PHYSICAL REVIEW RESEARCH 2, 023003 (2020)
Chi al quan um op ics in pho onic saw oo h la ices
Edua do Sánchez-Bu illo ,1Chao Wan,1,2Da id Zueco,3,4and Alejand o González-Tudela5
1Max-Planck-Ins i u ü Quan enop ik, Hans-Kop e mann-S . 1, 85748 Ga ching, Ge many
2Fakul ä Physik a Ludwig-Maximilians-Uni e si ä , Schellings aße 4, 80799 Munich, Ge many
3Ins i u o de Ciencia de Ma e iales de A agón and Depa amen o de Física de la Ma e ia Condensada,
CSIC-Uni e sidad de Za agoza, Calle Ped o Ce buna 12, 50009 Za agoza, Spain
4Fundación ARAID, Paseo Ma ía Agus ín 36, 50004 Za agoza, Spain
5Ins i u o de Física Fundamen al IFF-CSIC, Calle Se ano 113b, Mad id 28006, Spain
(Recei ed 4 July 2019; accep ed 2 Ma ch 2020; published 2 Ap il 2020)
Chi al quan um op ics has become a bu geoning ield due o i s po en ial applica ions in quan um ne wo ks
o quan um simula ion o many-body physics. Cu en implemen a ions a e based on he in e play be ween
local pola iza ion and p opaga ion di ec ion o ligh in nanopho onic s uc u es. In his manusc ip , we p opose
an al e na i e pla o m based on coupling quan um emi e s o a pho onic saw oo h la ice, a one-dimensional
model wi h an e ec i e lux pe plaque e in oduced by complex unnelings. We s udy he dynamics eme ging
om such s uc u ed pho onic ba h and ind he condi ions o ob ain quasipe ec di ec ional emission when he
emi e s a e esonan wi h he band. In addi ion, we ind ha he pho ons in his ba h can also media e complex
emi e -emi e in e ac ions unable in ange and phase when he emi e s ansi ion equencies lie wi hin a band
gap. Since hese e ec s do no ely on pola iza ion, we p opose an implemen a ion based on ci cui QED o
obse e his physics.
DOI: 10.1103/PhysRe Resea ch.2.023003
I. INTRODUCTION
Designing non ecip ocal op ical de ices a he classical
and quan um le el has emained a challenge o many yea s
(see, e.g., Re s. [1–4], and e e ences he ein). On he classical
le el, he sea ch was mainly ocused on sys ems b eaking
Lo en z ecip oci y such as magne o-op ical ma e ials. On
he quan um side nanopho onic sys ems ha e eme ged as
a powe ul candida e due o he link be ween he pola iza-
ion and p opaga ion di ec ion ha appea s hanks o he
subwa eleng h-ligh con inemen [5,6]. Using his connec ion
and he in insic pola iza ion o classical and quan um emi -
e s (QEs) many expe imen s ha e shown chi al ligh -ma e
coupling in wa eguides [7–12], and ha nessed i o achie e,
e.g., op ical isola ion a he single-pho on le el [2]. Chi al
quan um op ical sys ems [13] ha e also been p oposed o
build spin-pho on in e aces [14] and di ec ional ampli ie s
[15,16] in quan um ne wo ks, o enginee many-body spin o
pho onic s a es [17–21], o emi nonclassical ligh [22,23], o
o induce exo ic sel -o ganiza ion pa e ns [24], among o he
phenomena.
These exci ing p edic ions ha e igge ed a ace o obse e
his non ecip ocal penomena also beyond he op ical egime.
Fo example, linea op ical ci cula o s ha e been designed in
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access publica ion unded by he Max Planck Socie y.
he mic owa e egime [25–28] and non ecip ocal ansmis-
sion o pho ons has been achie ed using wo qubi s [29]. Re-
ga ding he possibili y o ob aining chi al ligh -ma e in e -
ac ions, he e a e al eady heo e ical p oposals o ob ain hem
based on using complex nonlocal ligh -ma e in e ac ions
[17–19] o wi h mo ing a oms [30], bu hei expe imen al
ealiza ion emains elusi e. We explo e he e an al e na i e
whe e QEs couple locally o a pho onic ba h wi hou ime-
ese al symme y. In pa icula , we use he so-called pho onic
saw oo h la ice [see Fig. 1(a)], also labeled as chain [31]o
iangle la ice [32,33]. This is a one-dimensional model wi h
closed loops, which allows o complex couplings (φ= 0in
Fig. 1) be ween he bosonic deg ees o eedom de ining an
e ec i e magne ic lux pe loop [34]. We p edic ha one can
ob ain quasipe ec di ec ional emission [13] when he QEs
a e esonan wi h he bands o he sys em. Ra he han selec -
ing a momen um using he des uc i e in e e ence induced
by nonlocal couplings [17–19] o b eaking he symme y by
means o an ex e nal lase in a phononic sys em [35], i is he
asymme ic na u e o he band s uc u e he one esponsible
o he chi ali y. The p ope ies o he ba h lead o o he
phenomena such as he eme gence o a subla ice-dependen
di ec ional emission o QE in e ac ions unable in ange and
phase when hei equencies lie in a band gap.
