Unified Framework and Engineering Pathways for Time Crystals: From Floquet Phases to Open System Limits, Quasiperiodic Drives, and Topological Protection

Unied F amewo k and Enginee ing Pa hways o Time C ys als:
F om Floque Phases o Open Sys em Limi s, Quasipe iodic
D i es, and Topological P o ec ion
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
Abs ac
Cons uc unied ime c ys al heo y pene a ing closed and open quan um sys ems, pe-
iodic and quasipe iodic d i es, and opological cons ain s. Using g oup ep esen a ion and
non-equilib ium s a is ics as ounda ion, p opose ope a ional o de pa ame e s, empo al co -
ela ions, and spec al c i e ia; a ma hema ical le el ia high- equency Floque Magnus ex-
pansion, LiebRobinson bounds, and spec al pai ing s uc u e, p o e igidi y and obus ness
o p e he mal disc e e ime c ys als on exponen ially long imescales; unde diso de ed many-
body localiza ion (MBL) and Lindblad open sys ems espec i ely p o ide exis ence and s abili y
heo ems, ex ending o mul i- equency quasipe iodically d i en " empo al quasic ys als"; u -
he demons a e opological o de p o ec ion mechanism o disc e e ime c ys als, cons uc ing
opological ime c ys als using su ace code logical ope a o s as o de pa ame e s; nally owa d
expe imen s, p opose enginee ing schemes and measu emen p o ocols o supe conduc ing qubi
a ays, Rydbe g gases, apped ions, and liquid c ys al/phononpola i on pla o ms. This pa-
pe 's heo ems and schemes sys ema ically abso b ecen key ad ances: equilib ium ime c ys al
no-go heo ems, igo ous deni ion and  s expe imen al obse a ion o Floque disc e e ime
c ys als, p e he mal exponen ial li e ime uppe bounds, la es ealiza ions o dissipa i e and
opological ime c ys als, and expe imen al e idence o con inuous/spa io empo al ime c ys-
als and mul i- equency empo al quasic ys als.
Keywo ds
: Disc e e ime c ys al; spon aneous b eaking o ime ansla ion symme y; Floque
p e he maliza ion; many-body localiza ion; Lindblad open sys ems; empo al quasic ys al; opolog-
ical ime c ys al; su ace code logical ope a o s; Rydbe g gas; supe conduc ing qubi s
1 In oduc ion & His o ical Con ex
Time c ys al idea o igina es om undamen al ques ion "can ime ansla ion symme y spon a-
neously b eak". Al hough ini ial con inuous ime c ys al p oposals a ac ed wide a en ion, igo ous
no-go heo ems exclude such phases in g ound s a es o canonical ensembles o b oad classes wi h
sho - ange in e ac ions, shi ing ocus owa d non-equilib ium d i en sys ems.
In pe iodically d i en many-body quan um sys ems, ElseBaue Nayak p o ided igo ous de -
ini ion o disc e e ime c ys al (DTC): sys em unde undamen al pe iod
T
Floque symme y
spon aneously selec s longe subha monic pe iod
mT
, exhibi ing " igid" agains pe u ba ions sub-
ha monic esponse, accompanied by long- ange empo al co ela ions and cha ac e is ic spec al
lines. Subsequen heo e ical wo k e ealed unied amewo k o igidi y, c i icali y, and ealizabil-
i y, p o iding clea bluep in o expe imen s on die en pla o ms.
1
Expe imen ally, wo miles one wo ks in 2017 espec i ely obse ed disc e e ime c ys alline o de
in apped ions and oom- empe a u e diamond spin sys ems, es ablishing DTC obse abili y and
obus ness; hese wo esul s simul aneously published in same issue o
Na u e
. Subsequen ly, supe -
conduc ing quan um p ocesso s con med "eigens a e ime c ys alline o de " a spec aldynamical
le el, emphasizing eigens a e o de ing s uc u e unde Floque MBL/p e he mal backg ound. Open
sys em di ec ion, s ongly in e ac ing Rydbe g gases obse ed dissipa i e ime c ys als (bo h con-
inuous and disc e e ypes), demons a ing con ollable ealiza ion and spec al gap p o ec ion o
open sys em limi cycles. A opological le el, wo-dimensional supe conduc ing qubi a ays e-
alized "long-li ed p e he mal opological ime c ys al", whose subha monic esponse signican ly
exhibi ed only by non-local logical ope a o s, accompanied by non-ze o opological en anglemen
en opy. Mo e mac oscopic spa io empo al symme y b eaking also de eloping: con inuous media
(liquid c ys als, supe uid
3
He, e c.) based on nonlinea dissipa i epumping coupling obse ed
con inuous/spa io empo al ime c ys als and con ollable coupling wi h mechanical modes; in 2025
ealized disc e e " empo al quasic ys al" unde mul i- equency d i e.
