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Spin esonance unde opological d i ing ields
To ci e his a icle: A A Reynoso e al 2017 New J. Phys. 19 063010
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Rela ed con en
Popula ion in e sion in a s ongly d i en
wo-le el sys em a a -o esonance
Youyuan Zhang, E ik Lö s ed and Kao u
Yamanouchi
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E ec i e geome ic phases and
opological ansi ions in SO(3) and SU(2)
o a ions
Hen i Saa ikoski, José Pablo Bal anás, J
En ique Vázquez-Lozano e al.
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Mølme –Sø ensen en angling ga e o
ca i y QED sys ems
Hi oki Takahashi, Ped o Ne ado and
Ma hias Kelle
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New J. Phys. 19 (2017)063010 h ps://doi.o g/10.1088/1367-2630/aa723a
PAPER
Spin esonance unde opological d i ing fields
A A Reynoso
1,2
, J P Bal anás
3
, H Saa ikoski
4
, J E Vázquez-Lozano
3
, J Ni a
5
and D F us aglia
3
1
Ins i u o Balsei o and Cen o A ómico Ba iloche, Comisión Nacional deEne gía A ómica, 8400 Ba iloche, A gen ina
2
Consejo Nacional de In es igaciones Cien íficas y Técnicas (CONICET), A gen ina
3
Depa amen o de Física Aplicada II, Uni e sidad de Se illa, E-41012 Se illa, Spain
4
RIKEN Cen e o Eme gen Ma e Science (CEMS), Sai ama 351-0198, Japan
5
Depa men o Ma e ials Science, Tohoku Uni e si y, Sendai 980-8579, Japan
E-mail: [email p o ec ed]
Keywo ds: opological ansi ions, geome ic phases, magne ic esonance, s ong d i ing, wo-le el sys ems, Floque physics
Abs ac
We s udy he dynamics o a localized spin-1/2 d i en by a ime-pe iodic magne ic field ha unde goes
a opological ansi ion. Despi e he s ongly non-adiaba ic e ec s domina ing he spin dynamics, we
find ha he field’s opology appea s clea ly imp in ed in he Floque spin s a es h ough an e ec i e
Be y phase eme ging in he quasiene gy. This has ema kable consequences on he spin esonance
condi ion sugges ing a whole new class o expe imen s o spo opological ansi ions in he dynamics
o spins and o he wo-le el sys ems, om nuclea magne ic esonance o s ongly-d i en supe -
conduc ing qubi s.
1. In oduc ion
The manipula ion o spin s a es by guiding fields in mesoscopic ci cui s o e s se e al possibili ies, om elec on
spin esonance [1] o elec on spin in e e ome y [2], among o he s. This includes he p ospec s o spin con ol
by geome ic means, i.e.,by making use o geome ic phases [3] ha a ise unde he ac ion o magne ic ex u es
wi h a sui able opology [4]. Recen ly, we ha e shown ha guiding magne ic ex u es unde going a opological
ansi ion can imp in his p ope y in complex spin dynamics [5]. Mo e conc e ely, we epo ed elec on
anspo simula ions in spin in e e ome e s subjec o hyb id spin–o bi /magne ic ex u es showing a phase
disloca ion in he conduc ance as he dis inc signa u e o a opological ansi ion. This esul is in iguing as he
complexi y o he spin dynamics nea he c i ical poin does no smoo h he way o anin ui i e pic u e o he
ansi ion in e ms o geome ic spin phases.
He e we add ess he p oblem o a localized spin subjec o he ac ion o ime-pe iodic d i ing fields which a e
he ime-dependen equi alen o he opological field ex u es s udied in [5] o spin ca ie s
6
. This leads us o
wo k wi hin he Floque amewo k, an a ea ha has ecen ly become e y ac i e due o he possibili y o
gene a ing no el ( opological)phases o ma e [8–11]. We show ha he opological cha ac e is ics o he
guiding field ha e s iking consequences on he spin esonance condi ion, leading o a defini e inflec ion o he
Bloch–Siege shi [12]a he c i ical poin whe e he field unde goes a opological ansi ion. We he e o e
p opose co esponding esonance expe imen s as a p oo o concep o opological ansi ions in he dynamics
o spins-1/2 o any o he wo-le el sys em as, e.g., s ongly-d i en supe conduc ing qubi s (SCQs). This
app oach has he ad an age o o e come he di ficul ies p esen in spin-ca ie implemen a ions as hose a ising
om diso de , mul ichannel anspo and dephasing in hyb id spin–o bi /magne ic ex u es. Addi ionally, we
de elop a sui able heo y ha cap u es he opologicalsigna u e le by he guiding field in e ms o an eme gen ,
e ec i e Be y phase in he quasiene gy o Floque spin s a es (FSSs).
In sec ion 2we in oduce he concep o opological d i ing fields in he Floque heo y. In sec ion 3we
p esen ou main esul s conce ning he opological imp in s on spin esonances. In sec ion 4we p esen a
OPEN ACCESS
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ACCEPTED FOR PUBLICATION
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PUBLISHED
6 June 2017
O iginal con en om his
wo k may be used unde
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Commons A ibu ion 3.0
licence.
Any u he dis ibu ion o
his wo k mus main ain
a ibu ion o he
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and DOI.
6
A classical equi alen was add essed in [6]. Fo a s udy on ela ed opological aspec s in d i en sys ems, see [7].
© 2017 IOP Publishing L d and Deu sche Physikalische Gesellscha
heo y e ealing complemen a y opological ea u es in he FSSs. B ie concluding ema ks appea in sec ion 5.
Technical de ails a e discussed in appendices A–C.
2. Spin dynamics unde opological d i ing fields
Conside a localized spin-1/2 guided by a ime-dependen magne ic field consis ing o wo coplana
componen s: a o a ing d i ing
ww=+() ( ˆˆ
)
B
B
xycos sin
110 0
and a cons an
=ˆ
B
B
x
22
(see figu e 1).
This is a Floque p oblem desc ibed by he ime-dependen Sch ödinge equa ion
Yñ=()∣ () () 01
wi h Floque ope a o


