Atom-Atom Interactions in Ultracold Quantum Gases

Transcription

1 Atom-Atom Interactions in Utracod Quantum Gases Caude Cohen-Tannoudji Lectures on Quantum Gases Institut Henri Poincaré, Paris, 5 Apri 7 Coège de France 1

2 Lecture 1 (5 Apri 7) Quantum description of eastic coisions between utracod atoms The basic ingredients for a mean-fied description of gaseous Bose Einstein condensates Lecture (7 Apri 7) Quantum theory of Feshbach resonances How to manipuate atom-atom interactions in a utracod quantum gas

3 A few genera references 1 L.Landau and E.Lifshitz, Quantum Mechanics, Pergamon, Oxford (1977) A.Messiah, Quantum Mechanics, North Hoand, Amsterdam (1961) 3 C.Cohen-Tannoudji, B.Diu and F.Laoë, Quantum Mechanics, Wiey, New Yor (1977) 4 C.Joachin, Quantum coision theory, North Hoand, Amsterdam (1983) 5 J.Daibard, in Bose Einstein Condensation in Atomic Gases, edited by M.Inguscio, S.Stringari and C.Wieman, Internationa Schoo of Physics Enrico Fermi, IOS Press, Amsterdam, (1999) 6 Y. Castin, in Coherent atomic matter waves, Lecture Notes of Les Houches Summer Schoo, edited by R. Kaiser, C. Westbroo, and F. David, EDP Sciences and Springer-Verag (1) 7 C.Cohen-Tannoudji, Cours au Coège de France, Année C.Cohen-Tannoudji, Compéments de mécanique quantique, Cours de 3 ème cyce, Notes de cours rédigées par S.Haroche 9 T.Köher, K.Gora, P.Juienne, Rev.Mod.Phys. 78, (6) 3

4 1 - Introduction Outine of ecture 1 - Scattering by a potentia. A brief reminder Integra equation for the wave function Asymptotic behavior. Scattering ampitude Born approximation 3 - Centra potentia. Partia wave expansion Case of a free partice Effect of the potentia. Phase shifts S-Matrix in the anguar momentum representation 4 - Low energy imit Scattering ength a Long range effective interactions and sign of a 5 - Mode used for the potentia. Pseudo-potentia Motivation Determination of the pseudo-potentia Scattering and bound states of the pseudo-potentia Pseudo-potentia and Born approximation 4

5 Interactions between utracod atoms At ow densities, -body interactions are predominant and can be described in terms of coisions. We wi focus here on eastic coisions (athough ineastic coisions and 3-body coisions are aso important because they imit the achievabe spatia densities of atoms). Coisions are essentia for reaching therma equiibrium At very ow temperatures, mean-fied descriptions of degenerate quantum gases depend ony on a very sma number of coisiona parameters. For exampe, the shape and the dynamics of Bose Einstein condensates depend ony on the scattering ength Possibiity to contro atom-atom interactions with Feshbach resonances. This expains the increasing importance of utracod atomic gases as simpe modes for a better understanding of quantum many body systems Purpose of these ectures: Present a brief review of the concepts of atomic and moecuar physics which are needed for a quantitative description of interactions in utracod atomic gases. 5

6 Notation Two atoms, with r r 1 (mass )V m, interacting with a -body interaction potentia In ecture 1, we ignore the spins degrees of freedom. They wi be taen into account in ecture. Hamitonian p (1 )p H = + + Vr r (1.1) 1 m m Change of variabes R = ( r + r ) P = p + p Center of mass variabes 1 /G G 1 M = m + m = m Tota mass 1 /= r r p = ( p p ) Reative variabes 1 /1 /r μ = mm ( m + m ) = m Reduced mass 1 1 ( = + μ ) H P M p + V ( r) G (1.) /Hamitonian of a free partice with mass M H /CM H re Hamitonian of a fictitious partice with mass μ, moving in V(r) 6

7 Finite range potentia Simpe case where V ( r) = for r > b b is caed the range of the potentia One can extend the resuts obtained in this simpe case to potentias decreasing fast enough with r at arge distances. For exampe, for the Van der Waas interactions between atoms decreasing as C 6 / r 6 for arge r, one can define an effective range See for exampe Ref. 5 b VdW μ /C 6 = 1 4 (1.3) 7