Abo e ad ances join ly poin owa d unied ques ion:
How o dene, de e mine, and en-
ginee ime c ys al exis ence and obus ness unde gene al condi ions?
Below p o ides
sys ema ic answe .
2 Model & Assump ions
2.1 Local Quan um Model (Closed Sys em)
Conside spin/boson/ e mion sys em on
d
-dimensional
la ice
Λ
, local in e ac ions wi h ni e ange o as decay. Pe iodic d i e
H( ) = H0+
X
α=1
Vα( ), Vα( +T) = Vα( ),|Vα| ≤ J.
Floque uni a y
F=Te−iRT
0H( )d
gene a es disc e e ime ansla ion
Z
. High- equency limi
(
ω= 2π/T ≫J
) denes p e he mal egion; diso de case
H0
suppo s MBL.
2.2 Lindblad Open Sys em Model
Densi y ma ix
ρ
sa ises
˙ρ=L (ρ) = −i[H( ), ρ] + X
µLµρL†
µ−1
2{L†
µLµ, ρ},L +T=L ,
single-pe iod quan um channel
E=TexpRT
0L d 
desc ibes Poinca é sec ion dynamics.
2.3 Symme y and O de Pa ame e s
I sys em possesses ni e in e nal symme y g oup
G
and disc e e ime ansla ion
Z
, s eady-s a e/eigens a e symme y can spon aneously b eak o
subg oup
H⊂G×Z
. Take local obse able
O
odd unde
G
ans o ma ion, dene empo al o de
pa ame e
CO(n) = lim
L→∞
1
|ΛL|X
x∈ΛLOx(nT)Ox(0),
exhibi ing s ic
m
-pe iodic spacing in
n
wi h non- i ial long- ime limi , cha ac e izing
m
-
subha monic DTC o de . This deni ion equi alen o ep esen a ion heo y pe spec i e.
3 Main Resul s (Theo ems and S a emen s)
Theo em 1
(1: Exis ence and Rigidi y o P e he mal Disc e e Time C ys al)
.
Le
H( )
be local
pe iodically d i en sys em, piecewise d i e composed o nea -
π
global symme ic "pulses" exis s,
2
making Floque uni a y w i able as
F=U∗e−iH∗TX U†
∗+ ∆,
whe e
X2=⊮
is
Z2
in e nal symme y gene a o ,
H∗
commu es wi h
X
and quasilocal,
|∆| ≤
Ce−cω/J
. Then any local ope a o
O
odd unde
X
ans o ma ion exhibi s s able
2T
subha monic
locking, main aining cohe ence o polynomial ime
≲τ∗∼ecω/J
; o small pe u ba ion
δH
ha e
dis (CO,C(0)
O)≤K|δH|
. P oo based on quasiconse ed quan i ies om high- equency Floque 
Magnus unca ion and exponen ial slow hea ing heo em.
Theo em 2
(2: Spec al Pai ing and Eigens a e O de in MBLDTC)
.
On one-dimensional s ongly
diso de ed localized chain, Floque ope a o wi h nea -
π
global symme ic kick admi s quasilocal
uni a y
U
making
F≃˜
X e−iH
MBL
T,
whe e
˜
X
is quasilocal
Z2
symme y,
H
MBL
diagonalized by se o
l
-bi s. Spec um exhibi s
π
pai -
ing (eigens a es die by
π
), ypical eigens a es ha e pa i y degene acy, inducing s a e-independen
2T
subha monic esponse (eigens a e o de ).
Theo em 3
(3: Sucien Condi ion o Open Sys em Dissipa i e Time C ys al)
.
Fo pe iodic
Lindblad semig oup
L
single-pe iod channel
E
, i all eigen alues wi hin spec al adius sa is y
|λj|<
1
, while unique modulus g oup consis s o
m
phases
{e2πik/m}m−1
k=0
wi h co esponding Jo dan blocks
non-spli able, hen almos all ini ial s a es con e ge long- ime o pe iod-
mT
limi cycle a ac o
amily, mani es ing as
m
-subha monic dissipa i e ime c ys al; limi cycle s uc u ally s able agains
small pe u ba ions unde Liou illian spec al gap. This c i e ion consis en wi h Pe onF obenius
ype esul s o quan um channels.
