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Resea ch
Ci e his a icle: Li J, Chen X, Ruschhaup A.
2022 Fas anspo o Bose–Eins ein
condensa es in anha monic aps. Phil. T ans.
R. Soc. A 380: 20210280.
h ps://doi.o g/10.1098/ s a.2021.0280
Recei ed: 11 July 2022
Accep ed: 20 Sep embe 2022
One con ibu ion o 15 o a heme issue
‘Sho cu s o adiaba ici y: heo e ical,
expe imen al and in e disciplina y
pe spec i es’.
Subjec A eas:
quan um physics, a omic and molecula
physics, quan um enginee ing
Keywo ds:
quan um con ol, Bose–Eins ein condensa es,
a omic anspo , sho cu s o adiaba ici y
Au ho o co espondence:
Jing Li
e-mail: [email p o ec ed]
Fas anspo o
Bose–Eins ein condensa es
in anha monic aps
Jing Li1,XiChen
2,3 and And eas Ruschhaup 1
1Depa men o Physics, Uni e si y College Co k, Co k, T12 H6T1
I eland
2Depa men o Physical Chemis y, Uni e si y o he Basque
Coun y UPV/EHU, Apa ado 644, 48080 Bilbao, Spain
3EHU Quan um Cen e , Uni e si y o he Basque Coun y UPV/EHU,
48940 Leioa, Spain
JL, 0000-0002-7565-3933;XC,0000-0003-4221-4288;
AR, 0000-0002-6044-993X
We p esen a me hod o anspo Bose–Eins ein
condensa es (BECs) in anha monic aps and in he
p esence o a om–a om in e ac ions in sho imes
wi hou esidual exci a ion. Using a combina ion
o a a ia ional app oach and in e se enginee ing
me hods, we de i e a se o E mako -like equa ions
ha ake in o accoun he coupling be ween he
cen e o mass mo ion and he b ea hing mode. By
an app op ia e in e se enginee ing s a egy o hose
equa ions, we hen design he ap ajec o y o
achie e he desi ed bounda y condi ions. Nume ical
examples o cubic o qua ic anha monici ies a e
p o ided o as and high- ideli y anspo o BECs.
Po en ial applica ions a e a om in e e ome y and
quan um in o ma ion p ocessing.
This a icle is pa o he heme issue ‘Sho cu s
o adiaba ici y: heo e ical, expe imen al and
in e disciplina y pe spec i es’.
1. In oduc ion
The accu a e manipula ion o ul acold a oms is a
key p e equisi e o implemen quan um echnologies
wi hin a omic, molecula and op ical science [1]. In
pa icula , he anspo o indi idual a oms and
o he mal o Bose-condensed clouds using mo ing
aps has been demons a ed in many expe imen s
[2–12] o di e en goals in quan um in o ma ion
p ocessing and me ology. In all quan um echnologies,
2022 The Au ho s. Published by he Royal Socie y unde he e ms o he
C ea i e Commons A ibu ion License h p://c ea i ecommons.o g/licenses/
by/4.0/, which pe mi s un es ic ed use, p o ided he o iginal au ho and
sou ce a e c edi ed.
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p ese ing quan um cohe ence and achie ing high inal ideli ies in sho imes is o c ucial
impo ance. One possibili y is called sho cu s o adiaba ici y (STA) [13,14] which p o ides a
oolbox o con ol bo h he in e nal and ex e nal deg ees o eedom o a quan um sys em in
as e - han adiaba ic imes.
Va ious sho cu s o adiaba ic anspo ha e been p oposed: Lewis–Riesen eld in a ian -
based in e se enginee ing [15–19], enhanced STA scheme [20–22], he Fou ie op imiza ion [23],
as - o wa d scaling me hod [24,25] and he coun e -diaba ic d i ing [26] ha e been heo e ically
pu o wa d, and expe imen ally demons a ed o a ious sys ems [7,10,11,27]. The possibili y o
ope a e wi h sho imes no only educes he sensi i i y o low- equency noise, bu also allows
o imp o ed measu emen s a is ics in he o al ime a ailable o he expe imen .
Di e en app oaches o anspo ing pa icles ha e been implemen ed. Neu al a oms
ha e been anspo ed as Bose–Eins ein condensa es (BECs) [2], he mal a omic clouds [28]o
indi idually [5], using magne ic o op ical aps. The commonly used aps o ul acold a oms
based on elec omagne ic ields a e ne e pe ec ly ha monic. The weak cubic anha monici y
plays a ole when a BEC is anspo ed pe pendicula o he a om chip su ace [29]. The
qua ic anha monici y is signi ican when app oxima ing he po en ial o an op ical weeze s o
anspo [7,16]. Thus cancelling he anha monic con ibu ions o he apping po en ial is i al
o use ul con ol schemes and is al eady a di icul echnical challenge o a s a ic ap [30].
