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. Ba din, Johannes Bausch, And eas Beng sson,
Alexande Bilmes, Sam Blackwell, Se gio Boixo, Gina Bo oli, Alexand e
Bou assa, Jenna Bo ai d, Leon B ill, Michael B ough on, Da id A. B owne,
B e Buchea, Bob B. Buckley, Da id A. Buell, Tim Bu ge , B ian Bu ke ,
Nicholas Bushnell, An hony Cab e a, Juan Campe o, Hung-Shen Chang,
Yu Chen, Zijun Chen, Ben Chia o, Desmond Chik, Cha ina Chou, Jahan
Claes, Agne a Y. Cleland, Josh Cogan, Robe o Collins, Paul Conne , William
Cou ney, Alexande L. C ook, Ben Cu in, Sayan Das, Alex Da ies, Lau a
De Lo enzo, D ip o M. Deb oy, Sean Demu a, Michel De o e , Agus in Di Paolo,
Paul Donohoe, Ilya D ozdo , And ew Dunswo h, Clin Ea le, Thomas Edlich,
Alec Eickbusch, A i Moshe Elbag, Mahmoud Elzouka, Ca he ine E ickson,
La a Fao o, Edwa d Fa hi, Vinicius S. Fe ei a, Leslie Flo es Bu gos, Eb ahim
Fo a i, Aus in G. Fowle , B ooks Foxen, Suhas Ganjam, Gonzalo Ga cia,
Robe Gasca, Élie Genois, William Giang, C aig Gidney, Da Gilboa, Raja
Gosula, Alejand o G ajales Dau, Die ich G aumann, Alex G eene, Jona han A.
G oss, S e e Habegge , John Hall, Michael C. Hamil on, Monica Hansen,
Ma hew P. Ha igan, Sean D. Ha ing on, F ancisco J. H. He as, S ephen
Heslin, Paula Heu, Osca Higgo , Go don Hill, Je emy Hil on, Geo ge Holland,
Sab ina Hong, Hsin-Yuan Huang, Ashley Hu , William J. Huggins, Le B.
Io e, Se gei V. Isako , Jus in I eland, E an Je ey, Zhang Jiang, Cody
Jones, S ephen Jo dan, Chai ali Joshi, Pa ol Juhas, D i Ka i, Hui Kang,
Ami H. Ka amlou, Kos yan yn Kechedzhi, Julian Kelly, T up i Khai e, Tanuj
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
92 Bibliog aphy
Kha a , Mos a a Khez i, Seon Kim, Paul V. Klimo , And ey R. Klo s, B yce
Kob in, Pushmee Kohli, Alexande N. Ko o ko , Fedo Kos i sa, Robin
Ko ha i, Bo isla Kozlo skii, John Ma k K eikebaum, Vladisla D. Ku ilo ich,
Na han Lac oix, Da id Landhuis, Tiano Lange-Dei, B andon W. Langley, Pa el
Lap e , Kim-Ming Lau, Loïck Le Gue el, Jus in Led o d, Joonho Lee, Kenny
Lee, Yu i D. Lensky, Shannon Leon, B ian J. Les e , Wing Yan Li, Yin Li,
Alexande T. Lill, Wayne Liu, William P. Li ings on, Adi ya Locha la, E ik
Luce o, Daniel Lundahl, Aa on Lun , Sid Madhuk, Fionn D. Malone, Ashley
Maloney, Sal a o e Mand à, James Manyika, Leigh S. Ma in, O ion Ma in,
S e en Ma in, Came on Max ield, Ja od R. McClean, Ma McEwen, Seneca
Meeks, An hony Meg an , Xiao Mi, Ke in C. Miao, Amanda Mieszala, Reza
Mola i, Sebas ian Molina, Shi in Mon aze i, Alexis Mo an, Ramis Mo assagh,
Wojciech M uczkiewicz, O e Naaman, Ma hew Neeley, Cha les Neill, Ani
Ne sisyan, Ha mu Ne en, Michael Newman, Jiun How Ng, An hony Nguyen,
Mu ay Nguyen, Chia-Hung Ni, Mu phy Yuezhen Niu, Thomas E. O’B ien,
William D. Oli e , Alex Op emcak, K is o e O osson, And e Pe ukho , Alex
Pizzu o, John Pla , Rebecca Po e , O ion P i cha d, Leonid P. P yadko, Ch is
Quin ana, Ganesh Ramachand an, Ma hew J. Reago , John Redding, Da id M.
