A icle h ps://doi.o g/10.1038/s41467-023-36611-z
Quan um Rabi dynamics o apped a oms
a in he deep s ong coupling egime
Johannes Koch
1
, Ge am R. Hunanyan
1
,TillOcken els
1
,En iqueRico
2,3,4
,
En ique Solano
3,4,5,6
& Ma in Wei z
1
The coupling o a wo-le el sys em wi h an elec omagne ic field, whose ully
quan ized e sion is he quan um Rabi model, is among he cen al opics o
quan um physics. When he coupling s eng h becomes la ge enough ha he
field mode equency is eached, he deep s ong coupling egime is app oa-
ched, and exci a ions can be c ea ed om he acuum.He ewedemons a ea
pe iodic a ian o he quan um Rabi model in which he wo-le el sys em is
encoded in he Bloch band s uc u e o cold ubidium a oms in op ical
po en ials. Wi h his me hod we achie e a Rabi coupling s eng h o 6.5 imes
he field mode equency, which is a in he deep s ong coupling egime, and
obse e a subcycle imescale aise in bosonic field mode exci a ions. In a
measu emen eco ded in he basis o he coupling e m o he quan um Rabi
Hamil onian, a eezing o dynamics is e ealed o small equency spli ings
o he wo-le el sys em, as expec ed when he coupling e m domina es o e
all o he ene gy scales, and a e i al o la ge spli ings.Ou wo kdemon-
s a es a ou e o ealize quan um-enginee ing applica ions in ye unexplo ed
pa ame e egimes.
The mo i a ion o de elop he quan um Rabi model1,2,whichisalso
e med he single-mode spin-boson model, mos ly s ems om he
ques o ob ain a comple e quan um desc ip ion o he in e ac ion o
ma e and ligh 3–6, and cu en ly his opic is also highly ele an in he
con ex o quan um in o ma ion echnologies7–13. Fo inc eased cou-
pling be ween ma e and ligh , as he coupling s eng h becomes
s onge han he decohe ence a e, he so-called s ong coupling
egime is eached, wi h mixed s a es o he wo-le el sys em and he
field mode becoming ele an , as can be desc ibed in e ms o he
Jaynes–Cummings model de eloped ea lie 14. The quan um Rabi
model, in addi ion o he co- o a ing, also includes he coun e -
o a ing e ms o he in e ac ion Hamil onian, which has s iking
consequences as he coupling s eng h app oaches he eigen-
equency o he oscilla o , a egime ha is no accessible o na u al
ligh -ma e in e ac ions. On he heo e ical side, an analy ic solu ion
o he ull quan um Rabi model has mo e ecen ly been ound15.
Expe imen ally, implemen a ions o he quan um Rabi model using
Josephson qubi , me ama e ial, and spin-mo ion cold a om se ings
ha e eached alues o he a io o coupling gand bosonic mode e-
quency ωo up o 1.4316–19. A ecen ion apping expe imen epo ing
a quan um phase ansi ion in he g ound s a e dynamics o he
quan um Rabi model has ope a ed in a egime wi h g/ω≈3.5520.
Using an expe imen al app oach based on implemen ing he
quan um Rabi model in he B illouin zone o apped cold a oms, we
demons a e a coupling a io o g/ω≈6.5. A egime whe e he coupling
e m domina es o e all o he ene gy scales can be expe imen ally
accessed in a wide pa ame e ange. Ou app oach uses a wo-le el
sys em p o ided by wo Bloch bands in an op ical la ice and a bosonic
mode p o ided by he quan ized a omic ib a ion in a supe imposed
op ical dipole ap po en ial. Fo sho in e ac ion imes, p edic ions
Recei ed: 8 Feb ua y 2022
Accep ed: 9 Feb ua y 2023
Check o upda es
1
Ins i u ü Angewand e Physik, Uni e si ä Bonn, Wegele s . 8, 53115 Bonn, Ge many.
2
EHU Quan um Cen e , Uni e si y o he Basque Coun y UPV/EHU,
P.O. Box 644, 48080 Bilbao, Spain.
3
Depa men o Physical Chemis y, Uni e si y o he Basque Coun y UPV/EHU, Apa ado 644, 48080 Bilbao, Spain.
4
IKERBASQUE, Basque Founda ion o Science, Plaza Euskadi 5, 48009 Bilbao, Spain.
5
Kipu Quan um, G ei swalde S aße 226, 10405 Be lin, Ge many.
6
In e na ional Cen e o Quan um A ificial In elligence o Science and Technology (QuA is ) and Depa men o Physics, Shanghai Uni e si y, 200444
Shanghai, China. e-mail: [email protected];ma in.wei [email protected]
Na u e Communica ions | (2023) 14:954 1
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1234567890():,;
o he quan um Rabi model in he in es iga ed pa ame e egime a e
expe imen ally alida ed. Fo long in e ac ion imes, upon ha he
edge o he B illouin zone is eached we obse e he onse o he
dynamics o a p oposed gene alized pe iodic e sion o his model21.
Resul s
Backg ound and expe imen al ealiza ion
A schema ic o he ele an ib a ional modes o ul acold a oms in he
implemen ed po en ial landscape is shown in Fig. 1a, see he le hand
side o an illus a ion o he bosonic mode ep esen ed by he
quan ized a omic ib a ion in he ha monic apping po en ial. The
igh hand side shows he supe imposed pe iodic la ice po en ial
se ing o implemen he wo-s a e sys em in i s Bloch band s uc u e,
and a omic wa epacke s now e ol e in ime in he combined po en ial,
see he bo om schema ic. Be o e discussing he deg ee o coupling
be ween hese wo quan ized modes o ypical expe imen al pa a-
me e s, we b iefly desc ibe ou expe imen al implemen a ion, see
Fig. 1b o a schema ic o he used se -up. A ha monic apping
po en ial o a cold cloud o ubidium a oms (87Rb) is gene a ed by a
ocused lase beam de i ed om a CO
2
-lase ope a ing nea a wa e-
leng h o 10.6 μm, which due o i s la ge de uning om he a omic
esonances allows o he c ea ion o deep po en ials while keeping he
sca e ing a e low enough o p e en spu ious hea ing o he a omic
ensemble. The addi ional la ice po en ial, o spa ial pe iodici y λ
4,
whe e λ= 783.5 nm deno es he wa eleng h o he d i ing lase beams,
is gene a ed by he dispe sion o Dopple -sensi i e Raman
ansi ions22,23,seeFig.1c o he used le el scheme. These e ec i e
ou -pho on p ocesses couple a oms in momen um s a es |−2_k+q〉
and |2_k+q〉, whe e q deno es he a omic quasimomen um and k=2π
λ.
