Topological phase transitions in superradiance lattices

2.8 MB·English

by  Da-Wei Wang

Topological phase transitions in superradiance lattices
Da-Wei Wang,1,2,∗ Han Cai,2 Luqi Yuan,2,3 Shi-Yao Zhu,4,5 and Ren-Bao Liu1,†
1Department of Physics and Centre for Quantum Coherence,
The Chinese University of Hong Kong, Hong Kong, China
2Institute for Quantum Science and Engineering and Department of Physics and Astronomy,
Texas A&M University, College Station, TX 77843, USA
3Department of Electrical Engineering, and Ginzton Laboratory, Stanford University, Stanford, CA 94305, USA
4Beijing Computational Science Research Centre, Beijing 100084, China
5Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
(Dated: August 6, 2015)
5 Topologicalphasesofmattersareoffundamentalinterestandhavepromisingapplications. Fasci-
1 natingtopologicalpropertiesoflighthavebeenunveiledinclassicalopticalmaterials. However,the
0 manifestation of topological physics in quantum optics has not been discovered. Here we study the
2 topological phases in a two-dimensional momentum-space superradiance lattice composed of timed
g Dickestates(TDS)inelectromagneticallyinducedtransparency(EIT).Byperiodicallymodulating
u thethreeEITcouplingfields,wecancreateaHaldanemodelwithin-situtunabletopologicalprop-
A erties,whichmanifestthemselvesinthecontrastbetweendiffractionsignalsemittedbysuperradiant
TDS. The topological superradiance lattices provide a controllable platform for simulating exotic

5 phenomena in condensed matter physics and offer a basis of topological quantum optics and novel
photonic devices.

]
s
c
i ThequantumHalleffect(QHE)[1]revealsanewclassof (a) k k (c) sublatticeoffset
t # #
p matter phases, topological insulators (TIs), which have 2 c c
o been extensively studied in solid-state materials [2–7] 2 2
s. and recently in photonic structures [8–12], time-periodic kp k1 ek "
c systems [13–16] and optical lattices of cold atoms [17]. 1 m
si The first TI is the Haldane model [2] proposed in 1988, k k " kp k2
y ! 3 3
which shows that the QHE is an intrinsic topological
h m
p property of the energy bands. It inspired the discov- (b) e e kp k1
[ ery of the quantum spin Hall effect [3–5] and topo- # # " kp
3 logical superconductors [7]. The Haldane model con- k c p 2 m
v sists of a honeycomb tight-binding lattice with complex 1,2,3 k e kp k3
9 next-nearest-neighbour (NNN) hopping, which breaks p,$ k!
m
9 the time-reversal symmetry and induces an energy gap g
0 between two bands that have opposite Chern numbers.
4 Inversion symmetry breaking by an on-site potential can
0 also open band gaps. The interplay of these two types of Figure 1: Realisation of the Haldane model in a superradi-
1. symmetry breaking leads to transitions between phases ance lattice of timed Dicke States in electromagnetically in-
0 ducedtransparency. (a)Schematicconfigurationoftheprobe
with Chern numbers 0 and ±1.
5 fieldkp,EITcouplingfieldsalongk1/2/3anddiffractionfields
1 Notwithstanding its foundational role in topological alongk+ ≡kp+k1−k3 andk− ≡kp+k1−k2. (b)Theen-
: condensed matter physics, the Haldane model has never ergy level diagram of the EIT coupling, probe and scattering
v
i been realized in solid-state systems. Floquet modulation fields. (c)HoneycombstructureoftheSL.|ek(cid:105)and|mk(cid:105)cor-
X of a circular polarised light in graphene was proposed to respond to the two sublattices. The NNN hopping Ω31 (red
solid arrow), Ω (blue dot arrow) and Ω (green dash dot
r induce the complex NNN tunnelling [13–15]. However, 23 12
a arrow) enclose a nonzero effective magnetic flux in momen-
the required light is soft X-ray which would directly ex-
tum space. The arrow on the circle denotes the direction of
cite electrons and be absorbed. The Haldane model of
excitation current when the Chern number C =1.
cold atoms in optical lattices [17] were recently realised
in experiments.
been an important topic in quantum optics. A single
Our recent study shows that Scully’s timed Dicke
photon with wave vector k can excite a spatially ex-
states (TDS) [18] can form a superradiance lattice (SL) p
tended N-atom ensemble from the ground state |G(cid:105) =
in momentum space [19]. Here we propose a quantum
|g ,g ,...,g (cid:105) to the TDS [18]
opticsrealisationofanin-situtunableHaldanemodelus- 1 2 N
ing two-dimensional SLs in a simple electromagnetically
N
1 (cid:88)
induced transparency (EIT) configuration [20]. Since |e (cid:105)= √ eikp·rj|g ,g ,...,e ,...,g (cid:105), (1)
Dicke’s seminal paper in 1954 [21], superradiance has kp N 1 2 j N
j=1
2
We consider periodically modulated Rabi frequencies Ω
3 3 2 l
2 (l=1,2,3) of coupling fields,
C!"1
" 1 Ωl =Ωs+2Ωdcos(νdt+φl), (3)
0 3 where Ωs and Ωd are the static and dynamic compo-
3
nents of the Rabi frequency, ν is the modulation fre-
d

# 3 quency and φl is the modulation phase. Here we have
assumedthesizeoftheatomicensemblebemuchsmaller
C!#1 1 than c/νd (where c is the speed of light) and hence ne-
3 3 glected the position dependence of the modulation. On
"
2
the other hand, the ensemble size is much larger than

2 c/ν and the number of atoms N (cid:29) 1 such that the
(a) 2 (b) c
TDS in the SL are approximately orthogonal to each
other, i.e., (cid:104)e |e (cid:105) ≈ δ . The periodic modulation in-
k(cid:48) k kk(cid:48)
Figure 2: Topological phases of the superradiance lattice. duces Floquet sidebands [2] with energy separation (cid:126)ν .
