Fractionalized topological defects in optical lattices

Transcription

1 PAPER OPEN ACCESS Fractonalzed topologcal defects n optcal lattces To cte ths artcle: Xng-Ha Zhang et al 205 New J. Phys Vew the artcle onlne for updates and enhancements. Related content - Fractonalzed flux, Majorana fermons and non-abelan anyons n topologcal superflud on optcal lattces Ya-Je Wu and Su-Peng Kou - Topologcal aspects n spnor Bose Ensten condensates Masahto Ueda - Lght-nduced gauge felds for ultracold atoms N Goldman, G Juzelnas, P Öhberg et al. Recent ctatons - Majorana modes and topologcal superfluds for ultracold fermonc atoms n ansotropc square optcal lattces Ya-Je Wu et al - Majorana modes and s-wave topologcal superfluds n ultracold fermonc atoms Ya-Je Wu et al - Fractonalzed flux, Majorana fermons and non-abelan anyons n topologcal superflud on optcal lattces Ya-Je Wu and Su-Peng Kou Ths content was downloaded from IP address on 2/09/208 at 5:22

2 do:0.088/ /7/0/0309 OPEN ACCESS RECEIVED 26 June 205 REVISED 2 September 205 ACCEPTED FOR PUBLICATION 2 September 205 PUBLISHED 2 October 205 PAPER Fractonalzed topologcal defects n optcal lattces Xng-Ha Zhang, Wen-Jun Fan,2, Jn-We Sh and Su-Peng Kou,3 Department of Physcs, Bejng Normal Unversty, Bejng, 00875, People s Republc of Chna 2 Luoyang Electronc Equpment Test Center of Chna, Luoyang, 47000, People s Republc of Chna 3 Author to whom any correspondence should be addressed E-mal: spkou@bnu.edu.cn Keywords: fractonalzed topologcal defects, dslocaton, optcal lattce Content from ths work may be used under the terms of the Creatve Commons Attrbuton 3.0 lcence. Any further dstrbuton of ths work must mantan attrbuton to the author(s) and the ttle of the work, journal ctaton and DOI. Abstract Topologcal objects are nterestng topcs n varous felds of physcs rangng from condensed matter physcs to the grand unfed and superstrng theores. Among those, ultracold atoms provde a playground to study the complex topologcal objects. In ths paper we present a proposal to realze an optcal lattce wth stable fractonalzed topologcal objects. In partcular, we generate the fractonalzed topologcal fluxes and fractonalzed skyrmons on two-dmensonal optcal lattces and fractonalzed monopoles on three-dmensonal optcal lattces. These results offer a new approach to study the quantum many-body systems on optcal lattces of ultracold quantum gases wth controllable topologcal defects, ncludng dslocatons, topologcal fluxes and monopoles.. Introducton Topologcal exctatons (.e., doman walls, strngs and monopoles) exst n a wde varety of systems n condensed matter physcs, such as superfluds, superconductors and lqud crystals. Among them, BECs of dlute atomc gases [, 2] are the ones of deal testng grounds for nvestgatng topologcal exctatons. A varety of topologcally nterestng structures, such as vortces, knotted textures, skyrmons, and monopoles, have been fascnatng for qute a few decades and recently have also been thoroughly studed n BECs of ultracold atoms [3 8]. In general, topologcal exctatons can be characterzed by an nteger number. An example of typcal 2D topologcal defects s quantzed vortex n sngle component BECs, of whch the crculaton of supercurrent M velocty v s s quantzed by = dl v 2 p s (M s the mass of partcle) owng to analytcty of sngle-valued order parameter (OP). A topologcal confguraton wth < (we call t fractonalzed vortex) always traps a phase strng that nduces a (sngular) branch-cut n the BECs and then s confned by a lnear potental. In a specal spnor BEC, deconfned topologcal exctatons wth half-quantzed number may exst. In threedmensons, an mportant topologcal defect s Drac monopole, a pont source of quantzed flux. In [9], the monopole