Electrons in Surface Acoustic Waves as Spin Qubits

Elec ons in Su ace Acous ic Wa es as Spin Qubi s Mikel Olano EHU/UPV Donos ia In e na ional Physics Cen e A p ojec submi ed o he deg ee o PhD in Physics o Ad anced Nanos uc u es 2025 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 (cc) 2025 Mikel Olano A anbu u (cc by 4.0) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 ii e a au e a joanen gai un guz iak ba ak bes ea en inbidi ik sen i uko ez duen momen u a e Mikel A egi REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 Abs ac Solid s a e coupled quan um do s (QDs) ha e been long p oposed as a pla o m o con ol spin qubi s and pe o m quan um compu a ion [ 54 , 14 ]. One o he mos popula me hods o c ea e hese a ays consis s in c ea ing a wo-dimensional elec on gas (2DEG) in he junc ion be ween wo semiconduc o s wi h a simila band gap, being able o c ea e elec on deple ed zones by applying elec ical ol ages wi h con olled ga es. As a possible solu ion o he sho ange in e ac ions ha can usually be ob ained in hese scena ios “ lying” qubi s [ 25 , 5 ] ha e been p oposed o anspo elec ons while pe o ming he one and wo-pa icle in e ac ions. Su ace acous ic wa es (SAWs), which can be c ea ed by in e digi al ansduce s (IDTs) in he su ace o a piezoelec ic ma e ial [ 21 , 82 ], ha e been p oposed and demons a ed as he ca ie s o single elec ons [ 7 , 33 ]. This Thesis co e s he main in e ac ions in he anspo o single elec ons om s a ic o mo ing do s, including he spin-o bi in e ac ion and he possible spin- lip p ocesses ha may happen due o he hype ine in e ac ion be ween he nuclea spin ba h p esen in GaAs and he elec on’s spin. The nume ical analysis o he ans e p ocess shows he possibili y o ea ing he e olu ion o he sys em wi h ew low-ene gy s a es, which educes he numbe o e ms o be aken in o accoun o desc ibe and op imize i . A he end, he Coulomb in e ac ion be ween wo pa icles in di e en do s is b ie ly discussed, mainly h ough he heo e ical exp essions ha en e in i s de ini ion and a nume ical app oxima ion o a pa icula impac pa ame e . REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 Con en s 1 In oduc ion 1 1.1 Quan um Simula ion &Compu a ion ................. 1 1.2 Quan um Do s ............................. 8 1.3 Su ace Acous ic Wa es ......................... 11 1.4 P oposal and Hypo hesis ........................ 12 2 E ec i e model and nume ical me hods 15 2.1 Nume ic de ini ion o he Hamil onian ................ 17 2.2 Nume ical ime e olu ion ........................ 20 3 The ans e p ocess 25 3.1 Po en ials ................................ 26 3.2 Va ying he impac pa ame e ..................... 29 3.3 Few-le el app oach ........................... 33 3.3.1 Simpli ied e sion and pa ame e a ia ions ......... 35 4 The spin o bi in e ac ion 43 4.1 Well-sepa a ed minima ......................... 45 4.2 E ec s on ans e p obabili y ..................... 47 4.3 En anglemen .............................. 51 5 The hype ine in e ac ion 55 5.1 App oxima ed exp essions ....................... 56 5.2 Spin- lip p ocess ime-e olu ion .................... 65 6 Two-pa icle in e ac ion 71 6.1 The coulomb in e ac ion ........................ 72 6.2 Fu he co ec ions ........................... 75 6.2.1 Real space exp essions ..................... 76 6.2.2 Momen um space exp essions ................. 78 6.3 Nume ical conside a ions and esul s ................. 81 7 Conclusions 83 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 i Con en s Appendices A Coulomb e m calcula ion 87 Bibliog aphy 91 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 1 In oduc ion Con en s 1.1 Quan um Simula ion &Compu a ion . . . . . . . . . . . 1 1.2 Quan um Do s . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Su ace Acous ic Wa es . . . . . . . . . . . . . . . . . . . 11 1.4 P oposal and Hypo hesis . . . . . . . . . . . . . . . . . . 12 1.1 Quan um Simula ion & Compu a ion Quan um compu a ion is a ield ha has d awn he a en ion o a g ea amoun o scien is s o he las hal cen u y. As elec onic compu e s a e eaching a omic sizes o hei ansis o s [ 22 ], hei quan um na u e s a s o become inc easingly di icul o manage. The incapabili y o scaling bo h algo i hms and compu e s ha we e powe ul enough o simula e quan um phenomena p ope ly, physicis s and ma hema icians s a ed hinking in an al e na e manne o s udying such sys ems, ocusing on he use o quan um s a es o hei bene i o unde s and be e he many-a om eali y. One o he g ea ields ha was opened ollowing his pa h is quan um simula ion, which allows o s udy Hamil onians ha a e oo complex o be compu ed in a classical compu e [ 31 ]. One can encode a Hamil onian in a p og ammable ashion in a quan um sys em such ha he ime e olu ion is 1 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 8 1.2. Quan um Do s in o de o ha e a decen ideli y. This is no necessa y in measu emen -based compu a ions, hough, ha uses massi ely en angled s a es and sequences o adap ed single-qubi measu emen s o pe o m quan um compu a ion. The emaining pla o m ha has shown g ea po en ial o pe o ming quan um compu a ion a e spin qubi s in quan um do s, which a e his hesis’ choice o s udy. 1.2 Quan um Do s Fo quan um compu a ional pu poses, quan um do s (QDs) a e small olumes in semiconduc ing ma e ials whe e a po en ial c ea es a con ining ene gy ha single o mul iple elec ons can ill. Due o he h ee-dimensional con inemen ha gi es ise o quan ized le els which can be illed, hey a e also e e ed as “a i icial a oms” [ 4 ]. One o he mos popula echniques o c ea e a po en ial ha can be con olled s a s by c ea ing a wo-dimensional elec on gas (2DEG). This is done by c ea ing insula ing op and bo om laye s on a semiconduc o . Me allic ga es a e a e wa ds placed on op o he insula ing ma e ial, such ha applied ol ages le a de ini e amoun o elec ons a el o he do , con olled by he ba ie c ea ed wi h he Coulomb in e ac ion be ween he QD and he 2DEG [ 13 ]. The spa ial zone in which elec odes allow elec ons a e called “sou ce” and “d ain”, which can be a ied allowing elec ons in hei conduc ion band o en e he do . Va ying he s eng h o he ga e po en ial, he equilib ium popula ion on he QD changes. I he e is no in e media e s a e ha allows o “sou ce” elec ons o each he “d ain”, his popula ion can be an in ege wi h minimum luc ua ions, which can be in e p e ed as a ini e numbe o elec ons in he do [ 67 ]. Jus like in egula a oms, he i s elec on o en e he QD will ill one o he s a es ha can be de ined by he Sch ödinge equa ion o a ee pa icle in a h ee dimensional “box” de ined by he po en ial wi h an e ec i e mass m∗ ha depends on he cu a u e o he ma e ial’s lowes ene gy pa o he conduc ion band [ 57 ]. Once he i s elec on en e s he QD, an addi ional ba ie is c ea ed in o de o allow a second elec on in he same do , which is p ecisely due o he REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 1. In oduc ion 9 Coulomb in e ac ion be ween hem. This ene gy is usually cha ac e ized by he QD’s capaci ance C , being i s alue U = e2/C [ 38 ], and i ’s he main cause o wha is called he Coulomb blockade. I he ene gy di e ence be ween ei he he sou ce o he d ain and he quan um’s do nex ene gy le el is no big enough, hen no o he elec on will en e he QD. Lowe ing he ga e ol age and main aining he same alue o bo h he d ain and he sou ce can lead o an addi ional elec on o en e he do . I ei he one o he la e al s uc u e’s po en ial is highe han he second elec on ene gy bu he o he one emains below, hen a s eady cu en o single elec ons will low be ween hem. Depending on how many s a es he QD allows inside, one can see a cu en om sou ce o d ain which is p opo ional o he numbe o s a es in he anspo window c ea ed by he bias. This quan iza ion o elec on mo ion due o Coulomb blockade can also be seen in sys ems o coupled quan um do s [ 53 ], whe e cu en s h ough bo h do s can be ei he allowed o s opped by he numbe o elec ons in bo h do s. One may block he s eam o single pa icles by ha ing a s a e occupied in one o he do s, independen ly o he accessible s a es o he o he one. A common way o isualize his e ec is o plo ∂I/∂Vsd o di e en alues o ga e and d ain-sou ce ol age [ 67 , 46 ]. The images show zones in which he alue o he de i a i e is close o ze o, whe e he numbe o elec ons in he double-do sys em is cons an . Due o i s dependence o his alue on he independen pa ame e s, hese zones ha e a homboid shape and a e called “Coulomb diamonds”, which a e su ounded by posi i e and nega i e alues indica ing exchange o elec ons be ween he wo QDs. Being able o a y he unneling ba ie be ween he wo do s one can also access s a es ha belong o bo h o hem, which a e o en called “molecula ” s a es, due o hei c ea ion om wo coupled a i icial “a oms”. Apa om Coulomb blockade he e is ano he in e es ing e ec called “Pauli blockade” [ 38 , 63 ] ha can occu in double quan um do s o which sou ce and d ain a e connec ed by sequen ial unneling h ough he wo do s. This is based on he e ec ha he spin s a e o an elec on in he double QD has on he s a es ha a e a ailable o unneling [ 48 ]. I one occupies one o he do s wi h wo elec ons REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 10 1.2. Quan um Do s in i s lowes s a e, he o al spin con igu a ion ha hey acqui e is he single s a e. Once one o he elec ons lea es he double do , he newly en e ing elec on can only be in he single s a e o go “ h ough” he al eady occupied do , since he ene gy o he iple s a es will be ene ge ically una ailable. This can allow con olled spin loads in quan um compu e s, since one can load elec ons on one side making su e ha he i s popula ed s a e belongs o he co ec o al spin s a e. Bo h he Pauli spin blockade and ex e nal magne ic ields can be used o ini ialize spin s a es in semiconduc o quan um do s [ 13 ]. Once he elec ons a e inse ed in he QDs om he su ounding Fe mi sea, hey can be anspo ed be ween di e en do s [ 59 ] o made in e ac wi h each o he i he po en ials a e ei he de uned o he po en ial ba ie be ween hem is dec eased. The e ec ha a change o he po en ial’s shape can ha e on he in e ac ion allows o a con olled spin in e ac ion be ween he wo elec ons ia he exchange coupling Hex = −J12  S1· S2 and can be u he con olled by an ex e nal magne ic ield. This was sugges ed o be a way o pe o ming quan um compu a ion wi h quan um do s [ 54 , 14 ] and s ill emains as a possible and de eloped candida e. Figu e 1.1: Ske ch showing he p inciple o Pauli blockade in a double quan um do . a) The ela i e posi ion o sou ce and d ain allows elec on passages h ough single s, b) blocking he single iple ansi ion. One o he bigges ad an ages ha quan um-do pla o ms ha e, is he al eady de eloped silicon-based indus y, ha has imp o ed he ep oducibili y o quan um REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 1. In oduc ion 11 do s wi h he same cha ac e is ics [ 86 ]. The closeness wi h which hese can be packed is g ea ly impac ed by hese ype o echniques, which allows o many qubi s ha a e in e connec ed one o each o he . Howe e , long- ange in e ac ions a e s ill an issue since he exchange in e ac ion is sho - anged, and mos quan um do in e ac ions a e jus possible be ween nea es neighbo s. Fu he mo e, hei possible in e ac ions wi h nuclea spin ba hs (s ong in he case o GaAs and mo e dilu ed o pu i ied 28 Si) pose challenges ha need o be a ge ed o ealizing quan um compu a ion, as is decohe ence a ising om cha ge noise. 