PHYSICAL REVIEW A 104, 053701 (2021)
Dis an emi e s in ul as ong wa eguide QED:
G ound-s a e p ope ies and non-Ma ko ian dynamics
Ca los A. González-Gu ié ez ,*Juan Román-Roche , and Da id Zueco
Ins i u o de Nanociencia y Ma e iales de A agón (INMA), CSIC–Uni e sidad de Za agoza, Za agoza 50009, Spain
(Recei ed 15 June 2021; accep ed 12 Oc obe 2021; published 1 No embe 2021)
S a ing om he pa adigma ic spin-boson model (SBM), we in es iga e he s a ic and dynamical p ope ies o
a sys em o wo dis an wo-le el emi e s coupled o a one-dimensional Ohmic wa eguide beyond he o a ing
wa e app oxima ion. Employing s a ic and dynamical pola on Ansä ze we s udy he e ec s o ini e sepa a-
ion dis ance on he beha io o he pho on-media ed Ising-like in e ac ion, qubi equency eno maliza ion,
g ound-s a e magne iza ion, and en anglemen en opy o he wo-qubi sys em. Based on p e ious wo ks we
de i e an e ec i e app oxima e Hamil onian o he wo-impu i y SBM ha p ese es he exci a ion-numbe
and hus acili a es he analy ical ea men . In pa icula , i allows us o in oduce non-Ma ko iani y a ising
om delay- eedback e ec s in wo dis an emi e s in he so-called ul as ong coupling (USC) egime. We
es ou esul s wi h nume ical simula ions pe o med o e a disc e ized ci cui -QED model, inding pe ec
ag eemen wi h p e ious esul s, and showing in e es ing dynamical e ec s a ising in ul as ong wa eguide
QED wi h dis an emi e s. In pa icula , we e isi he Fe mi wo-a om p oblem showing ha , in he USC
egime, ini ial co ela ions yield wo di e en e olu ions o symme ic and an isymme ic s a es e en be o e he
emi e s become causally connec ed. Finally, we demons a e ha he collec i e dynamics, e.g., supe adiance o
sub adiance, a e a ec ed no only by he dis ance be ween emi e s, bu also by he coupling, due o signi ican
equency eno maliza ion. This cons i u es ano he dynamical consequence o he USC egime.
DOI: 10.1103/PhysRe A.104.053701
I. INTRODUCTION
In mos scena ios, pho ons a e weakly coupled o ma -
e , and hey a el as ela i e o he ma e dimensions. In
wa eguide QED (wQED), howe e , hey a e con ined in one-
dimensional wa eguides o enhance he ligh -ma e coupling
[1–3]. Also, well-sepa a ed emi e s can be coupled o he
guide, such ha pho ons ake an app eciable amoun o ime
a eling be ween hem. In hese cases and depending on he
expe imen al condi ions and a chi ec u es [3–11], neglec ing
s ong ligh -ma e co ela ions and/o e a da ion e ec s is
no always jus i ied. When aken in o accoun , emi e dynam-
ics becomes non-Ma ko ian, i.e., he sys em exhibi s memo y.
This is in e p e ed as a back- low o in o ma ion be ween
he a eling pho ons and he emi e s. Non-Ma ko iani y
occu s al eady a he le el o a single emi e in e ac ing
wi h s uc u ed en i onmen s o ese oi s [12–19], in he
p esence o bound s a es [20], o in he ul as ong cou-
pling (USC) egime [10,11]. In his pa icula egime, highe
o de p ocesses beyond he c ea ion (annihila ion) o one
pho on by annihila ing (c ea ing) one ma e exci a ion, play
a undamen al ole. En e ing he USC egime implies ha
he o a ing-wa e-app oxima ion (RWA) o he in e ac ion
b eaks down and he a omic ba e pa ame e s ge eno malized,
by ei he he Bloch-Siege shi [21], he e ec i e qubi -qubi
couplings [22,23] in ca i y QED, o eno maliza ion due o
he coupling o he con inuum elec omagne ic (EM) ield in
wQED [24]. The g ound s a e also becomes non i ial [25].
*ca losgg@uniza .es
Apa om non-Ma ko iani y, USC in wQED has in e es ing
consequences, such as he localiza ion-delocaliza ion ansi-
ion [26,27], pa icle p oduc ion [28], nonlinea op ics a he
single pho on limi [29,30], o acuum ligh emission [31].
In his wo k, we will conside his egime o coupling.
When se e al emi e s a e p esen , adia ion e a da ion e -
ec s ac as an al e na i e sou ce o non-Ma ko iani y. This
has been ecen ly s udied ou side o he USC egime show-
ing ha e en in a simple sys em o wo sepa a ed wo-le el
emi e s, he collec i e dynamics can be signi ican ly a ec ed
by non-Ma ko ian in e e ence caused by adia ion-delayed
eedback be ween hem [32–38]. In his case e a da ion e -
ec s become impo an when he a omic li e ime γ−1∼x/ g,
o a sepa a ion dis ance x. In he USC egime his condi ion
depends on he coupling and equency eno maliza ion, as
we will see below. The case o h ee dis an emi e s has
also been epo ed ecen ly in [39]. This e a ded back-ac ion
can lead o new collec i e s a es in which he emission a e
o pho ons by he emi e s can be enhanced o inhibi ed
beyond he usual Ma ko ian limi . This e ec , e med su-
pe dupe adiance [36], allows decay a es la ge han 2γ,
o wo emi e s wi h indi idual decay a e γ, which has
been p edic ed o scale linea ly wi h he numbe o qubi s
and nume ically con i med up o 100 emi e s coupled o a
one-dimensional wa eguide [37]. Non-Ma ko ian ea u es in
he emission spec um o a d i en wo-qubi sys em we e also
in es iga ed ecen ly, oge he wi h an expe imen al p oposal
in ci cui QED using ansmon qubi s and Josephson-Junc ion
a ays [40]. Ano he in e es ing p oposal o obse ing modi-
ica ion o collec i e phenomena has been laid ou in [41], by
emula ing he emi e dynamics in op ical wa eguide a ays.
2469-9926/2021/104(5)/053701(14) 053701-1 ©2021 Ame ican Physical Socie y
CARLOS A. GONZÁLEZ-GUTIÉRREZ e al. PHYSICAL REVIEW A 104, 053701 (2021)
In gene al, memo y e ec s can be iewed as a p oblem o
quan um eedback, whe e he sys em is ed wi h a quan um
signal a e some ime delay. A gene al heo y o quan um
eedback using enso ne wo ks was de eloped in [42] and
la e gene alized o mul iple delays in [43]. E icien nume -
ical me hods o ea mo e complex sys ems ha e also been
p oposed in [44,45], and he capabili ies o eedback o gene -
a e uni e sal s a es o quan um compu ing was demons a ed
in [46].
