Two-body scattering in a trap and a special periodic phenomenon sensitive to the interaction Y. Z. He and C. G. Bao ∗ State Key Laboratory of Optoelectronic Materials and Technologies, Sun Yat-Sen University, Guangzhou, 510275, P.R. China Two-body scattering of neutral particles in a trap is studied theoretically. The control of the initial state is realized by using optical traps. The collisions inside the trap occur repeatedly; thereby the effect of interaction can be accumulated. Two periodic phenomena with a shorter and a much longer period, respectively, are found. The latter is sensitive to the interaction. Instead of measuring the differential cross section as usually does, the measurement of the longer period and the details of the periodic behavior might be a valid source of information on weak interactions among neutral particles. PACS numbers: 34.10.+x, 34.90.+q, 34.50.Cx, 37.10.-x The determination of interactions among microscopic particles is an important topic in physics. Historically, the measurement of the cross sections of 2-body scat- tering provides an important source of information. For neutral atoms and molecules the determination is more difficult because the interaction is in general weak and the initial momentum of the incident particle is difficult to be precisely controlled. However, if the scattering occurs in a trap, new phenomena previously unknown might emerge. Due to the recent progress in the tech- niques of trapping atoms (molecules) by using optical traps [1–3], trapped scattering might be eventually ex- perimentally realized. In a previous paper a model was proposed to study the trapped 2-body scattering theo- retically [4]. Instead of a single collision, numerous re- peated collisions have been found. Thereby the effect of interaction can be accumulated and enlarged, and might be eventually detected. This favors the determination of very weak interaction. The emphasis of that paper is placed on the study of the spin-flip phenomenon and the determination of the parameter g$_{0}$ (the strength of the channel with total spin zero) of the $^{52}$Cr atoms. This paper is also dedicated to trapped scatterings; however the emphasis is placed on the study of a specific period- icity emerging from the repeated collisions. It turns out that the associated period is sensitive to the interaction. Therefore, in addition to the measurement of cross sec- tions, the observation of the special periodic phenomenon might be also a valid way for the determination of inter- action. The model, related theoretical derivation, and numerical results are given below. We propose a device containing two deep optical traps initially. One is close to the origin, while the other one is far away. Each trap provides a harmonic potential. Thus the total potential was U$_{p}$ ( r ) = 1 2 M ω 2 $_{p}$( | r − a | 2 + | r − b | $^{2}$), where a and b are two given vector (norms a ≫ b ), and M is the mass of the particle involved. where ϕ ( µ ) nl ( r ) is a normalized eigenstates of the Hamilto- nian − 1 2 µ ∇ 2 r + 1 $_{2}$µr 2 with the eigenenergy 2 n + l + 3 / 2. L and l are coupled to the total orbital angular momentum where P$_{1}$$_{,}$$_{2}$ implies an interchange of 1 and 2, and η = ω$_{p}$/ω . We consider the case that a is lying along the X − axis, while b along the negative Z − axis. When R = ( r 1 + r $_{2}$) / 2 and r = r 2 − r 1 for the c.m. and relative motions, respectively, are introduced, and the h.o. states of the new trap are selected as base functions, the initial state can be expanded as Let ℏ ω and √ ℏ / M ω be used as units of energy and length. The normalized initial state is Each trap contains a particle in the lowest harmonic oscillator (h.o.) state. The two particles are assumed to be identical bosons (the generalization to fermions is straight forward), the interaction is assumed to be spin- independent, and ω$_{p}$ is large enough so that the particles are well localized initially and the overlap of their wave functions is negligible. Suddenly the two deep traps are cancelled. Instead, a broader new trap located at the origin U$_{evol}$ ( r ) = 1 2 M ω $^{2}$r2 is created, ω < ω$_{p}$ . Since the initial state is not an eigenstate of the new Hamiltonian, the system begins to evolve. The evolution is affected not only by U$_{evol}$ ( r ) but also by the interaction V ( | r i − r j | ). In what follows the details of the evolution is studied, two-body collisions occurring repeatedly are found, and the effect of interaction is demonstrated. Ψ$_{I}$ = 1 √$_{2}$ (1+ P$_{1}$$_{,}$$_{2}$ )( η π ) 6 4 exp[ − η 2 ( | r 1 − a | $^{2}$+ | r 2 − b | $^{2}$)] (1) Ψ$_{I}$ = ∑ NLnlJM B JM $_{NLnl}$[ ϕ (2) $_{NL}$( R ) ϕ (1 / 2) nl ( r )]$_{JM}$ (2) ∗ Corresponding author: stsbcg@mail.sysu.edu.cn arXiv:1001.1467v1 [cond-mat.other] 10 Jan 2010