The coupling leads o a spli ing be ween bands, see Fig. 1d o he
esul ing a omic dispe sion, and we in he ollowing es ic he dis-
cussion o he lowes wo bands. A he band c ossing (a q=0)wea e
le wi h he eigens a es o he wo-le el sys em o he quan um Rabi
Hamil onian, wi h |g〉=1
ffiffi2
p(|−2_k〉+|2_k〉)and|e〉=1
ffiffi2
p(|−2_k〉−|2_k〉)
espec i ely, whose coupling o he bosonic mode p o ided by he
ib a ional dynamics we a e in e es ed in. While in he gene al case he
sys em is desc ibed by a pe iodic a ian o he quan um Rabi model
(see “Me hods”and e . 21), which in e es ingly also maps on a Hamil-
onian ealizable in a fluxionium supe conduc ing qubi se ing24,wein
he p esen wo k concen a e on in e ac ion imes sho enough o
emain in he fi s B illouin zone such ha bo h he quan um Rabi and
he pe iodic quan um Rabi models coincide. The a omic dynamics in
his egime is de e mined by he quan um Rabi Hamil onian
^
H=_ω^
ay^
a+_ωq
2σz+i_gσx
^
ay^
a
,ð1Þ
whe e ^
ayand ^
aco espond o c ea ion and annihila ion ope a o s o he
bosonic field, wi h as usual ^
x=ffiffiffiffiffiffiffiffi
_
2mω
qð^
a+^
ayÞ,^
q=ffiffiffiffiffiffiffiffiffiffiffiffi
_mω
2
qð^
ay^
aÞ,andσ
x
and σ
z
a e Pauli ma ices ha ac on wo-componen spino s wi h he
componen s desc ibing cou se-g ain a omic wa e unc ions in uppe
and lowe bands, espec i ely, σx=∣nb=0ihnb=0∣∣nb=1ihnb=1∣and
σz=∣nb=1ihnb=0∣+∣nb=0ihnb=1∣,wi h heBlochbandindexn
b
(see
“Me hods”). Fu he , _ωqis he ene ge ic spacing be ween he bands a
heposi iono hec ossing(Fig.1d), which can be adjus ed by he dep h
o he la ice po en ial, and g=kffiffiffiffiffiffi
2_ω
m
qis he coupling cons an . This
magni ude o he coupling is well-unde s ood in e ms o he ene gy
ans e be ween momen um pic u e s a es ∣2_k+qiand ∣2_k+qi
being o o de ΔE=ðq+2_kÞ2=2mðq2_kÞ2=2m=2_kq=m≈ffiffiffi
n
p_g
wi h he abo e alue o he coupling cons an o a ypical alue o
q≈ffiffiffi
n
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m_ω=2
pin he ha monic oscilla o po en ial. Typical expe i-
men al pa ame e s a e a ap oscilla ion equency ω
2π2350,750
½
Hz, o
which we a i e a g
2πbe ween 2290 and 3090 Hz, so ha he deep
s ong coupling limi is well ulfilled, meaning ha he wo mo ional
Fig. 1 | Expe imen al schema ic. a A oms a e exposed o he combined po en ial
ob ained by supe imposing a ha monic apping po en ial (le ) gene a ed by a
ocused CO
2
-lase beam and a la ice po en ial o spa ial pe iodici y λ
4( igh ). The
ele an oscilla o y modes, o equency ω o he oscilla ion in he ha monic
apping po en ial and ω
q
o oscilla ion a he fi s band gap o he la ice, a e
indica ed. Fo a oms mo ing in he combined po en ial, he wo modes a e e y
s ongly coupled. bSchema ic o he expe imen al se -up along wi h he op ical
equency componen s in he op ical la ice beams used o syn hesize he ou -
pho on la ice po en ial o pe iodici y λ
4,seec o he coupling scheme. dDispe sion
ela ion o ubidium a oms in he la ice (blue) e sus he a omic quasimomen um
along wi h he dispe sion o ee a oms in s a es ∣2_k+qiand ∣2_k+qi(o ange
do ed). A he posi ion o he c ossing, a oms in he lowe and uppe band co -
espond o s a es |g〉and |e〉 espec i ely o he wo-le el qubi sys em.
A icle h ps://doi.o g/10.1038/s41467-023-36611-z
Na u e Communica ions | (2023) 14:954 2
modes p esen in he sys em, see also Fig. 1a, exchange ene gy wi h each
o he as e han he empo al pe iod. The qubi equency spacing ωq
2π
canbe unedbe ween0Hzand5.5kHz.In hecoupling egimeo g≫ω
ha we s udy, s iking dynamics is obse ed a his poin .