Anha monici ies can ha e an impo an impac on he dynamics as obse ed in a om cooling
[31], collec i e modes [32] o wa e packe dynamics [33]. In mos cases, he anha monic aps
a e conside ed as a pe u ba ion o a ha monic one. Pe u ba ion heo y has been used o design
sho cu p o ocols o expansion/comp ession [34] and anspo [35]. O cou se, he esul s a e
limi ed by he p emises o pe u ba ion heo y, i.e. by small anha monici ies. Conside ing a non-
pe u ba i e scena io is hus o much in e es .
In his pape , we p opose o in e se enginee apid and obus anspo o an in e ac ing BEC
in anha monic aps using a a ia ional app oach. The me hod elies on a a ia ional o mula ion
o he dynamics o de i e a se o coupled E mako -like and New on-like equa ions, om which
he ap ajec o y is in e ed in e pola ing be ween he desi ed bounda y condi ions. In §2, we
explain he a ia ional o malism. In §3, we wo k ou he explici solu ions o qua ic and cubic
anha monici ies o he con ining po en ial, and illus a e he e iciency o he me hod wi h a ious
nume ical examples. In §4, we will discuss he esul s.
2. Model, Hamil onian and me hod
Fo a ciga -shaped ap wi h s ong ans e se con inemen , e.g. ω⊥>> ω, i is app op ia e o
conside a one-dimensionless o mula by eezing he ans e se dynamics o he espec i e
g ound s a e and in eg a ing o e he ans e se a iables [36]. The e ec i e a omic in e ac ion
is deno ed by g=2asω⊥N/ωaho, wi h as he in e a omic sca e ing leng h and aho =¯
h/(mω).
The esul ing dimensionless o m o G oss–Pi ae skii equa ion (GPE) [37] can be w i en as
i∂ψ(x, )
∂ =−1
2
∂2
∂x2+V(x, )+g|ψ(x, )|2ψ(x, ), (2.1)
whe e
V(x, )=1
2[x−x0( )]2+κ
3![x−x0( )]3+λ
4![x−x0( )]4, (2.2)
whe e ψ(x, ) is he axial wa e unc ion o he condensa e wi h no maliza ion condi ion
+∞
−∞ |ψ(x, )|2dx=N. The a ac i e and epulsi e in e ac ions a e deno ed by g<0andg>0,
espec i ely. The axial ha monic ap equency is ω. The po en ial cen e x0( ) is ime-dependen
o anspo . No e ha he po en ial in equa ion (2.2) consis s wo ypes o anha monici ies, one
is cubic (κ≥0) and he o he is qua ic (λ≥0) anha monici y, which is shown in igu e 1.
To apply he a ia ional app oach, we i s de ine an ansa z o he wa e unc ion wi h a ew
ee pa ame e s and e alua e he Lag angian densi y. The minimiza ion o he o al Lag angian
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(a)(b)
Figu e 1. Cubic (a)andqua ic(b) anha monic po en ials ( ed solid lines) compa ed wi h he ha monic coun e pa s (blue
dashed lines). The g ound s a es a e plo ed o di e en po en ials. (Online e sion in colou .)
wi h espec o he ee pa ame e s p o ides equa ions o mo ion o he ee pa ame e s [38]. This
app oach is equi alen o a momen me hod [39].
We assume a gene al Gaussian ansa z,
ψ(x, )=A( )exp−(x−xc( ))2
2a( )2exp[ib( )(x−xc( ))2+ic( )(x−xc( )) +iφ( )], (2.3)
whe e he ime-dependen pa ame e s A( ), a( ), b( ), c( )andφ( ) ep esen , espec i ely, he
ampli ude, wid h, chi p, eloci y and global phase. The wa e unc ion cen e o mass is xc( ). In
he ollowing, we omi in hose a iables o simpli ica ion. The no maliza ion condi ion yields
A=N/(a√π).
