Nonlinear mode competition and symmetry-protected power oscillations intopological lasers

Simon Malzard and Henning SchomerusDepartment of Physics, Lancaster University, Lancaster, LA1 4YB, United Kingdom

Topological photonics started out as a pursuit to engineer systems that mimic fermionic single-particle Hamiltonians with symmetry-protected modes, whose number can only change in spectralphase transitions such as band inversions. The paradigm of topological lasing, realized in three recentexperiments, offers entirely new interpretations of these states, as they can be selectively amplifiedby distributed gain and loss. A key question is whether such topological mode selection persistswhen one accounts for the nonlinearities that stabilize such systems at their working point. Here weshow that topological defect lasers can indeed stably operate in genuinely topological states. Thesecomprise direct analogues of zero modes from the linear setting, as well as a novel class of statesdisplaying symmetry-protected power oscillations, which appear in a spectral phase transition whenthe gain is increased. These effects show a remarkable practical resilience against imperfections,even if these break the underlying symmetries, and pave the way to harness the power of topologicalprotection in nonlinear quantum devices.

I. INTRODUCTION

Topological quantum devices aim to evoke states thatdisplay a unique response to external stimuli. The un-derlying concepts were originally developed in a fermioniccontext, where they provide unified insights into diversephenomena that range from the quantum-Hall effect tothe emergence of quasiparticles with unconventional sta-tistics [1, 2]. The robustness of the ensuing propertiesmakes it desirable to replicate them in other types of sy-stems. This can be a considerable challenge since true to-pological protection requires symmetries that constrain asystem’s behaviour. In fermionic systems the underlyingsymmetries originate from the anticommutation relationsand therefore are exact, even in presence of interactions[3]. In bosonic systems, however, much looser constraintsapply. The first generation of bosonic analogues of fermi-onic topological effects therefore required impressive featsof precision engineering to replicate the relevant single-particle physics for photons [4–8], cold atomic gases [9],exciton polaritons [10–13] and sound [14, 15].

Only in very recent times it has been realized that theliberty afforded by bosonic systems also offers opportu-nities for topological effects that transcend the electronicsetting. Examples include squeezed light [16] and we-akly interacting bosonic systems characterized by a Bo-goliubov theory [17–23]. The foundations for genuinelybosonic devices were laid by taking symmetries from thefermionic single-particle context and generalising them toevents that change the particle number, which classicallyrepresent gain and loss [24–27].

A key paradigm to test these ideas is the concept of atopological laser [26]. Lasing in topological defects hasbeen realized in photonic crystals [28], which relied onconventional mode selection, and lasing based on the ana-logy to topological insulators has also been proposed [29].Separately, distributed gain and loss have been employed(beyond the conventional setting of distributed feedbacklasers), e.g., in PT-symmetric lasers [30–34], which ex-

1.00.7

A

IA

IB=0

IA

IB=0

B

(a) SSH array linear mode selectionzero mode

bulk modes

(b) Topological mirrors

α β

α central region β

n=1 2 3 ∙∙∙−1 1

Ḡ

GA

Re ω

Im ω

−1 1Ḡ

GA

Re ω

Im ω

zero mode

bulk modes

FIG. 1. Topological mode selection in laser arrays consis-ting of single-mode resonators grouped in dimers (enumeratedby n). The intra-dimer couplings κ and inter-dimer couplingsκ′ are chosen to produce interfaces between regions of topo-logically different band structures. (a) In the Su-Schrieffer-Heeger (SSH) model, the alternating couplings define a phaseα (κ > κ′ ) and a phase β (κ < κ′). The displayed defectstate arises from two consecutive weak couplings, forming aninterface between the two phases. (b) The defect region canbe extended, leading to a variant where the phases α and βfunction as selective mirrors that confine a defect state with alarger mode volume. In both cases, the resulting defect stateshave preferential weight on the A sublattice (red) and can beselected by distributed gain and loss. As illustrated in theright panels, in the linear regime the defect state acquires theeffective gain GA from the A sublattice, while the other mo-des acquire the average gain Ḡ in the system (GA = Ḡ+ 0.1,κ, κ′ = 1, 0.7). We demonstrate that this mode selection me-chanism extends to the nonlinear conditions at the workingpoint of a laser, where it stabilizes robust zero modes and alsoenables alternative topological operation regimes with poweroscillations.

ploit a spectral phase transition between conventionalmodes that then acquire different weights on lossy andamplified regions. Both aspects are combined in the con-cept of topological mode selection [26, 27]. This aimsto utilize an additional distinctive feature of topologi-cal modes besides their pinned energy, namely that they

2

display anomalous expectation values—typically, an une-qual weight on two subspaces associated with the under-lying symmetries. This anomalous response permits tomanipulate the life-time of the topological modes by ap-propriate distributions of gain and loss, which results ina topological mechanism of mode selection (see Fig. 1).Over the past months, three variants of these lasers haveindeed been realised [35–37], confirming the viability ofthis idea in practice. The topological mode selection re-alised in these experiments directly addresses a preci-sely predetermined mode that exists without any furtherspectral phase transitions, and from the outset makesoptimal use of the provided gain.

Here, we show that these systems indeed offer bothgenuine as well as unique topological operation condi-tions when one accounts for the nonlinearities that areindispensable to stabilize active systems at their workingpoint. In particular, besides confirming the possibilityof lasing in a precisely-defined nonlinear counterpart ofconventional zero modes, we uncover topological phasetransitions into states exhibiting topologically protectedpower-oscillations, not yet observed in experiments. Asin conventional incarnations of topology in electronic andphotonic systems, these phase transitions can be associ-ated to structural rearrangements of spectral features,which now pertain to the excitation spectrum of the sy-stem.

These findings address key practical and conceptualchallenges for the implementation and interpretation ofthe topological mode-selection mechanism to lasers. Onfirst sight it would appear that nonlinearities should de-grade the effectiveness of this mechanism. Even whenstarting under ideal linear conditions, the nonlinearitiesinduce spatially varying loss and gain, which depends onthe intensity profile of the mode across the system. Theresulting effective gain has the potential to disfavour thetopological mode, in particular when its mode volume issmall. Nonlinearities can also induce dispersive effectsthat explicitly break the assumed symmetries. Further-more, even under ideal conditions where all symmetriesare realized exactly, the nonlinearities can render a topo-logical state instable, and result in spontaneous symme-try breaking and nonstationary operating regimes. Con-ceptually, we identify operation conditions that expandthe practical scope of topological quantum devices andutilize nonlinear phenomena that enrich the underlyingtopological physics.

To establish these conclusions we evaluate the nonli-near aspects for a paradigmatic, flexible resonator arran-gement and identify conditions under which topologicallasing is possible. The model and its symmetries areintroduced in Sect. II. Section III describes the modecompetition for the gain and loss distributions for whichtopological mode selection was originally proposed. Wefind that this indeed supports stationary lasing in topo-logical modes over wide ranges of parameter space, butalso opens up a phase transition to an operation regimethat exhibits topologically protected power oscillations.

Section IV describes modified gain distributions and asetup in which the zero mode has a larger mode vo-lume, which both offer additional means to control theoperation conditions. Section V considers the role ofdisorder and symmetry-breaking nonlinearities, which inlarge parts of parameter space turn out to be surprisinglytame, but also can induce phase transitions to additionaloperation regimes. Paired with general considerations onthe stability of nonlinear systems, these results allow usto draw conclusions about the scope of topological effectsin classical nonlinear wave dynamics, which are describedin our concluding Sect. VI.