defect was created n a spn texture of spnor BEC by adabatcally modfyng external magnetc felds. However, the monopole wth fractonal quantum number (we call t fractonalzed monopole) has never been addressed. On the other hand, researchers recognzed that topologcal defects (such as the vortces, dslocatons, monopoles) wll have nontrval quantum propertes [0 2] due to the nterplay between defect topology and the topology of the orgnal states (the topologcal nsulators, topologcal superconductors). Moreover, t s predcted that a Majorana-Fermon zero mode may be trapped around the core of a quantzed vortex n topologcal superconductor and obeys non-abelan statstcs [3 6], whch may be appled to realze topologcal quantum computaton [7 20]. Here we propose a sgnfcantly smpler expermental protocol to embed topologcal confguratons to realze optcal lattces wth manpulated lattce defects dslocatons. In partcular, the dslocatons may nduce fractonalzed flux/monopole on 2D/3D optcal Peerls-lattce (see detaled dscussons below).we demonstrate that an analogue of the fractonalzed flux/monopole n BEC of an atomc gas can exst as a ground 205 IOP Publshng Ltd and Deutsche Physkalsche Gesellschaft

3 Fgure. (a) An llustraton of 2D square Peerls-lattce. On blue/red lnks, the hoppng parameters are te fx of 3D cubc Peerls-lattce. On blue/red/green lnks, the hoppng parameters are te fx te fy te fz. te f y. (b) An llustraton state confguraton. As a result, these results offer a new method to study the quantum many-body system on an optcal lattce of ultracold quantum gases wth varous fractonalzed topologcal defects. 2. Model Hamltonan In ths paper we study the one-component Bose Hubbard (BH) model on 2D square Peerls-lattce and that on 3D cubc Peerls-lattce, for whch the Hamltonans are gven by [2, 22] å( j j ) ájñ H =- t b b + h.c. -m n U + ån n -, 2 where b and b are the bosonc creaton and annhlaton operator on ste, respectvely. n = b b s the partcle number operator on ste. U s the strength of the repulsve nteracton. á, jñ represent all nearest neghborng lnks. Fgure (a) s an llustraton of 2D square Peerls-lattce. For the BH model on a 2D square Peerls-lattce, the hoppng parameters t j are not real numbers but have unform phase factors along the x-drecton t, e = te f x + x (the blue lnks n fgure (a)) and the y-drecton t, e = te f y + (the red lnks n fgure (a)) where f = ( f, y x fy) s the Peerls phase [23]. When a partcle moves around a plaquette, there s no extra phase factor of ts wavefuncton due to cancelaton effect as e fx e fy e- fx e- fy º. We call ths type of lattces wth complex hoppng parameters but no extra flux as Peerls-lattce. Smlar to 2D case, the hoppng parameters t j for the 3D cubc Peerls-lattce (fgure (b)) defned by t, e = te f x +, t te, t te x, e = fy z y, e = f + + are characterzed by a z vector f = ( fx, fy, fz). Peerls lattce s a lattce wth unform magnetc vector potental, whch s physcally trval and can be gauged away on flat space. These models can be solved by the mean-feld (MF) approxmaton [2, 22], b bj» b ábjñ+áb ñbj -áb ñ áb jñ, where y =áb ñs the superflud (SF) OP. Usng ths MF approach, we derve the propertes of the BH model on Peerls-lattces. For the case of strong couplng lmt, U t, the Bose gas turns nto the Mott nsulator (MI) phase; for the case of weak couplng lmt, U t, the ground state s the SF phase, y = y 0 e j 0, where y0 = y = n0 and j 0 s an arbtrary real number from 0 to 2 p. Because the Peerls phase wll never change the ampltude of the OPs, we get the same global phase dagram as that of the BHs wthout Peerls phase. 