The Lag angian densi y which co esponds o equa ion (2.1) eads [38]
L=i
2∂ψ
∂ ψ∗−∂ψ∗
∂ ψ−1
2
∂ψ
∂x
2
−g
2|ψ|4−V(x)|ψ|2. (2.4)
Inse ing he ansa z (2.3) in o equa ion (2.4), we ind an e ec i e Lag angian [38] by in eg a ing
he Lag angian densi y o e he whole coo dina e space, L=+∞
−∞ Ldx. The Eule –Lag ange
minimiza ion is pe o med o e Land wi h espec o he ee pa ame e s and he condi ions
δL/δξ =0whe eξ=a,b,co xc. Fou coupled equa ions esul o (˙
a,˙
b,˙
xc,˙
c), a e gi en by
˙
a=2ab, (2.5)
˙
b=1
2a4−1
2[1 −4κ(x0−xc)+18λ(x0−xc)2]−2b2+gN
2√2πa3, (2.6)
˙
c=(1 +18λa2)(x0−xc)−2κ(x0−xc)2+12λ(x0−xc)3−κa2(2.7)
and ˙
xc=c, (2.8)
which can be condensed in o wo second-o de coupled equa ions o he wid h aand he
wa epacke cen e xc,
¨
a=1
a3−a[1 −4κq+18λq2]+gN
√2πa2−9λa3(2.9)
and
¨
xc=(1 +18λa2)q−2κq2+12λq3−κa2, (2.10)
whe e q=x0−xcis he displacemen be ween he cen e o he ha monic e m and he
wa epacke . In equa ion (2.9), he cen e o mass mo ion xcis s ongly coupled wi h he wid h a
o he wa e unc ion h ough he anha monic e ms o he con ining po en ial. When we conside
an adiaba ic anspo such as q=0, one can see ha he cubic anha monici y κis s ongly
coupled wi h he wid h ain equa ion (2.10). Al e na i ely, he qua ic anha monici y λwill c ea e
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b ea hing mode due o he s ong coupling wi h he wid h ain he case o q=0. The nonlinea i ies
in oduced by a om–a om in e ac ions do no gene a e any coupling wi h anha monici ies, as is
known o ha monic aps [16].
This sys em gene alizes he s uc u e ound o ha monic aps ia in a ian -based in e se
enginee ing [16]. In he absence o anha monici ies (κ=0andλ=0), he wo coupled equa ions
(2.9) and (2.10) educe o an E mako equa ion [40] and a New on equa ion [16] o a single a om
(o ion) o a BEC. By con as , equa ion (2.10) o he ajec o y o he cen e o mass xccan be
gene ically eco e ed om he Eh en es heo em, and is he e o e immune o he p ecise shape o
he ansa z. In wha ollows, we shall exploi hese coupled equa ions o in e se enginee sho cu
o adiaba ic anspo o BECs.
3. In e se enginee ing
In his sec ion, we ocus on he as and high ideli y anspo o a BEC om a s a iona y s a e a
ini ial posi ion x0(0) =0 o a a ge s a e wi h x0( )=din a ini e ime . The desi ed dis ance o
po en ial is d. We will conside he cases o cubic (see §3a) and qua ic (see §3b) anha monici ies
indi idually. In pa icula , he ajec o y x0( ) o he po en ial cen e can be designed by using
in e se enginee ing me hods applied o he se o equa ions (2.9) and (2.10). Fu he mo e, we will
p o ide nume ical examples ha con i m he e ec i eness o he me hod.
(a) Cubic anha monici y
Le us conside a po en ial wi h cubic anha monici y [18],
V(x, )=1
2(x−x0)2+1
3!κ(x−x0)3. (3.1)
When κ=0andλ=0, we subs i u e he condi ion ¨
xc=¨
x0−¨
qin o he coupled di e en ial
equa ions (2.9) and (2.10), which can be simpli ied in o
¨
a=1
a3−a+gN
√2πa2+4κaq (3.2)
and
¨
q=¨
x0−q+κa2+2κq2. (3.3)
The second equa ion may be ega ded as a second-o de di e en ial equa ion o q.We equi e
ha bo h ini ial and inal s a es a e s a iona y s a es. Fi s , we can calcula e he ini ial and inal
condi ions o he unc ion qwhich a e q(0) =q( )=Q. By imposing ¨
x0−¨
q=0 in equa ion (3.3),
one ob ains
Q=
1−1−8a2
0κ2
4κ, (3.4)
whe e a0deno es he ini ial and inal wid hs, which a e equal. No e ha he di e ence Qis caused
by he asymme ici y o he cubic anha monic po en ial. Subs i u ing equa ion (3.4) in o equa ion
(3.2), we can ob ain he ini ial wid h a0as well as he inal wid h by imposing ¨
a=0,
1
a3
0−a0+gN
√2πa2
0+2κa0Q=0. (3.5)
The alue o a0is nume ically ob ained by sol ing equa ion (3.5), which is dependen on he
alues o he nonlinea i y gand anha monici y s eng h κ. The wid h a0inc eases when he
sys em has ei he epulsi e in e ac ion o cubic anha monici y.