II. NONLINEAR TOPOLOGICAL LASERARRAYS

A. Modelling laser arrays with saturable gain

The general design of the topological laser arrays stu-died in this work is shown in Fig. 1. The arrays canbe interpreted as chains of identical single-mode resona-tors, denoted by dots, which are coupled evanescentlyto their nearest neighbors. Given this structure of thecoupling it is convenient to divide the system into twoalternating sublattices A and B, and group neighbouringpairs of A and B sites into dimers. Denoting the corre-sponding wave amplitudes on the nth dimer as An andBn, their dynamical evolution is then governed by thecoupled-mode equations

idAndt

= [ωA,n + VA,n(|An|2)]An + κnBn + κ′nBn−1,(1a)

idBndt

= [ωB,n + VB,n(|Bn|2)]Bn + κnAn + κ′n+1An+1,(1b)

where ωs,n (s = A,B) are the bare resonance frequenciesof the isolated resonators, κn is the intra-dimer couplingbetween the A and B site in the nth dimer, and κ′n is theinter-dimer coupling between the B site in the (n − 1)stdimer and the A site in the nth dimer [38]. The effectivecomplex potentials [39]

VA,n(|An|2) = (i+ αA)(

gA1 + SA|An|2

− γA), (2a)

VB,n(|Bn|2) = (i+ αB)(

gB1 + SB |Bn|2

− γB)

(2b)

model nonlinear saturable gain of strength gs and back-ground loss γs, where the real constants Ss and αs are theself-saturation coefficient and the linewidth-enhancement(or anti-guiding) factor, respectively.

3

B. Topological features and mode selection in thelinear regime

For identical passive resonators ωs,n ≡ ωAB with va-nishing gain and loss (gs = γs = 0) and an alternatingcoupling sequence κn ≡ κ, κ′n ≡ κ′, the array is an in-carnation of the celebrated Su-Schrieffer-Heeger (SSH)model [40, 41]. This model displays a symmetric bandstructure with a gap of size ∆ = 2|κ−κ′| around the cen-tral frequency ωAB , which induces two topological phasesα (where κ > κ′) and β (where κ < κ′). At a physicalinterface between these phases one encounters a localizeddefect mode [see Fig. 1(a)] whose frequency Ω0 = ωABis pinned to the centre of the gap. Due to its topologicalorigin this mode persists for more complicated interfaceconfigurations, which we will exploit to change its modevolume as shown in Fig. 1(b). There, the terminatingdimer chains operate as topological mirrors while the de-fect mode extends uniformly over the central part of thesystem.

Exact zero-mode quantization requires that the systemis terminated on a fixed sublattice (here the A sublat-tice), which then contains one more site than the othersublattice (here B), so that overall the system supportsan odd number of modes. An intimately related princi-pal feature of this topological mode is that it only occu-pies the majority sublattice A (so that B = 0), in con-trast to all other states in the system which have equalweight on both sublattices (|A| = |B|; see the Appen-dix for a short proof of these spatial features). Thedefect mode can therefore be addressed by sublattice-dependent gain and loss, which results in a simple androbust mode-selection mechanism that employs the topo-logical origin of the mode. Assuming that the gain andloss are linear (Ss = 0) and do not break the symme-try of the frequency spectrum (αs = 0), the topologicalmode then acquires the effective gain GA = gA − γAon the A sublattice, while all bulk modes acquire theaverage effective gain Ḡ = (gA + gB − γA − γB)/2 (seethe right panels in Fig. 1). The topological protectionpersists because the effective non-hermitian Hamiltonianexhibits a non-hermitian charge-conjugation symmetry(H−ωAB)∗ = −σz(H−ωAB)σz (with the Pauli matrix σzoperating in sublattice space), which stabilizes any com-plex eigenvalues Ωn positioned on the axis Re Ωn = ωAB[26, 27, 42–45].

C. Nonlinear extension of charge-conjugationsymmetry

While this linear mechanism describes an initial com-petitive advantage of the defect mode if GA > Ḡ, andallows it to dynamically switch on if GA > 0 and theintensity is still small, this does not describe the quasi-stationary operation regime where the medium saturatesin response to a much larger intensity. This saturationis critical for the stabilization of any laser at its working

point, where the medium provides just as much energyas the lasing mode loses through radiative and absorptiveprocesses.

The nonlinear modification (2) of the model includesthe saturation dynamics and allows us to address the keychallenges for topological lasing described in the intro-duction. (i) The self-saturation quantified by Ss makesthe effective gain or loss nonuniform across the wholesystem, while at the same time favouring modes witha large mode volume. (ii) The linewidth-enhancementfactor αs induces symmetry-breaking terms in the gainand loss, which we can compare with linear symmetry-breaking via disorder in the bare resonance frequenciesωs,n. (iii) The system admits a dynamical counterpartof the nonhermitian charge-conjugation symmetry thatextends to the nonlinear case [46], described next, whichwill allow us to identify a range of topological and nonto-pological operation regimes.

This dynamical counterpart of the nonhermitiancharge-conjugation symmetry can be phrased in termsof the following general property of the coupled-mode equations (1). For any set of parameters(ωs,n, κn, κ

′n, gs, γs, αs, Ss) and an arbitrarily chosen refe-

rence frequency ωAB , any solution Ψ(t) =

(A(t)B(t)

)can

be mapped onto a solution

Ψ̃(t) = exp(−2iωABt)(

A∗(t)−B∗(t)

)(3)

of the model with parameters replaced by (ω̃s,n = 2ωAB−ωs,n, κn, κ

′n, gs, γs,−αs, Ss). This property turns into a

dynamical symmetry if ωs,n = ωAB , αs = 0 (hence,the same conditions as observed for the non-hermitiancharge-conjugation symmetry in the linear case).

To define the resulting operation regimes we hence-forth set the reference frequency to ωAB ≡ 0, whichcan always be achieved through a gauge transformationΨ(t)→ Ψ(t) exp(−iωABt). We then can distinguish self-symmetric stationary states Ψ(t) = Ψ̃(t) = const(t),which we interpret as nonlinear topological zero modes[Z], time-dependent versions of such self-symmetric states[S], as well as the notable class of twisted modes where

Ψ(t + T/2) = Ψ̃(t) [T]. The twisted modes are auto-matically periodic, Ψ(t + T ) = Ψ(t), and hence lead tostable power oscillations of period T/2. Finally, we canalso encounter stationary and time-dependent lasing mo-des that spontaneously break the dynamical symmetry,which automatically occur in pairs Ψ(t), Ψ̃(t) [P].

To discriminate between these types of modes we uti-lize the correlation functions

C(t) = |〈Ψ(tmax)|Ψ(tmax + t)〉|, (4a)C̃(t) = |〈Ψ̃(tmax)|Ψ(tmax + t)〉|. (4b)

For periodic modes tmax will be chosen such that C(0) =Imax coincides with the intensity maximum over a pe-riod. For stationary modes tmax is arbitrary and Imax

4

(a)

0 0.5γAB

0

0.5gA −

γ AB

T1&2

Z

T1

0

50

0 0.5

gA<γ AB Z T1T1&2

(b)

I A,B

gA − γAB

0

20

0 0.5

Z

T1

T2

(c)

T

gA − γAB0

Imax

0 0.5

Z

T1

T2

(d)t=T/2

t=0

C~ (t)

gA − γAB

FIG. 2. Topological lasing regimes for the SSH array of Fig. 1(a) pumped on the A sublattice (finite gain gA at fixedgB = 0, with amplitudes scaled such that SA = SB = 1) under conditions that preserve the symmetries in the linear case(ωs,n = ωAB , αs = 0), demonstrating operation in topological states over the whole parameter range. (a) Phase diagram ofstable quasistationary operation regimes depending on the gain gA and background losses γA = γB ≡ γAB , where lasing requiresgA > γAB . Over the whole gray region labelled Z, the system establishes stationary lasing in a topological zero mode. In theorange region, this is replaced by operation in a twisted topological mode T1 displaying power oscillations. In the pink regionan additional twisted state T2 exists, whose selection then depends on the initial conditions. The remaining panels analyze thelasing characteristics for varying gain gA along the line γAB = 0.1 (blue arrow in the phase diagram). (b) Sublattice-resolvedintensities IA (red) and IB (blue), including shaded intensity ranges for the power oscillations of T1 and dashed lines indicatingthe corresponding ranges for T2. (c) Amplitude oscillation period T (equalling twice the period of power oscillations for twisted

states, see Fig. 3). (d) Correlation function C̃(t) at t = 0, T/2, where C̃(T/2) = Imax reveals the topological nature of thestates (see text). As illustrated for the examples in Fig. 3, all states inherit the intensity profile of the linear defect mode fromFig. 1(a).

is to be interpreted as the stationary intensity. Self-symmetric modes are characterized by coinciding corre-lation functions, C̃(t) = C(t). For twisted modes thecorrelation functions alternate with an offset T/2, hence

C̃(t) = C(t+T/2) and in particular C̃(T/2) = Imax. Forsymmetry-breaking modes, the two correlation functionsdo not bear any simple relation but are constrained byC̃(t) < Imax for all t.