º-¶¶() () H iand Hamil onian

wws ws ws=++() ( ) ()H
2cos sin 2,2
xyx
100
2
whe e we ha e in oduced he La mo equencies

w
m=B
1,2 1,2 , wi h μ he gy omagne ic a io. The gene al
solu ion o equa ion (1) eads

eyYñ= - ñ
⎜⎟
⎛
⎝
⎞
⎠
∣() ∣() () exp i , 3
wi h quasiene gy εand pe iodic FSS yñ
∣
() sa is ying he eigen alue equa ion
yeyñ= ñ()∣ () ∣ () () .4
The FSSs ha e he o m
yyñ= ñ= -
qj
q
qj
q
+
-
-
-
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
∣∣ ()
cos e
sin
,sin e
cos
,5
2
i
2
2
i
2
wi h ime-dependen
q
()
and
j
()
. The spin dynamics esul ing om equa ion (4)can be a he complex
depending on he pa ame e se ing in he Hamil onian (2). As a ule, he ins an aneous quan iza ion axis o he
FSSs (5)does no poin along he di ec ion defined by he ins an aneous field
=+() ()
B
BB
12
. Ins ead, i
ypically p esen s a fini e p ojec ion ou o he field’s plane cha ac e ized by a
q
p¹() 2 o some in (5).An
excep ion o his ule is ound in he limi o adiaba ic spin dynamics whe e he ins an aneous La mo equency
o spin p ecession,
w
ww w wwº+ +() ( ) ( ) cos sin
21 0
210
2, is much la ge han he o a ing field’s
equency
w
0[13,14]. In his limi , he FSSs s ay (an i)aligned wi h he ins an aneous guiding field o all . These
adiaba ic FSSs educe o
yyñ= ñ= -
hh
--
⎜⎟ ⎜ ⎟
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠
∣∣ ()
1
2
e
1,1
2
e
1,6
ii
wi h
h
ww=+D() [ ( )
]
a c an sin cos
00
and ww
D
=2
1
.
The quasiene gy o a FSS na u ally spli s in o a dynamical and a geome ical con ibu ion as