Now we use in e se enginee ing acco ding o he ollowing s eps. We may ecall ha he ini ial
and inal s a es a e s a iona y s a es wi h wid h a0wi hou exci a ions a he inal ime. Then we
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can se up he bounda y condi ions o wid h aacco ding o equa ion (3.2)
a(0) =a0,a( )=a0(3.6)
and
¨
a(0) =0, ¨
a( )=0. (3.7)
Since ha he chi p and eloci y e ms sa is y b(0) =b( )=0andc(0) =c( )=0 in equa ions (2.5)
and (2.8), espec i ely, one can ind he condi ions om equa ions (2.6) and (2.7) ha
˙
a(0) =0, ˙
a( )=0 (3.8)
and
˙
q(0) =0, ˙
q( )=0, (3.9)
In addi ion, he bounda y condi ions o x0in equa ion (3.3) a e imposed by
x0( )=d,˙
x0( )=0. (3.10)
Then we se a nin h-o de polynomial o a( )=9
n=0an nand ix he pa ame e s by sa is ying
all he bounda y condi ions o equa ions (3.6)–(3.10). An example o he designed unc ion ais
shown in igu e 2a. Once we ob ain he unc ion a, one can easily ge he unc ion qin equa ion
(3.2). Finally, x0and xccan be exp essed easily in e ms o he wid h aand qwhich is shown in
igu e 2b. No e ha we ix alues o g,κand inal ime in he example. Figu e 2ashows he
wa epacke unde goes a sligh b ea hing and inally e u ns o he ini ial wid h du ing he non-
adiaba ic p ocess. This b ea hing phenomena is due o he coupling e m be ween anha monici y
κand wid h ain equa ion (3.2): wi h κ=0, he solu ion o equa ion (3.2) will be a cons an wid h a.
Figu e 2billus a es ha he ap ajec o y oscilla es om he ini ial posi ion and hen e u ns o
he desi ed posi ion a x0=d. The co esponding ime-e olu ion |ψSTA(x, )|2is shown in igu e 2c.
To check he pe o mance o he STA ajec o ies, we de ine he ideli y a he inal ime as
F=|ψSTA( )|Φ |2, (3.11)
whe e ψSTA( ) is ob ained om he di ec nume ical simula ion (spli -ope a o me hod) o
equa ion (2.1) using he STA ajec o y o x0( ). The desi ed g ound s a e Φis ob ained by he
imagina y ime-e olu ion echnique. Φ0, deno es he ini ial and inal g ound s a es, espec i ely.
No ing ha we ake he g ound s a e Φ0as an ini ial s a e when we do he ime-e olu ion o ge
he inal s a e ψSTA( ). The ideli y o he example in igu e 2ca he inal ime is F>0.999. The
high pe o mance o ideli y in sho ime wi h bo h a ac i e and epulsi e in e ac ions is plo ed
in igu e 3. The oscilla ions a e due o he ac ha he Gaussian ansa z (2.3) is no he solu ion o
BECs wi h a omic in e ac ions. I is epo ed ha ideli y is imp o ed by using a soli on ansa z in
he a ac i e nonlinea sys em [17]. Thus in his case, he s ong a ac i e in e ac ion will lead
o he pe iod oscilla ions (see do ed-g een line g=−2) due o he Gaussian ansa z we applied in
a ia ional app oach. Fo he case o epulsi e in e ac ion, he pe iod is g ea e han he a ac i e
one.
(b) Qua ic anha monici y
In his sec ion, we shall concen a e on he as anspo o BEC in qua ic anha monici y.