8 1 - Introduction Outine of ecture 1 - Scattering by a potentia. A brief reminder Integra equation for the wave function Asymptotic behavior. Scattering ampitude Born approximation 3 - Centra potentia. Partia wave expansion Case of a free partice Effect of the potentia. Phase shifts S-Matrix in the anguar momentum representation 4 - Low energy imit Scattering ength a Long range effective interactions and sign of a 5 - Mode used for the potentia. Pseudo-potentia Motivation Determination of the pseudo-potentia Scattering and bound states of the pseudo-potentia 8

9 Scattering by a potentia. A brief reminder Shrödinger equation for the reative partice (with E>) Δ+ V ( r) ψ( r) = Eψ( r) E = μ μ μ Δ+ ψ ( ) = ( ) ψ ( ) r V r r (1.4) Green function of Δ + Δ+ G ( r) = δ ( r) The boundary conditions for G wi be chosen ater on (1.5) (1.6) Integra equation for the soution of Shrödinger equation ϕ ( ) r Soution of the equation without the right member : Δ+ ϕ ( ) r = (1.7) ( ) ( ) 3 ψ r = ϕ r + d r G ( r r ) V ( r ) ψ ( r ) (1.8) 9

10 Choice of boundary conditions We choose for ϕ a pane wave.with wave vector ϕ ( ) r i.= e = e κ r /i r κ = (1.9) and we choose, for the Green function G, boundary conditions corresponding to an outgoing spherica wave (see Ref., Chap.XIX) 1 e G + ( r r ) = 4π r ir r r (1.1) We thus get the foowing soution for Schrödinger equation r r (+ )1 3 e ( )(+ )i ir ψ r = e d ψ 4π r.v r r r r If V has a finite range b, the integra over r is restricted to a finite range and we can write: If r b, r r r r. n with n= r / r ir + i. r e ψ ( r) e f(, κ, n) r μ 3 - in. r + f(, κ, n) = ψ 4π d r e V( r ) ( r ) (1.11) (1.1) 1

11 Scattering state with an outgoing spherica wave Asymptotic behavior for arge r + The state ψ )r is a soution of the Schrodinger equation behaving exp.for arge r as the sum of an incoming pane wave ( ir) and of (,, ()exp()/ an outgoing spherica wave f κ n i r r Scattering ampitude f (,, ).μ (3 )+ ) r (i f κ n = d e ψ 4π r V r r -/We have pu t = n = r,, )r( κ n is the ampitude of /f the outgoing spherica wave in the direction of = n = r r. It depends ony on and on the poar anges θ and ϕ of with respect to (1.13) Differentia cross section Comparing the fuxes aong and, one gets : (,, / dσ dω = κ n )f (1.14) 11

12 Born approximation In the scattering ampitude, the potentia V appears expicity (,, )-.μ (3 )( r + )i f κ n = d e ψ 4π r V r )r(to owest order in V, one can thus repace ψ + r by the zeroth exp(. ) order soution of the Schrodinger equation ir (,, )().μ (3 r )i f κ n = d e 4π r V r This is the Born approximation In this approximation, the scattering ampitude is proportiona to the spatia Fourier transform of the potentia (1.15) (1.16) 1

13 Low energy imit The presence of V(r ) in the scattering ampitude (,,)-. μ ()(3 r + f κ n = r V r ψ r 4π d e restricts the integra over r to a finite range r < b i. r If b 1, one can repace e by 1. The scattering ampitude μ 3 + f(, κ, n) = d ψ 4π r V( r ) ( r ) then no onger depends on the direction of the scattering vector It is sphericay symmetric even if V( r) is not.,(,, )When f κ n a (.e ψ + ) ir ir a r e a 1 r r a is a constant, caed scattering ength, which wi be discussed in more detais ater on )i (1.17) (1.18). (1.19) 13

