Manipulation of Single Electron Spin in a GaAs Quantum Dot through the Application of Geometric Phases: The Feynman Disentangling Technique

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1 Wilfrid Laurier Universiy Scholars Laurier Mahemaics Faculy Publicaions Mahemaics 2010 Manipulaion of Single Elecron Spin in a GaAs Quanum Do hrough he Applicaion of Geomeric Phases: The Feynman Disenangling Technique Sanjay Prabhakar Universiy of Albany, Sae Universiy of New York, sprbhakar@wlu.ca James Reynolds Universiy of Albany, Sae Universiy of New York Akira Inomaa Universiy of Albany, Sae Universiy of New York Roderick V.N. Melnik Wilfrid Laurier Universiy, rmelnik@wlu.ca Follow his addiional works a: hp://scholars.wlu.ca/mah_faculy Recommended Ciaion Prabhakar, Sanjay; Reynolds, James; Inomaa, Akira; Melnik, Roderick V.N., "Manipulaion of Single Elecron Spin in a GaAs Quanum Do hrough he Applicaion of Geomeric Phases: The Feynman Disenangling Technique" (2010). Mahemaics Faculy Publicaions. 33. hp://scholars.wlu.ca/mah_faculy/33 This Aricle is brough o you for free open access by he Mahemaics a Scholars Laurier. I has been acceped for inclusion in Mahemaics Faculy Publicaions by an auhorized adminisraor of Scholars Laurier. For more informaion, please conac scholarscommons@wlu.ca.

2 Manipulaion of single elecron spin in a GaAs quanum do hrough he applicaion of geomeric phases: The Feynman disenangling echnique Sanjay Prabhakar, 1,2 James Raynolds, 1 Akira Inomaa, 3 Roderick Melnik 2,4 1 College of Nanoscale Science Engineering, Universiy a Albany, Sae Universiy of New York, Albany, New York 12203, USA 2 M2NeT Laboraory, Wilfrid Laurier Universiy, Waerloo, Onario, Canada N2L 3C5 3 Deparmen of Physics, Universiy a Albany, Sae Universiy of New York, Albany, New York 12222, USA 4 BCAM, Bizkaia Technology Park, Derio, Spain Received 28 May 2010; revised manuscrip received 3 Ocober 2010; published 4 November 2010 The spin of a single elecron in an elecrically defined quanum do in a wo-dimensional elecron gas can be manipulaed by moving he quanum do adiabaically in a closed loop in he wo-dimensional plane under he influence of applied gae poenials. In his paper we presen analyical expressions numerical simulaions for he spin-flip probabiliies during he adiabaic evoluion in he presence of he Rashba Dresselhaus linear spin-orbi ineracions. We use he Feynman disenanglemen echnique o deermine he non-abelian Berry phase we find exac analyical expressions for hree special cases: i he pure Rashba spin-orbi coupling, ii he pure Dresselhause linear spin-orbi coupling, iii he mixure of he Rashba Dresselhaus spin-orbi couplings wih equal srengh. For a mixure of he Rashba Dresselhaus spin-orbi couplings wih unequal srenghs, we obain simulaion resuls by solving numerically he Riccai equaion originaing from he disenangling procedure. We find ha he spin-flip probabiliy in he presence of he mixed spin-orbi couplings is generally larger han hose for he pure Rashba case for he pure Dresselhaus case, ha he complee spin-flip akes place only when he Rashba Dresselhaus spin-orbi couplings are mixed symmerically. DOI: /PhysRevB PACS number s : w I. INTRODUCTION Geomeric phases abound in physics heir sudy has araced considerable aenion since he seminal work of Berry. 1,2 In recen years a number of researchers have shown heir ineres in he geomeric phases associaed wih single few-spin sysems for poenial applicaions in he field of quanum compuing noncharge based logic. 