Relative phase for atomic de Broglie waves A tutorial

Transcription

1 Relatve phase for atomc de Brogle waves A tutoral Claude Cohen-Tannoudj HYPER Symposum Fundamental Physcs and Applcatons of cold atoms CNES, Pars, 04 November 00

2 Purpose of ths lecture Introduce the basc concepts and the tools needed for nterpretng nterference experments performed wth matter waves Correlaton functons of the atomc feld operators Relatve phase between matter waves A more detaled dscusson can be found n the Proceedngs of the Cargese Euroschool 000 on Bose-Ensten condensates and atom lasers (A. Aspect and J. Dalbard eds, C.R.Acad.Sc. Tome, Sere 4, N 3, 00) Great analogy wth smlar problems n Optcs. But new effects appear as a result of atom-atom nteractons, as shown by recent experments whch wll be brefly descrbed

3 Atomc nterferometry wth non-condensed atoms The wave functon of each atom s splt nto coherent parts by the atomc equvalent of a beam spltter n optcs : π/ optcal, mcrowave or Bragg pulse Example of the Bordé-Ramsey nterferometer Senstvty to gravtatonal felds, to rotatons Interest of ultracold atoms : Large spatal coherence lengths 3

4 Atomc nterferometry wth Bose-Ensten condensates Larger sgnals Larger spatal coherence lengths Coherence propertes of a sngle condensate - Mean-feld descrpton of the condensate - Correlaton functons - Spatal coherence length of the condensate Coherence propertes of a several condensates - States of condensates wth a well-defned relatve phase - Number of atoms n each condensate and relatve phase : conjugate varables - BEC n an optcal lattce. Effect of atom-atom nteractons - More general states of condensates 4

5 Sngle condensate Good approxmate descrpton as an ensemble of N ψ r! bosons, all n the same quantum state ( ) ( r ) ψ! s not the ground state of the sngle partcle Hamltonan ˆ! ˆ (! h= p /m+ V ˆ) ext r, but the soluton of the Gross-Ptaevsk equaton contanng n addton the effect on each boson of the mean-feld due to the N other bosons "!!!!! + Vext ( r) ψ ( r) + g( N ) ψ ( r) ψ ( r) = µψ ( r) m 4π " g = a a: scatterng length µ : chemcal potental m ψ can be obtaned from a varatonal calculaton of the ground state of the full Hamltonan taken as a product of N dentcal wave functons ψ() ψ().. ψ(n) 5

6 Matter wave The possblty to assocate a 3-D wave functon wth a condensate of N partcles results from a varatonal approxmaton. Mean-feld theory An exact descrpton would requre a 3N-D wave functon. The nterpretaton of certan effects requre to go beyond the mean-feld approxmaton Macroscopc matter waves (N >> ) A macroscopc number of atoms are n the same quantum state descrbed by the wave functon ψ Ths 3-D wave functon s called a macroscopc or gant matter wave 6

7 Fock-space descrpton - Bass { ψ } of sngle partcle states ncludng the soluton of G.P. equaton denoted ψ - Occupaton numbers n n, n,... n... n Fock space Bass { } Each state of the bass s defned by the number n of bosons n each state ψ - The condensate s n a Fock state / ( ) ( + ˆ ) N n = N, n = 0 for = N! a 0 + aˆ ( ˆ a) :creates (destroys) one boson n ψ ˆ +! Ψ r and Ψˆ! r ˆ +! + *! Ψ ( r) = aˆ ( ) ˆ!! ( ) ˆ ψ r Ψ r = aψ( r) ˆ + Ψ ( r! ) ( Ψ ˆ ( r! )) :creates (destroys) one boson n r! + aˆ, ˆ ˆ ( ), ˆ + a j = δ j Ψ r! Ψ ( r! ) = δ ( r! r! ) Feld operators ( ) ( ) 7

8 Correlaton functons All physcal observables are proportonal to correlaton functons of the boson feld operators, whch are the average values of normally ordered products of feld operators. Equal number of creaton and annhlaton operators (the number of bosons does not change) Analogy wth the correlaton functons of quantum optcs!!,! : spatal coherence!!,!! : spataldensty ( ) ˆ + = Ψ ( ) Ψˆ ( ˆ ) () G r r r r ( ) ˆ + = Ψ ( ) Ψˆ ( ) () G r r r r () G ( r!! ˆ + ( ) ˆ +, r ) = Ψ r! Ψ ( rˆ ) Ψˆ ( r! ) Ψˆ ( rˆ) Probablty to fnd one partcle n r!! and another one n r 8