Rhodes, Gab ielle Robe s, Elio Rosenbe g, Emma Rosen eld, Ped am
Roushan, Nicholas C. Rubin, Nega Saei, Daniel Sank, Kannan Sanka agoma hi,
Ke in J. Sa zinge , Hen y F. Schu kus, Ch is ophe Schus e , And ew W.
Senio , Michael J. Shea n, Aa on Sho e , Noah Shu y, Vladimi Sh a s,
Sh addha Singh, Volodymy Si ak, Jind a Sk uzny, Spence Small, Vadim
Smelyanskiy, W. Cla ke Smi h, Rolando D. Somma, So ia Sp inge , Geo ge
S e ling, Doug S ain, Jo dan Sucha d, Aa on Szasz, Alex Sz ein, Douglas
Tho , Al edo To es, M. Me To unbalci, Abee Vaishna , Jus in Va gas,
Se gey Vdo iche , Gui e Vidal, Benjamin Villalonga, Ca he ine Vollg a
Heidweille , S e en Wal man, Shannon X. Wang, B ayden Wa e, Ka e Webe ,
T a is Weidel, Theodo e Whi e, K is i Wong, B yan W. K. Woo, Cheng
Xing, Z. Jamie Yao, Ping Yeh, Bicheng Ying, Juhwan Yoo, Nou eldin Yos i,
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 93
G ayson Young, Adam Zalcman, Yaxing Zhang, Ning eng Zhu, and Nicholas
Zob is . Quan um e o co ec ion below he su ace code h eshold. Na u e,
638(8052):920–926, Decembe 2024.
[2]
Ma in Hayhu s Appel, Alexande Gho bal, Noah Sho e , Leon Zapo ski,
San anu Manna, Saimon Filipe Co e da Sil a, U s Haeusle , Clai e Le Gall,
A mando Ras elli, Do ian A. Ganglo , and Me e A a ü e. A many-body
quan um egis e o a spin qubi . Na u e Physics, Janua y 2025.
[3]
F ank A u e, Kunal A ya, Ryan Babbush, Da e Bacon, Joseph C. Ba din,
Rami Ba ends, Rupak Biswas, Se gio Boixo, Fe nando G. S. L. B andao,
Da id A. Buell, B ian Bu ke , Yu Chen, Zijun Chen, Ben Chia o, Robe o
Collins, William Cou ney, And ew Dunswo h, Edwa d Fa hi, B ooks Foxen,
Aus in Fowle , C aig Gidney, Ma issa Gius ina, Rob G a , Kei h Gue in,
S e e Habegge , Ma hew P. Ha igan, Michael J. Ha mann, Alan Ho,
Ma kus Ho mann, T en Huang, T a is S. Humble, Se gei V. Isako , E an
Je ey, Zhang Jiang, D i Ka i, Kos yan yn Kechedzhi, Julian Kelly, Paul V.
Klimo , Se gey Knysh, Alexande Ko o ko , Fedo Kos i sa, Da id Landhuis,
Mike Lindma k, E ik Luce o, Dmi y Lyakh, Sal a o e Mand à, Ja od R.
McClean, Ma hew McEwen, An hony Meg an , Xiao Mi, K is el Michielsen,
Masoud Mohseni, Josh Mu us, O e Naaman, Ma hew Neeley, Cha les Neill,
Mu phy Yuezhen Niu, E ic Os by, And e Pe ukho , John C. Pla , Ch is
Quin ana, Eleano G. Rie el, Ped am Roushan, Nicholas C. Rubin, Daniel
Sank, Ke in J. Sa zinge , Vadim Smelyanskiy, Ke in J. Sung, Ma hew D.
T e i hick, Ami Vainsenche , Benjamin Villalonga, Theodo e Whi e, Z. Jamie
Yao, Ping Yeh, Adam Zalcman, Ha mu Ne en, and John M. Ma inis.
Quan um sup emacy using a p og ammable supe conduc ing p ocesso . Na u e,
574(7779):505–510, Oc obe 2019.
[4]
R. C. Ashoo i. Elec ons in a i icial a oms. Na u e, 379(6564):413–419,
Feb ua y 1996.
REGISTRO TELEMÁTICO
Sa e en E egis o
O oko a / Regis o Gene al
de En adas
02/09/2025 13:23
EHU2025E042506

94 Bibliog aphy
[5]
C. H. W. Ba nes, J. M. Shil on, and A. M. Robinson. Quan um compu a ion
using elec ons apped by su ace acous ic wa es. Physical Re iew B,
62(12):8410–8419, Sep embe 2000.
[6]
Cha les H. Benne , Gilles B assa d, Claude C épeau, Richa d Jozsa, Ashe
Pe es, and William K. Woo e s. Telepo ing an unknown quan um s a e ia
dual classical and eins ein-podolsky- osen channels. Physical Re iew Le e s,
70(13):1895–1899, Ma ch 1993.
