This is the accepted manuscript made available via CHORUS. The article has been
published as:

Photonic Topological Insulating Phase Induced Solely by
Gain and Loss
Kenta Takata and Masaya Notomi
Phys. Rev. Lett. 121, 213902 — Published 20 November 2018
DOI: 10.1103/PhysRevLett.121.213902

Photonic Topological Insulating Phase Induced Solely by Gain and Loss
Kenta Takata1, 2 and Masaya Notomi1, 2, 3
1
NTT Nanophotonics Center, NTT Corporation,
3-1 Morinosato-Wakamiya, Atsugi 243-0198, Kanagawa, Japan
2
NTT Basic Research Laboratories, NTT Corporation,
3-1 Morinosato-Wakamiya, Atsugi 243-0198, Kanagawa, Japan3
Department of Physics, Tokyo Institute of Technology,
H-55, Ookayama 2-12-1, Meguro 152-8550, Japan(Dated: October 11, 2018)

We reveal a one-dimensional topological insulating phase induced solely by gain and loss control in
non-Hermitian optical lattices. The system comprises units of four uniformly coupled cavities, where
successive two have loss, the others experience gain and they are balanced under two magnitudes.
The gain and loss parts are effectively dimerized, and a bulk bandgap, topological transition, midgap
topological edge and interface states in finite systems can all be achieved by controlled pumping.
We also clarify non-Hermitian topological invariants and edge states in gapless conditions.

Controlling optical properties with external signals isa major destination in photonics research [1], and it islargely associated with tailoring the refractive index. Recent studies have revealed that the imaginary part of
the refractive index, namely gain and loss, can do much
more than just tuning the optical intensity. The key concept, parity-time (PT ) symmetry [2, 3], was introducedfor obtaining real spectra of quantum systems with nonHermitian components. Its analogy in optics [4, 5] corresponds to complex refractive index profiles with symmetric real parts and antisymmetric imaginary parts, i.e.
n(r) = n∗ (−r). Such a system can show an exceptional
point (EP) [6] where its eigen-detuning sharply changes
from real to imaginary values [7, 8] (spontaneous PT
symmetry breaking). There are many interesting phenomena related to PT symmetry including power oscillation [4, 9], double refraction [4, 10], Bloch oscillation[11, 12], mode-locking [13], coherent absorption [14–16],
fast light [10, 17, 18], and unidirectional reflectivity [19–21]. Furthermore, nonlinearity-induced isolation [22, 23],
single-mode lasing [24, 25], and beam steering [26] wereachieved under controlled pumping.
To widen the scope of non-Hermitian optics [27, 28],
there are growing attempts to incorporate topologicalfeatures to photonic systems with gain and loss. While
Hermitian photonic topological phases [29–31] are basedon celebrated discoveries such as the quantum Hall effect[32, 33] and topological insulators [34, 35], non-Hermitian
topological optics originates from the theoretical question as to whether or not stable topological quantum
states exist in non-Hermitian systems [36–40]. For photonics based on classical electromagnetic waves, however,it has been clarified that there exist topological stateseven when their eigenvalues are not real [41]. Researchers
applied Su-Schrieffer-Heeger (SSH) photonic lattices [42]with relevant loss and experimentally confirmed their
topological interface states [43] and topological transition [44]. Moreover, a photonic topological bound statewith global PT symmetry was observed [45]. Even thelasing of such topological modes has recently been shown
to be feasible [46–49].
Then, another question may arise. Can we create
a topological insulating phase solely from gain and loss
control? In previous studies of non-Hermitian optics,
the emergence of nontrivial topologies was attributed toHermitian factors, namely the magneto-optic effect [49],and lattice and coupling profiles of host systems [37, 50–53]. Even though such systems are armed with nonHermiticity, they take over original Hermitian topological characters predetermined by fabrication. Moreover,gain and loss in conventional non-Hermitian systems[27, 28, 54] only close frequency bandgaps. Thus, they
were considered to destroy topological insulating phases.
In contrast, our aim is to generate a topological bandgap
solely by adding static gain and loss to a topologicallytrivial structure. Then, achieved topological features, including a well-defined topological number, should originate purely from non-Hermitian factors. Here, the gain
and loss are readily tunable by multi-channel current injection or properly masked optical pumping in various
laser systems [18, 26, 46–48]. We will hence have fullmanipulability over the topological properties in opticalcircuits, such as a topological transition, and the number
and position of topological states, simply by changing thegain and loss.
Here, we show theoretically a one-dimensional photonic lattice with the gain- and loss-induced reconfigurable topological insulating phase. We consider unit
cells of four uniformly coupled resonators, with loss introduced into two successive cavities and gain introduced
into the other two. The system then forms a pair ofdimers by effective decoupling between cavities with gain
and loss. This can result in a bulk bandgap, topological transition and midgap edge states for a wide rangeof parameters. Topological interface states can also be
achieved at a controlled boundary between the nontrivial and trivial lattices. Our scheme is unique in nonHermiticity-based midgap topological states protected by