Theo em 4
(4: Mul i-F equency D i e and Tempo al Quasic ys al)
.
Fo quasipe iodic d i e wi h
k
mu ually i a ional equencies
{ωi}
, i
miniωi≫J
en e s p e he mal egion, quasilocal uni a y
U
and eec i e Hamil onian
H⋆
exis such ha on
Zk
empo al la ice
U(n) = U e−iH⋆PiniTig(n)U†+Oe−cminiωi/J ,
whe e
g
is ni e index subg oup ep esen a ion o
Zk
. This implemen s spon aneous b eaking o
Zk
, o ming " empo al quasic ys al" wi h mul i-subha monic spec al lines.
Theo em 5
(5: Topological Time C ys al: Non-Local O de Pa ame e and P o ec ion)
.
Unde
wo-dimensional s abilize code (su ace code) d i e implemen ing logical
π
ip and Hamil onian
enginee ing, Floque uni a y in code subspace equi alen o
F
logical
≈XLe−iH
op
∗T,
whe e
XL
is logical non-local symme y. Any local ope a o insignican ly exhibi s subha monic
o de , while non-local logical closed s ing/memb ane ope a o s exhibi igid
mT
subha monic e-
sponse, quan i a i ely suppo ed by opological en anglemen en opy. This mechanism expe imen-
ally e ied on p og ammable supe conduc ing a ays.
P oposi ion 6
(6: Con inuous/Mac oscopic Spa io empo al Time C ys al)
.
In nonlinea dissipa i e
pumping coupled con inuous media (e.g., nema ic liquid c ys als) and supe uid
3
He sys ems, Hop 
Tu ing coope a i e ins abili y o ms s able limi cycles ipes simul aneously b eaking space and ime
ansla ion (spa io empo al c ys al), wi h con ollable coupling o mechanical modes.
3
4 P oo s
4.1 Exponen ially Long Time Bound in P e he mal Region (Theo em 1)
Fo local d i en sys em adop Floque Magnus expansion
F= exp{−iT Pn≥0Ωn}
, unca e a op-
imal o de
n∗∼ω/J
ob aining quasilocal eec i e Hamil onian
H∗=Pn≤n∗Ωn
. Rigo ous esul s
p o ide ene gy abso p ion and unca ion e o exponen ially small wi hin ime
τ∗∼exp(cω/J)
,
i.e.
|F−e−iH∗T| ≤ Ce−cω/J ,
d
d ⟨H∗⟩≤C′e−cω/J .
In oduce nea -
π
symme ic kick
UX
(global
Z2
ip) in piecewise d i e, combined wi h
X
and
H∗
nea -commu a i i y and exis ence o quasilocal uni a y ans o ma ion
U∗
, ob ain s uc u al
decomposi ion and subha monic locking; e o con olled o
≤τ∗
, de i ing exponen ial longe i y
and igidi y.
4.2 MBL
π
Spec al Pai ing and Eigens a e O de (Theo em 2)
Unde s ong diso de quasilocal uni a y
U
exis s making
UH0U†
diagonalized by
l
-bi s. Nea -
π
kick in
U
ep esen a ion p oduces quasilocal
˜
X
, commu ing wi h
H
MBL
. On spec um each s a e
|ψ⟩
wi h
˜
X|ψ⟩
cons i u es
π
pai ed subspace, inducing s a e-independen
2T
subha monic esponse.
This "eigens a e o de " consis en wi h p ocesso expe imen spec aldynamical consis ency.
4.3 Open Sys em Limi Cycle Spec al Condi ion (Theo em 3)
Decompose single-pe iod CPTP channel
E
spec um in o gene alized eigenmodes. I spec al adius
pe iphe y con ains only modulus
1
pu e phase eigen alues o
m
phases, wi h spec al gap elsewhe e,
hen
En
i e a ion p ojec s a bi a y ini ial s a e o
m
-cycle a ac o clus e , con e gence a e gi en
by Liou illian spec al gap. This esul belongs o posi i e ope a o quan um channel Pe on
F obenius heo y (E ansHøegh-K ohn e al.), compa ible wi h ecen algeb aic cha ac e iza ion o
limi cycles/synch oniza ion.