14 Another interpretation of the outgoing scattering state Another expression for this state (see refs. 4 and 8) ψ = ϕ + Lim V ψ T = p μ ε / + E T + iε + For ε non zero but very sma, ψ appears as the state obtained at t = by starting from the free state ϕ at t = and by switching on sowy V on a time interva on the order of ε Ingoing scattering state ( )ir 1 ( ) ( ψ r = e d ψ 4π r V r r r r 1 ψ = ϕ + Lim V ψ ε + E T iε If one starts from such a state at t = and if one switches off V sowy on a time interva on the order of ε, one gets the free state ϕ at t = + /3 e. r r )i / (1.) (1.1) 14

15 Definition S - Matrix (,)S = ϕ S ϕ = Lim ϕ U t t ϕ ji 1,j i t1 j i t + : U evoution operator in interaction representation + One can show that = ψ ψ Quaitative interpretation S ji j i (1.) (1.3) V is switched on sowy (time scae ħ/ε) between - and, and then switched off sowy (time scae ħ/ε) between and + One starts from ϕ i at t = - and one oos for the probabiity ampitude to be in ϕ j at t = + From t.= to t = the initia free state ϕ is transformed i + into ψ Since the evoution operator is unitary, and since ψ i j, + transforms into ϕ from t = to t = +,ψ ψ is the j j i ampitude to find the system in the free state ϕ at = + if one starts from ϕ at =.t j t i 15

16 1 - Introduction Outine of ecture 1 - Scattering by a potentia. A brief reminder Integra equation for the wave function Asymptotic behavior. Scattering ampitude Born approximation 3 - Centra potentia. Partia wave expansion Case of a free partice Effect of the potentia. Phase shifts S-Matrix in the anguar momentum representation 4 - Low energy imit Scattering ength a Long range effective interactions and sign of a 5 - Mode used for the potentia. Pseudo-potentia Motivation Determination of the pseudo-potentia Scattering and bound states of the pseudo-potentia 16

17 Centra potentia V depends ony on r 1D radia Schrödinger equation One oos for soutions of the form ϕ ( r) = R ( r) Y ( n) n= r / r If we put m m u( r) R( r) = r with the boundary condition u ( ) = (1.4) (1.5) (1.6) one gets for u the foowing 1D radia equation ) d + 1 μ () ( + V r = u r ) dr r ( 1D Schrödinger equation for a partice moving in a potentia which is the sum of V and of the centrifuga barrier (+ 1 ) μ r (1.7) (1.8) 17

18 Case of a free partice (V=) The soutions of the Schrödinger equation are: )( ) ()( (ϕ r = j r Y n m m ) π where the j are the spherica )Besse functions of order ()r ()1 sin(π )j r ()!j r (r r + 1 r r For arge r we thus have sin(/))( ) ( )r π ϕ r Y n m r m π (/r )(/)i r π i r π ( () = e e Y n m π ir = outgoing spherica wave + ingoing spherica wave These functions form )an ()orthonorma set (see Appendix) ϕ ϕ = δ ( δ δ ( m m ) mm (1.9) (1.3) (1.31) (1.3) 18

19 Expansion of a pane wave in free spherica waves Pane wave ()/. ( r = π e = δ 3 r )i (1.33) The factor (π) -3/ is introduced for the orthonormaization One can show that: ().()/* =+ ()( 3 3 )( )()ir π e = π 4π i Y n Y κ j r m m = m= *() m=+ 1 ()()( ) = i Y κ ϕ r m m = m= / /m with κ = (1.34) ({ () ϕm } The transformation from the orthonorma basis { } he orthonorma basis is given by the matrix ) ()* 1 ( ϕ = δ Y κ m m ) to t (1.35) 19

20 Effect of a potentia. Phase shifts We come bac to the Schrödinger equation with V. Consider, for r arge, an incoming wave exp[-i(r-π/)]. Since the refection coefficient of V is 1 (conservation of the norm), the refected outgoing wave has the same moduus and has just accumuated a phase shift with respect to the V= case. The superposition of the waves is thus a shifted sinusoid. We concude that there is a set of soutions of the Schrödinger equation with V which behave for arge r as: sin/()( ) ( ) π + δ ϕ r r Y n (1.36) m r m π r One can show that these functions are orthonormaized (see Appendix) ϕ ϕ = δ ( δ δ m m ) mm (1.37) They don t form a basis if there are aso bound states in the potentia V.

