Commun. Teor. Pys. (Beng, Cna) 39 (3) pp. 44{48 c Internatonal Academc Publsers Vol. 39, No., January 5, 3 Atomc Tunnellng Dynamcs of Two Squeezed Bose Ensten Condensates LI Jn-Hu and KUANG Le-Man y Department of Pyscs, Hunan Normal Unversty, Cangsa 48, Cna Department of Pyscs, Cenzou Teacer's Collega, Cenzou 43, Cna (Receved June 3, ) Abstract In ts paper, tunnellng dynamcs of squeezed Bose{Ensten condensates (BEC's) n te presence of te nonlnear self-nteracton of eac speces, te nterspeces nonlnear nteracton, and te Josepson-lke tunnellng nteracton s nvestgated by usng te second quantzaton approac. Te nuence of BEC squeezng on macroscopc quantum self-trappng (MQST) and quantum coerent atomc tunnellng s analyzed n detal. It s sown tat te MQST and coerent atomc tunnellng between two squeezed BEC's can be manpulated troug cangng squeezng ampltude and squeezng pase of BEC squeezed states. PACS numbers: 3.75.F, 5.3.Jp, 4.5.Hz Key words: atomc Bose{Ensten condensates, quantum tunnellng, squeezed states Te expermental realzaton of Bose{Ensten condensates (BEC) of weakly nteractng alkal atoms [ ] as provded a route to study neutral superud n a controlled and tunable envronment and to mplement novel geometres for te connecton of several Josepson unctons so far unattanable n carged systems. Te exstence of a Josepson current between two weakly lnked BEC's s a drect manfestaton of macroscopc quantum pase coerence. Recently, an oscllatng atomc current as been expermentally observed n a one-dmensonal array of Josepson unctons realzed wt an atomc BEC. [3] Teoretcal studes of BEC Josepson eect began n Smerz and coworkers' work [4 5] wc ndcates tat quantum coerent atomc tunnellng between two BEC's nduces two types of nterestng eects. One s an atomc Josepson effect, wc s a generalzaton of te snusodal Josepson eects famlar n superconductors. Te oter s macroscopc quantum self-trappng (MQST), wc s a knd of self-locked populaton mbalance between two BEC's. Recently, one of te present autors [6;9] developed a second quantzaton approac to study MQST and quantum coerent atomc tunnellng, and nvestgated te nuence of decoerence nduced by non-condensed atoms. [9] It as been known tat te MQST and atomc tunnellng depend upon not only te trap parameters and te total atoms, but also ntal states of te system. Usually, an atomc BEC n a trap can be treated as a coerent state or a number state. Recently, te Yale group [] as expermentally prepared squeezed states of BEC's by controllng relatve strengts of te tunnellng rate between traps and atom-atom nteractons wtn eac trap. Te realzaton of squeezed BEC's may enable substantal gans n senstvty for atom nterference-based nstruments as well as fundamental studes of quantum pase transtons. Questons tat naturally arse are, wat are te eects of te squeezng parameters n squeezed BEC's on te MQST and quantum coerent atomc tunnellng? Te purpose of ts paper s to nvestgate atomc tunnellng dynamcs of squeezed BEC's. We consder a zero-temperature two-speces Bose condensate system n wc te atoms nteract va aa, bb, and ab elastc collsons, and tere s a Josepson-lke couplng term denoted by a y b and ab y. In te formalsm of te second quantzaton, Hamltonan of suc a system can be wrtten as ^H = Z dxn ^y (x) ; m r + V (x) + U ^y (x) ^(x) ^(x) + ^y (x) ; m r + V (x) + U ^y (x) ^(x) ^(x) + U ^y (x) ^y (x) ^(x) ^(x) o + [ ^y (x) ^(x) + ^(x) ^y (x)] () were ^(x) and ^y (x) are te atomc eld operators wc annlate and create atoms at poston x, respectvely, and satsfy te commutaton relaton [ ^(x) ^y (x )] = (x ; x ). In Eq. (), atoms are con- ned n armonc potentals V (x) ( = ) of frequences!. Interactons between atoms are descrbed by a nonlnear self-nteracton term U = 4 a sc =m and a term tat Te proect supported n part by te Natonal \973" Researc Plan, te Natonal Natural Scence Foundaton, and te EYTF of te Educatonal Department of Cna y Correspondng autor