3 5 One ineresing proposal is he noion ha he spin of a single elecron rapped in an elecrosaically defined wo-dimensional 2D quanum do can be manipulaed hrough he applicaion of gae poenials by moving he cener of mass of a quanum do adiabaically in a closed loop inducing a non-abelian marix Berry phase. 6 A recen work shows ha he Berry phases can be changed dramaically by he applicaions of gae poenials may be deeced in an inerference experimen. 7 In he presen paper, we sudy he non-abelian uniary operaor of he spin saes during he adiabaic moion of a single-elecron spin. The non-abelian naure here sems from he spin-orbi SO coupling of an elecron in wo dimensions. The evoluion operaor which gives rise o he Berry phase is no easy o evaluae as i conains noncommuing operaors. In 1951, Feynman 8 developed an operaor calculus for quanum elecrodynamics, in which he devised a way o disenangle he evoluion operaor involving noncommuing operaors. In 1958, Popov 9 applied he operaor calculus, combined wih group-heoreical consideraions, o he spin roaion for a paricle wih a magneic momen in an exernal magneic field o obain exac ransiion probabiliies beween he iniial final spin saes. In a way similar o Popov s we employ he Feynman echnique o disenangle he evoluion operaor for a quanum do wih he Rashba Dresselhaus spin-orbi couplings derive analyical expressions for spin ransiion probabiliies. In paricular, we obain exac closed form expressions for hree specific cases: i he pure Rashba spin-orbi coupling, 10 ii he pure linear Dresselhaus spin-orbi coupling, 11 iii he symmeric combinaion of he Rashba Dresselhaus spin-orbi couplings. This approach provides us a convenien numerical scheme for an arbirary mixing of he wo ypes of spin-orbi couplings via a Riccai equaion. 12 An ineresing resul we find is ha he spin-flip probabiliy for he case of an arbirary mixure of he Rashba he Dresselhaus spin-orbi couplings is generally greaer han ha for he case eiher he Rashba or he Dresselhaus ineracion acs alone. Furhermore, we see ha he complee spin precession occurs only when he Rashba spin-orbi coupling he Dresselhaus spin-orbi coupling are equal in srengh. The work of Berry eaches ha if parameers conained in he Hamilonian of a quanal sysem are adiabaically carried around a closed loop an exra geomeric phase Berry phase is induced in addiion o he familiar dynamical phase. 1,2 A slow variaion in such parameers along a closed pah C will reurn he sysem o is original energy eigensae wih an addiional phase facor exp i n C. More specifically, he sae acquires phases afer a period of he cycle T as n T = exp i 0 T E n d exp i n C n. However his equaion applies only o nondegenerae saes. The deailed numerical analyical calculaions of Berry phase n C for he Hamilonian of a quanum do in 2D plane for differen nondegenerae eigensaes are explained in Ref. 13. The sysem of ineres here a single spin in a 2D /2010/82 19 / The American Physical Sociey

3 PRABHAKAR e al. elecrically defined quanum do is degenerae 14,15 for which Eq. 1 is no direcly applicable. In he formulaion developed by Wilczek ohers 2,16 for degenerae cases, he geomeric phase facor is replaced by a non-abelian uniary operaor U ab acing on he iniial saes wihin he subspace of degeneracy. The evoluion equaion of he sae is modified in he form, n,a = exp i 0 E d U ab n,b, b a b are he labels for degeneracy. The non- Abelian uniary operaor can be expressed in he form, U ab = T exp i A ab Ṙd, 0 T signifies he ime ordering, A ab = i n,a R n,b, R R being a vecor he gradien in parameer space, respecively. In general, he geomeric phase ransformaion U ab of Eq. 3 in parameer space conains noncommuing operaors ime-dependen parameers. I is possible o view he parameer-dependen evoluion in he subspace of degeneracy