d
(a) The value of α (defined in Eq.(5)) as a function of the While Ω is the intra-sideband nearest-neighbour hop-
s
modulation phases of the coupling fields φ2 and φ3. φ1 = 0. ping, Ω induces the inter-sideband hopping. We choose
d
The signs “±” are for the Chern numbers C = ±1. The
ν (cid:29)Ω ,∆ sothattheFloquetsidebandsarewellsep-
dashedlineseparatesthetwotopologicalphases. (b)Thetwo d s/d c
aratedinenergy. Thusbysecond-orderperturbation,the
Chernnumberscorrespondingtotwotopologicallydistinctive
effective Hamiltonian of the NNN intra-sideband transi-
configurations of the three phase angles φ , φ and φ on a
1 2 3
unit circle. tion mediated by Ωd is (see Supplementary Information)
3
(cid:88) (cid:88)
where ei and gi are the excited and ground states of the H(cid:48) = (cid:126)Ωlj(|ek+kl−kj(cid:105)(cid:104)ek| (4)
ith atom at position r , respectively. The TDS stores k l(cid:54)=j=1
i
a light momentum (cid:126)k via phase correlations between +|m (cid:105)(cid:104)m |),
p k−k1 k−k1+kl−kj
atoms excited at different positions. This momentum
wheretheNNNhoppingcoefficientΩ =2iΩ(cid:48)sin(φ −φ )
canbetransferredbacktoasinglephotonviadirectional lj l j
with Ω(cid:48) = Ω2/ν . The fact that the NNN hopping co-
emission [18]. By coupling |e(cid:105) to another state |m(cid:105) with d d
efficient is purely imaginary is crucial to the topological
three coherent plane wave fields, we construct a honey-
phases. The loop transitions via NNN hopping accumu-
comb SL of TDS in momentum space [19], as shown in
late nonzero phases, as shown in Fig.1 (c). H(cid:48) opens a
Fig.1. The SL Hamiltonian with rotating-wave approxi- √
band gap (cid:15) = 4 3(cid:126)|α|Ω(cid:48) where α is a dimensionless
mation is (see Supplementary Information) tr
quantity determined by the summation of Ω ,
(l+1)l
H =(cid:126)∆2c (cid:88)k (|ek(cid:105)(cid:104)ek|−|mk−k1(cid:105)(cid:104)mk−k1|) (2) α=−(cid:88)3 sin(φl+1−φl)=4(cid:89)3 sinφl+12−φl. (5)
(cid:88)(cid:88)3 l=1 l=1
− (cid:126)Ω [|e (cid:105)(cid:104)m |+h.c.],
l k k−kl TheChernnumbersoftheupperandlowerbandsC and
k l=1
C(cid:48) are C = −C(cid:48) = sign(α) (see Supplementary Infor-
where k = k +r(k −k )+s(k −k ) with integers mation). In Fig. S.2 (a), we plot α with φ = 0 and
p 2 1 3 2 1
r and s, Ω is the Rabi frequency of the coupling field 0 ≤ φ < 2π. The topological property of this SL Hal-
l 2,3
along k , |m (cid:105) are defined the same as in Eq.(1) but for dane model can be represented by the distribution of φ
l k l
|m(cid:105) states, and ∆ = ω −ν is the detuning of the on a unit circle. There are two distinct topological con-
c em c
transition frequency ω between |e(cid:105) and |m(cid:105) from the figurations, counter-clockwise φ , φ and φ for C =−1,
em 1 2 3
angular frequency ν of the EIT coupling fields. The and clockwise φ , φ and φ for C = 1, as shown in
c 1 2 3
states |e (cid:105) and |m (cid:105) form two sublattices of a honey- Fig.S.2 (b). The time reversal t → −t in Eq.(S.4) is
k k−k1
comb structure in momentum space [see Fig. 1(c)]. Ω equivalent to φ →−φ , which leads to C →−C.
l l l
causes the nearest-neighbour hopping. As we will show Unlike TIs [5, 6], topological superradiance lattices
below, periodically modulating Ω can introduce NNN (TSLs) have no outer edges in the semiclassical limit
l
hopping with controllable phases. On the other hand, of the coupling fields (see Supplementary Information).
thedetuning∆ causesanenergyoffsetbetweenthetwo Neither do TSLs have Fermi surfaces. Nonetheless, the
c
sublattices,breakingtheinversionsymmetry. Thusthree TSLhasitsuniquetopologicalpropertiesthatareobserv-
HaldanephaseswithChernnumbers±1and0canbein- able. The TDS have directional superradiance emission.
situ realised. Of all the TDS in the SL, only those |e (cid:105) with c|k|−ω
k eg
We first show how to induce complex NNN hopping. within the energy bands (ω is the transition frequency
eg
3
1 1.1000 00.1000.1 00.1000.1
(a(a11)) (a2) (a3)
(a) 0.08000 0.08000
(a) 0.06000 0.06000
0.04000 0.04000
0.02000 0.02000
0 0 0
0 0 0
!3 0 (b1) 1.1000 (b2) 00.1000.1 (b3) 00.1000.1
0.08000 0.08000
(b) 0.06000 0.06000
0.04000 0.04000
0.02000 0.02000
0 0 0
1 0 0 0
Figure 4: Real-space propagation of a light pulse through
!