1.3 Su ace Acous ic Wa es Acous ic wa es ha ad ance along he su ace o a ma e ial can be c ea ed by inducing bo h a e ical and a longi udinal o ce on he ou e laye s o he ma e ial [ 65 , 37 , 81 ]. This c ea es s ess on he su ounding a oms a he su ace, which p opaga es in he longi udinal di ec ion in which he ini ial o ce has been applied. Due o he lack o bulk ma e ial abo e he su ace, he s ess c ea ed by he o ce has o be ze o in i , while i pene a es in o he ma e ial wi h exponen ially dec easing s eng h de ined by he s eng h o he a omic o molecula bonds and he applied o ce. The s eng h o he SAW dec eases wi h dis ance om he su ace wi h a ypical pene a ion dep h on he o de o he wa es’ wa eleng h λ [ 80 ]. Since SAWs ha e lowe phase eloci y han bulk wa es hey a e no e y well coupled, hus enabling low-loss p opaga ion o e long dis ances. The piezoelec ic e ec e e s o he induc ion o elec ic pola iza ion unde mechanical de o ma ion. Depending on he a angemen o he la ice (which mus no ha e an in e sion cen e o show his e ec ) and he di ec ion in which he a oms a e displaced, each ma e ial has a di e en piezoelec ic esponse o mechanical s ess. The e ec is e e sible, which means ha applying an elec ic ield o he ma e ial he a oms will be displaced unde i . The ela ion be ween he mechanical s ess and he elec ical ield induced by i c ea es a coupled mechanical- elec omagne ic equa ion ha allows o wa es in which bo h s ess and ield e ol e a he same ime. The coupling be ween he elec ic ields ha a e c ea ed by REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 12 1.4. P oposal and Hypo hesis he mechanical s ess and i s e ec on u he comp ession in neighbou ing a oms allows o c ea ing di e se shapes o he su ace acous ic wa es ha can be c ea ed h ough he ol ages applied by deposi ed me allic in e digi al ansduce s (IDTs) on he su ace o he piezoelec ic ma e ial. These ga es a e posi ioned in a way such ha applying ime- a ying pulses co esponds o summing se e al plane wa es ha o m a wa e packe wi h he desi ed shape [ 84 , 82 ]. Se e al uses o su ace acous ic wa es p oduced by IDTs ha e been hough o since hei disco e y, and cu en ly he e a e se e al scien i ic and indus ial echniques ha ake ad an age o his phenomenon [ 49 , 52 ]. Fu he mo e, hei use ulness in quan um compu a ion has long been p edic ed [ 5 ], whe e apped single elec ons can pe o m single- and wo-qubi in e ac ions be ween s a ic and “ lying” spin qubi s [ 50 ] and hei p ospec s a y in many ields su ounding quan um compu a ion and me ology [ 21 ]. 1.4 P oposal and Hypo hesis The sys em his Thesis s udies is based on an idea ha combines quan um do s and su ace acous ic wa es in o de o add ess he connec i i y issues ha a g id o quan um do s may ha e. The limi a ion ha his pla o m poses is he in e ac ion be ween elec ons ha a e no nea es neighbou s. Usually, i one wan s an in e ac ion be ween non-nea es neighbou quan um do s, i has o be media ed by he in e media e qubi (s), which g ea ly slows he compu a ional p ocess and lea es li le oom o scaling o e o co ec ing. Some sugges ions p opose shu ling p ocesses as a mean o dis ibu e en anglemen o couple dis an qubi s [ 34 ], using mic owa e esona o s o long-dis ance coupling [ 56 ] and using “ loa ing” ga es [ 78 ]. Based on he p oposal by CHW Ba nes [ 5 ], Ch is ophe Bäue le’s g oup has been specially p oli ic in expe imen ally demons a ing he use o SAWs o ini ialize and ope a e wi h single spin qubi s [ 7 , 16 ], as well as achie ing he ini ializa ion, anspo and eadou o wo-elec on spin s a es[ 7 ]. Thei imp o emen s in SAW de ini ion and cha ac e iza ion ha e also been p ominen [ 82 ]. The me hod ha hese expe imen s use o injec elec ons in he mo ing minima is based on sending REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 1. In oduc ion 13 a pulse a he same ime ha he a ge ed minimum o he SAW passes he s a ic quan um do . As p ecise as i is, his does no ensu e he elec on o be in any pa icula s a e once i is ans e ed. Thei heo e ical ea men o said p ocess is also sca ce. T ans e s o dis an quan um do s ha a e no aligned a e also limi ed by he ime he elec on akes o unnel om side o side o he double channel scheme hey p opose in [ 75 ]. Mo eo e , he single-elec on anspo and he wo-elec on in e ac ion ha hey ha e shown un il now ha e no ye p oduced a necessa y en angling ga es be ween elec ons. Figu e 1.2: Simpli ied scheme o he p oposed pla o m. In o de o sol e he issues ha hese g oups ha e shown in he de elopmen o his pla o m, he p oposal ha his Thesis s udies consis s in elec onic spins as qubi s o he sys em ha a e anspo ed (as in Ba nes’ p oposal) bu mo ing hem om and o s a ic do s o pe o m single-qubi ope a ions. Con olling his p ocess and c ea ing a ays o s a ic quan um do s a a pa icula dis ance om he SAW, elec ons ha ha e been injec ed may in e ac wi h incoming elec ons in a minimum c ea ed by he SAW. Al hough his p oposal only includes s a ic do s in a ow nea he channel whe e he SAW p opaga es, mo e lexible se ups may be possible REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 14 1.4. P oposal and Hypo hesis wi h a ying a chi ec u es o mediums. The ad an age o ha ing s a ic do s in which single-qubi ope a ions can be mo e con olled adds o he bene i s o le ing se e al s a ic elec ons in e ac wi h he same mo ing one, hus enabling long- ange unable in e ac ions be ween elec ons ha a e no necessa ily nea es neighbo s. The pu pose o his Thesis is o p esen a oy model ha co e s he main in e ac ions in he ans e p ocess o a single elec on in a s a ic QD in o a SAW minimum and ice e sa. Being GaAs he e os uc u es o high in e es o hei piezoelec ic p ope ies, he main in e ac ions ha a ec he spin s a e o he mo ing quan um do , i.e., he Rashba and D esselhaus spin-o bi in e ac ions [ 40 ] and he hype ine in e ac ion wi h he Ga and As magne ic nuclei [ 27 ] a e aken in o accoun . Looking a he cu en si ua ion in which his p oposal lies, his Thesis p oposes h ee ques ions and hypo heses: • Assuming ha a single minimum po en ial c ea ed by a SAW can in e ac wi h a la e ally placed quan um do , which a e he op imal cha ac e is ics o ha e a high- ideli y ans e ? • I he men ioned ans e can happen, Can he ans e p ocess be desc ibed by conside ing only a low-dimensional subspace and how small can i be chosen? • Wha is he e ec o he spin-o bi and he hype ine in e ac ions in his ans e p ocess? • Wha a e he necessa y condi ions o c ea e an en angling ga e be ween he mo ing and he s a ic qubi ? Can his be as e han be ween neighbo ing QDs? REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 2 E ec i e model and nume ical me hods Con en s 2.1 Nume ic de ini ion o he Hamil onian .......... 17 2.2 Nume ical ime e olu ion ................. 20 The elec on whose quan um s a e’s ime e olu ion is desc ibed in his hesis li es in a h ee-dimensional space. The e a e wo ini ial conside a ions o desc ibing i s s a e in his i s pa . Fi s , a comple e desc ip ion o i s spa ial ep esen a ion includes bo h he en elope unc ion ha desc ibes i s p obabili y dis ibu ion in space and he shape ha he elec on wa e unc ion has a ound he nuclei o he ma e ial depending on he band o which i belongs. Secondly, ollowing he a chi ec u es ha p opose a igh po en ial in one o he spa ial di ec ions ( ωz>> ωx, ωy ) o ap he elec on in a wo-dimensional space, he en elope unc ion can be desc ibed as a p oduc o a unc ion in he z-di ec ion and ano he ha includes x and y. One can conside he s a e o be in a pa icula ene ge ic s a e in he z-di ec ion (usually he g ounds a e) ha will no be able o couple o o he eigens a es because he e is no change in he Hamil onian o his di ec ion. Fo a gene al quan um s a e o he elec on | Φ ⟩ , i s eal space ep esen a ion will be ⟨ |Φ⟩=ϕ( )u( )(2.1) 15 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 16 2. E ec i e model and nume ical me hods whe e u ( ) ep esen s he shape o he elec on a ound he nuclei and ϕ ( )is he en elope unc ion. As men ioned abo e, he ac o iza ion ϕ( ) = ϕ2D(x, y)·ϕz(z)(2.2) can be applied because o he ene gy sepa a ion be ween he eigens a es in he z-di ec ion and i s in a iabili y o e ime. The s a e o a single elec on in a quan um well can be app oxima ed as a ee pa icle wi h an e ec i e mass ha depends on he conduc ion band cu a u e o he ma e ial. In ou case, he po en ial’s shape in he z-di ec ion will be conside ed o be well ep esen ed by a ha monic po en ial wi h a cu a u e o m∗ω2 z/ 2, gi ing a Gaussian la e al con inemen o he elec on: ϕz(z) = s1 2π∆−1 ze−z2/4∆2 z.