4.4 Mul i-F equency Tempo al Quasic ys al G oup Rep esen a ion and P e he -
mal P o ec ion (Theo em 4)
Time ansla ion g oup o quasipe iodic d i e is
Zk
. Unde join high- equency limi , cons uc
ni e index subg oup ep esen a ion
g⊂Zk
, making eec i e e olu ion on la ice decompose as
p oduc o
e−iH⋆PiniTi
and
g(n)
; la e 's ni e image induces incommensu a e mul i-subha monic
peaks, o ming " empo al quasic ys al", consis en wi h gene al heo y o "mul iple ime ansla ion
symme y p o ec ion".
4.5 Topological Time C ys al Non-Local O de and P o ec ion (Theo em 5)
Unde pe iodic enginee ing o su ace code Hamil onian, Floque uni a y in code subspace mani-
es s as supe posi ion o logical
π
ip and eec i e Hamil onian. Non-local logical ope a o s (closed
s ings/memb anes) cons i u e o de pa ame e s, whose subha monic locking insensi i e o local
noise; opological en anglemen en opy p o ides independen e idence. Supe conduc ing a ay ex-
pe imen s obse e subha monic peaks signican only o logical ope a o s wi h non-ze o opological
en anglemen en opy.
4
5 Model Apply (Rep esen a i e Pla o ms and Obse ables)
5.1 Supe conduc ing Qubi A ay (Topological DTC)
De ice: wo-dimensional squa e a ay, su ace code s abilize s
(As, Bp)
+ pe iodic "logical
π
kick".
Obse ables: au oco ela ion and Fou ie spec um o non-local
ZL, XL
; opological en anglemen
en opy
γ
. Expec ed signal: equency domain exhibi s
ω/2
subha monic peak, signican only in
logical channel;
γ > 0
appea s coope a i ely wi h logical subha monic esponse.
5.2 Rydbe g A om Gas (Dissipa i e Time C ys al)
De ice: oom- empe a u e apo cell, con inuous op ical pumping and decohe ence channel; Obse -
ables: uo escence/ ansmission in ensi y au oco ela ion, popula ion pola iza ion Poinca é map
ajec o y; Expec ed signal: s able limi cycle and pa ame e phase diag am c ossing (diso de 
limi cyclemul is abili y), con e gence a e con olled by Liou illian spec al gap.
5.3 T apped Ions (P e he mal DTC)
De ice: long- ange in e ac ion chain, high- equency d i e supp esses hea ing; Obse ables: single-
body/many-body spin co ela ion unc ions and Fou ie spec a; Expec ed signal:
2T
subha monic
peak wi h li e ime exponen ially g owing wi h
ω
, disce nible dependence on ini ial s a e ene gy
densi y.
5.4 Liquid C ys al and Supe uid
3
He (Con inuous/Spa io empo al Time C ys-
al)
De ice: pho oinduced d i e o nema ic liquid c ys al; magnon condensa ion and ee su ace coupling
in
3
He; Obse ables: spa io empo al pa e ns coexis ing s ipesoscilla ions; cohe en equency and
phase locking; Expec ed signal: Hop Tu ing coope a i e mode and unable mechanical coupling.
5.5 Mul i-F equency D i e Tempo al Quasic ys al
De ice: quan um simula o simul aneously d i en by mul icolo mic owa e/lase ; Obse ables: in-
commensu a e subha monic peak amily and mul iple " igidi ies"; Expec ed signal: peak posi ions
locked only de e mined by ni e image o
Zk
ep esen a ion, no d i ing wi h small pe u ba ions.
6 Enginee ing P oposals (Realizabili y and E o Budge )
6.1 P e he mal DTC Pulse Design and F equency Window
Adop piecewise d i e
F=e−iHZτze−iHXτx
, implemen ing nea -
π
ip a
τx
; choose
ω
sa is ying
e−cω/J ≲ε/10
ensu ing
τ∗∼ecω/J
co e s
102∼103
pe iods. Ga e noise
ε≪e−cω/J
main ains
subha monic locking; decohe ence ime
T2≫τ∗
.
6.2 Topological Time C ys al Logical Ga eE o Co ec ion Coexis ence
Embed logical
π
ip in o su ace code pe iodic sequence, ensu ing s abilize measu emen and phase
accumula ion close loop wi hin
τ∗
; non-local eadou educes local noise sensi i i y. Re e ence ecen
de ice me ics (logical o de signican only in non-local channel obse ing non-ze o opological
en anglemen en opy).