III. IDEAL TOPOLOGICAL LASING

A. Operation regimes

We first consider lasing under the ideal conditions un-der which topological mode selection was originally con-ceived. This requires laser arrays with exact nonher-mitian charge-conjugation symmetry (ωs,n ≡ ωAB = 0,αs = 0) and gain confined to the A sublattice (gA finiteand variable by the pumping, while gB = 0). We consi-der resonator-independent self-saturation coefficients andscale the amplitudes A and B such that SA = SB = 1.The operation of the system then depends on the balancebetween the gain and the linear background losses, whichwe here assume to be resonator-independent and denoteas γA = γB ≡ γAB .

Figure 2 provides an overview of the resulting opera-tion regimes. Over a large part of the parameter space,the laser operates in a stable zero mode, which is quicklyapproached over time irrespective of initial conditions.In the phase diagram (panel a), this region is indicatedby the label Z. When the gain/loss ratio is increased the

zero mode becomes unstable and is replaced by a twistedmode T1, which results in lasing with power oscillations.As we show in Sec. III B, this change comes about in atopological phase transition. Upon a small further incre-ase of the gain/loss ratio the mode T1 starts to competewith a second twisted mode T2. Both modes sustain sta-ble lasing with power oscillations of different amplitudeand period, where the choice of mode depends on theinitial conditions.

Panels (b-c) in Fig. 2 examine the key characteris-tics of the lasing modes as one varies the gain for fixedbackground losses γAB = 0.1, which covers all describedregimes. The gain-dependence of the sublattice-resolvedintensities IA = |A|2 and IB = |B|2 can be interpretedas light-light curves. IA displays a characteristic kinkas one crosses the laser threshold gA = γAB and entersstationary operation in the zero mode Z, while IB ini-tially remains negligible. Upon increasing the gain, thestationary operation regime is replaced by lasing in thetwisted mode T1, which from its onset displays a finiteperiod T while the amplitude of its power oscillations (ofperiod T/2) increase smoothly (see the shaded intensityranges). The second twisted mode sets in with a slightlysmaller period, but covers very similar intensity ranges(indicated by the dashed white lines).

Panel (d) in Fig. 2 verifies the symmetry-protectednature of these states throughout the whole range of gain.For the self-symmetric zero mode Z, this is evidenced byits characteristic property C̃(0) = Imax. The twisted

modes are not self-symmetric, C̃(0) < Imax, but display

their hallmark property C̃(T/2) = Imax.

These features are further corroborated by the exam-

5

0

10

0 50

I A,B

t

0

40

I A,B

0

40

(b)

I A,B

0

10

0 50

C,C~

t

0

40

C,C~

0

40

(c)

C,C~

Z

T2

T1

(a)

FIG. 3. Topological wave features of representative lasingstates at parameters indicated by blue dots in Fig. 2 (gA = 0.2for Z, gA = 0.5 for T1 and T2). (a) Intensity distributionsover the array, shown both as spikes and as disks with areaproportional to intensity, substantiating the relation of thesestabilized states to the linear defect state from Fig. 1(a). (b,c)Time-dependence of the sublattice-resolved intensities IA(t)and IB(t) (red and blue) and of the correlation functions C(t),

C̃(t) (orange and brown). The alternating correlations C̃(t+T/2) = C(t) verify the twisted nature of the states T1 and

T2, while C(t) = C̃(t) = const verifies that the state Z is atopological zero mode.

ples of modes shown in Fig. 3. As shown in panel (a), allmodes clearly inherit their profile from the linear defectstate of Fig. 1(a). For the stationary mode, the inten-sity IB on the B sublattice is vanishingly small. For thetwisted modes, IB is of the order of the associated poweroscillations. Note that the intensities on both sublatticesoscillate out of phase (see panel b), and so do the cor-

relation functions C(t) and C̃(t) (panel c), as requiredby their twisted nature. Furthermore, the period T/2 ofthe power oscillations in IA,B(t) is indeed half of thatof the amplitude oscillations exhibited by the amplitudecorrelation functions.

B. Topological excitations and phase transition

The fact that we find a well-defined set of lasing mo-des professes that the described topological states arevery stable, at least as long as one stays away from thephase transition between the operation regimes. Thiscan be ascertained by a linear stability analysis, whichresults in a complex Bogoliubov excitation spectrum ωn.For a time-periodic mode, a similar analysis can be car-ried out based on a Bogoliubov-Floquet propagator Fof excitations over an oscillation period, whose eigenva-lues are written as λn = exp(−iωnT ). These excitationspectra reveal intriguing topological features, which areillustrated in Fig. 4 for the three example modes of Fig. 3,and in Fig. 5 at the phase transition between the opera-tion regimes. We here describe the resulting phenome-nology, while the technical details are recapitulated inthe Appendix. The key feature of this discussion is the

-0.1

0

-2 2

Z

ω-

ω+

Im ω

Re ω

T2

λ0,t

T1

λ0,t

λ′0 λ′tλ′0 λ′t

FIG. 4. Stability excitation spectra of the representativestates illustrated in Fig. 3. For the stationary state Z this re-presents the Bogoliubov spectrum ω, which separates into ex-citations ω± that preserve or break the symmetry. This sepa-ration further verifies its zero-mode character (see text), whileImω < 0 [apart from the U(1) Goldstone mode at ω = 0] af-firms that the state is stable. For the periodically oscillatingstates T1 and T2, this represents the Bogoliubov-Floquet sta-bility spectrum λ (top, green) and the spectrum λ′ of thehalf-step propagator (bottom, red). Both spectra are confi-ned by the unit circle in the complex plane, demonstratingthat these states are stable. The symmetry-protected excita-tions pinned to λ′ = ±1 further verify the twisted nature ofthese states.

concept topological excitations that are pinned to sym-metry protected positions, in analogy to Majorana zeromodes in fermionic systems with charge-conjugation sym-metry [1, 2] and zero modes in periodically driven systems[47, 48].

As the coupled-mode equations describe the dynamicsof a complex wave, the stability spectra contain twiceas many excitations as there are states in the linear sy-stem. These excitations are constrained by an indepen-dent spectral symmetry, forcing them to obey Reωn = 0or to occur in pairs ωn, ω̃n = −ω∗n (equivalently, theBogoliubov-Floquet eigenvalues λn are either real or formcomplex-conjugated pairs λ̃n = λ

∗n). A state is sta-

ble if all physical excitations decay, Imωn < 0 (hence|λn| < 0). An exception is the U(1) Goldstone mode pin-ned at ω0 = 0 (λ0 = 1), which arises from the arbitrarychoice of the global phase of the wavefunction Ψ. Thisphase can diffuse due to quantum noise, which results inthe finite linewidth of the emitted laser light. Further-more, in the Floquet case an additional pinned eigenvalueλt = 1 arises from the arbitrary choice of the referencetime t0 for any solution Ψ(t + t0). Let us now examinehow this general picture is modified by topological exci-tations.