(2.3) whe e ∆ z e e s o he wid h o he s a e in he z-di ec ion. In he o he wo di ec ions, one can sol e he Sch ödinge equa ion o he Hamil onian ha includes he kine ic and elec ic po en ial con ibu ions and has he eigens a es a each poin in ime. F om now on, any ime he posi ion ope a o o a s a e’s posi ion ep esen a ion is men ioned, i e e s o i s exp ession in he XY plane unless explici ly s a ed. Le us i s wo k on he ans e p ocess be ween he s a ic and mo ing QDs desc ibed in he p oposal. The ini ial quan um s a e o he elec on will e ol e o e ime wi h he changes ha occu in he po en ial landscape o which i is subjec ed. He e, we conside a po en ial ha is he sum o wo po en ials wi h a single minimum each, one emaining s a ic, ep esen ing a quan um do , while he o he is c ea ed by a p opaga ing SAW and hus o ms a mo ing po en ial minimum ha passes nea he quan um do . The ini ial exp ession ha is going o be conside ed o he wo-dimensional en elope unc ion ha desc ibes he elec on’s quan um s a e will jus ake in o accoun he e ec o he po en ials in i s e olu ion. Le us conside wo di e en po en ials wi h a single global minimum each, one o hem ep esen ing a s a ic quan um do and he o he one he pa icula minimum c ea ed by he SAW o REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 2. E ec i e model and nume ical me hods 17 which he elec on mus be ans e ed. Assuming ha he ha monic app oxima ions a ound hei minima in s ( o s a ic) and m ( ) = m (0) + SAW · ( o he mo ing) a e simila , hen he wo-dimensional Hamil onian H0( ) = p p p2 2m∗+Vs( − s) + Vm( − m( )) (2.4) has o la ge sepa a ion o po en ials almos doubly degene a e low-ene gy eigens a es and, in pa icula , an almos degene a e wo-dimensional g ounds a e subspace, he spli ing o which depends on he spa ial sepa a ion | m ( ) − s| be ween he po en ials. Ou aim is o ind a de e minis ic way o ans e an ini ial s a e loca ed in one o he minima o he o he in a sho pe iod o ime. This sugges s ha he e olu ion mus include a non-adiaba ic p ocess. Since he e is in e es in knowing he in e media e s a es ha a e popula ed du ing i s e olu ion, he desc ip ion o he s a e’s e olu ion on he eigens a e basis o he sys em will be done a some poin . The e o e, independen ly o he me hod ha is used o e ol e he sys em, one needs o know he solu ions o he eigen alue p oblem a di e en poin s in ime du ing he e ie al/injec ion o he elec on. 2.1 Nume ic de ini ion o he Hamil onian The desc ip ion o he sys em and i s e olu ion has been ansla ed in o a nume - ical p oblem by disc e izing he necessa y quan i ies, namely bo h posi ion and momen um, and ime. Each o he spa ial di ec ions has N poin s, whe eas he ime di e en ial mus mee some equi emen s men ioned la e . When pe o ming he disc e iza ion, we map ou ope a o s ( he po en ials and he kine ic ene gy) o he disc e e space. I he spa ial poin s a e de ined such ha ≡ ij = [ xi, yj ] whe e bo h subsc ip s i, j ∈ { 1 , N} , hen he elec ic po en ial is a diagonal ma ix wi h Vij,ij = V ( xi, yj ). Spa ial de i a i es ( o he momen um and kinec ic ene gy ope a o s) include o -diagonal e ms Tij,i′j′ = 0 o i = i′, j  = j′ ha a e de ined by hei cen e di e ence exp essions wi h pe iodic bounda y condi ions. Since he spa ial g id is a N2 -sized objec , bo h ope a o s a e de ined as a N2×N2 size REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 24 2.2. Nume ical ime e olu ion po en ial. This is why, e e y ime a ans e p obabili y is calcula ed, one mus use he “mo ing” e sion o he s a e o calcula e he alue. 10-1 10-2 10-3 10-4 -12 -10 -8 -6 -4 -2 0 Figu e 2.2: P obabili y o ind he analy ic e ol ed s a e a e e ol ing i o = 167 ps . The ime di e en ials a e ela i e o he ime uni u = 4.17 ps. Once we ha e he case well de ined, i ’s ime o see which o he e olu ions akes he smalles dis ance om he solu ion. Knowing wha he inal s a e would look like a e mo ing he s a e in he minimum o he mo ing po en ial o SAWT = 500 nm, one can e ol e he sys em wi h di e en me hods and see how he absolu e alue o he o e lap changes wi h ∆ . T ying he e olu ion o mulas gi en in equa ions 2.10 and 2.13, igu e 2.2 shows how well he p obabili y o ind he igh s a e is main ained a he end o he p ocess o di e en imes eps. The condi ion p e iously de ined o accep a ∆ as alid al eady happens o he alue ∆ = 4 . 17 · 10 −2ps . Ne e heless, o be ce ain ha mo e complex e olu ions will be co e ed as well, an o de o magni ude lowe di e en ial will be chosen, his is, ∆ = 4 . 17 · 10 −3ps . REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 3 The ans e p ocess Con en s 3.1 Po en ials ........................... 26 3.2 Va ying he impac pa ame e ............... 29 3.3 Few-le el app oach ..................... 33 3.3.1 Simpli ied e sion and pa ame e a ia ions ....... 35 Looking a ou p oposal, one o he main ques ions ha a ise is whe he i is possible o no o popula e he mo ing po en ial while he ini ial s a e is comple ely localized in he s a ic do and ice e sa. Since we a e in e es ed in an a chi ec u e wi h a single SAW p opaga ing channel connec ed o a ious QDs along he way, hey will ha e o be loca ed wi h a la e al dis ance o he channel. O he wise, any elec on ha would come wi h an incoming SAW pulse would s ongly in e ac wi h elec ons loca ed in he s a ic do s, which we may wan o a oid i wo-pa icle ga es a e o be a oided. This is a di e en app oach compa ed o he one seen in he expe imen s desc ibed in mos o he ci ed pape s [ 33 , 16 ], ha use a po en ial pulse imed such ha he s a ic elec on en e s in he wan ed minimum o he SAW. This is also one o he easons why he wo-pa icle in e ac ion is done while he elec ons a e in he channel and no in he s a ic do s. This chap e will i s desc ibe he po en ials ha ha e been used o he 25 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 26 3.1. Po en ials nume ical calcula ions, hen he e olu ion ha hey p o oke o he sys em whe e he elec on s a s loca ed in he s a ic QD. Gi en he simplici y o he app oach, one can expec o see a leas some hing simila o he p ocess ha is desc ibed in he in oduc ion: a one- ime ans e be ween he s a ic and he mo ing do , in ol ing he g ounds a e and he i s exci ed s a e o he s a ic Hamil onian a each poin in ime. I his is no he case, and he e is a need o in oducing mo e s a es o he p ocess o happen, an explana ion is due: one needs o ha e clea he e ec ha di e en pa ame e s ha e in he wan ed e olu ion. 3.1 Po en ials As p e iously s a es, he po en ials ha a e going o be conside ed will ha e a single global minimum, which ecen ad ances in he c ea ion o po en ials wi h IDTs allow [ 82 ]. They will also sha e he alue o he second de i a i e a ound i , such ha ∂2 iV|min(V)=1 2m∗ω2 i(3.1) whe e he subindex e e s o he spa ial di ec ion i∈ {x, y} , m∗ = 0 . 067 me is he e ec i e mass o he elec on in GaAs and ωi = 3 meV is he chosen ene gy gap o bo h po en ials, which is consis en wi h GaAs QD models [ 50 ]. Due o he possibili y o changing he shape o he s a ic do ia me allic ga es, i will be conside ed as exac ly equal o he mo ing minimum excep o an ene gy gap o 10 −4 , jus so he s a es a e no comple ely degene a e. The main pa icula unc ion ha has been chosen is he Gaussian unc ion Vgauss =V0·exp−kx,gauss(x−x0)2−ky,gauss(y−y0)2;(3.2) wi h kx/y,gauss =−m∗ωx/y 2V0 .(3.3) The second p oposed unc ion o he elec ic po en ial is he squa ed cosine unc ion unca ed a e hal a pe iod: REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 3. The ans e p ocess 27 Vcosq =V0·cos2kx,cosq (x−x0)·cos2ky,cosq (y−y0)· ·Hx(π/2−kx,cosq ·|x−x0|)·Hy(π/2−ky,cosq ·|y−y0|); (3.4) whe e now kx/y,cosq =qkx/y,gauss.(3.5) and he H x/y a e hea iside unc ions o ensu e he unca ion. -500 -400 -300 -200 -100 0 100 200 300 400 500 -40 -30 -20 -10 0 Figu e 3.1: Gaussian (’gauss’) and squa ed cosine (’cosq ’) elec ic po en ial unc ions y dependence o a cons an x posi ion. Figu e 3.1 shows a cu o he wo dimensional unc ion o a cons an alue o x while bo h po en ials a e cen e ed a ound he cen e o he g id. Fo e e ence, he e is a do ed line indica ing he ha monic unc ion wi h he same minimum alue and second de i a i e a ound i . As can be seen, hey o e lap o he poin whe e hey become indis inguishable. No ice ha he Gaussian po en ials wings sp ead longe han he squa ed cosine’s, whose alue inc eases as e . REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 28 3.1. Po en ials The las p oposal o a po en ial unc ion is inspi ed in he unc ion ha de ines he po en ial c ea ed by he SAW in [ 50 ]. I has a pa ame e by which one can une he numbe o minima ha he incoming po en ial may ha e. V [∆ ] = V0·exp−kx,gauss(x−x0)2·cos2(y−y0)· · anh(y+π∆ /ky,cosq )− anh(y−π∆ /ky,cosq )/2(3.6) -500 -400 -300 -200 -100 0 100 200 300 400 500 -40 -30 -20 -10 0 Figu e 3.2: Cu o a cons an x alue o he unc ion de ined in (3.6) using di e en alues o he ∆ pa ame e . The di e ence be ween he hype bolic angen s can be unde s ood as a so hea iside unc ion ha allows a di e en numbe o minima along he mo emen o he SAW. Figu e 3.2 shows he same cu as he one done o he o he wo po en ials. The alue o he pa ame e ∆ goes om 0 . 5, whe e he minimum in he cen e is much bigge han he ones in he wings, o 1 . 5, whe e he e a e h ee equally s ong minima. The alues used in his chap e ange om 0 . 5 o 0 . 8, whe e he minima on he side s a o be no iceable by he elec on in he s a ic do . REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 3. The ans e p ocess 29 Ou o he h ee p oposed po en ials he gaussian has been chosen as he e e ence po en ial o i s con inui y (including i s de i a i es) and simplici y. 3.2 Va ying he impac pa ame e Looking o he ans e p ocess o happen, one mus se some bounda ies in he pa ame e s. Le us de ine he impac pa ame e as he la e al dis ance be ween he cen e o he s a ic do and he mo ing one when hey a e a apa om each o he , gp = ( m (0) − s ) ·ˆ i (whe e ˆ i is he uni ec o in he x di ec ion). I i is possible o ha e he ansi ion be ween he wo s a es wi hou in ol ing any o he , hen i mus be when he wo po en ials a e sepa a ed and he e is a double well s uc u e du ing he whole e olu ion. In his scena io, he eigens a es main ain a ce ain esemblance o he e en and odd combina ions o he local s a es a ound each po en ial. In he case o he Gaussian po en ial, he closes dis ance a which he e a e s ill wo di e en minima is 145 . 1 nm o he pa ame e s chosen. The ollowing s a e e olu ions will be done o se e al impac pa ame e s, s a ing om his minimum o bigge ones. S a ing om a poin in space in which he indi idual po en ial eigens a es ha e a negligible o e lap, he elec on loca ed in he s a ic do is e ol ed o a inal poin a he same dis ance be ween he wo do s as he ini ial by he po en ial, which i e a i ely changes a a eloci y o 3 . 