The ex is o ganized as ollows. We i s in oduce he
model in Sec. II. We hen desc ibe he single-qubi dynamics
when he emi e is on esonance wi h he pho onic band in
Sec. III, cha ac e izing he eme gence o di ec ional emission.
We conside he complemen a y si ua ion in Sec. IV, when
he QEs a e o esonance wi h he band (i.e., in a band
gap), and we cha ac e ize bo h he single-QE bound s a es
and he e ec i e qubi -qubi in e ac ions media ed by he
2643-1564/2020/2(2)/023003(10) 023003-1 Published by he Ame ican Physical Socie y
EDUARDO SÁNCHEZ-BURILLO e al. PHYSICAL REVIEW RESEARCH 2, 023003 (2020)
FIG. 1. (a) Two QEs (in blue), wi h ene gy gap , coupled
wi h s eng h g o a saw oo h la ice desc ibed by a bipa i e la ice
o coupled esona o s. The ed/g een la ice si es deno e he A/B
subla ices wi h couplings JAA,JAB,andJABeiφ. (b) Bands ωu/l(k)
(solid/dashed) o JAA =JAB, o six equally spaced φ om φ=0
(da k g een) o φand φ=π/2 ( ed).
ba h. A ealis ic implemen a ion o he sys em based on
supe conduc ing echnologies is p esen ed in Sec. V. We end
up wi h he conclusions in Sec. VI. We lea e some echnical
aspec s o he Appendices.
II. MODEL
The global Hamil onian o he sys em composed by he
QE’s and he pho onic ba h eads (¯h=1)
H=HST +
M

j=1
σ+
jσ−
j+Hin ,(1)
whe e is he equency o each QE desc ibed as a wo-le el
sys em wi h ladde ope a o s σ+/−
j o he j h QE, Mis he
numbe o QEs, Hin desc ibes he in e ac ion, and HST is he
Hamil onian o he saw oo h la ice. The la e eads
HST =ωB
N

n=1
(a†
nan+b†
nbn)−JAA
N

n=1
(a†
nan+1+H.c.)
−JAB
N

n=1
(a†
nbn+e−iφa†
n+1bn+H.c.),(2)
being anand bn he annihila ion ope a o s o he n h aand b
modes [ he pho onic ba h is bipa i e and is desc ibed by wo
subla ices Aand B; see Fig. 1(a)], N he numbe o modes
pe subla ice, ωB he ene gy o each esona o , ha we ake
as he ene gy e e ence: ωB≡0, JAA he coupling be ween he
Asi es, JAB he hopping s eng h be ween he aand bmodes,
and φ he phase di e ence in each loop.
Fo he in e ac ion e m Hin [las e m o Eq. (1)], we con-
side poin like and dipole- ield coupling unde he o a ing-
wa e app oxima ion, alid when he coupling s eng h is small
wi h espec o he o he ene gy scales o he sys em [36]:
Hin =g⎛
⎝
Ma

j=1
σ+
janj+
M

j=Ma+1
σ+
jbnj⎞
⎠+H.c. (3)
He e gis he coupling cons an , Mais he numbe o qubi s
coupled o A(so Mb≡M−Maa e coupled o B), and njis
he posi ion o he j h qubi .
Since we a e in e es ed in p edic ions in he he mody-
namic limi , N→∞, we ake pe iodic bounda y condi-
ions o he ba h and in oduce plane-wa e modes ˆak/ˆ
bk≡
1/√NN
n=1e−iknan/bn, in e ms o which HST [Eq. (2)] eads
HST =
k
(ˆa†
kˆ
b†
k)hST(k)ˆak
ˆ
bk,(4)
wi h hST(k)
hST(k)=−2JAA cosk (k,φ)
∗(k,φ)0
,(5)
and (k,φ)=−JAB(1 +e−i(k+φ)). We in oduce he ˆ·no a-
ion o dis inguish he ope a o s in eal/momen um space.
We diagonalize hST(k) such ha HST =k[ωu(k)ˆu†
kˆuk+
ωl(k)ˆ
l†
kˆ
lk], whe e ˆukand ˆ
lka e ela ed o ˆakand ˆ
bkby means
o a uni a y:
ˆuk
ˆ
lk=cos(θk)e−iϕk−sin(θk)
sin(θk)e−iϕkcos(θk)ˆak
ˆ
bk,(6)
We p o ide he explici exp essions o θkand ϕkin
Appendix A. The bands ωu/l(k) ead
ωu/l(k)=−JAA cosk±J2
AA cos2k+4J2
AB cos2[(k+φ)/2].
(7)
A celeb a ed ea u e o he saw oo h la ice is he appea -
ance o la bands, se ing JAB/JAA =√2 and φ=0[37]. He e
we a e, howe e , in e es ed in he implica ions o a non i ial
phase, φ= 0, leading o nonsymme ic bands [see Fig. 1(b)],
due o he explici b eaking o ime e e sal [HST = H∗
ST,
which implies hST(k)= hST(−k)]. This canno happen in a
pho onic sys em wi hou loops since one can ge id o he
phase φby means o local ans o ma ions o he bosonic
ope a o s. Besides, he e appea s ano he gap be ween bo h
bands unless φ=±π/2 [see again Fig. 1(b)] whe e he wo
bands ouch a a single poin k=±π/2. These singula band
gaps lead o exo ic phenomenology in highe dimensions
[38–40]. This is no he case he e, since he coupling s eng h
o he uppe /lowe band ωl/u(k), de ined by he unc ions
θk,ϕ
k, u n his poin in o a i ial band c ossing, as we explain
in he nex sec ion.