3. Dslocaton and fractonalzed fluxes on 2D Peerls-lattce å ( ) () To descrbe the topologcal propertes of the dslocatons, we defne the Burgers vector a vector expressng the drecton and magntude of slp caused by a dslocaton, B = du [24, 25], wth u the dsplacement feld L vector and Γ the Burgers crcut defned on an deal lattce. For the dslocaton n fgure 4(a), the correspondng Burgers vector around the dslocaton O s -e y. Now we show how dslocatons on Peerls lattce nduce fractonal fluxes. We focus on the case f x = 0 and f y = fas fx s trval when B s n the y-drecton. Smlarly, we can consder a closed crcut G around the dslocaton on the dslocated lattce and fnd ts mage on an deal lattce. We could get that du =-B G [26]. As the magnetc vector potental s unform on a Peerls lattce, we fnd that the net flux nduced by the dslocaton s F= a d l =-f B. 2 G

4 Fgure 2. Illustraton of f flux on a Peerls square lattce wth dslocatons. The phases of hoppng terms on blue lnks are 0 whle that on red lnks are f. Each row of lattce stes s labeled wth a n y. Fgure 3. Illustraton of f flux and phase strng on a Peerls square lattce wth dslocatons after the gauge transformaton. The phases of hoppng terms on blue lnks are 0 whle that on red lnks are f. On a 2D square lattce, the Peerls phases fy can also be elmnated by a gauge transformaton b be- ny f and b b eny wth n y the y-coordnate of the ste. However, wth the exstence of topologcal defect dslocaton, the Peerls phases cannot be elmnated where the dslocaton exsts. We can perform a smlar gauge transformaton b b e -nyf, b b e ny f, () 2 where n y s llustrated n fgure 2. After ths gauge transformaton, the gauge phases n the hoppng terms on the blue lnks are elmnated whle the gauge phases on red lnks turn to f, as s llustrated n fgure 3. The red lnks can be vewed to be a phase strng, whose ends are f fluxes. For dslocatons wth hgher Burgers vector B =-m, the gauge phases on red lnks are mf and the ends of the strng are mf fluxes. Near the dslocatons the phase factors of the local SF OPs on dfferent stes are not unform due to the nonzero fractonal flux on the dslocaton. When a partcle moves around a dslocaton wth the Burgers vector B, an extra phase factor wll be obtaned: e F = exp[( f d u)]. A unversal relatonshp between the nduced L flux number (the crculaton of supercurrent velocty, ) and the Burgers vector of the dslocaton s gven by F B = = a dl = dj = f where the effectve vector potental s defned by a = j. L s the 2p 2p L 2p L 2p closed loop around the flux. Thus, the par of dslocatons have opposte topologcal fluxes: one s -f B, and the other s - f B. The total nduced topologcal flux s zero due to the cancelaton effect. For the case of f B ¹ 0, 2 p, we have the nduced topologcal fractonalzed flux around the dslocatons. In general, there s y twsted phase away from the dslocaton as determned by Im ln = j. Here j y s defned by f B j = Im ln z - z and z = x + y represents the locaton of the flux. It s obvous that the å( 2 ) ( 0) p

5 O (a) B n (b) O 0.08 n/n φ (c) (d) t/u Fgure 4. (a) An llustraton of the dslocatons on a 2D square Peerls-lattce, for whch the end ste s O. The Burgers vector B s ndcated by the bg black arrow. On blue/red lnks, the hoppng parameters are te fx te f y. (b) Partcle densty dstrbuton of nteractng bosons near the end (O ste) of the dslocaton wth t U = 0.2 and m U =.5. (c) The IMF numercal results of the phase factors of nteractng bosons near the dslocatons (O ste) wth t U = 0.05, m U =.5 and f = ( 0, p). j denotes the condensed phase on ste. (d) The local partcle densty varaton Dn n 0 va t/u on the end of the dslocaton (O ste) when m U =.5. remove ste lne connected one par of dslocatons on a Peerls-lattce plays the role of a phase branch-cut of partcle s wave-functon. Then, we study the propertes of nteractng bosons on the Peerls-lattce