0 µm·ns−1 . The e, one sol es he eigen alue p oblem o he mo ing minimum and mul iplies he solu ion wi h he momen um displacemen ope a o wi h he alue p0 = m∗ SAW o ob ain he a ge s a e as in 2.14. I he g ounds a e o he mo ing po en ial a he end o he p ocess is | Ψ 0,m ( T ) ⟩ , hen he a ge s a e would be: |Ψ a ⟩= exp im∗ SAW ℏyΨ0,m(T)⟩(3.7) Figu e 3.3 shows he ideli y o ob ain he a ge s a e a e he e olu ion wi h he h ee di e en po en ial shapes. Se e al maxima a e close o one in he cases o he Gaussian and he minima ain unc ion wi h he single minimum, while a single maxima can be ound in he case o he cosine squa e be o e i me ges wi h he REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 30 3.2. Va ying he impac pa ame e 150 155 160 165 170 175 0 0.2 0.4 0.6 0.8 1 Figu e 3.3: P obabili y o ind he a ge s a e 3.7 a e e ol ing he ini ial elec on loca ed in he s a ic QD o he h ee a o emen ioned po en ials in 3.2,3.4 and 3.6. Fo he las po en ial, he pa ame e ’s alue is ∆ = 0.5. s a ic po en ial a he closes app oach. I is clea hen ha he po en ial’s la e al ex ension is key o ha e he possibili y o ans e ing be ween hese wo s a es a di e en impac pa ame e s. Mo eo e , one can also no ice ha he maxima a e close be ween hem he sho e gp’s alues a e, which makes hem na owe . The igh mos peak ( he wides o hem) occu s o he same alue o gp o he h ee ele an po en ials, sugges ing some common e ec . As expec ed, o gp alues bigge han ce ain limi he e is no ans e p obabili y be ween he wo s a es. I can also be in e es ing o see how mul iple minima in he incoming po en ial can a ec his ans e p obabili y. Figu e 3.4 shows his by a ying he pa ame e in he de ini ion o he las p oposed po en ial V [∆ ]. The case whe e he pa ame e alue is ∆ = 0 . 5has a global minimum much bigge han he la e al ones and i essen ially ac s as he Gaussian po en ial as i can be seen in igu e 3.3. E en REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 3. The ans e p ocess 31 145 150 155 160 165 170 175 0 0.2 0.4 0.6 0.8 1 Figu e 3.4: P obabili y o ind he a ge s a e a e e ol ing he ini ial elec on loca ed in he s a ic QD wi h he po en ial de ined in 3.6 o di e en alues o he pa ame e ∆ = 0.5. changing i s alue o ∆ = 0 . 6has a sizeable e ec in he ans e po babili y, lowe ing he igh mos maximum o abou 0.8. Whe eas he inc ease o ∆ = 0.8 does no dec ease his maximum as much, he lowe he alue o he impac pa ame e he mo e he peaks’ alues dec ease compa ed o he p e ious case. As i could ha e been p edic ed, he exis ence o se e al minima in he incoming dec eases he ans e p obabili y o he p ocess. Independen ly o whe e he es o he wa e unc ion has ended, i is clea ha a s ong single minimum is necessa y o ha e a success ul ans e p ocess in his a chi ec u e. Ha ing a 2D wo-minima po en ial means ha , looking a he nea es exci ed s a es, ou eigens a es a e close o degene acy when he wo do s a e apa . Once he po en ials a e close , wo o hem ge close o he wo- old g ounds a e basis, which leads o possible exci a ions du ing he e olu ion o he sys em. Knowing al eady REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 32 3.2. Va ying he impac pa ame e he gp alues o a ious high- ideli y poin s, one can ask whe he he exci a ion occu s only in he wo- old g ounds a e o i some pa o he e ol ed s a e exci es u he . The p obabili y o inding he s a e in he wo i s ins an aneous eigens a es o he sys em can be calcula ed a a ious poin s along i s pa h. The emaining p obabili y belongs o he p opo ion o he s a e ha has been exci ed. 0 20 40 60 80 100 120 140 160 0 0.5 1 1.5 10-3 Figu e 3.5: To al p obabili y o inding he ime-e ol ed s a e ou o he wo old ins an aneous g ounds a e in ime o wo di e en alues o he g azing pa ame e . Figu e 3.5 shows his o he le mos peak in ideli y ha has been p e iously pic u ed, gp = 145 . 9 nm and he igh mos one a gp = 160 . 5 nm . Simila beha io s ha e been ound o he es o he peaks in igu e 3.3. The e a e wo imes a which his p obabili y inc eases, dec easing sho ly a e o lea e a pa ial o his peak as a eminiscen p obabili y o ind highe -ene gy s a es. No ice ha since hese p ojec ions ha e been done in he ins an aneous basis and no he ue eigens a es o he sys em ( aking in o accoun he momen um gi en by he mo ing po en ial) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 3. The ans e p ocess 33 some o he inal p obabili y o inding he s a e in an exci ed s a e belongs o he ac ual eigens a e o he mo ing po en ial, p e iously called a ge s a e. In ei he way, he p obabili y o inding he s a e in he wo- old g ound s a e along he pa h is always highe han 0 . 999 in he case o he a hes maximum, sugges ing ha a ew-le el desc ip ion can be used o desc ibe he e ec and be e unde s and he p ocess. 3.3 Few-le el app oach The e olu ion o he s a e gi es us some guidance on he e ec ha can be expec ed om he sys em, bu i does no show wha he mechanisms a e and why his occu s. T ying o exp ess he e olu ion wi h a known basis can gi e he missing in o ma ion on wha a e he op imal condi ions o ou pu pose. The need o he eigens a es o he sys em o e alua e ou esul s mo i a es hei use as a ime-dependen base in his nex sec ion. Using a basis de ined by a Hamil onian ha changes a each poin in ime, i s e ol ing eigens a es can be inse ed in o he ime-dependen Sch ödinge equa ion (TDSE) o ob ain an exp ession o he coupling be ween hem a each ins ance, a oiding he need o w i e he e ol ed s a e a ime in he basis a + ∆ . The single-pa icle Hamil onian a some pa icula ime in he ins an aneous eigens a e basis can be w i en as ˆ H=X i|Ψi⟩εi⟨Ψi|(3.8) being | Ψ i⟩ he eigens a es a ime , wi h ene gies εi . The TDSE applied o a gene al s a e looks like iℏ∂ ∂ |Ψ⟩=ˆ H|Ψ⟩ ⇒ iℏ∂ ∂ X n cn|Ψn⟩=X i |Ψi⟩εi⟨Ψi|X n cn|Ψn⟩ ⇒ ⇒iℏX n˙cn|Ψn⟩+cn˙ |Ψn⟩=X n cnεn|Ψn⟩, (3.9) which p ojec ed o a pa icula ou going eigens a e | Ψ m⟩ REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 40 3.3. Few-le el app oach Looking a igu e 3.3, one can al eady see ha he igh mos maximum, being he wides , seems o be he bes poin o op imize he elec on e ie al p ocess wi h espec o he g azing pa ame e . A sho e dis ances, he alues o ∂2 gpF a e highe a he ex eme poin s, making hem less eliable o any noise ha he sys em may ha e. E en in his bes case, a change in ∆ gp∼ 3 nm al eady dec eases he ideli y o ze o. 150 160 170 180 190 200 0 0.2 0.4 0.6 0.8 Figu e 3.9: Values o γ depending on gp o di e en alues o he po en ial’s s eng h. The analysis o he change in he po en ial’s s eng h is a mo e complica ed issue. One mus ake in o accoun ha bo h e ms ha en e in he desc ip ion o he g ounds a e’s e olu ion change wi h i . Mo eo e , he s eng h wi h which he wo old g ounds a e is coupled o he exci ed s a e du ing he e olu ion is also changed. In gene al, one can ensu e ha bigge s eng hs ( igh e po en ials) will dec ease he possibili y o exci ing any s a e li ing in he wo old g ounds a e. On he o he hand, one would need o ge he do s close o ha e he possibili y o ul illing he condi ion in (3.22) , as can be seen in igu e 3.9. A sho e dis ances, he change REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 3. The ans e p ocess 41 in ene gy occu s as e , implying ha he condi ion in (3.21) is me as e be ween ideli y maxima, inc easing he e o in ans e p obabili y o smalle e o s in gp . Gene ally speaking, i one would like o a oid e o s esul ing om ha ing oo big o a change in ideli y a ying he g azing pa ame e , lowe eloci ies (which mos o he ime equi es a change o ma e ial) and lowe po en ial s eng hs a e he solu ion. Ob iously, his dec eases he numbe o pa icles ha can be e ie ed/injec ed pe ime uni , which may no be use ul o quan um compu a ion pu poses. REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 42 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 4 The spin o bi in e ac ion Con en s 4.1 Well-sepa a ed minima ................... 45 4.2 E ec s on ans e p obabili y ............... 47 4.3 En anglemen ......................... 51 Since he elec on is mo ing in a space wi h a b oken symme y, i s spin will e ol e unde he in luence o a spin-o bi coupling ha can be desc ibed by he Rashba [ 15 ] and D esselhaus [ 26 ] spin-o bi coupling HSO =αR(ˆpxσy−ˆpyσx) + βD(−ˆpxσx+ ˆpyσy). (4.1) This in e ac ion couples he spin and o bi al deg ees o eedom o he pa icle in a gene ally non- i ial way. Al hough he e ha e been some s udies desc ibing he e olu ion o such sys ems o mo ing po en ials, [ 40 , 23 ] as well as using hem o pe o m single-qubi ga es, [ 32 , 30 ] he loading p ocess o a s a ic do , which includes also mo emen in he ans e se di ec ion o he elec on ca ying po en ial has no been ye ackled. No e ha , e en i he s eng h o he spin- o bi in e ac ion is smalle han he ones esponsible o he ans e o he s a e (and he e o e can be conside ed a pe u ba i e e ec on ene gy), he quali a i e di e ence be ween he s a es can po en ially change he ime de i a i es o he 43 REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 44 4. The spin o bi in e ac ion eigens a es o he sys em in a signi ican manne . Reo de ing he e ms acco ding o hei momen um di ec ion p opo ionali y ˆ HSO =−(βDσx−αRσy)·ˆpx−(αRσx−βDσy)·ˆpy=−σa·ˆpx−σb·ˆpy,(4.2) ha can be seen as a momen um shi on he o iginal Hamil onian, since (ˆpx−m∗σa)2+ (ˆpy−m∗σb)2 2m∗=p 2m∗−σa·ˆpx−σb·ˆpy+1 2m∗(σ2 a+σ2 b)(4.3) whe e he las e m is jus a global shi in ene gy. Knowing his, one can be emp ed o use he displacemen ope a o o ob ain he solu ions o he o al Hamil onian, bu as σa = σb a e usually di e en , he wo-momen a displacemen ope a o does no gi e us he wan ed Hamil onian ˆ H( ) = ˆ H0( )+ ˆ HSO =ˆ Dp(m∗σa, m∗σb)ˆ H0( )ˆ D† p(m∗σa, m∗σb)−1 2m∗(σ2 a+σ2 b)(4.4) whe e he shi in momen um would be ob ained by applying ˆ Dp , ha ing he o m ˆ Dp(m∗σa, m∗σb) = eim∗(σax+σby)/ℏ.