计算自旋轨道耦合Spin-2 BEC基态的带拉格朗日乘子的正规梯度流法.pdf

2023,43A(5):16071619http:/Og;?Spin-2 BEC?.KFf?5F6?(O?-:?&?H“O?410081):T?O.KFf?5F6(GFLM)?g;?Spin-2-Odv(BEC)?.uKXm?X,)?.k(o9o|)v(kKX?(J./cNXek?g;?Spin-2 BEC?1?,?y?k?5,?g;?Spin-2 BEC?G?G?g;?Xp=z?C5.c:-Odv;?;?5F6;.KFf.MR(2010)Ka:65N06;65N12;65Z05a:O242.1zI:A?:1003-3998(2023)05-1607-131-Odv(BEC)?-.1?,f3?C?G?.dOd3 19241925 c9,13,14,u 1995 cg3n?y1,10,12.3?,f3Uu)g.1998 c,|14f,g3g23Na N?)?BEC,Sggd?-21.31,?f3mk?,l?)L?g(?.gf F(F?)?BEC,?Spin-F BEC,3 2F+1?,v?2F+1?1x15,16,22.AO/,dufgf$?pK?)g;?(SOC)?A,Cc,Lin?182035?f?BEC?,p?g;?BEC.3|e,T?.Tc,Spin-F BEC 2F+1?Gross-Pitaevskii (CGPEs)5x.?g;?Spin-2 BEC?.,(x,t)=(2(x,t),1(x,t),0(x,t),1(x,t),2(x,t)Tvjz?CGPEsit2=H+0()21Fz()2+1F()1+25A00()2(iy1 x1),(1.1)vF:2022-08-26;?F:2023-03-23E-mail:78:Ig,7(11971007)Supported by the NSFC(11971007)1608n?Vol.43 Ait1=H+0()1Fz()1+1(62F()0+F()2)25A00()1 iy(2+620)x(620 2),(1.2)it0=H+0()0+621F+()1+F()1+25A00()062iy(1+1)x(1 1),(1.3)(x,0)=0(x),x Rd,=2,1,0,1,2.(1.4)H=122+V(x),()=2X=2|(x,t)|2,Fz()=2(|2|2|2|2)+|1|2|1|2,F+()=F()=2(21+12)+6(10+01),A00(x,t)=1522(x,t)2(x,t)21(x,t)1(x,t)+20(x,t),p t m,i J,x Rd(d=2,3)(k?I,g;?r,o,2,1,0gg?,g?gp,V(x)?,?kN1.V(x)?N,Vhar(x)=122xx2+2yy2,d=2,2xx2+2yy2+2zz2,d=3,L-(=x,y,z)?.V(x)?1,Vopt()=IEsin2(q),=x,y,z,p E=122 q2L,IJ-1r?jz.(x,t)venC5N(t):=N(,t):=2X=2ZRd|(x,t)|2dx 1,t 0;(1.5)|M(t):=M(,t):=2X=2ZRd|(x,t)|2dx M,M 2,2;(1.6)fUE(t):=E(,t):=ZRd?2X=2?12|2+V(x)|2)+022+12?|F+()|2+|Fz()|2?+22|A00()|2?i(y)Tfx (x)Tfy?dx E(,0)=:E0,t 0,(1.7)No.5?:Og;?Spin-2 BEC?.KFf?5F61609 N Lo,M Lo|,TOL?=.w,U E()LU Ekin(),U Epot(),pU Espin()g;?U Esoc(),=E()=Ekin()+Epot()+Espin()+Esoc(),Ekin()=12ZRd 2X=2|2!dx,Epot()=ZRd2X=2V(x)|2#dx,Esoc()=ZRd?i(y)Tfx (x)Tfy?dx,Espin()=ZRd?02()2+12?|F+()|2+|Fz()|2?+22|A00()|2?dx,fx fy 5?,NL/fx=01000106200062062000620100010,fy=0 1000106200062062000620100010.?BEC?-K.g;?Spin-2 BEC?g(x)E(g)=minSE(),(1.8)8 S S=(2,1,0,1,2)T|N()=1,M()=M,E()j2(i j)2ki(,tn)k2kj(,tn)k2 0.eb?k(,tn)(=2,1,0,1,2),K?,d(2.12)(2.13)3)c0=m2 m1Mm0m2 m21,c1=m0M m1m0m2 m21.(2.14)?d(2.11)?n=ec0+c112=exp?m2 m1M+(m0M m1)m0m2 m2112?,=2,1,2.(2.15)3l3 V(x)e,?|x|+,?.?)P,?O?m?k.,g Dirichlet.T?/./e?l,n/aq?.k.(a,b)(c,d)?1?,-hx=(b a)/J 0 hy=(d c)/K 0,p?J,K?.P?:xj:=a+jhx,j=0,1,J,yk:=b+khy,k=0,1,K.?nj,k=(n,2j,k,n,1j,k,n,0j,k,n,1j,k,n,2j,k)(xj,yk,tn)?Cq,n t=tn?),nj,k.X-f n,.KFf?F6(2.5)(2.8)?l,j,k n1,j,kt=122h,j,k n1,j,k+n1n1,j,k+Gn1,j,k,(3.1)n,j,k=n,j,k,=2,2.