As illustrated in Fig. 4, the symmetries of the to-pological states allow us to systematically deconstructtheir excitation spectrum. For the stationary zero modeZ, we can distinguish excitations that preserve the self-symmetry, denoted as ω+,n, from excitations that breakthe self-symmetry, denoted as ω−,n. The latter containthe Goldstone mode ω−,0 = 0, and in our setting des-cribe the more slowly decaying excitations. Notably, inthe considered system both sets of spectra contain an odd

6

-0.1

0

-2 2

Z side of transitionω−,★ω~−,★

ω0λ=e−iωT

Im ω

Re ω

T1 side

λ−,★

λ~−,★λ0

away

λt

λf λ0

FIG. 5. Topological phase transition between the zero-mode regime Z and the twisted mode T1, at gA = 0.291 alongthe line γAB = 0.1 (see Fig. 2). At the transition two Bogo-liubov excitations ω−,? = 2π/T and ω̃−,? = −ω−,? are mar-ginally stable, where T is the period of the emerging twistedmode T1. Along with the U(1) Goldstone mode, they all maponto Floquet-Bogoliubov excitations λ = 1 for this emergingmode. Away from the transition, these excitations split intotwo degenerate excitations λ0 = λt = 1 associated with theU(1) and time translation freedoms, and a decaying excita-tion λf related to the amplitude stabilization of the poweroscillations. (Note that at the transition another pair of exci-tations is almost unstable, which will give rise to the twistedmode T2.)

number of excitations (equalling the number of resona-tors in the laser array).

As further illustrated in the figure, for the twisted sta-

tes T1 and T2 we can relate the eigenvalues λn = λ′n2

tothe eigenvalues of a twisted half-step propagator F ′ thatcharacterizes propagation of excitations over half a pe-riod T/2 (see the Appendix for the exact definition of thispropagator). This reduced spectrum contains a modepinned at λ′t = 1, which arises from time-translation in-variance, and a mode pinned to λ′0 = −1, which origina-tes from the U(1) Goldstone mode. This configuration ofexcitations for propagation over half a period constitutesa distinctive topological signature of the twisted modes.

The described features become are further illumina-ted when one inspects the phase transition between thezero-mode regime and the twisted state T1. In the ge-neral setting of nonlinear optical systems [49, 50], thistransition corresponds to a Hopf bifurcation, which herehowever occurs in a symmetry-constrained setting. Fi-gure 5(a) shows the Bogoliubov spectrum at the transi-tion, where a pair of symmetry-breaking excitations withω̃−,? = −ω−,? crosses the real axis and thereby destabi-lizes the zero mode. This pair of excitations combinesto display the oscillatory time dependence of the emer-ging twisted state T1, whose initial oscillation frequencyis given by 2π/T = |ω−,?|. Different combinations ofthese two excitations amount to a time translation ofthese resulting oscillations. Notably, at the transition theBogoliubov-Floquet spectrum of this emergent state isgiven by λn = exp(−iωnT ), as is illustrated in Fig. 5(b).

Note that upon this mapping the destabilizing exci-tations ω̃−,? = −ω−,? map to λ−,? = λ̃−,? = 1. Forthe twisted mode, they therefore constitute two excitati-ons that right at the transition are both degenerate withthe U(1) Goldstone mode. Departing from the transition

0 0.5γAB

0

0.5

gA −

γ AB

T1,2

ZP 0

30

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0C~ (t)

gA − γAB

0

20

0 0.5

T

gA − γAB

0

4

0 50

I A,B

t

0

20

I A,B

0

20

I A,B

0

30

I A,B

0

4

0 50

C,C~

t

0

20

C,C~

0

20

C,C~

0

30

C,C~

-0.1

0

-2 2

ω-

ω+Im

ω

Re ω

λ′ λ

λ′ λ

λ

Z

T2

T1

P

FIG. 6. Role of reduced gain imbalance, obtained underthe same conditions as in Figs. 2-4 (see also Fig. 14), but withfinite gain gB = 0.1 on the B sublattice. For γAB < gB theparameter space now also contains a region (dark orange) sup-porting additional pairs of symmetry-breaking modes P. Asillustrated for the marked example, these modes have substan-tial weight on the B sublattice, while their independent corre-lation functions C(t) and C̃(t) show that they spontaneouslybreak the symmetry. For such modes the Bogoliubov-Floquetspectrum contains many eigenvalues close to the unit circle,indicating their high sensitivity under parameter changes. Asshown in the top panels for the cross-section now placed atγAB = 0.2, the remaining parameter space supports the samerobust topological lasing modes as observed for gB = 0 (twis-ted modes T1 and T2 and stationary topological modes Z, asillustrated by the marked examples).

into the twisted-state regime [Fig. 5(c)], these excitati-ons split into two separate real eigenvalues λt and λf .Of these, λt describes the time-translation freedom andtherefore remains degenerate with the U(1) Goldstonemode. The eigenvalue λf , on the other hand, is asso-ciated with perturbations of the finite amplitude of thepower oscillations. These perturbations decay due to thenonlinear feedback, so that |λf | < 1, guaranteeing thatthe oscillations are stable. This mechanism gives riseto the aforementioned topological excitations λ′0 = −1,λ′t = 1 in the half-step propagator, which remain a ro-bust signature of the twisted state even when one movesfar away from the transition, as we already have seen inthe examples of Fig. 4.

IV. MODIFIED OPERATION CONDITIONS

To verify the versatility and resilience of the laser arraywe consider two ways to modify the mode competitionbetween the different states in the system. To facilitatethe comparison with ideal conditions, Fig. 14 in the Ap-pendix provides a condensed summary of Figs. 2, 3 and4.

7

0 0.5γAB

0

0.5gA −

γ AB

T1,2

Z

T2 only

T1 only

0

50

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0C~ (t

)

gA − γAB

0

20

0 0.5

T

gA − γAB

0

10

0 50

I A,B

t

0

50

I A,B

0

50

I A,B

0

10

0 50

C,C~

t

0

50

C,C~

0

50

C,C~

-0.1

0

-2 2

ω-

ω+Im

ω

Re ω

λ′ λ

λ′ λ

Z

T2

T1

FIG. 7. Role of increased mode volume, obtained for thelaser array with topological mirrors illustrated in Fig. 1(b).Here we consider ideal lasing conditions with variable gaingA and background loss γA = γB ≡ γAB , at vanishing gaingB = 0 on the B sublattice. The representation of the data isthe same as in Fig. 6. The resulting operation regimes closelyresemble those of the SSH laser array under correspondingconditions (cf. Figs. 2-4, summarized in Fig. 14), with a phaseof stationary zero-mode lasing supplemented by phases withone or two twisted modes displaying power oscillations. Theintensities of these modes have increased, which reflects theirlarger mode volume, as illustrated in more detail for the threeexamples marked Z, T1 and T2.

A. Modified gain distribution

Figure 6 examines the role of the gain distribution viathe addition of finite gain gB = 0.1 on the B sublattice,which amounts to a reduction of the gain imbalance. Inthe linear model, the additional gain does not affect thedefect state, which sees the effective gain GA, but incre-ases the effective gain Ḡ of all the other states in thesystem (see Fig. 1). In the nonlinear model, the additi-onal gain modifies the operation regimes in parts of theregion γAB < gB , where the losses are not strong enoughto suppress modes with substantial weight on the B su-blattice. Besides additional twisted modes, this regionthen become populated by oscillating pairs of symmetry-breaking modes P. As shown for an example in the figure,these modes extend over the whole system and displaysubstantial weight on both sublattices. The Bogoliubov-Floquet spectrum of any two partner modes are identical,but they cannot be further deconstructed as for the to-pological states. The position of the eigenvalues closeto the unit circle reflects a reduced robustness of thesesymmetry-breaking modes against parameter variations.