(4.5) The Zassenhaus o mula gi es an in ini e se ies ha desc ibes he o al displace- men ope a o as a mul iplica ion o sepa a e displacemen s. F om he second o de on, one inds ha he commu a o be ween σa and σb appea s in all indi idual displacemen s. This exp ession looks like [σa, σb]=2i(α2 R−β2 D)σz. (4.6) which is p ecisely he eason o (4.4) . Since σa and σb do no commu e in gene al, he sum in he exponen is no he same as doing each displacemen sepa a ely and he exponen ia ion o such a ma ix is a nume ically expensi e ask. The condi ion o which he displacemen s can be applied sepa a ely is |αR| = |βD| , which is eally speci ic bu has been ea ed as possible [ 40 ] and expe imen ally REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 4. The spin o bi in e ac ion 45 achie able [ 23 ]. Taking in o accoun hese limi a ions, his wo k will deal wi h he pa icula case whe e he commu a o is ze o. Choosing x and y along GaAs’s [110] and [ ¯ 1 10] o ien a ions, he expe imen al s eng h o bo h he Rashba and D esselhaus in e ac ion can be uned o αR = 300 nm ·ns−1 [ 40 , 23 ]. Di e en pa ame e choices can be made, bu as long as his same o de o magni ude is chosen, he physical meaning and e ec s will be, b oadly speaking, he same as hose desc ibed he e. 4.1 Well-sepa a ed minima The p e ious sec ion has pa ame ized he e olu ion o he spinless elec on such ha i can be desc ibed by he wo- old g ounds a e. Fo su icien ly sepa a ed do s, his is spanned by he g ounds a es o he ee Hamil onian wi h a single po en ial. Wi h he inclusion o he spin-o bi in e ac ion, he subspace o in e es is doubled, each o he ee s a es ha ing he spin deg ee o eedom ied o hei o bi al s a e. In he ini ial and inal con igu a ions o he comple e Hamil onian he elec ons a e loca ed a ound each pa icula po en ial and a e no coupled be ween hem, so hey can be sepa a ely desc ibed. Le us examine he e ec o he spin-o bi in e ac ion in his con igu a ion. The s a ic spinless s a e | Ψ s⟩ is he eigens a e o he Hamil onian ˆ H0,s = p / 2 m∗ + Vs . Conside ing he case o αR = βD makes σa = σb≡σ and he displacemen ope a o simpli ies o ˆ Dp(σ) = eim∗σ(ˆx+ˆy)/ℏ, (4.7) which changes he pa ial Hamil onian o ˆ Dp(σ)ˆ H0,s ˆ Dp(σ)†=(ˆpx+m∗σ)2+ (ˆpy+m∗σ)2 2m∗+Vs−m∗σ2(4.8) and he co esponding eigens a e o (ˆ H0,s +ˆ HSO)|Ψs, ms⟩=εs,SO|Ψs, ms⟩,(4.9) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 46 4.1. Well-sepa a ed minima whe e ms ep esen s he coupled spin s a e. The case o he mo ing do ’s eigens a e is a bi di e en . Since he ins an aneous eigens a e o he mo ing po en ial shi s in ime in he y di ec ion, i can be unde s ood as a ime-dependen basis s a e in ou desc ip ion, |Ψm( )⟩=ei( SAW· )·ˆpy/ℏ|Ψm⟩(4.10) simila o he a ge s a e in he p e ious chap e . The way o ob ain he eigens a e o he mo ing po en ial wi h he spin-o bi in e ac ion is a i ing o an exp ession ha includes bo h he displacemen in momen um and ime. Applying he momen um ope a o o he ime displacemen , one ob ains ˆ Dp(σ)ei( SAW· )·ˆpy/ℏ=ei(m∗σ(ˆx+ˆy)+( SAW· )·ˆpy−m∗( SAW· )σ/2)/ℏ(4.11) whe e he Bake -Campbell-Hausdo o mula has been used, along wi h he usual commu a ion ela ion [ ˆy, ˆpy ] = iℏ . The inal displacemen ope a o ha we a e looking o includes he wo i s addends in he igh hand side exponen ial, which lea es us wi h he ela ion ˆ D ,σ =eim∗σ(ˆx+ˆy)/ℏei( SAW· )·ˆpy/ℏeim∗( SAW· )σ/(2ℏ)(4.12) ha applied o he ins an aneous eigens a e o he mo ing po en ial as in equa ion 4.9 (ˆ H0,m( ) + ˆ HSO)|Ψm, ms( )⟩=εm,SO|Ψm, ms( )⟩,(4.13) ob ains he eigens a e o he mo ing elec on unde spin-o bi in e ac ion a ime . This allows us o compu e Ω mm om 3.11 o desc ibe a non-adiaba ic p ocess ela ed o he elec on’s spin p ecession unde hese ci cums ances: Ωmm =⟨Ψm, ms( )|˙ Ψm, ms( )⟩(4.14) =im∗ 2ℏ⟨Ψm, ms( )|σ|Ψm, ms( )⟩(4.15) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 4. The spin o bi in e ac ion 47 which has he e ec o an e ec i e magne ic ield in he di ec ion o σ/√2αR and s eng h Bσ = m∗ αR/√2µBℏ , which has been obse ed expe imen ally. This means ha i one chooses he eigens a es o he o al Hamil onian wi h a spin in he σ = αR ( σx + σy )di ec ion, he e will be wo uncoupled subspaces by he spin alue and wo s a es in each subspace, shi ed in ene gy depending on whe he hey mo e o no . 4.2 E ec s on ans e p obabili y Following he same p inciples ha ha e been desc ibed in sec ion 2 he ene gies and eigens a es o he Hamil onian ˆ H ( ) = ˆ H0 ( ) + ˆ HSO ha e been ob ained a se e al poin s in ime o he ajec o y o he mo ing po en ial. The coupling e ms be ween di e en eigens a es can be calcula ed wi h he exac same exp ession as be o e, bu now one needs o also include he ene gy shi ha comes by ha ing mo ing eigens a es wi h a spin-o bi in e ac ion. The new in e ac ion is no s ong enough o signi ican ly inc ease he ans e o exci ed s a es in he icini y o he ans e p obabili y maximum loca ed a 160 . 5 nm , which is whe e we a e in e es ed in ope a ing ou sys em. As p e iously easoned, in he (now doubled) wo- old g ounds a e he e a e wo uncoupled subspaces i he eigens a es’ spins a e di ec ed in he σ di ec ion, such ha ˆ H( ) = H+( ) 0 0H−( )!(4.16) whe e he subsc ip s in H± e e o he spin alues and bo h objec s a e 2 × 2 ma ices ha will e ol e he ini ial s a e o he elec on. To e alua e how hese new exp essions will a ec he e olu ion o he elec on depending on i s spin, le us compa e hem wi h H0 . Taking in o accoun ha he new e ms a e smalle in s eng h and hey come om he Hamil onian’s ime-de i a i e pa , we can w i e H±( ) = H0( )±ℏ δΩs( )δΩsm( ) δΩsm( )δΩm( )!(4.17) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 48 4.2. E ec s on ans e p obabili y whe e he second ma ix encapsula es he changes o he e olu ion added by he SOI. The subsc ip s ha e been decided depending on whe he he eigens a es is ini ially s a ic ( s ) o mo ing ( m ). I s diagonal elemen s a e ela ed such ha : δΩm=−δΩs+ Ω0 mm.(4.18) W i ing i as in equa ion 3.15, changes appea as an addi ion o he ene gy gap and a new pe pendicula e m: H±= [ z( )±δ z( )]σz+ y( )σy±δ x( )σx(4.19) whe e δ z( ) = ℏδΩs−Ω0 mm/2(4.20) δ x( ) = ℏδΩsm.(4.21) When he po en ials ge close o each o he , he o bi al pa s can be mo e ela ed o he e en and odd combina ions o he localized s a es, depending on he il be ween he wo po en ials. Mo eo e , he diagonal pa s o he non-adiaba ic con ibu ion a e expec ed o be s ic ly eal since all s a es’ spins a e equi ed o poin in he di ec ion se by he spin-o bi in e ac ion. As igu e 4.1 shows, he ini ial and inal diagonal elemen s o he non-adiaba ic pa a e cons an , coinciding wi h he s eng h p e iously men ioned as an e ec i e magne ic ield ac ing in he mo ing pa icle. The middle pa , whe e one expec s o ha e e en and odd combina ions o o bi al pa s in he eigens a es, shows hal o he s eng h ound in he beginning and he end o he p ocess. This indica es p ecisely ha he e is a pa o he eigens a es ha keeps mo ing in he same di ec ion as he po en ial. Since he elec on is no only mo ing o wa d bu also la e ally, one would also expec o see imagina y componen s in he non-diagonal elemen s o he nonadiaba ic con ibu ion. This is shown in he lowe pa , whe e bo h he eal and he imagina y pa s o he non-diagonal elemen s in he non-adiaba ic REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 4. The spin o bi in e ac ion 49 0 20 40 60 80 100 120 140 160 -4 -3 -2 -1 0 1 Figu e 4.1: All elemen s ha a e included in he Hamil onian due o spin-o bi in e ac ion 4.17. con ibu ion a e depic ed. The eal pa pa ially co esponds o he nonadiaba ic ansi ion o he o bi al pa , which can also be ound in he ee-elec on e olu ion p ocess. I will also ha e a con ibu ion o he spin-o bi in e ac ion coming om he la e al mo emen o he elec on. On he o he hand, he imagina y pa may exclusi ely come om he spin-o bi in e ac ion. This sugges s ha an e ec i e ime-dependen magne ic ield can be acked du ing he e olu ion o he sys em, which implies a possibili y o dis u bing he p ocess o ans e ing he elec on om one do o he o he , gi en ha he o bi al and spin pa s a e en angled. Once he e olu ion ma ix is de ined o a pa icula pa h o he mo ing po en ial, one can e ol e he sys em wi h di e en ini ial spin s a es and see he change in ideli y o he ans e p ocess o analyze i s dependence on he e ec i e ime- dependen magne ic ield c ea ed by he spin-o bi in e ac ion. Gi en ha du ing mos o he in e ac ion he e ec i e ield is di ec ed in he σ di ec ion, i can be REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 56 5.1. App oxima ed exp essions Once his is done and aking in o accoun he scale di e ence in in e ac ion s eng h be ween he hype ine and he e ms go e ning he e olu ion o he elec on, one can e ol e he spin dynamic be ween he elec on and he nucleus’s spin by e ol ing he s a e de ined in ha subspace in he in e ac ion pic u e as a pe u a i e e ec . Ha ing he uppe bound o he spin- lip a e o a single nucleus, one can ob ain an app oxima e alue o he o e all p obabili y o lipping he elec on’s spin once i goes in and ou o he s a ic do . 5.1 App oxima ed exp essions The hype ine in e ac ion desc ibes he ene gy ha exis s be ween wo magne ic momen s, which can be desc ibed as dis ibu ions o poin -like. I is commonly used o desc ibe he elec on’s spin in e ac ion wi h he nucleus’ [ 9 ], which only depends on hei ela i e posi ion = e− n is gi en by ˆ Hh = 2ℏµBγnh −3I·(L−S)+3 −5(I· )(S· ) + (8π/3)δ( )I·Si(5.1) whe e µB is Boh ’s magne on, γn is he nuclea gy omagne ic a io. The S e e s o he elec on’s spin, whe eas I ep esen s he nucleus’. The i s hamil onian e m e e s o he in e ac ion o an elec on ha is o bi ing a nucleus, he second one akes in o accoun he s a ic case a a dis ance and he las one, he con ac in e ac ion, sol es he inde e mina e case o wo magne ic momen s a he same poin in space. In semiconduc o s, he in e ac ion o he elec on spin wi h he ( ully o pa ially) andomly o ien ed la ice nuclei leads o as decohe ence o i s spin s a e [ 70 ] (on a ime scale o ns, depending on he ma e ial). The e ec is educed i he wa e unc ion is sp ead ou o e mo e nuclei (coupling mo e weakly o each one) o i i is mo ing [ 27 , 40 ] ( he eby e ec i ely coupling mo e weakly o a la ge numbe o nuclei). Nuclea spins, howe e , ha e longe decohe ence imes due o dis ances be ween hem and nuclea magne ic momen s eng hs. The same way he nuclea spin ba h can be an incon enien ea u e o he sys em as i a ec s he cohe ence o he elec on’s spin s a e, his same e ec can be used a o ably. Due o he hype ine REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 5. The hype ine in e ac ion 57 in e ac ion ha exis s be ween he elec on and he nuclei in i s icini y, he e ha e been p oposals o c ea e quan um memo ies by pola izing he nuclea spins h ough shu ling o elec ons [ 77 , 85 , 47 , 28 ] an ecen ly demons a ed expe imen ally [ 2 ]. Taking ad an age o he knowledge ha he model p esen ed in his a icle gi es on he e olu ion o he elec on’s wa e unc ion o bo h spin di ec ions, i is in e es ing o analyze he possible use ha i may ha e as nucleus spin pola ize and e alua ing how good he app oxima ion o ins an aneous loading done in [ 28 ] is. The in en ion o his sec ion is o gi e an app oxima e numbe on he pola iza ion a e o nuclei in a quan um do by choosing a ep esen a i e case o es ima e he o e all beha io . In many semiconduc o s, including GaAs, he mos ele an e ec s a e well desc ibed assuming he elec on lies in a s- ype conduc ion band, which means ha he e is no o bi al momen um con ibu ion in he in e ac ion [ 70 ]. The e o e, he Hamil onian e m o he elec on’s in e ac ion wi h he su ounding nuclei can be simpli ied as: ˆ Hh =X α ˆ SαX n Anδ( − n)ˆ Iα n.