Quan um con ol o con inuous sys ems ia nonha monic
po en ial modula ion
Pio T. G ochowski1,2,3, Hannes Pichle 1,2, Cindy A. Regal4,5, and O iol Rome o-Isa 1,2,6,7
1Ins i u e o Quan um Op ics and Quan um In o ma ion o he Aus ian Academy o Sciences, A-6020 Innsb uck, Aus ia
2Ins i u e o Theo e ical Physics, Uni e si y o Innsb uck, A-6020 Innsb uck, Aus ia
3Depa men o Op ics, Palacký Uni e si y, 17. lis opadu 1192/12, 771 46 Olomouc, Czech Republic
4JILA, Na ional Ins i u e o S anda ds and Technology and Uni e si y o Colo ado, Boulde , Colo ado 80309, USA
5Depa men o Physics, Uni e si y o Colo ado, Boulde , Colo ado 80309, USA
6ICFO - Ins i u de Ciencies Fo oniques, The Ba celona Ins i u e o Science and Technology, 08860 Cas ellde els (Ba celona), Spain
7ICREA, Passeig Lluis Companys 23, 08010 Ba celona, Spain
We p esen a heo e ical p oposal o p epa -
ing and manipula ing a s a e o a single
con inuous- a iable deg ee o eedom con-
ined o a nonha monic po en ial. By u iliz-
ing op imally con olled modula ion o he po-
en ial’s posi ion and dep h, we demons a e
he gene a ion o non-Gaussian s a es, includ-
ing Fock, Go esman-Ki ae -P eskill, mul i-
legged-ca , and cubic-phase s a es, as well
as he implemen a ion o a bi a y uni a ies
wi hin a selec ed wo-le el subspace. Addi-
ionally, we p opose p o ocols o single-sho
o hogonal s a e disc imina ion, algo i hmic
cooling, and co ec ing o nonlinea e olu-
ion. We analyze he obus ness o his con-
ol scheme agains noise. Since all he p e-
sen ed p o ocols ely solely on he p ecise mod-
ula ion o he e ec i e nonha monic po en ial
landscape, hey a e ele an o se e al expe -
imen s wi h con inuous- a iable sys ems, in-
cluding he mo ion o a single pa icle in an
op ical weeze o la ice, o cu en in ci cui
quan um elec odynamics.
1 In oduc ion
The p epa a ion o a con inuous- a iable sys em in
a non-Gaussian quan um s a e is o pa amoun im-
po ance in a ious aspec s o quan um science. This
anges om undamen al es s o quan um mechan-
ics [1–5], h ough he design o quan um senso s [6–
9], o quan um in o ma ion p ocessing [10–18]. The
gene a ion o non-Gaussian s a es equi es a nonlin-
ea esou ce, o en in oduced h ough coupling o
an auxilia y deg ee o eedom, e.g., a wo-le el sys-
em [5,12,15,19–22]. On he o he hand, some
con inuous- a iable sys ems al eady possess in insic
nonha monici y in he po en ial o a canonical a i-
able (see Fig. 1). No able examples include he mo-
Pio T. G ochowski: [email p o ec ed]
Figu e 1: (a) Examples o con inuous- a iable nonlinea sys-
ems ha can be op imally con olled wi hou he need o
auxilia y sys ems—single a oms in op ical weeze s and lux-
unable ansmons. (b) Time-e ol ed p obabili y densi y
P(x, τ) = |ψ(x, τ)|2, du ing he s a e p epa a ion p o ocol
wi h he snapsho s o Wigne unc ions W(x, p)a he begin-
ning and he end o he p o ocol. The po en ial is Gaussian
and i s op imally con olled posi ion u(τ)is depic ed ia a
solid black line. The op panel shows a compa ison be ween
a weak and long sinusoidal d i e esonan wi h he g ound-
i s exci ed s a e ansi ion and an op imized, much as e
con ol. The bo om panel p esen s an op imal con ol lead-
ing o he GKP s a e. (c) Exci ed s a es occupa ion numbe s
|cn(τ)|2=|⟨ψn(x)|ψ(x, τ)⟩|2 o ψ0(x)→ψ1(x)Fock ex-
ci a ion p o ocols o (b) op. The le panel co esponds o
he slow Rabi lop, while he igh one shows he as op i-
mal con ol. U ilizing mo e s a es wi hin he nonha monic
po en ial accele a es and makes he con ol mo e e sa ile.
ion o a pa icle in a ap o ini e dep h [23,24]
and he cu en in an elec ic ci cui wi h a Josephson
junc ion [25]. These nonha monici ies in he po en ial
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a Xi :2311.16819 4 [quan -ph] 5 Aug 2025
a e ypically used o de ine a qubi wi hin con inuous-
a iable sys ems [26–30]. In con as , we in es iga e
whe he his in insic nonlinea i y su ices o imple-
men quan um p o ocols beyond he wo-dimensional
subspace. Since i is weake han he nonlinea i y p o-
ided by auxilia y wo-le el sys ems, we aim o assess
how a op imal con ol can compensa e o his lim-
i a ion.
Mo e speci ically, we de elop p o ocols o gene -
a e a ple ho a o s a es, including Fock, Go esman-
Ki ae -P eskill (GKP) [10], mul i-legged-ca [1,31],
and cubic-phase [10,32], using op imal con ol [33–35]
o he po en ial’s posi ion and dep h. Fu he mo e,
we employ his con ol mechanism o design p o ocols
ha implemen a bi a y uni a ies in selec ed sub-
spaces, enable single-sho o hogonal s a e disc imi-
na ion [36,37], spa ial s a e ans e , co ec o non-
Gaussian e olu ion, and pe o m algo i hmic cool-
ing [38,39]. These p o ocols could be implemen ed
wi h single a oms in op ical weeze s [40–44] and la -
ices [23,45] o ex end he mo ional s a es a ailable
in p o ocols wi h i ine an pa icles, e.g., e mionic
quan um p ocesso s [18]; o wi h lux- unable ans-
mons [25,28,46] o minimally in asi e s a e manip-
ula ion [c . Fig. 1(a)]. P e ious wo ks, including con-
ol o Bose-Eins ein condensa es [47–56], as a om
anspo [45,57,58], ion shu ling [59], ca s a e c e-
a ion h ough wo-pho on d i ing [60], and la ice in-
e e ome y [61] ha e demons a ed simila concep s.