Tempo al e olu ion o sys em exci a ions
To begin wi h, we ha e cha ac e ized he empo al e olu ion o he
bosonic exci a ion numbe , as o e i y he p esence o quan um Rabi
physics in he deep s ong coupling egime. Fo his, bo h he a omic
dynamics in posi ion space was moni o ed by spa ial imaging o he
a omic cloud ollowing i s manipula ion in he combined la ice and
ha monic dipole apping po en ial, as well as he a omic dynamics in
momen um space by ime-o -fligh imaging. In his way, he expec a-
ion alue o he bosonic exci a ion numbe 〈N〉,whe e
_ωNhi+1
2
=mω2
2x2
+1
2mhq2i, can be de e mined. Fo hese mea-
su emen s, a oms a e ini ially p epa ed a a momen um cen e ed a
|−2_k〉, co esponding o a quasimomen um o q= 0 in he B illouin
zone, see Fig. 1d, and N
hi
=0. The blue do s in Fig. 2a show he
obse ed empo al dynamics o he a omic exci a ion numbe in ou
sys em o a qubi spli ing ωq
2π=586 6ðÞHz, which is o he o de o he
ha monic apping equency ω
2π=346 7ðÞHz. The obse ed inc ease o
he exci a ion numbe hNiwi h ime shows ha he deep s ong cou-
pling egime is eached. In gene al, he expe imen al da a is in good
ag eemen wi h heo y based on nume ically in eg a ing he Sch ö-
dinge equa ion using he Hamil onian o Eq. 1, o he la ge coupling
s eng h o g
ωffi6:5 used in he expe imen (blue line). Remaining
di e ences isible especially o sho e in e ac ion ime a e a ibu ed
o he limi ed spa ial esolu ion o he imaging sys em o 6.5 μm
(“Me hods”), causing sys ema ic unce ain ies in he de e mina ion o
he momen x2
. Fo compa ison, he o ange da a poin s co espond
o da a o he la ge qubi spacing o ωq
2π=5200 50ðÞHz, a which o he
used alue o g
2π=2275ð23ÞHz he dispe si e deep s ong coupling
egime, defined as ω
q
≥g21, is eached. He e, he inc ease in exci a ion
numbe occu s mo e slowly. Nex , we ha e eco ded expe imen al
da a o di e en a ios o he ela i e coupling s eng h g
ω.Fo his, he
apping equency ωwas uned, and Fig. 2b gi es co esponding da a
eco ded a he fixed in e ac ion ime o =3
8
π
ω e sus he ela i e
coupling s eng h g
ωbo h o a qubi equency o ωq
2π=590 6ðÞHz (blue
do s) and ωq
2π=5850 60ðÞHz (o ange iangles), espec i ely. The da a
shows ha he exci a ion numbe inc eases wi h he ela i e coupling
s eng h g
ω, and he achie ed la ge alues o up o abo e 70 exci a ion
quan a, which a e achie ed a he used sho subcycle in e ac ion ime,
i.e., being much sho e han he pe iod 2π
ω, gi es e idence ha we
ope a e in he egime o he coupling s eng h g a exceeding he
oscilla o equency ω. This can be seen analy ically when o sake o
simplici y as a lowe bound he o mula o he maximum alue o he
exci a ion numbe in he slow qubi app oxima ion ωqffi0 o which a
displaced ha monic oscilla o model applies, o g
ω≥ffiffiffiffiffi
N
hi
p
2is used (see
“Me hods”). Fo a quan i a i e compa ison, gi en bo h ha we ope a e
a a non anishing alue o ω
q
and ha a he used in e ac ion imes he
maximum o he bosonic exci a ion numbe is no ye eached, we
ha e o ely on a compa ison o a nume ic solu ion o he Hamil onian;
see he good ag eemen o he expe imen al da a wi h co esponding
heo y in he wo di e en egimes.
Dynamics o eal and momen um space mean alues
Nex , we ha e analyzed he a ia ion o he mean displacemen xhio
he a omic cloud om he ap cen e e sus ime. Fo his mea-
su emen , a oms a e p epa ed a momen um o ∣2_kiand a e
e olu ion in he combined la ice and ha monic apping po en ial
imaged in eal space. Co esponding expe imen al da a is shown in
Fig. 3a as a unc ion o in e ac ion ime o di e en alues o he
qubi equency ω
q
, as uned by adjus ing he la ice dep h. Fo a
anishing dep h o he la ice po en ial, i.e., in he slow qubi limi o
ω
q
→0, we obse e he onse o a ha monic oscilla ion in he ha -
monic apping po en ial, while o inc easing la ice dep h, co e-
sponding o a non- anishing alue o he qubi spacing ω
q
, he
obse ed displacemen is educed, and he e olu ion o s onge
la ice po en ials becomes nonha monic. F om he obse ed dis-
placemen o he ω
q
= 0 da a, we can eadily de e mine he a io o
he coupling o he oscilla ion equency, which equals g
ω=xm,0
xho , whe e
xm,0 =2_k
mωis he ampli ude o he classical oscilla ion in he absence o
a la ice po en ial and xho =ffiffiffiffiffiffi
2_
mω
q≈0.82 μm he size o he ha monic
oscilla o g ound s a e wa epacke . F om his, we ob ain g
ω=5:6ð6Þ,
which is nea he abo e-desc ibed esul o he coupling a io. To
pu he deep s ong coupling condi ion g
ω≫1 di e en ly, only in his
limi he spli ing o wa epacke s in he bosonic mode expec ed o
he non i ial case o a non anishing qubi spli ing can exceed he
wa epacke size. Tha is, only in he deep s ong coupling egime we
can expec o obse e dis inguishable dynamics no only in he qubi
occupa ion, bu also in he bosonic field modes. The ime e olu ion
o he obse ed mean displacemen hxio he da a wi h non anishing
qubi spacing depic ed in Fig. 3a quali a i ely ag ees wi h simula ions
Fig. 2 | C ea ing sys em exci a ions. a Va ia ion o he numbe o exci a ion
quan a N
hi
in he po en ial e sus in e ac ion ime o a coupling g
2π=2275 23ðÞHz
and a bosonic mode equency ω
2π=346ð7ÞHz, co esponding o a ela i e coupling
s eng h g
ω=6:58ð7Þ, i.e., a in he deep s ong coupling egime. The used qubi
spacings we e ωq
2π=586ð6ÞHz (blue do s) and 5200(50) Hz (o ange iangles).
The lines a e heo y. A oms o his measu emen a e p epa ed in he momen um
s a e |−2_k〉in he cen e o he ha monic apping po en ial. bVa ia ion o he
numbe o exci a ions on he ela i e coupling s eng h g
ω.He eafixed in e ac ion
ime o 3
8
π
ωwas used, and blue do s and o ange iangles co espond o qubi
spacings ωq
2π=590 6ðÞHz and 5850(60) Hz, espec i ely. The isible e o ba s deno e
he s a is ical unce ain ies.
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Na u e Communica ions | (2023) 14:954 3
o he quan um Rabi model in he deep s ong coupling egime
depic ed as lines o la ge a omic displacemen s. A mo e de ailed
analysis o he eal space da a in ou sys em again is limi ed by he
fini e ins umen al esolu ion o he imaging sys em.