(3.2)0,j,k=(xj,yk,t=0),Gn1,2=?n1 2n1 V(x)0n1+21Fn1z?n12 1Fn1+n11No.5?:Og;?Spin-2 BEC?.KFf?5F6161325An1002n1+(iyn11+xn11),Gn1,1=n1 n1 V(x)0n1+1Fn1zn11 1(62Fn1+n10+Fn1n12)+25An1001n1+iy(n12+62n10)+x(62n10 n12),Gn1,0=n1 V(x)0n1n10621(Fn1+n11+Fn1n11)25An1000n1+62iy(n11+n11)x(n11 n11),Gn1,1=n1+n1 V(x)0n1 1Fn1zn11 1(62Fn1n10+Fn1+n12)+25An1001n1+iy(n12+62n10)x(62n10 n12),Gn1,2=n1+2n1 V(x)0n1 21Fn1zn12 1Fn1n1125An1002n1+(iyn11 xn11).-f n?n=12(bnmax+bnmin),(3.3)p bnmax9 bnminLbnmax=max1jJ1max1kK1(Vjk+0njk),bnmin=min1jJ1min1kK1(Vjk+0njk).L n1,?1lFp“C,?n1,p,q=1JKJ1Xj=0K1Xk=0n1,j,keip(xja)eiq(ykc),(3.4)p p=2pba,q=2qdc,p=J/2,J/2+1,J/21,q=K/2,K/2+1,K/21.?,?f 2?Cq 2h.Dhxn1,j,k=iJ/21Xp=J/2K/21Xq=K/2pn1,p,qeip(xja)eiq(ykc),Dhyn1,j,k=iJ/21Xp=J/2K/21Xq=K/2qn1,p,qeip(xja)eiq(ykc),2hn1,j,k=J/21Xp=J/2K/21Xq=K/2(2p+2q)n1,p,qeip(xja)eiq(ykc).(3.5)L(3.1)?Fp“C?d,pqn1,pqt=2p2+2q2+n1!d,pq+n1n1,pq+Gn1,pq.(3.6)1614n?Vol.43 Az?d,pq=11+t(2p2+2q2+n1)?n1,pq+tn1n1,pq+tGn1,pq?.(3.7)L(3.7)_Fp“C=?(3.1)?).4?$GFLM?g;?Spin-2 BEC 3e?.NX(1 0)cNX(1 0),?0(x)=0g(x),=2,1,0,1,2,g(x)|BEC?.it(x,t)=?122+V(x)+0|(x,t)|2?(x,t)?Cq ho(x)=1e(x2+y2)/2.AO,?O D=(10,10)2,?J=K=128,V(x)=0.5(x2+y2).1(15)?0=100,1=1,2=2,|M=0,g;?r =0.A GFLM Te?).1 1?)?2 1 U(E),o(N)o|(M)S“g?Cz 1 GFLM?1?)?(J.*?:(1)?)g=g(=1,2),=y,n(J?,dg;?Spin-2 BEC v?CGPEs(1.1)(1.3),?o|M=0,?)?p?CGPEs?C,d?)v g=g(=1,2);(2)g;?r =0,?)?/G.2 LOL,U!oo|S“g?Cz.*?:(1)U?,?-G?;(2)o N 1,o?;(3)o|M 0,o|?.,?9?(JL:A GFLM?g;?Spin-2 BEC?1?.No.5?:Og;?Spin-2 BEC?.KFf?5F61615 2(.NXe(J)?0=100,1=1,2=2,M=0,gOg;?r =1.5,=2,=2.3,=2.4,=3,=4?).3 2 =1.5?)?4 2 =2?)?5 2 =2.3?)?6 2 =2.4?)?7 2 =3?)?8 2 =4?)?1616n?Vol.43 A 3 8 2?k.NXeg;?Spin-2 BEC?.*?:(1)?)g g(=1,2)”;(2)?|M=0,Xg;?r?O,?)?YdYCY2CY;(3)?)yY(Y),?Og;?r?,()O.3(.NXe(J)?0=100,1=1,2=2,gO M=1,=1.5 M=1.2,=2|e?).9 3 M=1,=1.5?)?10 3 M=1.2,=2?)?9 10 3?g;?Spin-2 BEC?).*?:(1)M 6=0,?)y?YYu)-;(2)?)g g(=1,2)”.d?:GFLM k?.NXeg;?Spin-2 BEC?);|M=0,Xg;?r?O,?)YY.4(cNXe(J)?0=100,1=1,2=25,M=0 gOg;?r =1.5,=2,=3.2,=3.3,=3.5,=4?).11 4 =1.5?)?12 4 =2?)?No.5?:Og;?Spin-2 BEC?.KFf?5F61617 13 4 =3.2?)?14 4 =3.3?)?15 4 =3.5?)?16 4 =4?)?11 16 4?g;?Spin-2 BEC?).*?:(1)?)g g(=1,2)”;(2)?|M=0,Xg;?r?O,?yYY;(3)?)yY(Y),?Og;?r?,()O.AO/,3dYY=C?.G?ey?Y.5(cNXe(J)?0=100,1=1,2=25,gO M=0.6,=1.5 M=0.5,=2|e?).17 5 M=0.6,=1.5?)?1618n?Vol.43 A 18 5 M=0.5,=2?)?17 18 5?g;?r?g;?Spin-2 BEC?).*?:(1)M 6=0,?)