The po en ial eads
V(x)=1
2(x−x0)2+1
4!λ(x−x0)4. (3.12)
Since λ=0andκ=0, he coupled E mako -like and New on-like equa ions (2.9) become
¨
a=1
a3−a(1 +18λq2)+gN
√2πa2−9λa3(3.13)
and
¨
q=¨
x0−(1 +18λa2)q−12λq3. (3.14)
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0 1 2 3 4
0.935
0.940
0.945
0.950
0.955
0.960
0.965
(a)(b)
a( )
0 1 2 3 4
0
1
2
3
4
5
xc, x0
1.0
«ψ«
2
0.5
0
–4 –2 0
0
1
2
3
4
2
x
468
(c)
Figu e 2. Cubic anha monici y. (a) Wid h a( ) wi h espec o ime. (b) Designed ajec o ies o ap cen e x0( ed solid) and
cen e o mass xc(do -dashed blue). The es o pa ame e s a e Q=4.5 ×10−3,a0=0.95, g=0.5, κ=0.02, =4, and
he dis ance d=5o anspo .(c) The co esponding ime e olu ion |ψSTA(x, )|2. (Online e sion in colou .)
2 4 6 8 10 12 14
0.80
0.85
0.90
0.95
1.00
F
Figu e 3. Cubic anha monici y: ideli y wi h espec o he inal ime o a ac i e a omic in e ac ions g=−0.1 ( ed
solid), g=−0.5 (dash-do ed blue), g=−2 (do ed g een), epulsi e in e ac ion g=0.5 (dashed pu ple), he anha monic
s eng h κ=0.02. (Online e sion in colou .)
The i s equa ion (3.13) p edic s he b ea hing mode and he oscilla ions in he wid h o he wa e
packe du ing he anspo .
Ou in e sion s a egy will be di e en om he one ollowed p e iously o cubic
anha monici y because he displacemen qappea s quad a ically in equa ion (3.13). We shall
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0
1
2
3
4
5
0 1 2 3 4
0.970
0.975
0.980
0.985
0.990
0.995
1.000
01234
(a)(b)
a( )
xc, x0
Figu e 4. Qua ic anha monici y. (a) The wid h awi h espec o ime. (b) Sho cu o he designed ajec o y o cen e o
mass xc(dash-do ed blue) and ap cen e x0( ed solid). Pa ame e s: a0=0.995329, g=0.01, λ=0.06, =4andd=5.
(Online e sion in colou .)
design q( ) wi h a polynomial as q( )=M
n=0qn n. The bounda y condi ions o unc ion qin
equa ion (3.14)
q(0) =0, q( )=0
and ˙
q(0) =0, ˙
q( )=0.⎫
⎬
⎭
(3.15)
Then we inse he polynomial unc ion qin o he coupled equa ions (3.13) and (3.14) o
pa ame ically sol e he unc ions o wid h aand xcwi h he condi ions a(0) =a0,˙
a(0) =0and
xc(0) =0, ˙
xc(0) =0. Howe e , we need addi ional bounda y condi ions o achie e he inal s a e a
inal ime , wi h
a( )=a0,˙
a( )=0
and x0( )=d,˙
x0( )=0,⎫
⎬
⎭
(3.16)
whe e a0is he ini ial and inal wid h calcula ed by equa ion (3.13) by imposing ¨
a=0. The numbe
o he bounda y condi ions abo e is eigh , he e o e one can choose M=7. Howe e , we wan o
demand he ollowing condi ions,
q
4=0andq3
4=0, (3.17)
o make he dis ance di e ence qbe ween he cen e o po en ial and he cen e o wa epacke
coincide a hese wo imes. Al e na i e bounda y condi ions would be also possible. Acco ding
o he abo e bounda y condi ions (3.15)–(3.17), we ob ain he unc ions q,a,xcand x0. An example
o he esul ing ap ajec o y and dynamics is shown in igu e 4. No e ha his s a iona y
alue makes i di e en om he anspo o cold a oms in pu ely ha monic aps, since he
nonlinea i y and anha monic e m a e in ol ed. On he o he hand, we shall also emphasize
ha he wid h aoscilla es ( igu e 4a) du ing he anspo , calcula ed om equa ion (3.13): his
oscilla ion is again due o he coupling e m be ween qua ic anha monici y λand wid h ain
he sense ha wi h λ=0, he solu ion o equa ion (3.13) will be again a cons an wid h a.We
a e now in a posi ion o design he sho cu s o adiaba ic anspo p o ocol. Figu e 4bshows
he ajec o ies o he cen e o mass o wa e packe and ap cen e, by using in e se enginee ing
and bounda y condi ions, men ioned be o e. A he ini ial and inal imes, he ajec o ies coincide
wi h each o he , which means he e is no displacemen de ia ion, gua an eeing he high ideli y
(F=0.9999) o he anspo .
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–2 –1 0 1 2
0.88
0.90
0.92
0.94
0.96
0.98
1.00
g
F
Figu e 5. Fideli y o qua ic λ=0.06 (blue-do ed line), and cubic κ=0.02 ( ed line) wi h espec o g.O he pa ame e s
a e =3π,N=1andd=5. (Online e sion in colou .)