No. Atomc Tunnellng Dynamcs of Two Squeezed Bose{Ensten Condensates 45 corresponds to te nonlnear nteracton between derent condensates U = 4 a sc =m, were asc s an s-wave scatterng lengt of condensate and a sc tat between condensates and. For smplcty, trougout ts paper we let = and assume tat a sc = a sc = a sc, and V (x) = V (x). It s well known tat te above Hamltonan can be reduced to a two-mode boson Hamltonan troug expandng te atomc eld operators over sngle-partcle states ^(x) = ^a N (x) + ~ R (x), were ^a y = dx N (x) ^y (x) create partcles wt dstrbutons N (x) and [^a ^a y ] =. Te rst term n te mode expanson acts only on te condensate state vector, wereas te second term ~ (x) accounts for noncondensed atoms. Substtutng te mode expansons of te atomc eld operators nto te Hamltonan (), retanng only te rst term representng te condensates, we arrve at te followng two-mode approxmate Hamltonan ^H =! (^a y ^a + ^a y ^a ) + q(^a y ^a + ^a y ^a ) + g(^a y ^a + ^a y ^a ) + ^a y ^a ^a y ^a () were q, and g are couplng constants wc caracterze te strengt of nteratomc nteracton n eac condensate, te nterspeces nteracton, and Josepson-lke couplng, respectvely. In general, te two-mode Hamltonan () cannot be exactly solved, but for weak nonlnear nteractons a closed analytcal soluton can be obtaned under te rotatng wave approxmaton suggested by Alodanc et al. [] In order to obtan an approxmate analytc soluton of te Hamltonan (), we ntroduce a new par of bosonc operators ^A and ^A troug te expressons: ^a = ( ^A e gt ; ^A e ;gt )= p and ^a = ( ^A e gt + ^A e ;gt )= p, were ^A and ^A are slowly varyng operators, satsfyng te usual bosonc commutaton relatons, [ ^A ^A ] = and [ ^A ^Ay ] = wt ^Ay beng te ermtan conugaton of ^A. Ten under te rotatng wave approxmaton te Hamltonan () reduces to te followng approxmate Hamltonan ^H A =! ^N + g( ^Ay ^A ; ^Ay ^A ) + 4 q[3 ^N ; ( ^Ay ^A ; ^Ay ^A ) ] + ^N ; ^Ay ^A ^Ay ^A : (3) In order to solve te approxmate Hamltonan (3) we ntroduce two Fock spaces of ( ^A ^A ) and (^a ^a ) n wc te bases are dened by n m) = n m = p n!m! ^Ayn p n!m! ^a yn ym ^A ) (4) ^aym (5) were n and m take non-negatve ntegers. Obvously, te number states n m) are egenstates of te Hamltonan wt egenvalues E(n m) =!(n + m) + g(n ; m) + (3q + )(n + m) 4 ; 4 q(n ; m) ; nm : (6) Consder two squeezed coerent states dened n Fock spaces of ( ^A ^A ) and (^a ^a ), respectvely, = D^a ( ) ^S^a ( ) ( = ) (7) u z ) = D ^A (u )S ^A (z ) ) ( = ) (8) were D^a ( ) and D ^A (u ) are dsplacement operators, ^S^a ( ) and ^S ^A (z ) are squeezng operators n te ( ^A ^A ) and (^a ^a ) representaton, respectvely. Generally, a drect-product state of two squeezed coerent states n te (^a, ^a ) representaton labelled by s transferred to an entangled state n te ( ^A ^A ) representaton. However, a drect-product state of two squeezed coerent state wt te same squeezng parameter but derent dsplacng parameters n te (^a ^a ) representaton s transferred to a drect-product state of two squeezed coerent states wt derent dsplacng and squeezng parameters n te ( ^A ^A ) representaton = u u ;) (9) ) = u u ; () were te dsplacng parameters on te rgt-and sde of te above equatons are dened by u k = p [ + (;) k+ ] (k = ) u k = p [ + (;) k+ ] (k = ) : () Now we assume tat te two BEC's are ntally n te squeezed coerent state, wc s an egenstate of ^a and ^a. Ten te wave functon of te two speces condensate system at tme t can be explctly ex-
46 LI Jn-Hu and KUANG Le-Man Vol. 39 pressed as (t) = X n n l l = Y = u n ;l L n ;l l (u ) n cos r exp ; N ; e l (l )! tan r l! p n! e ;E(n m)t n m () were we ave let N = + = u + u, and = r e. It s easy to ceck tat te coerent-state soluton n Refs. [5] and [7] can be recovered wen te squeezng parameter approaces zero n te squeezed coerent state soluton (). We are concerned wt te MQST. Te MQST effect s caracterzed by te nonzero tme mean value of te fractonal populaton mbalance between te two condensates dened by p(t) = (N (t) ; N (t))=n, were N (t) = ^a y ^a. Makng use of te squeezed coerent state soluton (), troug a lengty but stragtforward calculaton we get p(t) = X n n l l l l = Y = " (l )!(l )! l!l!n! exp(;n ) cos r(n + sn r) tan r l +l u +(n ;l ;l ) # cos( + ) cosf( ; ) + [4g + (q ; )(n ; n )t]g L n ;l l (u )L n ;l l (u ) It s easy to nd tat wen te tunnellng nteracton and nonlnear nteractons satsfy te condton 4g = K(x ; q), were K = m ; n s an nteger, we can obtan a nonzero tme-averaged value of te populaton mbalance p = X n l l l l = Y = (l )!