Adiabatically induced coherent Josephson oscillations of ultracold atoms in an asymmetric two-dimensional magnetic lattice

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1 Edth Cowan Unversty Research Onlne ECU Publcatons Pre Adabatcally nduced coherent Josephson oscllatons of ultracold atoms n an asymmetrc two-dmensonal magnetc lattce Ahmed Abdelrahman Edth Cowan Unversty Peter Hannaford Kamal Alameh Edth Cowan Unversty Ths paper was publshed n Optcs Express and s made avalable as an electronc reprnt wth the permsson of OSA. The paper can be found at the followng URL on the OSA webste: Systematc or multple reproducton or dstrbuton to multple locatons va electronc or other means s prohbted and s subject to penaltes under law. Ths Journal Artcle s posted at Research Onlne.

2 Adabatcally nduced coherent Josephson oscllatons of ultracold atoms n an asymmetrc two-dmensonal magnetc lattce A. Abdelrahman 1 *, P. Hannaford 2 and K. Alameh 1 1 Electron Scence Research Insttute, Edth Cowan Unversty 270 Joondalup Drve, Perth WA 6027 Australa 2 Centre for Atom Optcs and Ultrafast Spectroscopy, and ARC Center of Excellence for Quantum Atom Optcs,Swnburne Unversty of Technology, Melbourne, Australa 3122 *a.abdelrahman@ecu.edu.au Abstract: We propose a new method to create an asymmetrc twodmensonal magnetc lattce whch exhbts magnetc band gap structure smlar to semconductor devces. The quantum devce s assumed to host bound states of collectve exctatons formed n a magnetcally trapped quantum degenerate gas of ultracold atoms such as a Bose-Ensten condensate (BEC) or a degenerate Ferm gas. A theoretcal framework s establshed to descrbe possble realzaton of the excton-mott to dschargng Josephson states oscllatons n whch the adabatcally controlled oscllatons nduce ac and dc Josephson atomc currents where ths effect can be used to transfer n Josephson qubts across the asymmetrc twodmensonal magnetc lattce. We consder second-quantzed Hamltonans to descrbe the Mott nsulator state and the coherence of multple tunnelng between adjacent magnetc lattce stes where we derve the self consstent non-lnear Schrödnger equaton wth a proper feld operator to descrbe the excton Mott quantum phase transton va the nduced Josephson atomc current across the n magnetc bands Optcal Socety of Amerca OCIS codes: ( ) Bose-Ensten condensates; ( ) Quantum nformaton and processng. References and lnks 1. L. Peraks, Condensed-matter physcs: Excton developments, Nature 33, 417 (2002). 2. A. Abdelrahman, P. Hannaford, M. Vaslev, and K. Alameh, Asymmetrc Two-dmensonal Magnetc Lattces for Ultracold Atoms Trappng and Confnement, n progress, arxv: v1 [quant-ph (2009). 3. L. V. Butov, A. C. Gossard, and D. S. Chemla, Towards Bose-Ensten condensaton of exctons n potental traps, Nature 47, 417 (2002). 4. S. Ghanbar, T. D Keu, A. Sdorov, and P. Hannaford, Permanent magnetc lattces for ultracold atoms and quantum degenerate gases, J. Phys. B 39, 847 (2006). 5. B.V. Hall, S. Whtlock, F. Scharnberg, P. Hannaford, and A. Sdorov, A permanent magnetc flm atom chp for Bose-Ensten condensaton, J. Phys. B: At. Mol. Opt. Phys. 39, 27 (2006). 6. M. Sngh, M. Volk, A. Akulshn, A. Sdorov, R. McLean, and P. Hannaford, One dmensonal lattce of permanent magnetc mcrotraps for ultracold atoms on an atom chp, J. Phys. B: At. Mol. Opt. Phys. 41, (2008). 7. S. Ghanbar, T. D. Keu, and P. Hannaford, A class of permanent magnetc lattces for ultracold atoms, J. Phys. B: At. Mol. Opt. Phys. 40, 1283 (2007). (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24358

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4 1. Introducton Recently, magnetc lattces have been created usng permanent magnetc materals fabrcated wth specfc patterns to create perodcally dstrbuted non-zero magnetc local mnma n one and two-dmensonal confguratons [2,4. Ultracold atoms and Bose-Ensten condensates (BECs) prepared n so-called low magnetc feld seekng-states, can be trapped n such perodcally dstrbuted magnetc local mnma to create the magnetc lattces confguratons [6, 23. The magnetc lattces are recognzed as quantum devces wth abltes to coherently access and manpulate the quantum states of the trapped ultracold atoms n a smlar scenaro to optcal lattces [30. These advances n the feld of ultracold atoms open the way to study many nterestng fundamental problems n dfferent area of physcs such as condensed matter [27 and quantum nformaton processng [30. Magnetc lattces created by trappng ultracold atoms are used to smulate such envronments and crtcal quantum phase transtons because they can provde adabatc control over nterestng quantum phenomena such as quantum tunnelng, superfludty and formaton of bound states of long lved quas-partcles, e.g., excton BECs [24, 25. Bound states are naturally hard to detect and t s dffcult to access ther ndvdual quantum states due to the short lfetme and ther fragle couplng to a desrable envronment [1, 3. One of the nterestng problems s to detect and manpulate the phase coherence sgnature n a macroscopc quantum system. A well known example of the drect manfestaton of a macroscopc phase coherence s the Josephson effect between two superfluds or two superconductors [31, 32. Theoretcally, the use of trapped ultracold atoms n varyng perodcally dstrbuted potental felds has been proposed to smulate the Josephson effect [29, 33, 34, 36 and expermentally has been realzed [35, 37, 38. In ths artcle we propose a new approach to create an asymmetrc two-dmensonal magnetc lattce whch allows the trapped ultracold atoms to smulate collectve exctatons smlar to those formed n condensed matter systems. The ultracold atoms can be trapped to mantan an asymmetrc two-dmensonal lattce confguraton n whch our proposed method offers the possblty of havng a magnetc band gap structure. We also show the possblty of usng ths type of 2D magnetc lattce to smulate the adabatcally controlled oscllatons of collectve exctatons such as the dschargng Josephson states and the excton-mott quantum phase transton formed by coherent tunnelng of cold atoms across the lattce s stes. In Secton 2 we descrbe the method we use to create our asymmetrcal magnetc lattce; n Secton 3 we descrbe the tunnelng mechansms; and n Secton 4 we establsh a theoretcal framework to descrbe the adabatcally nduced atomc Josephson currents and the excton-mott phase transton usng trapped ultracold atoms n the asymmetrcal magnetc lattce. 2. The Asymmetrc Two-Dmensonal Magnetc Lattce Magnetc lattces are realzed by perodcally dstrbutng magnetc feld mnma across the surface of a permanent magnetc materal. The dstrbuted feld mnma follow the lattces patterns n whch they are specfcally engneered so that the magnetc mnma are located n a free space at an effectve workng dstance, d mn, from the surface. Dependng on the fabrcated patterns, the dmenson of the perodcty can be selected, where one and two-dmensonal magnetc lattces have been acheved and produced by trappng ultracold atoms usng patterned permanent magnetc materals [6, 23. We have developed a new method to create a two-dmensonal magnetc lattce whch allows symmetrcal and asymmetrcal confguratons of the dstrbuted magnetc feld mnma. To realze a two-dmensonal magnetc lattce square-hole matrces are patterned on a surface of a magneto-optc flm of thckness τ btm by mllng an m m array of blocks such that each block s an array of n n square holes, where n represents the number of holes of wdth α h and (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24360

5 (a) (b) (d) (e) (c) (f) Fg. 1. (a) Schematc dagram of an magnetc lattce surrounded by an unperturbed area. (b) The lattce parameters are specfed by the hole sze αh αh, the spacng αs between the holes and the magnetc layer thckness τbtm. (c) 3D plot of the magnetc feld (along the z-axs) of the dstrbuted stes across the x y plane smulated at a dstance dmn from the surface. (d) Magnetc densty plot of the smulated magnetc lattce stes n the z x plane along the center of the lattce. The traps (dark color) are located at the effectve z-dstance, dmn, from the surface of the magnetc flm (brght color). (e-f) Magnetc feld densty and contour plots across the x y plane at dmn wth no external magnetc bas felds, respectvely. Smulaton nput parameters: αs = αh = 7 µm Mz = 2.80 kg, τbtm = 2 µm and τ p wall = 2 µm. separated by αs wthn each block as shown n Fgs. 1(a),1(b). The depths of all holes are equal and extend through the magnetc thn flm down to the substrate surface