as a non-abelian local gauge ransformaion. Correspondingly A ab in Eq. 4 may be seen as a non- Abelian gauge connecion or he Yang-Mills fields. Alhough i is no sraighforward o consruc he non- Abelian gauge connecion, we consider he following observaion insrucive for he case he parameer space coincides wih he configuraion space. Suppose he Hamilonian of a sysem is given by H = 1 2m P A 2 + V r. The energy eigenequaion H n =E n n remains invarian under he local posiion-dependen gauge ransformaion, n n = Ū n, A A = ŪAŪ + i Ū Ū. If we choose such a gauge ha he ransformed vecor poenial vanishes, ha is, A =0, hen he ransformaion operaor is o be of he form, Ū = exp i c A dr. 7 In oher words, his ransformaion will gauge away he vecor poenial from Hamilonian 5. Conversely, if he sae wih he vanishing gauge is aken o be he iniial sae, he final sae wih an arbirary gauge A is obained by he inverse gauge ransformaion, n =Ū 1 n. Moreover, if he inverse gauge process is ime-dependen via he variaion in posiion, hen he evoluion operaor is given by U = Ū 1 = T exp i A ṙd. 0 This observaion will be useful for our discussion on he Berry phase associaed wih he spin-orbi coupling. A marix elemen of he evoluion operaor gives he ransiion ampliude propagaor from an iniial sae o he final sae, which is usually evaluaed by approximaion. For insance, he propagaor for he spin-orbi ineracion has been calculaed semiclassically in a differen conex by Feynman s pah inegral represened in coheren saes. 17 In Sec. II, we rea he phase ransformaion Eq. 3 as a gauge ransformaion, employ Feynman s disenangling echnique, raher han Feynman s pah inegral, o evaluae he ime-ordered exponenial for he spin-orbi coupling Hamilonian. Use of Feynman s disenangling mehod in Popov s version 18 enables us o obain analyical numerical resuls for he spin ransiion probabiliies wihou approximaion. In Sec. III, we plo he spin-flip probabiliy versus he roaion angle, compare he daa for he pure Rashba, he pure Dresselhaus, mixed cases. Secion IV is devoed in deriving analyical expressions of he non- Abelian Berry phase he adiabaic evoluion operaor as a 2 2 marix for he pure Rashba he pure Dresselhaus coupling. II. SPIN TRANSITION PROBABILITIES VIA FEYNMAN DISENTANGLING METHOD To discuss he revoluion of spin ha induces a geomeric phase, we consider a GaAs quanum do formed in he plane of a 2D elecron gas 2DEG, he cener of mass of which moves adiabaically along a closed pah under he influence of applied poenials. 6 The single-elecron Hamilonian in 2DEG in he xy plane may be wrien in he form, H = 1 2m P2 + H SO, m is he effecive mass. The firs erm is he kineic energy in wo dimensions. Evidenly, P 2 = P x 2 + P y 2. The second erm is he SO coupling Hamilonian in linear approximaion, H SO =2 P y S x P x S y 2 P x S x P y S y Here S is he spin operaor whose componens obey he SU 2 algebra see, e.g., Ref. 19, S +,S =2S 0, S 0,S = S, 11 S =S x is y S 0 =S z. The spin-orbi Hamilonian 10 consiss of he Rashba coupling whose srengh is characerized by parameer he linear Dresselhaus coupling wih. These coupling parameers are dependen on he elecric field E of he quanum well confining poenial i.e., E= V/ z along z direcion a he inerface in a heerojuncion as

