Topological Bulk Lasing Modes Using an Imaginary Gauge Field
TTopological Bulk Lasing Modes Using an Imaginary Gauge Field
Stephan Wong and Sang Soon Oh ∗ School of Physics and Astronomy, Cardiff University, Cardiff CF24 3AA, UK (Dated: December 18, 2020)Topological edge modes, which are robust against disorders, have been used to enhance the spatialstability of lasers. Recently, it was revealed that topological lasers can be further stabilized usinga novel topological phase in non-Hermitian photonic topological insulators. Here we propose a pro-cedure to realize topologically protected modes extended over a d -dimensional bulk by introducingan imaginary gauge field. This generalizes the idea of zero-energy extended modes in the one-dimensional Su-Schrieffer-Heeger lattice into higher dimensional lattices allowing a d -dimensionalbulky mode that is topologically protected. Furthermore, we numerically demonstrate that thetopological bulk lasing mode can achieve high temporal stability superior to topological edge modelasers. In the exemplified topological extended mode in the kagome lattice, we show that largeregions of stability exist in its parameter space. Keywords:
I. INTRODUCTION
In an attempt of ultimate control of the flow of light,photonic topological insulators (PTIs) [1] have enabledexciting devices such as unidirectional waveguides andtopological lasers that are robust against perturbationsand defects. In particular, the realization of robust topo-logical optical systems has drawn attention for advancedphotonics by reducing propagation loss in optical de-vices [2, 3], for example, quantum computers [4, 5], pho-tonic neural networks [6] and near-zero epsilon devices [7–9].Recently, considerable effort has been made to studynon-Hermitian PTIs by engaging topological edge modesto enable a lasing regime with optical non-linearity [10,11], distribution of gain and loss [12–15] or non-reciprocalcouplings [16–18]. For example, the one-dimensional(1D) Su-Schrieffer-Heeger (SSH) model [19] has been uti-lized to generate edge states with gain/loss and imple-ment topological lasing devices [11–13]. A cavity made oftopologically distinct PTIs has been proposed to enhancethe lasing efficiency by using unidirectional topologicallyprotected edge modes [15, 16].However, the edge-mode-based topological lasers arenot appropriate for high power lasers due to the local-ized nature of the edge modes. As an alternative, topo-logical bulk lasers have been proposed to achieve broad-area emission by using extended topological modes basedsolely on the parity symmetry at the Γ-point in a two-dimensional (2D) hexagonal cavity [17] or by using animaginary gauge field in a 1D PT -symmetric SSH lat-tice to delocalize the zero-energy boundary mode overthe 1D-bulk [18].Since temporal instability can deteriorate the perfor-mance of lasing devices, it is important to study the dy-namics and the temporal stability of the topological las-ing modes [20]. Indeed, although the spatial stability of ∗ Email: ohs2@cardiff.ac.uk the topological lasing mode is guaranteed based on topo-logical band theory, its temporal behaviour is not neces-sarily stable due to the non-linear nature of the laser rateequation [21].In this work, we generalize the topological extendedmode on the 1D SSH lattice to higher dimensional lat-tices. In particular, we demonstrate a topological ex-tended mode on a 2D bulk by using a kagome lattice witha rhombus geometry and an imaginary gauge field. Thetopological bulk laser is studied with a gain and loss con-figuration similar to the PT -symmetric configurations.We show that the topological extended mode lases andhas large stable regions in its parameter space. We thusdemonstrate that a phase-locked broad-area topologicallasers can be realized in a 2D kagome lattice.The structure of this manuscript is as follows: in sec-tion II, we recall the result in Ref. [18] where a topologicalextended mode is achieved using the 1D SSH lattice [19]along with an imaginary gauge field. Then section IIIgeneralizes this result to higher dimensions. An explicitexample is carried out on the kagome lattice with rhom-bus geometry. Section IV is dedicated to the study of thenon-Hermitian kagome lattice in the active setting wherethe temporal dynamics of the topological protected modeis studied. II. EXTENDED TOPOLOGICAL MODE IN 1DLATTICE