9 Correlaton functons n a Fock state / ( ) ( + ˆ ) n = N,0,0,... = N! a 0 + Usng a ˆ, ˆ a j = δ j, one easly gets ()!! *!! N + ( ) ( ) ( + ) * (, )! ˆ ˆ ˆ ˆ ( ) ( ) 0 N!! G r r = N ψ r ψ r a aa a 0 = Nψ( r) ψ( r )!! In general, ψ s real, so that G () ( r, r ) s also real - Sngle phase throughout the condensate - The spatal coherence length of the condensate s equal to the spatal extent of the condensate (MIT, NIST, Munch)!!!! () G ( r, r ) = N( N ) ψ( r) ψ( r ) (3)!!!!!! G ( r, r, r ) = N( N )( N ) ψ( r) ψ( r ) ψ( r ) 3 N( N ) # N N( N )( N ) # N f N $ For k % N, functons N ( k ) G can be wrtten as a product of k * ψ and Nψ. A gant matter wave Nψ ( r! ) can be assocated wth the condensate N 9

10 ψ r! ψ Two well separated condensates r! ψ, ψ : Solutons of the G.P. equaton ψ ( r! ) = ψ ( r! ) = 0 Can one fnd a quantum state descrbng the condensates wth a well defned relatve phase? ()!! G r, r ˆ +! = Ψ r Ψˆ! r must be dfferent from zero ( ) ( ) ( ) Can one take a Fock state n = N, n = N,0,0,...? ˆ! ˆ!!! N, N Ψ + r Ψ r N, N = ψ ( r) ψ ( r ) N, N aˆ + aˆ N, N = 0 ( ) ( ) * Two condensates n a Fock state N, N cannot pssess a well defned relatve phase Next dea Take N bosons, all n the same lnear combnaton of ψ and ψ 0

11 Phase states Lnear combnaton of ψ and ψ θ µ, µ, θ = µ ψ + µ e ψ µ, µ real µ + µ = θ:relatve phase Creaton operator of a boson n the state µ, µ, θ aˆ aˆ e µµθ = µ + µ a + + θ + N bosons all n the same state µ, µ, θ Phase state N, µ, µ, θ, or more smply N, θ ( ) / + θ + ˆ N, θ = N! µ a + µ e a 0 = N! N! n = N n = N n n nθ n n + + n n nθ µ µ e ( aˆ ) ( aˆ ) 0 = µ µ e n, n n = 0, n = N n n! n! n = 0, n = N n n! n! N Lnear combnaton of several Fock states n, n wth n + n = N fxed

12 Dstrbuton of n and n n a phase state N! n P n = µ µ µ = µ n = N n n ( ) ( ) ( ) n! n! Bnomal dstrbuton n = Nµ n = Nµ = N n ( ) n = Nµ µ = Nµ µ = n Dfference n= n n between n and n n= N, N, N 4,... N +, N n = n n n = 4 n = 4Nµ µ n µµ = % N $ N N f

13 Propertes of phase states - Well defned value of N - Well defned relatve phase θ - N, θ s not a product of a state of mode by a state of mode. Quantum correlatons between the modes - ˆ ρ = N, θ N, θ has off dagonal elements between states correspondng to dfferent values of n= n n The extent of off-dagonalty n n s of the order of N 3

14 Correlaton functons n a phase state + () () ( k ) Usng a ˆ, ˆ a j = δ j, one fnds that G, G,... G (wth k N) are * products of,4, k functons ψ and ψ wth!!! ψ ( r) = n ψ ( r) + n e ψ ( r) θ Analogy wth the results obtaned n quantum optcs for quas-classcal optcal felds, for whch the quantum correlaton functons are equal to products of the correspondng classcal felds Two nterferng gant matter waves n ψ ( r! ) and! n e θ ψ ( r) can thus be assocated wth the two condensates 4