[7]
B. Be and, S. He melin, S. Takada, M. Yamamo o, S. Ta ucha, A. Ludwig,
A. D. Wieck, C. Bäue le, and T. Meunie . Fas spin in o ma ion ans e be-
ween dis an quan um do s using indi idual elec ons. Na u e Nano echnology,
11(8):672–676, Aug 2016.
[8]
R. C. Bialczak, M. Ansmann, M. Ho heinz, E. Luce o, M. Neeley, A. D.
O’Connell, D. Sank, H. Wang, J. Wenne , M. S e en, A. N. Cleland, and
J. M. Ma inis. Quan um p ocess omog aphy o a uni e sal en angling ga e
implemen ed wi h josephson phase qubi s. Na u e Physics, 6(6):409–413, Ap il
2010.
[9]
S.M Blinde . The hype ine in e ac ion hamil onian. Jou nal o Molecula
Spec oscopy, 5(1):17–23, 1961.
[10]
Dole Blu s ein, Simon J. E e ed, Alexand a A. Geim, Sophie H. Li, Hengyun
Zhou, Tom Mano i z, Sepeh Ebadi, Madelyn Cain, Ma cin Kalinowski,
Dominik Hanglei e , J. Pablo Bonilla A aides, Nishad Maska a, I is Cong,
Xun Gao, Ped o Sales Rod iguez, Thomas Ka olyshyn, Giulia Semeghini,
Michael J. Gullans, Ma kus G eine , Vladan Vule ić, and Mikhail D. Lukin.
Logical quan um p ocesso based on econ igu able a om a ays. Na u e,
626(7997):58–65, Decembe 2023.
[11]
M. Bo n and V. Fock. Beweis des adiaba ensa zes. Z. Physik, 51(3–4):165–180,
Ma ch 1928.
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 95
[12]
Colin D. B uzewicz, John Chia e ini, Robe McConnell, and Je emy M. Sage.
T apped-ion quan um compu ing: P og ess and challenges. Applied Physics
Re iews, 6(2):021314, 05 2019.
[13]
Guido Bu ka d, Thaddeus D. Ladd, And ew Pan, John M. Nichol, and Jason R.
Pe a. Semiconduc o spin qubi s. Re . Mod. Phys., 95:025003, Jun 2023.
[14]
Guido Bu ka d, Daniel Loss, and Da id P. DiVincenzo. Coupled quan um do s
as quan um ga es. Physical Re iew B, 59(3):2070–2078, Janua y 1999.
[15]
Yu. A. Bychko and É. I. Rashba. P ope ies o a 2D elec on gas wi h li ed
spec al degene acy. So ie Jou nal o Expe imen al and Theo e ical Physics
Le e s, 39:78, Janua y 1984.
[16]
Ch is ophe Bäue le, D Ch is ian Gla li, T is an Meunie , Fabien Po ie ,
Pa ice Roche, P eden Roulleau, Shin a o Takada, and Xa ie Wain al.
Cohe en con ol o single elec ons: a e iew o cu en p og ess. Repo s on
P og ess in Physics, 81(5):056503, ap 2018.
[17]
Ve onica Ce le i, W A Coish, Oli e Gywa , and Daniel Loss. Recipes o
spin-based quan um compu ing. Nano echnology, 16(4):R27–R49, Feb ua y
2005.
[18]
J. I. Ci ac and P. Zolle . Quan um compu a ions wi h cold apped ions.
Physical Re iew Le e s, 74(20):4091–4094, May 1995.
[19]
Daniel Compa a . Gene al condi ions o quan um adiaba ic e olu ion. Phys.
Re . A, 80:012106, Jul 2009.
[20]
D. G. Co y, M. D. P ice, W. Maas, E. Knill, R. La lamme, W. H. Zu ek, T. F.
Ha el, and S. S. Soma oo. Expe imen al quan um e o co ec ion. Physical
Re iew Le e s, 81(10):2152–2155, Sep embe 1998.