Here, we briefly recall the procedure used to delocal-ize the topologically protected (zero-energy) mode in theSSH lattice, as presented in Ref. [18].The SSH lattice, shown in Fig. 1(a), is a 1D latticemade of an array of N s sites. It has a unit cell composedof two sites ( A and B ) and the lattice is characterized byalternating intra- and inter-unit cell couplings given bythe real scalars t and t , respectively.In the non-Hermitian configurations, where an ex-tended mode has been proposed [18, 22], an imaginary a r X i v : . [ phy s i c s . op ti c s ] D ec (a)(b) A AAB B ...... t t t t (c) FIG. 1. (a) Schematic of a non-Hermitian SSH lattice madeof an array of N s sites. The usual Hermitian SSH latticecorresponds to the case where h = 0. (b) Spectrum of thefinite-size SSH lattice in the Hermitian and non-Hermitiansetting. The lattice starting from a site A and terminating ata site A. (c) Normalized field profile | ψ n | of the zero-energymode from the finite-size SSH lattice in the Hermitian andnon-Hermitian setting. The parameter are chosen such that N s = 19, E A = E B = 0, t = 0 . t = 0 .
06, and h = h . gauge potential, A = − ih e , is introduced. In the pres-ence of gauge field, the Peierl’s phase modifies the hop-ping terms by a factor: e i (cid:82) A · dl , with dl = dx i e i , thedirection of the hopping. The couplings are thereforemodified and become asymmetric. Note that the usualpre-factors are here absorbed in A , or equivalently in h . The coupling constants get a e h factor term whenhopping from left to right, and a e − h factor term whenhopping from right to left. The coupled-mode equationsare then written as follow: i da n dt = E A a n + t e − h b n + t e h b n − , (1) i db n dt = E B b n + t e h a n + t e − h a n +1 , (2)with a n and b n the modal amplitudes on the A and Bsites at the n -th unit cell, respectively. E σ is the on-siteenergy on the site σ . For a finite system, the introduction of the imaginarygauge field will not affect the spectrum [22]. This is madeexplicit here because the system of equations above canbe solved with a suitable gauge transformation: a n = e hn ˜ a n , (3) b n = e hn e − h ˜ b n . (4)where a n and b n are solutions of the coupled-mode equa-tions if ˜ a n and ˜ b n are solutions of the Hermitian SSHcoupled-mode equations, namely when h = 0.For a SSH lattice starting and terminating on an A site,it is known that the zero-energy mode of the HermitianSSH lattice reads: ˜ a n = ˜ r n ˜ a , (5)˜ b n = 0 , ∀ n, (6)where ˜ r = − t t defined as ˜ a n +1 = ˜ r ˜ a n and satisfies thedestructive interference condition on the B sites t +˜ rt =0. The solution for the non-Hermitian SSH lattice is thenwritten as: a n = r n a , (7) b n = 0 , ∀ n, (8)where r = − t t e h defined as a n +1 = ra n and satisfies thedestructive interference condition on the B sites t e h + rt e − h = 0.The main effect of this imaginary gauge field is tochange the localization property of the modes withoutaffecting the spectrum. In particular, one can delocal-ize the topological protected edge mode over the whole1D bulk, while keeping its topological protection fromthe chiral symmetry of the Hermitian topologically pro-tected (zero-energy) mode. The exponentially increasingor decaying factor is removed by appropriately choosingthe gauge field h : h = h := −