2 an atom ensemble driven into a trivial or topological super-
radiancelattices. DiffractionofaweakGaussianprobepulse
in x-y plane for (a) a trivial SL with Ω = 0 and (b) a
1.0 d
TSL with C = 1. φ = 0, φ = 4π/3, φ = 2π/3 and
(b)C#0 C#1 C#0 Ω(cid:48) = 0.01. The deriv1ed linear2and nonline3ar susceptibili-
0.8 ties are χ(1) = i0.1410, χ = χ = −i0.0522 in (a) and
+ −
χ(1) = i0.1446, χ = −i0.1057, χ = −i0.0021 in (b). The
+ −
0.6 parameters are such that Ωs = 3, ∆p = ∆c = 0, γe = 1
and γ = 0.1. The square denotes the area occupied by
m
"
atoms. The three figures in each group are for scaled time
0.4
t = 1,80,100 successively. We assume the group velocity be
uniform everywhere.
0.2
is analogue to tuning the Fermi surface in a fermionic
0.0
system.
-3 -2 -1 !0/ 1 2 3 For the sake of simplicity, we set the wavevectors of
c tr
the three EIT coupling fields to be k = −k xˆ, k =
√ √ 1 c 2
(c) k (xˆ− 3yˆ)/2 and k = k (xˆ+ 3yˆ)/2, and the probe
c 3 c
field wavevector k = −k . In this case we have only
p 1
K K# K K# K K# three superradiant TDS with wavevectors kp, k+ =−k3
and k = −k , and the diffraction fields are along k ,
!0 !" !2" − 2 ±
c c tr c tr asshowninFig.1(a). Wesetallfieldsonresonance, i.e.,
∆ = 0 and ∆ ≡ ω −ν = 0. The excitation |e (cid:105)
c p eg p kp
Figure3: Topologicalphasetransitionssignaturedbysuper- flows to |e (cid:105) and emits photons along k . We denote
k± ±
radiancecontrast. (a)Thecontrastη(definedinequation(6)) thesteadystateprobabilityamplitudesofstates|e (cid:105)as
asafunctionofthemodulationphasesφ andφ withφ =0. k±
2 3 1 c and define the superradiance contrast
The red dot lines are zero points. ∆ = ∆ = 0. Ω(cid:48) = 0.01. k±
p c
(b) The topological phase transition between C = 1 and 0.
|c |2−|c |2
∆p = ∆c/2. Ω(cid:48) = 0.01 (red), 0.1 (green), 0.5 (blue) and 1.0 η = k+ k− . (6)
(black). (c) The SL band structures in the topological phase |ck+|2+|ck−|2
transition. Ω(cid:48) =0.1. K andK(cid:48) arethetwovalleypoints. For
(b) and (c), φ =0, φ =4π/3 and φ =2π/3. For all three For C = 1, the excitation current flows along |e (cid:105) →
1 2 3 kp
figures, Ωs =3, γe =1 and γm =0.1. |ek+(cid:105) → |ek−(cid:105) [23], as shown in Fig.1 (c). Since each
TDS |e (cid:105) or |m (cid:105) has decoherence rate γ or γ , respec-
k k e m
tively, the excitation decays while flowing and it is more
between |e(cid:105) and |g(cid:105)) can satisfy both energy and mo- probable in state |e (cid:105) than in state |e (cid:105). We therefore
k+ k−
mentumconservation,andhavedirectionalemissionink have η >0. Similarly, for C =−1, η <0. Thus the sign
[18]. WecallthesestatessuperradiantTDSandtheother changeofthesuperradiancecontrastsignaturesthetopo-
ones subradiant TDS. We can regard these superradiant logical phase transition, as seen in Fig.3 (a). The super-
TDS as an inner edge of a honeycomb lattice of subra- radiance contrast in Fig.3 (a) is consistent with Fig.S.2
diant TDS [23]. The topological orders lead to different (b) except for the two diagonal corners where the topo-
lightemissionsfromdifferentsuperradiantTDS.Alterna- logicalcurrentsareweakandthelocaleffectinsideaunit
tively, we can also tune the probe field frequency to test cell dominates (see Supplementary Information).
the topological band properties at certain energy, which The topological phase with C =0 for non-zero Ω(cid:48) can
4
be reached by breaking the inversion symmetry. Sub- [19] and offers a platform for high-dimensional topologi-
stantially easier than in graphene electrons, a sublattice cal physics [27].
offset in SLs can be introduced by choosing ∆ (cid:54)=0. We
c
set φ = 4π(l−1)/3 and thus C = 1 when ∆ = 0. The
l c
energy gap in absence of sublattice offset is (cid:15) =18(cid:126)Ω(cid:48).
tr
Topologicalphasetransitionsoccurat(cid:126)|∆ |=(cid:15) [2](see Funding Information
c tr
Supplementary Information). In Fig.3 (b), we plot η as
a function of (cid:126)∆c/(cid:15)tr. For Ω(cid:48) = 0.01, the phase transi- Science Foundation Grants No. PHY-1241032 (IN-
tion is smeared out by the relatively large γe. The phase SPIRE CREATIV) and PHY-1068554; Robert A. Welch
transition gets more apparent for larger Ω(cid:48). Depending Foundation (Grant No. A-1261); Herman F. Heep and
on Ω(cid:48), η has peaks or kinks near the phase transition Minnie Belle Heep Texas A&M University Endowed
point because the two bands touch at the middle of the Fund; Hong Kong RGC/GRF; CUHK VC’s One-Off
bandgap,wheretheprobefieldprobes,asshowninFig.3 Discretionary Fund; Natural Science Foundation of
(c). China U1330203.
TheTSLshaveuniquefeaturesintransientlightprop-
agation under pulse probe. In Fig.4, we compare the
The authors thank Marlan O. Scully for helpful dis-
pulse propagation in a trivial SL with zero Ω(cid:48) and in a
cussions.