21 Partia wave expansion of the outgoing scattering state Consider the inear superposition of the states ϕ with the same m coefficients as those appearing in the expansion (1.34) of the pane wave exp(on ).the, each state being mutipied by the phase factor iδ We wi show that such a state is nothing but the outgoing state ψ (mutipied by for having an orthonormaized state) ) ϕ(m (/+ 3 π m=+ + 1 ( = = m ) = )( ) iδ ψ i Y κ e ϕ = m=+ = m= ϕ () m Before demonstrating this identity, et us discuss its physica meaning.the outgoing scattering state is obtained by switching ( on sowy V on the free state. Each spherica wave ϕ of the )m expansion of is transformed into ϕ, but in addition it acquires m iδ a phase factor e which depends on and which thus varies from one spherica wave of the expansion to another one.m * e iδ ϕ m m (1.38) 1

22 Demonstration For arge r, the inear superposition introduced in (1.38) behaves as: ( π + δ ) ( π ) m= + *i r 1 ()( ) ( ) i r e e i Y κ Y n m m = m= π ir exp/exp/ Using i r i r we get: ( δ 1) / =+ * sin(/)i i m 1 ()( ) ( ) π κ r i Y Y n + m m = m= π r i r ( ) ( ) ( δ π δ π i + = 1 + e 1) // π e e ir e (1.39) The contribution of the first term of the bracet is nothing but the asymptotic expansion of the pane wave in spherica waves. The second term gives an outgoing spherica wave 3.ir ()ir (,, /)e π e + f κ n (1.4) r This demonstrates that the state given in (1.38) is an outgoing scattering state and gives in addition the expression of the ampitude f (1.4) (1.41)

23 Scattering ampitude in terms of the phase shifts (, =+, )m 4π sin(δ )( *)i f κ n = e δ Y n Y κ m m = m= 1 ()sin(cos(1.43) ) i δ = + 1 e δ P θ (cos) = where P θ is a Legendre poynomia and where θ. is the ange between n and κ Integrating f over the poar anges of κ gives the scattering cross section ) ()()4π ()sin( σ = (σ σ = + 1 δ (1.44) ) = Scattering of identica partices Quantum interference between different paths )()()θ π - θ f θ (f θ + ε f π θ ) ε =+ 1 1 for bosons (fermions)(π sin()( σ = π θ dθ θ + ε π θ tota /f f (1.45) ) 3

24 Partia wave expansion of the ingoing scattering state ψ is given by a inear superposition of the states ϕ anaogous m + to the one introduced for ψ each state being now mutipied by exp() exp().the phase factor iδ instead of iδ () =+ =+ 1 * δ δ ψ = m ()() i κ e ϕ = m i i Y ϕ e ϕ m m m m = m= = m= The demonstration of this identity is simiar to the one given above for the outgoing scattering state (1.46) If we start from this state and if we switch off V sowy, it transforms ( into the free state. Each wave ϕ is transformed into ϕ, but m m iδ ) in addition its phase factor changes from e to 1 w hich corresponds +iδ to acquiring a phase factor e Finay, when we go from t = to t = + switching on and then )() switching off V sowy, we start from ϕ and we end in ϕ acquiring m m + iδ + iδ + iδ, a goba phase factor e e = e(4

25 S Matrix in the anguar momentum representation S = S = ψ ψ + We use the expansion of ψ + and ψ in spherica waves m=+ ( m =+ + ψ ϕ ) i e δ ϕ ψ ϕ ϕ = = m= m m = m= = m = = = m = ) iδ e( m m This gives a first expression of S m=+ m =+ + S = ψ ψ = ϕ ϕ ϕ ϕ (On the other hand, a change of basis gives for S = m= = m = )() + iδ iδ e e m m m m ( ) δ δ δ mm =+ m =+ (S S ϕ ϕ S ϕ ϕ = = Comparing the expressions obtained for we get, S )() + i δ () ϕ S ϕ = e(δ δ δ m m mm (1.48) (1.49) ))m () () ( m m m m which shows that the S-matrix is diagona exin p(the anguar )momentum representation, with diagona eements iδ ceary showing the unitarity of S (1.47) (1.5) (1.51) 5