4. E ec o nonlinea i y
In his sec ion, we shall check he ideli y o ou esul s by sol ing he GPE nume ically (wi hou
app oxima ion) wi h he designed sho cu s. Figu e 5 demons a es ha he as anspo o
BECs is pe ec wi h a ious anha monic aps aking in o accoun he a ac i e and epulsi e
in e ac ions. The Gaussian ansa z is also alid o he a ia ional app oxima ion in ou model in
he p esence o an a omic in e ac ion g=0. In igu e 5, he ideli y o cubic anha monici y κ=0.02
is plo ed o di e en a omic in e ac ions g. The ideli y is abo e 0.99 o a omic in e ac ion
|g|<1.2, i.e. as anspo o BEC in cubic anha monic aps can be achie ed o bo h a ac i e
and epulsi e in e ac ions. The ideli y d ops wi h in e ac ions |g|≥1.2. This is no su p ising as
one would expec ha he Gaussian a ia ional app oach (2.3) wo ks be e o small in e ac ion g.
Fo example, he nonlinea i y g=2 is in he ange whe e we would no expec he ansa z o wo k.
Fo g=−2, he ideli y will oscilla e wi h espec o he inal ime which is shown in igu e 3.In
igu e 5, he ideli y o qua ic anha monici y λ=0.06 is plo ed o di e en a omic in e ac ions
g. The ideli y is always g ea e han 0.99, i.e. as anspo o BEC in qua ic anha monic aps
can be achie ed o bo h a ac i e and epulsi e in e ac ions.
5. Conclusion
In summa y, we p esen an e icien way o design high- ideli y and as anspo o BEC in
anha monic aps by combining he a ia ional app oach and in e se enginee ing me hods. The
sho cu s o adiaba ic anspo o he BEC a e demons a ed wi h nume ical examples in qua ic
and cubic anha monici y aps. I is concluded ha pe ec anspo can be achie ed in cubic
anha monic aps in he p esence o bo h a ac i e and epulsi e in e ac ions. Ou me hod
p esen ed he e is di e en om he p e ious ones [18], in which he anha monic po en ial is
conside ed as a pe u ba ion. The sho cu ajec o y can be u he op imized by using op imal
con ol heo y, o ins ance, by aking in o accoun noise and e o in aps posi ion and equency
[41]. The echnique may be ex ended o h ee-dimensional Gaussian-beam op ical aps [42],
he spin-o bi coupled BECs [43], s ongly in e ac ing bosons (Tonks–Gi a deau gas) [44]and
supe luid Fe mi gas [45]. The anspo o soli on ma e wa es will also be epo ed in u u e
wo k. We expec ou sho cu design o as anspo o ha e po en ial applica ions no only in
a om in e e ome y [46] bu also in quan um in o ma ion p ocessing.
Da a accessibili y. This a icle has no addi ional da a.
Au ho s’ con ibu ions. J.L.: me hodology, esou ces, so wa e, w i ing—o iginal d a , w i ing— e iew and
edi ing; X.C.: concep ualiza ion, supe ision, w i ing— e iew and edi ing; A.R.: supe ision, w i ing—
e iew and edi ing.
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All au ho s ga e inal app o al o publica ion and ag eed o be held accoun able o he wo k pe o med
he ein.
Con lic o in e es decla a ion. We decla e we ha e no compe ing in e es s.
Funding. J.L. and A.R. acknowledge ha his publica ion has emana ed om esea ch suppo ed in pa by
a g an om Science Founda ion I eland unde g an numbe 19/FFP/6951 ("Sho cu -Enhanced Quan um
The modynamics"). This wo k has been inancially suppo ed by EU FET Open G an EPIQUS (899368),
QUANTEK p ojec (KK-2021/00070), he Basque Go e nmen h ough g an no. IT1470-22, and he p ojec
g an PID2021-126273NB-I00 unded by MCIN/AEI/10.13039/501100011033 and by "ERDF A way o making
Eu ope" and "ERDF In es in you Fu u e". X.C. acknowledges he Ramón y Cajal p og am (RYC-2017-22482).
Acknowledgemen s. We a e g a e ul o D. Rea, C. Whi y and M. Odelli o commen ing on he manusc ip . J.L.
app ecia ed he discussions om J. G. Muga and D. Gué y-Odelin a ea ly s age o wo k.
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