Howe e , he ole o nonha monic po en ials in quan-
um in o ma ion p ocessing emains unde explo ed,
despi e hei na u al ele ance—pa icula ly o ex-
ploi ing, o example, he mo ion o single a oms in
scalable op ical weeze a ays [39].
The p epa a ion o con inuous- a iable modes in
quan um non-Gaussian s a es and hei subsequen
manipula ion ha e been an objec o inc easing in-
e es ac oss se e al pla o ms, including single neu-
al a oms [44], apped ions [62–65], supe conduc -
ing ci cui s [66–68], p opaga ing ligh a he elecom-
munica ion wa eleng h [69,70], e c. The con ol
scheme we p esen o e s a ealiza ion o a la ge a i-
e y o ele an asks and can be applied o a gene ic
con inuous- a iable pla o m ha in ol es nonha -
monici y. Ou p o ocols ake ad an age o he con-
ollabili y o he ex e nal po en ial landscape, na u-
ally o e ed by, e.g., op ical [39,44,45], elec ic [24],
and magne ic [71–73] po en ials, and hei hyb id
combina ions [62–65,74]. Wi h such schemes, a high
in o ma ion capaci y o bosonic deg ee o eedom is
ac i a ed ia p epa a ion and con ol o high-ene gy
and high-quali y quan um non-Gaussian s a es. Ou
p o ocols a e pa icula ly in e es ing and use ul o
sys ems wi hou well-con olled access o nonlinea
esou ces, e.g., in e nal deg ees o eedom o aux-
ilia y supe conduc ing ci cui s. Howe e , hey can
also be combined wi h such couplings o simul ane-
ous con ol o bo h bosonic and spin deg ees o ee-
dom, opening new possibili ies o , e.g., he use o
hype en anglemen [75].
The pape is s uc u ed as ollows. Sec ion 2in-
oduces de ails abou he physical model and op-
imiza ion echniques, while Sec ion 3p esen s se -
e al p o ocols ha can be achie ed h ough po en ial
modula ion, including s a e p epa a ion, co ec ing
o non-Gaussian e olu ion, implemen a ion o uni-
a ies, single-sho measu emen s, cooling, and spa ial
s a e ans e . Fu he , Sec ion 4analyzes he easi-
bili y o ou p oposal, add essing he speed limi s o
he p o ocols, hei obus ness agains decohe ence,
and how couplings o o he modes can be inco po a ed
in o op imiza ion. Sec ion 5concludes he pape wi h
a inishing discussion and an ou look.
2 Con ol o po en ial landscape
In he ollowing Sec ion, we p esen wo exempla y e-
aliza ions o one-dimensional con inuous- a iable sys-
ems o which ou p oposal can be applied—a single
neu al a om apped ia op ical o ces, ei he in a
weeze o in a la ice, and a lux- unable ansmon.
In he Subsec ion 2.1, we p o ide de ails o he in-
ol ed Hamil onians in hese pla o ms, in oduce a
uni e sal desc ip ion o ou con ol scheme, and ad-
d ess how much nonha monici y is needed o he p e-
sen ed p o ocols. The Subsec ion 2.2 is dedica ed o
de ails on he con ol op imiza ion o ou scheme.
2.1 Hamil onians o di e en sys ems
As a s a ing poin , we ocus on a one-dimensional
con inuous- a iable sys em desc ibed by wo conju-
ga e quad a u e ope a o s, which may co espond o
an a bi a y ealiza ion o a single mode, e.g., he mo-
ion o a single a om in an op ical weeze [44,76,77]
o a la ice [45,78], o phase and cha ge ope a o s in
a lux- unable ansmon [25,79,80]. In he o me
case, cohe en dynamics is d i en by a Hamil onian,
ˆ
H=ˆ
P2
2m+ [1 + a( )] Vhˆ
X−U( )i,(1)
whe e mis he mass o an a om, ˆ
Xis he posi ion
ope a o , ˆ
Pis he momen um ope a o , is ime, and
unc ions a( )and U( )con ol he dep h and posi ion
o he po en ial. He e, he apping po en ial V(ˆ
X)
is Gaussian o he weeze ,
V(ˆ
X) = V01−e−2ˆ
X2
w2
0,(2)
and a squa ed sine o he la ice,
V(ˆ
X) = V0sin22π
λˆ
X,(3)
whe e w0is he weeze wais , λis he op ical wa e-
leng h, and V0is ei he he weeze o he la ice
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dep h. Fo he ansmon case, he Hamil onian eads
ˆ
H= 4ECˆn2+EJcos πϕx( )
Φ0s1 + d2 an2πϕx( )
Φ0
×1−cos ˆφ−d an πϕx( )
Φ0.(4)
He e, he phase di e ence ope a o is
ˆφ=ˆφ1+ ˆφ2
2,(5)
whe e ˆφi=2πˆ
Φi
Φ0deno es he phase di e ence ac oss
he i h Josephson junc ion, and Φ0=h/2eis he mag-
ne ic lux quan um. The cha ge numbe ope a o is
de ined as
ˆn=ˆ
Q
2e,(6)
and he cha ge ene gy is
EC
h=e2
2hC ,(7)
whe e Cis he o al capaci ance. The o al Josephson
ene gy is
EJ=EJ1+EJ2,(8)
wi h EJi ep esen ing he Josephson ene gy o he i h
junc ion. The asymme y be ween he wo junc ions
is cha ac e ized by
d=EJ2−EJ1
EJ2+EJ1
.(9)
Finally, ϕx( )deno es he ime-dependen ex e nal
magne ic lux h eading he SQUID loop.
