper we first study the $M$-body local correlation functions, $g_M$, of the one-dimensional (1D) strongly repulsive Bose gas within the Lieb-Liniger model using the analytical method proposed by Gangardt and Shlyapnikov [1,2]. In the strong repulsion regime the 1D Bose gas at low temperatures is equivalent to a gas of ideal particles obeying the non-mutual generalized exclusion statistics (GES) with a statistical parameter $α=1-2/γ$, i.e. the quasimomenta of $N$ strongly interacting bosons map to the momenta of $N$ free fermions via $k_i\approx αk_i^F $ with $i=1,\ldots, N$. Here $γ$ is the dimensionless interaction strength within the Lieb-Liniger model. We rigorously prove that such a statistical parameter $α$ solely determines the sub-leading order contribution to the $M$-body local correlation function of the gas at strong but finite interaction strengths. We explicitly calculate the correlation functions $g_M$ in terms of $γ$ and $α$ at zero, low, and intermediate temperatures. For $M=2$ and $3$ our results reproduce the known expressions for $g_{2}$ and $g_{3}$ with sub-leading terms (see for instance [3-5]). We also express the leading order of the short distance \emph{non-local} correlation functions $\langleΨ^\dagger(x_1)\cdotsΨ^\dagger(x_M)Ψ(y_M)\cdotsΨ(y_1)\rangle$ of the strongly repulsive Bose gas in terms of the wave function of $M$ bosons at zero collision energy and zero total momentum. Here $Ψ(x)$ is the boson annihilation operator. These general formulas of the higher-order local and non-local correlation functions of the 1D Bose gas provide new insights into the many-body physics.

cond-mat.quant-gas↗

arXiv2014

Manipulation of p-wave scattering of cold atoms in low dimensions using the magnetic field vector

It is well-known that the magnetic Feshbach resonances of cold atoms are sensitive to the magnitude of the external magnetic field. Much less attention has been paid to the direction of such a field. In this work we calculate the scattering properties of spin polarized fermionic atoms in reduced dimensions, near a p-wave Feshbach resonance. Because of spatial anisotropy of the p-wave interaction, the scattering has nontrivial dependence on both the magnitude and the direction of the magnetic field. In addition, we identify an inelastic scattering process which is impossible in the isotropic-interaction model; the rate of this process depends considerably on the direction of the magnetic field. Significantly, an EPR entangled pair of identical fermions may be produced during this inelastic collision. This work opens a new method to manipulate resonant cold atomic interactions.

cond-mat.quant-gas↗

arXiv2013

Universality in s-wave and higher partial wave Feshbach resonances: an illustration with a single atom near two scattering centers

It is well-known that cold atoms near s-wave Feshbach resonances have universal properties that are insensitive to the short-range details of the interaction. What is less known is that atoms near higher partial wave Feshbach resonances also have remarkable universal properties. We illustrate this with a single atom interacting resonantly with two fixed static centers. At a Feshbach resonance point with orbital angular momentum $L\ge1$, we find $2L+1$ shallow bound states whose energies behave like $1/R^{2L+1}$ when the distance $R$ between the two centers is large. We then compute corrections to the binding energies due to other parameters in the effective range expansions. For completeness we also compute the binding energies near s-wave Feshbach resonances, taking into account the corrections. Afterwards we turn to the bound states at large but finite scattering volumes. For p-wave and higher partial wave resonances, we derive a simple formula for the energies in terms of a parameter called "proximity parameter". These results are applicable to a free atom interacting resonantly with two atoms that are localized to two lattice sites of an optical lattice, and to one light atom interacting with two heavy ones in free space. Modifications of the low energy physics due to the long range Van der Waals potential are also discussed.

cond-mat.stat-mech↗

arXiv2012

Universal Bound States of Two Particles in Mixed Dimensions or Near a Mirror

Some novel TWO-body effects analogous to the well-known THREE-body Efimov effect are predicted. In the systems considered, particle A is constrained on a TRUNCATED or BENT one-dimensional line or two-dimensional plane, or on one side of a flat mirror in three dimensions (3D). The constraining potential is fine-tuned such that particle A's ground state wave function is a constant in the region in which it is constrained. Particle B moves in 3D and interacts with particle A resonantly. An infinite sequence of giant two-body bound states are found in each case.

cond-mat.stat-mech↗