ò
eyy =á ñ=-∣()∣ ¯()
T E
T
1d, 7
sTss s
s
0g
whe e

òò
yy wqjh=á ñ= -
¯∣()∣ () () ()ETH s
T
1d2sin cos d 8
sTss T
00
Figu e 1. Guiding field ex u e (dashed a ows)unde going a opological ansi ion a D=1, om a o a ing field (D<1) o an
oscilla ing one (D>1). The field consis o a o a ing d i ing
(
)
B
1
plus a cons an
B
2(solid a ow), wi h
DºBB
21
.
2
New J. Phys. 19 (2017)063010 A A Reynoso e al
is he mean ene gy and
òò
yy q
j
=á
¶
¶ñ= + ¶
¶=- W∣∣ ( ) ()
s
id
1
21cos d 1
29
sTss Ts
g00
is he geome ic phase o he FSS wi h
=
s
, wi h Ws he co esponding solid angle (
p[
]m
od 4
)sub ended o e
he Bloch sphe e du ing he spin e olu ion in one ime pe iod pw=T2
0
. Fo he adiaba ic FSSs (6), his
geome ic phase educes o a Be y phase [13]. O he wise, i adop s he name o (non-adiaba ic)Aha ono –
Anandan (AA)phase [15].
We no ice in (2) ha he magne ic field unde goes a opological ansi ion a
D
=1
: a o a ing field o
D
<1(domina ed by
(
)
B
1
and enclosing he poin o anishing field in i s ound ip, figu e 1(a)) u ns in o an
oscilla ing field o
D
>1(domina ed by
B2
and lea ing he poin o anishing field ou o he loop, figu e 1(c)).
A
D
=1
, he o al field
()
B
anishes whene e w=- cos 1
0
, see figu e 1(b). Such opological cha ac e is ics
lea e an imp in in he adiaba ic FSSs (6): in a ime pe iod T hey acqui e a geome ic (Be y)phase
p=
B o
D
<1while
=0
B o
D
>1(ob ained a e se ing q=cos 0 in equa ion (9)). Topological ansi ions o his
kind ha e been expec ed o show up in anspo expe imen s wi h spin ca ie s elying on Be y-phase
in e e ence e ec s [4]. Howe e , his easoning u ns ou o be o e simplified: in he icini y o he ansi ion
poin
D
=1
,figu e 1(b), he adiaba ic condi ion can no be sa isfied since he guiding field anishes o e e se
i s di ec ion and he spins a e unable o ollow i . S ill, a mo e gene al app oach has shown ha field ex u es can
lea e a opological signa u e in elec on spin anspo simula ions e en a om he adiaba ic egime [5].A
nume ical explo a ion o he ime-dependen sys em modeled by equa ion (1) e eals se e al opological
imp in s le by he guiding fields away om he adiaba ic egime, as we discuss in sec ions 3and 4.
3. Topological imp in s on spin esonance
A esonan ans e o ene gy be ween he d i ing field
(
)
B
1
and he spins occu s whene e hei expec a ion
alue along he uni o m field
B2
anishes in a ime-pe iod a e age, i.e., whene e
s=⟪⟫ 0
xT
. Wi hin he
Floque o malism, his esonan condi ion eads [16]
e
w
¶
¶=()0, 10
2
which can be ob ained by applying he Hellmann–Feynman heo em [17] o equa ion (4)and gi es he posi ion
o single- and mul iple-pho on p ocesses. The solid lines in figu e 2show he posi ion o he esonances
acco ding o equa ion (10)(see appendix A o de ails on he nume ical me hod). Fo a weak d i ing
w
w1
10 , esonances appea a in ege alues o
w
w
20
due o he combined ac ion o he oscilla ing
componen s pa allel and no mal o he uni o m field
B2
[16]. Fo la ge d i ing ampli udes, he esonances shi
hei posi ion o smalle (non in ege ) alues o
w
w
20
due o he Bloch–Siege e ec [12]. E en ually, he
esonance cu es de elop inflec ion poin s o ganized along he c i ical line
D
=1
. These inflec ions in he
Figu e 2. Solid lines: posi ion o he esonances as a unc ion o he guiding field’s se ing de eloping a Bloch–Siege shi as he
d i ing s eng h
ww
10
inc eases. The inflec ions along he diagonal ww=
12
(c i ical line D=1)indica es a change in he guiding
field’s opology. Dashed lines: poin s o anishing mean ene gy. Do ed lines: posi ion o he esonances o a s anda d linea d i ing
wi h i ial opology. The ci cles indica e he S enholm’s poin s [18].
3
New J. Phys. 19 (2017)063010 A A Reynoso e al
esonance p ofile a e an ac ual signa u e o he opological ansi ion unde gone by he guiding field’s ex u e a
D
=1
, imp in ed in he esonan FSSs. The iden ifica ion o such opological ea u es in he Bloch–Siege shi
is he main esul o his pape .
Fo a compa ison, he do ed lines in figu e 2show he posi ion o he esonances in he case o a s anda d
linea d i ing owning a i ial opology