In he leading o de , each o hese se ups is ha -
monic wi h equency ω, yielding na u al ime τ=ω ,
canonical leng h and momen um scales,
ˆx=ˆ
X
X0
=ˆn
n0
=1
√2(ˆa†+ ˆa),
ˆp=ˆ
P
P0
=ˆφ
φ0
=1
√2i(ˆa†−ˆa),(10)
whe e ˆa†c ea es a single exci a ion and [ˆx, ˆp] = i. Fo
an a om apped in a weeze , we ha e
X0= (ℏ2w2
0/4mV0)1/4,
P0=ℏ/X0,
ω=q4V0/mw2
0,(11)
while o he op ical la ice,
X0= (ℏ2λ2/8π2mV0)1/4,
P0=ℏ/X0,
ω=p8π2V0/mλ2.(12)
Fo he lux- unable ansmon, he canonical leng h
and momen um scales ead
n0= (8EC/EJ)1/4,
φ0= 1/n0,
ω=p8ECEJ/ℏ.(13)
In he canonical a iables ˆx,ˆp, each o he Hamil-
onians (1),(4)can be ew i en as
ˆ
H
ℏω=1
2ˆp2+ [1 + a(τ)] [ˆx−u(τ)] ,(14)
whe e =V/ℏωis he dimensionless po en ial. The
con ol o he sys em is pe o med h ough op imal
modula ion o posi ion, u(τ), and dep h, a(τ), o he
po en ial. While o an a om in an op ical weeze ,
hey a e independen con ols ealized h ough, e.g.,
acous o-op ic modula o [77], o lux- unable ans-
mons, hey a e cons ained h ough a single con ol
unc ion, he in ensi y o he ex e nal lux [79],
a(τ) = s1+4pEC/2EJu2(τ)
1+4pEC/2EJu2(τ)d−2−1,
u(τ) = d
2η an πϕx(τ)
Φ0
.(15)
Such modula ions ha e been ealized in a ious o he
expe imen al pla o ms [45,54,55,61,81], whe e he
choice o ei he posi ion o dep h con ol depends
on easibili y wi hin a speci ic se up. Fo example,
Paul aps allow o bo h posi ion and dep h con ol
ia a ying cu en h ough elec odes gene a ing he
elec ic ield [54], while in op ical la ices, phase con-
ol allows e y p ecise posi ion con ol [45]. As o
op ical weeze s, he use o mul iple adio equency
ones makes a powe ul knob o he ime-dependen
con ol o bo h posi ion and dep h [44].
In gene al, we conside he shape o he po en ial
o be symme ic in x, and we ind i use ul o exp ess
i as
(x) = 1
2η2 0(ηx)≈1
2x2−1
6η2x4(16)
when expanded up o he second o de in he small
pa ame e η. No e ha , as he po en ial (x)is be-
yond quad a ic in x, he ensuing dynamics is nonlin-
ea , i.e., i does no p ese e he Gaussiani y o he
s a es. Fo he examples conside ed in his wo k, he
weeze po en ial hen eads as
0(x) = 3
21−exp −2
3x2,(17)
while o he la ice i is
0(x) = sin2x, (18)
and o he lux- unable ansmon,
0(x) = 1
2[1 −cos (2x)] (19)
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No e ha we ha e de ined he uni s so ha he unc-
ional o ms o hese po en ials ma ch up o he sec-
ond leading o de , allowing o a uni ied discussion o
nonha monici y. He e, ηis he measu e o he non-
ha monici y o he po en ial, compa ing he canonical
leng h o he cha ac e is ic leng h scale o he po en-
ial, and ansla es di ec ly in o an ene gy le el spli -
ing,
ω21 −ω10
ω=−1
2η2,(20)
whe e ωij is a ansi ion equency be ween i h and
j h le els. Fo ins ance, o an a om in a weeze , i
is
η=√3X0
w0
;(21)
o an op ical la ice, i is a Lamb-Dicke pa ame e
η=2πX0
λ;(22)
o he lux- unable ansmon, i is a mono onic unc-
ion o ansmon anha monici y α,
α
ω= EC
8EJ
=1
2η2;(23)
and o Ke oscilla o s, i can be associa ed wi h Ke
nonlinea i y κ∼ωη2/24 [82]. The ypical alues o
physical pa ame e s o he examples in oduced in
his sec ion lead o nonha monici ies no la ge han
η∼0.5(see Tab. 1) and can be uned down a
leas an o de o magni ude depending on he speci ic
se up. Hence, he e we will analyze such a pa ame e
egime. Ne e heless, many o he expe imen al pla -
o ms a e cha ac e ized by much lowe alues, such
as apped ions [24,83] o le i a ed mechanical os-
cilla o s [84,85]. The e, lowe nonha monici y is as-
socia ed wi h ei he la ge po en ial leng h scales o
la ge masses o apped objec s. In such cases, he
addi ional enhancemen o nonha monici y is needed,
e.g., h ough s a e exci a ion [85–87], o gene a e non-
Gaussian s a es.
Op ically apped a om Supe conduc ing ci cui
m101-102uEC/h MHz-GHz
w0,λ102-103nm EJ101-104EC
V0/kBµK-mK d0-1
Table 1: Typical physical pa ame e s o a oms held in op ical
weeze s o op ical la ices and lux- unable ansmons.
2.2 Con ol op imiza ion
Be o e p esen ing speci ic p o ocols, le us discuss pos-
sible me hods o designing con ols u(τ)and a(τ).
Wi hin each o he subsequen p o ocols, we aim o
achie e some speci ic goal—be i s a e p epa a ion,
uni a y implemen a ion, o o he s— ha can be quan-
i ied h ough a ewa d unc ion ha ough o be
maximized. A se o me hods o designing ime-
a ying con ols o he maximiza ion o such a e-
wa d unc ion is collec i ely called quan um op imal
con ol (QOC) [33,34,88]. In oduced in he eigh -
ies and since hen b oadly de eloped, QOC has be-
come one o he main quan um con ol ools, among
o he op imiza ion app oaches, such as adiaba ic pas-
sages [89], sho cu s o adiaba ici y [90], and com-
posi e pulses [91]. Va ious echniques ha e been es-
ablished o design he con ol pulses, elying on bo h
g adien - ee and g adien -based app oaches. The la -
e assume he abili y o di e en ia e he ewa d unc-
ion and include, among o he s, GRAPE (g adien as-
cen pulse enginee ing)- [92] and K o o -like [93,94]
echniques o op imizing con ols ha a e piecewise
cons an in ime. On he o he hand, he o me
ely only on he e alua ion o he ewa d unc ion
i sel . The examples a e nume ous, including Nelde -
Mead [95], e olu ion s a egies [96], simula ed anneal-
ing [97], and many o he s. No ably, ecen yea s ha e
b ough a apid su ge in he use o machine lea n-
ing app oaches o quan um con ol p oblems [98].