TSL. For a weak probe pulse, the linear susceptibility is
χ(1) ∝ c and the linear absorption is Imχ(1). The two
kp
multi-wave-mixing signals along k correspond to the
±
nonlinear susceptibilities χ ∝ c and can be under-
± k±
stood as a result of optical grating [19]. We simulate the ∗ Electronic address: [email protected]
pulse propagation for the three modes along k using † Electronic address: [email protected]
p,±
coupled wave equations [24] (see Supplementary Infor- [1] K.vonKlitzing,G.DordaandM.Pepper,PhysicalReview
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Supplementary I
EFFECTIVE HAMILTONIAN
and k replaced by k −k . Similar coupling also ex-
p p 2
The three-level atoms we use to construct the two- ists for the other two modes. |m ,n ,n + 1,n (cid:105)
kp−k2 1 2 3
dimensional (2D) superradiance lattice (SL) have a is in turn coupled to |e ,n − 1,n + 1,n (cid:105) or
kp+k1−k2 1 2 3
ground state |g(cid:105), an excited state |e(cid:105) and a third state |e ,n ,n + 1,n − 1(cid:105) via excitation by modes
kp+k3−k2 1 2 3
|m(cid:105). Theopticalfieldsthatcouple|e(cid:105)and|m(cid:105)havethree k or k . These states form a honeycomb lattice with
√ 1 3
modes with wave vectors k =−k xˆ, k =k (1xˆ− 3yˆ) discrete momentum coordinates, as shown in Fig. S.1,
√ 1 c 2 c 2 2
and k = k (1xˆ+ 3yˆ). The interaction Hamiltonian called the superradiance lattice [1]. The two sublattices
3 c 2 2
with rotating-wave approximation is of the SL correspond to TDS for |e(cid:105) and |m(cid:105). We de-
notethecouplingfieldfrequencyasν andthetransition
c
3 N frequency between |e(cid:105) and |m(cid:105) as ω . The energy dif-
(cid:88)(cid:88) em
Hc =−(cid:126) κlaleikl·rj|ej(cid:105)(cid:104)mj|+H.c., (S.1) ferencebetweenthetwosublatticeis∆c =ωem−νc. We
l=1j=1 set the zero energy at the middle of the energies of the
two sublattices. Then the energies of the |e(cid:105) and |m(cid:105)
where κ and a are the vacuum coupling strength and
l l sublattices are ∆ /2 and −∆ /2, respectively.
c c
annihilation operator of the lth mode, respectively. N
is the number of atoms and r is the position of the jth
j
atom.
Although the coupling strengths in the SL are site-
Initially the atomic ensemble is in the ground state
dependent, if we use coherent coupling fields with large
|G(cid:105)=|g ,g ,...,g (cid:105). A weak probe field with wave vec-
1 2 N averagephotonnumbers(cid:104)n (cid:105)(cid:29)1, theRabifrequencyof
tor k can prepare the atomic ensemble in the timed l
p the lth mode can be approximated as the classical Rabi
Dicke state (TDS) (cid:112)
frequency Ω =κ (cid:104)n (cid:105). The SL with quantized photon
l l l
numbershasedgeswheneitherofthethreecouplingfield
N
|e (cid:105)= √1 (cid:88)eikp·rj|g ,g ,...,e ,...,g (cid:105) (S.2) photon numbers reduces to zero. The total lattice has a
kp N 1 2 j N triangularboundary. However,sincetheTDShavefinite
j=1
life time, which we suppose to be τ on average, once we
by the collective absorption of a single photon. The createanexcitation|e (cid:105)intheSL,theaveragedistance
kp
combinational quantum states of the atoms and three this excitation can travel is in the order of Ω τ. In this
l
coupling modes can be written as |e ,n ,n ,n (cid:105), where paper we have Ω τ (cid:28) (cid:104)n (cid:105) and the edges can never be
kp 1 2 3 l l
n (l = 1,2,3) is the photon number in mode k . This reached. Interesting edge effects might exist in the few-
l l
state is coupled to |m ,n ,n +1,n (cid:105) with coupling photon limit and it will be discussed elsewhere. Since
√ kp−k2 1 2 3
strength −(cid:126)κ n +1. The state |m (cid:105) is defined the photon numbers are correlated to the momenta k’s
2 2 kp−k2
the same as in equation (S.2) with e replaced by m in the TDS, we can simplify the notations by dropping
j j
6
the photon numbers. The Hamiltonian of these TDS is ThedynamicsofthesystemisaFloquetproblem[2,3].
Based on the Floquet theorem, the wave function can be
H =(cid:126)∆c (cid:88)(|e (cid:105)(cid:104)e |−|m (cid:105)(cid:104)m |) written as
2 k k k−k1 k−k1
k (S.3) |Ψ(cid:105)=e−i(cid:15)t/(cid:126)|ψ(t)(cid:105), (S.10)
3
(cid:88)(cid:88)
− (cid:126)Ω [|e (cid:105)(cid:104)m |+h.c.],
l k k−kl where (cid:15) is the quasi-eigenenergy and
k l=1
(cid:88)
where k = kp+r(k2−k1)+s(k3−k2) with integers r |ψ(t)(cid:105)= einνdt|ψn(cid:105), (S.11)
and s. n
The next-nearest-neighbour (NNN) hopping terms are
isaperiodicwavefunctionwithperiod2π/ν . Substitut-
inducedbytheperiodicmodulationofthethreecoupling d
ing equations (S.6)-(S.11) to the Schr¨odinger equation,
field Rabi frequencies,
i(cid:126)∂|Ψ(cid:105)/∂t=H|Ψ(cid:105), we obtain
Ω =Ω +2Ω cos(ν t−kl ·r+φ ), (S.4)
whereΩ anldΩ sarethedstaticdandddynamiclcomponents i(cid:126)∂∂te−i(cid:15)t/(cid:126)(cid:88)einνdt|ψn(cid:105)
s d n
oφf tihsetRheabmiofrdeuqluaetniocniesp.hνadseisotfhtehme oltdhulfiaetilodn. frkelquisentchye. =(H0+H+1eiνdt+H−1e−iνdt)e−i(cid:15)t/(cid:126)(cid:88)einνdt|ψn(cid:105).