2

where a is the spatial interval between the four-cavity
units and k is the Bloch wavenumber. The eigenfrequency detuning ω(k) with reference to ω0 is given by,
s
r
ka
1
√
,
(2)
A ± A2 − B 2 − 16κ4 sin2
ω(k) = ±22where A = 4κ2 − g12 − g22 and B = 2g1 g2 . We also findanalytic forms of |ψB,s i (s: eigenstate index), which are
given in Supplementary Materials [61]. In the following
analysis, the gain and loss are measured with respect tothe cavity coupling, i.e. κ = 1. We focus on the casewhere g1 > 0 and g1 ≥ |g2 | for studying the bulk properties, because the spatial and/or time reversal can map thesystem with this condition to that with the other parameter range. When g1 = g2 = 0, the system has a gaplessfour-fold cosinusoidal band structure with a degeneracyat ω(0) = 0, because of the reduced first Brillouin zone
[Fig. 1 (b)].
The system band structure is classified into five patterns via the value of the inside of the double radical
sign of Eq. (2) for k = 0 [Fig. 1 (c)]. With g1 , g2 ≥ 0 for
simplicity, the divided phase regions are, (I) B = 0, (II)A + B > 0, A − B ≥ 0, (III) A + B > 0, A − B < 0, (IV)A+B ≤ 0, A2 −B 2 −16κ4 < 0 and (V) A2 −B 2 −16κ4 ≥ 0.
Here, the phase boundaries are symmetric to g1 = g2 .
The systems in phase (II) and (III) have complete frequency bandgaps, while those in phase (I), (IV) and (V)

(a)

(c)

g >0κg >0

κ

1

-g1

-g2

1

κ

κ

κ

κ
-g2

-g1



(V)

g2>0



(IV)

g2<0

(b)

g2

their isolation from bulk states. Although, we also clarify system topological features in non-Hermitian gaplessconditions, which will be relevant with defect and edge
states in gapless systems [37, 55–59].
Theoretical model.– The system comprises periods offour single-mode cavities with uniform couplings κ [Fig.
1 (a)]. We introduce an on-site imaginary potential profile (ig1 , −ig2 , −ig1 , ig2 ) to the cavities, where its positive
and negative coefficients mean gain and loss, respectively.
Here, we assume that κ, g1 and g2 are sufficiently small
compared to the cavities’ resonant frequency ω0 and thecavity-mode Q-factor is high, so that we can safely neglect the effect of the imaginary index profile and radiation loss on κ, as expected in semiconductor lasers[18, 22, 23, 25, 26, 46–48, 60]. Within the linear analysis, the coupled mode equation describing the system is
equivalent to the Schrödinger equation i∂t |Ψi = Ĥ |Ψi,where |Ψi = ({Ψn })T is the vector of the slowly-varying
complex cavity-mode amplitudes (n: cavity index) andĤ is a tight-binding lattice Hamiltonian. Considering the
Bloch theorem and a dynamical factor e−iωt , the analysis
reduces to an eigenvalue problem for the four-component
eigenvector |ψB i under the Bloch Hamiltonian Ĥ(k),


ig1
κ
0 κe−ika κ −ig2 κ
0 
,
(1)
Ĥ(k) = 
 0
κ −ig1κ κeika 0
κ
ig2

(III)
Gapped(II)
Gapped(I)

Bloch Wavenumber (π/a)