The choice o app oach elies on cons ain s gi en
by a speci ic expe imen al se up, including how accu-
a ely and as we can sol e he dynamics and wha
he limi a ions o con ol pulses a e, such as max-
imal Fou ie bandwid h and in ensi y. We choose
o u ilize he d essed chopped andom-basis ech-
nique (dCRAB) [99] ha is based on he Nelde -Mead
g adien - ee me hod. He e, he con ol unc ion is
expanded in o a Fou ie basis wi h andomized e-
quencies and a high- equency cu o co esponding
o app oxima ions o expe imen ally accessible band-
wid hs, i.e.,
u(τ) =
Np
X
k=1
akcos νkτ+bksin νkτ, (24)
whe e akand bka e pa ame e s o be op imized, νk
a e equencies ha a e p obabilis ically d awn om
he uni o m dis ibu ion in he ange [0, νmax][100],
νmax is he equency cu o , and Npis he numbe
o equency componen s. We ha e chosen his pa -
icula g adien - ee me hod as we conside se e al
di e en ewa d unc ions, i na u ally implemen s a
equency cu o o he con ol unc ion, and he e
a e a ailable well-de eloped open-sou ce packages, in-
cluding he Quan um Op imal Con ol Sui e [35],
which we u ilize o he op imiza ion. Howe e , de-
pending on he speci ics o he expe imen al imple-
men a ion, ou esul s can be achie ed wi h an al e -
na i e app oach, such as, e.g., GRAPE.
Gene ally speaking, con ollabili y o he sys em
desc ibed by Eq. (1)depends sensi i ely on he shape
and dep h o he po en ial. Fo shallow po en ials,
he ini e numbe o bound s a es can limi he se
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o achie able s a e ans o ma ions. Howe e , in he
case o su icien ly deep o unbounded po en ials—
such as he qua ic po en ial— ull con ollabili y can
be es ablished unde ce ain con ol schemes [101].
The chosen po en ial landscape is hen c ucial in de-
e mining he accessible Hilbe space and, conse-
quen ly, he e ec i eness o op imal con ol p o ocols.
Fo all he simula ions, he spli -s ep me hod [102]
was used o simula e he dynamics on a spa ial g id.
In he case o one-dimensional calcula ions, h ough-
ou op imiza ion uns, he nume ical pa ame e s ead:
numbe o spa ial g id poin s Nx= 28, whe e he g id
spanned x∈[−13,13], and numbe o ime g id poin s
Nτ= 500. A e op imizing he pulses, he dynam-
ics was un and u he op imized wi h inc eased ac-
cu acy o make su e he esul s a e con e ged. The
op imiza ion has been pe o med wi h QuoCS lib a y
e sion 0.0.44 [35]. The numbe o Fou ie compo-
nen s used o dCRAB a ied om 20 o 50, de-
pending on a pa icula p o ocol, and was andom-
ized om a uni o m dis ibu ion on a bandwid h ha
also a ied be ween di e en p o ocols. The choice
o hese op imiza ion pa ame e s was ine- uned o
each o he p o ocols, howe e , i was no exclusi e—
di e en choices usually yielded simila alues o spe-
ci ic ewa d unc ions, bu wi h di e en o al op i-
miza ion i e a ions. No e ha u he op imiza ion o
all he p o ocols owa ds lowe in ideli ies is possible
and would in ol e u he ine- uning o op imiza ion
pa ame e s.
3 P o ocols
In he ollowing sec ion, we discuss se e al p o o-
cols ha can be implemen ed ia a nonha monic po-
en ial modula ion. I includes a e sa ile Gaussian
(Sec. 3.2) and non-Gaussian s a e p epa a ion, bo h in
a single- (Sec. 3.1) and double-well (Sec. 3.5) po en ial
landscape, as well as he implemen a ion o uni a ies
(Sec. 3.4), single-sho measu emen s (Sec. 3.6), cool-
ing (Sec. 3.7), and co ec ing s a e e olu ion o non-
linea e olu ion due o nonha monici ies (Sec. 3.3).
3.1 S a e p epa a ion
As he i s ype o p o ocol, we conside s a e p epa-
a ion. The ini ial s a e o he sys em is assumed o
be he g ound s a e ψ0(x)o he nonha monic po en-
ial (x), well app oxima ed by he wa e unc ion
ψ0(x)≈1
π1/4e−x2/2,(25)
which is he g ound s a e o he leading ha monic ap-
p oxima ion, x2/2. Fo his p o ocol, we choose o
con ol only he posi ion o he po en ial u(τ), o max-
imize he ideli y
F=|⟨ψ(x, τ =τmax)|ψT(x)⟩|2(26)
Figu e 2: Examples o a s a e p epa a ion p o ocol in ol -
ing Gaussian [(a), (c), (e), (g)] and cosine [(b), (d), ( ),
(h)] single-well po en ials, wi h η= 0.25 and τmax/2π= 6,
using only op imized posi ion o he po en ial u(τ)in he
case o he Gaussian po en ial. The p o ocols include sec-
ond exci ed Fock [(a-b)], ca [(c-d), s= 9], cubic-phase
[(e- ), κ= 2, = 0.7], and GKP [(g-h), = 0.6,
(d1, d2, d3) = (−√4π, 0,√4π),(h1, h2, h3) = (1,2,1)] s a e
p epa a ion. P o ocol ideli ies Fa e: (a) 99.8% (b) 99.8%
(c) 92.7% (d) 93.4% (e) 97.2% ( ) 94.9% (g) 99.8% (h)
99.8%. See he cap ion o Fig. 1 o subplo and cu e leg-
end de ails.
wi h he a ge s a e ψT(x). We conside he p epa-
a ion o se e al s a es, including Fock, ini e GKP,
cubic-phase, and ca s a es wi h high ideli y in a
one-dimensional geome y. As hese s a es a e non-
Gaussian, he ole o nonha monici y o he po en ial
is e iden —i i was quad a ic, and dynamics linea ,
he Gaussiani y o he s a e would be p ese ed du -
Accep ed in Quan um 2025-08-04, click i le o e i y. Published unde CC-BY 4.0. 5
ing he e olu ion. In Fig. 1(b), we p esen an exam-
ple o he exci a ion p o ocol, whe e we specialize o
a Gaussian po en ial wi h η= 0.25. The a ge s a e
is a ini e GKP s a e,
ψGKP(x) = N
3
X
i=1
hie /2ψ0[e (x−di)] (27)
wi h (h1, h2, h3) = (1,2,1),(d1, d2, d3) =
(−√4π, 0,√4π), he squeezing pa ame e = 0.6, he
no maliza ion cons an N, and he whole p o ocol
is se o ake τmax/2π= 6. The p esen ed p o ocol
yields ideli y F ≈ 99.8%. The u he examples o
he s a e p epa a ion a e p esen ed in Fig. 2, bo h
o he Gaussian po en ial and o cosine po en ial
wi h he cons ain (15),η= 0.25 and d= 0.8.
