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Atom-Atom Interactions in Ultracold Gases Claude Cohen-Tannoudji

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Atom-Atom Interactions in Ultracold Gases Claude Cohen-Tannoudji Atom-atom interactions in ultracold gases Claude Cohen-Tannoudji To cite this version: Claude Cohen-Tannoudji. Atom-atom interactions in ultracold gases. DEA. Institut Henri Poincaré, 25 et 27 Avril 2007, 2007. cel-00346023 HAL Id: cel-00346023 https://cel.archives-ouvertes.fr/cel-00346023 Submitted on 12 Dec 2008 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Atom-Atom Interactions in Ultracold Quantum Gases Claude Cohen-Tannoudji Lectures on Quantum Gases Institut Henri Poincaré, Paris, 25 April 2007 Collège de France 1 Lecture 1 (25 April 2007) Quantum description of elastic collisions between ultracold atoms The basic ingredients for a mean-field description of gaseous Bose Einstein condensates Lecture 2 (27 April 2007) Quantum theory of Feshbach resonances How to manipulate atom-atom interactions in a ultracold quantum gas 2 A few general references 1 – L.Landau and E.Lifshitz, Quantum Mechanics, Pergamon, Oxford (1977) 2 – A.Messiah, Quantum Mechanics, North Holland, Amsterdam (1961) 3 – C.Cohen-Tannoudji, B.Diu and F.Laloë, Quantum Mechanics, Wiley, New York (1977) 4 – C.Joachin, Quantum collision theory, North Holland, Amsterdam (1983) 5 – J.Dalibard, in Bose Einstein Condensation in Atomic Gases, edited by M.Inguscio, S.Stringari and C.Wieman, International School of Physics Enrico Fermi, IOS Press, Amsterdam, (1999) 6 – Y. Castin, in ’Coherent atomic matter waves’, Lecture Notes of Les Houches Summer School, edited by R. Kaiser, C. Westbrook, and F. David, EDP Sciences and Springer-Verlag (2001) 7 – C.Cohen-Tannoudji, Cours au Collège de France, Année 1998-1999 http://www.phys.ens.fr/cours/college-de-france/ 8 – C.Cohen-Tannoudji, Compléments de mécanique quantique, Cours de 3ème cycle, Notes de cours rédigées par S.Haroche http://www.phys.ens.fr/cours/notes-de-cours/cct-dea/index.html/ 9 – T.Köhler, K.Goral, P.Julienne, Rev.Mod.Phys. 78, 1311-1361 (2006) 3 Outline of lecture 1 1 - Introduction 2 - Scattering by a potential. A brief reminder • Integral equation for the wave function • Asymptotic behavior. Scattering amplitude • Born approximation 3 - Central potential. Partial wave expansion • Case of a free particle • Effect of the potential. Phase shifts • S-Matrix in the angular momentum representation 4 - Low energy limit • Scattering length a • Long range effective interactions and sign of a 5 - Model used for the potential. Pseudo-potential • Motivation • Determination of the pseudo-potential • Scattering and bound states of the pseudo-potential • Pseudo-potential and Born approximation 4 Interactions between ultracold atoms At low densities, 2-body interactions are predominant and can be described in terms of collisions. We will focus here on elastic collisions (although inelastic collisions and 3-body collisions are also important because they limit the achievable spatial densities of atoms). Collisions are essential for reaching thermal equilibrium At very low temperatures, mean-field descriptions of degenerate quantum gases depend only on a very small number of collisional parameters. For example, the shape and the dynamics of Bose Einstein condensates depend only on the scattering length Possibility to control atom-atom interactions with Feshbach resonances. This explains the increasing importance of ultracold atomic gases as simple models for a better understanding of quantum many body systems Purpose of these lectures: Present a brief review of the concepts of atomic and molecular physics which are needed for a quantitative description of interactions in ultracold atomic gases. 5 Notation Two atoms, with mass m, interacting with a 2-body interaction (GG) potential Vr12− r In lecture 1, we ignore the spins degrees of freedom. They will be taken into account in lecture 2. Hamiltonian pp22(GG) H =++−12Vr r (1.1) 22mm 12 Change of variables GGGG/ G G RrrGG=+()122 Ppp =+1 2Center of mass variables Mmm=+=2 mTotal mass GGG12 G G G/ rrr=− p = pp − 2 Relative variables 12/ ( 1 2)/ μ =+=mm12() m 1 m 2 m 2 Reduced mass G / / 22G G HP=+G 22 M pμ + Vr( ) (1.2) ( ) HCM Hrel Hamiltonian of a free Hamiltonian of a “fictitious” particle particle with mass M with mass μ, moving in V(r) 6 Finite range potential G Simple case where Vr( ) = 0 for r> b b is called the range of the potential One can extend the results obtained in this simple case to potentials decreasing fast enough with r at large distances. For example, for the Van der Waals interactions between atoms 6 decreasing as C6 / r for large r, one can define an effective range / 14 ⎛⎞2μC 6 (1.3) bVdW = ⎜⎟2 ⎝⎠= See for example Ref. 5 7 Outline of lecture 1 1 - Introduction 2 - Scattering