5

6.3 Open Sys em CTC GainDecohe ence Ra io
Open limi cycle ia con ollable pumpingdecohe ence a io
G/κ
, spec al gap
∆
Liou
se s eco e y
ime
τ ∼1/∆
Liou
and pe u ba ion obus ness; Rydbe g pla o m p o ides oom- empe a u e
easible pa ame e domain.
6.4 Mul i-F equency Tempo al Quasic ys al D i e A angemen
In
Zk
amewo k design d i e phase and equency "cop ime", a oiding acciden al in ege pe iod
ecu ence; expe imen al spec al lines use incommensu a e peak posi ions as nge p in .
7 Discussion (Bounda ies, Risks, and Rela ed Wo k)
Bounda ies and isks
: (i) Ou side high- equency window hea ing and chaos may ex inguish
p e he mal DTC; (ii) MBL s abili y limi ed in high dimensions and long- ange in e ac ions; (iii)
open sys em enginee ing noise and non-Ma ko iani y may induce phase wande ing and mul is able
compe i ion.
Alignmen wi h exis ing wo k
: This pape 's Theo ems 12 compa ible wi h ElseBaue 
Nayak deni ion, abso bing Yao e al. amewo k on igidi y and c i icali y; exponen ial longe i y
consis en wi h Mo iKuwaha aSai o and AbaninDe RoeckHoHu enee s p e he mal uppe bounds;
Theo em 3 consis en wi h Rydbe g dissipa i e ime c ys al obse a ions and posi i e ope a o spec-
al heo y; Theo em 5's logicalnon-local o de ma ches supe conduc ing a ay " opological ime
c ys al" da a; mac oscopic con inuous/spa io empo al ime c ys als and He-3 coupling esul s p o-
ide c oss-scale phenomenology.
Me hodological pa allel (
Z2
holonomy and "
π
" nge p in )
: DTC
π
equencyspec al
pai ing and subha monic igidi y cha ac e izable ia
Z2
quan ized nge p in . Rela ed
Z2
holon-
omy/squa e oo de e minan holonomy ideas ansplan able as c i e ion in ela i e cohomology
and modula connec ion con ex s, iden i ying spec al ansi ion and non- i ial phase winding o
"
π
" ip ype.
8 Conclusion
Es ablished unied ime c ys al heo y and enginee ing pa hways: cons uc ing common s uc u e
among closed sys em p e he maliza ion, MBL, open sys em limi cycles, and opological p o ec ion;
ex ending o mul i- equency empo al quasic ys als and c oss-scale con inuous/spa io empo al ime
c ys als; p oposing ep oducible expe imen al p o ocols suppo ing mul iple pla o ms. This ame-
wo k in eg a es "g oup ep esen a ionspec um" wi h "p e he maliza ion/localiza ion/dissipa ion
p o ec ion" dynamical p inciples, p o iding solid ounda ion owa d obus ime- equency de ices,
con ollable syn hesis o non-equilib ium phases, and opologicallogical s o age.
Acknowledgemen s, Code A ailabili y
Thank collec i e con ibu ions in ime symme y b eaking, Floque enginee ing, and open sys em
quan um dynamics elds. Fo mula de i a ion, spec al semig oup nume ical e ica ion, and peak
wid hlocking me ic  ing e e ence sc ip s ep oducible ollowing appendix algo i hm ins uc-
ions; o iginal sc ip s a ailable upon easonable eques .
6
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11. X. Wu
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Dissipa i e Time C ys al in a S ongly In e ac ing Rydbe g Gas
, Na . Phys.
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12. L. Xiang
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,
Long-li ed Topological Time-C ys alline O de on a Quan um P ocesso
,
Na . Commun.
15
, 10036 (2024).
13. D. E. E ans, R. Høegh-K ohn,
Spec al P ope ies o Posi i e Maps on C*-Algeb as
, J.
London Ma h. Soc.
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Cycles unde Dissipa ion
, SciPos Phys.
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Long-Li ed In e ac ing Phases P o ec ed by Mul iple Time-
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, Phys. Re . X
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A P e he mal DTC Cons uc ion and Exponen ial Li e ime
A.1 P e he mal Uppe Bound and Eec i e Hamil onian
Unde
ω≫J
condi ion Floque Magnus expansion
Ω0=1
TZT
0
H( )d , Ω1=1
2TiZZ0< 1< 2<T
[H( 2), H( 1)] d 1d 2, . . .
7
op imal o de
n∗∼αω/J
unca ion denes
H∗
, wi h
|F−e−iH∗T| ≤ Ce−cω/J ,
d
d ⟨H∗⟩≤C′e−cω/J ,
cons an s depending only on locali y (implemen ed ia LiebRobinson bound).