III. QE RESONANT WITH ωu/l(k)
We s udy he spon aneous decay o one qubi when
lies wi hin he bands. The s a e a ime eads
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CHIRAL QUANTUM OPTICS IN PHOTONIC SAWTOOTH … PHYSICAL REVIEW RESEARCH 2, 023003 (2020)
FIG. 2. Decay a e γD=−2ImD
e(solid lines) and ene gy shi
δωD()=ReD
e(+i0+) (dashed lines) as a unc ion o  o a
qubi coupled o subla ice A(blue) and B( ed), wi h JAB =JAA and
φ=π/3 (a) and φ=π/2 (b). The e ical black lines s and o he
band limi s.
|D( )=e−iH σ+| ac, whe e | acis he acuum s a e and
Ds ands o he subla ice he QE is coupled o. As he
numbe o exci a ions N≡N
n=1(a†
nan+b†
nbn)+σ+σ−is
a conse ed quan i y unde he o a ing-wa e app oxima ion
(3), he s a e |D( )can be spanned in he single-exci a ion
subspace:
|D( )=cD
e( )σ++
kcD
u(k, )ˆu†
k+cD
l(k, )ˆ
l†
k|0.
(8)
We can hen calcula e he dynamics nume ically o semi-
analy ically using he esol en me hod [36], whe e cD
e( )
is ob ained as cD
e( )=L−1[Ge(z)], being L he Laplace
ans o m and Ge(z) he QE G een Func ion. The la e eads
Ge(z)=1/[z+D
e(z)], being he sel -ene gy D
e(z)(see
Appendix B)
D
e(z)=
k
α=u,l
|0|αkHin σ+|0|2
z−ωα(k).(9)
Wi hin he Ma ko ian app oxima ion zcan be eplaced
by z=+i0+when doing he in e se Laplace, yielding
cD
e( )≃e−iD
e(+i0+) , whe e D
e(+i0+)=δωD−iγD/2.
The e o e, δωDand γDco espond o he eno maliza ion o
he exci ed s a e equency and line wid h, espec i ely. In
Fig. 2we plo hese quan i ies o φ=π/3 and φ=π/2
[Figs. 2(a) and 2(b), espec i ely]. Bo h δωDand γDdepends
on he subla ice he QE is coupled. Besides, as usual wi h
his kind o sys em, γDdi e ges in he band limi s; he e is
FIG. 3. Coupling G±,A(k) be ween a qubi placed a subla ice
Aand ω±(k) o φ=π/2andJAA =JAB. As seen, he couplings
G±,A(k) a e no discon inuous a k=π/2.
an excep ion o φ= π/2, whe e γAac ually ends o 0 when
ma ches he he uppe (lowe ) band edge o ωl(k)[ωu(k)]
o φ∈[0,π/2) [(π/2,π]]. Simila beha io was also
ound in wo-dimensional pho onic c ys als wi hou subla ice
symme y [41].
When φ=π/2, he sel -ene gy (bo h he eal and he
imagina y pa s) is a smoo h unc ion o ene gies in he
band. This p edic s ha he spon aneous emission by a QE
will unde go an exponen ial decay. Consequen ly, he singula
band gap he e does no imply exo ic dynamics, con a ily o
he case o o he pho onic ese oi s in highe dimensions
[38–40]. This is be e unde s ood s udying how he QE
couples o he bands o φ=π/2. We de ine wo new bands
ω±(k), oge he wi h he co esponding qubi -band couplings
G±,D(k) (see Appendix C)as
ω±(k)≡ωu(k)i k<π/2
ωl(k)i k>π/2,(10)
G±,D(k)≡Gu,D(k)i k<π/2
Gl,D(k)i k>π/2,(11)
being Gu/l,D(k) he coupling o he uppe /lowe band when
he QE is coupled o he subla ice D(see again Appendix C).
E en i he esul s do no depend on he subla ice he QE
couples o, we conside o he sake o simplici y ha he QE
is coupled o A. We plo bo h ω±(k) and G±,A(k)inFig.3
o φ=π/2. As seen, hese bands ω±(k) do no ha e any
kink: They a e smoo h unc ions and jus c oss a k=π/2.
The couplings a e also smoo h. In conclusion, he appa en
nonanaly ical beha io is ac ually an a i ac o he de ini ion
o he bands.