by a potential. A brief reminder • Integral equation for the wave function • Asymptotic behavior. Scattering amplitude • Born approximation 3 - Central potential. Partial wave expansion • Case of a free particle • Effect of the potential. Phase shifts • S-Matrix in the angular momentum representation 4 - Low energy limit • Scattering length a • Long range effective interactions and sign of a 5 - Model used for the potential. Pseudo-potential • Motivation • Determination of the pseudo-potential • Scattering and bound states of the pseudo-potential 8 Scattering by a potential. A brief reminder Shrödinger equation for the relative particle (with E>0) ⎡⎤==22G GG k 2 ⎢⎥−Δ+Vr()ψψ() r = E () r E = (1.4) ⎣ 22μ ⎦ μ 2 G 2μ GG ⎡⎤Δ+krψψ() = Vrr() () ⎣⎦ =2 Green function of Δ + k2 G G ⎡⎤2 (1.5) ⎣Δ+kGr⎦ ( )() =δ r The boundary conditions for G will be chosen later on (1.6) Integral equation for the solution of Shrödinger equation G : ϕ0 (r ) Solution of the equation without the right member 2 G ⎡⎤Δ+krϕ =0 (1.7) ⎣⎦0 () G G GGGG ψ rrrGrrVrr=+ϕψd3 ′ −′′′(1.8) () 0 ( ) ∫ ( ) ( ) ( ) 9 Choice of boundary conditions We choose for ϕ0 a plane. wave .with wave vector k GGG G GG G / ikr ikκ r (1.9) ϕ0 ()rk==ee κ =k and we choose, for the Green function G, boundary conditions corresponding to an outgoing spherical wave (see Ref.2, Chap.XIX) GG G G 1 eikr− r′ Gr()−=− r′ GG (1.10) + 4π rr− ′ We thus get the following solution for Schrödinger equation GG ikr− r′ ( ) G .G ( ) ( ) ++GGikr 1 3 e G ψ GGrrV=−ed′ GG r′′ψ r(1.11) kk4π ∫ rr− ′ If V has a finite range b, the integral over r’ is restricted to a finite range and we can write: G GGG G G If rbrrrrn ,.− ′ −=′ with nrr / G G ikr + GGik. r Ge ψκG ()rfkn e − (, ,) k r (1.12) GG G G 2μ 3 -.ikn r′ G + G fk(,,)κψ n=− derVr′ ()()′′G r 4π =2 ∫ k 10 Scattering state with an outgoing spherical wave Asymptotic behavior for large r ( ) + G The state ψ G r is a solution of the Schrodinger equation behaving k exp G .G for large r as the sum of an incoming plane wave ikr and of ( ,GG, )exp( )/ () an outgoing spherical wave fkκ n ikr r Scattering amplitude f ( , , ) - G .G ( ) ( ) GG 2μ 3 ik′′ r GG+ fkκψ n = − d r′ e Vr′′G r (1.13) 2 ∫ k G4π = GG/ We have put kknk′ ==r r ( ,G ,G ) fkκ nis the amplitude of the outgoing spherical wave in the G GG/ direction of kknkrr′ ==GG. It depends only on k and on the polar angles θϕand of kk′ with respect to Differential cross section Comparing the fluxes along k and k’, one gets : / ( ,G ,G )2 ddσκΩ= f kn (1.14) 11 Born approximation In the scattering amplitude, the potential V appears explicitly ( , , ) - G .G ( ) ( ) GG 2μ 3 ik′′ r GG+ f knκψ=− der′ Vrr′′G (1.15) =2 ∫ k 4π ( ) + G G ′ To lowest order in V, one can thus replace ψ k r by the zeroth exp( G .G ) order solution of the Schrodinger equation ikr ( ). ( ,G ,GG) 2μ GG G ( ) f knκ =− de3r′ ik− k′′ r Vr′ (1.16) 4π =2 ∫ This is the Born approximation In this approximation, the scattering amplitude is proportional to the spatial Fourier transform of the potential 12 Low energy limit The presence of V(r’) in the scattering amplitude - . ( , , ) G G ( ) ( ) G G 2μ 3 ik′′ r GG+ f knκψ=− der′ Vrr′′G (1.17) 4π =2 ∫ k restricts the integral over r’ to a finite range r’< b G If kb 1 , one can replace e−ik′. r′ by 1. The scattering amplitude GG 2μ 3 GG+ ′′G ′ fk(,,)κψ n=− 2 d rVr()() r (1.18) 4π = ∫ k G then no longer depends on the directionG of the scattering vector k′. It is spherically symmetric even if Vr()is not. , ( ,G ,G ) When k →→0 f knκ − a . ( ) G G ikr (1.19) + G ikr e a ψ G ra e − →−1 k rr a is a constant, called “scattering length”, which will be discussed in more details later on 13 Another interpretation of the outgoing scattering state Another expression for this state (see refs. 4 and 8) / ++1 2 ψ GG=+ϕψμLim VTpG =2 (1.20) kkε →0 k + E − Ti+ ε + G For εψ non zero but very small, k appears as the state obtained at tt==0 by starting from the free stateϕ G at −∞ and by k / switching on slowly V on a time interval on the order of = ε Ingoing scattering state G G −−ikr r′ G .G −−( GG) ikr 1 3 e ( ) ( G) ψ GGrrV=−ed′ GG r′′ψ r kk4π ∫ rr− ′ (1.21) −−1 ψϕGG=+Lim V ψG kkε →0 k + E − Ti− ε If one starts from such a state at t = 0 and if one switches off V slowly on a time interval on the order of = / ε, one gets G the free stateϕk at t =+∞ 14 S - Matrix Definition ( , ) GG G G SS==ϕϕLim ϕ Utt ϕ, ji kkji kj21 ki (1.22) t1 →−∞ t →+∞ : 2 U evolution operator in interaction representation − + One can show that S = ψ G ψ G (1.23) ji kkj i Qualitative interpretation V is switched on slowly (time scale ħ/ε) between -∞ and 0, and then switched off slowly (time scale ħ/ε) between 0 and +∞ One starts from ϕi at t = -∞ and one looks for the probability amplitude to be in ϕ at t = +∞ j , From tt= −∞to = 0 the initial free state ϕ is transformed .
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