26 1 - Introduction Outine of ecture 1 - Scattering by a potentia. A brief reminder Integra equation for the wave function Asymptotic behavior. Scattering ampitude Born approximation 3 - Centra potentia. Partia wave expansion Case of a free partice Effect of the potentia. Phase shifts S-Matrix in the anguar momentum representation 4 - Low energy imit Scattering ength a Long range effective interactions and sign of a 5 - Mode used for the potentia. Pseudo-potentia Motivation Determination of the pseudo-potentia Scattering and bound states of the pseudo-potentia 6

27 Centra potentia. Low energy imit Suppose first V=. The centrifuga barrier in the 1D Schrödinger equation prevents the partice from approaching near the region r= ) ) (= μr μ μ r ()/ r = + 1 μ r r () r = + ( 1 db If the range b of the potentia is sma enough, i.e. if.b a partice with cannot fee the potentia Ony = wave wi fee V " s-wave scattering" d μ () ( + V r = u r ) dr db (1.5) (1.53) 7

28 Scattering ength sin()for r arge enough, u = u varies as r + δ sin (). Let v be the function r + δ extending for a u r Let P be the intersection point of v with the r-axis which is the cosest from the origin.by definition, the scattering ength a is the imit of the abscissa of P when (see figure) Expansion of v in powers of r near r = v r r r Abscissa of : a ) sin()sin()cos(= + δ δ δ + r 1 tan()p δ tan()im δ π () π = δ (+ ) (1.54) (1.55) 8

29 Limit = Scattering ength (continued) d dr μ () ( r u r )V = Far from r=o, the soution of the S.E. is a straight ine and )v r r (a (1.57) The abscissa of Q is equa to a (1.56) Scattering cross section ()4π ()sin()()σ = δ σ π + 1 = 4 = ()() δ a σ = 4πa = () For identica bosons σ = 8π a = sin(δ ) (1.58) (1.59) (1.6) 9

30 Scattering ength for square potentias Square potentia barriers()v r Square barrier of height /V and width b = μ V a )u (r b )v (r r a r For r)> b (and = r = v r )u r (a For ()< b and ()r = u r = u r The curvature of (u is )positive and u r = = We concude that the scattering ength is aways positive and smaer than the range b of the potentia a b When V (hard sphere potentia) a b 3

31 For and (Scattering ength for square potentias (continued) Square potentia wes V r a V )v (r r a u r b r = /V μ r)r )u > b ( = = v r r a For )< b and ()= u r = ( u r The curvature of (u is )r negative and u r = = If V is sma enough so that there is no bound state in the potentia we, the curvature of u for r < b is sma and a is negative When increases, the curvature of u for r < bincreases in absoute vaue and a Then a switches suddeny to +.V and decreases. This divergence of a corresponds to the appearance of the first bound state in the potentia we 31

32 Square potentia wes (continued) Variations of a with figure taen from Ref.5 When the depth of the potentia we increases, divergences )of occur for a vaues of V such that b = n +(1π /a corresponding to the appearances of successive bound states in the potentia we. These divergences of a which goes from to + are caed "zero-energy" resonances 3

33 Long range effective interactions and sign of a The scattering ength determines how the ong range behavior of the wave functions is modified by the interactions. To understand how the sign of a is reated to the sign of the effective ong range interactions, it wi be usefu to consider the partice encosed in a spherica box with radius R, so that we have the boundary condition u R )=( (1.61) eading to a discrete energy spectrum In the absence of interactions (V=), the normaized eigenstates and the eigenvaues of the 1D Schrödinger equation are: sin(/))() 1 π π ψ = N r R 1 π = (N μ =,,r E N N R r R (/N r R )(1.6)..N inπsfigure corresponding to N=3 R r 33