When calcula ing he exac QE dynamics (no shown),
we ind band-edge- ela ed phenomena such as ac ional and
powe -law decays in he long- ime limi , simila ly o hose
appea ing in o he pho onic ma e ials [38,41–52]. We ocus
ins ead on he emi ed pho ons, which displays e y dis inc-
i e ea u es om o he pho onic ba hs. We plo in Fig. 4(a) a
snapsho o he pho on popula ion in eal space o a si ua ion
whe e he emission is highly di ec ional, which co esponds
o a QE coupled o he Bsubla ice wi h pa ame e s JAB =
0.2JAA,φ=1.5, and =−0.5JAA. We emphasize ha , e en
hough he ba h b eaks he ±ksymme y o any φ= 0, he
023003-3
EDUARDO SÁNCHEZ-BURILLO e al. PHYSICAL REVIEW RESEARCH 2, 023003 (2020)
FIG. 4. (a) Emi ed pho on in posi ions as a unc ion o ime,
a†
nan( )andb†
nbn( ), when he qubi is coupled o B o JAB =
0.2JAA,φ=1.5, =−0.5JAA,g=0.1JAA, and ba h size N=200.
The qubi is a n=ne. (b) Di ec ionali y a io RB
L o JAB =0.2JAA as
a unc ion o and φ.
deg ee o di ec ionali y depends s ongly on he pa ame e s,
especially . Tha is, φ= 0 is a necessa y bu no su icien
condi ion o chi al emission.
Le us u he unde s and he o igin and possibili ies o he
di ec ional emission in his sys em by conside ing ha he QE
is esonan wi h ωl(k) and aking he limi when gis small
enough such ha we a e in he Ma ko egime. In his egime,
he dynamics is domina ed by he esonan k-modes de ined
by ωl(kR/L)=, whe e kR/Lco espond o igh - and le -
mo ing pho ons, espec i ely [see Fig. 1(b)]. Fu he mo e,
wi hin each di ec ion he exci a ions spli be ween pho ons
p opaga ing in he Ao Bsubla ices. Thus, he decay a e γD
in oduced as he imagina y pa o he sel -ene gy [Eq. (9)]
can be sepa a ed in o ou con ibu ions:
γD=D
a(kR)+D
a(kL)+D
b(kR)+D
b(kL),(12)
whe e D
α(k) deno es he decay a e in o he αsubla ice a
momen um k o a QE coupled o he Dsubla ice, eading
A(B)
a(b)(k)=|sin(θk)|4(|cos(θk)|4)
| l(k)|,(13)
A
b(k)=|sin(θk)|2|cos(θk)|2
| l(k)|=B
a(k) (14)
whe e l(k) is he g oup eloci y in he lowe band, l(k)=
∂kωl(k) and θkwas de ined in (6) (see also Appendix A). We
de ine a global di ec ionali y a io:
RD
L/R=αD
α(kR/L)
αD
α(kR)+D
α(kL)(15)
wi h α=a,b, ha ell us he a io o ligh emi ed in he
le / igh side in bo h subla ices, and a local one which
dis inguishes be ween subla ices RD
R/L,a/bwi h he same ex-
p essions bu wi hou summing in α.
In Fig. 4(b) we plo RB
Las a unc ion o and φ>0 o
a unneling JAB =0.2JAA. We obse e ha we can ind non e-
cip ocal emission, ha is, RD
α>1/2 o any φ= 0. Howe e ,
in o de o ind RB
L≈1 one has o ake he limi JAB/JAA 1,
φπ/2, and 0. In his egime, he slope o ωl(k) a ound
k=π/2 is d as ically di e en o he esonan momen a kR
and kL, yielding a densi y o s a es which is much la ge o kL
han o kR. Besides, θkis such ha he nume a o s o B
α(k)
a e much la ge o kL han o kR, which ac ually ein o ces
he p e ious a gumen . No ice ha he di ec ional emission
can be kep o mo e han 95% o a ia ions o φand 
a ound 10% o he op imal alue.
I he QE couples o he Asubla ice ins ead, he global
emission is no di ec ional: RA
L/R=1/2. Howe e , locally in
each subla ice, he emission can be made e y di ec ional,
RA
L,b,RA
R,a≈1. We illus a e his ac in Fig. 5, whe e we see
ha once again we ge chi ali y o φ→π/2 and →0.
This is possible because in ha case he coe icien s θkR/L
a e such ha he di ec ionali y is e e sed o emission in o
he amodes: A
a(kR)A
a(kL), compensa ing he e ec o
he densi y o s a es. We show a snapsho o wa e packe s
emi ed in o opposi e di ec ions in Fig. 6: As seen, he qubi
emi s mainly in o igh -/le -p opaga ing a/bmodes. To ou
knowledge, his is he i s ime his subla ice-dependen
chi ali y has been epo ed in he li e a u e.
IV. QES OUTSIDE OF THE BAND:
TUNABLE COMPLEX INTERACTIONS
We ocus now on he egime whe e /∈ωl/u(k), such ha
he physics is domina ed by he bound s a es (BSs) [52–66].