wth dslocatons by numercal approach. Ths model can be solved by the nhomogeneous mean-feld (IMF) approxmaton [2, 22], b bj» b ábjñ + áb ñbj - áb ñáb jñ, wherey =ábñs the local superflud (SF) OP. Wthout translaton nvarance, we have one local SF OP on each lattce ste that conssts of the varable space of { y } on all stes. { y } can be solved self-consstently locally from the local Hlbert space up to 0 one-component bosons. In the MI phase, the SF OPs vansh, y = 0, and the partcle number at each ste remans unt for the ground state. As a result, we fnd that dslocatons have no sgnfcant effect n the MI phase. In contrast, n the SF phase, we fnd that near the dslocatons O, due to the decrease of the connectng lnk numbers, the local partcle densty decreases. See the llustratons n fgure 4(b). The changes n the local partcle densty n near the dslocatons are strongly enhanced by the on-ste nteracton. Fgure 4(d) shows local partcle densty varaton Dn n 0 near the the dslocatons va t/u changes where D n = n - n 0. From fgure 4(d), one can see that the on-ste repulsve nteracton wll reduce the nfluence of the lattce defects on the local partcle densty varaton. In partcular, from numercal calculatons, we found that the local SF OPs show nontrval topologcal propertes. For example, for f = 0, p, there ndeed exsts a p flux around the par of dslocatons, of whch the nduced flux number (the crculaton of supercurrent velocty, ) s 2. The phase patterns of topologcal vortces near the dslocaton j are gven n fgure 4(c). The nduced topologcal flux by the dslocatons n the Peerls-lattce s protected by the topologcal propertes of the dslocatons. The fluctuatons of the local hoppng parameters wll never change the topologcal flux. In experments, people may detect the phase dstrbuton n the SF order to observe the effect from the dslocatons [27]. 4. Fractonalzed skyrmons on 2D Peerls-lattce Based on the BH model n large-u lmt, U (the hard-core boson lmt), we dscuss the fractonalzed skyrmon on 2D Peerls-lattce. The Hamltonan of hard-core bosons on Peerls-lattce can be wrtten as å j j ájñ å H =- t b b - Vn, 3 () where bˆ satsfes the non-double-occupancy constrant, b ˆ b ˆ. V denotes the trap potental. The above representaton of the many-body quantum states s equvalent to the followng operator dentty + - between the spn and boson operators b Sˆ, b Sˆ ««, b b «Sˆz + 2[28, 29]. Wth ths mappng, the hard-core bosons Hamltonan, equaton (3) becomes that of the XY model wth a magnetc feld appled along the z-drecton: H =- t ( S ˆ + S ˆ - + z S ˆ + S ˆ - ) - V S ˆ +. In the spn ordered state (that s just the å j j j j å ( 2) 4

6 Fgure 5. (a) The IMF numercal results of the spn vectors of a quarter-skyrmon (Q = 4) nduced by dslocaton for the nteractng r Bosons, wth t U = 0.0, V U ( ; r 0 ) = - 2 (b) s the IMF numercal results of S ˆ z (or n ) va r = r for a quarterskyrmon. BEC state for the hard-core bosons), we can use the sem-classcal representaton to denote the local OPs as ( ) ( ) Sˆ x, Sˆ y, Sˆ z N = N x, N y, N z 2 2 = ( sn q cos j, sn q sn j, cos q). 2 In the spn ordered state, a dslocaton may nucleate a skyrmon wth a fractonalzed topologcal charge (we call t fractonalzed skyrmon).wedefne the topologcal charge (Pontryagn ndex) of fractonalzed skyrmons to be d2 f B º ò r N xn yn cos 4 ( ) = ( - q ) p 4p where q denotes the z-drecton of spn polarzaton far from the core of (fractonalzed) skyrmon whch s determned by the fllng number of hardcore bosons, cos q = n0-2. Let s ntroduce a complex feld w = w() z + w2() z by a steregraphc n () z n2 () z projecton from north pole, w() z =, w z n z 2() = [30, 3]. A typcal (fractonalzed) skyrmon s - 3() -n3 () z z- z soluton wth topologcal charge s gven by