Gapless(IV) (V)g1

FIG. 1. (Color online) (a) Schematic of considered system.
Upper and lower lattices for g2 > 0 and g2 < 0 are topologically nontrivial and trivial, respectively. (b) Folded cosinusoidal band structure for g1 = g2 = 0. (c) Phase diagram forthe system band structure. κ = 1, g1 , g2 ≥ 0.

are gapless. System band structures for phase (IV) and(V) are shown elsewhere (Fig. S1 [61]). Note that the
diagram for g2 < 0 is obtained by the mirror inversion ofFig. 1 (c) with regard to g2 = 0.
We see from Eq. (2) that ω(k) is real as long as A > 0
and A2 − B 2 − 16κ4 sin2 (ka/2) > 0. Such real eigenvalues
are obtained because Ĥ(k) has a pseudo-Hermiticity [62],Ŝ(k)Ĥ(k)Ŝ(k) = Ĥ† (k). Here, the k-dependent linear
operator Ŝ(k) = Ŝ(k)−1 = σ̂x ⊗ (cos k2 )Iˆ2 + σ̂y ⊗ (sin k2 )Iˆ2
means a half-period translation. σ̂x,y,z are Pauli matrices and Iˆ2 is the 2×2 identity matrix. The pseudoHermiticity guarantees an associated antilinear symmetry [63]. Although the system does not respect PT symmetry, the bulk antilinear symmetry can instead give atleast partially real spectra by balancing net gain and loss.
Meanwhile, the operation to which Ĥ(k) shows the invariance is implicit due to its complexity.
Ĥ(k) also satisfies a pseudo-anti-Hermiticity [36, 37,61], Ĥ(k) = −η̂ Ĥ† (k)η̂, where η̂ = Iˆ2 ⊗ σ̂z =diag(1, −1, 1, −1) and η̂ −1 = η̂ † = η̂ in our model. It is
known that this symmetry can lead to a nontrivial topology via chirality in terms of pairwise eigenvalues, ω(k)and −ω ∗ (k). Since η̂ is local (diagonal), the resultant
topological protection covers all the system parameters.
We notice that the symmetry is equivalent to a particlehole symmetry, −Ĥ(k) = η̂ Ĥ∗ (−k)η̂.
Bulk properties.– Figure 2 shows the real and imaginary band structures and eigenmode distributions fordifferent g1 and g2 values. When g1 < 2 and g2 = 0[phase (I) with A > 0], Dirac-like dispersion in Re ω(k)[Fig. 2 (a)] appears around ω(0) = 0, with cancelled net
gain and loss [Im ω(k) = 0, Fig. 2 (b)]. This implies
a topological transition point and reflects the antilinear
symmetry. The gapless feature is attributed to B = 0 in
Eq. (2). The band structure also has two EPs meaning
the spontaneous antilinear symmetry breaking. The in-3Bulk mode patterns in the insulating conditions confirm this non-Hermitian effect. Before crossing the EPs inphase (II), the eigenmodes spread over the entire unit cellto cancel the net gain and loss for real ω(k) [k = 0.05π/a,Fig. 2 (f)]. After the symmetry breaking, however,they eventually turn into couples of states with localization at the gain cavities (n = 1, 4) and loss cavities(n = 2, 3) for complex ω(k) (k = 0.95π/a). In phase(III) (|g2 | > 2κ − g1 ), the complete antilinear symmetry
breaking makes the pairs of upper and lower real bands
overlap and gives the split imaginary bands [Fig. 2 (g),(h)]. Thus, only the dimerized eigenstates are allowed allover the Brillouin zone [Fig. 2 (i)]. Here, reversing thesigns of g1 and g2 does not affect the band structure [Eq.
(2)], while a topological transition between systems withg2 > 0 and g2 < 0 is expected at the gap closing with
ω(0) = 0. The non-Hermiticity-based bandgap broadens
continuously as (g1 , g2 ) gets toward the inside of phase(II) and (III) from their boundaries (Fig. S2 [61]).
We introducePthe normalized
global Berry phase [64] inHidkhhψB,s | ∂k |ψB,s i (s = 1, . . . , 4),k-space, W = s 4π
as our topological number. W denotes the topological feature of the entire system [65], thus it resolves
the problem that the Zak phase [66] of each bandis not discretized in non-Hermitian systems with EPs.
Here, |ψB,s ii is the left eigenstate that forms a duality with |ψB,s i, namely Ĥ(k) |ψB,s i = ω(k) |ψB,s i andĤ† (k) |ψB,s ii = ω ∗ (k) |ψB,s ii [67]. The biorthonormal basis ({|ψB,s i}, {|ψB,s ii}) enables the normalizationhhψB,s |ψB,t i = δs,t and the extraction of pure geometricphases from non-Hermitian eigenvectors. W also reflects
the 4π periodicity of the eigenvectors [52, 61]. We obtain integer values of W = 1 for g2 > 0 and W = 0 for
g2 < 0, under g1 > 0 (Fig. S3 [61]). W hence confirms
the non-Hermiticity-induced nontrivial photonic topology and topological transition between the two conditions in Fig. 1 (a). Interestingly, the discrete change in
W holds in the gapless phases, (IV) and (V). We discuss