4 MANIPULATION OF SINGLE ELECTRON SPIN IN A = e a RE, = c 2me 2/3 2 E 2/3, 12 a R =4.4 Å 2 c =26 ev Å 3 for he GaAs quanum do. 14 The quanum well confining poenial i.e., E= V/ z along z direcion is no symmeric in III-V ype semiconducor. 15 I means, he formaion of quanum do a he inerface of III-V ype semiconducor in he plane of 2DEG is asymmeric. Now we look for he evoluion operaor 8 for he case of spin-orbi coupling. I has been known ha he linear spinorbi erm in Eq. 9 can be gauged away. 20,21 In fac, Hamilonian 9 may be expressed as H = 1 2m P A 2 V 0, A =2m S y + S x S x S y V 0 = m If he semiclassical momenum P = mṙ is used for he adiabaic evoluion, hen he spin-orbi gauge connecion is relaed o he SO Hamilonian 10, A ṙ = H SO. 16 Assuming ha he spin-orbi coupling is adiabaically inroduced ino he iniial sae, we obain via Eq. 8 he evoluion operaor of he form, U = T exp i H SO d, 0 17 which we shall evaluae by uilizing he Feynman disenangling mehod. This form of he evoluion operaor is commonly employed for Berry s phase associaed wih he spinorbi ineracion. 6,15 Before disenangling, we noe ha he SO Hamilonian 10 may also be expressed as wih H SO = H + S + + H S H = P y P x i P y P x Suppose he quanum do orbis around a closed circular pah of radius R 0 in he x-y plane under he influence of gae poenials, so ha r=r 0 cos,sin,0. Then he semiclassical momenum P = mṙ has componens, P x = R 0 m sin, P y = R 0 m cos, P z =0. 20 Subsiuion of Eq. 20 ino Eq. 19 yields H = R 0 m e i i e. 21 Since S + S do no commue, he evaluaion of he imeordered exponenial for he evoluion operaor 17 is cumbersome. We now urn o a discussion of he Feynman disenangling echnique is applicaion o he presen problem. For he case he Hamilonian is given by H = A + B + C +, 22 A, B, C,... are noncommuing operaors,,,,... are ime-dependen parameers, Feynman 8 devised an operaor calculus by which he ime-ordered exponenial can be disenangled in he form U = e a A e b B e c C, 23 a,b,c,... are ime-dependen coefficiens which can be deermined by solving relevan differenial equaions. This procedure is referred o as he Feynman disenangling mehod. 18 Here we apply Feynman s mehod for disenangling he ime-ordered exponenial in Eq. 17 wih Hamilonian 10. Firs we rewrie Hamilonian 10 as H SO = S + + H + S + + H S, 24 is a ime-dependen funcion o be deermined appropriaely. According o Feynman s procedure, he evoluion operaor may be pu ino he form, U = e a S + exp 1 d H + S + + H S, 25 a = 1 d, S + = e as +S + e as + = S +, S = e as +S + e as + = S 2aS 0 a 2 S Subsiuing Eqs ino Eq. 25 choosing such ha he coefficien of S + in he inegr vanishes, we ge U = e a S +T exp 1 d 2aH S 0 + H S, 29 in which he erm conaining S + is disenangled. In a similar fashion, we disenangle he ime-ordered exponenial involving he muually noncommuing operaors S 0 S by leing U = e a S +e b S 0T exp 1 d 2aH S 0 + H S, 30 b = 1 d,

5 PRABHAKAR e al. FIG. 1. Color online Transiion probabiliy, w 1/2, 1/2 vs for hree cases: a pure Rashba =0, b pure Dresselhaus =0, c mixed nonzero spin-orbi ineracions. The orbial radius is 60 nm. The hree curves represen he following elecric field srenghs: V/cm solid line, V/cm dashed line, V/cm doed-dashed line, respecively. S 0 = e bs 0S 0 e bs 0 = S 0, S = e bs 0S e bs 0 = S e b Again choosing = 2aH, we reduce he evoluion operaor 25 ino he compleely disenangled form, U = e a S +e b S 0e c S, a = 1 H + a 2 H d, b = 2 a H d, c = 1 H e b d Alhough he ime-ordered exponenial is disenangled, he evaluaion of he evoluion operaor remains incomplee unil he coefficiens a, b, c are deermined. In general, he inegral in Eqs or he equivalen differenial equaions are difficul o solve. In Sec. IV, we shall deermine he coefficiens he evoluion operaor for he pure Rashba, he pure Dresselhaus coupling. As i is seen in Appendix A, he spin ransiion probabiliy depends only on a. Therefore he full form of he evoluion operaor is no needed. To deermine a, we conver he inegral Eq. 35 ogeher wih Eq. 19 ino a Riccai equaion of he form, R=mR 0 /, da d = R f + f a 