?Yu)-;(2)?)g g(=1,2)”.d?:GFLM k?cNXeg;?Spin-2 BEC?);|M=0,Xg;?r?O,?)y,?Y.z1 Anderson M H,Ensher J R,Matthewa M R,et al.Observation of Bose-Einstein condensation in a diluteatomic 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gases,zweite abhandlung.Sitzungsberichte derPreussischen Akademie der Wissenschaften,1925,1:31415 Ho T L.Spinor Bose condensates in optical traps.Physical Review Letters,1998,81:74274516 Kawaguchi Y,Ueda M.Spinor Bose-Einstein condensates.Physics Reports,2012,520:25338117 Liu W,Cai Y.Normalized gradient flow with Lagrange multiplier for computing ground states of Bose-Einstein condensates.SIAM Journal on Scientific Computing,2021,43(1):B219B24218 Lin Y J,Compton R L,Jimenez-Garcia K,et al.Synthetic magmetic fields for ultracold neutral stoms.Nature,2009,462:62863219 Lin Y J,Compton R L,Perry A R,et al.Bose-Einstein condensates in a uniform light-induced vectorpotential.Physical Review Letters,2009,102:130401No.5?:Og;?Spin-2 BEC?.KFf?5F6161920 Lin Y J,Jimenez-Garcia K,Spielman I B.A spin-orbit-coupled Bose-Einstein condensates.Nature,2011,471:838621 Stamper-Kurn D M,Andrews M R,Chikkatur A P,et al.Optical confinement of a Bose-Einstein condensate.Physical Review Letters,1998,80:2027203022 Stamper-Kurn D M,Ueda M.Spinor Bose gases:Symmetries,magnetism,and quantum dynamics.Reviewof Modern Physics,2013,85:1191124423 Yuan Y,Xu Z,Tang Q,Wang H.The numerical study of the ground states of spin-1 Bose-Einstein con-densates with spin-orbit-coupling.East Asian Journal on Applied Mathematics,2018,8(3):598610A Normalized Gradient Flow with Lagrange Multipliers forComputing Ground States of Spin-Orbit Coupled Spin-2Bose-Einstein CondensatesYuan Yongjun(LCSM(MOE)&School of Mathematics and Statistics,Hunan Normal University,Changsha 410081)Abstract:In this paper,a normalized gradient flow with Lagrange multipliers is designed to computeground states of spin-orbit coupled Spin-2 Bose-Einstein condensates.By excavating the implicitrelation between projection coefficients,the difficult that the existed conditions(the conservation oftotal mass and magnetization)of the model problem is insufficient to determine all the projectioncoefficients,is overcome.Extensive numerical experiments are done to compute the ground states ofspin-orbit coupled Spin-2 BECs with cyclic/ferromagenetic interactions.As a result,the effectivenessof the two algorithms is verified,and the phase transformation law about how the stripe pattern groundstates and the square-lattice pattern ground states of spin-orbit coupled Spin-2 BECs change to eachother with the spin-orbit coupling parameter,are revealed.Key words:Bose-Einstein condensate;Ground state;Normalized gradient flow;Lagrange multiplier.MR(2010)Subject Classification:65N06;65N12;65Z05