In the remainder of parameter space we encounter thesame topological operation regimes as in the ideal case,with the boundary between zero modes and twisted mo-des now shifted to larger losses. The modes themselvesdisplay the same features as before, as illustrated for va-riable gain gA along the line γAB = 0.2. The thresholdto stationary lasing again gives rise to a marked increaseof intensity on the A sublattice, while the power oscil-

0 0.5γAB

0

0.5

gA −

γ AB

T1,2

ZP 0

40

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0C~ (t)

gA − γAB

0

20

0 0.5

T

gA − γAB

0

5

0 50

I A,B

t

0

30

I A,B

0

30

I A,B

0

40

I A,B

0

5

0 50

C,C~

t

0

30

C,C~

0

30

C,C~

0

40

C,C~

-0.2

0

-2 2

ω-

ω+Im

ω

Re ω

λ′ λ

λ′ λ

λ

Z

T2

T1

P

FIG. 8. Interplay of mode volume and gain imbalance.Same as Fig. 7, but for finite gain gB = 0.1 on the B sublat-tice, and the cross-section through parameter space shiftedto γAB = 0.2. Compared to the corresponding conditions inthe SSH laser array (Fig. 6), a larger range of parametersnow supports a multitude of additional states. At the repre-sentative point marked P, this includes a pair of symmetry-breaking oscillating states, whose power oscillations are mo-dulated. The features of these symmetry-breaking states arenot very robust, as indicated by their Bogoliubov-Floquet sta-bility spectra, which display many slowly decaying excitati-ons. These modifications are restricted to the range of pa-rameters that previously displayed the twisted states T1 andT2 (now only seen for large enough gain), but does not affectthe operation in the zero-mode Z. Along the cross-sectionγAB = 0.2, we enter only briefly enter this modified regime,in a region where there is only one extra, twisted, state, whichdestabilizes the zero mode.

lations of the twisted states at larger gain display verysimilar periods and relative amplitudes as before. Thethree marked examples verify that these topological mo-des still inherit their mode profile from the linear defectstate, and display the required topological correlationsand excitations that can only change in phase transiti-ons.

B. Modified mode volume

Figures 7 and 8 examine the modified set-up ofFig. 1(b), where the defect region is extended. In thelinear system, the terminating regions act as selectivemirrors for a zero mode with an increased mode volume,which remains confined to the A sublattice. Moreover,because of its increased length the system also supportsa larger number of extended states that compete for thegain. In Fig. 7 the gain on the B sublattice is set togB = 0, while in Fig. 8 we have gB = 0.1.

In the ideal case gB = 0 (Fig. 7), the resulting opera-tion regimes closely resemble those of ideal lasing in theSSH laser array (Figs. 2-4). The parameter space is divi-

8

ded into a region with a topological zero mode Z and re-gions with one or two twisted modes T1 and T2. Each ofthese modes can now be involved in the topological phasetransition with the stationary zero mode, with a crosso-ver point gA ≈ 0.59, γAB ≈ 0.17. The modes continue toshow all the required topological signatures in their cor-relation functions and stability excitation spectra. Ho-wever, they all now display a larger mode volume, whichis inherited from the profile of the zero mode in the linearcase (cf. Fig. 1(b)). As a consequence, the output powerof these modes (quantified by the intensities IA and IB)has increased.

Compared to the situation in the SSH laser array inFig. 6, the modification of the gain imbalance examinedin Fig. 8 now affects a much larger range of parameters,reaching up to γAB . 2gB . This can be attributed notonly to the larger number of competing states, but alsoto the larger propensity of the zero mode to hybridizewith such states in the central region, which on its ownwould constitute a topologically trivial system. In thisregime we indeed encounter a very large number of ad-ditional solutions, which are all close to instability andtherefore very sensitive to parameter changes, as demon-strated by the Bogoliubov-Floquet spectrum of the statemarked P. Furthermore, an additional twisted mode ap-pears close to the phase boundary of the zero mode, andindeed drives its instability along parts of this boundary(see the properties of the modes along the cross sectionat γAB = 0.2). In the remaining range of parameters, thesystem operates in analogous ways as before, with topo-logical modes that display a larger output power whencompared to the SSH laser array with analogously redu-ced gain imbalance (Fig. 6).

V. ROBUSTNESS OF OPERATIONCONDITIONS

Typical bosonic systems are subject to fabricationimperfections and residual internal and external dyna-mics, which may or may not break the assumed sym-metries. For our laser arrays (1) with saturable gain(2), these deviations manifest themselves as linear sta-tic perturbations in the bare resonator frequencies ωs,nand the couplings κn, κ

′n, and the symmetry-breaking

nonlinearities quantified by the linewidth-enhancementfactors αs. We therefore consider the case of couplingdisorder (with perturbations κn = κ̄(1 + Wrn), κ

′n =

κ̄′(1 + Wr′n)) and onsite disorder (with perturbationsωA,n = ωAB + Wrn, ωB,n = ωAB + Wr

′n), where rn,

r′n are independent random numbers uniformly distribu-ted in [−1/2, 1/2], and compare the effects with the caseof a finite linewidth-enhancement factor αA = αB .

(a)

0 0.5W

0

0.5

gA −

γ AB

T1&2

ZT1

T1&

2′P

1&2′

T1,P2′ (b)

0 0.5W

0

0.5

gA −

γ AB

T1&2

Z

T2

T1′&2 P1′

T2,P1′

T1

(c)

0 0.5W

0

0.5

gA −

γ AB

T1&2

Z

T1T1′(d)

0 0.5W

0

0.5

gA −

γ AB

T1&2

Z

T2T1

FIG. 9. Disorder-driven phase transitions for the SSH la-ser array as in Figs. 2-4, but with fixed γAB = 0.1 and variablestrength W of coupling disorder. Each panel corresponds toone randomly selected disorder configuration, with perturbedcouplings κn = κ̄(1 +Wrn), κ

′n = κ̄

′(1 +Wr′n) obtained froma fixed realizations of uniformly distributed random numbersrn, r

′n ∈ [−1/2, 1/2]. Zero-mode lasing persists at all disorder

strengths. Twisted states remain robust for weak to moderatedisorder, while phase transitions to other operating regimescan appear when the disorder is very strong.

A. Coupling disorder

As a notable feature, the spectral and nonlinear dy-namical symmetries of the considered laser arrays re-main preserved if all perturbations are restricted to thecouplings. This type of disorder does not affect thesymmetry-protected spectral position of the defect modein the linear model, and also preserves the classificationof topological states in the nonlinear extension with sa-turable gain.

As shown forW = 0.1 in Fig. 15 in the Appendix, smallto moderate levels of coupling disorder have a practicallynegligible effect on the main operation regimes of the la-ser array. Such levels should be easily attainable in manyapplications, as they are well within the requirements toengineer any bandstructure effects in the first place. Onlyat much larger strengths the fundamental effects of dis-order become discernible. As shown in Fig. 9, this canresult in disorder-strength-dependent phase transitionsthat modify the operation regimes in parts of parameterspace, with the details generally depending on the disor-der realization. Here, we have fixed the background lossesto γAB = 0.1, and instead vary the disorder strength forfour fixed, randomly selected coupling profiles. In all ca-ses, new operation regimes emerge only for very strongdisorder W & 0.3 − 0.5, so that the parameter space re-mains dominated by the zero mode and the two twistedstates.