l d n
modulation wavevector. We choose the atomic ensemble (S.12)
much smaller than 1/|kld| such that the position depen- The terms with the same time evolution phase factors
dence of the phase −kld · r can be neglected. We set should be equal on both sides. We therefore have
−kl ·r=0 for simplicity. On the other hand, the size of
d
the atomic ensemble shall be much larger than 1/ν and ((cid:15)−(cid:126)nν )|ψ (cid:105)=H |ψ (cid:105)+H |ψ (cid:105)+H |ψ (cid:105).
c d n 0 n +1 n−1 −1 n+1
the total number of atoms N (cid:29)1 such that (S.13)
The quasi-eigenenergy can be obtained by diagonalizing
N
(cid:104)ek(cid:48)|ek(cid:105)= N1 (cid:88)ei(k−k(cid:48))·rj ≈δkk(cid:48). (S.5) theFoarbotvheeHsaakmeilotfonsiimanp.licity, we assume the separation
j=1
between the Floquet sidebands is much larger than the
We assume the atoms be randomly distributed and their bandwidth, ν (cid:29) Ω ,∆ , where the perturbation the-
d s,d c
numberislargeenoughtocoverallpossiblepointsinthe ory can be applied [4, 5]. When the probe field is near
real space Brillouin Zone. The SL can be regarded as resonance, ∆ = ω −ν (cid:28) Ω where ω is the tran-
p eg p s eg
infinite. sition frequency between |e(cid:105) and |g(cid:105) and ν is the probe
p
We expand the Hamiltonian into static, positive- and field frequency, only the Floquet band with n = 0 in
negative-frequency components, Eq.(S.13) is relevant. For states with eigenfrequencies
near (cid:15)=0, the effective Hamiltonian can be obtained by
H =H0+H+1eiνdt+H−1e−iνdt, (S.6) standard second-order perturbation as
where
(cid:15)|ψ (cid:105)=H |ψ (cid:105), (S.14)
0 eff 0
H0 =(cid:126)∆2c (cid:88)(|ek(cid:105)(cid:104)ek|−|mk−k1(cid:105)(cid:104)mk−k1|) where
k
(cid:88)(cid:88)3 (S.7) Heff =H0+H(cid:48), (S.15)
−(cid:126)Ω [|e (cid:105)(cid:104)m |+h.c.],
s k k−kl
with NNN Hamiltonian
k l=1
1
H(cid:48) = (H H −H H ). (S.16)
3 (cid:126)ν +1 −1 −1 +1
H+1 =−(cid:126)Ωd(cid:88)(cid:88)eiφl[|ek(cid:105)(cid:104)mk−kl|+h.c.], (S.8) d
Explicitly,
k l=1
3
(cid:88) (cid:88)
3 H(cid:48) = (cid:126)Ω
(cid:88)(cid:88) lj
H−1 =−(cid:126)Ωd e−iφl[|ek(cid:105)(cid:104)mk−kl|+h.c.]. (S.9) k l(cid:54)=j=1
k l=1 (cid:0) (cid:1)
|e (cid:105)(cid:104)e |+|m (cid:105)(cid:104)m | ,
k+kl−kj k k−k1 k−k1+kl−kj
Note that H are not Hermitian themselves, but H (S.17)
±1 +1
is the Hermitian conjugate of H−1. The phase factors where Ωlj = Ω(cid:48)[ei(φl−φj)−ei(φj−φl)] = 2iΩ(cid:48)s√in(φl−φj)
e±iφl in H±1 play the crucial role for the complex NNN with Ω(cid:48) = Ω2d/νd. The crucial factor i = −1 comes
hopping terms in the Haldane model. fromthequantuminterferencebetweenthetwopathways
7
e 9 !
k (a) K
r
" " #
3 1 #d K K
e y 0
k
" " p
3 1
e K K
k
! r
"
(a) (b) K #9 !
x
Figure S.2: Complex next-nearest-neighbor hopping in- (b) K 27 !
duced by Rabi frequency modulation. (a) The Floquet side- r
bands of the SL and the NNN transition Ω13 (dashed arrow) #
induced by the interference between two second-order inter- K K
sideband transitions (solid arrow). Phases are labelled for
each step. (b) The effective NNN transitions in a unit cell.
TheNNNhoppingΩ31 (redarrow),Ω23 (bluedotarrow)and y 0
Ω (green dash dot arrow) enclose a nonzero effective mag-
12
netic flux.
K K
r
showninFig.S.2(a). ThelooptransitionsviaNNNhop-
"
ping accumulate nonzero phases due to these complex
K #27 !
factors, as shown in Fig.S.2 (b).
The effective Hamiltonian is greatly simplified in the
x
real-spacerepresentation. Wedenotethereal-spacebasis
states as Figure S.3: Effective magnetic fields in a Brillouin zone
for topological and trivial SL’s. (a) A topological SL with
|e (cid:105)=|g ,g ,...,e ,...,g (cid:105),
rj 1 2 j N (S.18) ∆c =0. (b) A trivial SL with ∆c =36Ω(cid:48). The arrows show
|m (cid:105)=|g ,g ,...,m ,...,g (cid:105), the vector h xˆ+h yˆand the colours show h . The x and y
rj 1 2 j N axis are in uxnit of ky−1. The phase of the couzpling field k is
c √ l
The effective Hamiltonian can be written as φ =(l−1)4π/3andα=3 3/2. NotethattheK/K(cid:48) points
l
at the boundaries of the first Brillouin zone are equivalent.