2, 38 f = i + i e i f = i i e i Solving Eq. 38 for a, we can obain he spin ransiion probabiliies, w s,s. In paricular, he ransiion probabiliies from spin 1/2 o 1/2 are calculaed by w 1/2,1/2 = 1 1+ a 2, w 1/2, 1/2 = a 2 1+ a 2. III. NUMERICAL ANALYSIS 41 As i is shown in Appendix B, exac soluions of he Riccai Eq. 38 can be obained only for special cases, which include hose for i he Rashba limi =0, ii he Dresselhaus limi =0, iii he symmeric mixure of he wo couplings =. The spin-flip probabiliies obained in Appendix B for exacly solvable cases wih = are: i The Rashba limi 0, =0 : 2sin2 R w 1/2, 1/2 = 4R R 2 1+4R ii The Dresselhaus limi =0, 0 : 2sin2 D w 1/2, 1/2 = 4R R 2 1+4R iii The symmeric Rashba-Dresselhaus R-D limi = 0 : sym w 1/2, 1/2 = sin 2 2 R sin cos For an arbirarily mixed R-D coupling, he Riccai Eq. 38 is no exacly solvable. Therefore numerical analysis is needed. In he below we rea he mixed R-D coupling he symmeric R=D coupling = separaely. Comparison of he Rashba coupling, he Dresselhaus coupling, he mixed R-D coupling.figures 1 3 plo he spin

6 MANIPULATION OF SINGLE ELECTRON SPIN IN A FIG. 2. Color online Transiion probabiliy w 1/2, 1/2 vs for he following cases: a pure Rashba =0, b pure Dresselhaus =0, c mixed nonzero. The orbi radius is chosen o be 250 nm he following values of he elecric field are considered: V/cm solid line, V/cm dashed line, V/cm doed-dashed line. flip probabiliy w 1/2, 1/2 versus he roaion angle = in he uni of 2 for he orbi radius R 0 =60 nm, 250 nm, 500 nm, respecively. The plos of a, b, c in hese figures correspond o a he pure Rashba case =0, b he pure Dresselhaus case =0, c he mixed R-D case 0, 0, respecively. The hree differen values of he elecric field E=1 10 5, , V/cm, are chosen for he curves in each figure, solid line, dashed line, doed-dashed line, respecively. The symmeric case R=D will be examined separaely wih Figs The curves for a he pure Rashba case b he pure Dresselhaus case are obained from he exac resuls As i is obvious from hese equaions, he spin-flip probabiliy increases as he elecric field increases via he coupling parameer bu remains o be less han uniy. Anoher observaion we can make from hese plos is ha he periods of spin flip for he pure Rashba coupling he pure Dresselhaus coupling are differen. This is also expeced from he analyical resuls The curves in Figs. 1 c, 2 c, 3 c show he spin-flip probabiliy for c he mixed R-D case boh are no zero no equal. Noe ha hey are no he resuls from he exac formula 44 for he symmeric R-D coupling. Since he Riccai Eq. 38 for arbirary nonzero is no solvable, we carry ou numerical simulaions by using numerical soluions of Eq. 38 in Eq. 41. The spin-flip probabiliy for he mixed case is generally larger han he pure cases. Furhermore, i does no reach uniy if. In oher words, he complee spin-flip is no likely o occur during he enire period of he adiabaic moion along he closed orbi. In he viciniy of he symmery poin =, he ransiion probabiliy becomes very close o uniy a cerain angles. Figure 4 gives a furher comparison sudy of he ransiion probabiliy for he pure Rashba, he pure Dresselhaus, he mixed case. In Fig. 4 a, when he elecric field is weak, he curve for he mixed case appears o be a superposiion of hose for he wo pure cases. As he elecric field increases, he superposiion effec becomes obscure as is seen in Fig. 4 b. As he Riccai equaion is nonlinear in naure, here is no reason o expec ha he mixed case is a superposiion of he wo pure cases. I is ineresing o observe ha he mixed case has a beer chance o achieve he spin flip han he pure cases during he period of evoluion. Analysis of he symmeric R-D coupling. The symmeric mixure of he Rashba Dresselhaus couplings has been FIG. 3. Color online Transiion probabiliy w 1/2, 1/2 vs for he following cases: a pure Rashba =0, b pure Dresselhaus =0, c mixed nonzero. The orbi radius was chosen o be 500 nm he following values of he elecric field were chosen: V/cm solid line, V/cm dashed line, V/cm doed-dashed line