(5.2) whe e An is he in e ac ion s eng h o he n - h nucleus wi h he elec on. The ope a o ha couples a single nucleus wi h he elec on is he e o e o he o m ˆ Sˆ In=X α ˆ Sαˆ Iα n=ˆ Szˆ Iz n+1 2(ˆ S−ˆ I+ n+ˆ S+ˆ I− n).(5.3) whe e he plus and minus signs a e he usual ladde ope a o s ha inc ease o dec ease he spin’s alue in he σ di ec ion. 5.2 includes all magne ic nuclei in gene al, which can be educed o an e ec i e numbe Ne using he in e se pa icipa ion a io ( IPR ). Since ou in e es lies in he pola iza ion o nuclei ha a e in he icini y o a s a ic quan um do ’s cen e , le ’s use he wa e unc ion o he eigens a e in he s a ic do a ime = 0 ps , p e iously named | Ψ s (  , 0) ⟩ . Le us de ine he o al numbe o magne ic nuclei on ou h ee-dimensional g id as Nn . Using he de ini ion o he in e se pa icipa ion a io (which es ima es 1 /Ne whe e Ne is he numbe o pa icles REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 58 5.1. App oxima ed exp essions in e ac ing wi h a s a e de ined by a sp ead p obabili y unc ion), we can se he numbe o in e ac ing nuclei as Ne =Nn X i=1 |ϕs( i)|4 2 0−1(5.4) whe e he index i e e s o he poin s in he h ee-dimensional g id ha de ines he s a e’s p obabili y densi y in eal space, whe eas 0 e e s o he olume uni a ound a single a om. The no maliza ion o he s a e is ensu ed by he condi ion PNn i=1 |ϕs ( i ) |2 0 = 1. Ne has a alue be ween 7 . 9 · 10 5 and 3 . 9 · 10 6 o ∆ z a ying om 2 nm o 10 nm . Du ing he nex calcula ions, he s onges po en ial s eng h will be conside ed, pu ing an uppe limi on he in e ac ion’s e ec . The o e all Hamil onian his a has conside ed he kine ic ene gy o he elec on, he elec ic po en ial c ea ed by he SAW ha allows i s anspo and he spin- o bi e m ha de ines a p e e en ial di ec ion o he spin s a e o he elec on. Including he hype ine in e ac ion one can w i e ˆ HT( ) = ˆ T+ˆ Ve( ) + ˆ HSO +ˆ Hh =ˆ H0( ) + ˆ Hh (5.5) as he o al Hamil onian. Since he z componen o he hype ine in e ac ion is diagonal in he basis de ined by ˆ H0 ( )i can be sepa a ed o m he es o he hype ine e ms and added o his o iginal Hamil onian ˆ H( ) = ˆ H0( ) + ˆ Sz Ne X i=1 Aiδ( − i)ˆ Iz i(5.6) lea ing he spin- lipping e ms as he only in e ac ion Hamil onian ˆ Hsp =ˆ S+ Ne X i=1 Aiδ( − i)ˆ I− i+ˆ S− Ne X i=1 Aiδ( − i)ˆ I+ i.(5.7) The uni a y ha e ol es he sys em om an ini ial ime o in he Sch ödinge pic u e wi hou he hype ine in e ac ion is de ined as ˆ U( , 0) = Te−iR 0ˆ H( ′)d ′/ℏ,(5.8) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 5. The hype ine in e ac ion 59 no o be con used wi h he o al uni a y ha e ol es he s a e o ime , since his would ha e o ake in o accoun he hype ine in e ac ion. The ma ix ep esen a ion o he Hamil onian in ou educed basis has been de ined up un il now in he basis o | Ψ s, ms⟩ and | Ψ m, m′ s⟩ , he ins an aneous eigens a es o he Hamil onian wi hou he hype ine in e ac ion. Ins ead o his, we could change he basis such ha i ollows he e ol ed s a e. To do his, one needs o ind a ma ix ha , applied o he p e ious coe icien s, always p ojec s he s a e o he i s componen . Taking in o accoun ha o he a o emen ioned condi ions on he spin-o bi pa ame e s he e is no possible spin-exchange in he z di ec ion, le us call he ime-e ol ed s a e |Φj( )⟩±≡ |Ψj( ),±⟩ =c(j) s,±|Ψs,±⟩+c(j) m,±|Ψm,±⟩,(5.9) whe e he ime dependence comes om he ime-dependen basis s a es | Ψ i⟩ and he co esponding componen s ci . The j unde sc ip makes e e ence o he ini ial s a e o he elec on, which de e mines he alues o he componen s du ing he e olu ion o he elec on. An o hogonal s a e ha li es in he same Hilbe space and is no popula ed du ing he e olu ion is |¯ Φ( )⟩±=c(j)∗ m,±|Ψs,±⟩−c(j)∗ s,±|Ψm,±⟩ (5.10) which will be he second s a e o a new basis ha is use ul o us. The ma ix ha would mee he needed equi emen s is he basis change ma ix om ou p e ious basis o he new one: MΦ¯ Φ sm j ± = c(j)∗ s,±c(j)∗ m,± c(j) m,±−c(j) s,±!.(5.11) Since he e olu ion o he o bi al s a e is independen o each ms alue, his basis change ma ix has o be applied o bo h di ec ions, which lea es us wi h he comple e basis change ma ix MΦ¯ Φj sm =     MΦ¯ Φ sm j + 0 0MΦ¯ Φ sm j −      .(5.12) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 60 5.1. App oxima ed exp essions A e i is applied, he ime-e olu ion uni a y will jus be a diagonal ma ix gi ing each popula ed s a e i s co esponding phase. Fo simplici y, he no a ion o s a es o ope a o s will no change, bu all elemen s a e now calcula ed wi h he popula ed o bi al s a es as a ime-e ol ing basis, ins ead o he ins an aneous eigens a es o he o al Hamil onian wi hou he hype ine in e ac ion. In he in e ac ion pic u e, one can de ine he e olu ion o he sys em wi h a simpli ied ope a o i he e ec o a pa o he o al on he sys em is known. Since all ou p e ious conside a ions make he uni a y ˆ U ( , 0) manageable, i is con enien o desc ibe he Hamil onian and he ime e ol ed s a e in he in e ac ion pic u e as ˆ ˜ Hsp( ) = ˆ U†( , 0) ˆ Hsp ˆ U( , 0) (5.13) lea ing he simpli ied in e ac ion as ime dependen , while he s a es e ol e o ake in o accoun he phase hey acqui e wi h ¯ H ( ): |˜ Ψ( )⟩=ˆ U†( , 0)|Ψ( )⟩.(5.14) Since we a e in e es ed in he e olu ion o he o bi al s a e o he injec ed elec on, i s spin and hose o he su ounding nuclei, he gene al s a e o he sys em will include hose deg ees o eedom. A gene al s a e can be w i en as a supe posi ion o s a es desc ibed by |Φj, ms,m( )⟩=|Φj( , )⟩ms⊗|m1. . . mNe ⟩(5.15) whose i s e m is he elec on’s s a e ha has e ol ed o e ime and he second pa is a pa icula con igu a ion o Ne nuclei a ound he s a ic do a ime . The ex eme magne iza ion o he sys em would be ob ained o all alues mi ha ing maximal alues all up o down (in ou case, +3 ℏ/ 2o − 3 ℏ/ 2). The e a e only wo ex emal s a es ha can change one ou o he magne ic momen o Ne s a es o change i s pa icula con igu a ion. In he opposi e case, a minimum magne iza ion, one can ind he highes amoun o possible s a es wi h same o al Iz . REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 5. The hype ine in e ac ion 61 Following 5.14, one can w i e an expanded e sion o i , which can be help ul o unde s and bo h he e ms ha ha e al eady come as well as he ones ha will come o wa d. The s a e in he in e ac ion pic u e is |˜ Ψ( )⟩=X j,ms eiR 0εj,ms,m( ′)d ′/ℏ˜cj,ms,m( )|Φj, ms,m( )⟩(5.16) whe e he “pseudoene gy” ha e ol es he phase o each o he s a es has componen s co esponding o bo h he o bi al and hype ine (O e hause in his case, since i only in ol es he z componen ) in e ac ion. I s sepa a ion based on he pa ame e s o e which i depends can be w i en as εj,ms,m( ) = ε0,SO(j, ms, ) + εo (ms,m, )(5.17) whe e he i s e m is he diagonal e m o he uni a y ime e olu ion ob ained a e changing he basis wi h he ma ix in 5.11 co esponding o he e ol ing s a e | Φ( ) ⟩ . The second e m, hough, in ol es applying he O e hause e m o he hype ine in e ac ion o he spin s a e o he basis elemen s as: εo =⟨Φj, ms,m( )|ˆ Sz  Ne X n=1 Anδ( − n)ˆ Iz n |Φj, ms,m( )⟩= =ms·  Ne X n=1 φn( )·mn , (5.18) whe e he e m ha weigh s he sum inside he pa en hesis is φj,n( ) = An⟨Φj( )|δ( − n)|Φj( )⟩(5.19) Following [ 70 ], one can calcula e he alue o he wo componen s ha de ine he hype ine in e ac ion’s s eng h be ween he elec on and a pa icula nucleus: An=4µ0µB 3 µI,n In (5.20) ⟨Φj( )|δ( − n)|Φj( )⟩=ZΦ∗ j( , )δ( − n)Φj( , )d3 (5.21) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 62 5.1. App oxima ed exp essions whe e n e e s o he posi ion and species o he nucleus. Following 2.1, he ampli ude o he elec on wa e unc ion a he posi ion o he pa icula nucleus de e mines he s eng h o he hype ine in e ac ion, which can be pa ame ized wi h a (in gene al species-dependen ) cons an ηn such ha ⟨Φj( )|δ( − n)|Φj( )⟩=ηnZϕ∗ j( , )δ( − n)ϕj( , )d3 =ηn|ϕ( n)|2.(5.22) Wi h his de ini ion o he weigh inside he sum in 5.18, le us de ine a magne iza ion numbe as ⟨M( )⟩= Ne X n=1 φn( )·mn.(5.23) 0 50 100 150 200 250 300 -2 -1 0 1 2 3 Figu e 5.1: Weigh s inside he magne iza ion ac o measu ed in he cen e o he s a ic do . The wo lines indica e each an elec onic e olu ion: he solid line o he ini ially mo ing elec on and he dashed o he ini ially s a ic one. The dash and do ed plo is a scaled down pseudo-ene gy gap be ween he mo ing and he s a ic s a e, which is ele an o he e olu ion o he elec on-nucleus spin sys em e olu ion. REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 5. The hype ine in e ac ion 63 Taking in o accoun he alues o abundance, nuclea magne ic momen and con inemen cons an s ound in Table 1 in [ 70 ], one can calcula e a weigh ed a e age o he alue o An wi h a alue o Aa = 1 . 