We ha e in mo e de ail analyzed he momen um space da a
ob ained by he a -field ime o fligh imaging, om which bo h he
quasimomen um qand he band index n
b
o Bloch bands, wi h n
b
=0
and 1 o momen a measu ed in he absence o a apping po en ial o
p=2_k+qand p=+2_k+q espec i ely, can be de i ed (see
“Me hods”). Figu e 3b, c show bo h he a ia ion o he mean a omic
quasimomen um q
and he mean Bloch band occupa ion hσxi,
which in he basis o he band eigens a es can be w i en as
^
σx=∣nb=0
nb=0
∣∣nb=1ihnb=1∣, wi h ime along wi h heo y.
Rema kably, a small la ice dep h, co esponding o a low alue o
ω
q
, we obse e a empo ally nea ly cons an alue o he Bloch band
occupa ion hσxi. This is unde s ood as signal p epa a ion, de ec ion
and he sys em Hamil onian— he la e in he unusual egime o g≫ω
being domina ed by he in e ac ion e m—all a e diagonal in he same
basis, he eigenbasis o he Pauli ma ix σ
x
. In con as , o la ge
la ice dep h, i.e., wi h inc eased ω
q
, an oscilla o y beha io is
obse ed, as a ibu ed o a omic wa epacke s localized in he ap
cen e pe o ming Rabi oscilla ions be ween he momen um eigen-
s a es ∣±2_k espec i ely. This is mos clea ly isible o he da a
shown by he ed squa es o ωq
2πffi3600 40ðÞHz, o which wi h ω
q
>ω
he dispe si e deep s ong coupling egime is eached. In gene al, we
obse e ha he a e age alue o he Bloch band occupa ion hσxi
educes o la ge la ice dep h, as has been p edic ed in ea lie
wo k25.
One also finds ha nea he la ges in es iga ed in e ac ion imes
he expe imen al da a (da a poin s) isible in Fig. 3b, c s a s o de ia e
om he heo y cu es (lines), which we e de i ed based on he
quan um Rabi model, as unde s ood om ha he edge o he B il-
louinzonea =π
2ωis eached, upon which i becomes ele an ha ou
sys em ealizes a pe iodic a ian o he quan um Rabi model. This is
mos clea ly seen o he da a se s eco ded wi h he smalles qubi
spacings. Theo y p edic ions based on he pe iodic quan um Rabi
model, which quali a i ely ep oduce he expe imen al da a also nea
he band edge, a e shown by semi- anslucen lines.
P epa ing a oms in qubi eigens a es
In u he measu emen s, we ha e p epa ed a oms in he qubi g ound
s a e |g〉and exci ed s a e |e〉, espec i ely, o med by he Bloch bands
and s udied he empo al a ia ion o he qubi popula ion. As
desc ibed abo e, he qubi s a es co espond o cohe en supe -
posi ions o he momen um pic u e s a es |±2_k〉, espec i ely, and o
p epa e hese s a es B agg ansi ions we e d i en wi h coun e -
p opaga ing momen um ans e using Raman beams wi h he co e-
sponding phase di e ence imp in ed. We again s a a a anishing
bosonic mode quan um numbe (hNi= 0). The ini ial s a es |g,0〉,|e,0〉
p epa ed in his way co espondingly ha e di e en pa i y12.Fo
de ec ion, gi en ha he qubi occupa ion is encoded in he ela i e
phase o wo wa epacke s, a he end o he measu emen a oms we e
Fig. 3 | Re ealing he ime e olu ion o sys em pa ame e s. a Va ia ion o he
mean alue o he a omic cloud posi ion x
hi
on he in e ac ion ime. On he igh
hand scale, his quan i y is gi en in uni s o he ha monic oscilla o leng h x
ho
.
Expe imen al pa ame e s we e a coupling g
2π= 2275 23ðÞHz and a bosonic mode
equency ω
2π=346ð7ÞHz. Fu he , he used qubi spacing ωq
2πwas 0 Hz (blue do s),
586(6) Hz (o ange iangles), 1660(20) Hz (g een upside-down iangles), and
3600(40) Hz ( ed squa es) espec i ely. The sys em is ini ially p epa ed in he
momen um s a e |−2_k〉. Theo y esul s o he quan um Rabi and he pe iodic
quan um Rabi models a e ep esen ed by non- anspa en and semi- anspa en
lines, espec i ely. bTime e olu ion o he obse ed mean a omic quasimomen um
q
and c he Bloch band occupa ion σx
o co esponding alues o he qubi
spacing. The isible e o ba s deno e he s a is ical unce ain ies.
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Na u e Communica ions | (2023) 14:954 4
fi s adiaba ically mo ed away om he bandgap by chi ping he ou -
pho on la ice po en ial, which emaps he uppe and lowe bands |e〉
and |g〉 o he ba e s a es |2_k〉and |−2_k〉, espec i ely, and hen
obse ing he band popula ion, which allows o de e mine he popu-
la ion in he co esponding qubi s a es. Expe imen al esul s o he
a ia ion o he measu ed qubi popula ion di e ence σz
=hð∣eihe∣
∣gihg∣Þi wi h ime a e shown in Fig. 4a. He e he blue do s and yellow
iangles co espond o an ini ial popula ion in he g ound s a e o a
qubi equency ω
q
o ω
q
→0 and 1050(10) Hz, espec i ely, and he ed
squa es and g een iangles o p epa a ion in he exci ed s a e o he
co esponding qubi equencies. In all cases, apid decay o he
popula ion di e ence hσziwi h ime is obse ed, in ag eemen wi h
heo e ical p edic ions, as unde s ood om he s ong coupling o he
qubi s a es wi h he bosonic field mode leading o a highly en angled
na u e o he sys ems eigens a es12. We a ibu e he isible de ia ion
om heo y o he la ges in es iga ed in e ac ion imes o nea
700 μs, a which he end o he B illouin zone is eached, o non-
adiaba ic ansi ions occu ing in he used expe imen al band-
mapping eadou scheme.