ww
(
)ˆ
y2sin
10
( o be compa ed wi h he opological d i ing o
equa ion (2)). These esonances p esen a usual Bloch–Siege shi mee ing he ho izon al axis a he S enholm’s
poin s (ci cles)[18].
Mo eo e , he dashed lines in figu e 2show he poin s whe e he mean ene gy anishes, unning igh along
he esonances (solid lines). This sugges s ha he condi ion o anishing mean ene gy is a good app oxima ion
o he esonance condi ion (10) o ou opological d i ing. Consequen ly, in sec ion 4we ocus on he ela ed
opological ea u es unde lying a he le el o he quasiene gy, he mean ene gy and he geome ic phase
in oduced in equa ions (9)–(7). We no ice ha his app oxima ion is no alid in gene al cases: i applies o ou
opological d i ing bu no o linea d i ings. In he la e case, he esonan condi ion (10)coincides wi h he
anishing-mean-ene gy one only o weak d i ings (
w
w1
10 ).
4. Unde lying opological ea u es
In figu e 3(a)we depic he quasiene gy (7)as a unc ion o he guiding field’s se ing in equa ion (2)(see
appendix A o de ails on he nume ical me hod). Mo e specifically, and wi hou loss o gene ali y, we plo

e+
()Tcos as a unc ion o
7
w
w
10
and
w
w
20
. The e we obse e a phase disloca ion along he
D
=1
diagonal
as a sign o he quasiene gy’s esponse o he opology o he guiding field. This beha io esembles he
p edic ions o he conduc ance o semiconduc ing Rashba loops subjec o simila field configu a ions
(de e mined by he o al phase acqui ed by spin ca ie s)[5]. I also ecalls wha expec ed o adiaba ic spin
dynamics acco ding o [4]. Howe e , as we show in he ollowing, he spin dynamics is ac ually domina ed by
non-adiaba ic e ec s.
In figu es 3(b)and (c)we depic , espec i ely, he mean ene gy (in e ms o he dynamical phase

+
¯
E
T
)
and he AA geome ic phase by plo ing

+
(¯)ETcos
and
+
cos
gas a unc ion o he field’s pa ame e s (see
equa ions (9)–(7), and u he de ails on nume ics in appendix A). The e we no ice he complex pa e ns
displayed nea he c i ical line
D
=1
, indica i e o a s ongly non-adiaba ic spin dynamics. This is clea ly
illus a ed by he FSS spin ex u es, figu e 4, adop ing in ica e shapes wi h apidly changing solid angle in he
c i ical egion. Tha demons a es he unsui abili y o an adiaba ic ea men o unde s anding he phase
disloca ion in he quasiene gy (figu e 3(a)) as a esponse o he change in he guiding field’s opology.
To gain insigh in o he opological cha ac e is ics s amped on he spin dynamics, we in oduce a modified
e sion o a ea men fi s p oposed in [19] o he s udy o (Be y)adiaba ic phases in spin ca ie s, he e
adap ed o he case o non-adiaba ic spin dynamics (see appendix B). As a s a ing poin , le us ew i e he
Floque ope a o as he sum o diagonal (d)and nondiagonal (nd)p ojec ions on o he non-adiaba ic FSS
Figu e 3. (a)Response o he quasiene gy
e
+
o he guiding field’s se ing in e ms o