w () z = ( 0 ) x. Here ξ s the radus of ts core. For the skyrmons wth nteger, the functon w s not only analytc, but also meromorphc. However, for the skyrmons wth fractonalzed topologcal charge, the functon w s not analytc and there exsts a dslocatonnduced branch-cut n the functon w. Then, we study the propertes of hard-core bosons on the Peerls-lattce by numercal approach. From the numercal calculatons, we found that for the hard-core boson on a unform Peerls-lattce, the dslocaton nduces a vortex-lke spn confguraton on XY plane (a meron wth a narrow core). When we add a smooth trap potental, the stuaton s dfference. We studed the hard-core bosons on three dfferent types of trap potental, harmonc trap potental (V ( x) ~ x 2 ), lnear trap potental (V ( x) ~ x) and square-root trap potental (V ( x) ~ x). All these trap potentals have mnmum values at O ste (the dslocaton). The numercal results show that for the hard-core bosons on the Peerls-lattces wth dfferent trap potentals, the dslocaton-nduced fractonalzed skyrmons have the same topologcal charge as = f B. These topologcal confguratons may be 4 pctured as the smlar pcture: nsde the core r - r, p 0 < x the spn s polarzed along z-drecton (that means n ); outsde t r - r 0 > x, the spn s on XY plane (that means n 2or q = p ). See the numercal 2 results n fgure 5(a), n whch the spn vectors of a fractonalzed skyrmon wth = 4 (we call t a quarterskyrmon) nduced by dslocaton are shown for the nteractng bosons, wth t U = 0.0, V U = r 2 (. r 0 ) Fgure 5(b) shows S z ˆ (n ) va r = r. From these fgures, one can see that the core s radus dffer for quarter-skyrmons n dfferent potental traps. 5. Fractonalzed flux-tubes and fractonalzed monopoles on 3D Peerls-lattce In ths secton, we study the propertes of BH models on 3D cubc Peerls-lattce wth dslocatons. We take a dslocated 3D Peerls-lattce characterzed by f = ( 0, fy, 0) as an example. For 3D cubc Peerls-lattce wth an edge dslocaton, of whch the Burgers vector B s set to be along y- drecton, B = ( 0,, 0). Now, the edge dslocaton wll trap a topologcal fractonalzed flux-tube along t. See the llustraton n fgure 6(a). For smplfy, we don t plot the lnks along z-drecton to emphasze the flux-tube on x y plane. In the x y plane, we get a 2D square Peerls-lattce wth a 2D edge dslocaton. When a partcle moves on the x y plane around an edge dslocaton, an extra phase factor s e F = exp[( f du)] where the nduced 5 L

7 Fgure 6. (a) An llustraton of 3D cubc Peerls-lattce wth an edge dslocaton. (b) An llustraton of a dslocaton-nduced π-fluxtube. (c) The numercal results of the phase pattern of the π-flux-tube on a 3D cubc Peerls-lattce wth a dslocaton. Fgure 7. (a) An llustraton of 3D cubc Peerls-lattce wth half-dslocaton. (b) An llustraton of a half-monopole as end of a π-fluxtube. (c) The numercal results of the phase pattern of a half-monopole on a 3D cubc Peerls-lattce wth a half-dslocaton. flux number and the Burgers vector of the dslocaton s also gven by F=-f B. Thus, as shown n fgure 6(b), the edge dslocaton may nduce a fractonalzed flux-tube. Then, we study the propertes of nteractng bosons on the Peerls-lattce wth dslocatons by the numercal approach. The phase pattern of a π-flux-tube on 3D Peerls-lattce s shown n fgure 6(c). The result ndcates that there ndeed exsts the dslocaton-nduced fractonalzed flux-tube along z-drecton. Next, we study the fractonalzed monopoles nduced by dslocatons on 3D cubc Peerls-lattce. There are two methods to generate fractonalzed monopoles. One method s to consder the nonunform Peerls phases: on upper half system (for y > 0), f = ( 0, fy, 0 ), on lower half system (for y < 0), f = ( 0, 0, 0 ). Thus the dslocaton wll nduce flux-tube on upper