(g)

1.5
1.0
0.5
0.0
0.5
1.0
1.5
1.0

ɗB,3
0.5 0.0

ɗB,2

Im Ȧ

ɗB,2

(b)

ɗB,4
ɗB,1
0.5

1.0
(e)

ɗB,4
ɗB,3

0.5 0.0

Im Ȧ

Re Ȧ
Re Ȧ

(d)

1.5
1.0
0.5
0.0
0.5
1.0
1.5
1.0

ɗB,1
0.5

1.0

1.0
ɗB,2, ɗB,4
0.5
0.0
0.5
ɗB,1, ɗB,3
1.0
1.0 0.5 0.0 0.5 1.0
Bloch wavenumber (ʌ /a)

(h)
Im Ȧ

For g1 , g2 6= 0 under phase (II), a bandgap opens due to
B 2 > 0 [|g2 | ≤ 2κ−g1, Fig. 2 (c)], while the real eigenvalues remain around k = 0 [Fig. 2 (d)]. Here, the effective
couplings between gain and loss cavities become weaker
than that between the two gain cavities and that between
the loss cavities [61], as anticipated from the mode localization by PT symmetry breaking [7, 8, 27, 28]. Thisdecoupling results in dimerization of the successive cavities with gain and those with loss [61].

(a)

Re Ȧ

tensity distributions for |ψB,s i [Fig. 2 (c)] show that thefields are evenly distributed in the gain and loss cavities(n = 1, 3) before this transition (k = 0.05π/a), while theeigenmodes with the broken symmetry (k = 0.95π/a) exhibit localization at either of them, and hence complex
ω(k). Note that g1 > 2 and g2 = 0 [phase (I), A < 0] givea Dirac cone in Im ω(k), with a degeneracy at ω(0) = 0(Fig. S1 [61]).

0.4
0.2
0.0
0.2
0.4
1.0
0.6
0.4
0.2
0.0
0.2
0.4
0.6
1.0
1.0
0.5
0.0
0.5
1.0

ɗB,1, ɗB,4(c) k ʌa
1

s2

3
4
k ʌa1
s2
3
4
1 2 3 4

ɗB,2, ɗB,3
0.5 0.0

0.5

ɗB,1, ɗB,4

1.0n(f) k ʌa1s2

ɗB,2, ɗB,3
0.5 0.0

ɗB,1, ɗB,4

0.5

1.0

3
4
k ʌa1
s2
3
4
1 2 3 4n(i) k ʌaɗB,2, ɗB,3

1.0 0.5 0.0 0.5 1.0
Bloch wavenumber (ʌ /a)

1
s2
3
4
k ʌa1
s2
3
4
1 2 3 4n00.25 0.5

FIG. 2. (Color online) Band structures and mode patterns of system for different gain and loss profiles. κ = 1.
(a) Re ω(k), (b) Im ω(k) and (c) |ψB,s,n |2 / hψB,s |ψB,s i for
g1 = 1, g2 = 0, phase (I). (d) Re ω(k), (e) Im ω(k) and(f) |ψB,s,n |2 / hψB,s |ψB,s i for g1 = 1, g2 = 0.5, phase (II).
(g) Re ω(k), (h) Im ω(k) and (i) |ψB,s,n |2 / hψB,s |ψB,s i for
g1 = 2, g2 = 1, phase (III). s and n: eigenstate and cavity indices.