In he single-exci a ion subspace, he BS wa e unc ion o one
emi e coupled o he Dsubla ice eads
D
m=
ncD
m,a(n)a†
n+cD
m,b(n)b†
n|0+cD
m,eσ+|0,(16)
whe e m=−1,0,1 deno es he di e en BSs ha can appea
in he uppe /middle/lowe band gap, espec i ely. Thei wa e
unc ions and ene gies a e ob ained om H|D
m=ED
m|D
m
wi hED
m/∈ωl,u(k) (seeAppendixD).The e alwaysexis sa BS
|D
∓1below [abo e] ωl[u](k), because he sel -ene gy always
di e ges a hese band edges [62,64], such ha he in e ac ion
wi h he ba h is able o push one s a e ou o he band;
howe e , in he middle band gap, an ex a BS |D
0eme ges
when >0, |φ|<π/2o <0 and φ∈(−π,−π/2)∪
(π/2,π) o D=Bo A, espec i ely (see Appendix D). The
eason o his condi ion is he ini e alue o D
e(z) in one o
he band edges, as shown in Fig. 2(a), which de ines a c i ical
de uning o he exis ence o he BS.
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CHIRAL QUANTUM OPTICS IN PHOTONIC SAWTOOTH … PHYSICAL REVIEW RESEARCH 2, 023003 (2020)
FIG. 5. Di ec ionali y a io RA
L,a/bin panels (a) and (b), espec-
i ely, o JAB =0.2JAA. No ice ha he scale o bo h panels is
di e en : (0,0.5) in (a) and (0.5,1) in (b).
To illus a e he main ea u es o hese BSs, we plo
in Fig. 7 hei wa e unc ion coe icien s in momen a and
posi ions o he in e band BS |D
0. (1) Con a ily o wha
happens in emission, he absolu e alue o he wa e unc ion
|cD
m,α (n)|is always symme ically dis ibu ed a ound he QE
no ma e he band gap o pa ame e s conside ed; (2) as
i occu s wi h o he pho onic la ices [52–66], he BS a e
exponen ially localized a ound he emi e wi h a localiza ion
FIG. 6. Emi ed wa e packe when he qubi is coupled o A o
JAB =0.2JAA,φ=1.5, =−0.1JAA,g=0.1JAA,N=1000, and
JAA =200.
FIG. 7. In e band BS |B
0 o a qubi coupled o B o JAA =
JAB =1, φ=2.094, =−0.01, and g=0.1. In panel (a) we plo
he squa e modulus o he wa e unc ions in momen a in bo h
subla ices, |cB
0,a/b(k)|2, whe eas we show he absolu e alue ( ed
dashed line) and he eal (black do s) and imagina y (g een do s)
pa s o he coe icien s in posi ions o he bmodes in panel (b). The
wa e unc ions in kspace a e cen e ed a k=π/3, so he pe iodici y
in posi ions is 6. No ice ha he qubi is placed a n=ne.
leng h which can be uned: he close ED
mlies o one o he
band edges, he less localized he BS is; and (3) he no el y is
ha he BSs acqui e a unable complex phase cD
m,α (n)∝eiϕD
mn.
Fo small g,ϕD
mma ches he posi ion o he band edge closes
o , which u ns ou o depend on JAB/JAA and φ, as seen
in Fig. 1(b), e.g., he minimum o he uppe band uns om
0 oπ/2, so he phase o he in e band BS can be uned in
his ange p o ided  ends o his band edge. In he example
o Fig. 7, he momen um o he closes band edge occu s a
kedge ≃π/3, so |cB
0,α (k)|2is dis ibu ed a ound π/3 [Fig. 7(a)]
and cB
0,α (n) has pe iodici y 2π/(π/3) =6 [Fig. 7(b)].
When many emi e s couple o he ba h, he BS media e
in e ac ions be ween QEs, which can be ha nessed o simu-
la e spin models wi h unable in e ac ions, allowing o he
s udy o many-body physics, phase ansi ions, e c. (e.g., see
Re s. [38,38,40,62,64,66–68]).
In he Ma ko ian app oxima ion, ha is, when he coupling
s eng h be ween he qubi s and he ba h is small compa ed o
he gap be ween and he bands ( ecall ha we a e consid-
e ing he egime whe e is no embedded in he pho onic
modes) and also compa ed o he wid h o bo h bands, an
e ec i e Hamil onian o he qubi s can be de i ed [67,68]:
Hqb =
i<jJDiDj
ij σ+
iσ−
j+H.c.,(17)
023003-5

EDUARDO SÁNCHEZ-BURILLO e al. PHYSICAL REVIEW RESEARCH 2, 023003 (2020)
anan+1
bn
anan+1
bn
FIG. 8. On he le -hand side, we show a plaque e o he saw-
oo h la ice coupled o h ee qubi s. On he igh -hand side, he p o-
posal o he implemen a ion. The qubi s a e he blue pa s. In black,
we show he qubi - esona o couplings. Bo h he a/b esona o s and
he esona o - esona o couplings a e in ed and g een.
being Di he subla ice he i h qubi is coupled o. He e JDiDj
ij
is p opo ional o he wo-qubi sel -ene gy (see Appendix E)
and i inhe i s he shape o he BS wa e unc ion wi h ene gy
, i.e., JAA/BB
ij ∝cA/B
a/b( ij) and JAB
ij ∝cA
b( ij). This has se e al
implica ions. Fi s , qubi -qubi e ec i e in e ac ions decay ex-
ponen ially wi h he dis ance be ween qubi s, being he ange
o he in e ac ions unable; in pa icula , i is a dec easing
unc ion o he gap be ween and he closes band and i
di e ges when his gap closes. Besides, we can also une i s
phase, which, as in he saw oo h ba h, is physically ele an
(see Appendix D), because he geome y o he e ec i e spin
la ice has closed loops. This allows us o une bo h he pe i-
odici y o he in e ac ions [see, o ins ance, Fig. 7(b), whe e
pe iodici y is 6], which may gi e ise o quali a i ely di e en
spin phases. E en iche many-body dynamics will appea in
he nonpe u ba i e egime eplacing spins by pola i ons [65].