We ha e also de e mined he a ia ion o he mean exci a ion
numbe hNion he in e ac ion ime, as shown in Fig. 4b o bo h
a oms ini ially in he qubi g ound s a e ∣g(blue do s) and he
exci ed s a e ∣ei(o ange iangles) espec i ely. He e an enhance-
men o he exci a ion numbe o a oms ini ially in he uppe qubi
s a e wi h espec o ha when p epa ing in he lowe s a e is
obse ed. Gi en ha he qubi s a es a e supe posi ion s a es his
demons a es a dependence o hNion he phase o he ini ial s a e,
which gi es e idence ha also a he la ges in e ac ion imes
in es iga ed in Fig. 4a, b quan um cohe ence is p ese ed. The di -
e ence is smalle han heo e ical p edic ions, and we a ibu e he
educed con as o he impe ec esolu ion o ou imaging sys em,
which educes he dis inguishabili y o he di ac ion peaks.
Figu e 4c shows he a ia ion o he di e ence o he exci a ion
numbe hNibe ween when p epa ing he qubi in he g ound and
exci ed s a es espec i ely on bo h ime and qubi equency ω
q
. This
gene alizes he esul s shown in Fig. 4b o di e en qubi equency
spacings. While o small spacings he sensi i i y o he exci a ion
numbe on he ini ial s a e o he qubi is small, a qubi equencies
abo e ωq
2π≈1 kHz a clea di e ence is isible. While o he simple case
o ω
q
= 0 he a omic wa epacke supe posi ion oscilla ing in he
apping po en ial can be exp essed by he Sch ödinge ca s a es
1
ffiffi2
pð∣ig
ωi±∣ig
ωiÞ, espec i ely, o a non anishing qubi equency ω
q
he quan um s a es become much mo e complex en angled s a es.
The expe imen al findings o Fig. 4c demons a e he phase-
dependen beha io o he quan um Rabi dynamics in he deep
s ong coupling egime, see also he good ag eemen o he da a wi h
heo y (Fig. 4d) o compa ison.
Discussion
Ou expe imen demons a es ha quan um Rabi physics a unp e-
ceded high coupling s eng h can be ealized wi h ul acold a oms in
op ical la ices using solely he spa ial deg ees o eedom. In ou
app oach, a wo-le el sys em has been encoded in he occupa ion o
Bloch bands, in e ac ing wi h a bosonic mode implemen ed by
Fig. 4 | P epa ing a oms in qubi s a es. a Time e olu ion o he qubi exci a ion
σz
ollowing p epa a ion o a oms in lowe (o ange iangles) and uppe ( ed
squa es) s a es o he wo-s a e sys em o a qubi spacing ωq
2π=1050 10ðÞHz. Fo
compa ison, he blue do s and g een upside-down iangles co espond o mea-
su emen s wi h ω
q
= 0. The lines a e heo y. g
ω=6:50 5ðÞin all measu emen s.
bVa ia ion o he mean exci a ion numbe <N > on he in e ac ion ime o a oms
ini ially p epa ed in he g ound (blue do s) and he exci ed qubi s a es (o ange
iangles), espec i ely, o ωq
2π=4660 50ðÞHz, along wi h heo y (lines).
cExpe imen al da a o he di e ence in he mean exci a ion numbe hNi∣eihNi∣gi
obse ed when p epa ing a oms ini ially in qubi exci ed and g ound s a es,
espec i ely, on bo h in e ac ion ime and qubi spacing ep esen ed in colo code.
dCo esponding heo y expec a ions.
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ha monic mo ion in a dipole ap. The cha ac e is ic dynamics a hese
pa ame e egimes has been mapped ou .
Fo he u u e, ex ensions o his wo k can include quan um
in o ma ion p ocessing based on qubi s encoded in he ib a ional
dynamics o cold a oms in enginee ed supe posi ions o pe iodic
la ices and slowly a ying dipole apping po en ials. This is emi-
niscen o wo k done in he phase space o supe conduc ing qubi
sys ems, albei wi h s onge coupling s eng hs24,26. Fo coupling o
di e en qubi s, digi al echniques, al e na ing be ween igh ly
confined in e ac ion and qubi manipula ion pe iods ollowing he e
demons a ed echniques can be en isioned27. I also will be in e -
es ing o ex end he p esen wo k o longe in e ac ion imes, as o
s udy p edic ions o he pe iodic quan um Rabi model and obse e
collapse and e i al pa e ns o he ini ial s a e21. O he in e es ing
u u e wo k includes he sea ch o phase ansi ions o he spin-
boson model28,29.
Me hods
Expe imen al se -up and p ocedu e
Ou expe imen al appa a us, see also he schema ics shown in Fig. 1b
o he main ex , is a modified e sion o a se -up used in ea lie
wo ks30. Inside a acuum appa a us, cold ubidium a oms (87Rb) col-
lec ed in a magne o-op ic ap a e loaded in o he dipole apping
po en ial induced by a beam ocused o 42 μm diame e de i ed om a
CO
2
-lase ope a ing nea 10.6 μm wa eleng h. The a oms a e e a-
po a i ely cooled o quan um degene acy by lowe ing he dep h o he
dipole po en ial. In he final s ages o he cooling, a magne ic quad-
upole field is ac i a ed, which allows o gene a e a spin-pola ized
Bose-Eins ein condensa e in he m
F
=1componen o heF=1
hype fine g ound s a e. To keep in e ac ion e ec s small, he e we wo k
wi h small condensa e numbe s o ypically 2500 a oms, as achie ed
by amping he dep h o he apping po en ial in e apo a i e cooling
o lowe alues han needed o achie e condensa ion o educe he
numbe o confined a oms. Subsequen ly, he dipole apping po en-
ial is amped up adiaba ically wi hin 250 ms o each he desi ed
alues o he apping equency ω, see also he main ex , o simu-
la ion o he quan um Rabi model. Typical beam powe s a e 32 W,
30 mW, and 100 mW du ing loading, he final s age o e apo a i e
cooling, and quan um Rabi manipula ion phases, espec i ely. In he
la e phase, a oms emain well confined in he cen e o he Gaussian
beam, in a ange whe e he dipole apping po en ial can well be
desc ibed as a ha monic po en ial. The anha monici y, defined as he
di e ence o he dipole po en ial expec ed o be imp in ed by a
Gaussian lase beam and a ha monic po en ial, o he expe imen ally
ele an pa ame e egime is below 0.25%. The es ima ed decohe ence
a e om pho on sca e ing om he mid-in a ed apping lase beam
in he quan um Rabi manipula ion phase is 1.7 × 10−5/s, i.e., is negligible.