Speci ically, we addi ionally show he second exci ed
Fock s a e,
ψ2(x) = 1
√2
1
π1/42x2−1e−x2/2,(28)
he squeezed cubic-phase s a e,
ψcub(x) = eiκx3e /2ψ0(e x),(29)
and he ca s a e,
ψca (x) = 1
√2[ψ0(x+s/2) + ψ0(x−s/2)],(30)
whe e κis he cubici y, is he squeezing pa ame e ,
and sis he sepa a ion be ween he cohe en s a es.
The speci ic alues o he s a e pa ame e s and ideli-
ies a e shown in he cap ion.
3.2 Gaussian ope a ions
As we conside nonlinea dynamics, a ele an ques-
ion is whe he i is s ill possible o implemen Gaus-
sian ope a ions wi hin his amewo k. In ou se ing,
his ypically in ol es p o ocols ha ansien ly gen-
e a e non-Gaussian s a es du ing he e olu ion, bu
yield a inal s a e ha is Gaussian. A ele an exam-
ple is he p epa a ion o squeezed acuum s a es, o
which we demons a e wo dis inc ealiza ions. The
i s p o ocol employs only displacemen modula ion,
while he second elies solely on dep h modula ion.
Bo h cases a e shown in Fig. 3and achie e high inal
ideli ies. The second example ep esen s a gene al-
ized e sion o he equency-jump p o ocol, in which
a sudden change in po en ial dep h exci es b ea h-
ing in he wa e unc ion, esul ing in squeezing. In
ou case [Fig. 3(b)], he achie ed squeezing su passes
wha would be expec ed om a single- equency-jump
p o ocol, which is undamen ally limi ed by he a io
o ini ial and inal equencies.
Figu e 3: Gaussian squeezing o he ini ial g ound s a e ia
(a) displacemen and (b) modula ion. The po en ial is aken
o be Gaussian wi h η= 0.15. P o ocols ideli ies ead: (a)
99.8% (b) 99.6%. See he cap ion o Fig. 1 o subplo and
cu e legend de ails.
Figu e 4: (a) Tempo al e olu ion o a ca s a e in a pe ec ly
ha monic po en ial. I o a es in he phase space and ully
e i es e e y ap pe iod. (b) E olu ion o a ca s a e in a
weeze cha ac e ized by η= 0.25. (c) E olu ion o a ca
s a e in a weeze cha ac e ized by η= 0.25, howe e , wi h
op imally con olled po en ial posi ion and dep h. (d) Fi-
deli ies be ween a ca s a e o a ing in a pe ec ly ha monic
po en ial and in Gaussian po en ials wi h di e en nonha -
monici ies wi hou (shades o g een) and wi h ( ed) op imal
con ol. See he cap ion o Fig. 1 o subplo and cu e leg-
end de ails.
3.3 S a e s abiliza ion
No e ha und i en e olu ion is nonlinea and, hence,
he p epa ed s a e ge s dis o ed du ing he e olu-
ion. In con as , when e ol ed in he pe ec ly ha -
monic po en ial, he s a e unde goes o a ion in he
phase space wi h he ap pe iod. In a ealis ic se up,
such a s a e p ese a ion in he o a ing ame is de-
Accep ed in Quan um 2025-08-04, click i le o e i y. Published unde CC-BY 4.0. 6
s oyed by he nonha monici y o he po en ial. The
emedy lies in ei he dec easing he nonha monici y
o , al e na i ely, mimicking pe ec phase space o a-
ion h ough op imal con ol [60]. We p esen he la -
e me hod, which consis s o wo op imiza ion s eps.
The i s one in ol es using he a e age ideli y wi h
espec o he pe ec ly o a ing s a e du ing a mul-
iple o he ap pe iod τmax =n2πas a ewa d unc-
ion,
F o =1
τmax Zdτ|⟨ψ(x, τ)|ψ o (x, τ)⟩|2,(31)
whe e
|ψ o (τ)⟩=e−iˆa†ˆaτ |ψ(0)⟩(32)
is he ini ial s a e unde going a phase-space o a ion
in a pe ec ly ha monic ap. A e op imizing F o ,
he nex s ep in ol es maximizing he inal ideli y
wi h he ini ial s a e a e a single ap pe iod,
F=|⟨ψ(x, τmax)|ψ(x, 0)⟩|2.(33)
The second s ep gua an ees high ideli y, should he
p o ocol be applied consecu i ely many imes, while
he i s one ensu es ha he s a e is no dis o ed a
any ime. In Fig. 4, we p esen an example o an op i-
mally con olled ca s a e s abiliza ion. The op imiza-
ion is conduc ed o a single ap pe iod, τmax = 2π,
wi h bo h posi ion and dep h con ol o a Gaussian
po en ial. Subsequen ly, he op imized con ol is ap-
plied six imes consecu i ely o e six ap pe iods and
compa ed wi h he ba e e olu ion in a Gaussian po-
en ial. The esul s showcase ha he p esen ed op i-
mal s abiliza ion o he s a e su passes ha o a Gaus-
sian po en ial wi h a ela i ely low nonha monici y
η= 0.05, wi hin his pa icula ime scale.
3.4 Uni a ies
The nex ype o p o ocol in ol es he implemen a-
ion o a speci ic uni a y wi hin a selec ed subspace.
Fo simplici y and ele ance o se e al p oposals and
ealiza ions o bosonic qubi s [13–17,62,103,104], we
analyze an example o a wo-le el subspace. This sub-
space can be spanned by any wo o hogonal s a es
ψ±(x), including wo lowes -lying ib a ional s a es
ψ0(x)and ψ1(x)(Fock basis), wo mu ually displaced
GKP s a es (GKP basis), ou -, and wo-legged-ca
bases. Fo he p o ocol conside ed, he ewa d unc-
ion is he subspace a e age ga e ideli y [105]
Fˆ
U=1
6[T (ˆ
Mˆ
M†) + |T ˆ
M|2],(34)
whe e
ˆ
M=ˆ
Pˆ
UTˆ
Uˆ
P, (35)
Figu e 5: (a,b) Two o hogonal ou -legged-ca s a es span-
ning a wo-le el subspace e ol ed in a posi ion- and dep h-
con olled single Gaussian well, ealizing a σxope a ion.