level. The magnetc structure s magnetzed n ts remanently-magnetzed state, wth the magnetzaton drecton perpendcular to the x y plane. The gaps between the blocks contanng no holes are assumed to be greater than, or equal to αs. The thckness of the gaps s an mportant desgn feature snce t ntroduces a control over an extra degree of confnement whch s realzed through the creaton of magnetc feld walls encrclng the n n matrces, and solatng them from one another. It also can control the magnetc bottom, Bmn, and the dstance from the surface, dmn, of the stes at the center of the magnetc lattce. The presence of the holes on the surface of the permanent magnetc thn flm results n a magnetc feld wth a maxmum at the openng of the holes decreasng steeply outwards from the surface n the z drecton and creatng magnetc feld mnma that are located at effectve workng dstances, dmn, above the plane of the thn flm as shown n Fgs. 1(d) 1(f). These mnma are localzed n confnng volumes representng the magnetc potental wells that contan a certan number of quantzed energy levels occuped by the ultracold atoms. In our desgn, we assumed that the wdth of the holes αh and the separaton of the holes αs are equal, # $15.00 USD (C) 2009 OSA Receved 2 Nov 2009; accepted 5 Dec 2009; publshed 18 Dec December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24361

6 α h = α s α, to smplfy the mathematcal dervatons and analyses whch are smlar to those reported n [4, Detaled Analyss of the Dstrbuton of the Magnetc Feld Mnma The spatal magnetc feld components B x, B y and B z can be wrtten analytcally as a combnaton of a feld decayng away from the surface of the trap n the z-drecton and a perodcally dstrbuted magnetc feld n the x y plane produced by the magnetc nducton, B o = µ o M z /π, at the surface of the permanently magnetzed thn flm. We defne a surface reference magnetc feld as B re f = B o (1 e βτ ), where β = π/α, and τ = τ btm denotes the magnetc flm thckness and a plane of symmetry s assumed at z = 0. The analyss of the surface magnetc feld ncludes components of external magnetc bas felds along the x,y and z drectons, B x bas, B y bas and B z bas, respectvely. Takng nto account the surface effectve feld, B re f, and the characterstc perodcty nterval α, analytcal expressons can be derved to descrbe the perodcally dstrbuted local mnma across the x y plane of the magnetc thn flm for the case of an nfnte magnetc lattce as follows B x = B re f sn(βx)e β[z τ B re f 3 sn(3βx)e 3β[z τ B x bas (1) B y = B re f sn(βy)e β[z τ B re f 3 sn(3βy)e 3β[z τ B y bas (2) B z = B re f [cos(βx) + cos(βy) e β[z τ B [ re f cos(3βx) + cos(3βy) e 3β[z τ B z bas (3) Only cold atoms prepared n low magnetc feld seekng states,.e. atom s magnetc moment orented ant-parallel to the local magnetc feld n the trap, are attracted to the local magnetc mnma, where at certan values of the effectve dstance d mn, namely larger than α/2π, the cold atoms effectvely nteract wth the local magnetc mnma are loaded nto the lattce stes. Thus the hgher order terms n these equatons can be neglected for d mn > α/2π reducng Eqs. (1),(2),(3) to the followng smplfed set of expressons ( B x = B o 1 e )e βτ β[z τ sn(βx) + B x bas (4) ( B y = B o 1 e )e βτ β[z τ sn(βy) + B y bas (5) ( [ B z = B o 1 e )e βτ β[z τ cos(βx) + cos(βy) + B z bas (6) Hence the magntude B of the magnetc feld at d mn above the surface of the magnetc flm can be wrtten as { [ B = B 2 x bas + B2 y bas + B2 z bas + 2B2 re f 1 + cos(βx)cos(βy) e 2β[z τ + 2B re f [ )} 1/2 e (sn(βx)b β[z τ x bas + sn(βy)b y bas + cos(βx) + cos(βy) B z bas (7) A detaled analyss of the effect of the characterstc parameters, such as α s, α h, s reported elsewhere [2. (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24362