a geometric picture of W , which is based on coupling
parameters relevant with dimerization and can illustrate
the topological transition of both the SSH and our model,
in Supplementary Materials [61] (Fig. S4 and S5).
Edge states.– Figure 3 shows the topological edge
states in our finite systems with 40 cavities for g1 >0, g2 > 0. Here, there is relatively weaker effective coupling between gain and loss cavities at each edge [Fig. 3(a)], like the edge-state generation condition in the Hermitian SSH model [42, 50]. Displayed eigenfrequencies
show a pair of midgap states for both phase (II) [Fig. 3(b)] and (III) [Fig. 3 (d)]. For Fig. 3 (d), we consider
an offset absorption potential iγ for every cavity, which
only shifts all the eigenfrequencies by iγ and cancels Im ωof a midgap state. Each eigenmode with Re ω = 0 is localized at the left or right edge. Their Im ω reflect theimaginary potential around the relevant edge cavities. A
remarkable difference is whether the localization is unitbased [phase (II), Fig. 3 (c)] or cavity-based [phase (III),
Fig. 3 (e)], corresponding to whether the bulk without
the imaginary offset is in the exact phase [Im ω(0) = 0] or
broken phase [Im ω(0) 6= 0]. In phase (III), a topologicaledge mode has the largest Im ω and can be the only statethat oscillates (Im ω = 0) via the loss offset, γ < 0 [Fig.
3 (d)]. Note that the edge states are also found wheneach side is terminated by a loss cavity (g1 < 0, g2 < 0).
However, they disappear if a cavity on one edge has gain
and one on the other undergoes loss (g1 g2 < 0), as indicated by W = 0. In Supplementary Materials [61], we

4

g2>0



●

1.0

●

0.5

●

0.0
-0.5

■

●

●

●

●
●
● ● ● ●

●

-1.5
10

●

●

■

0.6● ● ●

Im ω

-1.0

● ● ● ●

0
●
●

- 0.6
10

●

●
■
●
●
●●●● ■●
●●●

●

●

20

20

●
●

30

Normalized Intensity

1.5

30

g2

1.0

●
● ● ● ●
● ● ● ●

0.5
0.0
- 0.5
● ●
● ● ● ●
● ● ● ●

- 1.0
10

■

0.5
■
0
- 0.5●● ●●● ●●● ●■-1
10 20 30

Im ω

■

20
Index of eigenstates

30

(a)

0.6

g1>0

Im ω > 0

0.3
0

1

10

20

30

40

Im ω > 0

0.0

■

●

1

10

1

20

30

40

Im ω = 0

0.5
1

10

20

30

40

30

40

1

Im ω < 0
0.5
0

●

- 0.5

0.3

0

(c)

1.0
0.5

0.6

0

Topological Trivial
Gain or loss offset: γ
κ
κ
κ
κ
κ
κ

-g2 -g1 g2>0 -g1 -g2
g2
g1

κ


(b)

(e)

(d)

Normalized Intensity

Re ω

-g1

(c)

(b)

Re ω

-g2

g1

κ

κ

κ

κ

1

10

20

Cavity index

FIG. 3. (Color online) (a) Illustration of our finite nonHermitian topological lattice. (b), (d) Sorted and selected
real eigenvalues for forty-cavity systems. Insets: corresponding imaginary eigenvalues. Squares: topological edge states.
(c), (e) Intensity distributions for the edge states. (b), (c)g1 = 1, g2 = 0.5, γ = 0. (d), (e) g1 = 2, g2 = 1, γ ∼ −1.569.