15 Statstcal mxtures of phase states Statstcal mxture of states N, θ wth N fxed and wth dfferent values of θ dstrbuted wth a probablty densty W ( θ ) π ˆ ρ = d θ W( θ) N, θ N, θ ˆ N n+ n ρ = µ µ n = 0 n = 0 N 0 N! N! n! n! n! n! n π ( n n ) θ dθ W( θ) e n 0, n n, n wth n n n n N + = + = ( n n ) ( n n ) = n n = n n + n n = ( n n ) '()(* '()(* = n = n ( ) + n n + n = n + n n n = n n ( n ) θ π d W e = d W e ( ) π n n θ θ θ θ 0 0 ( n ) θ / 5

16 Relatve phase and off-dagonalty n n = n n The densty operator descrbng the statstcal mxture of phase states contans terms of the form ( θ) π ( n n ) θ / dθ W e n 0, n n, n The broader W ( θ ), the smaller the factor multplyng the off-dagonal element of ˆρ between states wth dfferent values of n= n n and n = n n = π ], the ntegral over θ gves δ nn, and ˆρ s dagonal, not only n N = n+ n, but also n n= n n n= n n thus appears as the conjugate varable of the relatve phase θ If W ( θ ) s flat [ W ( θ ) / 6

17 Interference between condensates The experment of M.I.T. Condensate cut n parts by a laser Imagng Probe laser M.R.Andrews, C.G.Townsend, H.-J. Mesner, D.S. Durfee, D.M.Kurn, W.Ketterle, Scence, 75, 637 (997) 7

18 Problem of the orgn of the relatve phase If the condensates are kept a long enough tme before beng released, they lose any possble memory of ther ntal phase (when they were formng a sngle condensate) and they should not have a well defned relatve phase. Why do they gve rse to nce nterference frnges? Smlar problem n optcs; nterference between two ndependent laser beams Ths problem has been nvestgated by several authors J. Javananen, S.M. Yoo Y. Castn, J. Dalbard T. Wong, M.J. Collett, D. Walls I. Crac, C. Gardner, N. Naraschewsk, P. Zoller Analytcal calculatons, Monte-Carlo smulatons 8

19 Emergence of a relatve phase as a result of the detecton process Intal state of the condensates : Fock state N, N n= n n s well defned. No relatve phase Frst detecton process The detected photon can come from «mode» or «mode» After ths detecton, the state vector becomes ψ = α N, N + β N, N After the second detecton process, we have : ψ = λ N, N + µ N, N + ν N, N The off-dagonalty of ˆρ = ψ ψ n n= n n ncreases wth the number of detectons and the relatve phase becomes more and more well-defned From one expermental realzaton to another, the fnal value of the relatve phase θ s dfferent 9

20 BEC n an optcal lattce The trappng potental V ext s a spatally perodc array of potental wells : spatally modulated lght shfts produced by a far off-resonant laser standng wave We gnore n a frst step atom-atom nteractons M potental wells The N atoms condense n the ground state of the sngle partcle Hamltonan for such a perodc potental Expermentally, one starts wth a BEC n an ordnary magnetc or dpole trap, and one swtches on adabatcally the perodc optcal potental 0

21 Matter wave assocated wth the condensate The ground state of the sngle partcle Hamltonan s the lowest energy Bloch state n the perodc potental V ext Ths Bloch state s a lnear combnaton of the ground state wave functon ϕ of each ndvdual potental well M ψ ( r! ) = ϕ ( r! ) M = The ϕ s are supposed normalzed and quas-orthogonal If we ntroduce the operator a ˆ + creatng a boson n the state ϕ ( r! ) and the operator M ˆ+ + ˆ satsfyng ˆ, ˆ+ A = a A A M = = the state of the condensate can be wrtten ( ) N ˆ + ψ cond = A 0 N!