7 (a) (b) (c) (d) (e) (f) (g) (h) () Fg. 2. (a) Magnetc feld densty plot of the smulaton of a 9 9 magnetc lattce where the magnetc local mnma are located at d mn above the holes. The central stes of the lattce exhbt larger dstances from the magnetc thn flm surface than the edge stes. (b-d) Contour and magnetc feld densty plots of two adjacent edge stes showng the separatng dstance along the gravtatonal feld z-axs and the magnetc potental tlt, δ B, and the magnetc tunnelng barrers, B, between the two stes. The magnetc feld s calculated across the two stes as ndcated by the dashed lne. (e) Comparson of the magnetc feld local mnma along the z-axs at the center and the edge stes. (f) Two dfferent magnetc bands at the center and at the edge of the asymmetrcal magnetc lattce. (g) Smulated B z mn for dfferent values of α s where the wdth of the holes s fxed, α h = 2 µm. (h) The effect of applyng a varyng external magnetc bas feld on the non-zero local mnma along the z-axs. () Tunnelng barrers can adabatcally be controlled va applyng the magnetc bas feld B z bas along the z-axs. Smulaton results show the effect of B z bas on B and δb. Smulaton nputs: α s = α h = 7 µm M z = 2.80 kg, τ btm = 2 µm and τ n wall = -0.5 µm. (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24363

8 (a) (b) (c) (d) Fg. 3. (a-b) Schematc representaton of the magnetc band structure (the pyramd-lke dstrbuton) where each band contans a set of magnetc mnma. (c-d) Schematc representaton of the two-modes confguraton and a possble scenaro of coupled spns between two magnetc bands hostng two dfferent degrees of freedom of spns such a two-components spnor BECs or ultracold Fermons Magnetc Band Structure n the Asymmetrc Two-Dmensonal Magnetc Lattce The asymmetrc property of the two-dmensonal magnetc lattce s created due to the exstence of dfferent levels of the non-zero magnetc local mnma n a fnte magnetc lattce. As shown n Fg. 2, each neghborng magnetc quantum wells,.e., lattce stes, are separated by a tunnelng barrer B and ther magnetc bottoms are dsplaced n the gravtatonal feld z-drecton by a ttlng potental δb. Both B and δb can be controlled by applyng an external magnetc bas feld along the negatve drecton of the z-axs, as shown n Fg. 2(). The stes n the center of the magnetc lattce exhbt the deepest magnetc mnma, where the values of the magnetc mnma, B mn, are dstrbuted n space downward to the edges of the lattce, see Fg. 2(f). The dstrbuton has a pyramd shape n the x y plane where each level has ts B z mn value and the values of the B mn are spaced along the z-axs by a tltng potental δb. The potental tlt creates a gap between each two sets of magnetc mnma dstrbuted n two adjacent bands. Ths confguraton exhbts a scenaro smlar to that of the energy band gap structure n semconductor devces. We denote the pyramd-lke dstrbuton by a magnetc band gap structure n our proposed two-dmensonal magnetc lattce. A schematc representaton for the pyramd-lke dstrbuton s shown n Fg. 3. Intally, wth no applcaton of external magnetc bas felds, B x bas and B y bas, all stes have magnetc mnma close to zero. Once the external magnetc bas feld s appled the values of the magnetc mnma ncrease and dffer from neghborng stes by the ttlng magnetc potental δb as shown n Fg. 2(d). The amount of tlt δb can be calculated from δb = B,l mn B+1,l+1 mn (8) where = 0,1,2,...,n s the non-zero local mnmum ndex at each band represented by l = 0,1,2,..,N assumng there are N magnetc bands startng from the center of the lattce. The magnetc band gap s gven by the dfference between the maxmum of the tunnelng barrer, B max, and the magnetc bottom of the lattce ste, B mn, and s denoted by the tunnelng barrer heght B = Bmax,l B,l mn (9) The non-zero local mnmum values determne the depth Λ depth of the harmonc potental wells n whch Λ depth of an ndvdual potental well can be expressed as Λ depth (r) = µ Bg F m F B(x) = µ [ Bg F m F B,l k B k max(x) B,l mn (x) (10) B (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24364

9 (a) (b) (c) (d) Fg. 4. (a) Scannng Electron Mcroscope (SEM) and (b) Atomc Force Mcroscope (AFM) mages of the fabrcated two-dmensonal magnetc lattce. (c) Magnetc feld mnma Bzmn smulated across the surface of the magnetc lattce, where we smulate the applcaton of n-stu bas feld along the x-axs, Bx bas = 3 G.(d) Magnetc Force Mcroscope (MFM) mage showng the phase across the surface wth the same condtons appled n the smulaton. where x = {x, y, z}. The gf s the Lande g-factor, µb s Bohr magneton, F s the atomc hyperfne state wth the magnetc quantum number mf and kb s the Boltzmann constant. In Fg. 4, we show the measurement results for a fabrcated 2 2 blocks of 9 9 asymmetrc two-dmensonal magnetc lattce. The quantum devce s fabrcated usng the dual electronfocused on beams technology and maged wth scannng electron mcroscope and the atomc force mcroscope, as shown n Fgs. 4(a) 4(b). The structure s desgned n such a way that allows n-stu magnetc feld bas to be appled along the x-axs. The measurement output and the smulaton results are shown n Fgs. 4(c) 4(d). 