V. IMPLEMENTATION
As he physics o Eq. (2) does no ely on pola iza ion,
i can be ob ained in pla o ms beyond op ical ones [7,8,10–
12] such as cold a oms in s a e-dependen la ices [69–71],
whe e such complex loops ha e al eady been enginee ed
[72], o ci cui QED pla o ms [73–79]. In Fig. 8we show
a p oposal o design his geome y wi h complex couplings
using supe conduc ing qubi s as implemen ed o a single loop
in Re . [80]. Fo he qubi - esona o in e ac ion, we assume
ha he qubi is capaci i ely coupled o an LC esona o ,
whe eas he esona o s a e induc i ely coupled ia a SQUID-
ype loop based on Josephson junc ions [81,82]. The junc ions
a e con olled ia an ex e nal lux φex ( ) leading o he ime-
dependen coupling Hamil onian be ween wo esona o s:
Haux =
i=1,2
ωia†
iai+J( )(a†
1+a1)(a†
2+a2).(18)
Se ing ω1=ω,ω2=ω+δ, and J( )=Jcos(δ +φ),
and assuming J,δ ω, we can a e age ou he ime depen-
dence o J( ), ge ing he Hamil onian we wan o simula e;
c . Eq. (2)[83]. Fo he implemen ed loop o Re . [80], hey
measu e couplings o he o de o 4.1 MHz (i.e.. J/ω ∼10−3).
Thus, conca ena ing se e al o hese cells and placing selec-
i ely he qubi s, as done in Re s. [84,85] o simple coupled-
ca i y a ays, one can explo e he phenomena p edic ed in his
pape .
Thanks o he unabili y o cQED implemen a ions,
one can explo e di e en egimes o pa ame e s (e.g., see
Re s. [78,79]); in pa icula , i is possible o s udy he Ma ko-
ian egime, on which he esul s p esen ed he e a e based.
I would be in e es ing also o s udy o he egimes, e.g., he
ul as ong coupling, which has been ound in wa eguide
QED [86], and whose consequences in chi al ligh -ma e
in e ac ions ha e been ecen ly explo ed [87].
VI. CONCLUSIONS
We ha e s udied he p ope ies o QEs coupled o a min-
imal one-dimensional model b eaking ime- e e sal symme-
y: he pho onic saw oo h la ice. When he emi e s a e
esonan wi h he band hey decay in an asymme ic ash-
ion in o le -/ igh -mo ing modes. We iden i ied egimes o
quasipe ec di ec ionali y, o mo e exo ic ones in which he
emi e decays in bo h di ec ions bu o a di e en subla ice.
Thus, hese sys ems can be an al e na i e way o explo ing
chi al quan um op ics wi hou pola iza ion o mo ing emi -
e s. Besides, when he emi e equency lies in a band gap we
ha e ound he eme gence o BSs whose no only hei spa ial
ange, bu also hei complex phase can be uned h ough
he sys em pa ame e s. Since hese BSs media e in e ac ions
be ween emi e s when many o hem couple o he ba h, ou
se up p o ides access o he simula ion o a la ge class o spin
models wi h complex in e ac ions. Fu he mo e, we discussed
an implemen a ion o obse e such phenomenology based on
supe conduc ing echnologies.
ACKNOWLEDGMENTS
E.S.B. acknowledges ERC Ad anced G an QUENO-
COBA unde he EU Ho izon 2020 p og am (G an
Ag eemen No. 742102). A.G.T. and D.Z. acknowledge sup-
po om CSIC Resea ch Pla o m PTI-001. A.G.T. acknowl-
edges unding om he na ional p ojec PGC2018-094792-B-
I00 om Minis e io de Ciencia e Inno ación.
APPENDIX A: DIAGONALIZATION
OF THE SAWTOOTH LATTICE
In his Appendix, we gi e some de ails on he diagonal-
iza ion o he saw oo h Hamil onian [Eq. (2)]. The bosonic
ope a o s which diagonalize he model, ˆuk/ˆ
lk[see Eq. (6)] a e
ela ed o ˆakand ˆ
bkby means o a uni a y ans o ma ion Pk.