So a , all p e ious wo ks ely on neglec ing coun e -
o a ing e ms in he ligh -ma e coupling. In his wo k we
p esen a i s s ep in gene alizing he e ec o dis ance
o he USC coupling egime, combining bo h sou ces o
non-Ma ko iani y. This is done by conside ing an Ohmic
en i onmen wi h non la spec al densi y, and by aking in o
accoun delay-memo y e ec s due o ini e dis ance sepa-
a ion be ween emi e s. Using he pola on ans o ma ion,
a me hod ha has been shown o be use ul o discussing
he dynamics o wQED in he USC egime, we i s e iew
he g ound-s a e p ope ies o wo dis an emi e s as done
in [47]. We in oduce a disc e e model o suppo ou nu-
me ical calcula ions, ma ching he Ohmic spec al densi y
used h oughou he pape . We discuss and cha ac e ize he
localiza ion-delocaliza ion quan um phase ansi ion, inding
closed o mulas o he en anglemen en opy. Then, we dis-
cuss he dynamics o he emi e s in he low-ene gy sec o . A
de ailed discussion o he Fe mi wo-a om p oblem in he USC
egime is also gi en. As well, we compu e he e ec i e decay
a es o ini ially co ela ed symme ic (an isymme ic) ini ial
s a es. In sho , we p esen a o malism o he s udy o he
di e en sou ces o non-Ma ko iani y wi h a ious emi e s
in he nonpe u ba i e egime o wQED.
The es o he pape is o ganized as ollows. The nex
sec ion deals wi h he model, he pola on o malism, and
in oduces he disc e e model o he wa eguide. Then he
g ound s a e is discussed. Sec ion IV is he main sec ion o
his wo k. We discuss he low-ene gy sec o o he dynamics.
The pape ends wi h he conclusions, while se e al echnical
aspec s a e assigned o he Appendixes.
II. THEORETICAL MODEL
We s a e iewing he spin-boson model (SBM) o Nq
spin-1/2 pa icles (o qubi s) loca ed a posi ions xjin
one spa ial dimension. This model is also known as he
Nq-impu i y SBM, and i is conside ed a pa adigma ic Hamil-
onian o he unde s anding o decohe ence, dissipa ion, and
he physics o open quan um sys ems [24,48]. Applica ions o
his model include p oblems om undamen al physics such
as he quan um o classical ansi ion [48], elec onic anspo
in biological complexes [49], and quan um simula o s [50].
The gene al SBM is desc ibed by he ollowing Hamil onian
(¯h=1):
H=
2
Nq
j=1
σz
j+
k
ωka†
kak
+
Nq
j=1
σx
j
k
gk(ake−ikxj+H.c.),(1)
whe e σz
j=|ee|j−|gg|j,σx
j=|eg|j+|ge|j.The
SBM was o iginally in oduced o a single wo-s a e sys em
as a undamen al physical model o decohe ence [24]. I
desc ibes he in e ac ion o Nqspin-1/2 pa icles wi h he
su ounding en i onmen conside ed as a collec ion o ha -
monic oscilla o s a ini e empe a u e (hea ba h). In wQED,
i is he s anda d model o ei he ac ual spin-1/2 sys ems o
quan um emi e s wi h anha monic spec um a e pe o ming
he wo-le el app oxima ion. In he la e case, he wo-le el
app oxima ion can be ques ioned a enough s ong couplings
[51,52]. In his wo k, we ix ou a en ion on na u al o a -
i icial wo-le el emi e s, o which he SBM p o ides an
excellen desc ip ion in he USC egime, ou main in e es
he e. This ul as ong coupling egime o ligh and ma e
has been ecen ly achie ed in se e al expe imen al sys ems
in ol ing supe conduc ing ci cui s [53,54], semiconduc o s
[55], o ganic agg ega es [56], op omechanical sys ems [57],
and o he s [11,58].
The sys em-en i onmen coupling can be comple ely en-
coded in he so-called spec al unc ion o he en i onmen ,
de ined as [48]
J(ω)=2π
k|gk|2δ(ω−ωk),(2)
whe e gkis he coupling s eng h o he k h mode o e-
quency ωk. The explici o m o his spec al unc ion depends
on he physical ealiza ion o he co esponding ba h. In
his pape we will be in e es ed in modeling he en i on-
men as an Ohmic wa eguide o which gk∝√ωk,gi ing
ise o an Ohmic spec al unc ion J(ω)=παω in he con-
inuum limi , being α he coupling s eng h pa ame e (see
Appendix A1b). Fo he single SBM (Nq=1) and ze o em-
pe a u e i is well known ha , as he coupling inc eases, he e
exis s a c i ical alue whe e he sys em su e s a quan um
phase ansi ion called localiza ion ansi ion wi h anishing
spin magne iza ion σz=0[59]. Expe imen al ealiza ions
o he SBM h ough he coupling o a lux qubi o an open
1D ansmission line using a sha ed Josephson junc ion ha e
been achie ed in Re . [54]. In he case o s ong coupling, he
spon aneous emission a e om he qubi o he wa eguide
was measu ed o be γ∼2π×88MHz o a qubi equency
o ∼2π×3.99GHz. F om he mic oscopic desc ip ion
o he SBM we know ha α=γ/π (see Appendix A1d),
which esul s in a alue o α≈7×10−3.In his egime
esul s a e well desc ibed wi hin he RWA. On he o he hand,
o explo e he USC egime we equi e ha γ∼, which
was also achie ed in he same expe imen (wi h a di e en
de ice), measu ing alues o γ∼2π×9.24 GHz, and ∼
2π×7.68GHz, om which we ex ac he spin boson cou-
pling α≈0.38. This is a clea mani es a ion ha he sys em
has en e ed he nonpe u ba i e USC egime [54]. F om a
gene al iew, i seems ha expe imen s wi h open ansmis-
sion lines (p opaga ing pho ons) can explo e a wide ange
o coupling egimes in ega ds o he SBM, anging om
he unde damped (α<0.5) o he localized egime (α>1.0).
Ano he ecen expe imen in he con ex o he d i en SBM
[60] has shown ha alues o α∼0.8 a e wi hin he cu en
each.