3. Tunnelng Mechansms n the Two-Dmensonal Magnetc Lattce Expermentally, well spaced clouds of Bose-Ensten condensates n a double-well potental have been created [9, and a long tunnelng lfetme of 50 ms has been observed [10 wth expermental lfetmes n the range s [11. In ths proposal, weakly coupled mcroscopc clouds of BECs, wth a small number of trapped ultracold atoms, are allowed to tunnel between stes by adabatcally controllng the hoppng strength of the adjacent magnetc bands through applcaton of external magnetc bas felds along the negatve drecton of the z-axs. Atoms tunnel through the magnetc barrers from the hghest magnetc band to the neghborng lowest magnetc band followng the pyramd-lke dstrbuton of energy levels. Fgures 2(b) 2(c) shows smulaton results of two adjacent edge lattce stes where the gap can also be regarded as the dfference between the heghts of the two stes n the earth s gravtatonal felds. In each ndvdual ste there are a number of avalable energy levels due to the magnetc feld confnement and avalable degrees of quantum degeneracy. We consder n our scenaro a two-level quantum system confguraton n whch there s a vbratonal ground state φ,g j and a vbratonal excted state φ,e j n each sngle lattce ste at the th (or jth ) magnetc band, where 6= j {1,..., n} s the potental well ndex startng from the center ste. The stes are characterzed by a confgured magnetc bottom Bmn to trap alkal atoms whch are magnetcally prepared n a low magnetc feld seekng state. Our system conssts of n quantum wells (QWs) that are ndrectly coupled va the magnetc band gap. In the asymmetrcal QWs, we consder the two lowest energy state, n each ndvdual potental well,.e., an ndvdual lattce ste, are closely spaced and well separated from the other hgher levels wthn the lattce ste. Ths s a pcture of a tow-level quantum system wth neglgble nteracton between the many bosons dstrbuted n the two energy levels of the system, permttng the two-mode approxmaton of the many-body problem n our proposed magnetc lattce [9, 10, 13. # $15.00 USD (C) 2009 OSA Receved 2 Nov 2009; accepted 5 Dec 2009; publshed 18 Dec December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24365

10 The general second-quantzed Hamltonan, n terms of bosonc creaton, Φ (x), and annhlaton, Φ(x), feld operators, for a system of N nteractng boson of mass M confned by an external magnetc potental B(x) at zero temperature s gven by ( ) Ĥ sys = dx Φ (x) h2 2M 2 +B(x) Φ(x) dx Φ (x)[ d x Φ ( x)u nt (x x) Φ( x) Φ(x) (11) The feld operators obey the usual canoncal commutaton rules. The second term n ths equaton represents the many-bosons nteracton n the usual zero-range approxmaton, where n the s-wave lmt the nter-atomc potental reduces to U nt (x x) g = 4π h 2 a s /M. The couplng constant g s determned from the scatterng length a s. For an n n asymmetrc magnetc lattce wth n magnetc bands and no tunnelng between stes,.e., the case of uncoupled magnetc bands, the ndvdual lattce ste of the two energy levels k {0,1} allows a localzed sngle wave functon of the condensate to be n the ground state, takng the form φ [k,m (x x ) = x,k,m (12) whch can be approxmated by the egenfuncton of an sotropc smple harmonc oscllator. The superscrpt m accounts for the quanta of angular momentum n the z-drecton, where the value of the ndex m depends on the value of k such that for k = 0 there s no angular momentum,.e., m = 0, and for k = 1, m { 1,0,1} n the three dmensons [14,15. In the followngs, we gve a general descrpton for the Hamltonan and the state vectors n such away that t can also be used to descrbe a spnor Bose-Ensten condensate or ultracold fermons. In both scenaros the proposed magnetc lattce can be used to smulate Josephson oscllatons and excton as well as bexcton formaton [26. There are two modes n each ndvdual lattce ste. The lower mode, φ [0,m, s essentally symmetrc and