half system, but not on lower half system. The flux-tube s termnated at y = 0, or we get a fractonalzed monopole at y = 0. To characterze the fractonalzed monopole, we defne the monopole number, = 2 p y By ds = S 2 p a dl = f where the effectve magnetc feld s defned by = a. S s the closed L 2p surface around the monopole. Thus, we have a fractonalzed monopole attachng a fractonalzed flux-tube (a true Drac strng). Here, the sngularty s physcal, as t can not be gauged away. The other method s to consder a half-dslocaton on a 3D cubc unform Peerls-lattce: the dslocatons exst n lower half system. See the llustraton n fgure 7(a). As a result, the dslocaton-nduced flux-tube termnates n the mddle of the system. As shown n fgure 7(b), the end of the flux-tube wll play the role of a fractonalzed monopole. We use the second method to generate a fractonalzed monopole and the correspondng BH model s calculated by numercal approach. From numercal calculatons, we found there ndeed exsts a fractonalzed 6

8 Fgure 8. The optcal system and the ntensty dstrbuton of 2D optcal lattce wth edge dslocaton. The phase shft f0 = p,0and topologcal charge m =. monopole around the dslocaton, of whch the monopole number s = 2. We call ths topologcal object half-monopole. The phase pattern near the half-monopole s gven n fgure 7(c). Now, the half-monopole can be vewed as the end of a π-flux-tube whle the Drac monopole s the end of an unphyscal flux tube the Drac strng. 6. The physcal realzaton 6.. Optcal Peerls-lattce Frstly, we show how to realze the optcal Peerls-lattce. In recent experments, tunable Peerls phases n a D Zeeman lattce have been realzed usng a combnaton of rado-frequency and Raman couplng [32]. Now, a hot topc s to realze the spatally dependent Peerls phases that leads to an effectve (staggered) magnetc flux through one unt cell of the optcal lattce. Thus, to engneer unform complex hoppng ampltudes wth Peerls phases along the y-drecton [23], t te, + e = f, the Raman-asssted hoppngs are used by a par of far-detuned y runnng-wave laser beams wth dfferent frequences, w0 and w 0 +D[33 36]. A lnear potental s added along the y-drecton, for whch the energy dfference between two nearest neghbor stes s D. Thus the Ramanasssted hoppngs are restored along the y-drecton wth the resonance energy w =DHowever,. along the x-drecton, there s no such Raman-asssted hoppngs. As a result, we can obtan unform complex hoppng ampltudes wth Peerls phases along the y-drecton, t, e = t, t, e = te f y +. x + By smlar approach, we can y realze a 3D optcal Peerls-lattce, for example t = t, t = te f y + +, t, + e = t. z, ex, ey D optcal lattces wth edge dslocatons Secondly, we dscuss the physcal realzaton of the 2D optcal lattce wth dslocatons. An optcal vortex s a beam of lght whose phase vares n a screw thread lke manner along ts axs of propagaton. The optcal vortex waves possess a phase sngularty whch occurs at a pont or a lne where the physcal property of the wave becomes nfnte or changes abruptly. We use the propertes of vortces to realze the dslocatons [37 42]. An deal optcal vortex propagatng n the z-drecton may be wrtten n the cylndrcal coordnate as E ( r, j, z) = A( r, z) exp( mj) exp( - kz), where A ( r, z) s ampltude functon, k = 2p ls the wave number, and m s known as the topologcal charge. The optcal system s shown n fgure 8: a plane wave and an optcal vortex wave nterfere n the x z plane wth wavelength l, and two plane waves form an optcal standng wave along the y-axs wth wavelength l2. We use dfferent frequences for the laser felds n order to elmnate any resdual nterferences between the standng waves and the optcal vortex. The optcal vortex wave propagates along the z-axs. The plane