34 V = R V a > a The dotted ine is the sinusoid outside the range of the potentia It has a shorter waveength than for V=, and thus a arger wave number. The inetic energy in this region, which is aso the tota energy, is arger R V a < a R The dotted ine is the sinusoid outside the range of the potentia It has a onger waveength than for V=, and thus a smaer wave number. The inetic energy in this region, which is aso the tota energy, is smaer 34

35 Correction to the energy to first order in a a R )/ For the state ψ, we have R = N λ N Because of the interactions, these N haf waveengths occupy now a ength /R a so that ()/λ = R /N λ = R/ a N ) = π λ = π λ = R R a ///) a E = μ E = E = E R R a (E 1 + N N N (N N R a π Finay, we have δ = = = N E E E E a (1.63) N N N N 3 R μ R Long range effective interactions are - repusive if a > /(- attractive if a < 35

36 1 - Introduction Outine of ecture 1 - Scattering by a potentia. A brief reminder Integra equation for the wave function Asymptotic behavior. Scattering ampitude Born approximation 3 - Centra potentia. Partia wave expansion Case of a free partice Effect of the potentia. Phase shifts S-Matrix in the anguar momentum representation 4 - Low energy imit Scattering ength a Long range effective interactions and sign of a 5 - Mode used for the potentia. Pseudo-potentia Motivation Determination of the pseudo-potentia Scattering and bound states of the pseudo-potentia 36

37 Mode used for the potentia V(r) Why not using the exact potentia? The interaction potentia is very difficut to cacuate exacty. A sma error in V can introduce a very arge error on the scattering ength deduced from this potentia. Mean fied description of utracod quantum gases require in genera a first order treatment of the effect of V (Born approximation). But Born approximation cannot be in genera appied to the exact potentia Approach foowed here The motivation here is not to cacuate the scattering ength. This parameter is supposed nown experimentay. We are interested in the derivation of the macroscopic properties of the gas from a mean fied description using a singe parameter which is a. The ey idea is to repace the exact potentia by a pseudo-potentia simper to use than the exact one and obeying conditions: - It has the same scattering ength as the exact potentia - It can be treated with Born approximation so that mean fied descriptions of its effects are possibe 37

38 Determination of the pseudo-potentia Derivation à a H.Bethe and R.Peiers (Y.Castin, private communication) We add to the 3D Schrödinger equation of a free partice (V=) a term proportiona to a deta function Δ μ ψ )+ δ ( ) (= μ ψ ( ) r C r r (1.64) To determine the coefficient, we impose to the soution of this equation to coincide with the extension to a r of the asymptotic in()/()/c behavior r + δ of the true wave function sr u r r In particuar, for sma enough, one shoud have: )()ψ r (B r a r (1.65) /r )(/)()Inserting (1.65) into 1.64 and using Δ 1 = 4π δ r,r we get an equation containing a deta function mutipied by a coefficient 4π (C a B μ which must vanish. This gives the coefficient C appearing in (1. 64) 4π 4π C = g B where g = a = a μ m (1.66) 38

39 Determination of the pseudo-potentia (continued) It wi be more convenient to express C=gB, not in terms of the coefficient B appearing in the wave function ψ = B(r-a)/r of equation (1.65), but in terms of the wave function ψ itsef. We use for that d B = rψ d r r = Equation (1) can be rewritten as: Δ ψ( r) + Vpseudoψ( r) = ψ( r) μ μ d where Vpseudoψ( r) = gδ( r) [ rψ( r) ] dr (/)is caed the pseudo-potentia. The term d d reguarizes pseudo r r ( ) /the action.v of δ r when it acts on functions behaving as 1 r near r = For functions which are reguar in r =, V has the pseudo ):same (effect as δ )()/r ()()()( )ψ r = u r r with u V ψ r = g u δ r (pseudo )()()( ()g ψ r reguar in r = V ψ r = gψ δ r pseudo (1.67) (1.68) (1.69) (1.7) 39