73 · 10 −8meV . Figu e 5.1 uses his a e aged alue o show he s eng h o he weigh s inside he mean magne iza ion’s sum o di e en elec onic e olu ions. The du a ion o he in e ac ion is limi ed in his case such ha he second mo ing minimum is in he ini ial posi ion o he i s one a ime = T . As a e e ence, his calcula ion has been done o a nucleus ound in he middle o he s a ic do , which can be conside ed o be an uppe limi o hese alues. Rega ding he magne ic momen dis ibu ion wi hou any ex e nal magne ic ield, one can ind an expec ed alue o ⟨PNe n=1 ·mn⟩ = 0, bu a non-ze o dis ibu ion wid h o ⟨PNe n=1 m2 n⟩ ∝ qNe . This gi es a good measu e o he limi o look a while a ying he mean magne iza ion when e ol ing he s a es. One has o ake in o accoun ha his in e ac ion couples di e en o bi al s a es inside each elec onic spin subpace oo. To measu e he p obabili y o his e ec o happen, one has o compa e φjj′,n( ) = 4µ0µB 3 µI,n In ηnϕ∗ j( n, )ϕj′( n, )(5.24) o he ene gy gap be ween hese wo o bi al s a es εm−εs . Taking in o accoun an app oxima e numbe o qNe ∼ 10 3 nuclei ha can c ea e a magne ic momen di e ence, one can compa e |φjj′,n ( ) |· 10 3 o he ene gy gap ha i mus o e come o popula e he o he s a e. F om igu e 5.2 one sees ha he maximum alue o he coupling e m coincides wi h he ene gy gap’s maximum alue and is 10 −6 imes smalle . This gi es an app oxima ed 10 3 ac o by which he ene gy gap is bigge o ou wo s expec a ion. The e o e, his ans e p obabili y is hugely educed and will be igno ed. As a las no e, he non-ze o alue o his coupling e m be o e he middle-poin o he e olu ion comes om he emaining p obabili y o he ou going elec on o emain in he s a ic do , which is unca ed a ime T coming om a cha ge measu emen o see whe e he elec on is a his poin . REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 64 5.1. App oxima ed exp essions 0 50 100 150 200 250 300 0 0.5 1 Figu e 5.2: Absolu e alue o he O e hause e m coupling di e en o bi al e ms measu ed a he cen e o he s a ic do and he scaled-down ene gy gap. Fo he expanded exp ession o 5.13, he iden i y ma ix composed by he basis elemen s can be inse ed be ween he uni a ies and he Hamil onian, a e which he phases a i e as in 5.16: ˆ ˜ Hsp( ) = X j,ms,m j′,m′ s,m′ eiR 0(ε−ε′)d ′/ℏ|Φj, ms,m( )⟩Hj′,m′ s,m′ j,ms,m⟨Φj′, m′ s,m′( )|(5.25) whe e bo h “pseudoene gies” and he ma ix elemen s Hj′,m′ s,m′ j,ms,m depend bo h on all he pa ame e s o e which his sum is done. Looking a he alues ound in igu e 5.1, he spin- lip can be conside ed o be pe u ba i e compa ed o he e ms ha e ol e he elec onic s a e. The e o e, he subspace de ined by he ou possible ime-e ol ed ini ial s a es and hei o hogonal coun e pa s is su icien o desc ibe he necessa y Hilbe space. Since he elec on’s o bi al e olu ion is de ined by he e ms calcula ed p e iously, e ol ing he ini ial s a es | Φ j, ms⟩ is enough o REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 5. The hype ine in e ac ion 65 ha e all he necessa y in o ma ion abou he s eng h o he hype ine in e ac ion be ween he elec on and any nucleus along i s pa h. 5.2 Spin- lip p ocess ime-e olu ion Now ha he in e ac ion’s s eng h be ween a single nucleus and he elec on has been calcula ed, i can be use ul o model he e olu ion o he whole sys em by gene alizing wi h wha would happen o a single nucleus wi h i s en i onmen ’s e ec . The new basis elemen s can be de ined as |ms,m⟩=|ms, mk⟩⊗|m¯ k⟩(5.26) whe e he in e ac ing nucleus’ spin has been sepa a ed om he es o he nuclei. The no a ion o he s a e desc ibing he N−1nuclei ha emain indica es which nucleus is missing and he o iginal con igu a ion om which i comes. In oducing his no a ion o he exp ession in 5.25 one ge s ˆ ˜ Hsp( ) = X χ,χ′ eiR 0(εχ−ε′ χ)d ′/ℏ|ms, mk⟩⊗|m¯ k⟩Hχ,χ′⟨m′ ¯ k|⊗⟨m′ s, m′ k|(5.27) whe e he pa ame e s on he sum ha e changed acco dingly o he new no a ion and summa ized in χ . The e m inside he sum can be summa ized as Hχ,χ′=φj,kδm¯ k,m′ ¯ k(I−S+δms,m′ s+1δmk,m′ k−1+I+S−δms,m′ s−1δmk,m′ k+1)(5.28) whe e I± ( mk ) = qI(I+ 1) −mk(mk±1) and a simila exp ession o he elec on spin S± ( ms ) = qS(S+ 1) −ms(ms±1) . The i s del a unc ion simpli ies he sum in 5.27 since | m ¯ k⟩ = | m ′ ¯ k⟩ is a necessa y condi ion o he Hamil onian elemen . Also, ollowing he exp ession o S± , i s alues a e always 1as long as hey appea . Taking in o accoun hese simpli ica ions, one can ew i e 5.27 as ˆ ˜ Hsp( ) = X mk,m˜φ↑↓ kI−| ↑, mk⟩⊗|m¯ k⟩⟨m¯ k|⊗⟨↓, mk+ 1|+ + ˜φ↓↑ kI+| ↓, mk⟩⊗|m¯ k⟩⟨m¯ k|⊗⟨↑, mk−1| (5.29) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 72 6.1. The coulomb in e ac ion 6.1 The coulomb in e ac ion The ene gy gap be ween he wo o al spin subspaces comes om he selec i i y ha coulomb in e ac ion imposes on spins added o he ac ha elec ons a e indis inguishable e mions. All he in ol ed calcula ions o de e mine he s eng h o his in e ac ion can be sepa a ed in e ms ha do no necessa ily imply any o hose conside a ions. Fo wo dis inguishable cha ge dis ibu ions wi h an o al elec ic cha ge o e , he Coulomb in e ac ion eads Cijlm =kee2Zd 2 1d 2 2Ψ∗ i( 1)Ψ∗ j( 2)1 | 1− 2|Ψl( 1)Ψm( 2),(6.1) in eal space, whe e he subindices e e o he label ha each dis inguishable pa icle has. In he case o wo in e ac ing pa icles on wo di e en s a es hose indices can ake wo di e en alues, each e e encing one pa icle in a pa icula s a e. Con enien ly enough, hese indices will be labeled s and m . The e a e N = 2 4 combina ions o which hese wo s a es can ill ou posi ions (in his case, indices). The alue o he in eg al, hough, has a smalle numbe o possible alues due o he symme y on pa icle exchange. I one exchanges  1 and  2 he exp ession emains cons an , which means ha alues can be ca ego ized o he numbe o epea ed indices. In he case o wo pa icles in wo s a es •Cmmmm •Csmmm =Cmsmm =Cmmsm =Csmms •Cssmm =Csmsm =Csmms =Cmssm =Cmsms =Cmmss •Csssm =Cssms =Csmss =Cmsss •Cssss which means ha he e a e only i e alues o he in eg al ha need o be e alua ed in o de o ha e all possible in o ma ion in his subspace. In he case o using he same po en ial s eng hs, he i s and las e m a e also he same. Once REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 6. Two-pa icle in e ac ion 73 he e ms a e calcula ed, one can c ea e a i s app oach o he Hamil onian c ea ed by hese pa icles as Hdis =X ij |i0⟩εij⟨j0|+X ijlm |i0j0⟩Cijlm⟨l0m0|.(6.2) The ec o s deno e |i0j0⟩=|i0( 1)⟩⊗|j0( 2)⟩ ⟨l0m0|=⟨l0( 1)|⊗⟨m0( 2)|. (6.3) The i s sum co esponds o he single-pa icle ene gies whose alues a e calcula ed in chap e 4. The wo-pa icle s a es a e, in his case, ba e mul iplica ions be ween single-pa icle s a es, since hey a e dis inguishable. Taking in o accoun now ha e mions ha e an isymme ic wa e unc ions ha ca y spin in o ma ion, one can iden i y he symme ic spa ial combina ions be ween exchanged pai s as belonging o he an isymme ic spin con igu a ion, i.e., he single . Following he same logic, he odd combina ion o spa ial s a es co esponds o he iple ha is sha ed by s a es ha ing bo h spins up and bo h down. On he o he hand, since he doubly occupied s a es a e coupled o he single s a e de ined by he wo eigens a es in he wo di e en do s, hey mus ha e an an isymme ic spin con igu a ion. All hese s a es can be ep esen ed as •Single : |s↑s↓⟩ ≡ |s0s0⟩⊗ 1 √2(| ↑↓⟩−| ↓↑⟩)(6.4) |m↑m↓⟩ ≡ |m0m0⟩⊗ 1 √2(| ↑↓⟩−| ↓↑⟩)(6.5) 1 √2|m↑s↓⟩−|m↓s↑⟩≡Nsm|s0m0⟩+|m0s0⟩⊗1 √2(| ↑↓⟩−| ↓↑⟩)(6.6) •T iple : 1 √2|m↑s↓⟩+|m↓s↑⟩≡Nsm|s0m0⟩−|m0s0⟩⊗1 √2(| ↑↓⟩+| ↓↑⟩)(6.7) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 74 6.1. The coulomb in e ac ion The o al spin o he sys em is ze o o he single and one o he iple , and since he e is no spin changing in e ac ion in he Hamil onian, he popula ions in he single and iple subspaces do no change. I is he ene gy di e ence be ween he wo las s a es ha mo i a es ou esea ch. One can ini ialize a s a e |m↑s↓⟩ by picking up an elec on wi h a spin in a pa icula di ec ion wi h a su ace acous ic wa e and mo e i nea by a s a ic do in which he e is an elec on wi h he opposi e spin di ec ion. Thei in e ac ion will c ea e a o a ion in he subspace c ea ed by he iple and he single s a es o Sz = 0 which will depend on he ene gy gap be ween hese wo s a es. I he e olu ion o he sys em can be kep be ween he wo lowes ene gy s a es, he inal s a e will look like |Ψ(T)⟩=1 √2       0 0 1 e−iRT 0J( ′)d ′       (6.8) up o a global phase, whe e J ( ) = ( ET ( ) −ES ( )) /ℏ and he e o e he gap be ween he wo subspaces along he ajec o y de ines he e olu ion i he e is no ans e o exci ed s a es. The o a ion may lea e he s a e wi h he spins lipped i he in eg al o he ene gy gap is π , which is commonly known as a SWAP ga e. I ins ead, he solu ion o he in eg al is hal ha , π/ 2, he ga e is e e ed o as √SWAP and he inal s a e is pa ially en angled. The √SWAP ga e ( oge he wi h he single-qubi ga es) is uni e sal, and i can be used o implemen he s anda d wo-qubi quan um ga es necessa y o quan um compu a ion [ 8 ]. Bu ka d e al. [ 14 ] p oposed a model o his p oblem o calcula e he single- iple ene gy gap depending on ex e nal ields using he Hei he -London and Hund-Mulliken echniques. The wo-pa icle in e ac ion is a pa icula ly icky in eg al o do o gene ic dis ibu ions, mainly due o he di icul y ela ed o he compu a ion o he alue a | 2− 1| = 0. The gaussian unc ions come in handy in his si ua ion, allowing a ans o ma ion o a iables such ha REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 6. Two-pa icle in e ac ion 75  += ( 1+ 2)/2  −= ( 1− 2)/2(6.9) which allows he sepa a ion o he in eg al in wo pa s, i s in eg a ing o e  + conside ing  − cons an , which eads Cijlm =K+2π qde (A+)Zd −exp1 2B+⊤A+−1B++C+1 qx−2+y−2= =K−Zd −exp−x−y−A− x− y−!+B−⊤ x− y−!1 qx−2+y−2 (6.10) Fo mo e in o ma ion abou he cons an and a iables in he in eg al, check Appendix A. Changing o cylind ical coo dina es, he Jacobian unc ion ha en e s in he in eg al is | −| , which akes ca e o he di iding dis ance. The emaining in eg al o e he dis ance can be done by hand, he e o e lea ing he in eg al o e he azimu al angle as a nume ically sol able p oblem, which can be done eally as Cijlm =K−Zd dθexp−Aθ 2+Bθ =K−√π 2Z2π 0dθexpBθ2/4Aθ √Aθ (6.11) Once he calcula ion o bo h he single- and wo-pa icle in e ac ions a e done, one can look o he ene gy gap be ween he wo lowes ene gy eigens a es, which is he in e ac ion s eng h. I his is done o he whole du a ion o he p ocess, he in eg al o e ha gi es he angle a which he inal s a e has e ol ed. 6.2 Fu he co ec ions Al hough using Gaussian unc ions as a ep esen a ion o he in-plane wa e unc ions can be a use ul app oach o modelling some dependencies, in ou case he non- adiaba ic ime-e olu ion and he icini y o he po en ials a some poin s equi e u he s eps o be able o p ope ly desc ibe he sys em’s e olu ion. In o de o do his, we would like o know he eigens a es o he sys em including he Coulomb REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 76 6.2. Fu he co ec ions in e ac ion so ha we can measu e he s eng h o bo h he ene gies and non- adiaba ic e ms ha may cause he sys em o be exci ed. The e o e, we wan o ha e a desc ip ion (as analy ic as possible) o he Coulomb in e ac ion in 2D aking in o accoun ha ou s a es li e in a 3D space. The e a e app oaches ha ha e used less poin s in k-space o de ine he eigens a es o he sys em [ 50 ], so he desc ip ion o he in e ac ion should be gi en also in his space. 