We further illustrate these emerging regimes in Fig. 10,which corresponds to the disorder configuration ofFig. 9(a) with W = 0.5. As seen in this example, cou-pling disorder of this level can make all states visiblyasymmetric and push the power-oscillating twisted stateT1 into regions that previously supported the stationaryzero mode Z, which however still dominates large partsof parameter space. Even though here this twisted statehas a period similar to T2 in the clean case, it traces backto the state T1 when the disorder strength is adiabati-cally reduced. The state labelled T′2, on the other hand,appears in a disorder-strength-dependent phase transi-tion, and therefore cannot be traced back to any state in

9

0 0.5γAB

0

0.5gA −

γ AB

P1,P2′

ZT1,T2′

0

40

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0

P2′

P1

C~ (t)

gA − γAB

0

20

0 0.5

T2′

P1

(P2′aper.)

×2T1

Z

T

gA − γAB

0

5

0 50

I A,B

t

0

20

I A,B

0

40

I A,B

0

40

I A,B

0

20

I A,B

0

5

0 50

C,C~

t

0

20

C,C~

0

40

C,C~

0

40

C,C~

0

20

C,C~

-0.1

0

-2 2

ω-

ω+Im ω

Re ω

λ′ λ

λ

λ′ λ

Z

T1

P1

P2′

T2′

FIG. 10. Effect of strong coupling disorder for the SSHlaser array as in Figs. 2-4, with the disorder configurationof Fig. 9(a) at W = 0.5. For this realization the regime ofzero-mode lasing is slightly reduced in favour of the power-oscillating twisted mode T1, while the twisted state T2 hasbeen replaced by another twisted mode T′2, which appearsin a disorder-strength-dependent phase transition. As gain isfurther increased, T1 undergoes a period-doubling bifurcationto a symmetry-breaking pair of states P1, while T

′2 is replaced

by an aperiodic pair P′2 (for which the Floquet-Bogoliubovstability spectrum is not defined). All modes display visibledistortions of their mode profile, and the symmetry-breakingpairs display noticeable amplitude on the B sublattice.

(a)

0 0.5W

0

0.5

gA −

γ AB

Z

T1&2T2

T1

(b)

0 0.5W

0

0.5

gA −

γ AB

Z

T1&2T2

T1

(c)

0 0.5W

0

0.5

gA −

γ AB

Z

T1&2T1

X1 X2

(d)

0 0.5W

0

0.5

gA −

γ AB

Z

T1&2 T2

T1

FIG. 11. Robustness against onsite disorder in analogyto Fig. 9, but for randomly selected disorder configurationswith perturbed bare frequencies ωA,n = ωAB +Wrn, ωB,n =ωAB +Wr

′n, rn, r

′n ∈ [−1/2, 1/2]. While this type of disorder

breaks the symmetries, the states can typically be trackedto large values of disorder. The mode originating from thezero mode Z persists at all disorder strengths, and at weak tomoderate disorder extends into regions of larger gain. Thishappens at the expense of the originally twisted modes, whichin panel the configuration of (c) are replaced by new power-oscillating modes X1, X2 when the disorder becomes strong.

the clean system. Both twisted states become vulnera-ble to symmetry-breaking instabilities as one approachesconditions where the gain/loss ratio is large, gA � γAB .In the given disorder realization, the twisted mode T1undergoes a period-doubling bifurcation into a pair ofsymmetry-breaking modes P1, which goes along with anoticeable increase of weight on the B sublattice. Thesecond twisted mode T′2 also bifurcates into a pair of

0 0.5γAB

0

0.5

gA −

γ AB

T1,2

ZT 2

0

50

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0C~ (t

)

gA − γAB

0

20

0 0.5

T

gA − γAB

0

10

0 50

I A,B

t

0

40

I A,B

0

40

I A,B

0

10

0 50

C,C~

t

0

40

C,C~

0

40

C,C~

-0.1

0

-2 2

Im ω

Re ω

λ

λ

Z

T2

T1

FIG. 12. Effect of strong onsite disorder in analogy toFig. 10, for the disorder realization of Fig. 11(a) at W =0.5. Even though the disorder breaks the symmetry, all statescan be traced back to their disorder-free predecessors. Thestationary lasing regime originating from the zero mode Zis barely affected. The mode originating from T1 is pushedinto a smaller part of parameter space, so that the instabilityphase transition now involves the modes originating from Zand T2. The power-oscillations of the originally twisted statesare modulated to clearly display the period T of underlyingamplitude oscillations. The mode profiles of all states are onlyslightly distorted.

symmetry-breaking modes, but these turn out to be ape-riodic.

B. Onsite disorder

For onsite disorder, the strict classification of states bysymmetry breaks down, and only the distinction betweenstationary states and power-oscillating states (as well asaperiodic and chaotic states) persists in a precise sense.However, as shown for W = 0.1 in Fig. 16 in the Ap-pendix, the effects of small to moderate levels of onsitedisorder are again barely noticeable, just as in the casefor coupling disorder. Furthermore, as shown in Fig. 11,even for relatively strong disorder the states can typi-cally be traced back to their symmetry-respecting pre-decessors, which allows us to retain the previous label-ling. The disorder tends to expand the regime of stati-onary lasing originating from mode Z at the expense ofthe power-oscillating modes, while only occasionally lea-ding to transitions into new operation regimes. Figure 12illustrates this resilience against strong disorder for thedisorder configuration of Fig. 11(a) with W = 0.5. Forthis disorder configuration the stationary lasing regimeoriginating from mode Z is barely affected. Amongstthe power oscillating states, the mode originating fromT1 is pushed into a smaller part of parameter space, sothat the instability phase transition now involves the mo-des originating from Z and T2. The main visible conse-quence of broken symmetry is a modulation of the power-oscillations, which now acquire the same period T as the

10

0 0.5γAB

0

0.5gA −

γ AB

T1,2

Z

0

50

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0C~ (t

)

gA − γAB

0

20

0 0.5

T

gA − γAB

0

10

0 50

I A,B

t

0

40

I A,B

0

40

I A,B

0

10

0 50

C,C~

t

0

40

C,C~

0

40

C,C~

-0.1

0

-2 2

Im ω

Re ω

λ

λ

Z

T2

T1

FIG. 13. Effect of nonlinear symmetry breaking onthe modes of the SSH laser array shown in Fig. 2-4 (see alsoFig. 14), obtained by setting the linewidth-enhancement fac-tor to αA = αB = 0.5. Most properties of the states are onlyslightly modified. The twisted correlation function C̃(T/2)are slightly smaller than Imax, while small independent modu-lations appear in the time-dependence of C(t), C̃(t). For thestate originating from T2, this results in noticeable modulati-ons of the power oscillations, whose period is doubled. Thereare also noticeable changes in the stability spectra (green),which can no longer be deconstructed as in the case of exactsymmetry.

complex-amplitude oscillations, while the two correlationfunctions C and C̃ exhibit different oscillation amplitu-des. Notably, the spatial intensity profiles of the statesare still only slightly modified—indeed, they are affectedmore weakly than in the case of coupling disorder.

C. Symmetry-breaking nonlinearities

Similarly to the case of weak coupling and onsite dis-order we find that the lasing regimes are also highly resi-lient against realistic symmetry-breaking nonlinearities,giving rise to practically negligible effects for αA = αB =0.1. As shown in Fig. 13, even at much larger symmetry-breaking nonlinearities αA = αB = 0.5 only small mo-difications are observed. The effects of the nonlineari-ties are still small enough to preserve the division intostationary and power-oscillating states, even though thebroken symmetry once more prevents the precise topo-logical characterization of these states. The symmetry-breaking terms again modulate the power oscillations,which is displayed more clearly for the mode originatingfrom T2. The Bogoliubov spectra show that the statesremain highly stable as long as one stays away from theclearly defined phase transitions.