show detuned edge states for g1 = g2 , striking robustness of the edge states to disorder, and edge states in thegapless phases (Fig. S6, S7, S8 and S9).
The topological vortex charge
R of non-Hermitian bulk
1eigen-detuning [57], V = 2π
∂k Arg[ω(k)]dk, can beevaluated for our model [61]. Here, finite fractionalcharges V = ±1/2 are found around two degenerate EPs
with ω = 0 in the gapless phase (IV) [Fig. S10 (a)]. They
indicate anomalous edge states with non-Hermitian chirality emerging at Re ω = 0, and such states are compatible with both the topologically nontrivial (W = 1)
and trivial (W = 0) systems. In phase (V), V = ±1/2
is held by each eigenstate of the non-Hermitian flatband
with Re ω = 0 [Fig. S10 (b)].
We can discuss the origin of the topological edge stateswith a particle-hole symmetry [68] equivalent to ourpseudo-anti-Hermiticity. Because our lattice Hamiltonian Ĥ with a finite number of cavities and the open
boundary condition is a symmetric matrix (Ĥ = ĤT ),its global pseudo-anti-Hermiticity, Ĥ = −η̂ ′ Ĥ† η̂ ′ , easily reduces to a particle-hole symmetry, −Ĥ = η̂ ′ Ĥ∗ η̂ ′ .
Here, η̂ ′ = diag(1, −1, 1, −1, . . . , 1, −1, 1, −1) is again local, η̂ ′−1 = η̂ ′† = η̂ ′ and Ĥ∗ is the complex conjugate ofĤ. It indicates that the number of states with Re ω = 0
at each edge can change only by two [69, 70]. Thus, a single isolated edge state with Re ω = 0 on each side, basedon (1, 0, 0, . . . , 0)T and (0, . . . , 0, 0, 1)T for g1 = g2 = 0, is
topologically protected by this symmetry under proper
bandgap-opening conditions [71].

- 1.0
10

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

●

■

0.5
0
-0.5
-1
-1.5
10

Im ω

-g1

-g2

κ

Re ω

g1>0

Gain or loss offset: γ

κ

κ

κ

●●

■
●●

●

●

●●

●●

■

20

30

20
Index of eigenstates

30

Normalized Intensity(a)1

Im ω = 0

0.5
0

1

10

1

20

30

40

30

40

Im ω < 0

0.5
0

1

10

20

Cavity index

FIG. 4. (Color online) (a) Interface between topologically
nontrivial (left) and trivial (right) lattices, with 20 cavities
for each. (b) Sorted and selected Re ω of the system. Inset:
corresponding Im ω. Left and right squares: eigenvalues for
topological interface and edge states. (c) Intensity profiles forthe interface (upper) and edge (lower) modes. g1 = 1.5, g2 =3, γ ∼ −2.526.

Interface states.– We apply our non-Hermitian scheme
for a controllable midgap topological interface state (Fig.
4). Here, we prepare butting of topologically nontrivial
and trivial lattices, effectively forming an “long-long defect” at their boundary, as in an SSH system [72] [Fig.
4 (a)]. Both lattices are adjusted to be in phase (III),and larger gain is applied to the interface cavity of thenontrivial array, than its edge (g2 > g1 ). In consequence,a topological interface state with Re ω = 0 and strong
boundary localization obtains the largest Im ω as theabovementioned edge state. Including a global loss biasiγ, we can hence expect single-mode lasing (Im ω = 0) of
such a state. The system eigenfrequency profile [Fig. 4(b)] confirms a pair of midgap states and cancellation of
Im ω in one of them. The intensity distributions of theeigenmodes with ω = 0 and Re ω = 0, Im ω < 0 certainly
indicate topological interface and edge states confined at
the right (n = 20) and left (n = 1) sides of the nontrivial lattice, respectively [Fig. 4 (c)]. A topologicalbound state with global PT symmetry, which systematically satisfies Re ω = Im ω = 0, can also be demonstrated(Fig. S11 [61]).
In conclusion, we have shown that the topologicalinsulating properties of the one-dimensional resonatorarray can be controlled by the gain and loss. Our
scheme is experimentally feasible by modifying the existing laser arrays [46–48] and valid for coupled waveguides [44, 45]. Moreover, it can be explored as an extension of PT -symmetric systems in photonics [10, 18, 22–
26, 43], phononics [73–75], and circuit electronics [76, 77].
It would pave the way for various possibilities of nonHermitian topological photonics, such as reconfigurable
lasing modes, non-Hermiticity-based topological pumping [41, 78], superstructures [79] and Floquet systems[40, 80]. Developing the topological controllability in two
dimension is another important direction.

5
We thank Hiroshi Yamaguchi, Kensuke Inaba and Koji
Muraki for fruitful discussions. We acknowledge financial
support from the Japan Science and Technology Agency(JST) through the CREST program under Grant No.
JPMJCR15N4.

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