12 ln (cid:18) t t (cid:19) . (9)Figure 1(b) shows that the spectrum of the finite-size Hermitian and non-Hermitian SSH lattice are indeedidentical. However, Fig. 1(c) shows, for t < t , that thefield profile | ψ n | of the zero-energy mode, which is local-ized on the left edge for the Hermitian case, is extendedover the 1D bulk for the non-Hermitian case.Finally, it is worth noting that although the zero-energy mode is topologically protected, its localizationproperty depends on the coupling constants, and is there-fore sensitive to their perturbations. However, for rea-sonably small perturbations, i.e. small enough so thatthe band gap does not close, the delocalization is notdestroyed: the amplitudes remain of the same order ofmagnitude over the bulk but are simply not equal any-more. B CA t C t (b) IJJII (a)
IJJII -g-gg t t FIG. 2. (a) Schematic of a d -dimensional lattice in a quasi-1D lattice made of an array of N s ( d − e h , e h (cid:48) , e h (cid:48)(cid:48) correspond to the imaginarygauge field introduced for delocalising the topological modein the non-Hermitian kagome lattice. III. EXTENDED TOPOLOGICAL MODE INd-DIMENSIONAL LATTICE: EXAMPLE ON THEKAGOME LATTICE
We now generalize this notion of delocalized (or ex-tended) topological mode over a whole d -dimensionalbulk. A. General framework
The strategy follows the previous section, namely tofind an exact solution of the (topologically protected)boundary state, then use non-Hermiticity to change thelocalization property of the chosen mode.In order to find an exact solution, the procedure fol-lows Ref. [23, 24]. One needs to consider a d -dimensionallattice as a stack of ( d − e (cid:107) -direction) and with open boundary condition (OBC)in the remaining 1-dimensional ( e ⊥ -direction) bound-ary [23–25] (see Fig. 2 for the example of the kagomelattice). The lattice can then be considered as a quasi-1Dlattice with the unit cell composed of two lattice-sites, I and J , except that here the lattice-sites represent ( d − I lattice-sites [23, 24]. Therefore, these lattices naturally supportthe exact disappearance of the wave-function amplitudeof n wave functions on the J lattice-sites with n the num-ber of degrees of freedom on the I lattice-site.From the quasi-1D formalism, the coupled-mode equa-tion is conveniently written as: i d Ψ dt = H lattice Ψ (10)where Ψ = ( ψ I, , ψ J, , . . . , ψ I,N ) T with ψ I,n and ψ J,n being the modal amplitudes on the I and J lattice-sites in the n -th stacked unit cell, respectively. N is the indexof the last unit cell.The Hamiltonian of the lattice reads: H lattice = H I H I ← J · · · ˜ H † I ← J H J ˜ H † J → I · · · H J → I H I · · · ... ... ... . . . (11)with H I and H J being the Hamiltonian of the lattice I and J , respectively. H I ← J and H J → I are, respectively,the intra- and inter-unit cell couplings between the I and J lattices. For the Hermitian case, ˜ H † I ← J = H † I ← J and˜ H † J → I = H † J → I For the general d -dimensional lattice, the eigenvalueproblem H lattice Ψ = E Ψ yields, for n = 1 , . . . , N : H I,k ψ I,n + H I ← J ψ J,n + H J → I ψ J,n +1 = Eψ I,n , (12) H J,k ψ J,n + H † I ← J ψ I,n + H † J → I ψ I,n +1 = Eψ J,n . (13)The condition for destructive interference on the J -lattices, ψ J,n = 0, is given by: H † I ← J ψ ( i ) I,n + H † J → I ψ ( i ) I,n +1 = 0 . (14)From Eq. 12, the solution with vanishing amplitude onthe J -lattice therefore gives the additional condition: H I ψ I,n = Eψ I,n , (15)namely ψ I,n must be an eigenmode of the Hamiltonianon the lattice I , labelled ψ ( i ) I,n , with corresponding energy E = E ( i ) I .Since we are looking at edge states, one can ask for so-lutions which exponentially decay or increase, or equiva-lently solutions that satisfy [23, 25]: ψ ( i ) I,n +1 = r i ψ ( i ) I,n (16)with r i being a scalar term representing the decayingamplitudes of the mode inside the quasi-1D lattice.The solution of the Hamiltonian H lattice with energy E ( i ) I is therefore of the form: ψ ( i ) I,n = r ni ψ ( i ) I, (17)with r i satisfying the destructive interference conditionon the J lattice: H † I ← J + r i H † J → I = 0 (18)In summary, the modes with eigenenergy E = E ( i ) I ,such that H I ψ ( i ) I = E I ψ ( i ) I , are the modes which are ex-ponentially localized on one edge and with non-vanishingamplitudes only on the lattice-sites I , with adjacent π -phase difference and with mode distribution correspond-ing to ψ ( i ) I on the lattice-site I .The delocalization of the edge modes is realized byintroducing an imaginary gauge field such that | r i | = 1.The non-Hermiticity allows the change of the localizationproperties while keeping the spectrum unchanged. (a)(b)(c) FIG. 3. (a) Spectrum of the kagome lattice in the rhombusgeometry in the Hermitian and non-Hermitian setting. Thenormalized field profile | ψ n,m | of the zero-energy mode of thekagome lattice in the rhombus geometry is plotted in (b) forthe Hermitian case, and in (c) for the non-Hermitian case.Here, there are N s = 11 sites both in the I lattice and thequasi-1D lattice. t = 0 . t = 0 .
06. The Hermitian settingcorresponds to the case h = h (cid:48) = h (cid:48)(cid:48) = 0, whereas the non-Hermitian setting is for h = h (cid:48) = h , h (cid:48)(cid:48) = 0. B. Extended topological mode in 2D kagome lattice
As a concrete example we will look at the case of thekagome lattice as shown in Fig. 2. The kagome latticeis characterized by unit-cells composed of three sites A,B, and C and the coupling strengths between sites aredifferent for intra-unit cell ( t ) and inter-unit cell ( t )couplings.Applying the previous results to the kagome latticein the rhombus geometry (see Fig. 3(b) for the geom-etry of the lattice), we have H I = H SSH , H J = E B .˜ H † I ← J = H † I ← J = ( t , t , . . . , t , t ) T and ˜ H † J → I = H † J → I = ( t , t , . . . , t , t ) T are ( N s × N s is the number of sites on the I lattices. E B is the on-site energy at the B sites.The rhombus geometry is interesting since in the e (cid:107) direction, the I lattices, which are equivalent to the SSHlattice, start with and are terminated by the same site(here site A). In this configuration, the chiral symmetryof the SSH lattice guarantees the presence of the zero-energy mode. Therefore a boundary state of the kagomelattice with eigenenergy E ( i ) I = E (0) I = 0 is topologicallyprotected by virtue of the chiral symmetry in the SSHlattice (lattice I ). The corresponding zero-energy modeof this kagome lattice is written as: ψ (0) I,m = r m , ψ (0) I, (19)with (cid:104) ψ (0) I, (cid:105) n = r n , a , being the n -th component of thezero-energy mode, ψ (0) I, , of the SSH lattice where theinterference conditions (Eq. 18) give r , = − t t , and r , = − t t . a n,m is the modal amplitude on the A siteat the n -th unit cell in the m -th lattice I .Choosing different intra- and inter-unit cell couplingconstants, t < t or t > t , the zero-mode is thenexponentially localized, respectively, on the bottom-leftor upper-right edge of the SSH lattice with vanishingamplitudes on the B and C sites: it is a (topological)corner mode [26–28].Figure 3(a) shows the spectrum of the kagome latticein the rhombus geometry and demonstrates the existenceof the zero-energy mode, as explained previously. Fig-ure 3(b) plots the normalized field distribution | ψ n,m | ofthe zero-energy mode, for t < t . This shows that themode is indeed localized on the bottom-left corner.We use an imaginary gauge field to change the local-ization property of this corner mode [18, 22]. Figure 2(b)sketches the