7 PRABHAKAR e al. FIG. 4. Color online Transiion probabiliy w 1/2, 1/2 vs for he following cases: pure Rashba =0: doed-dashed line, pure Dresselhaus =0: dashed line, mixed nonzero : solid line. The orbi radius was chosen o be 250 nm he following values of he elecric field were chosen: a E= V/cm b E= V/cm. discussed in connecion wih he persisen spin helix. 22,23 Bernevig e al. 22 found an exac SU 2 symmery in he symmeric mixure prediced he persisen spin helix which is a helical spin densiy wave wih conserved ampliude phase. Recenly spin lifeime enhancemen of wo orders of magniude near he symmery poin = has been repored experimenally. 24 The coupling parameers of he Rashba Dresselhaus ineracions are given by Eq. 12 for he GaAs quanum do. The wo parameers become equal a E= V/cm. For he siuaion in which he wo couplings have equal srengh i.e., =, he Riccai Eq. 38 is exacly solved he corresponding ransiion probabiliy is given by Eq. 44. In Fig. 5, he spin-flip probabiliy versus he angle of roaion along he orbi of radius 60 nm is ploed a E= V/cm for he pure Rashba case open circles, he pure Dresselhaus case dashed line, he symmeric case solid line. We see ha he symmeric Rashba-Dresselhaus spin-orbi coupling definiely achieves a spin flip during he adiabaic process as he wo pure cases have less chances. Figure 6 plos he ransiion probabiliy of he symmeric R-D case for hree differen radii of he orbi of he quanum do: 60 nm solid line, 175 nm dashed line, 250 nm doed-dashed line. I shows ha he chance of being in he spin-flip sae is enhanced by increasing he orbi radius. I is imporan o noice ha he complee spin flip akes place only in he symmeric R-D coupling. This may be an indicaion of he persisen spin helix. Alhough he assumed orbi of moion is circular, we can regard he moion for a small angle of roaion as linear. Le = 0 or =3 /2. If is small, hen sin cos +1, he exac formula 44 may be approximaed by sym w 1/2, 1/2 = sin 2 2 R. 45 As varies from 0 o / 2 2 R, he spin-flip probabiliy moves from zero o uniy, ha is, he spin complees a full precession. For insance, if R=60 nm by leing m=q=1, he range 0 2 R /2 corresponds o he porion of he solid curve for 0 /2 0.2 in Fig. 5. Le s = / 2 2 R. Then he R s is he disance he elecron progresses while he spin precesses by 2. Therefore, we may be able o idenify his disance wih he spin diffusion lengh L s as L s = R 0 / = FIG. 5. Color online Transiion probabiliy w 1/2, 1/2 vs for =. Physically, his siuaion occurs for elecric field srengh given by E= V/cm. Here he solid line represens for boh Rashba Dresselhaus spin-orbi coupling effecs as he dashed line represens only for Dresselhaus spin-orbi coupling effec open circles represens only for Rashba spin-orbi coupling effec. Here we choose 60 nm orbi radius. FIG. 6. Color online Transiion probabiliy w 1/2, 1/2 vs for =. Physically, his siuaion occurs for elecric field srengh given by E= V/cm. The following orbi radii were chosen: 60 nm solid line, 175 nm dashed line,, 250 nm doeddashed line