In p ac ice, decohe ence will be de e mined by sca e ing om he
Raman beams and a omic in e ac ion e ec s.
The me hod used o gene a e a high spa ial ha monic la ice
po en ial o pe iodici y λ
4, whe e λ’783:5 nm (which is de uned
3.5 nm om he ubidium D2-line) deno es he wa eleng h o he
d i ing lase beams, elies on ou -pho on Raman p ocesses22. The
ansi ions a e d i en in a h ee-le el configu a ion wi h wo s able
g ound s a es |1〉and |2〉and one spon aneously decaying exci ed
s a e |3〉by a beam o equency ω
la
and wo coun e p opaga ing
supe imposed beams o equencies ωla +Δωla and ωla Δωla . The
m
F
=−1andm
F
= 0 componen s o 5S1=2,F= 1 cons i u e he used
g ound s a es and he 5P3=2mani old se es as he exci ed s a e o he
h ee-le el configu a ion. A omic momen um is exchanged wi h he
d i ing ligh field in uni s o ou -pho on momen a, which is a ac o
wo abo e ha o he ele an p ocesses in a usual s anding wa e
la ice induced by wo-pho on p ocesses. Co espondingly, he spa-
ial pe iodici y o he induced po en ial is a ac o wo smalle and
equals λ
4
22. In he expe imen , we ypically use a magne ic bias field o
1 G o emo e he degene acy o Zeeman suble els and a equency
di e ence ΔωZ
2π’1:6 MHz.
In he expe imen al sequence, ollowing he amping up o he
CO
2
-lase beam in ensi y o he desi ed ha monic apping equency,
a oms a e p epa ed nea he fi s a oided c ossing o he la ice band
s uc u e (Fig. 1d) by means o B agg di ac ion. Fo he used la ice
wi h spa ial pe iodici y λ
4usual B agg di ac ion, ans e ing
momen um in uni s o wo-pho on momen a, can be used o p epa e
a oms a he posi ion o he fi s band c ossing. Fo he expe imen al
da a shown in Fig. 4, wi h qubi s a es |g〉and |e〉 espec i ely as he
ini ial s a es, wo simul aneously pe o med B agg pulses wi h oppo-
si e di ec ions o he momen um ans e we e used wi h he ela i e
phase o he pulses allowing o se he desi ed qubi ini ial s a e. Fol-
lowing p epa a ion, a oms we e le in he desi ed combined po en ial
o la ice and ha monic apping o quan um Rabi manipula ion o a
a iable in e ac ion ime.
Subsequen de ec ion o he a omic cloud was pe o med a e
ex inguishing bo h he la ice and he dipole apping beams. Fo his,
abso p ion imaging o he a omic sample was employed on o a sCMOS
came a. Du ing he expe imen s desc ibed in he main ex , bo h
measu emen s p obing he eal-space dis ibu ion a e ca ied ou by
p obing di ec ly ollowing he expe imen , as well as a -field ime-o -
fligh imaging p obing he momen um dis ibu ion we e pe o med.
Fo an analysis o measu emen s o he ms displacemen x2
o he
a omic cloud om he ap cen e , he expe imen al image da a was
fi s decon olu ed by he poin sp ead unc ion o he imaging sys em
(o nea 6.5 μm ins umen al esolu ion) de e mined in an indepen-
den measu emen be o e analysis o his momen om a se ies o
measu emen . Example images a e decon olu ion a e shown
in Fig. 5a.
The momen um pmeasu ed in he absenceo a apping po en ial
maps on o he quasimomen um qand he band index nb20,1 g!
2_k,2_k
mapping he basis s a es o ou qubi s a e ia
p=q+2_k2nb1
. Example ime o fligh images o ob ain he
momen um pand subsequen ly quasimomen um qandbandindexn
b
a e shown in Fig. 5b.
Theo e ical me hods
The single-pa icle Hamil onian o a cloud o ul acold a oms is
desc ibed by he sum o a ha monic pa , which includes he kine ic
Fig. 5 | Examples o ob ained image da a. a Se ies o eal space images o he
sys em ini ially p epa ed in he qubi exci ed s a e o he quan um Rabi Hamil o-
nian, wi h ωq
2π=2380 40ðÞHz and g
ω=6:5ð5Þ, a e decon olu ion o he ob ained aw
abso p ion imaging da a o he a omic ensemble accoun ing o he poin sp ead
unc ion o he used op ics. F om le o igh he in e ac ion ime inc eases in s eps
o 50 μs. Despi e he limi ed ins umen al op ical esolu ion, a spli ing up o he
a omic cloud is obse ed. bSe ies o ime-o -fligh images o he a omic cloud, o
he same sys em s a e as in a. The numbe s on he le -hand side ep esen he
measu ed a omic momen um p, while he on he igh -hand side he co esponding
quasimomen um qis gi en.
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ene gy o he a oms and he ha monic ap, and a pe iodic po en ial,
^
H=
^
p2
2m+mω2
2
^
x2+V
2cos 4k^
x
,ð2Þ
whe e ^
p=i_d
dx and ^
xa e momen um and posi ion o an a om o mass
m, espec i ely. I we w i e his Hamil onian in he Bloch basis unc ion
x∣ϕnðqÞ
=eiqx=_ei2kxei4nkx and we p ojec o he wo lowes ene gy
bands, i is ecas in o
^
H=
^
q2
2m+mω2
2
^
x2+2_k
m
10
01
^
q+V
4
01
10
:ð3Þ
Defining c ea ion and annihila ion ope a o s ^
a=ffiffiffiffiffiffi
mω
2_
pð^
x+i
mω
^
qÞ
and ^
ay=ffiffiffiffiffiffi
mω
2_
pð^
xi
mω
^
qÞ, while o a ing he qubi (band index) wi h he
uni a y ope a o U=1
ffiffi2
p11
11
,anddefining he Pauli ma ices in
he o a ed basis as
σx=∣n=0ihn=0∣∣n=1ihn=1∣
σz=∣n=1ihn=0∣+∣n=0ihn=1∣
he o al sys em Hamil onian is he one gi en in Eq. (1).