(cid:88)
H = h(r )·σ (S.19)
eff j j
j
BERRY CONNECTION AND BERRY
where the effective magnetic field h(r ) = (h ,h ,h )
j x y z CURVATURE
with
3 Thetopologicalpropertiesofthewavefunctionsarede-
(cid:88)
h =−(cid:126)Ω cos(r ·k ), (S.20)
x s j l terminedbytheBerryconnectionAandBerrycurvature
l=1 B,
3 A=i(cid:104)ψ|∇|ψ(cid:105), (S.23)
(cid:88)
h =(cid:126)Ω sin(r ·k ), (S.21)
y s j l
l=1
B=∇×A, (S.24)
h = (cid:126)∆c +2i(cid:126)(cid:88)3 Ω sin[r ·(k −k )], (S.22) where ∇ and ∇× are the gradient and curl operators
z 2 (l+1)l j l+1 l with respect to r.
l=1
Similartoaspin-1/2inamagneticfieldh,theeigenen-
and the pseudo spin σj = (σjx,σjy,σjz) with the Pauli ergies of the SL eigenstates of the effective Hamiltonian
matrices for the jth atom defined as σjx = |erj(cid:105)(cid:104)mrj|+ in equation (S.19) are
|m (cid:105)(cid:104)e |, σy = −i|e (cid:105)(cid:104)m | + i|m (cid:105)(cid:104)e | and σz =
rj rj j rj rj rj rj j
|e (cid:105)(cid:104)e |−|m (cid:105)(cid:104)m |. (cid:15) =±h (S.25)
rj rj rj rj ±
8
where ± are for the upper and lower bands and h is the e e e
magnitude of h. The eigen wavefunction in the upper r r r!
band can be written as
e
1 (cid:18) h+h (cid:19) r!
|ψ+(r)(cid:105)= (cid:112)2h(h+hz) hx+ihzy . (S.26) K K" K K"
(a) (b)
This wavefunction is well defined except for the south
pole of the Bloch sphere where h = (0,0,h ) with h =
z z
−h. The eigen wavefunction near the south pole can be FigureS.4: TheeigenstatesatK andK(cid:48)pointsfortopolog-
written as ical and trivial SL’s. (a) For a topological SL, the eigenstate
|e(cid:105) at r (i.e., K point) is in the upper band while it is in
(cid:18) (cid:19) +
1 h −ih thelowerbandatr (i.e.,K(cid:48) point). (b)ForatrivialSL,|e(cid:105)
|ψ(cid:48) (r)(cid:105)=eiφ(r)|ψ (r)(cid:105)= x y , −
+ + (cid:112)2h(h−hz) h−hz is the eigenstate in the upper band at both r+ and r−. Red
(Blue) lines denote the bands with more components in |e(cid:105)
(S.27)
(|m(cid:105)).
where the gauge transformation
h −ih
eiφ(r) = x y . (S.28) offset, ∆ , h is a constant in the Brillouin zone. h can
|h −ih | c z
x y only cover a patch on the Bloch sphere and the Chern
number is zero.
This way, the whole Bloch sphere is fully covered by two
The two bands of H have the smallest gap at the K
gauges [6]. eff
and K(cid:48) points r , where
The Berry connection in the upper band (except for ±
the south pole of the Bloch sphere) is
3
(cid:88)
eir±·kl =0, (S.33)
A =i(cid:104)ψ |∇|ψ (cid:105)
+ + +
l=1
1 (S.29)
=− 2h(h+h )(hx∇hy−hy∇hx), andhencehx =hy =0. Atthesesymmetrypointsofthe
z Brillouin zone the Hamiltonian is diagonal,
and near the south pole we can use (cid:126)∆ √
H(r )=[ c ±2 3Ω(cid:48)(cid:126)α]σ , (S.34)
A(cid:48) =A −∇φ(r). (S.30) ± 2 z
+ +
where α=−(cid:80)3 sin(φ −φ ).
The Berry curvature is l=1 l+1 l
Let us consider the specific case that ∆ = 0, φ =
√ c l
(l−1)4π/3, and thus α = 3 3/2. In this case H(r ) =
B =∇×A ±
+ + ±9(cid:126)Ω(cid:48)σ has opposite signs at r , as shown in Fig.S.3
=−2zˆIm(cid:104)∂ ψ |∂ ψ (cid:105) z ±
x + y + (S.31) (a). Itisobviousthatatr , theeigenstateintheupper-
+
zˆ band with eigenenergy 9(cid:126)Ω(cid:48) is |e(cid:105), while at r , |e(cid:105) is the
=− (cid:15) h ∂ h ∂ h , −
2h3 abc a x b y c eigenstate in the lower band, as shown in Fig.S.4 (a).
Near r , we can use equation (S.26) to describe the
where (cid:15) with a,b,c∈x,y,z is the Levi-Civita symbol. +
abc
wavefunction. However, equation (S.26) cannot describe
TheChernnumberisdefinedasthetotalBerrycurvature
all the wavefunctions in the upper band. It has a sin-
in the whole first Brillouin zone
‹ gularity at r− where the magnetic field h points to the
1 southpole. Wecanremovethesingularitybytuning∆ .
C = B · dS √ c
2π + When ∆ >4 3Ω(cid:48)α, the effective magnetic field h does
B‹Z (S.32) c
1 1 not experience the whole Bloch sphere [see Fig.S.3 (b)].