the second mode, φ [1,m, s antsymmetrc. Due to the ntally tlted n potental wells, the tunnelng of the condensate produces a superposton between the hgher magnetc band ground state mode φ [0,m and the adjacent lower band excted mode, φ [1,m j. The superposton ampltude strongly depends on the spatal separaton of the wells determned by α s where B = B max B mn α2 /2 [19. Dynamcally, nteractng bosonc cold atoms n n potental wells wth tght B(x) magnetc feld confnement can be descrbed by generalzng, for n stes, the quantzed Josephson or a two-mode Bose-Hubbard Hamltonan, assumng the ste number dstrbuton starts from the center of the magnetc lattce Ĥ BH = n j =1 + δb j ( J [k,m, j b [k,m ( b b j + b j b ) + b [k,m j + b [k,m j n j =1 b [k,m U [k,m, j ( ) n [ n 1 + n j [ n j ), k k {0,1} (13) where ˆb, j and ˆb, j are the bosonc creaton and annhlaton operators obeyng the canoncal commutaton rules, and ˆn, j = ˆb, j ˆb, j s the bosons number operator and δb s the tltng magnetc potental. The J [k,m, j and U [k,m, j are tunnelng and nter-lattce ste boson-boson nteracton parameters, respectvely, defned as ( ) J [k,m, j = dxφ [k,m, j (x x ) h2 2M 2 +B(x,t) φ [k,m, j (x x j ) (14) (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24366

11 (a) (b) (c) Fg. 5. (a) Schematc representaton of the two-mode system φ g, φ e. (b-c) Representaton of the Josephson oscllaton,.e., adabatcally nduced Josephson current, and the couplng between two dfferent states at each par of adjacent stes, respectvely. The schematc representaton shows only one ste of the propagatng Josephson currents. U [k,m, j = 3g 8 dx φ [0,0, j (x x j ) 4 (15) When δ B = 0, the Bose-Hubbard Hamltonan descrbes the nonnteractng magnetc bands where the condensate s assumed to occupy the ground state φ [0,0, j of an ndvdual lattce ste n each magnetc band wth a sngle-mode one-level confguraton regardless of the exstence of the excted mode. Ths s because the no-tunnelng condton results n nter-ste many boson nteracton whch creates a lattce structure descrbed n the Fock regme as wll be explaned n the followng secton. A typcal characterstc that can be encountered n such a stuaton s the Mott nsulator state [13, 27. Snce our nterest s n the adabatcally nduced transton process, we only summarze the case where the trapped ultracold atoms exhbt a superposton between the vbratonal ground state φ [0,m n the QW and the vbratonal excted state φ [1,m j n the QW j. Ths s a smultaneous b-drectonal Josephson transton between each par of adjacent magnetc bands propagatng from the center towards the edges of the lattce,.e. along x,y and x, y drectons, as schematcally represented n Fg. 5(b). Based on the confguraton of ths new type of magnetc lattce, we realze that the number of stes n creates nterestng confguratons at the center of the lattce. When n takes odd values,.e., n = 5,7,9,..., there s only one center ste screened by the surroundng four stes. In ths case of a sngle center ste, the nduced tunnelng nteracton s domnated by the equvalent Coulomb potentals between the four surroundng stes dstrbuted along the x y magnetc bands and takng a molecule-lke confguraton whch can be created only n bound states assumng that there are symmetrcal tunnelng ampltude n the four drectons. When n takes even values,.e., n = 4,6,8,..., there are four symmetrcal center stes whch exhbt a frst Brlloun zone dmensonalty and hence have Bloch nteractng wave functons and a feld operator expanded n localzed sngle-partcle wave functons of the form Φ(x) = 4 =1 b γ φ [k,m 1,γ (x) (16) where φ [k,m 1,γ represents the Bloch wave functon wth k = 0, and γ s the wave vector n the frst Brlloun zone. b m γ s the usual bosonc annhlaton operator. The center tunnelng mechansm (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24367