REGISTRO TELEMÁTICO
Sa e en E egis o
O oko a / Regis o Gene al
de En adas
02/09/2025 13:23
EHU2025E042506
96 Bibliog aphy
[21]
Pe Delsing, And ew N Cleland, Ma in J A Schue z, Johannes Knö ze , Géza
Giedke, J Ignacio Ci ac, Ka ik S ini asan, Ma celo Wu, K ishna Coimba o e
Bal am, Ch is ophe Bäue le, T is an Meunie , Ch is ophe J B Fo d, Paulo V
San os, Edga Ce da-Méndez, Hailin Wang, Hube J K enne , Emeline D S
Nys en, Ma hias Weiß, Geo R Nash, Lau a The ena d, Ca he ine Gou don,
Pauline Ro illain, Max Ma angolo, Jean-Y es Duquesne, Ge ha d Fische aue ,
We ne Ruile, Alexande Reine , Ben Paschke, Dmy o Denysenko, Di k
Volkme , Achim Wix o h, Hen ik B uus, Ma in Wiklund, Julien Reboud,
Jona han M Coope , YongQing Fu, Manuel S B ugge , Flo ian Reh eld , and
Ch is oph Wes e hausen. The 2019 su ace acous ic wa es oadmap. Jou nal
o Physics D: Applied Physics, 52(35):353001, July 2019.
[22]
Sujay B. Desai, Su abhi R. Madh apa hy, Angada B. Sachid, Juan Pablo
Llinas, Qingxiao Wang, Geun Ho Ahn, G ego y Pi ne , Moon J. Kim, Je ey
Boko , Chenming Hu, H.-S. Philip Wong, and Ali Ja ey. MoS
2
ansis o s wi h
1-nanome e ga e leng hs. Science, 354(6308):99–102, Oc obe 2016.
[23]
Flo ian De wile , Jiyong Fu, Shawn Mack, Pi min J. Weigele, J. Ca los Egues,
Da id D. Awschalom, and Dominik M. Zumbühl. S e chable pe sis en spin
helices in gaas quan um wells. Phys. Re . X, 7:031010, Jul 2017.
[24] Da id P. DiVincenzo. Topics in quan um compu e s, 1996.
[25]
Da id P. DiVincenzo. The physical implemen a ion o quan um compu a ion.
Fo sch i e de Physik, 48(9-11):771–783, Sep embe 2000.
[26]
G. D esselhaus. Spin-o bi coupling e ec s in zinc blende s uc u es. Phys.
Re ., 100:580–586, Oc 1955.
[27]
C. Eche e ía-A ondo and E. Ya. She man. Relaxa ion o lying spin qubi s
in quan um wi es by hype ine in e ac ion. Phys. Re . B, 87:081410, Feb 2013.
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 97
[28]
Yinan Fang, Ying-Dan Wang, Rosa io Fazio, and S e ano Chesi. Supe adian -
like dynamics o nuclea spins by nonadiaba ic elec on shu ling. Phys. Re .
B, 103:155301, Ap 2021.
[29]
Edwa d Fa hi, Je ey Golds one, Sam Gu mann, and Michael Sipse . Quan um
compu a ion by adiaba ic e olu ion, 2000.
[30]
D. Fe nández-Fe nández, Yue Ban, and G. Pla e o. Flying spin qubi s
in quan um do a ays d i en by spin-o bi in e ac ion. Quan um, 8:1533,
No embe 2024.
[31]
Richa d P. Feynman. Simula ing physics wi h compu e s. In e na ional Jou nal
o Theo e ical Physics, 21(6-7):467–488, June 1982.
[32]
Pé e Földi, Balázs Molná , Mihály G. Benedic , and F. M. Pee e s. Spin onic
single-qubi ga e based on a quan um ing wi h spin-o bi in e ac ion. Physical
Re iew B, 71(3), Janua y 2005.
[33]
Ch is ophe J. B. Fo d. T anspo ing and manipula ing single elec ons
in su ace-acous ic-wa e minima. physica s a us solidi (b), 254(3):1600658,
Feb ua y 2017.
[34]
Taka umi Fuji a, Timo hy Alexande Baa , Ch is ian Reichl, We ne Wegschei-
de , and Lie en Ma k Koen aad Vande sypen. Cohe en shu le o elec on-spin
s a es. npj Quan um In o ma ion, 3(1), June 2017.
[35]
I.S. G adsh eyn and I.M. Ryzhik. Table o in eg als, se ies and p oduc s.
Academic P ess, Bu ling on, 7 edi ion, 2007.
[36]
Lo K. G o e . A as quan um mechanical algo i hm o da abase sea ch.
In P oceedings o he wen y-eigh h annual ACM symposium on Theo y o
compu ing - STOC '96. ACM P ess, 1996.
[37]
Y.V. Gulyae . Re iew o shea su ace acous ic wa es in solids. IEEE
T ansac ions on Ul asonics, Fe oelec ics and F equency Con ol, 45(4):935–
938, July 1998.
REGISTRO TELEMÁTICO
Sa e en E egis o
O oko a / Regis o Gene al
de En adas
02/09/2025 13:23
EHU2025E042506