gauge potential considered where e h , e h (cid:48) and e h (cid:48)(cid:48) represent the phase factors in the couplings betweenthe sites A and B, A and C, and B and C, respectively.Conditions on h , h (cid:48) and h (cid:48)(cid:48) are imposed given an imagi-nary gauge field A = ( A , A ). From the Peierls’ phasecorresponding to e h we have A = − ih , with dl = e .Similarly, e h (cid:48) gives A = − ih (cid:48) , using dl = e . Thesetwo conditions on A mean that for e h (cid:48)(cid:48) we must have: h (cid:48)(cid:48) = − h + h (cid:48) , (20)using dl = e − e .With the imaginary gauge field, the cou-pling matrices are then modified as ˜ H † I ← J =( t e h (cid:48) , t e h (cid:48)(cid:48) , . . . , t e h (cid:48) , t e h (cid:48)(cid:48) ) and ˜ H † J → I =( t e − h (cid:48) , t e − h (cid:48)(cid:48) , . . . , t e − h (cid:48) , t e − h (cid:48)(cid:48) ). The interferenceconditions now yield: r , = − t t e h , (21) (a)(b) FIG. 4. Normalized field profile | ψ n,m | of the zero-energymode of the kagome lattice in the rhombus geometry with (a) h = h , h (cid:48) = h (cid:48)(cid:48) = 0 and (b) h (cid:48) = h , h = h (cid:48)(cid:48) = 0. The otherparameters are the same as in Fig. 3. and r , = − t t e h (cid:48) . (22)Delocalization over the e -direction is achieved by re-quiring | r , | = 1, namely choosing h = h . Similarly, de-localization over the e -direction is realized with h (cid:48) = h so that | r , | = 1. One can notice that there is no furthercondition on h (cid:48)(cid:48) to delocalize the mode in the quasi-1Dlattice. This is because of the vanishing amplitudes onthe B and C sites.Figure 4(a),(b) show the normalized field profile | ψ n,m | of the zero-energy mode using h = h , h (cid:48) = h (cid:48)(cid:48) = 0 and h (cid:48) = h , h = h (cid:48)(cid:48) = 0, respectively. In Fig. 4(a), the modeis localized on the bottom edge while being extended overthe e -direction. On the other hand Fig. 4(b) shows thatthe mode is localized on the left edge while being ex-tended along the e -direction. It is worth noting that forthe values of h , h (cid:48) and h (cid:48)(cid:48) chosen for drawing Fig. 4, thespectrum has been changed compare to the Hermitiancase.However, provided Eq. 20 holds, the introduction ofthe imaginary gauge field will only affect the localizationproperty of the mode while keeping the spectrum un-changed. Therefore, in this case, the condition | r ,i | = 1does not correspond to band touching with the edge bandand the bulk band [23].Combining the two results obtained above for the delo-calization of the zero-energy mode, we have h = h (cid:48) = h and, from Eq. 20, h (cid:48)(cid:48) = 0. Figure 3(a) shows that the numerically calculated spectrum of the kagome lattice inthe rhombus geometry doesn’t change when the imag-inary gauge is introduced. Figure 3(c) plots the nor-malized field distribution of the zero-energy mode using h = h (cid:48) = h and h (cid:48)(cid:48) = 0. This topologically protectedzero-energy mode is therefore extended over the wholebulk of the kagome lattice: it is a topological bulk modein the 2D kagome lattice.Generalization to higher dimensional lattices can beachieved using a similar procedure: starting from a topo-logically protected mode in lower dimension to delocalizethis topological mode over that low dimensional bulk,and repeating this step with the higher dimension. IV. LASING IN THE NON-HERMITIANKAGOME LATTICE
The peculiarity of this extended topological mode isits vanishing amplitudes on the B and C sites but mostof all, that it is topologically protected over a 2D bulk,and has a π -phase difference between non-vanishing sites.This hints at the possibility to realize phase-locked broad-area topological lasers in 2D lattices. A. An active and non-Hermitian kagome lattice