12 allows the Bloch wave functon φ [k,m 1,γ va the standard egenvalue problem ( ) h2 2m 2 + B(x) φ [k,m 1,γ = E 1,γ φ [k,m 1,γ (17) Ths stuaton s smlar to the band structure n a two-dmensonal optcal lattce; however, n our asymmetrc two-dmensonal magnetc lattce, t s lmted to the four center stes when n s even [8. Ths s because there s no tltng potental between the four stes, δb = 0. When the tunnelng s allowed between the four stes only the excted states, φ [1,m 1 contrbute to the local tunnelng ampltude between the four stes where the superposton state s descrbed [ for the four center stes {a,b,c,d} va the couplng ampltude such that J [1,m 1 φ [1,m 1,b φ [1,m 1,c + φ [1,m 1,c φ [1,m 1,d + φ [1,m 1,d φ [1,m 1,a. φ [1,m 1,a φ [1,m 1,b + The vbratonal ground states, φ [0,m 1 contrbute to the nteracton mechansms between the four center stes and the stes of the surroundng frst magnetc band where the local nteracton of the excted mode, φ [1,m 1, between the four stes s assumed to be neglgble. As wll be descrbed n the followng secton, ths pcture can be generalzed to descrbe, regardless of the asymmetrcal feature, the Mott nsulator quantum phase transton across the n n twodmensonal magnetc lattce va the dschargng Josephson state. The quantum phase transton can be oscllatng adabatcally, a sgnfcant feature for quantum nformaton processng n such type of magnetc lattces. 4. Josephson Oscllatons and the Exctons Mott Phase Transton The two well known regmes, the Josephson regme and the Fock regme, can be appled to our approach to descrbe the outcome of the tunnelng process of the trapped ultracold atoms [16. Regardless of the asymmetrcal effect n our two-dmensonal magnetc lattce, one can stll descrbe the transtons between the lattce stes as an nduced Josephson current,.e., the superflud phase transton, between two recognzable states whch are the Mott nsulator state and the dschargng Josephson state n whch the adabatcally controlled Josephson current oscllaton depends on the couplng values. We descrbe brefly n the followng the requred condtons for such transtons. Mott Insulator State In the non-nteractng regme where the frst-mode φ [0,m, j s regarded as the ground state and as a soluton for the Schrödnger equaton wth neglgble nter-ste nteractons regardless of the exstence of the excted mode φ [1,m, j, the condensate s sad to have an ndvdual mode soluton of the form φ g, j = φ [0,m, j (x x) [19. Intally, there s an approxmately equal number of atoms n each ste N s N/n, where ths number s fxed untl tunnelng s allowed. The atoms are completely localzed at the lattce ste where we assume the couplng strength between the two levels s very weak such that N s U [0,0, j E [k,m, j = 2 [ E [1,m, j E [0,m, j = hω (18) where E [k,m, j s the energy dfference. Localzaton n our scenaro means that the frst symmetrc mode, φ g, j, s domnatng over all lattce stes and the expanson of the excted mode,, between each two adjacent magnetc bands s neglgble. Ths condton can be thought φ e, j (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24368

13 of as the Fock regme where the on-ste nteracton s greater than the hoppng strength, U [0,m U [1,m J [0 1,m j. It s a sgnature of the Mott nsulator state where all partcles are localzed n the ground state, neglectng the excted level, wth a defned number of atoms at each ste exhbtng no coherence nor a macroscopc wave functon and permttng a one-level approxmaton havng a one-level Bose-Hubbard Hamltonan for the n n asymmetrc magnetc lattce of the form Ĥ 1HB = n j =1 ) J [0,0, j (ˆb ˆb j + ˆb j ˆb + n j =1 U [0,0 ( ), j ˆn [ˆn 1 + ˆn j [ˆn j 1 2 where δb = 0, and Ĥ 1HB descrbes a fnte number of stes. In the case of an nfnte number of stes, Eq. (18) descrbes the sngle-band Bose-Hubbard Hamltonan for weakly nteractng bosons n a symmetrc magnetc lattce,.e., Ĥ 1HB Ĥ HB. Note that the operators ˆb, j, ˆb, j and ˆn, j are for the case of k = m = 0. The Fock space state vector takes the form ϕ = N+1 n, j =0 (19) c [0,0, j n,n j (20) Here the sze of the Hlbert space s roughly estmated to be (N + 1) and c [0,0, j represents the ground state ampltude. Excton Mott Quantum Phase Transton va the Induced Josephson Atomc Current The stuaton where the hoppng strength s ncreased adabatcally, can typcally be descrbed usng the self-consstent nonlnear Schrödnger equaton,.e., the Gross-Ptaevsk equaton (GPE) h ϕ (x,t) t [ = h2 2m 2 +B(x) + g ϕ (x,t) 2 ϕ (x,t) (21) whch represents the nonlnear generalzaton of the snusodal Josephson oscllatons occurrng n superconductng junctons, descrbed n some detal n [17,18,28. It also allows the noton of the generalzed tunnelng mechansms of the condensate between two stes to be