e
+
(
)
Tcos . The disloca ion along he diagonal
ww=
12
(c i ical line D=1)indica es a change in he guiding field’s opology. (b)Mean ene gy
+
¯
E
in e ms o

+
(¯
)
ETcos
. The
complex pa e n displayed nea he diagonal is indica i e o a non-adiaba ic spin dynamics. (c)Cosine o he AA geome ic phase +
g
displaying a complex pa e n complemen a y o ha one shown by he mean ene gy in panel (b). In all panels, he dashed lines indica e
he poin s o anishing mean ene gy =
+
¯
E
0. Spin ex u es o he Floque s a e y
ñ
+
∣
co esponding o field se ings A, B and C a e
shown in figu e 4. A s udy o he Floque s a e y
ñ
-
∣
p oduces equi alen esul s.
7
We choose he Floque solu ion wi h

e<<
w
+
0
2
0
which is equi alen o he solu ions wi h quasiene gy 
e
w+
+n0.
4
New J. Phys. 19 (2017)063010 A A Reynoso e al

basis(5), i.e.

=+() () ()
dnd
. By he sole defini ion o Floque s a e, i mus hold

º() 0
nd
(and,
he e o e,

º() (
)
d
). As a consequence, he FSSs a e cons ained o sa is y
wqjh q
j
-+ ¶
¶=() ( ) ( )
cos cos sin 0, 11
as shown in appendix B. F om equa ions (8)–(11)we can ew i e he quasiene gy (7)as (see appendix C)

ò
eq
jp=- ¶
¶-ℓ()
s
T
T2
1
cos d12
sT
0
hanks o he cancela ion o he e m p opo ional o qcos in equa ion (9). We ecognize wo con ibu ions o
he quasiene gy in equa ion (12): a smoo h dynamical e m p opo ional o
q1
cos ( om equa ion (11)we see
ha his e m does no di e ge o anishing qcos )plus a opological one de e mined by he in ege numbe
ò
p
j
=¶
¶
ℓ()
1
2d, 13
T
0
accoun ing o he windings ga he ed by he FSSs a ound he no h pole o he Bloch sphe e. This is illus a ed by
he spin ex u es o figu e 4(A)wi h
=
ℓ
1
(
D
<1)and figu e 4(C)wi h =
ℓ
0(
D
>1). We u he no ice
om equa ion (12) ha