In gene al, he SBM can no be diagonalized, bu a use ul
app oach based on he in oduc ion o a a ia ional displaced
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oscilla o basis can be employed o he s udy o s a ic and
dynamical p ope ies o such sys ems [19,31,47,61–63]. The
accu acy o he pola on app oach has been es ed in se -
e al wo ks o one-qubi [27,64–66] and mul iqubi sys ems
[67,68]. The a ia ional app oach is based on he ollowing
mul iqubi pola on ans o ma ion:
UP[{ k}]=exp −
Nq
j=1
σx
j
k
( ka†
keikxj− ∗
kake−ikxj),
=
Nq
j=1
Uj,(3)
whe e Uj=exp[−σx
jk( ka†
keikxj−H.c.)]. The ac o iza-
ioninEq.(3) in local ope a o s ac ing on he qubi s can
be done once iso opic p opaga ion o bosons (pho ons) is
assumed, i.e., | k|=| −k|. The g ound-s a e Ansa z can be
de ined as he applica ion o he uni a y in Eq. (3) o a nonen-
angled s a e o he qubi s and pho ons
|GS[ k,ζ
s]=UP
sj
ζs|s1,...,sNq⊗|0,(4)
whe e |s1,...,sNqis an a bi a y s a e o he qubi s and |0=
|0,...,0is he mul i acuum s a e o he pho onic wa eguide.
Wi hin his app oach he ask o inding he g ound-s a e en-
e gy o he SB sys em is comple ely equi alen o minimizing
he ene gy o e he ollowing e ec i e Nq-qubi Hamil onian:
HS=
2
Nq
j=1
σz
j−
i<j
Jijσx
iσx
j+2
k
k(ωk k−2gk),(5)
whe e he qubi equencies become eno malized acco ding
o he ule
=exp −2
k| k|2,(6)
and he pho on-media ed Ising in e ac ion
Jij =2
k
k(2gk−ωk k)cos [k(xi−xj)].(7)
The Hamil onian in Eq. (5) esembles he one desc ibing he
Ising model in he p esence o a ans e se magne ic ield.
In he case o a single-qubi , he g ound-s a e ene gy in he
pola on basis is EGS =− /2+k k[ωk k−2gk], which
can be minimized wi h espec o he a ia ional pa ame e s
{ k}. The co esponding ee-ene gy minimiza ion esul s in a
sel -consis en ela ion o he a ia ional pa ame e s [61]
k=gk
ωk+
.(8)
In he case o a single qubi , he eno maliza ion equency can
be compu ed explici ly, ob aining he well-known o mula in
he scaling limi /ωc1[24,48],
≈
ωcα/(1−α)
.(9)
I is wo h ecalling ha his eno maliza ion is esponsi-
ble o he localiza ion-delocaliza ion phase ansi ion ha
FIG. 1. (a) Spec al unc ion o he Ohmic wa eguide J(ω).
Open ci cles a e ob ained om he disc e e model simula ions wi h
pa ame e s: {N,ω
c,α}={1001,1.0,0.1}and he solid line wi h
he o mula J(ω)=παω. (b) Pic o ial illus a ion o wo wo-le el
emi e s sepa a ed a dis ance xin e ac ing wi h he guided modes o
an Ohmic one-dimensional wa eguide. (c) Ci cui -QED implemen-
a ion o he sys em (see Appendix A1 o u he de ails).
co esponds o he e omagne ic-an i e omagne ic phase
ansi ion in he Kondo model [69]. He e d ops o ze o as
he coupling inc eases, igge ing he ansi ion a he c i ical
coupling αc=1.
Disc e e mic oscopic model
Below we will ace si ua ions, mainly when we discuss
he dynamics o he sys em, o which an analy ical solu ion
is no possible. Then we app oxima e he en i onmen us-
ing a ini e numbe o modes Nby aking ad an age o he
disc e e model o a ansmission line esona o exposed in
Appendix A1b. In he con inuum limi (N→∞) his model
ep oduces he spec al linea unc ion o an Ohmic en i-
onmen wi h an app op ia e cu o equency: see Fig. 1(a),
whe e he poin s a e he spec al unc ion cons uc ed om
he disc e e model, eco e ing he linea dispe sion ela ion
ωk= g|k|. This ini e N-model has al eady been success-
ully used o es he dynamics o ew emi e s in e ac ing
wi h wa eguides wi hin [35], and beyond [62] he RWA, wi h
nume ical me hods based on ma ix-p oduc -s a e simula ions
(MPS), showing ema kable ag eemen wi h he ad an age o
low compu a ional cos . In Appendix A1, o comple eness,
we ske ch he main cha ac e is ics o he disc e e model in
he con ex o ci cui QED. O cou se, when possible, he
con e gence o his model o he Ohmic (con inuum) model
has been e i ied.
III. TWO-QUBIT SPIN-BOSON MODEL:
STATIC PROPERTIES
The g ound s a e o wo qubi s in an Ohmic en i onmen
was discussed in [47]. They used a a ia ional Ansa z equi -
alen o he a ia ional pola on ea men . In his sec ion,
we e iew g ound-s a e p ope ies ha will be needed when
discussing he dynamics o he sys em. This also is use ul
o benchma k he disc e e model o nume ical simula ions.
Fo he wo-qubi case, he ela i e dis ance be ween emi e s
053701-3
CARLOS A. GONZÁLEZ-GUTIÉRREZ e al. PHYSICAL REVIEW A 104, 053701 (2021)
FIG. 2. Ising-like coupling as a unc ion o he no malized ela-
i e dis ance be ween he emi e s, o di e en coupling pa ame e s.
Solid lines a e plo ed acco ding o Eq. (12), and emp y do s co e-
spond o nume ical esul s based on he disc e e Ohmic model wi h
pa ame e s {N,/ω
c}={1001,0.2}.
plays an impo an ole in he s a ic and dynamical p ope ies.
In his si ua ion he induced Ising in e ac ion o he e ec i e
Hamil onian modi ies he single-qubi beha io , gi ing ise o
collec i e e ec s. Fo Nq=2 he Hamil onian (5) eads
HS=
2
2
j=1
σz
j−Jσx
1σx
2+2
k
k(ωk k−2gk),(10)
whe e J(x)=2k k(2gk−ωk k) cos(kx),being x=x2−
x1 he dis ance be ween he qubi s. In his case he g ound-
s a e ene gy is EGS =−
2
+J2+2k k(ωk k−2gk),
and i s minimiza ion esul s in
k=gk
ωk
E+Jcos(kx)
E+Jcos(kx)+2
/ωk
,(11)
whe e E=2
+J2. The e ec i e Ising model is cha ac-
e ized by he a io J/ . In wha ollows, we compu e bo h
pa ame e s nume ically by using he ini e-Nmodel. Howe e ,
i is always appealing o ha e analy ical exp essions e en i
app oxima es. Taking he limi J→0inEq.(11) we eco e
he one-qubi ela ion in Eq. (8). In e es ingly, he induced
Ising coupling can be compu ed in he la ge coupling app ox-
ima ion o which k−→ gk/ωkas →0, which gi es he
esul [47]
J≈1
πωc
0
dωJ(ω)
ωcos(ωx/ g)
=αωcsinc(ωcx/ g),(12)
whe e we ha e used he Ohmic spec al unc ion J(ω)=
παω, and assumed a linea dispe sion ela ion ωk= g|k|
in he con inuum limi . This Ising coupling shows long-
ange damped oscilla ions as a unc ion o he dis ance. This
means ha , depending on he dis ance, e omagne ic (J>
0) o an i e omagne ic (J<0) in e ac ion can be induced.