=− (cid:15) h ∂ h ∂ h dxdy,
4π h3 abc a x b y c In this case, hz in equation (S.34) has the same sign for
BZ
r , |e(cid:105) is in the upper band at both r , as shown in
± ±
which counts the winding number of the effective mag- Fig.S.4 (b). The wavefunction can be described by the
netic field h wrapping around the Bloch sphere in the same gauge. Thus the single-valued Berry connection A
whole Brillouin zone [7, 8]. has no singularities on a closed surface and the Chern
In Fig.S.3, we plot h in the Brillouin zone for a topo- number is zero. Generally, the Chern number of the up-
logicalnontrivialSLwherethebandgapisopenedbythe per band can be written as
√
NNN hopping. If we go from the K point r =−4 3πyˆ
to the K(cid:48) point r− = 49√k3cπyˆ, h moves fro+m the n9okcrth C = 21(cid:104)sign(cid:16)∆c+4√3Ω(cid:48)α(cid:17)−sign(cid:16)∆c−4√3Ω(cid:48)α(cid:17)(cid:105).
pole to the south pole of the Bloch sphere. The Chern (S.35)
number is one. If the band gap is opened by the on-site Specifically, when ∆ =0, C =sign(α).
c
9
DYNAMIC EVOLUTION Foruniformlydistributedatoms,inthelimitN (cid:29)Ω /γ ,
s e
we can assume all points in the Brillouin zone are oc-
We can solve the dynamics of the atomic ensemble in cupied by atoms, so the summation can be written as
realspacewiththetotalHamiltonianincludingtheprobe integration in the first Brillouin zone,
field, √ ‹
N
c = c (r)e−ik·rdxdy, (S.44)
(cid:88)N k S BZ e
H = (h −(cid:126)ν˜ )|e (cid:105)(cid:104)e |−(h +(cid:126)ν˜ )|m (cid:105)|m |
t z p rj rj z p rj rj where S is the area of the first Brillouin zone.
j=1
+[(h −ih )|e (cid:105)(cid:104)m |+h.c.]
x y rj rj
−[(cid:126)Ω eikp·rj|e (cid:105)(cid:104)G|+h.c.], DIFFRACTION CONTRAST AND TOPOLOGY
p rj
(S.36)
with ν˜ = ∆ /2 − ∆ being the probe detuning with Inthissection,wequantitativelycomparethecontrast
p c p
respect to the middle of the band gap and Ω the probe between |c |2 and|c |2 intopologicalandtrivialSL’s.
p k+ k−
field Rabi frequency. The wavefunction in real space We assume ν˜ =0 and γ =0. Then Eq.(S.41) becomes
p m
h
|Ψ(cid:105)=c |G(cid:105)+(cid:88)N c (r )|e (cid:105)+c (r )|m (cid:105). (S.37) ce(r)= h2−i(cid:126)zh γ (cid:126)Ωpeikp·r. (S.45)
G e j rj m j rj z e
j=1
We assume Ω(cid:48),∆ (cid:28)Ω and thus c (r) is highly centred
c s e
Thedynamicequationsoftheprobabilityamplitudesare at K and K(cid:48) points, where we have hx =hy =0 and
(cid:20) (cid:21) (cid:126)Ω
c˙e(rj)= −(cid:126)i(hz−(cid:126)ν˜p)−γe ce(rj) ce(r)= hz−ip(cid:126)γeeikp·r. (S.46)
(S.38)
− i(h −ih )c (r )+iΩ eikp·rjc , AccordingtoEq.(S.44),theprobabilityamplitudeck can
(cid:126) x y m j p G
be approximately calculated by the integration of c (r)
e
in small areas near r ,
±
(cid:20) (cid:21) √ ¨
c˙m(rj)= (cid:126)i(hz+(cid:126)ν˜p)−γm cm(rj) c ≈ N(cid:126)Ωp[ei(kp−k)·r+ (h −i(cid:126)γ )−1dxdy
(S.39) k S z e
− i(h +ih )c (r ). ¨ r+ (S.47)
(cid:126) x y e j
+ei(kp−k)·r− (h −i(cid:126)γ )−1dxdy],
z e
where γe/m is the decoherence rate of |e/m(cid:105) states. r−
In the limit of weak probe field Ω (cid:28) γ , we have
p e
where the phase factors have been taken out of the in-
c ≈ 1. In the steady state, c˙ (r ) = c˙ (r ) = 0. From
G e j m j
tegration since they do not change appreciably in those
Eq.(S.39), we obtain
small areas.
c (r )= hx+ihy c (r ). (S.40) For a topological SL where Ω(cid:48) (cid:54)= 0 and˜∆c = 0,
m j hz+(cid:126)ν˜p+i(cid:126)γm e j hz has opposite signs at r±˜. We denote r+(hz −
i(cid:126)γ )−1dxdy = p+iq, then (h −i(cid:126)γ )−1dxdy =
Substituting Eq.(S.40) in Eq.(S.38), we obtain −pe+iq. Thus r− z e
√
c (r )= (cid:126)Ωpeikp·rj . (S.41) c ≈2i N(cid:126)Ωp{psin[(k −k)·r ]+qcos[(k −k)·r ]}.