22 Dstrbuton P(n ) of the number n of bosons n a gven ste M ˆ + + ( + ˆ+ A = aˆ ˆ ) = a + M b M = M ˆ+ + wth b ˆ ˆ ˆ+ = a b, b = M ( ) N ˆ + ψ cond = A 0 N! N N N! ( ) ( ) N n ( ) n ( ) N n + ˆ ˆ+ = M a b 0 N! M n = 0 n! N n! P(n ) = Square of the coeffcent multplyng the normalzed state wth n bosons n ste and N-n bosons n stes n N n N! P( n ) = n! ( N n)! M M Bnomal dstrbuton wth p=/m and q=-p ( ) n N p N/ M n N pq N = = = = M M

23 Dfference n=n -n between the numbers n and n of bosons n stes and A calculaton smlar to the calculaton of P(n ) gves for the jont probablty dstrbuton P(n,n ) for havng n bosons n ste and n bosons n ste (see Appendx) N! Pn (, n) = n! n! N n n! M M M ( ) Knowng P( n ), P( n ) and P( n, n ) n n N n n, one can calculate the mean value and the varance of n= n n ( ) n = 0 n = n n = N M The state of the condensate s a lnear superposton of states wth dfferent values of n= n n The dsperson n of the values of n vares as N 3

24 Spatal coherence Usng the general results derved above, we get : M * G r!, r! = Nψ r! ψ r! wth ψ r! = ϕ r! M = We take r! nsde ste, r! nsde ste. Snce!! ϕ r = 0 f, ϕ r = 0 f () ( ) ( ) ( ) ( ) ( ) we get : ( ) ( )!! N G r r r r M ϕ!! () (, ) = ( ) ϕ ( ) The matter waves nsde the varous potental wells are coherent The spatal coherence between the waves n stes and s related to the fact that the number of bosons n each potental well s not well defned 4

25 How to check the spatal coherence between the matter waves n the varous stes The total trappng potental s swtched off The varous matter waves ϕ expand freely. If they are coherent, they nterfere. One gets a -D or a 3-D nterference pattern dependng whether the optcal lattce s -D or 3-D -D experments n Yale (and NIST) C. Orzel, A. Tuchman, M. Fenselau, M.Yasuda, M. Kasevch, Scence 9, 386 (00) Seealso: Scence 8, 686 (998) 3-D experments n Munch M. Grener, O. Mandel, T. Esslnger, T. Hänsch, I. Bloch Nature 45, 39 (00) 5

26 Examples of nterference patterns 3-D Munch -D Yale 6

27 Effect of atom-atom nteractons Up to now, they have been gnored The nteractons between the n bosons n ste are descrbed by the term : ˆ 4π " V ˆ ( ˆ = gn n ) g = a m The nteractons between bosons n dfferent stes are much smaller and can be gnored Up to now, g has been supposed very small compared to the tunnel couplng J between adjacent stes, responsble for the delocalzaton of the atoms throughout the lattce What happens when g becomes larger than J? We frst consder the case where g» J. In such a case, the n s are good quantum numbers 7

28 Interacton energy E nt of a confguraton {n,n,..n,..n M } The number n of bosons n ste can be wrtten : N n = n + δ n n = M Snce M M n = N = n we have = = M = M δ n = 0 M N N Ent { n, n,... n,... nm} = g n( n ) = g + δn + δn = = M M M M M N = E0 + g δn + g ( δn) = E0 + g ( δn) M = = = ')* = 0 where N E0 = gn M 8

29 Confguraton wth the lowest nteracton energy The mnmum of E nt s obtaned for δn = 0 All stes contan the same number N/M of bosons Frst excted confguraton The smallest non-zero value of ( g/) ( δ n ) wth δ n = 0 s obtaned when one δ n s equal to, another one to -, all others beng equal to 0 Ent = E0 + g( + ) = E0 + g Second excted confguraton Two δ n s are equal to, two other δ n s are equal to -, all others beng equal to 0 Ent = E0 + g All energy splttngs are nteger multples of g 9

30 Conclusons. For large nteracton energes (g» J), the ground state of the N-boson system corresponds to a stuaton where all stes contan the same well-defned number of bosons The ground state s a product of Fock states, one for each ste.. The ground state no longer corresponds to a stuaton where all bosons are n the same state. It cannot be descbed by a Gross-Ptaevsk equaton 3. There s an energy gap g n the exctaton spectrum 4. All energy splttngs are nteger multples of g 5. There s no well-defned relatve phase between the matter waves n dfferent stes and j snce n n j s well-defned and equal to 0 Dsappearance of the nterference pattern when the optcal lattce s swtched off 30