In Fig. 2we show he beha io o he Ising-like coupling
as a unc ion o he emi e s ela i e dis ance o di e en
coupling s eng hs alues. Con inuous lines a e calcula ed
using he app oxima ion in Eq. (12), and open ci cles indi-
ca e esul s ob ained om nume ical simula ions based on he
mic oscopic disc e e model p esen ed in Appendix A1b.We
obse e a e y good ag eemen be ween he con inuous and
FIG. 3. Reno malized equency as unc ion o he coupling
s eng h o di e en ela i e dis ances be ween emi e s. O he pa-
ame e s a e he same as in Fig. 2.
he disc e ized mic oscopic Ohmic model. A mo e accu a e
bu cumbe some exp ession o he Ising coupling can also
be ob ained by using he exac exp ession o kin Eq. (8),
which includes small co ec ion e ms o he o mula gi en in
Eq. (12). We show his exp ession in Appendix A2.
Ob aining an exp ession o he eno malized qubi e-
quency o wo qubi s is no as simple as in he single-qubi
case. In gene al, i has o be ound by nume ical means. As-
suming ha Jisgi enbyEq.(12) and he e o e does no play
any ole on he ene gy minimiza ion, he op imal a ia ional
pa ame e s can be app oxima ed by k≈gkE/(ωkE+2
),
which a e eplacemen in o Eq. (6)gi es
=exp −2
k
g2
kE2
ωkE+2
2.(13)
This equa ion shows ha is now dis ance-dependen
h ough he Ising-like coupling. Fo in ini e sepa a ion dis-
ance (J=0), he eno malized equency in he scaling limi
is gi en by ≈(e/ωc)α/(1−α), which shows he quan um
phase ansi ion a α=1, in ag eemen wi h he single impu-
i y case in Eq. (9). Fo sligh ly sepa a ed qubi s in he limi
J/ 1, we can ake E→J, ob aining ha
≈(e2/ωcJ)α/(1−2α),(14)
p edic ing he c i ical coupling a α=1/2 in s a k con as
wi h a single qubi [47]. I is wo h men ioning ha his shi ed
c i ical coupling can also be in e ed by s udying he p ope -
ies o he dynamical Ke nel in he gene al non-Ma ko ian
case [62]. In Fig. 3we plo he eno malized equency
a ying he ela i e dis ance x. We show he lowe and up-
pe bounds, gi en by he exp essions o ze o and in ini e
sepa a ion dis ances, espec i ely. I is clea ha he e ec i e
in e ac ion be ween qubi s modi ies he localiza ion ansi ion
poin depending o hei ela i e posi ion. In pa icula , a
in e media e ela i e posi ions (e.g., x=8xin Fig. 3),
d ops ab up ly o ze o in a discon inuous way, and he local-
iza ion ansi ion occu s o alues o he coupling s eng h
α∈[0.5,1.0].
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DISTANT EMITTERS IN ULTRASTRONG WAVEGUIDE … PHYSICAL REVIEW A 104, 053701 (2021)
G ound-s a e and en anglemen en opy
The g ound s a e o he e ec i e wo-spin Hamil onian (10)
can be ound by s aigh o wa d diagonaliza ion. I eads
|GSS=cos θ|gg+sin θ|ee,(15)
whe e
cos θ= +E
( +E)2+J2(16)
and
sin θ=J
( +E)2+J2.(17)
The g ound s a e o he comple e sys em o wo-qubi s plus
he bosonic ba h is hen ob ained by applying he pola on
ans o ma ion o he p oduc o he g ound s a e |GSSand
he mul imode acuum s a e o he bosonic en i onmen ac-
co ding o he Ansa z (4)
|GS=UP|GSS|0
=(cos θσx
1+sin θσx
2)UP|e|g|0.(18)
A e applying he ans o ma ion we ge he ollowing en-
angled s a e in e ms o mul imode cohe en s a es o he
pho onic ba h
|GS=1
2(sin θ+cos θ)|−−|χ−−
+1
2(sin θ−cos θ)|−+|χ−+
+1
2(sin θ−cos θ)|+−|χ+−
+1
2(sin θ+cos θ)|++|χ++,(19)
whe e we chose he eigens a es o σx
j o ep esen ing he
s a e o he qubi s, i.e., |±j=(|ej±|gj)/√2, and he mul-
imode cohe en s a es a e gi en by
|χ±±=exp −i
k
2
ksin kx
k|±( k+ keikx ),(20)
|χ±∓=exp i
k
2
ksin kx
k|± k∓ keikx,(21)
wi h |αk=D(αk)|0, being D(αk) he displacemen op-
e a o . No ice ha in he deep-s ong coupling (localized
egime) when →0, he g ound s a e becomes |GS=
(|−−|χ−−+|++|χ++)/√2, which can be ecognized
as a Sch ödinge ca s a e o ligh and ma e analogous o he
one-qubi SBM [70]. Ha ing ob ained he g ound s a e, one
can also compu e ele an physical obse ables, o ins ance,
he qubi magne iza ion and he en anglemen en opy. In he
i s case we ind ha
σz
jGS =GS|σz
j|GS=−
cos 2θcos [φ(x)],(22)
whe e we ha e in oduced he auxilia y unc ion φ(x)=
4k 2
ksin kx. I is also in e es ing o compu e he linea
en anglemen en opy, de ined as SL=1−T 2
q, whe e he
FIG. 4. En anglemen en opy as a unc ion o he coupling
s eng h o inc easing dis ance be ween he emi e s. Pa ame e s a e
he same as in Fig. 3.