The la e eads
Pk=cos(θk)eiϕksin(θk)eiϕk
−sin(θk) cos(θk)=Nu(k) ∗(k,φ)Nl(k) ∗(k,φ)
Nu(k)(ωu(k)+2JAA cosk)Nl(k)(ωl(k)+2JAA cosk),(A1)
023003-6
CHIRAL QUANTUM OPTICS IN PHOTONIC SAWTOOTH … PHYSICAL REVIEW RESEARCH 2, 023003 (2020)
whe e Nu/l(k) is a no maliza ion ac o
Nu/l(k)=1
| (k,φ)|2+[ωu/l(k)+2JAA cosk]2.(A2)
APPENDIX B: SINGLE-QUBIT SELF-ENERGY
To compu e cD
e( ), we use he esol en ope a o me hod
[36], which ells us ha he p obabili y ampli ude can be
compu ed as
cD
e( )=− 1
2πi∞
−∞
dEGD
e(E+i0+)e−iE ,(B1)
whe e GD
e(z) is he single-qubi G een unc ion when i is
coupled o he subla ice D:
GD
e(z)=1
z−−D
e(z),(B2)
whe e D
e(z) he so-called sel -ene gy. In his sec ion, we
de i e he exp essions o he single-qubi sel -ene gy when
he qubi is locally coupled o Ao B.TheD
e(z) o ou
wo-band model eads
D
e(z)=
k
α=u,l
|0|αkHin σ+|0|2
z−ωα(k).(B3)
Conside ing Hin [Eq. (3)] o a single qubi ) coupled o A
o Band aking in o accoun he ela ion be ween (uk,lk) and
(ak,bk)[seeEqs.(6) and (A1)] and he exp essions o ωu/l(k)
[see Eq. (7)]:
A
e(z)=g2
2ππ
−π
dk z
z2+2zJAA cosk−| (k,φ)|2,(B4)
B
e(z)=g2
2ππ
−π
dk z+2JAA cosk
z2+2zJAA cosk−| (k,φ)|2.(B5)
We ake he e he he modynamic limi : N→∞. One can
sol e hese in eg als by means o he change o a iable
y≡eik. The in eg a ion domain is now he uni ci cle in he
complex plane:
A
e(z)=g2
2πidy zJAA
zJAA −J2
ABeiφ(y−y+)(y−y−),(B6)
B
e(z)=g2
2πidy JAAy2+zy +JAA
zJAA −J2
ABeiφy(y−y+)(y−y−),(B7)
whe e y±a e
y±=2J2
AB −z2±2J2
AB −z22−4z2J2
AA +J4
AB −2zJAAJ2
AB cosφ
2zJAA −J2
ABeiφ.(B8)
We de ine ymin/max as he minimum/maximum o {y−,y+}
wi h espec o he absolu e alues |y±|. Applying he
Cauchy’s esidue heo em and aking in o accoun ha (|y+|−
1)(|y−|−1) <1 o all z∈Cwi h Im(z)= 0:
A
e(z)=g2zsign(|y−|−|y+|)
zJAA −J2
ABeiφ(y+−y−),(B9)
B
e(z)=g2JAA
zJAA −J2
ABeiφ1
y+y−+y2
min +(z/JAA)ymin +1
ymin(ymin −ymax).
(B10)
Fo ins ance, i we conside ha is embedded in he lowe
band, i is s aigh o wa d o de i e Eqs. (13) and (14) om
Eqs. (B9) and (B10).
APPENDIX C: QUBIT-BAND COUPLINGS
He e we w i e he qubi -band coupling o bo h bands.
Le us conside he in e ac ion Hamil onian Hin [Eq. (3)]
o a single qubi . Fo he sake o simplici y, he qubi will
be coupled o A; i is because o his ha we will deno e he
in e ac ion Hamil onian as HA
in in his Appendix. We w i e his
Hamil onian in e ms o ukand lk[see Eq. (6)]:
HA
in =g
√Nσ+
k
eikx0[cos(θk)eiϕkˆuk+sin(θk)eiφkˆ
lk]+H.c.,
(C1)
whe e x0is he posi ion o he qubi and cos/sin(θk)a e
he ma ix elemen s o he uni a y ans o ma ion Pk[see
Eq. (A1)]. The la e de e mines he coupling s eng h o each
band: Gu,A(k)=|cos(θk)|2and Gl,A(k)=|sin(θk)|2,up o
he densi y o s a es, which is gi en by 1/|∂ωu/l(k)|.
APPENDIX D: BOUND STATES
In his sec ion, we discuss he exis ence condi ions o he
bound s a es (BSs) and we compu e hei wa e unc ions in
eal space.
As men ioned in he main ex , we ha e o impose he
eigen alue equa ion H|D
m=ED
m|D
mwi h he ene gy ED
m
ou side o he bands. This can be mapped in o inding he oo s
o he ollowing unc ion:
FD(E)≡E−−D
e(E),(D1)
wi h E/∈ωl,u(k)[64]. I can be easily p o ed om Eq. (9)
ha D
e(E) is a dec easing unc ion, so FD(E) is an inc easing
unc ion. Besides, limE→±∞ FD(E)=±∞. Then, acco ding
o he beha io o D
e(E) in he band edges, we can igu e ou
whe he he e exis s o no a bound s a e in each o he band
gaps:
(1) A BS exis s wi h Ebs <ω
l(k)[Ebs >ω
l(k)] o all he
alues o he pa ame e s i and only i FD(E)>(<)0 when
E ends o he minimum o ωl(k) [maximum o ωu(k)]. We
plo D
e(E) o Eou side o he band in Fig. 2and show ha
i di e ges in he he lowes /highes ene gy band edge, which
gua an ees he a o emen ioned condi ions, so he exis ence o
woBSbelowωl(k) and o e ωu(k), which we label as |D
−1
and |D
+1, espec i ely.