(c) Ca s a e p epa a ion using a double-well op imal con-
ol. The supe sc ip L(R) signi ies he s a e cen e ed in he
le ( igh ) po en ial well. (d,e) S a e ans e be ween he
wells—each o he ib a ional s a es, ψ0(x)and ψ1(x), is
ans e ed o he o he well, wi hou al e ing he ela i e
phase. See he cap ion o Fig. 1 o subplo and cu e leg-
end de ails.
ˆ
Pis a p ojec o on o a subspace, ˆ
UTis a a ge uni-
a y, and ˆ
Uis a uni a y gene a ed h ough Eq. (14).
Taking ad an age o he op imized posi ion o he po-
en ial u(τ)and addi ional sligh , op imized modu-
la ion o he dep h a(τ)( o inc ease he ideli y o
he p o ocol), we p esen examples o uni a ies, e.g.,
σxo Hadama d ope a ions, o he abo e-men ioned
subspaces. In Fig. 5(a,b), we show an exempla y case
o ou -legged-ca basis and σxope a ion pe o med
wi h a Gaussian po en ial, whe e he basis s a es a e
gi en by
ψ±
4LC(x) = N±[φβ(x) + φ−β(x)±φiβ(x)±φ−iβ(x)] ,
(36)
espec i ely [106,107]. He e, he cohe en s a e eads
φβ(x) = 1
π1/4exph−(x−√2 Re β)2/2 + i√2xIm βi,
(37)
N±a e he no maliza ion cons an s, and we use β=
2. Again, he p o ocol is pe o med wi h high ideli y,
Accep ed in Quan um 2025-08-04, click i le o e i y. Published unde CC-BY 4.0. 7
Fˆ
U≈99%. In Fig. 6, we show addi ional examples
o he Gaussian po en ial, including, among o he s,
he GKP basis, whe e he basis is spanned by wo
mu ually displaced GKP s a es,
ψ±=ψGKP(x±d3/4),(38)
whe e d3is de ined as in (27)and he squeezing and
displacemen need o be enough o he s a es no
o o e lap. Fu he uni a y implemen a ions, bu o
he cosine po en ial wi h he cons ain (15), a e p e-
sen ed in he Appendix A.
3.5 Double-well po en ial landscape
Up o now, we ha e analyzed quan um dynamics ak-
ing place in a single well o a po en ial landscape.
He e, we b ing ou a en ion o he nonha monic po-
en ial landscape case ha in ol es wo independen ly
con olled po en ial wells. Such a ealiza ion is a ail-
able o op ical la ices, op ical weeze s [42,77,108–
110], o a ious ci cui quan um elec odynamics se-
ups [111]. Fi s , we add ess he s a e p epa a ion
and single-pa icle uni a y implemen a ion wi h wo
po en ial wells. We assume ha we ha e wo inde-
penden ly con olled wells a ou disposal, such ha
he o al po en ial eads
(x) = X
i={1,2}
1
2η2 0[η(x−ui(τ))] (39)
wi h wo independen posi ion con ol unc ions,
ui(τ). He e, we assume ha 0con ains only a single
well wi h some cha ac e is ic wid h. Con olling he
ela i e dis ance be ween he wells amoun s o chang-
ing he ba ie heigh , which a ec s he coupling be-
ween he bound s a es in each o he wells. I can be
unde s ood as mode mul iplexing [112], accele a ed
h ough he op imal con ol. In Fig. 5(c), we show a
balanced ca s a e p epa a ion u ilizing wo Gaussian
po en ial wells,
ψ0(x±s/2) →1
√2[ψ0(x±s/2) + iψ0(x∓s/2)],
(40)
whe e sis he sepa a ion be ween he wells. No e ha
he p esen ed con ol is symme ic, u1(τ) = −u2(τ) =
u(τ), and he ins an aneous sepa a ion be ween he
wells is cons ained in he op imiza ion. The ideli y
o he p esen ed p o ocol yields F ≈ 99%. The ca
s a es p oduced in such a manne a e pe inen o
he undamen al es s o quan um mechanics, espe-
cially o massi e objec s. This p o ocol, along wi h
a as op imized anspo [45], should allow o a
as mac oscopic ca s a e c ea ion and in e e ome -
ic p o ocols.
The double-well po en ial has also been shown o
p o ide a pla o m o aul - ole an quan um com-
pu ing wi h so-called Ke -ca s [15]. The e, he com-
pu a ional subspace is spanned by supe posi ions o
Figu e 6: E olu ion o selec ed o hogonal s a es in a Gaus-
sian po en ial cha ac e ized by η= 0.25 and wi h op imally
con olled displacemen u(τ)and in ensi y a(τ). The solid
line shows u(τ)and he p o ocol las s τmax/2π= 9. (a-b)
σxuni a y wi hin a subspace spanned by ψ0and ψ1Fock
s a es. The ideli y eads Fˆ
U≈99.8%. (c-d) Hadama d
uni a y wi hin a subspace spanned by GKP s a es wi h
= 0.7,(d1, d2, d3) = (−√6π, 0,√6π). The ideli y eads
Fˆ
U≈96.4%. (e- ) σyuni a y wi hin a subspace spanned by
ou -legged-ca s a es s a es wi h β= 2. The ideli y eads
Fˆ
U≈96.2%. See he cap ion o Fig. 1 o subplo and cu e
legend de ails.
he g ound s a es o he espec i e wells, co espond-
ing o he (almos ) degene a e g ound-s a e mani old
o a ull double-well po en ial,
ψ±
KC(x) = 1
√2[ψ0(x+s/2) ±ψ0(x−s/2)].(41)
Wi h op imal con ol, one can again ealize a bi a y
uni a ies wi hin his subspace, u ilizing he subspace
a e age ga e ideli y (34). Examples o σx,σy, and
Hadama d uni a ies a e p esen ed in de ail in Ap-
pendix B. Fu he mo e, independen con ol o wo
wells can also be used o pe o m a s a e-p ese ing
anspo be ween he wells [113]. He e, he ini ial
s a e is p epa ed wi hin a wo-dimensional subspace
spanned by wo o hogonal s a es localized in he le
Accep ed in Quan um 2025-08-04, click i le o e i y. Published unde CC-BY 4.0. 8
(L) well,
ψL
±(x) = ψ±(x+s/2).(42)
The aim o he p o ocol is o ans e such a s a e o
a subspace localized in he igh (R) well,
ψR
±(x) = ψ±(x−s/2).(43)
Then, i we in oduce he ollowing ma ix,
ˆ
M′=ψL
+(x, τmax)ψR
+(x) ψL
+(x, τmax)ψR
−(x)
ψL
−(x, τmax)ψR
+(x) ψL
−(x, τmax)ψR
−(x),
(44)
whe e ψL
±(x, τ)is e ol ed acco ding o he po en-
ial (39), we can de ine he s a e ans e ideli y,
Fs =1
6[T (ˆ
M′ˆ
M′†) + |T ˆ
M′|2],(45)
as he ewa d unc ion. In Fig. 5(d,e), we show he
example o a s a e o he wo lowes ib a ional le els
o he le well ans e ed o he igh one wi h Fs ≈
99%.