(l )! exp(;n ) cos r(n + sn r) l!l! tan r l +l u +(n ;l ;l ) u +(n +K;l ;l ) L n ;l l (u )L n +K;l l (u ) L n +;l l (u )L n +K+;l l (u ) cos( + ) cos( ; ) (5) wc mples te exstence of te MQST penomenon. From Eq. (5) t can be seen tat te MQST depends upon not only varous couplng constants but also te dsplacng and squeezng parameters n te ntal state of te two condensates. Wen te squeezng parameter approaces zero, equaton (3) reduces to te followng expresson p(t) = u u N cos[4gt + (t)] exp n;n o sn (q ; ) t (6) wc mples tat te coerent-state result,.e., equaton (38) n Ref. [6] s recovered. Here (t) s gven by Eq. () n Ref. [6]. L n +;l l (u )L n +;l l (u ) (3) were k s a pase of u k, and we ave ntroduced te followng pases k = (l k + l k) + ( ; k )(l k ; l k) (k = ) : (4) Equaton (3) ndcates tat te fractonal populaton mbalance between two BEC's exbts complcated oscllatons wt tme evoluton n general, but t s a smple snusodal oscllaton wen q =. Obvously, te perod of tese oscllatons s ndependent of te dsplacng and squeezng parameters of te BEC ntal state. From Eq. (3) we can nd tat a self-trappng statonary state, n wc te populaton derence s tmendependent, can be obtaned wen te tunnellng nteracton vanses and nonlnear self-nteracton equals to nonlnear nterspeces nteracton,.e., g = and q =. Ts s te consequence of te competton between nonlnear self-nteracton and nonlnear nterspeces nteracton. Fg. Te beavor of te fractonal populaton mbalance as functon of te squeezng ampltude r for derent values of K wen te rst condensate s n a squeezed coerent state and te second condensate n a squeezed vacuum state wt =, =, and =. In wat follows we pay our attenton to te nuence of squeezng parameters on te MQST. In Fg. we plot curves of te fractonal populaton mbalance wt respect to te squeezng ampltude r for derent values of K wen
No. Atomc Tunnellng Dynamcs of Two Squeezed Bose{Ensten Condensates 47 te rst condensate ntally s n a squeezed coerent state and te second condensate n a squeezed vacuum state wt = 5 and =, respectvely. From Fg. we can see tat te weaker te squeezng ampltude r s, te more apparent te MQST becomes. Te MQST vanses wen te squeezng ampltude approaces nnty. In Fg. we plot curves of te fractonal populaton mbalance (5) wt respect to te squeezng pase for derent values of K wen te rst condensate ntally s n a squeezed coerent state and te second condensate n a squeezed vacuum state wt = 5 and =, respectvely. of eac peak depends upon te value of K,.e. ratos of related couplng constants. Te coerent atomc tunnellng current between te two condensates s dened by I(t) = _N (t) ; _N (t) wc can be obtaned from Eq. (3) wt te followng expresson X exp(;n I(t) = ; cos r n n l l l l = Y (l )!(l )! = ) l!l! tan r l +l u +(n ;l ;l ) cos( + ) 4g (q ; ) (q ; ) + ; n!n! n!(n ; )! (n ; )!n! sn n ( ; ) + [4g + (q ; )(n ; n )]t o L n ;l l (u )L n ;l l (u ) Fg. Te beavor of te fractonal populaton mbalance as functon of te squeezng pase for derent values of K wen te rst condensate s n a squeezed coerent state and te second condensate n a squeezed vacuum state wt r =, =, and =. Fgure ndcates tat te squeezng pase serously aects te MQST. Te curves of p wt respect to exbt a symmetrc double-peak structure. Te MQST penomenon s te most apparent at two peaks. Te egt L n +;l l (u )L n +;l l (u ) : (7) Numercal calculatons sow tat te tunnellng current gven by Eq. (7) canges perodcally wt te evoluton of tme, so te atoms perodcally transfer between te two condensates. However, te beavor of te atomc tunnellng current n te weak squeezng regme s derent from tat n te strong squeezng regme. In Fgs. 3 and 4 we dsplay te tme evoluton of te tunnellng current for derent values of te squeezng ampltude n te weak and strong squeezng regmes, respectvely, wen te rst condensate s n a squeezed coerent state and te second condensate n a squeezed vacuum state wt = and =. Fg. 3 Tme evoluton of tunnellng current for derent values of squeezng parameters n te weak squeezng regme, wen te rst condensate s n a squeezed coerent state and te second condensate n a squeezed vacuum state wt =, =, =, g = :5, and q ; =. Fg. 4 Tme evoluton of tunnellng current for derent values of squeezng parameters n te strong squeezng regme, wen te rst condensate s n a squeezed coerent state and te second condensate n a squeezed vacuum state wt =, =, =, g = :5, and q ; =.