educe densi y ma ix o he wo-qubi sys em can be ob-
ained by acing ou he ba h deg ees o eedom, i.e., q=
T en GS =T en |GSGS|. This measu e quan i ies he de-
g ee o mixedness o he wo-qubi g ound s a e wi h he
bosonic ba h. Rema kably, a a he gene al exp ession o he
linea en opy o a bi a y emi e sepa a ions can be ob-
ained, bu he exp ession is oo leng hy o be shown he e (see
Appendix A3). Howe e , we show he limi ing o mula o
in ini e dis ance
SL=1
43−2
2
−
4.(23)
I is well known ha en anglemen en opy e lec s he appea -
ance o a quan um phase ansi ion, so we expec o obse e
signals o he localiza ion ansi ion in he en anglemen as
he coupling is inc eased, in a simila ashion as obse ed
o in Fig. 3. Quan um phase ansi ions mani es as a
nonanaly ici y in he en anglemen con ained in he o al s a e
o he sys em and i s en i onmen [71]. We con i m his by
plo ing he en anglemen en opy as a unc ion o he cou-
pling in Fig. 4 o inc easing ela i e dis ance. The dashed line
in Fig. 4is gi en by Eq. (23), indica ing a smoo h beha io o
he en opy wi h no discon inui ies o la ge couplings. This is
expec ed om he one-qubi esul s o he eno malized e-
quency discussed in Sec. II o which αc∼1. The asymp o ic
en opy in his case is a comple e mixed s a e o he qubi s
wi h SL→3/4, which can be seen di ec ly om Eq. (23)
in he limi →0. Fo ini e sepa a ion we ha e a iche
beha io , as sudden decay o he en opy occu s a speci ic
couplings, e lec ing he in luence o he e ec i e qubi -qubi
in e ac ion on he localiza ion ansi ion. No ice also ha in
his case he s eady-s a e en opy as a unc ion o he coupling
is SL=1/2. This is because he Ising in e ac ion domina es
o e he magne ic ield p oduced by in he wo-spin e -
ec i e model, and he g ound s a e becomes degene a e wi h
educed densi y ma ix q=1/2. This is also clea om he
gene al exp ession o he linea en opy in Eq. (A20)in he
case o sin θ=cos θ=1/√2.
IV. DYNAMICS
No only he g ound s a e bu he i s exci ed s a es a e
well cap u ed by he pola on Ansa z [31,62,63,72]. I is con e-
nien o wo k in he pola on pic u e (1): HP=U†
PHUP≈He ,
053701-5
CARLOS A. GONZÁLEZ-GUTIÉRREZ e al. PHYSICAL REVIEW A 104, 053701 (2021)
which can be app oxima ed by [62]
He =
2
2
j=1
σz
j−2
2
j=1
σz
j
kk
k kei(k−k)xja†
kak
+
2
j=1
k2 (0)
k+( −ωk)σ−
ja†
keikxj+H.c.
+
k
ωka†
kak−Jσx
1σx
2+2
k
k(ωk k−2gk).(24)
He e (0)
kis he a ia ional pa ame e o a single qubi gi en
by Eq. (8), i.e., when he e omagne ic in e ac ion is neg-
ligible (J=0), and is a small co ec ion. The key ac
he e is ha He p ese es he exci a ion numbe . The e o e,
his e ec i e Hamil onian can be now ea ed using s anda d
RWA me hods. In he ollowing we will s udy he gene a ed
dynamics in he pola on ame in o de o ex ac ele an
in o ma ion o he sys em in he labo a o y pic u e.
As he Hamil onian He p ese es he numbe o o al ex-
ci a ions, we can p ojec he dynamics in he single exci a ion
sec o spanned by he basis: {|e|g|0,|g|e|0,|GSS|1k}.
This means ha pho onic exci a ions a e c ea ed om he
pola on g ound s a e o he wo-spin sys em. The s a e o he
sys em a any ime is pos ula ed ia he ollowing dynamical
pola on Ansa z [62]:
|( )P=c1|e|g+c2|g|e+
k
ψka†
k|GSS|0,(25)
whe e he ime-dependen p obabili y ampli udes equi e o
ul ill he no maliza ion condi ion i|ci|2+k|ψk|2=1.
These coe icien s sa is y he coupled se o equa ions,
i˙c1=−Jc2+
k
Gke−ikx1ψkcos θ, (26)
i˙c2=−Jc1+
k
Gke−ikx2ψkcos θ, (27)
i˙
ψk=−(˜
−ωk)ψk+Gk
j
eikxjcjcos θ
+2 k
j,k
kei(k−k)xjψkcos 2θ. (28)
He e we ha e de ined he unc ions Gk=2 (0)
k+( −
ωk) and ˜
= cos 2θ+Jsin 2θ. Using he disc e e
model, his se o equa ions a e in eg a ed nume ically and,
hen, ans o med back o he o iginal labo a o y pic u e.
A. Dynamics in he pola on ame
Fo u u e con enience and unde s anding, i is use ul
o s a analyzing he dynamics in he pola on pic u e. The
eason is wo old. I is easie , so some explici o mulas
can be ob ained, and i will be con enien o unde s and
he dynamics in he labo a o y ame. As well, i he cou-
pling is small enough, bo h pola on and pola on ame a e
equi alen . Mo ing o he o a ing ame o he pho on am-
pli udes, ˜
ψk=e−i(˜
−ωk) ψk, and pe o ming he in eg a ion,
i.e., acing ou he pho onic modes, and se ing ˜
ψk(0) =0, we
a i e o
i˙ci=−Jcj−2i2
cos2θ
πωc
0
dωJ(ω)
(ω+ )2
×
0
dτei(˜
−ω)τci( −τ)
−2i2
cos2θ
πei˜
x/ gωc
0
dωJ(ω)
(ω+ )2
× −x/ g
0
dτei(˜
−ω)τcj( −x/ g−τ),(29)
whe e i was assumed ha |Gk|2≈42
(0)
k
2 o small .
He e he p esence o non-Ma ko ian p ocesses is e iden . The
second and hi d con ibu ions in he .h.s bo h depend on
he spec al unc ion J(ω) o he wa eguide (en i onmen ).
