arXiv:1001.0225v1 [physics.atom-ph] 1 Jan 2010 V. A. Yurovsky School of Chemistry, Tel Aviv University, 69978 Tel Aviv, Israel M. Olshanii Department of Physics, University of Massachusetts Boston, Boston MA 02125, USA (Dated: October 27, 2018) This supplementary material contains a detailed derivation of the eigenenergies and eigenstates for a system consisting of two atoms in a circular, transversely harmonic waveguide in the multimode regime. PACS numbers: 67.85.-d, 37.10.Gh, 05.45.Mt, 05.70.Ln The waveguide system [1], as well as the ˇ Seba billiard [2], belongs to the class of problems where an integrable system of Hamiltonian ˆ H$_{0}$ is perturbed by a separable rank I interaction V |L〉〈R| (see e.g. [3]). Eigenstates | α 〉 of the interacting system are solutions of the Schr¨ odinger equation Fermi-Huang interaction (see Eq. (2) in [1]) can be writ- ten in the separable form [3] with the interaction strength and formfactors given by In what follows, we will restrict the Hilbert space to the states of zero z -component of the angular mo- mentum and even under the z ↔ − z reflection; the interaction has no effect on the rest of the Hilbert space. The non-interacting eigenstates are products of the transverse and longitudial wavefunctions. The transverse two-dimensional zero-angular-momentum har- monic wavefunctions, where E$_{α}$ are the corresponding eigenenergies. Let us expand the eigenstate | α 〉 over the eigenstates | ⃗ n 〉 (of en- ergy E$_{⃗}$ $_{n}$) of the non-interacting hamiltonian ˆ H$_{0}$ . The Schr¨ odinger equation leads to the following equations for the expansion coefficients V = 2 π ¯ h $^{2}$a$_{s}$ µ , |L〉 = δ$_{3}$ ( r ) , 〈R| = δ$_{3}$ ( r ) ∂ ∂r ( r · ) . (5) [ ˆ H$_{0}$ + V |L〉〈R| ] | α 〉 = E$_{α}$ | α 〉 , (1) 〈 ρ | n 〉 = 1 √$_{πa}$ ⊥ L (0) n ( ξ ) exp ( − ξ/ 2) , (6) ( E$_{α}$ − E$_{⃗}$ $_{n}$) 〈 ⃗ n | α 〉 = 〈 ⃗ n |L〉〈R| α 〉 ∝ 〈 ⃗ n |L〉 , (2) | α 〉 ∝ ∑ ⃗ n | ⃗ n 〉〈 ⃗ n |L〉 E$_{α}$ − E$_{⃗}$ n . (3) 〈 z | k 〉 = (2 /L ) 1 / $^{2}$cos 2 πkζ, 〈 z | 0 〉 = L − 1 / $^{2}$, (7) where the omitted factor in the second equality is inde- pendent of ⃗ n . As a result, any eigenfunction | α 〉 functionally co- incides with the Green function of the non-interacting Hamiltonian taken at the energy of the eigenstate E$_{α}$ , labeled by the quantum number n ≥ 0, are expressed in terms of the Legendre polynomials, L (0) n ( ξ ), where ξ = ( ρ/a ⊥ ) 2 and a ⊥ = (¯ h/µω ⊥ ) 1 / 2 is the transverse os- cillator range. The longitudial wavefunctions, labeled by l ≥ 0, are the symmetric plane waves satisfying periodic conditions with the period L : The omitted factor is determined by the normalization conditions. Substitution of this expression to the Schr¨ odinger equation (1) leads to the following eigenenergy equation: where ζ ≡ z/L − 1 / 2. The unperturbed spectrum is therefore given by Substituting the above non-interacting eigenstates and eigenenergies to Eq. (3) and using Eq. (1.445.8) in [8] for summation over l , one obtains Eq. (3) in [1] for the interacting eigenstates. The normalization factor C$_{α}$ is determined by the condition Similar expressions were obtained in Refs. [2, 4] for the case of the ˇ Seba billiard and its generalizations. In the case of the relative motion of two short-range- interacting atoms in a circular, transversely harmonic waveguide, the derivation uses ideas of the analogous model involving an infinite waveguide [5–7]. The unper- turbed Hamiltonian here is given by Eq. (1) in [1] and the E$_{nl}$ = 2¯ hω ⊥ n + ¯ h $^{2}$(2 πl/L ) $^{2}$/ (2 µ ) (8) ∑ ⃗ n 〈R| ⃗ n 〉〈 ⃗ n |L〉 E$_{α}$ − E$_{⃗}$ n = 1 V . (4) 2 π L ∫ 0 dz ∞ ∫ 0 ρ $^{2}$dρ 〈 α $^{′}$| ρ, z 〉〈 ρ, z | α 〉 = δ$_{αα}$ ′ . (9) Supplementary material to: Restricted Thermalization for Two Interacting Atoms in a Multimode Harmonic Waveguide