That this robustness persists both for symmetry-breaking onsite disorder and nonlinearities can be attri-buted to the spectral isolation of the defect mode in thelinear model. This isolation suppresses any matrix ele-ments of hybridization with extended modes in a per-turbative treatment. Note that in the linear case, this

spectral isolation is increased by the favourable gain im-balance, as seen from the position of the complex reso-nance frequencies in the Fig. 1. Furthermore, disordercan turn the extended modes into localized ones, therebydecreasing their mode volume.

VI. DISCUSSION AND CONCLUSIONS

The pursuit of topological effects in photonic systems ismotivated by the desire to achieve robust features in ana-logy to fermionic systems, which in the bosonic settingrequires a dedicated effort to evoke the required symme-tries. The concept of a topological laser emerged fromthe realization that anomalous expectation values facili-tate the selection of topological states by linear gain andloss. Our investigation of topological laser arrays showsthat these concepts seamlessly extend to the nonlinearsetting, which accounts for the effects that stabilize activesystems in their quasi-stationary operation regimes. Weuncovered large ranges in parameter space that favourtopological operation conditions, of which we encounte-red two types—stationary lasing in self-symmetric zeromodes, and lasing in twisted states displaying symmetry-protected power oscillations. The topological nature ofthese states can be ascertained by their characteristicspatial mode structure, and on a deeper level by dis-tinctive properties of their correlation functions and li-near excitation spectra. These features also uncover to-pological phase transitions in which zero modes and twis-ted states interchange their stability. Encouragingly, theoperation conditions can be tuned by changing the gainand loss distribution and the mode volume, while remai-ning remarkably robust under weak to moderate linearand nonlinear perturbations, even if these break the un-derlying symmetry.

These findings raise the prospect to explore the muchsimplified topological mode competition in a wide rangeof suitably patterned lasers with distributed gain andloss. The laser arrays considered here and in the expe-riments [35–37] realize the required symmetry by pro-viding two sublattices, a setting that directly extendsto two- and three-dimensional geometries, including sys-tems with flat bands [25]. Alternatively, one may alsoexploit orbital and polarization degrees of freedom insuitably coupled multi-mode cavities, or design photo-nic crystals with an equivalent coupled-mode representa-tion. All these systems then provide topological lasingmodes with highly characteristic spatial and dynamicalproperties, which are stabilized at a working point thatis spectrally well isolated from competing states in thesystem.

We gratefully acknowledge support by EPSRC viaProgramme Grant No. EP/N031776/1 and Grant No.EP/P010180/1.

11

0 0.5γAB

0

0.5gA −

γ AB

T1,2

Z

0

50

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0C~ (t

)

gA − γAB

0

20

0 0.5

T

gA − γAB

0

10

0 50

I A,B

t

0

40

I A,B

0

40

I A,B

0

10

0 50

C,C~

t

0

40

C,C~

0

40

C,C~

-0.1

0

-2 2

ω-

ω+Im

ω

Re ω

λ′ λ

λ′ λ

Z

T2

T1

FIG. 14. Overview of results for the ideal SSH laser,summarizing Figs. 2, 3, and 4, for reference and comparisonwith the condensed figures for other operation conditions inthe main text.

0 0.5γAB

0

0.5

gA −

γ AB

T1,2

Z

0

50

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0C~ (t)

gA − γAB

0

20

0 0.5

T

gA − γAB

0

10

0 50

I A,B

t

0

40

I A,B

0

40

I A,B

0

10

0 50

C,C~

t

0

40

C,C~

0

40

C,C~

-0.1

0

-2 2

ω-

ω+Im

ω

Re ω

λ′ λ

λ′ λ

Z

T2

T1

FIG. 15. Effect of weak coupling disorder on the SSHlaser array, obtained for a representative disorder realizationwith W = 0.1 (see Fig. 10 for the analogous results withW = 0.5). The results are virtually identical to those in theideal system (summarized in Fig. 14).

Appendix A: Summary of results for reference

For reference, Fig. 14 summarizes the results of Figs. 2,3, and 4 for ideal topological lasing in the same format asadopted in the figures for the other operation conditionscovered in this work.

Appendix B: Resilience against weak perturbations

As mentioned in the Sect. V, weak to moderateamounts of disorder have a negligible effect on the opera-tions regimes. This is illustrated for coupling disorder inFig. 15 and for onsite disorder in Fig. 16, where in bothcases W = 0.1.

0 0.5γAB

0

0.5

gA −

γ AB

T1,2

Z

0

50

0 0.5

I A,B

gA − γAB0

Imax

0 0.5

t=T/2

t=0C~ (t)

gA − γAB

0

20

0 0.5

T

gA − γAB

0

10

0 50

I A,B

t

0

40

I A,B

0

40

I A,B

0

10

0 50

C,C~

t

0

40

C,C~

0

40

C,C~

-0.1

0

-2 2

Im ω

Re ω

λ

λ

Z

T2

T1

FIG. 16. Effect of weak onsite disorder on the SSH laserarray for a representative disorder realization with W = 0.1(see Fig. 12 for the analogous results with W = 0.5). As in thecase of coupling disorder (Fig. 15), the results are virtuallyidentical to those in the ideal system (summarized in Fig. 14).

Appendix C: Bogoliubov theory

1. Preparations

In matrix form, the nonlinear evolution equations (1)can be written as

id

dtΨ(t) = HΨ(t) + V [Ψ(t)]Ψ(t), Ψ(t) =

(A(t)B(t)

),

(C1)

H =

(ωA KKT ωB

), V [Ψ] =

(VA 00 VB

), (C2)

where

Knm = δnmκn + δn,m+1κ′n (C3)

represents the couplings, while the resonance frequenciesand nonlinear potentials (corresponding to Eq. (2)) havebeen promoted to diagonal matrices,

ωA,nm = δnmωA,n, ωB,nm = δnmωB,n, (C4)

VA,nm = δnmVA,n, VB,nm = δnmVB,n. (C5)

Stationary states Ψ(t) = exp(−iΩnt)Ψ(n) with real fre-quency Ωn are determined as self-consistent solutions ofthe equation

ΩnΨ(n) = (H + V [Ψ(n)])Ψ(n), (C6)

while general periodic states of period T fulfill

Ψ(T ) = exp(−iϕ)Ψ(0) (C7)

with a real phase ϕ.As in the main text, we set the reference frequency

ωAB = 0 [a finite value can always be reinstated by mul-tiplying any solution by exp(−iωABt)]. The property

(H + V )∗|ωs,n,αs = −σz(H + V )σz|−ωs,n,−αs (C8)

12

with σz =

(1 00 −1

)then results in the mapping of so-

lutions

Ψ̃(t)|−ωs,n,−αs = σzΨ(t)|ωs,n,αs , (C9)

cf. Eq. (3). For αs = 0, ωs,n = 0, this becomes a state-ment for solutions within a fixed set of parameters.

In the purely linear case with effective potentials VA =i(gA − γA) ≡ iGA, VB = i(gB − γB) ≡ iGB , we encoun-ter the conventional non-hermitian charge-conjugationsymmetry σzH0σz = −H∗0 for the linear HamiltonianH0 = H + V [26, 27]. We can then exploit that Kis an (N + 1) × N -dimensional matrix (as there is onemore A site than B sites) to determine one zero mode

with KTA(Z) = 0, B(Z) = 0 [51]. This mode obeysH0Ψ

(Z) = iGAΨ(Z), which above threshold (GA > 0)

describes an exponentially increasing state, signifying thelack of feedback in the linear theory. In this linear case,the extended states still occur in pairs Ψn, Ψ̃n with ge-nerally complex Ωn = −Ω̃∗n, unless Re Ωn = 0, whichdescribes additional self-symmetric states that can occurvia a spectral phase transition [42–45]. Using optical re-ciprocity, H0 = H

T0 , the symmetry-breaking states are

constrained by the condition

0 = Ψ†n(σzH0 +H†0σz)Ψn = (Ωn + Ω

∗n)Ψ

†nσzΨn

= (Ωn + Ω∗n)(|A|2 − |B|2), (C10)

hence |A| = |B|. Furthermore, from

iΨ†n(GA +GB)Ψn = iΨ†n[GA +GB + σz(GA −GB)]Ψn

= Ψ†n(H0 −H†0)Ψn = (Ωn − Ω∗n)Ψ†nΨn, (C11)

we find that they all have the same life time, accordingto Im Ωn = (GA +GB)/2 ≡ Ḡ. This confirms the state-ments in Sect. II B and Fig. 1.