3.6 Single-sho disc imina ion
A e p esen ing s a e p epa a ion and uni a y im-
plemen a ion in a ious nonha monic po en ial land-
scapes, we mo e on o wo p o ocols ha pe o m
single-sho disc imina ion be ween wo o hogonal
s a es ψ±(x)(see Fig. 7), p o iding an al e na i e
o al eady exis ing me hods in ol ing auxilia y sys-
ems in, e.g., supe conduc ing ci cui s [36]. The i s
one in ol es a single nonha monic well and is pe -
o med h ough imp in ing opposi e momen um kicks
on o each o he s a es ia a po en ial displacemen ,
ψ±(x)→ψ±
k(x) = ψ±(x)e±ikx,(46)
whe e kis chosen such ha wo phase-imp in ed
s a es a e nono e lapping in phase space. A e
he phase imp in ing, a selec i e measu emen akes
place, which can be ealized in, e.g., an op ical weeze
se up h ough he subsequen elease o he ap, in
which ime-o - ligh e olu ion e eals spa ially sepa-
a ed de ec ion clicks o each o he s a es [44,114].
The ewa d unc ion o such a momen um-kick p o-
ocol in ol es a sum o equally weigh ed ideli ies,
Fmk =1
2X
j=±Dψj(x, τmax)ψj
k(x)E
2(47)
No e ha in con as o he uni a y implemen a ion,
he e he inal ela i e phase be ween he s a es does
no ma e , as we aim only o dis inguish he ini ial
s a es. As an example, in Fig. 7ψ±(x)a e aken o be
he wo lowes eigens a es o he well, and he p o ocol
is shown in a schema ic way. In Fig. 8, we show in de-
ail speci ic ealiza ions o his and o he momen um-
kick p o ocols, u ilizing a double-well Gaussian po-
en ial. They also in ol e o he choices o o hogonal
Figu e 7: Two p oposed disc imina ion p o ocols, based on
a phase-space sepa a ion. (i) The opposi e momen um kicks
a e imp in ed on o each o he s a es. (ii) S a es a e sepa-
a ed spa ially ia a second po en ial well.
Figu e 8: Examples o he implemen a ion o momen-
um kick p o ocols ia an op imally con olled single-well
Gaussian po en ial wi h η= 0.25,τmax/2π= 6, and
k= 2. (a-b) ψ+(x) = ψ0(x)and ψ−(x) = ψ1(x),
(c-d) ψ±(x)=[ψ0(x)±ψ1(x)]/√2, and (e- ) ψ±(x) =
[ψ0(x)±iψ1(x)]/√2. Momen um-kick ideli ies ead (a-b)
Fmk ≈98.5%, (c-d) Fmk ≈99.4%, and (e- ) Fmk ≈99.5%.
See he cap ion o Fig. 1 o subplo and cu e legend de ails.
s a es o he disc imina ion, namely eigens a es o
bo h σxand σyPauli ma ices. Such disc imina ion
p ocedu es hen co espond o σxand σymeasu e-
men s wi hin a gi en wo-le el subspace.
The second disc imina ion p o ocol in ol es a
Accep ed in Quan um 2025-08-04, click i le o e i y. Published unde CC-BY 4.0. 9
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Accep ed in Quan um 2025-08-04, click i le o e i y. Published unde CC-BY 4.0. 18
A Single-well wo-le el uni a y imple-
men a ions
In his sec ion, we p esen examples o he imple-
men a ion o selec ed uni a ies wi hin a wo-le el sub-
space spanned by ei he Fock, GKP, o ou -legged-ca
s a es. Fig. 6shows esul s o a single-well Gaussian
po en ial, while Fig. A1 conce ns a cosine one.
Figu e A1: E olu ion o selec ed o hogonal s a es in a co-
sine po en ial cha ac e ized by η= 0.2and wi h op imally
con olled displacemen u(τ)and in ensi y a(τ) ha a e con-
s ained h ough he lux con ol o a lux- unable ansmon
wi h d= 0.8. The solid line shows u(τ)and he p o-
ocol las s τmax/2π= 9. (a-b) Hadama d uni a y wi hin
a subspace spanned by |0⟩and |1⟩Fock s a es. The i-
deli y eads Fˆ
U≈99.99%. (c-d) σyuni a y wi hin a sub-
space spanned by GKP s a es, ψ±=ψGKP(x±d3/4) wi h
= 0.7,(d1, d2, d3) = (−√6π, 0,√6π). The ideli y eads
Fˆ
U≈89.5%. (e- ) σxuni a y wi hin a subspace spanned by
ou -legged-ca s a es s a es wi h β= 2. The ideli y eads
Fˆ
U≈94.0%. See he cap ion o Fig. 1 o subplo and cu e
legend de ails.
B Double-well wo-le el uni a y imple-
men a ions
In Fig. B1, we p esen examples o he implemen a-
ion o selec ed uni a ies wi hin a wo-le el subspace
spanned by Ke -ca s a es, ψ±(x) = [ψ0(x+s/2) ±
iψ0(x−s/2)]/√2wi h s= 9 ia a double-well Gaus-
sian po en ial wi h η= 0.25.
Figu e B1: E olu ion o selec ed o hogonal Ke -ca s a es
in a double-well Gaussian po en ial wi h op imally con olled
posi ion displacemen s u1(τ)and u2(τ). The solid lines show
u1(τ)and u2(τ)and he p o ocol las s τmax/2π= 6. (a-b)
σxuni a y. The ideli y eads Fˆ
U≈99.2%. (c-d) σxuni a y.
The ideli y eads Fˆ
U≈90.9%. (e- ) Hadama d uni a y.