8 MANIPULATION OF SINGLE ELECTRON SPIN IN A IV. ANALYTICAL EXPRESSION FOR THE NON-ABELIAN BERRY PHASE Applying he Feynman disenangling mehod, we have been able o reduce he ime-ordered evoluion operaor 17 o he disenangled form 34 wih he ime-dependen scalar funcions a, b, c obeying he inegral Eqs I is someimes convenien o express he evoluion operaor as a 2 2 marix in he spin represenaion of SU 2. Evidenly he SU 2 algebra Eq. 11 is saisfied by S + = , S 0 = , S = Using he properies S 2 =0 S 0 2 =1/4, we can wrie Eq. 34 as U = 1 a 0 1 eb/2 0 0 e b/2 1 0 c 1, 48 from which immediaely follows ha U = eb/2 + ace b/2 ce b/2 ae b/2 e b/2. 49 This is he desired marix represenaion for he Berry phase, is used for calculaing he spin-flip probabiliies in Appendix A. The expressions remain formal unil he ime-dependen funcions a, b, c are specified. Equaion 35 for a is equivalen o a Riccai equaion wihou whose soluion, Eqs canno be solved for b c. In Appendix B, we show ha he Riccai equaion can be solved exacly if he funcion h defined by 2 2 h = 2R / / sin 2 50 becomes ime-independen h =h 0. The las resricion Eq. 50 is fulfilled only when one of he following condiions is me: =0, =0, or =. This implies ha he funcion a can be deermined only for he pure Dresselhaus coupling, he pure Rashba coupling, he symmeric Rashba- Dresselhaus coupling. The resul we find for a is Here a = if e i 1 f n 1 e i, n 2 f = e i + i e i, = R n 1 n 2 f d. 0 n 1,n 2 = h 0 h For convenience, we choose n 1 n 2. A closed form expression for is given in Eq. B12. In calculaing he spin ransiion probabiliy, all we need is a. However, for compleing he evoluion operaor we have also o deermine oher funcions b c by solving Eqs for he already deermined funcion a. As has been menioned above, he Riccai equaion can be solved exacly for he pure Rashba coupling, he pure Dresselhaus coupling, he symmeric Rashba- Dresselhaus coupling. In he wo pure couplings, he phase funcion can be expressed in he form, =, 55 = R 2 for he Rashba coupling = R 2 for he Dresselhaus coupling. For he symmeric R-D coupling, i canno be simplified in he form of Eq. 55. Therefore, i is difficul o carry ou inegraion in Eqs This means ha we have analyical expressions of he adiabaic evoluion operaor 49 only for he pure Rashba he pure Dresselhaus cases. For he symmeric R-D coupling, even hough we have no analyical expressions for b c, we can calculae he spin-flip probabiliy since a is found in closed form. In wha follows we provide he resuls of inegraion for he pure Rashba coupling he pure Dresselhaus coupling. i The pure Rashba coupling 0, =0. In his case, Eqs. B2, B4, B5, B11 yield, if/ f = e i, h 0 = 1 2 R =, = R 2. Upon subsiuion of hese resuls ino Eq. B13 we arrive a From Eq. 36, using a = e i e i 1 n 1 e, n 2 n 1,n 2 = 1 2 R 1 2 R R 2. H = R e i, ogeher wih Eq. 56, we obain e b = n 1 n 2 2 e i n 1 e i n 2 2 c = 1 ei n 1 e i n ii The pure Dresselhaus coupling =0, 0. In his case, we have if/ f = ie i, h 0 = 1 2 R Hence we ge =, = R

9 PRABHAKAR e al. Use of Eq. 59 leads o a = ie i e i 1 n 1 e, n 2 n 1,n 2 = 1 2 R 1 2 R R 2. H = i R e i, e b = n 1 n 2 2 e i + n 1 e i n e i c = i n 1 e i. n Here,, are he ime-dependen Eulerian angles, s D,,, are he elemens of he Wigner D marix being he irreducible uniary represenaions of SU 2 group, d is Wigner s d funcion. The corresponding ransiion probabiliy along he z axis is s w = d 2. A2 In paricular, he ransiion probabiliy from spin 1/2 o 1/2 is w 1/2,1/2 = cos2 A3 2 because w 1/2, 1/2 = sin2 A4 2 V. CONCLUSION 1/2 d 1/2,1/2 = cos 2, d 1/2 1/2, 1/2 = i sin 2. A5 In he presen paper we have considered spin manipulaion via he non-abelian Berry phase induced by an adiabaic ranspor of a single spin along a circular pah in he 2D plane in he presence of he Rashba Dresselhaus spinorbi couplings. We have adoped he Feynman disenangling echnique o calculae he spin-flip probabiliy. We have shown