40 Scattering states of the pseudo-potentia We are ooing for soutions of equation (1.68) with E > For the centrifuga,barrier prevents the (partice ),from approaching r = and one (can show that ψ = so that, ). according to (1.7), V ψ r = V gives ony s-scattering pseudo pseudo,and one can write ψ ( r ) = ()/where ( )r r u can be.u )()()() r u u r u ()()u 1 d u Δ = Δ + = 4π u δ r + r r r (r dr The Schrödinger equation for u becomes: ()() ()( )u r ( )() 4π δ δ u r u r + + g r u = μ r μ r ( )Canceing (the term proportiona to δ and the term independant ),r of δ r we get equations: ) μ ()() ()u = g = a u r + u r = u π ( (1.71) (1.7) (1.73) 4

41 Scattering states of the pseudo-potentia (continued) The soution of the second equation can be written: )sir = nu r +(δ ( ) Inserting this soution into the first equation gives: anδ = ta On the other hand, the s-wave scattering ampitude is equa to: )1 siδ ni f =( e δ Using equation (1.75) giving tanδ finay gives after simpe agebra: )a f = 1 +(ia is proportiona to a A first order treatment of V pseudo pseudo thus gives the correct resut for ).V the scattering ampitude in the zero energy imit ( a =. This shows that Born approximation can be used with V for utracod atoms pseudo The conditions imposed above on V pseudo are thus fufied (1.74) (1.75) (1.76) (1.77) 41

42 The unitary imit From the expression of the scattering ampitude obtained above, we deduce the scattering ampitude for identica bosons ) 8π a σ = 1 +(a which is vaid for a. )The ow energy imit a 1 gives the we nown resut: ( σ( ) a 1 8πa There is another interesting imit, corresponding to high energy, )or strong interaction a 1 eading to resut independent of a: ( 8π σ ( ) a 1 This is the so caed unitary imit (1.78) (1.79) (1.8) 4

43 Bound state of the pseudo-potentia The cacuation is the same as for the /scattering states, except that we repace the positive energy μ by a negative one - κ / μ The equations derived from the Schrodinger equation are now: ) ()() ()u = a u r κ u r = (1.81) u ( The soution of the second equation (finite for r ) is: u r )=(e κ r which inserted into the first equation gives: κ =/1 a The pseudo-potentia thus has a bound state with an energy E = μa and a wave function: xp e r a (1.8) (1.83) (1.84) (1.85) 43

44 Energy shifts produced by the pseudo-potentia We come bac to the probem of a partice in a box of radius R. We have cacuated above the energy shifts of the discrete energy eves of this partice produced by a potentia characterized by a scattering ength a. To first order in a, we found: π δ = N E a N 3 μ R This resut was deduced directy from the modification induced by the interaction on the asymptotic behavior of the wave functions and not from a perturbative treatment of V. We show now that: ()( δe = ψ V ψ to first order in V N pseudo )N N pseudo (1.86) (1.87) which is another evidence for the fact that the effect of the pseudo potentia can be cacuated perturbativey, which is not the case for the rea potentia. For exampe, a hard core potentia (V= for r< a) cannot obviousy be treated perturbativey, but its scattering ength is a, and using a pseudo-potentia with scattering ength a aows perturbative cacuations. 44

45 Demonstration The unperturbed normaized eigenfunctions of the partice in the spherica box are: )sin(/)( 1 Nπ r R ψ r =()N )π R r () ψ r is reguar in r (= ) and ()N 1 1 ψ =(Nπ N 3 π R (1.47) (1.88) so that (()( )(V ψ r = gδ r ψ )() N N )pseudo (1.89) We deduce that ()( N π π δ = ψ = = N E g g a N )N 3 3 πr μr (1.9) which coincides with the resut obtained above in (1.63). 45

46 Concusion Eastic coisions between utracod atoms are entirey characterized by a singe number, the scattering ength Effective ong distance interactions are attractive if a< and repusive if a> Giving the same scattering ength as the rea potentia, the pseudo-potentia gives the good asymptotic behavior for the wave function describing the reative motion of atoms, and thus correcty describes their ong distance interactions In a diute gas, atoms are far apart. The pseudo-potentia is proportiona to a and can be treated perturbativey. A first order treatment of the pseudo-potentia is the basis of mean fied description of Bose Einstein condensates where each atom moves in the mean fied produced by a other atoms. Next step: can one change the scattering ength? 46