So he e is an una oidable sou ce o non-Ma ko iani y as
consequence o he memo y o he en i onmen , which canno
be conside ed as la (Ma ko ian) in gene al [73]. On he
o he hand, he hi d e m con ains an addi ional sou ce o
non-Ma ko iani y due o he ini e dis ance e ec be ween
emi e s, which also en e s in he nonlocal pa s o he Ke nel
and he e o e in alida es he Ma ko app oxima ion. These
wo non-Ma ko ian imescales a e di e en in gene al and
need o be dis inguished in he dynamical e olu ion. Un o -
una ely, his is no so simple, and we ha e o ge id o one
o such memo y ime scales in o de o s udy he beha io o
Eq. (29). A his poin we ollow he lines o he Weisskop -
Wigne heo y o spon aneous emission [74], and neglec he
memo y e ec s o he en i onmen in o de o e alua e he
con ibu ion o he las wo e ms in Eq. (29). We can hen
make he eplacemen ci( −τ)≈ci( ) and ex end he ime
in eg al o in ini y. By using he iden i y [75]
lim
→∞
0
dτei(˜
−ω)τ=πδ(ω−˜
)+iP1
˜
−ω,(30)
we a i e o he coupled delay-di e en ial equa ions
˙ci( )=iJcj( )−γ
2ci( )
−γ
2ei˜
x/ gcj( −x/ g)( −x/ g),(31)
wi h he spon aneous emission a e de ined by
γ=42
J(˜
)
( +˜
)2cos2θ, (32)
and he unc ion (τ) being he Hea iside s ep unc ion.
Ob iously, once he app oxima ion ci( −τ)≈ci( ) is made,
he only sou ce o non-Ma ko iani y is due o he ini e dis-
ance o he emi e s, which induces back-ac ion o he ield,
leading o a adia ion eedback phenomenon in he sys em. As
a consequence, hese equa ions esemble he delay dynamics
ob ained o wo dis an emi e s coupled o a wa eguide wi h
la spec al densi y o he ield modes [36]. Howe e , in his
case a cohe en coupling e m be ween he emi e s appea s
as a consequence o he Ising coupling Jinduced by he
pola on ans o ma ion. As his cohe en in e ac ion depends
on he dis ance be ween emi e s [see Eq. (12) and Fig. 2],
i con ibu es o he dynamics a small sepa a ion dis ances.
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DISTANT EMITTERS IN ULTRASTRONG WAVEGUIDE … PHYSICAL REVIEW A 104, 053701 (2021)
No ice also ha he spon aneous emission a e and he ela i e
phase gained by he e a ded adia ion ge s eno malized as
hey a e gi en in e ms o he pola on qubi equency .
Fo in ini ely dis an emi e s, he spon aneous emission a e
educes o he esul o a single emi e wi h γ=J( )=
πα [19,62].
I is wo h men ioning ha in de i ing he delay di e en ial
equa ions (31), we ha e neglec ed he Lamb-shi co ec ion
o he qubi equency, which is o mally gi en by
δL=2
π2
Pωc
0
dωJ(ω)
(ω+ )2(˜
−ω).(33)
A de ailed analysis o he consequences o collec i e spon-
aneous emission and Lamb shi is gi en in [35]. The
magne iza ion dynamics (o popula ion in e sion) in he po-
la on image can be exp essed in e ms o he qubi p obabili y
ampli udes,
σz
i( )P=cos2θ[2|ci( )|2−1] −sin2θ[2|cj( )|2−1].(34)
Le us now analyze he eme gence o non-Ma ko ian dy-
namics due o he back-ac ion induced by he delay e m in
Eq. (31) o di e en ini ial s a es o he emi e s. Na u ally,
we expec hese esul s o ag ee wi h hose ob ained wi hin he
RWA wi h a la en i onmen o e y small couplings [36].
We i s explo e he case o la ge bu no in ini e sepa a ion
dis ance be ween he emi e s o which J≈0. The s eady-
s a e solu ions in he pola on pic u e can be compu ed om
he s anda d inal alue heo em [76], by aking he Laplace
ans o m in Eqs. (31). We i s ake ini ial symme ic (an i-
symme ic) s a es in he pola on Ansa z, i.e., c1(0) =c2(0) =
1/√2o c1(0) =−c2(0) =1/√2. Fo hese ini ial condi ions
he Laplace ans o med ampli udes ead
˜c±(s)=1
√2[s+γ/2±γe(i −s)x/ g/2],(35)
whe e ˜c+=˜c1=˜c2, and ˜c−=˜c1=−˜c2. Fo a eno malized
qubi equency, and sepa a ion dis ance sa is ying he condi-
ion x/ g=2nπ, being n∈Z, he symme ic ini ial s a e
will decay o he pola on g ound s a e, i.e., lim →∞ c+( )=
0. Howe e , i he ini ial s a e is an isymme ic we ge
a bound s a e wi h ini e exci a ion p obabili y ampli ude
gi en by lim →∞ c−( )=(1 +γx/2 g)−1/√2[36]. This
gi es he s eady-s a e magne iza ion o he an isymme ic
s a e σz
−P=(1 +γx/2 g)−2−1, which is di e en om
−1 o ini e-dis ance emi e s [see Fig. 5(a)]. This bound s a e
in he con inuum (BIC) is he esul o a apped s a iona y
exci a ion be ween he wo emi e s ac ing as an e ec i e
ca i y o med by wo pe ec ly e lec i e mi o s [77,78].
B. Labo a o y ame
The p e ious analysis will help us discuss he ac ual dy-
namics. They can be ob ained by back- ans o ming wi h he
pola on uni a y. I u ns ou his can be done exac ly, as we
ha e es ic ed ou analysis o he single-exci a ion pola on
mani old. In pa icula , he magne iza ion in he labo a o y
FIG. 5. Dynamics o he magne iza ion σz o ini ial symme ic
|+
0and an isymme ic |−
0s a es. Dash-do ed lines indica e he
e olu ion in he pola on pic u e. To enable compa ison wi h he ue
dynamics, he pola on lines a e eno malized by /.F om(a) o
(d) he inc easing couplings a e α=0.01, α=0.1, α=0.5, and
α=1.0. O he pa ame e s a e L=40π,N=1001, x=2π g/.
pic u e eads
σz
i( )=P( )|U†
Pσz
iUP|( )P
=
σz
iP+4 cos θReci
k
kψ∗
k
−4sinθRecj
k
kψ∗
k
+4 cos 2θ
kk
k kψkψ∗
k,(36)
which shows he dynamical con ibu ions due o each emi e
and ield co ela ions.
In (36) we see ha he di e ences be ween bo h ames a e
O( k); he e o e in he weak coupling bo h ames gi e he
same esul s, as expec ed by cons uc ion. In he same egion,
one can pe o m he RWA app oxima ion al eady in he o ig-
inal spin boson model (1). This is illus a ed in Fig. 5(a) o
which we eco e RWA esul s.