In ou expe imen al sequence, he p epa ed ini ial s a es co e-
spond o qui e highly ene ge ic s a es o he sys em Hamil onian, as
can be seen in Fig. 6, which shows he nume ically de e mined
occupa ion p obabili y o he s a es o a oms p epa ed in a ∣2_k
momen um s a e and he bosonic acuum s a e, co esponding o
ypical ini ial condi ions. The ini ial s a e ∣2_kco esponds o a
supe posi ion o wo qubi s a es, such ha he e bo h eigens a es
wi h nega i e and posi i e pa i y (see Fig. 6a, b, espec i ely) a e
popula ed.
In wha ollows, we would like o unde s and he occupa ion
numbe , o numbe o pho ons, in an expe imen s a ing wi h he
g ound s a e o a ca i y mode wi h equency ωin a Rabi model, whe e
he equency o he qubi is se o ze o, i.e., ω
q
= 0, and he coupling
s eng h o he ca i y and qubi is gi en by g,
Hωq=0 =ωN+1
2
+gσxa+ay
=ωay+gσx
ω
a+gσx
ω
+ω
2g2
ω
ð4Þ
As i can be di ec ly seen om he second line, his Hamil onian is
diagonal wi h he displaced ca i y ope a o s
b=a+gσx
ω=a+α,
whe e he displacemen ope a o is gi en by
DαðÞ=eαayα?a=e∣α∣2
2eαayeα?a,
o a gene al αpa ame e .
The ac ion o he ime e olu ion o a displaced Hamil onian o a
ca i y mode on he acuum, i.e., no exci a ions, is gi en by
eiHðαÞ ∣0i=DðαÞeiHð0Þ DðαÞ∣0i=eiω
2eig2
ωDðαÞ∣eiω αi
=eiω
2eig2
ωeImð∣α∣2eiω ÞD½αðeiω 1Þ∣0i:
F om his exp ession, we can de i e he expec a ion alue in he
numbe o exci a ions o an in e ac ion ime
Nhi=∣α ðÞ∣2=4∣α∣2sin2ω
2
:ð5Þ
Co espondingly, wi h α=g
ω, he maximum numbe o he
expec a ion alue N
hi
is gi en by
Nmax =4∣α∣2=4g2
ω2:ð6Þ
Compa ison o fluxonium qubi sys em
Finally, we ema k ha he he e ele an Hamil onian Eq. 2maps on o
Hamil onians eached wi h supe conduc ing fluxonium sys ems. Spe-
cifically, see Eq. 1o he quasicha ge qubi sys em o e . 24,which
eads:
H=EC
Q
2e
2
+1
2ELφ2EJcos φφex
,ð7Þ
wi h Q as he cha ge and φas he supe conduc ing phase di e ence.
Fu he , E
C
deno es he cha ging ene gy, E
L
he induc i e ene gy, E
J
he
Josephson ene gy, and φ
ex
an ex e nal phase. By sepa a ing he ime-
dependen Sch ödinge equa ion ob ained wi h he Hamil onian o
Eq. 7in o slow and as a ying pa s espec i ely and subs i u ion o
Fig. 6 | Simula ion o he ini ial sys em s a e. a Popula ion dis ibu ion o
nega i e pa i y eigens a es o he sys em when ini ially p epa ed in he ∣2_k
momen um s a e, and he acuum field mode (<N > = 0), e sus he exci a ion
numbe n.Thecaseo ωq
2π= 0Hz is shown in blue colo , and he case o ωq
2π= 1000Hz
is shown in o ange. bCo esponding popula ion dis ibu ion o he posi i e pa i y
eigens a es.
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φ=4kx we find ha esul ing equa ion o mo ion can be w i en in
e ms o an e ec i e Hamil onian ha up o a basis ans o ma ion
di ec ly maps on o Eq. 3 o
EC=_=2_k2
m,EJ=_=ωq,EL=_=mω2
16_k2,_g=8ELE3
C
1
4,ð8Þ
Fo he specificcaseo anex e nalphaseφ
ex
=π,onealsofinds
ha wi h hese iden ifica ions Eq. 2and Eq. 1 o e . 24 a e akin. This
allows us o di ec ly compa e he ene gy scales gi en in e . 24 o he
pa ame e s used he e:
g
ω’1:91, ωq
ω’2:42:
The he e de i ed alue o he (no malized) coupling s eng h g/ω
o his supe conduc ing sys em is abo e ha o ea lie wo ks explici ly
s udying quan um Rabi physics in supe conduc ing sys ems, and
below he co esponding alues o bo h he ion apping wo k o e . 20
and o he p esen wo k.
Da a a ailabili y
Sou ce da a a e p o ided wi h his pape .
Re e ences
1. Rabi,I.I.On hep ocesso spacequan iza ion.Phys. Re . 49,
324 (1936).
2. B aak, D., Chen, Q.-H., Ba chelo , M. T. & Solano, E. Semi-classical
and quan um Rabi models: in celeb a ion o 80 yea s. J. Phys. A 49,
300301 (2016).
3. Na a , P. & Ciu i, C. P o ec ed quan um compu a ion wi h mul iple
esona o s in ul as ong coupling ci cui QED. Phys. Re . Le . 107,
190402 (2011).
4. Rome o,G.,Balles e ,D.,Wang,Y.,Sca ani,V.&Solano,E.Ul a as
quan um ga es in ci cui QED. Phys.Re .Le .108,120501(2012).
5. Kyaw, T. H., He e a-Ma i, D. A., Solano, E., Rome o, G. & Kwek, L.-C.
C ea ion o quan um e o co ec ing codes in he ul as ong
coupling egime. Phys. Re . B 91, 064503 (2015).
6. Rossa o, D. Z. e al. En angling pola i ons ia dynamical Casimi
e ec in ci cui quan um elec odynamics. Phys.Re .B93,
094514 (2016).
7. Fo n-Díaz, P., Lama a, L., Rico, E., Kono, J. & Solano, E. Ul as ong
coupling egimes o ligh -ma e in e ac ion. Re . Mod. Phys. 91,
025005 (2019).
8. Lang o d, N. e al. Expe imen ally simula ing he dynamics o
quan um ligh and ma e a deep-s ong coupling. Na . Commun.
8,1715(2017).
9. Ma ko ić, D. e al. Demons a ion o an e ec i e ul as ong cou-
pling be ween wo oscilla o s. Phys.Re .Le .121,040505(2018).