ha he problem can be solved exacly in hree cases: i he pure Rashba coupling, ii he pure Dresselhaus coupling, iii he symmeric combinaion of Rashba Dresselhaus couplings. For an arbirary combinaion of he wo couplings, we have carried ou numerical simulaions. We have ploed he spin-flip probabiliy versus he angle of he adiabaic roaion wih various values of he elecric field he radius of he circular pah in he 2D plane. We have observed ha a complee spin flip a complee spin precession occurs only when he srengh of he wo couplings becomes equal. The relaion beween he complee spin precession he persisen spin helix will be discussed in deail else. We have also obained analyical expressions of he non-abelian Berry phase for he pure Rashba case he pure Dresselhaus case. ACKNOWLEDGMENTS This work was suppored by he NRI INDEX cener, USA, NSERC, CRC program, Canada. APPENDIX A: THE SPIN TRANSITION PROBABILITIES Following Popov s procedure, 9,18 we show ha he spinflip probabiliy can be expressed in he form of Eq. 41. Since he ime evoluion of he spin sae can be achieved by a ime-dependen roaion, he ransiion ampliude for spin o is given by s U = D,,, = exp i + d. A1 For spin s=1/2, he roaion marix is given in he sard form, 18,19 D,, =, A6 = cos 2 exp + i, = i sin 2 2 exp i. 2 A7 Comparison of he evoluion operaor for he spin-1/2 ransiion expressed in he marix form, U = eb/2 + ace b/2 ae b/2 ce b/2 e b/2, A8 he roaion marix yields a 2 = an 2 2. A9 Again comparing his resul wih Eqs. A3 A4, we arrive a w 1/2,1/2 = Noe ha w 1/2,1/2 +w 1/2, 1/2 = a 2, w 1/2, 1/2 = a 2 1+ a 2. A10 APPENDIX B: SPECIAL SOLUTIONS OF THE RICCATI EQUATION Here we wish o solve under a special condiion he Riccai Eq. 38, da = R f + f a 2, d B

10 MANIPULATION OF SINGLE ELECTRON SPIN IN A f = e i + i e i, f = e i i e i. B2 This equaion conains he Rashba limi 0, =0, he Dresselhaus limi =0, 0, boh of which have exac soluions. Firs we le a =g X in Eq. B1. If we furher le g = if/ f, hen we see ha X obeys h = dx d = ir f X2 2h X 1, f = sin 2 1/2 B3 B R / / sin 2. B5 Now we consider a special case h is a consan, say, h 0. In his case, Eq. B3 can be expressed as n 1 n 2 are roos of ha is, Noe ha dx X n 1 X n 2 = ir f d, X 2 2h 0 X 1=0, n 1,n 2 = h 0 h B6 B7 B8 n 1 n 2 = 1, n 1 + n 2 =2h 0, n 1 n 2 =2 h B9 Upon inegraion, we obain wih he condiion X 0 =0, Riccai Eq. B1 is exacly solved, he resul being of he form, a = if e i 1 f n 1 e i. n 2 B13 Since 0 =0, i is eviden ha a 0 =0. Using Eq. B8 in Eq. B13, we obain a 2 = sin 2 /2 h sin 2 /2. B14 The ransiion probabiliies from spin 1/2 o 1/2 are given by 1 w 1/2,1/2 =1 h 2 0 sin2 B w 1/2, 1/2 = h 2 0 sin2, B which are characerized only by he consan h 0 he phase funcion. Alhough he above resuls are exac under he assumpion ha h =h 0 is a consan, hey are approximae resuls when h h 0. Finally, specifying he values of h 0, we shall obain he exac resuls for he Rashba, he Dresselhaus he symmeric cases. i The Rashba limi 0, =0. In his case, from Eq. B5 follows h 0 = 1 2 R. B17 X = The phase funcion is 1 ei n 2 n 1 e i. = R n 1 n 2 f d 0 B10 B11 Furhermore he righ-h side of Eq. B11 can be easily inegraed, so ha = R 2. Thus he spin-flip probabiliy is obained in he form, B18 which can be expressed in closed form, =2R h E 4, 2 E 4, 2 +, + B12 E,k is he ellipic funcion of he second kind defined by E,k = 0 1 k 2 sin 2 d. Consequenly, for he case h =h 0, he saring 2sin2 R w 1/2, 1/2 = 4 2 R R 2, R 2 B19 =. ii The Dresselhaus limi =0, 0. In his case, Eq. B5 leads o h 0 = The inegral of Eq. B11 yields 1 2 R. = R 2. Hence he spin-flip probabiliy is B20 B

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