appled to the n asymmetrc magnetc bands where the feld operator s expanded n terms of the statc ground state soluton, φ g, of the GPE Eq. (21) for each ndvdual uncoupled lattce ste, and n terms of the ampltude of the relatve populaton N s N s expressed as ψ (x,t) = N s(x,t)eθ(t), where θ s the correspondng phase [29. Thus we expand the feld operator as ϕ (x,t) = N n =1 ψ (x,t)φ g (22) Hence the evoluton of the ultracold atoms populaton N s (t) of the n stes can be obtaned by ntegratng the spatal dstrbutons ϕ (x,t) to obtan the tme dependent equaton [16, 18, 29 h ψ (x,t) t [ = Ω, j (x,t) +U [0,m, j N ψ ψ 2 n j J [0,m, j (x,t)ψ j (23) where Ω, j (x,t) = [ h 2 g dx φ 2M 2 +φ g B(x,t)φ g (24) (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24369

14 Equaton (23) s known as the Bose Josephson juncton (BJJ) tunnelng equaton whch can be re-derved n terms of the phase dfferences and the fractonal populaton dfferences between the lattce n stes. A smlar case of a double well s dscussed n [19. The transton from self-trappng to Josephson oscllatng states [10, 12 s possble n ths type of asymmetrcal magnetc lattce havng several uncoupled magnetc bands whch can be coupled va the nduced Josephson current or t can be descrbed as a Josephson dschargng state created va quantum cold collsons. Startng from the Mott nsulator state and by adabatcally tunng the hoppng strength va the applcaton of B z bas, the number of atoms n each ste N s vares n tme to a crtcal number of atoms that s perodcally oscllatng and predcted to produce the n magnetc nterband Josephson oscllatons. Schematc representatons are shown n Fg. 5. The quantum state of the asymmetrcal magnetc lattce at ths pont of the transton belongs to what s known as the Josephson regme, where the n-ste (nter-well) many boson nteracton of the two energy states k = 0 and k = 1 s very small compared to the off-ste many boson nteracton, namely the condton U [0,m U [1,m J [0 1,m j (25) The domnant transton at the crtcal number of atoms s, startng from the center of the lattce, the n magnetc band one-drectonal Josephson transton, φ [0,m φ [1,m j. Oscllatng bound states wll be created n both stuatons, on and off-stes leadng to excton Mott multtranstons. The on-ste bound state s created between the ground state and the excted state, e.g., two dfferent components of a spnor BEC trapped n a sngle ste. Ths s due to the dpole nteracton whch occurs when the condensate n the excted state perodcally couples to the oscllatng ground state at each ndvdual ste. The number of atoms may play an mportant role n such a scenaro where t mght be requred to have an equal number of atoms n both modes [ The fragmented condensate n the ground state of each ste couples perodcally va the Coulomb potental to the excted state n the neghborng ste exhbtng a recombnaton rate between two dfferent adjacent magnetc bands wth a plasma oscllaton frequency relatve to the strength of the Coulomb nteracton whch s expermentally realzed n a sngle bosonc Josephson juncton [10. Ths can also be thought of as a dfference n the chemcal potental between stes n the case of trapped ultracold fermons. 5. Concluson We have developed a new method to create an asymmetrc two-dmensonal magnetc lattce. In ths artcle we have proposed one of the applcatons that can be realzed usng ths type of quantum devce, where trapped ultracold atoms can smulate the crtcal quantum phase transton of collectve exctatons such as the excton-mott bound states and the oscllaton mechansms of the Mott nsulator to a dschargng Josephson state. In our approach, adabatcally controlled coherent tunnelng of the ultracold atoms wll nduce a dc and ac Josephson current, a sgnfcant feature that can be used to encode and transfer, across the 2D asymmetrcal lattce wth n n Josephson qubts for quantum nformaton processng. We have establshed a theoretcal framework that can be used to calculate the relevant parameters requred to descrbe the fundamental concepts. Acknowledgment We thank James Wang, from the Centre for Atom Optcs and Ultrafast Spectroscopy n Swnburne Unversty, for performng the atomc force mcroscope and magnetc force mcroscope measurements. (C) 2009 OSA 21 December 2009 / Vol. 17, No. 26 / OPTICS EXPRESS 24370