10. Ha oche, S. & Raimond, J.-M. in S ong Ligh -ma e Coupling: F om
A oms To Solid-s a e Sys ems (ed. Au è es, A. e al.) 1–36 (Wo ld
Scien ific, 2013).
11. Ciu i, C., Bas a d, G. & Ca uso o, I. Quan um acuum p ope ies o
he in e subband ca i y pola i on field. Phys.Re .B72, 115303 (2005).
12. Casano a, J., Rome o, G., Lizuain, I., Ga cía Ripoll, J. J. & Solano, E.
Deep s ong coupling egime o he Jaynes-Cummings model.
Phys.Re .Le .105, 263603 (2010).
13. Pe opad e, B., Fo n-Díaz, P., Solano, E. & Ga cía-Ripoll, J. J.
Swi chable ul as ong coupling in ci cui QED. Phys.Re .Le .105,
023601 (2010).
14. Jaynes, E. T. & Cummings, F. W. Compa ison o quan um and
semiclassical adia ion heo ies wi h applica ion o he beam mase .
P oc. IEEE 51,89–109 (1963).
15. B aak, D. In eg abili y o he Rabi model. Phys.Re .Le .107,
100401 (2011).
16. Da eau, A., Meng, Y., Schneeweiss, P. & Rauschenbeu el, A.
Obse a ion o ul as ong spin-mo ion coupling o cold a oms in
op ical mic o aps. Phys. Re . Le . 121,253603(2018).
17. L , D. e al. Quan um simula ion o he quan um Rabi model in a
apped ion. Phys. Re . X 8,021027(2018).
18. Yoshiha a, F. e al. Supe conduc ing qubi -oscilla o ci cui beyond
he ul as ong-coupling egime. Na . Phys. 13,44–47 (2017).
19. Baye , A. e al. Te ahe z ligh -ma e in e ac ion beyond uni y
coupling s eng h. Nano Le . 17,6340–6344 (2017).
20. Cai, M.-L. e al. Obse a ion o a quan um phase ansi ion in he
quan um Rabi model wi h a single apped ion. Na . Commun. 12,
1126 (2021).
21. Felice i, S. e al. Quan um Rabi model in he B illouin zone wi h
ul acold a oms. Phys. Re . A 95, 013827 (2017).
22. Ri ,G.,Geckele ,C.,Salge ,T.,Cennini,G.&Wei z,M.Fou ie
syn hesis o op ical po en ials o a omic quan um gases. Phys. Re .
A74, 063622 (2006).
23. Salge , T., Geckele , C., Kling, S. & Wei z, M. A omic Landau-Zene
unneling in Fou ie -syn hesized op ical la ices. Phys. Re . Le . 99,
190405 (2007).
24. Pechenezhskiy, I., Mencia, R., Nguyen, L., Lin, Y.-H. & Manucha yan,
V. The supe conduc ing quasicha ge qubi . Na u e 585,
368–371 (2020).
25. Wol , F. A., Kolla , M. & B aak, D. Exac eal- ime dynamics o he
quan um Rabi model. Phys. Re . A 85,053817(2012).
26. Manucha yan, V., Koch, J., Glazman, L. & De o e , M. Fluxonium:
single coope -pai ci cui ee o cha ge o se s. Science 326,
113–116 (2009).
27. Lama a, L., Pa a-Rod iguez, A., Sanz, M. & Solano, E. Digi al-analog
quan um simula ions wi h supe conduc ing ci cui s. Ad . Phys. X 3,
1457981 (2018).
28. Hwang, M.-J., Puebla, R. & Plenio, M. B. Quan um phase ansi ion
and uni e sal dynamics in he Rabi model. Phys.Re .Le .115,
180404 (2015).
29. Heyl, M. Dynamical quan um phase ansi ions: a e iew. Rep. P og.
Phys. 81,054001(2018).
30. Cennini, G., Ri , G., Geckele , C. & Wei z, M. Bose-Eins ein con-
densa ioninaCO
2
-lase op ical dipole ap. Appl. Phys. B 77,
773–779 (2003).
Acknowledgemen s
We acknowledge suppo by he DFG wi hin he p ojec We 1748-24
(642478) (M.W.), he ocused esea ch cen e SFB/TR 185 (277625399)
(M.W.) and he Clus e o Excellence ML4Q (EXC 2004/1 –390534769)
(M.W.), he QMiCS (820505) (E.S.) and OpenSupe Q (820363) (E.S.)
p ojec s o he EU Flagship on Quan um Technologies, he Na ional
Na u al Science Founda ion o China (NSFC) (12075145) (E.S.), he
Shanghai Go e nmen g an STCSM (2019SHZDZX01-ZX04) (E.S.), he
Spanish Go e nmen PGC2018-095113-B-I00 (MCIU/AEI/FEDER, UE)
(E.R.), he Basque Go e nmen IT986-16 (E.R.), and he FET Open Qu -
omo phic (E.S.) and EPIQUS (E.S.) EU p ojec s.
Au ho con ibu ions
J.K. and G.R.H. conduc ed he expe imen and analyzed he da a. T.O.
conduc ed ea ly expe imen al g oundwo k. E.R. did nume ical and
analy ical s udies and simula ions. E.S. and M.W. planned he p ojec . All
au ho s con ibu ed o he w i ing o he manusc ip and he da a
in e p e a ion.
Funding
Open Access unding enabled and o ganized by P ojek DEAL.
Compe ing in e es s
The au ho s decla e no compe ing in e es s.
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Addi ional in o ma ion
Supplemen a y in o ma ion The online e sion con ains
supplemen a y ma e ial a ailable a
h ps://doi.o g/10.1038/s41467-023-36611-z.
Co espondence and eques s o ma e ials should be add essed o
Johannes Koch o Ma in Wei z.
Pee e iew in o ma ion Na u e Communica ions hanks he anon-
ymous e iewe (s) o hei con ibu ion o he pee e iew o his wo k.
Rep in s and pe missions in o ma ion is a ailable a
h p://www.na u e.com/ ep in s
Publishe ’s no e Sp inge Na u e emains neu al wi h ega d o ju -
isdic ional claims in published maps and ins i u ional a filia ions.
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