e j h −(cid:126)ν˜ −i(cid:126)γ + h2x+h2y k S p + p +
z p e hz+(cid:126)ν˜p+i(cid:126)γm (S.48)
For k =−k , k =−k and k =−k , we have
p 1 + 3 − 2
IntheSLcoordinates,thewavefunctioncanbewritten
√
as |c |2−|c |2 2 3pq
η ≡ k+ k− ≈ . (S.49)
|Ψ(cid:105)=c |G(cid:105)+(cid:88)c |e (cid:105)+c |m (cid:105). (S.42) |ck+|2+|ck−|2 3p2+q2
G k k k−k1 k−k1
k For a trivial SL where Ω(cid:48) =˜0 and ∆c (cid:54)= 0, hz has
the same sign at r . We have (h −i(cid:126)γ )−1dxdy =
where the probability amplitude ˜ ± r+ z e
(h −i(cid:126)γ )−1dxdy =p+iq and
r− z e
1 (cid:88)N √
c =(cid:104)e |Ψ(cid:105)= √ e−ik·rjc (r ). (S.43) N(cid:126)Ω
k k N e j c ≈2 p(p+iq)cos[(k −k)·r ]. (S.50)
j=1 k S p +
10
Therefore, c =c and η =0. 1
k+ k−
ForatrivialSLwithΩ(cid:48) (cid:54)=0and∆ (cid:54)=0,itisdifficultto
c
analyticallycalculatetheresults. Nevertheless,asshown
in Fig.3 (b) in the main text, η reduces to a small value
in the C =0 phases especially for large Ω(cid:48).
0
COUPLED- WAVE EQUATIONS
In this section, we calculate the propagation of a
weak probe pulse. We set kp = −k1, k+ = −k3 and 1
k = −k . There will be directional emission in k by
− 2 ±
TDS |e (cid:105). The emitted photons in k interact with
k± ±
the atomic ensemble the same way as the probe field k ,
p
Figure S.5: The contrast of the NNN coupling strengths.
creatingTDSandresultingindirectionalemissioninthe
The contrast (|Ω |−|Ω |)/(|Ω |+|Ω |) where Ω (Ω )
13 12 13 12 13 12
other two modes. For example, emission at k excites
+ is the NNN hopping coefficient from |e (cid:105) to |e (cid:105) (|e (cid:105)).
the atom ensemble and leads to emission at k− and kp. The dot lines are zero points. kp k+ k−
Therefore, the three optical fields in k are coupled
p,±
via the atoms. In the following, we derive the coupling
equations. |kj| = νp/c ≡ kp. The coupled-wave Maxwell equations
Theexpectationvalueofthedipolemomentofthejth for the three modes are
atom at position rj is (cid:18) 1 ∂ (cid:19) ik
kˆ ·∇+ E = p (χ E +χ E +χ E ),
(cid:88) p vg ∂t p 2 0 p − + + −
(cid:104)Ψ|µˆ |Ψ(cid:105)= c (cid:104)G|µˆ |e (cid:105)+c.c.
j k j k (S.53)
k
(S.51)
=√µ (cid:88)ckeik·rj +c.c., (cid:18)kˆ ·∇+ 1 ∂ (cid:19)E = ikp (χ E +χ E +χ E ),
N + v ∂t + 2 0 + − − + p
k g
(S.54)
where µˆ is the dipole operator of the jth atom and
j
(cid:18) (cid:19)
µ = (cid:104)gj|µˆj|ej(cid:105). Since the atoms are homogeneously dis- kˆ ·∇+ 1 ∂ E = ikp (χ E +χ E +χ E ),
tributed, the polarization density as a function of posi- − v ∂t − 2 0 − − p + +
g
tions is, (S.55)
where kˆ are the unit vectors in the directions of
p/+/−
P(r)=NV √µN (cid:88)ckeik·r+c.c., (S.52) kp/+/− and vg is the group velocity of the pulses.
k
where V is the volume of the atomic ensemble. The LOCAL AND GLOBAL EFFECTS
probe field Rabi frequency is Ω = µE /(cid:126) with E be-
p p p
ing the probe field strength. The polarization density In the main text, we use the chiral topological current
contains the√Fourier components in k±, P±,p = (cid:15)0χ±Ep to explain the consistence between η and the topological
with χ = Nµc /V(cid:15) E and (cid:15) is the permittivity order. Thetopologicalcurrentisaglobaleffectforwhich
± k± 0 p 0
in vacuum. The notation P means the k Fourier we must treat the lattice as a whole. Since the three
±,p ±
component of polarization density generated by optical superradiant TDS |e (cid:105) are next nearest neighbours in
kp,±
field in k . Once modes k are excited and the corre- the lattice, the local effect induced by the transitions
p ±
sponding fields E are generated, they also polarize the inside a unit cell is also important. The direct NNN
±
atoms. For φ =0, φ =4π/3 and φ =2π/3, the three transition strength from |e (cid:105) to |e (cid:105) or |e (cid:105) is Ω
1 2 3 kp k+ k− 13
fieldshaveacyclicrelationbetweeneachother. Wehave or Ω , respectively. If the dynamics is dominated by
12
P = (cid:15) χ E , P = (cid:15) χ E , P = (cid:15) χ E and these direct transitions, we expect the contrast of the
p,+ 0 − + −,+ 0 + + p,− 0 + −
P =(cid:15) χ E . probabilityη shouldbesimilarto(|Ω |−|Ω |)/(|Ω |+
+,− 0 − − 13 12 13
Weassumethequasi-staticapproximationinwhichthe |Ω |), which is plotted in Fig.S.5.
12
atoms are assumed in steady sates, i.e., c is given by We plot the contrast η in Fig.S.6 for different Ω(cid:48)/γ .
k m
equation(S.44). Thisapproximationisjustifiedwhenthe The larger Ω(cid:48) is, the closer η is to Fig.S.5 and the more
pulse durationis muchlonger than the decoherence time important is the local effect, as shown in Fig.S.6 (a) and
1/γ . We also use the slowly-varying-envelop approxi- (b). ForΩ(cid:48) ≤γ ,theeffectofthedirectNNNtransitions
e m
mation. The fields in the three relevant modes k with is suppressed by the decoherence rate γ . The global
j m
j = p,+ and − are denoted as Ej(r)e−iνpt+ikj·r where topologicaleffectdominatesformostareasandηiscloser