023003-7
EDUARDO SÁNCHEZ-BURILLO e al. PHYSICAL REVIEW RESEARCH 2, 023003 (2020)
(2) The si ua ion is di e en in he middle band gap. In
Fig. 2(a) we obse e when he QE is coupled o he B la ice,
he sel -ene gy di e ges in bo h he uppe /lowe middle band
edges. Thus, a middle BS, |B
0, always exis s. On he o he
hand, i he emi e is coupled o A, he s a e |A
0exis s
i >0 when |φ|<π/2 because he sel -ene gy A
e(E)
anishes when E ends o he maximum o ωl(k). When
φ∈(−π,−π/2) o φ∈(π/2,π), he exis ence condi ion is
<0 (no shown).
Conce ning he wa e unc ions, i D=A, he coe icien s
ead
cA
m,a(n)=gcA
e
2ππ
−π
dkeikn|(Pk)11|2
ED
m−ωu(k)+|(Pk)12|2
EA
m−ωl(k),
(D2)
cA
m,b(n)=gcA
e
2ππ
−π
dkeikn(Pk)21(Pk)∗
11
EA
m−ωu(k)+(Pk)22(Pk)∗
12
EA
m−ωl(k),
(D3)
whe e EA
mis he ene gy o |A
m,bsand cA
eis ob ained imposing
he no maliza ion condi ion. Doing he ma h,
cA
m,a(n)=gce
2ππ
−π
dk eikn EA
m
EA
m2+JAA2EA
mcosk−| (k,φ)|2,
(D4)
cA
m,b(n)=−gce
2ππ
−π
dk eikn ∗(k,φ)
EA
m2+2JAAEA
mcosk−| (k,φ)|2.
(D5)
I he qubi is ins ead coupled o B,
cB
m,a(n)=gcB
e
2ππ
−π
dkeikn(Pk)11(Pk)∗
21
ED
m−ωu(k)+(Pk)12(Pk)∗
22
EB
m−ωl(k),
(D6)
cB
m,b(n)=gcB
e
2ππ
−π
dkeikn|(Pk)21)|2
EB
m−ωu(k)+|(Pk)22)|2
EB
m−ωl(k),
(D7)
which becomes
cB
m,a(n)=−gcB
e
2ππ
−π
dk eikn (k,φ)
EB
m2+2JAAEB
mcosk−| (k,φ)|2,
(D8)
cB
m,b(n)=gcB
e
2ππ
−π
dk eikn(EB
m+2cosk)
EB
m2+2JAAEB
mcosk−| (k,φ)|2.
(D9)
No ice ha all hese exp essions look simila o D
e(z)[see
Eqs. (B4) and (B5)], so we can calcula e he coe icien s in
e ms o complex in eg als [Eqs. (B6) and (B7)]. The change
o a iable is s ill y=eik i n⩾1, bu y=e−ik i n⩽−1.
In he i s case, he poles o he in eg al a e y±, while in he
second a e hei complex conjuga es y∗
±.
APPENDIX E: TWO-QUBIT SELF-ENERGY
We de i e he e he exp essions o he collec i e sel -
ene gy D12
c, which is p opo ional o he coe icien s o he
e ec i e Hamil onian (17). The compu a ion is o ally analo-
gous o he single-qubi sel -ene gy (see Sec. B). They ead
AA
c(z; 12)=g2
2ππ
−π
dk eik 12 z
z2+2zJAA cosk−| (k,φ)|2,(E1)
BB
c(z; 12)=g2
2ππ
−π
dk eik 12 (z+2JAA cosk)
z2+2zJAA cosk−| (k,φ)|2,(E2)
AB
c(z; 12)=−g2
2ππ
−π
dk eik 12 ∗(k,φ)
z2+2zJAA cosk−| (k,φ)|2,(E3)
whe e 12 =x2−x1is he ela i e posi ion o he qubi s.
No ice ha AA
c(z; 12), BB
c(z; 12), and AB
c(z; 12)a e
p opo ional o he bound-s a e coe icien s cA
m,a( 12),
cB
m,b( 12), and cA
m,b( 12) espec i ely, by changing he
bound-s a e ene gies ED
mby z[see Eqs. (D4), (D9), and
(D5)]. I is he e whe e i becomes e iden ha he e ec i e
in e ac ions a e media ed by he bound s a es.
Finally, we can compu e he accumula ed phase o a
closed loop in he e ec i e spin la ice. Fo example, aking
he pa ame e s o Fig. 7(JAA =JAB =1, φ=2.094, =
−0.01, and g=0.1) and choosing he closed pa h a→a→
b→a, his phase is a g[AA
c(;1)]+a g[AB
c(;1)]+
a g[AB
c(;−1)] ≃−1.22. As i is nonze o, he e ec-
i e models can simula e sys ems wi hou ime and pa i y
in a iance.
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