48 LI Jn-Hu and KUANG Le-Man Vol. 39 From Fgs. 3 and 4 t can be seen tat te ampltude of te tunnellng current can be manpulated by cangng te atomc squeezng strengt. Fgure 3 ndcates tat te ampltude of te tunnellng current ncreases wt te ncreasng of te atomc squeezng n te weak squeezng regme. Fgure 4 sows tat te ampltude of te tunnellng current decreases wt te ncreasng of te atomc squeezng n te strong squeezng regme. In partcular, wen te squeezng parameter approaces zero, equaton (7) reduces to Eq. (5) n Ref. [6] I(t) = ;u u f4g sn[4gt + (t)] + (q ; )[u sn((q ; )t ; 4gt ; (t)) + u sn((q ; )t + 4gt + (t))]g exp n;n sn (q ; )t o (8) wc mples tat te coerent-state result s recovered (Note tat te factor expf;n sn [ (q ; )t]g s lost, and (t) sould be replaced by (t) + 4gt on te rgt-and sde of Eq. (5) n Ref. [6]). Here (t) s gven by Eq. (3) n Ref. [6]. Terefore, te ampltude manpulaton can be realzed troug cangng te ampltude of te squeezng parameter. In summary, we ave studed te tunnellng dynamcs of two squeezed BEC's. Speccally, we ave nvestgated te MQST penomenon and te quantum coerent atomc tunnellng n a two-speces squeezed-state of BEC system n te presence of nonlnear self-nteracton of eac speces, te nterspeces nonlnear nteracton, and te Josepsonlke tunnellng nteracton wen te two BEC's ntally s n a drect-product state of two squeezed coerent states wt te same squeezng parameter but derent dsplacng parameters. We ave gven new nsgt to control and manpulate te MQST and te atomc tunnellng by cangng te ampltude and pase of te squeezng parameter. In partcular, we ave found te symmetrc double-peak structure n te pase manpulaton of te MQST and te ampltude manpulaton approac of te atomc tunnellng current n te weak and strong squeezng regmes. We ope tat te squeezng manpulaton of te MQST and atomc tunnellng can be appled to quantum controllng of condensates and cold atom quantum nformaton processng. [] Acknowledgment Te autors ws to tank LU Jng for s elp on numercal analyss. References [] F. Dalfovo, S. Gorgn, L.P. Ptaevsk, and S. Strngar, Rev. Mod. Pys. 7 (999) 463. [] A.J. Legget, Rev. Mod. Pys. 73 () 37. [3] F.S. Catalott, S. Burger, C. Fort, P. Maddalon, F. Mnard, A. Trombetton, A. Smerz, and M. Ingusco, Scence 93 () 843. [4] A. Smerz, S. Fanton, S. Govanazz, and S.R. Senoy, Pys. Rev. Lett. 79 (997) 495. [5] S. Ragavan, A. Smerz, S. Fanton, and S.R. Senoy, Pys. Rev. A59 (999) 6. [6] L.M. Kuang and Z.W. Ouyang, Pys. Rev. A6 () 364. [7] L.M. Kuang and Z.Y. Zeng, Cn. Pys. Lett. 5 (998) 73. [8] Bamb Hu and L.M. Kuang, Pys. Rev. A6 () 36. [9] L.M. Kuang, Z.Y. Tong, Z.W. Ouyang, and H.S. Zeng, Pys. Rev. A6 () 368. [] C. Orzel, A.K. Tucman, M.L. Fenselau, M. Yasuda, and M.A. Kasevc, Scence 9 () 386. [] A.P. Alodanc, S.M. Arakelan, and A.S. Crkn, JETP 8 (995) 34. [] S. Cu, Nature 46 () 6 C. Monroe, Nature 46 () 38.