A.2 Nea -
π
Kick and
Z2
In e nal Symme y
Le
UX= exp −iπ+ϵ
2X
i
σx
i!=Xexp −iϵ
2X
i
σx
i!,
wi h
H0
commu ing wi h
X
o exponen ial accu acy. Then
F=UXe−iH0T≈X e−iH∗T+O(ϵ) + O(e−cω/J ),
combined wi h quasilocal uni a y
U∗
yields Theo em 1's s uc u al decomposi ion. Subha monic
locking om
XOX =−O
odd ans o ma ion p ope y and e o supp ession.
B MBLDTC Spec al Pai ing and Eigens a e O de
B.1
l
-bi Diagonaliza ion and Quasilocal
˜
X
Uni a y
U
exis s making
UH0U†= ({τz
i})
; nea -
π
kick in his ep esen a ion becomes
˜
X=
UXU†≃Qi˜σx
i
, commu ing wi h
H
MBL
.
B.2
π
Spec al Pai ing
I
F≃˜
Xe−iH
MBL
T
, hen o eigens a e
|ψ⟩
F|ψ⟩=e−iET ˜
X|ψ⟩, F(˜
X|ψ⟩) = e−i(ET+π)|ψ⟩,
hus eigens a es die by
π
. A bi a y ini ial s a e expanded in pai ed subspace yields s a e-
independen
2T
subha monic esponse, consis en wi h supe conduc ing p ocesso spec aldynamical
obse a ions.
C Open Sys em Dissipa i e Time C ys al Spec al C i e ion
C.1 CPTP Mapping Pe onF obenius S uc u e
Assume
E
pe iphe al spec um con ains only
m
eigenphases
{e2πik/m}
, emaining spec um s ic ly
con ac ing. By posi i e ope a o spec al heo y (E ansHøegh-K ohn):
m
mu ually disjoin a -
ac o componen s exis ,
En
con e ges a bi a y ini ial s a e o pe iod-
m
limi cycle; spec al gap
∆
Liou
con ols con e gence a e and pe u ba ion esis ance.
C.2 In e acing wi h Algeb aic Synch oniza ion Theo y
Fo ime-independen Liou illian, can cha ac e ize pu ely imagina y eigen alues and pe sis en oscil-
la ion modes ia algeb aic symme yLie algeb a s uc u e, delimi ing condi ions o s able/me as able
synch oniza ion and mul i- equency commensu abili y; his pic u e consis en wi h abo e pe iph-
e al spec um g oup s uc u e.
8
D Mul i-F equency D i e and Tempo al Quasic ys al G oup Rep-
esen a ion P oo
D.1 Time T ansla ion G oup and Fini e Image
Quasipe iodic d i e makes ime ansla ion g oup
Zk
. Unde high- equency limi cons uc eec i e
e olu ion
U(n) = U e−iH⋆PiniTig(n)U†+O(e−cminiωi/J ),
whe e
g:Zk→G
is quo ien ep esen a ion o ni e g oup
G
. I
Im g
non- i ial, hen
CO(n)
exhibi s peaks on mul iple incommensu a e subha monic lines, dening empo al quasic ys al o de .
E Topological Time C ys al Logical O de and En anglemen En-
opy
E.1 Code Subspace Floque Uni a y
Unde su ace code Hamil onian
H
SC
and pe iodic sequence,
Flogical ≃XLe−iH
op
∗T
. Non-local
logical ope a o s exhibi pe iod-doubled igid esponse.
E.2 O de Pa ame e and Topological En anglemen En opy
Dene au oco ela ion
CWC(n)
o non-local loop ope a o
WC
and opological e m
γ
o subsys em
en opy
S(A)
. Expe imen s measu e non-ze o
γ
join ly wi h
WC
subha monic locking es ablishing
opological ime c ys al o de .
F Expe imen al E o Budge and Calib a ion
F.1 P e he mal Window Enginee ing
Gi en ha dwa e noise ampli ude
ε
, selec
ω
making
e−cω/J ≲ε/10
, wi h pulse a ea e o
|ϵ|≲ε
;
ensu e
T2≫τ∗∼ecω/J
.
F.2 Open Sys em Limi Cycle Noise Shaping
Realize single limi cycle phase egion ia
G/κ
and de uning scan measu ing
E
pe iphe al eigen alue
amily;
∆
Liou
and cohe encedecohe ence a io de e mine s abili y.
9