wave n x z plane s travelng at an angle θ to the z-axs. The ntensty of lght n recevng plane at z 0 can be wrtten as: I µ cos( k( x sn q + z0 cos q - z0) + mj + f0) + cos 2( k2 y), where k = 2 p l, k2 = 2 p l2, m s the topologcal charge, f0 s the phase shft from the optcal path dfference between the optcal vortex and the plane wave n the x z plane, j = tan - ( x y ). We wll set the nterferng plane at z0 = 0 n all our smulaton. The results are shown n fgure 8. The smulaton regon s 5 5 um. l = 500 nm, l 2 = 600 nm are used n our smulatons. From fgure 8, we can see that the dslocaton emerges at the phase sngularty pont. The smulaton results show that the dslocaton s senstve to the phase shft between the optcal vortex and the plane wave (not shown). The ntensty dstrbuton wll be reversed when the phase dfference s π, as shown n fgures 8(a), (b). So we need to adjust the phase shft properly to obtan the desgned dslocaton optcal lattce, whch can be realzed by tunable phase plate. The frnge spacng would ncrease duo to the decrease of the angle θ. We can also 7

9 Fgure 9. The optcal system and the ntensty dstrbuton of optcal lattce wth 3D edge dslocaton. Fgure 0. The optcal system and the ntensty dstrbuton of optcal lattce 3D screw dslocaton. a, b are ordnary flat mrror and c s a phase conjugate mrror. generate more dslocaton arms f the topologcal charge m > (not shown). The advantage of ths system s we can obtan the dslocatons relatve convenently, but t s dffcult to acheve the multple or perodc dslocatons Three-dmensonal optcal lattces wth edge dslocatons It s mpossble to acheve 3D optcal lattce wth perfect dslocaton smply usng the method descrbed n fgure 8. The dslocaton lne wll drft when penetratng through layers. We mproved the optcal system, and realzed a perodc 3D optcal lattce wth stable dslocatons. The optcal system s shown n fgure 9. The man structure s the same as 2D optcal lattce except two dfferences. Frst, we obtan the perodc array of the dslocaton accordng to the optcal vortex reflected from the new added mrror n z-drecton. The second and more mportant s that the plane wave n x z plane can only propagate along the x-drecton or be perpendcularly to the optcal vortex whch s dfferent to realze n the 2D stuaton. If not, one needs to add another proper plane wave, and can also obtan the 3D dslocaton optcal lattce. The ntensty of the 3D dslocaton optcal lattce can be wrtten as: I µ cos( 2k z) + cos( kz ) cos ( kx - mj + f0) + cos2( ky. 2 ) All the meanngs and value of the parameters are the same as the 2D case. Usng ths method, we get the perodc dslocaton optcal lattce. The style of the dslocatons s assocated wth the topologcal charge and the phase shft. When gven a certan topologcal charge and phase shft, we can get two styles of dslocaton emerged n z-drecton perodcally. The perod depends on the wavelength of the optcal vortex wave Three-dmensonal optcal lattces wth screw dslocatons The 3D smple cubc optcal lattce has been realzed and studed before. In our proposed experment setup to realze 3D optcal lattce wth screw dslocatons, most of the expermental condtons and desgns are the same as 3D smple cubc optcal lattce except for two dfferences. In order to realze the topologcal defects wth the screw dslocatons, we utlze an optcal vortex wave and a phase conjugate mrror. As shown n fgure 0, the blue cylnders are optcal vortex whch travels n z-drecton and reflected by a phase conjugate mrror. The red cylnders are standng waves orthogonal to each other n x y plane. When the topologcal charge of the optcal vortex wave m = 0.5, the 3D optcal lattce wth screw dslocatons s obtaned, shown n fgure 0 (rght). 8

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