As we inc ease he sys em-ba h coupling α, non-
Ma ko ian dynamics eme ges as consequence o he memo y
o he ba h and also om in e e ence caused by e a da ion
e ec s [36]. A his poin he collec i e wo-qubi dynamics
is no longe cap u ed by he e ec i e delay dynamics dic a ed
by Eqs. (31). We show hese esul s in Figs. 5(b),5(c), and
5(d) o a ixed sepa a ion dis ance be ween he emi e s. We
also expec ha o e y s ong couplings, he localiza ion
ansi ion akes place ( →0), and he wo-qubi sys em
eezes in a s a e o ze o magne iza ion. In his deep-s ong
coupling egime e a da ion e ec s a e no longe p esen in
he magne iza ion.
1. The Fe mi p oblem in he USC egime
Ano he in e es ing ac is he appa en causali y iola ion
ha can be obse ed in Fig. 5(c). Fo mode a e couplings
(α=0.01,0.1), whe e RWA s ill applies, he dynamics o
symme ic and an isymme ic ini ial s a es a e exac ly he
053701-7
CARLOS A. GONZÁLEZ-GUTIÉRREZ e al. PHYSICAL REVIEW A 104, 053701 (2021)
same be o e he e a da ion ime, i.e., independen exponen ial
decay o bo h emi e s a he same a e (Fe mi golden ule)
occu s. As we inc ease he coupling we can see om Fig. 5(c)
ha his is no longe he case. Symme ic and an isymme -
ic ini ial s a es s a o ha e di e en dynamics long be o e
hey each he e ical line indica ing he ligh -cone o e lap
be ween he wo emi e s. This appa en pa adox was i s
s udied by Fe mi in 1932, and i is known as he Fe mi p ob-
lem [79]. Fo una ely his pa adox can be explained in e ms
o co ela ions be ween spacelike dis an e en s a ising in he
ul as ong coupling egime and ha a e absen in usual RWA
models [80]. The Fe mi pa adox can be o mula ed explici ly
as ollows: in a sys em o wo dis an emi e s, whe e ini ially
one o hem is in he exci ed s a e and he o he is in i s g ound
s a e, he ollowing in e es ing ques ion a ises: is i possible o
exci e he second emi e h ough he spon aneous emission
o he i s one a a ime <x/ g? I u ns ou ha his is no
possible, and he e is no causali y iola ion in his p oblem,
as was poin ed ou by Fe mi [79], and be e jus i ied la e
by se e al au ho s [81–83]. Howe e , as was shown by Sabin
e al. [80], i nonlocal co ela ions a e sha ed ini ially by he
wo emi e s, he p obabili y o inding he second emi e in
he exci ed s a e and he i s one in he g ound s a e can be
di e en om ze o e en a imes <x/ g. We will show he e
ha he exis ence o hese co ela ions explains he di e en
dynamics expe ienced by symme ic and an isymme ic ini ial
s a es a imes be o e he e a da ion ime obse ed in Fig. 5.
We now mo e o con enience o he Heisenbe g pic u e, and
w i e he equa ion o mo ion o he magne iza ion using he
o iginal labo a o y Hamil onian (1)
σz
j( )=σz
j(0) +2
0
dτσy
j(τ)V(xj,τ),(37)
whe e we ha e de ined
V(xj, )=V0(xj, )+V1(xj, )+V2(xj, ) (38)
wi h he ield ope a o
V0(xj, )=
k
gk(ake−ikxje−iωk +H.c.) (39)
and he qubi ope a o
Vi(xj, )=−
k
0
dτ|gk|2σx
i(τ)eik(xj−xi)e−iωk( −τ)+H.c.
(40)
The expec a ion alue o he qubi magne iza ion is hen he
sum o di e en con ibu ions
σz
j( )=σz
j(0)+2
0
dτσy
j(τ)V0(xj,τ)
+2
0
dτσy
j(τ)V1(xj,τ)
+2
0
dτσy
j(τ)V2(xj,τ),(41)
whe e he las wo co ela ion e ms sugges some in luence
o back-ac ion o he espec i e dis an emi e . Howe e ,
i can be shown ha bo h con ibu ions a e independen
FIG. 6. Dynamics o σz
1,2s a ing om he ini ial s a e |eg.
F om (a) o (d) he pa ame e s a e he same as in Fig. 5.
o such dis an emi e [80]. To make his clea le us o-
cus in he i s emi e magne iza ion, i.e., σz
1( ).In his
case one can show ha he co ela ion σy
1( )V2(x1, )∝
d
dxσy
1( )σx
2( −x/ g)( −x/ g), showing ha his unc ion
is ze o o imes <x/ g.An analogous calcula ion o he
au oco ela ion unc ion o he i s emi e shows ha i is
in ac independen o he second emi e , so he only con i-
bu ion o he qubi magne iza ion is due o he second e m
in Eq. (41). We can examine his con ibu ion o ini ial sym-
me ic o an isymme ic s a es o he emi e s, which a e ini ial
en angled s a es in he pola on pic u e. Then
σy
1(τ)V0(x1,τ)=±
0|σy
1(τ)V0(x1,τ)|±
0,(42)
whe e |±
0= 1
√2(σx
1±σx
2)UP|g|g|0.An explici calcu-
la ion o hese qubi - ield co ela ions is complica ed as
consequence o he non i ial ac ion o he pola on ans o m,
bu one can see ha , in gene al, hey a e di e en om ze o
o all imes and ha e di e en alue o symme ic and an-
isymme ic ini ial s a es. This is in s ong con as wi h he
RWA case, whe e hese co ela ions a e o ally absen and
his con ibu ion is exac ly ze o, as he pola on ans o m
does no en e in he ini ial s a e. This analysis explains he
di e en dynamics expe ienced by bo h ini ial s a es when he
sys em en e s in he ul as ong coupling egime as shown in
Fig. 5(c). I is also wo h men ioning ha al hough we we e
in e es ed in he di e ence in ime e olu ion o symme ic
and an isymme ic ini ial s a es, he abo e analysis is in ac
independen o he ini ial s a e o he wo-qubi sys em. These
co ela ions eme ge as he coupling o he wo dis an emi e s
o he Ohmic wa eguide inc eases, b eaking down he usual
RWA. We ha e addi ionally explo ed he dynamics o he
ini ial s a e |e|g|0in he pola on image. This is shown in
Fig. 6, whe e he magne iza ion o each emi e is plo ed
o inc easing coupling and o he same pa ame e s used
in Fig. 5. Fo weak coupling we obse e exponen ial decay
(Fe mi golden ule) o he exci ed emi e and he subsequen
exci a ion o he nonini ially exci ed emi e a ime =x/ g.