The ideli y eads Fˆ
U≈95.9%. See he cap ion o Fig. 1 o
subplo and cu e legend de ails.
Accep ed in Quan um 2025-08-04, click i le o e i y. Published unde CC-BY 4.0. 19
C Selec i e s ealing p o ocols
In Fig. C1, we p esen examples o he implemen a ion
o selec i e s ealing p o ocols ia an op imally con-
olled double-well Gaussian po en ial wi h η= 0.25
and τmax/2π= 6.
Figu e C1: Spa ial sepa a ion o o hogonal s a es ψ±(x+
s/2) →ψ±(x∓s/2) wi h s= 9 o (a-b) ψ+(x) = ψ0(x)
and ψ−(x) = ψ1(x), (c-d) ψ±(x)=[ψ0(x)±ψ1(x)]/√2,
and (e- ) ψ±(x) = [ψ0(x)±iψ1(x)]/√2. Fideli ies ead (a-
b) Fˆ
U≈98.9%, (c-d) Fˆ
U≈98.9%, and (e- ) Fˆ
U≈97.4%.
See he cap ion o Fig. 1 o subplo and cu e legend de ails.
D Fou ie ans o ms o con ol unc-
ions
In his sec ion, we p esen Fou ie ans o ms o dis-
placemen con ol unc ions,
Fu=Zu(τ) exp−i˜ω
ωτ
,(62)
o all he p esen ed p o ocols. Speci ically, Figs. D-
1,D-5, and D-9 pe ain o he igu es om he main
ex , while Figs. D-2,D-6,D-A1,D-B1,D-8,D-C1,
and D-11 a e associa ed o he Supplemen al Ma e i-
als.
Fig. 1(b) op
Fig. 1(b)cen e
Fig. 1(b)bo om
2 4 6 8 10
Figu e D-1: Fou ie ans o ms o displacemen con ol unc-
ions u(τ) om Fig. 1in he main ex . The e ical axis is
in a bi a y uni s, no malized o he highes equency con-
ibu ion.
Accep ed in Quan um 2025-08-04, click i le o e i y. Published unde CC-BY 4.0. 20
Fig. 2(a)
Fig. 2(b)
Fig. 2(c)
Fig. 2(d)
Fig. 2(e)
Fig. 2( )
Fig. 2(g)
Fig. 2(h)
2 4 6 8 10
Figu e D-2: Fou ie ans o ms o displacemen con ol unc-
ions u(τ) om Fig. 2. The e ical axis is in a bi a y uni s,
no malized o he highes equency con ibu ion.
Fig. 3(a)
2 4 6 8 10
Figu e D-3: Fou ie ans o m o displacemen con ol unc-
ion u(τ) om Fig. 3(a). The e ical axis is in a bi a y
uni s, no malized o he highes equency con ibu ion.
Fig. 5(a,b)
Fig. 5(c)
Fig. 5(d,e)
Fig. 5(d,e)
2 4 6 8 10
Figu e D-5: Fou ie ans o ms o displacemen con ol unc-
ions u(τ) om Fig. 5in he main ex . The e ical axis is
in a bi a y uni s, no malized o he highes equency con-
ibu ion.
Fig. 6(a,b)
Fig. 6(c,d)
Fig. 6(e, )
2 4 6 8 10
Figu e D-6: Fou ie ans o ms o displacemen con ol unc-
ions u(τ) om Fig. 6. The e ical axis is in a bi a y uni s,
no malized o he highes equency con ibu ion.
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Fig. 8(a,b)
Fig. 8(c,d)
Fig. 8(e, )
2 4 6 8 10
Figu e D-8: Fou ie ans o ms o displacemen con ol unc-
ions u(τ) om Fig. 8. The e ical axis is in a bi a y uni s,
no malized o he highes equency con ibu ion.
Fig. 9
Fig. 9
2 4 6 8 10
Figu e D-9: Fou ie ans o ms o displacemen con ol unc-
ions u(τ) om Fig. 9in he main ex . The e ical axis is
in a bi a y uni s, no malized o he highes equency con-
ibu ion.
Fig. 11(a)
Fig. 11(b)
Fig. 11(c)
2 4 6 8 10
Figu e D-11: Fou ie ans o ms o displacemen con ol
unc ions u(τ) om Fig. 11. The e ical axis is in a bi a y
uni s, no malized o he highes equency con ibu ion.
Fig. A1(a,b)
Fig. A1(c,d)
Fig. A1(e, )
2 4 6 8 10
Figu e D-A1: Fou ie ans o ms o displacemen con ol
unc ions u(τ) om Fig. A1. The e ical axis is in a bi a y
uni s, no malized o he highes equency con ibu ion.
Fig. B1(a,b)
Fig. B1(a,b)
Fig. B1(c,d)
Fig. B1(c,d)
Fig. B1(e, )
Fig. B1(e, )
2 4 6 8 10
Figu e D-B1: Fou ie ans o ms o displacemen con ol
unc ions u(τ) om Fig. B1. The e ical axis is in a bi a y
uni s, no malized o he highes equency con ibu ion.
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Fig. C1(a,b)
Fig. C1(a,b)
Fig. C1(c,d)
Fig. C1(c,d)
Fig. C1(e, )
Fig. C1(e, )
2 4 6 8 10
Figu e D-C1: Fou ie ans o ms o displacemen con ol
unc ions u(τ) om Fig. C1. The e ical axis is in a bi a y
uni s, no malized o he highes equency con ibu ion.
E Dep h modula ions
Some o he p o ocols, p esen ed in bo h he main
ex and he supplemen a y ma e ials, u ilized addi-
ional op imal con ol o he po en ial’s dep h, a(τ).
In Figs. E-5,E-6, and E-8, hese con ol unc ions a e
shown.
Fig. 5(a,b)
-0.2
0.2
Fig. 5(d,e)
12345
-0.2
0.2
Figu e E-5: Dep h con ol unc ions a(τ) om Fig. 5.
Fig. 6(a,b)
-0.2
0.2
Fig. 6(c,d)
-0.2
0.2
Fig. 6(e, )
12345678
-0.2
0.2
Figu e E-6: Dep h con ol unc ions a(τ) om Fig. 6.
Fig. 8(a,b)
-0.2
0.2
Fig. 8(c,d)
-0.2
0.2
Fig. 8(e, )
12345
-0.2
0.2
Figu e E-8: Dep h con ol unc ions a(τ) om Fig. 8.
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