e =-()()() ()
ℓ
Tcos 1 cos , 14
s
d
0
wi h
ò
=
q
j¶
¶
d
T
d
0
0
1
2cos he smoo h con ibu ion o he dynamical phase. Thus, a pa i y ansi ion in he
winding numbe ℓa
D
=1
would explain he phase disloca ion shown by he quasiene gy in figu e 3(a)
( oge he wi h he esul s epo ed in [5])in e ms o an e ec i e Be y phase
pºℓ
B
e
ecalling he geome ic
phase acqui ed by he adiaba ic FSSs (6). S ill, he ac ual exis ence o complex spin ex u es wi h singula
de i a i e
j
¶
¶
(co esponding o a spin passing o e he poles o he Blochsphe e as in figu e 4(B))
complica es he analysis. In figu e 5we ep esen he winding pa i y (i.e., he pa i y o he winding numbe ℓ)as
a unc ion o he guiding field’s configu a ion. As expec ed, we find opposi e domina ing pa i ies on di e en
sides o he c i ical line: e en (da k) o
D
>1and odd (whi e) o
D
<1. In e es ingly, anomalous egions o
fluc ua ing pa i y appea along he lines co esponding o a anishing mean ene gy (dashed lines in figu e 5).
Simila winding-numbe cha ac e is ics ha e been iden ified ecen ly in spin ca ie s subjec o magne ic
ex u es [5,20]. The ul ima e eason o ha is he p oximi y o spin esonances (as shown in figu e 2)de eloping
complex spin ex u es. Such anomalies should no be necessa ily unde s ood as noisy phases in he pa ame e
space since hey ac ually show a defini e mosaic s uc u e in fine scales (inse in figu e 5). The p esence o
anomalies ul ima ely indica es ha he winding numbe ℓ, while o he wise use ul, is no an op imal indica o o
he opological ansi ion. The ques o an indica o ha ully cap u es he opological o igin o he phase
disloca ion in he quasiene gy a
D
=1
,figu e 3(a), emains open.
5. Concluding ema ks
We ha e s udied he opological imp in s le by an ex e nal d i ing in he dynamics o a localized spin-1/2. We
find ha he opological ansi ion unde gone by a guiding magne ic field eme ges in he esonance spec um o
he spin sys em as a dis inc inflec ion in he Bloch–Siege shi . This ema kable ea u e opens a doo o a new
amily o esonance expe imen s in wo-le el sys ems as an al e na i e o p e ious a emp s o obse e eal-space
opological ansi ions based on spin-ca ie in e e ome y in mesoscopic loops [5], he ealiza ion o which
p esen s echnical di ficul ies up o da e. The possibili ies un om nuclea magne ic esonance (NMR) o
s ongly-d i en SCQs. In NMR, hese e ec s can be demons a ed expe imen ally by shaping adio equency
pulses o gene a ing he sui able d i ing Hamil onian in he o a ing ame o he nuclea spins [21]. The
p edic ed e ec s migh be significan o he design o shaped pulses o obus con ol o quan um sys ems [22].
Figu e 4. Spin ex u e o he Floque s a e y
ñ
+
∣
ep esen ed as a pa h o e he su ace o he Bloch sphe e o di e en guiding field’s
se ings (see poin s A, B and C in figu es 3and 5).
5
New J. Phys. 19 (2017)063010 A A Reynoso e al
As o SCQs [23], we no ice ha high-o de mul ipho on in e e ome y has been demons a ed success ully
and he sys ems can eadily be adap ed o he opological d i ing fields p oposed he e [24]. All esul s ex end o
any o he wo-le el sys em liable o equi alen d i ings (as, e.g. adap ed e sions o Landau–Zene –S ückelbe g
in e e ome ic se ups [25]).
Fo he opological d i ing conside ed he e, we ound ha he esonance condi ion can be app oxima ed by
he condi ion o anishing mean ene gy. This has pe mi ed us o shi ou analysis owa ds he dynamical and
geome ical con ibu ions o he quasiene gy. As a consequence, we ended up wi h a desc ip ion o he
opological ansi ion imp in ed in he spin dynamics in e ms o an e ec i e Be y phase. This desc ip ion, e en
when inexac , cap u es he essen ial ea u es o he p oblem.
Acknowledgmen s
This wo k was suppo ed by P ojec No. FIS2014-53385-P (MINECO, Spain)wi h FEDER unds and by G an s-
in-Aid o Scien ific Resea ch (C)No. 26390014, o Specially P omo ed Resea ch No. H1505699, and o
Scien ific Resea ch on Inno a i e A eas No. JP15K21717 (Japan Socie y o he P omo ion o Science). AAR
hanks he hospi ali y o he Depa amen o de Física Aplicada II, Uni e sidad de Se illa. We hank G Ál a ez and
W D Oli e o use ul commen s on NMR and SCQs, espec i ely.
Appendix A. Nume ical me hod
We sol e equa ion (4)by he cus oma y p ocedu e o expanding in Fou ie se ies he Hamil onian (2),
=åw
() ()
H H e
nnn i0, and he FSS, yyñ=åñw
∣
() ∣ ()
e
nnn i0. This leads o he infini e se o equa ions:

yewy
å
ñ= - ñ
-∣( )∣
()() ()
Hn
mnm m n
0wi h nin ege . We sol e he eigen alue p oblem o εand yñ
{
∣}
()nby
es ic ing o a fini e se o equa ions wi h 
∣
∣nn
max . We chose
=
n
120
max and checked ha o he s onges
simula ed d i ing ampli udes he esul s a e una ec ed by he unca ion.
The mean ene gies a e ob ained om he Fou ie decomposi ion o he Floque s a es as

ew yy=- åáñ
¯∣∣∣
() ()
E
n
nnn
0
2
, whe e
pyy=åáñ∣∣∣
() ()
n2
nnn
g2
a e he co esponding AA geome ic phases.
Appendix B. Non-adiaba ic spin dynamics unde pe iodic d i ing
We in oduce a modified e sion o a ea men fi s p oposed in [19] o he s udy o adiaba ic (Be y)phases in
spin ca ie s, he e adap ed o he case o non-adiaba ic spin dynamics in pe iodic, ime-dependen fields
8
. Le us
ew i e he Floque ope a o o equa ion (1)as he sum o diagonal (d)and nondiagonal (nd)p ojec ions on o
he non-adiaba ic FSS basis (5), i.e.

=+() () ()
dnd
. To his aim, we define ins an aneous p ojec o s
Figu e 5. Winding pa i y, showing a ansi ion along he diagonal ww=
12
(c i ical line D=1). Anomalous egions o fluc ua ing
pa i y (inse )appea along he cu es o anishing mean ene gy (dashed), closely ela ed o esonances. Spin ex u es o he Floque
s a es co esponding o poin s A, B and C a e shown in figu e 4.
8
Fo a ecen s udy on non-adiaba ic spin-ca ie dynamics, see [26].
6
New J. Phys. 19 (2017)063010 A A Reynoso e al
on he co esponding subspaces gi en by
s
=

() ˆ()· () l1
2,B.1
wi h
qj qj q=++
ˆ() () ()
ˆ() ()
ˆ()
ˆ
lxyzsin cos sin sin cos
he uni ec o defining he ins an aneous
quan iza ion axis o he FSSs a ime . We s ess ha ˆ(
)
lgene ally di e s om he guiding field’s axis
hh=+
ˆ() ()
ˆ()
ˆ
nxycos sin in he non-adiaba ic egime. We u he no ice ha , by defini ion o Floque
s a es, i holds
=+º
++ --
() () ( ) ,B.2
d
=-=+º
+- -+
() ( ) 0. B.3
nd d
The cons ain imposed by equa ion (B.3)es ablishes a defini e link be ween he magne ic ex u e and he non-
adiaba ic FSS ex u e ha e en ually leads o he iden ifica ion o an e ec i e Be y phase imp in ed by he
guiding field. We can make he mos o he o mal iden i ies (B.2)and (B.3)by no icing ha he p ojec o s

do
no commu e wi h

¶¶
i(which he e o e mixes he FSS subspaces). By ollowing [19,27], we in oduce an
ope a o () esponsible o he ansi ions be ween he subspaces associa ed wi h

+and

-while


¶¶- iac s only wi hin each subspace. This is accomplished wi hou ambigui y by defining
 
=¶
¶-¶
¶-¶
¶
++--
⎜⎟ ⎜⎟
⎛
⎝
⎞
⎠
⎛
⎝
⎞
⎠()
ii i, B.4
which e ifies

¶¶- =

[
] i,0
and

 =

0
. By aking in o accoun he p ojec o s’p ope ies
(

=

2
and

+=
+-
), one ob ains

sm=-
¶
¶-
⎜⎟
⎛
⎝
⎞
⎠
() ( ·ˆ)(ˆ·) ( )
Bll i, B.5
d
ssm=- -º() [· (·
ˆ)(ˆ·)] ( ) BBll 0, B.6
nd
whe e we ha e d opped he dependence on when con enien o ease in no a ion. Mo eo e , he explici
e alua ion o