Fo α=0.1 [Fig. 6(b)] he spon aneous decay a e ge s eno -
malized, and we a e able o see he damped cohe en exchange
o he exci a ion be ween he wo emi e s un il i eaches he
expec ed equilib ium magne iza ion gi en by − /. Exci-
a ion o he second emi e be o e e a da ion ime is shown in
Fig. 6(c) o α=0.5 wi h no o ious e ec s on he dynamics
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DISTANT EMITTERS IN ULTRASTRONG WAVEGUIDE … PHYSICAL REVIEW A 104, 053701 (2021)
FIG. 7. Dynamics o he ins an aneous decay o inc easing cou-
pling α. (a) Symme ic ini ial s a e |+
0, and (b) an isymme ic
ini ial s a e |−
0. O he pa ame e s a e he same as in Fig. 5.
a x/ gand he expec ed localiza ion ansi ion o α=1.0
in Fig. 6(d), whe e localized and small ampli ude (2 /)
Rabi oscilla ions be ween he emi e s a e obse ed due o he
cohe en Ising in e ac ion mo ing he exci a ion be ween he
wo emi e s [see inse in Fig. 6(d)].
2. Time-dependen decay a e
The analysis o non-Ma ko ian dynamics allows a de-
sc ip ion in e ms o a mas e equa ion ha is local in
ime [73]. I u ns ou ha he ime-dependen gene a o
o his mas e equa ion can be w i en in he s anda d
Lindblad o m wi h ime-dependen decay a es. Following
his app oach, and om ou analysis in he pola on pic-
u e, we can de ine a ime-dependen decay a e o each
emi e as
γ(i)( )=−2Re˙ci( )
ci( ).(43)
In gene al, as in he case o p obabili y ampli udes in he
s ong coupling egime, an analy ical e alua ion o his
o mula is no possible, and we mus eso o nume ical cal-
cula ions based on he disc e e model. The ime e olu ion o
his quan i y will con ain bo h sou ces o non-Ma ko iani y,
he one due o he memo y o he ba h, and he in luence o
e a da ion e ec s. O cou se, as Eq. (43) is de ined in e ms
o he pola on p obabili y ampli udes, we need o ans o m i
back o he labo a o y ame. In Fig. 7we show he nume ical
calcula ion o he collec i e dynamics o he ime-dependen
decay a e o symme ic (γ+) and an isymme ic (γ−) ini ial
s a es.
As expec ed, o weak coupling, he decay a e s a s o
oscilla e a ound he Fe mi golden ule alue (almos wi hou
eno maliza ion) γ±=γ=πα ∼
=πα un il he eed-
back om he o he emi e suddenly accele a es he decay
p ocess, eaching a maximum alue o ∼4.4γ o he sym-
me ic s a e in Fig. 7(a). A his poin , he emi e s see each
o he , o ming a collec i e s a e and emi ing in a supe adian
manne beyond he Ma ko ian Dicke limi o 2γ o wo
emi e s loca ed a he same posi ion. This non-Ma ko ian
collec i e enhancemen o spon aneous emission has been
epo ed ecen ly wi hin he RWA limi in Re . [36], whe e
i was shown ha la ge alues can be eached a a pa icu-
la c i ical dis ance. We mus emphasize ha e en o weak
coupling he Ohmic en i onmen model used he e is no la ,
so i is easonable o expec ha ou calcula ion o he decay
a e does no ma ch exac ly he esul s o a la en i onmen .
Also, i is clea ha non-Ma ko ian p ocesses in ol ing e a -
da ion a e also p esen beyond he RWA limi s. On he o he
hand, o he an isymme ic ini ial s a e in Fig. 7(b) we ha e
simila beha io a imes <x/ g. Once hey a e causally
connec ed, in his case, he an isymme ic s a e is a da k s a e
and no decay is obse ed. Fo weak coupling his is compa -
ible wi h he appea ance o bound s a es in he con inuum
o sub adian ini ial s a es in RWA [36,77]. We also ha e
e i ied ha o sligh ly dis an emi e s in he weak coupling
we eco e he usual Ma ko ian dynamics o ini ial sym-
me ic (an isymme ic) s a es. As he coupling is inc eased
we obse e he supp ession o emission caused by he qubi
eno maliza ion equency, i.e., γ=πα accompanied by
an oscilla ing beha io accoun ing o emission and eabso p-
ion o adia ion, which explains he nega i e alues o he
decay a e. In he deep-s ong coupling egime, whe e α→1,
he sys em unde goes a localiza ion ansi ion wi h ze o mag-
ne iza ion and he wo-qubi sys em does no adia e anymo e.
I is wo h ecalling ha wha de e mines he na u e o he
collec i e s a e gene a ed a e bo h qubi s become causally
connec ed is he ac ual phase gained by he adia ion ield
du ing p opaga ion inside he wa eguide. This phase is e-
sponsible o he cons uc i e (des uc i e) in e e ence o he
delay dynamics [36,40]. Wi hin he RWA his phase is gi en
by x/ g, which is independen o he qubi - ield coupling.
In he USC egime he accumula ed phase is now coupling
and dis ance-dependen o ini e dis an emi e s as we saw in
Sec. III, h ough he eno malized qubi equency, d opping
o ze o in he localized phase o a pa icula c i ical coupling.
This is a pu ely USC phenomenon, in which non-Ma ko ian
in e e ence e ec s a e also a ec ed by he coupling o he
wa eguide as we obse e in Fig. 7. In pa icula supe adian
and sub adian beha io s a ise as app op ia e combina ions o
he symme y o he ini ial s a e, he coupling s eng h, and he
ela i e dis ance be ween he emi e s, e lec ing he complex
ela ion be ween he Hamil onian pa ame e s in his egime,
which has o be aken in o accoun in eal expe imen s aiming
o in es iga e non-Ma ko ian delay dynamics in wa eguide
QED [84]. As complemen a y in o ma ion we show in Fig. 8
he indi idual emission (abso p ion) a es, whe e he ini ial
s a e is |e|g|0in he pola on pic u e in co espondence
wi h Fig. 6, showing he an ico ela ed dynamics o bo h
emi e s. In his case one can clea ly see he in luence o
Rabi oscilla ions in he wo-qubi sys em o e he spon aneous
emission (abso p ion) a es a s ong coupling in he localized
egime.
V. SUMMARY AND CONCLUSIONS
Employing heo e ical ools based on he pola on ans o -
ma ion, we ha e add essed he p oblem o wo dis an emi e s
in e ac ing wi h a one-dimensional Ohmic wa eguide beyond
RWA. We ha e es ed ea ly esul s o he s a ic p ope ies
o he wo-impu i y SBM in he con inuum limi , compa ing
hose wi h a disc e ized ini e N-model based on ci cui QED,
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