In the nonlinear case, the relation between solutions atfixed parameters applies to stationary zero modes

Ψ(Z) = Ψ̃(Z), (C12)

which now must be stabilized at an exactly vanishingfrequency ΩZ = 0 [see Eq. (C6)], and twisted modes

Ψ(T )(T/2) = Ψ̃(T )(0). (C13)

For both cases, these definitions exploit the U(1) gaugefreedom to multiply any solution by an overall phasefactor exp(iχ). E.g., if a zero mode fulfills Ψ(Z)′ =

exp(−2iχ)Ψ̃(Z)′ then Ψ(Z) = ± exp(iχ)Ψ(Z)′ fulfillsEq. (C12), and the same redefinition applies for a twis-

ted mode Ψ(T )′(T/2) = exp(−2iχ)Ψ̃(T )′(0). Irrespectiveof these redefinitions, zero modes always display a rigidphase difference of ±π/2 between the amplitudes on theA and the B sublattice, while twisted modes always ful-fill Ψ(T ) = Ψ(0), i.e. they are periodic modes (C7) withguaranteed ϕ = 0.

2. Stability analysis

Given a reference solution Ψ(t) of the nonlinear waveequation (C1), we can analyse its stability by adding asmall perturbation

δΨ(t) = u(t)+v∗(t), u =

(uA(t)uB(t)

), v =

(vA(t)vB(t)

),

(C14)and linearizing in u and v. This yields the Bogoliubovequation

id

dtψ(t) = H[Ψ(t)]ψ(t), ψ(t) =

uA(t)uB(t)vA(t)vB(t)

, (C15)with the Bogoliubov Hamiltonian

H[Ψ] =(H + Γ ∆−∆∗ −H∗ − Γ∗

), (C16)

Γ =

(ΓA 00 ΓB

), ∆ =

(∆A 00 ∆B

), (C17)

where

ΓA,nm = δnm(i+ αA)

(gA

(1 + SA|An|2)2− γA

), (C18)

ΓB,nm = δnm(i+ αB)

(gA

(1 + SB |Bn|2)2− γB

), (C19)

∆A,nm = −δnm(i+ αA)SAgAA

2n

(1 + SA|An|2)2, (C20)

∆B,nm = −δnm(i+ αB)SBgBB

2n

(1 + SB |Bn|2)2. (C21)

For a stationary state fulfilling Eq. (C6), we seek so-

lutions of the form us = exp(−iΩnt − ωmt)u(m)s , vs =exp(iΩnt − ωmt)v(m)s (s = A,B), which follow from theeigenvalue equation

ωmψ(m) =

(H[Ψ(n)]− ΩnΣz

)ψ(m) (C22)

where here and in the following we use the Pauli-likematrices

Σx =

0 0 1 00 0 0 11 0 0 00 1 0 0

, Σz = 1 0 0 00 1 0 00 0 −1 0

0 0 0 −1

. (C23)For a periodic state (C7), we first integrate the Bogoli-ubov equation over a period, so that ψ(T ) = U(T )ψ(0),and then introduce the Bogoliubov-Floquet operator

F = exp(iΣzϕ)U(T ), (C24)

whose eigenvalues are denoted as λm = exp(−iωmT ).Here, the shift by the phase factor ϕ plays a similar role

13

as the appearance of Ωn in Eq. (C22). In both cases, asolution is stable if all eigenvalues fulfill Imωm ≤ 0, sothat the associated perturbations do not grow over time.

In general, the Bogoliubov Hamiltonian displays thesymmetry

(H[Ψ])∗ = −ΣxH[Ψ]Σx. (C25)

In the stationary case, this yields a spectrum ωm thatis symmetric under reflection about the imaginary axis,yielding pairs of eigenvalues ωm, ω̃m = −ω∗m and indi-vidual purely imaginary eigenvalues ωm = −ω∗m. Thisincludes a U(1) Goldstone mode

ψ(0) =

(Ψ(n)

−Ψ(n)∗), ω0 = 0, (C26)

which accounts for the free choice of the overallphase factor of a stationary solution. Analogously,the Bogoliubov-Floquet spectrum contains complex-conjugate pairs of eigenvalues λm, λ̃m = λ

∗m and indi-

vidual real eigenvalues λm = λ∗m. This again includes a

U(1) Goldstone mode

ψ(0) =

(Ψ(t)−Ψ∗(t)

), λ0 = 1 (C27)

reflecting the free choice of the overall phase of any solu-tion, and now also a time-translation Goldstone mode

ψ(t) =

(dΨ/dtdΨ∗/dt

), λt = 1 (C28)

that reflects the freedom to displace any solution Ψ(t) intime.

3. Topological modes

To account for the possible symmetries of the nonlinearevolution equation (C1) we adapt the general considera-tions of [46]. The mapping of solutions (C9) amounts tothe property

(H[Ψ])∗|ωs,n,αs = − ZH[Ψ̃]Z∣∣∣−ωs,n,−αs

, (C29)

Z =(σz 00 σz

). (C30)

Along with Eq. (C25), this property dictates that the Bo-goliubov excitation spectra of the two mapped solutionsΨ, Ψ̃ are identical. For αs = 0, ωs,n = 0, we can use thisto further deconstruct the excitation spectra of topolo-gical modes. For zero modes (C12), we can distinguish

symmetry-preserving excitations vA = uA, vB = −uB ,fulfilling

ω+,mu(+,m) = (H + 2Γ− V )u(+,m), (C31)

from symmetry-breaking excitations vA = −uA, vB =uB , fulfilling

ω−,mu(−,m) = (H + V )u(−,m), (C32)

where the latter includes the mode (C26), now expressedas u(−,0) = Ψ(n), ω−,0 = 0.

For twisted modes (C13), we can factorize theBogoliubov-Floquet propagator

F = ZU∗(T/2)ZU(T/2)= ZΣxU(T/2)ΣxZU(T/2)

= F ′2, (C33)

F ′ = ZΣxU(T/2), (C34)

which defines the twisted half-step propagator F ′. Itseigenvalues λ′m determine the stability spectrum asλm = (λ

′m)

2. The U(1) Goldstone mode (C27) fulfillsψ(0)(T/2) = −ZΣxψ(0)(0), so that the associated eigen-value λ′0 = −1, while the time-translation mode (C28)fulfills ψ(t)(T/2) = ZΣxψT (0), so that λ′t = 1.

4. A brief note on time evolution

The Bogoliubov Hamiltonian (C16) also naturally ap-pears in an efficient numerical integration scheme of thenonlinear wave equation (C1). For this we first introducethe wave equation in the doubled space,

iΦ̇ = H0Φ, Φ =(

ΨΨ∗

), (C35)

H0 =(H + V 0

0 −H∗ − V ∗). (C36)

Using the mid-point predictor

Φ(t+ dt) ≈ (1− idtH − idtV [Φ(t+ dt/2)])Φ(t) (C37)

and linearizing in the exact same way as in the stabilityanalysis, we then obtain

Φ(t+ dt) ≈ (1 + iHdt/2)−1[1− i(2H0 −H)dt/2]Φ(t),(C38)

which amounts to a second-order integrator akin to theCrank-Nicolson scheme.

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