p oduces

ss=- ¶
¶
(ˆ·) (
ˆ·) ( )
ll
i
2.B.7
We now ew i e

dand

nd in he FSS basis by in oducing he ins an aneous uni a y ope a o
=-
qjq
qjq
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
() ( )
cos e sin
sin e cos
B.8
2
i
2
2
i
2
such ha
yyñ= ñ=
+-
() ()
()∣ ()∣ ( )
1
0,0
1.B.9
We fi s no ice ha
 =
+-
-+
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟()
†
a
a
0
0,B.10
g
g
whe e

qj q=- ¶ ¶ + ¶ ¶() ( )
¯
a
s sin i 2
ss
gplays he ole o a geome ic mixing. Mo eo e ,
 


¶
¶-=+
+
¶
¶
+
¶
¶
-
⎜⎟
⎛
⎝
⎞
⎠
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟()
†
A
A
A
ii0
0i
,B.11
g
g
whe e

qj
=+ ¶
¶
() ( ) ( )A s 21cos B.12
s
g
is esponsible o he eme gence o non-adiaba ic geome ic phases (see appendix C). Back o he Floque
ope a o , we find
  

=
+
-
⎜⎟
⎛
⎝
⎞
⎠()
†
0
0,B.13
d
7
New J. Phys. 19 (2017)063010 A A Reynoso e al
wi h

m=-
¶
¶+
⎜⎟
⎛
⎝
⎞
⎠
() ()·ˆ() () ( ) s A Bl i. B.14
ss
g
The fi s e m in equa ion (B.14)is he e ec i e Zeeman spli ing esul ing om he ins an aneous p ojec ion o
he non-adiaba ic FSSs along he guiding field, which can be also expanded as

wyy yyáñ-áñ

()(∣ ∣ ∣ ∣ ∣ ∣) 2
ss22
wi h
=
s
. Simila ly, we find
  

=


⎛
⎝
⎜⎞
⎠
⎟()
†
0
0,B.15
nd
wi h

wyy yy yy yy=áñáñ-áñáñ-º
 
()(∣ ∣ ∣ ∣) () ( )
¯¯¯
¯
a
20. B.16
ss s s s s ss
g
The e alua ion o he eal and he imagina y pa s o equa ion (B.16)leads o he ollowing iden i ies,
espec i ely:
wqjh q
j
-+ ¶
¶=() ( ) ( )
cos cos sin 0, B.17
wjh
q
-+
¶
¶=() ( ) ( )
sin 0. B.18
These equa ions e eal he geome ic cons ain s sa isfied by he FSSs unde he ac ion o a d i ing ep esen ed
by
w
() and
h
()
.
Appendix C. Topological phases and quasiene gy
F om equa ion (B.14), we can now ew i e equa ions (9)–(7)as


e
=á ñ= -() ¯
E T
ss ss
gwi h

òò
mwqjh== -
¯()·ˆ() () () ()Es
T
s
T
Bld2sin cos d , C.1
sTT
00

òò
q
j
==+
¶
¶
() ( ) ( )A s
1d1
21cos d. C.2
sTsT
g0g0
F om equa ion (B.17), we no ice ha he mean ene gy (C.1) akes he o m

òqq
j
=-
¶
¶
⎜⎟
⎛
⎝
⎞
⎠
¯()Es
T
2cos 1
cos d. C.3
sT
0
The quasiene gy educes hen o

ò
eq
jp=- ¶
¶-ℓ()
s
T
T2
1
cos d, C.4
sT
0
hanks o he cancella ion o he e ms p opo ional o qcos in equa ions (C.2)and (C.3). We obse e ha he
quasiene gy (C.4)consis s o a dynamical con ibu ion plus a opological one de e mined by he pa i y o he
in ege numbe ò
pj=¶¶
ℓ
()
12 d
T
0, which accoun s o he windings ga he ed by he FSSs a ound he
no h pole o he Bloch sphe e. This mo i a es he in oduc ion o an e ec i e Be y phase
pºℓ
B
e
cap u ing
he opological ea u es o he FSSs.
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New J. Phys. 19 (2017)063010 A A Reynoso e al