31 Evoluton of the ground state when the atom-atom nteractons are progressvely ncreased When g s progressvely ncreased, by ncreasng the depth of the potental wells of the optcal lattce, the ground state of the N-boson system changes from a state where all bosons are n the same delocalzed state ψ to a state where they are well localzed n equal numbers n the varous potental wells. Analogy wth the superflud-mott nsulator transton n condensed matter physcs Expermentally, the transton s detected by a change of the nterference pattern whch s observed after a swtchng off of the optcal lattce. The nterference between the matter waves orgnatng from the varous stes dsappears when g s large enough 3

32 Expermental observaton 3-D Munch -D Yale 3

33 Collapse and revval of the nterference pattern The potental well depth can be ncreased n a tme - long compared to the oscllaton perod n the wells - short compared to the tunnelng tme between wells Startng from the superflud phase where n can take several values n each well, one reaches a stuaton where the n s have had no tme to change, but where the atom-atom nteractons have become predomnant In each well, n can take several dfferent values, so that there s a spatal coherence between the matter waves orgnatng from the varous wells Dfferent values of n Dfferent oscllaton frequences n G () (r,r ) Collapse of the nterference pattern Equally spaced oscllaton frequences (splttng g) Revval of the nterference pattern after a tme /g 33

34 Expermental observaton M. Grener, O. Mandel, T. Hänsch, I. Bloch, Nature 45, 39 (00) Ths experment shows the quantzaton of the matter feld Analogy wth the revval of the Rab oscllaton n cavty QED showng the quantzaton of the radaton feld 34

35 A few possble applcatons A stuaton s acheved where several potental wells contan a precsely known number of atoms, the same for each well Sub-Possonan number fluctuatons. Squeezng n n Ths opens nterestng perspectves. Massve entanglement va controlled cold collsons Theoretcal work of P. Zoller, I. Crac et al. Hesenberg lmted atomc nterferometry By applyng Ramsey sequences of pulses to the paths of an atomc nterferometer contanng well defned numbers of atoms n and n, one can reduce the quantum projecton nose varyng as / N / to values varyng as / N Theoretcal work of P. Bouyer and M. Kasevch 35

36 Another nterestng state not yet acheved Two orthogonal one-partcle states ϕ and ϕ and condensates n a state ψ = = = + = =, 0 0, n N n n n N If a generalzed π / pulse could be found for producng such a state from an ntal state n = N, n = 0, the dfference of phase shfts nduced on the components of ψ by an external perturbaton dfferent for ϕ and ϕ would be N tmes bgger than the one produced on N atoms prepared all n the state ( / ) ϕ + ϕ by an ordnary π / pulse The frst non-zero spatal correlaton functon n the state ψ would be N N ˆ + Ψ ( r! ) Ψˆ ( r! ) 36

37 Calculaton of P(n,n ) Appendx M ˆ + + ( + + ˆ+ A = aˆ ˆ ˆ ) = a + a + M b M = M ˆ+ + wth b ˆ ˆ ˆ+ = a b, b = M ( ) N, ˆ + ψ cond = A 0 N! N N N! ( ) ( ) N n n ( ) n ( ) ( ) n N n n + + ˆ ˆ ˆ+ = M a a b 0 N! M n = 0 n! n! N n n! P(n,n ) = Square of the coeffcent multplyng the normalzed state wth n bosons n ste, n bosons n ste and N-n -n bosons n stes, N N! N n n P( n, n) = ( M ) n! n! ( N n n)! N! M n! n! ( N n n)! N! = n! n! N n n! M M M ( ) n n N n n 37

38 nn Appendx (contnued) Mean value and varance of n = n -n n = n n = 0 n = n + n nn = ( n) + n nn ')* + N N = = M M M ( ) ( ) ( ) n, n n, n ( N! ) ( n ) ( n ) N ( n ) ( n ) N! = nn P( n, n) = n! n! N n n! M M M ( n ) ( n ) N ( n ) ( n ) = N( N ) M n, n!!! M M M '((((((((((((((((( )(((((((((((((((((* nn = N( N ) M n = n n = n ( ) = n n N n n N N N = + N ( N ) M M = M M M 38