6.2.1 Real space exp essions The eal space in eg al exp ession o he wo-pa icle in e ac ion be ween he wo-pa icle s a es | Ψ i⟩ and | Ψ j⟩ is de ined as Cij ≡ ⟨Ψi|C|Ψj⟩=Z∞ −∞ Ψ∗ i( 1, 2)Ψj( 1, 2) q(x2−x1)2+ (y2−y1)2+ (z2−z1)2d3 1d3 2.(6.12) In ou pa icula case, due o he ene gy le els in he z-di ec ion being much u he apa om each o he han he ones in x and y, we can w i e any ele an s a e o he e olu ion o he sys em as: ⟨ |Ψi⟩= Ψxy i(x1, x2, y1, y2)Ψz i(z1, z2)(6.13) as i was done o one-pa icle s a es in 2.2, whe e he z-di ec ion p obabili y dis ibu ion unc ion can be modeled as a gaussian unc ion wi h a ∆ z wid h o bo h pa icles: Ψz i=s1 2π∆−1 ze−(z2 1+z2 2)/4∆2 z.(6.14) so we can in oduce i in he exp ession ha we wan o e alua e (minus he no malizing cons an s w i en in he p e ious exp ession): Cij =Z∞ −∞ Ψxy i ∗Ψxy je−(z2 1+z2 2)/2∆2 z q(x2−x1)2+ (y2−y1)2+ (z2−z1)2d3 1d3 2=(6.15) =Z∞ −∞ Ψxy i ∗Ψxy jdx1dx2dy1dy2Z∞ −∞ e−(z12+z22)/2∆2 z q 2+ (z2−z1)2dz1dz2(6.16) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 6. Two-pa icle in e ac ion 77 wi h 2≡ (x1, x2, y1, y2)2= (x2−x1)2+ (y2−y1)2.(6.17) To simpli y he in eg al o e he z-coo dina es u he , le us de ine gxy ≡g(x1, x2, y1, y2) = Z∞ −∞ e−(z12+z22)/2∆2 z q 2+ (z2−z1)2dz1dz2.(6.18) To p oceed wi h he in eg al i is con enien o change a iables o zs = z1 + z2 and zd = z1−z2 which lea es i as: gxy = 2Z∞ −∞ e−(zs2+zd2)/4∆2 z q 2+z2 d dzsdzd= 4∆z√πZ∞ −∞ e−zd2/4∆2 z q 2+z2 d dzd= = 4∆z√πZ∞ −∞ e−z′ d 2 q ′2+z′ d 2dz′ d (6.19) and he e ′ = / 2∆ z . Now we can use he ac ha he unc ion inside he in eg al is e en and in eg a ed o e an e en ange a ound 0: gxy = 8∆z√πZ∞ 0 e−z′ d 2 q ′2+z′ d 2dz′ d.(6.20) This in eg al can be ound in [ 35 ], page 367, in eg al numbe 3.462 (25). The solu ion ha is gi en comes om a sequence o a iable changes. Fi s , le ’s call z = z′ d/ ′ such ha gxy = 8∆z√πZ∞ 0 e− ′2z 2 q1 + z 2dz .(6.21) The e a e wo ways o p oceeding he e. I he a iable change z = sinh ( )is con- side ed: gxy = 8∆z√πZ∞ 0e− ′2sinh2 d (6.22) which is a p e y compac o m o he in eg al and, mos impo an ly, has a mono onic endency, which makes he es ima ion o he e o o he app oxima ed REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 78 6.2. Fu he co ec ions in eg al easy e alua ing he alue o he unc ion. The o he possible change o a iable is = z2 , which lea es equa ion 6.21 as: gxy = 4∆z√πZ∞ 0 e− ′2 √1 + 2√ d . (6.23) This also appea s in [ 35 ], page 1023, in eg al numbe 9.211 (4). I is also simila o he exp ession in [ 50 ], Appendix B, equa ion B4. The main di e ence wi h ha exp ession is ha hey appea o ha e i in he denomina o o he Coulomb e m and he cons an s in he unc ion a e di e en . No ice, o example, ha hei α = − 1 / 2 alue is nega i e while in [ 35 ] he exp ession is gi en only o α > 0. This ime, he exp ession ha he book gi es can be ela ed o Bessel unc ions such ha : gxy = Ψ(1 2,1, ′2) = Γ(0)Γ(1) Γ(1/2) e ′2/2J0(i ′2/2) (6.24) combining equa ions 9.210 (2) and 9.215 (2) om he same sec ion. 6.2.2 Momen um space exp essions Ano he way o app oaching his in eg al is o change he space in which i is de ined such ha he ma ix elemen s ha need o be compu ed o sol e he eigen alue p oblem a e easie o compu e. Le us de ine ou new wo-pa icle basis as {|k1k2⟩} , whe e ⟨ |k1k2⟩=e−i(k1· 1+k2· 2)(6.25) desc ibe he h ee-dimensional o m o pa icle 1 ha ing momen um k1 and k2 in he case o pa icle 2. Inse ing he uni a y de ined by his basis in he Coulomb in e ac ion ope a o ˆ Ck=X k1k2k′ 1k′ 2|k′ 1k′ 2⟩Ck1k2k′ 1k′ 2⟨k1k2|(6.26) asks o he Fou ie ans o m o he Coulomb in e ac ion o a pa icula com- bina ion o momen a: REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 6. Two-pa icle in e ac ion 79 Ck1k2k′ 1k′ 2=Ze−i(k1−k′ 1) 1−i(k2−k′ 2) 2 | 2− 1|d3 1d3 2.(6.27) This exp ession ge s simpli ied by pe o ming a change o a iable such ha 2− 1= d 2+ 1= s (6.28) lea ing (6.27) as Ck1k2k′ 1k′ 2=1 2Ze−i[(k1−k′ 1)+(k2−k′ 2)] s/2d3 sZe−i[(k1−k′ 1)−(k2−k′ 2)] d/2 d d3 d= =δh(k1−k′ 1)+(k2−k′ 2)i 2Ze−i[(k1−k′ 1)−(k2−k′ 2)] d/2 d d3 d (6.29) whe e he 1 / 2 ac o comes as he Jacobian o he a iable change and he del a unc ion in he las exp ession ensu es he momen um conse a ion in all di ec ions. De ining q= k1−k′ 1 , one can w i e he emaining in eg al as he limi o he Yukawa po en ial wi h he scaling pa ame e going o ze o: Ze−iq· d d d3 d= lim λ→0Ze−λ de−iq· d d d3 d.(6.30) Changing o sphe ical coo dina es and doing he usual a iable changes, one can ob ain an exp ession depending only on he dis ance d ha eads lim λ→0Ze−λ de−iq· d d d3 d= lim λ→0 2π qi Z∞ 0e(iq−λ) d−e−(iq+λ) dd d(6.31) which can be in eg a ed and e alua ed lea ing a inal exp ession lim λ→0Ze−λ de−iq· d d d3 d= lim λ→0 2π qi 2qi q2+λ2=4π q2(6.32) which depends only on he modulus o he momen um exchanged be ween he in ol ed plane wa es. Going back o wha we in end o ob ain, one can w i e (6.26) as ˆ Ck=X k1k2q|k1−q k2+q⟩2π q2⟨k1k2|.(6.33) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 80 6.2. Fu he co ec ions Simila ly o wha was done in sec ion 6.2.1, in o de o ob ain an exp ession ha can be used o a wo-dimensional p obabili y dis ibu ion one needs o inse he in o ma ion o he hi d di ec ion in his las equa ion. B acke ing i wi h he z componen ha is common o he wo pa icles ˆ C2D,k =⟨Ψz|ˆ Ck|Ψz⟩(6.34) one can ob ain such exp ession. Fi s , one mus ind he Fou ie ans o med o m o he z componen o he s a es, since i is equi ed by he a iable change. Following he con en ion ha ma lab’s ’ ’ unc ion gi es us, le us de ine ou no malized z componen o he wa e unc ion as Ψkz=s2 π∆ze−∆2 z(k2 z1+k2 z2),(6.35) which can be inse ed in he p e ious exp ession gi ing ˆ C2D,k = 4∆2 zZe−∆2 z[(k1z+qz)2+(k2z−qz)2]e−∆2 z(k2 1z+k2 2z) q2 x+q2 y+q2 z dk1zdk2zdqz= = 4∆2 zZe−2∆2 z[k2 1z+k2 2z+qz(k1z−k2z)+q2 z] q2 x+q2 y+q2 z dk1zdk2zdqz= = 4∆z π 2Ze−2∆2 z[k2 2z−k2zqz+q2 z−q2 z/4] q2 x+q2 y+q2 z dk2zdqz= = 2πZ+∞ −∞ e−∆2 zq2 z q2 x+q2 y+q2 z dqz (6.36) whe e he simpli ica ions ha e been made using he in eg al numbe 3.323 (2) in page 339 on [ 35 ]. The las e m ha has been w i en in he simpli ica ion appea s as pa o an al e na i e exp ession o he e o unc ion (’e ’), in eg al numbe 8.252 (4) in page 898 o he same e e ence: ˆ C2D,k = 2π2e∆2 zq2 xy qxy [1 −e (∆zqxy)] (6.37) whe e qxy = qq2 x+q2 y is he no m o he in-plane exchanged momen um be ween he co esponding plane wa es. This exp ession is ill-de ined o qxy = 0, whe e he limi goes o + ∞ . REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 6. Two-pa icle in e ac ion 81 6.3 Nume ical conside a ions and esul s Ha ing he explici exp essions o he wo-dimensional Coulomb ene gy ha he pa icles will ha e de ines he nume ical Hamil onian’s en ances o he g id poin s ha one is conside ing. Being a wo-pa icle s a e, he o al amoun o poin s ha de ine a eal-space ep esen a ion o he wa e unc ion is N4 , which in ou case has a alue o e 4 million. The opea o s, mo eo e , would ha e o be N4×N4 objec s. I is clea ha some dimensionali y educ ion has o be done o his o he wise un ac able p oblem. 0 20 40 60 80 100 120 140 160 0 0.02 0.04 0.06 0.08 0.1 Figu e 6.1: Single - iple ene gy gap o he wo-pa icle eigens a es o he Coulomb in e ac ing Hamil onian wi h wo po en ials. He e is whe e he desc ip ion in he momen um space can simpli y hese issues. Since he eal-space ansla ion co esponds o he mul iplica ion o a plan wa e in k-space one can de ine he mo ing po en ial by he necessa y g idpoin s a ound | k | = 0 and hen ansla e i as needed, allowing he use o a lowe numbe o g id REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 88 A. Coulomb e m calcula ion whe e K+=AiAjAlAmeγx+γy+γxy A+=−2 αxαxy αxy αy! B+= 2 βxx−+βxyy−−˜αx βxyx−+βyy−−˜αy! C+=αxx−2+αyy−2+ 2αxyx−y−−2˜ βxx−−2˜ βyy− (A.5) and Ai e e s o he no malizing ac o o s a e i . No e ha bo h B+ and C+ ha e e ms in ol ing x− and y− , which a e being in eg a ed a e wa ds. The cons an s αu ,..., ˜ βu a e gi en below in Eqs. (A.10-A.15). Fo he momen , one can in eg a e o e + : Cijlm =K+2π qde (A+)Zd −exp1 2B+⊤A+−1B++C+1 qx−2+y−2 =K−Zd −exp−x−y−A− x− y−!+B−⊤ x− y−!1 qx−2+y−2, (A.6) whe e he new pa ame e s appea ing on he igh hand side a de ined as K−=K+2π qde (A+)expαy˜α2 x+αy˜α2 y−2αxy ˜αx˜αy α2 xy −αxαy A−=1 αxαy−αxy2 α′ xα′ xy α′ xy α′ y!+ αxαxy αxy αy!= α′′ xα′′ xy α′′ xy α′′ y! α′ x=αyβx2+αxβxy2−2αxyβxβxy α′ y=αxβy2+αyβxy2−2αxyβyβxy α′ xy =βxy(αyβx+αxβy)−αxy(βxβy+βxy2) B−=−2 αxαy−αxy2 β′ x β′ y!−2 ˜ βx ˜ βy! β′ x=αxy(βx˜αy+βxy ˜αx)−αyβx˜αx−αxβxy ˜αy β′ y=αxy(βxy ˜αy+βy˜αx)−αyβxy ˜αx−αxβy˜αy (A.7) which a e all independen o ( x−, y− ). Rew i ing he emaining 2d in eg al in e ms o cylind ical coo dina es ( , θ ), he exp ession o he Jacobian coincides wi h he denomina o and enables he in eg a ion o e : REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 A. Coulomb e m calcula ion 89 Cijlm =K−Zd dθ exp−(α′′ xcos2θ+α′′ xy2 sin θcosθ+α′′ ysin2θ) 2+ B−(1)cos θ+B−(2)sinθ =K−Zd dθ exp−Aθ 2+Bθ (A.8) whe e Aθ and Bθ jus depend on he angle a iable in oduced by cylind ical coo dina es. A las possible simpli ica ion can be made, lea ing an in eg al ha is nume ically easy o sol e: Cijlm =K−√π 2Z2π 0dθexpBθ2/4Aθ √Aθ(A.9) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 90 A. Coulomb e m calcula ion These a e he exp essions o p e iously used cons an s, all o hem being sums o e indices u∈ {i, j, l, m} αx=X u kux αy=X u kuy αxy =X u kuxy (A.10) αk x=X u kuxxu αk y=X u kuyyu αkx xy =X u kuxyxu αky xy =X u kuxyyu (A.11) γx=X u kuxxu2 γy=X u kuyyu2 γxy =X u kuxyxuyu (A.12) ˜αx=αk x+αky xy ˜αy=αk y+αkx xy (A.13) βx=kix +kmx −kjx −klx βy=kiy +kmy −kjy −kly βxy =kixy +kmxy −kjxy −klxy (A.14) βk x=kixxi+kmxxm−kjxxj−klxxl βk y=kiyyi+kmyym−kjyyj−klyyl βkx xy =kixyxi+kmxyxm−kjxyxj−klxyxk βky xy =kixyyi+kmxyym−kjxyyj−klxyyk ˜ βx=βk x+βky xy ˜ βy=βk y+βkx xy (A.15) REGISTRO TELEMÁTICO Sa e en E egis o O oko a / Regis o Gene al de En adas 02/09/2025 13:23 EHU2025E042506 Bibliog aphy [1] Rajee Acha ya, Dmi y A. Abanin, Laleh Aghababaie-Beni, Igo Aleine , T ond I. Ande sen, Ma kus Ansmann, F ank A u e, Kunal A ya, Ab aham As aw, Niki a As akhan se , Juan A alaya, Ryan Babbush, Da e Bacon, B ian Balla d, Joseph C. 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