The passive design presented above can be extended topresent a topological laser by using semiconductor ringresonators with gain ( g ) and loss ( − g ), and auxiliaryrings for the imaginary gauge field [18].The gain and loss configuration is chosen based onthe SSH sublattices since we are looking for topologicalmodes in the kagome lattice that originate from the topo-logical mode in the SSH sublattice. In the literature, it iswell known that the SSH lattice in a PT -symmetric con-figuration [11, 12] (gain on site A, E A = ig , and loss onsite C, E C = − ig ) can preserve the topological protectionof the zero-energy. Particularly, if the SSH lattice is inthe unbroken PT -symmetric phase, i.e. g < | t − t | , thenthe zero-energy mode is still guaranteed, from its pseudo-anti-Hermiticity [11, 14]. When the energy is complex,we refer to the zero-energy as the real part of the energybeing zero.Since the only non-vanishing terms of the extendedtopological mode are located in the A sites, we expectlasing coming from these sites. Therefore we set gain onsite A, E A = ig , and lossy rings on the B and C sites,( E B = E C = − ig ), as shown in Fig. 2(b).Figure 5(a) shows the spectrum in the complex plane,numerically calculated with E A = ig , E B = E C = − ig , h = h (cid:48) = h and h (cid:48)(cid:48) = 0. This demonstrates that thezero mode is present. Because of the gain on the Asites, i.e. where the zero-energy mode is non-vanishing,the zero-energy mode has higher gain compared to othermodes. Figure 5(b) displays the normalized field pro-file | ψ n,m | of the zero-energy mode. This shows that, (a)(b) FIG. 5. (a) Spectrum of the active kagome lattice in therhombus geometry. (b) Normalized field profile | ψ n,m | of thezero-energy mode. We have N s = 11 sites both in the I latticeand the quasi-1D lattice, t = 0 . t = 0 . g = 0 .
03 and h = h (cid:48) = h , h (cid:48)(cid:48) = 0. although active design has been considered, the delocal-ization of the zero-energy mode is not altered. B. Temporal dynamics of the zero-energy mode
In the frequency analysis, we have seen it is possibleto have an active non-Hermitian kagome lattice with anextended topological mode. It is now interesting to seewhether this mode presents temporal instabilities.The previous analysis provides a simple physical modelof the active non-Hermitian kagome lattice. It has beenshown that temporal instabilities in the laser array mayprevent phase-locking and reduce the laser quality oreven dominate and suppress the topological signature ofthe corresponding lasing mode [18, 21]. Therefore timedomain modelling of the mode dynamics is essential fordetermining whether lasing is stable.We consider the laser rate equation for modelling thegain in the active rings A [18, 29, 30]. Because the zero-energy mode has vanishing amplitude on the ring res-onators B and C, linear loss is chosen for those rings.The laser rate equation in the kagome lattice, with h = h (cid:48) = h and h (cid:48)(cid:48) = 0, is then: i da n,m dt = 1 τ p (1 − iα ) Z n,m a n,m + t e − h b n,m + t e − h c n,m + t e h b n,m − + t e h c n − ,m , (23) i db n,m dt = − ig B b n,m + t e h a n,m + t c n,m + t e − h a n,m − + t c n − ,m +1 , (24) i dc n,m dt = − ig C c n,m + t e h a n,m + t b n,m + t e − h a n +1 ,m + t b n +1 ,m − , (25) τ s dZ n,m dt = p A − Z n,m − (1 + 2 Z n,m ) | a n,m | , (26)where a n,m , b n,m and c n,m are the modal amplitudes onthe site A, B and C and in the ( n, m )-th unit cell, respec-tively. n and m stand for the unit-cell index in the SSHlattice and quasi-1D lattice, respectively. Z n,m is thenormalized excess carrier density in the active ring A, τ p and τ s are the photon and spontaneous carrier lifetimes,respectively, α the linewidth enhancement factor, p A thenormalized excess pump current in the ring A, and g B and g C the linear loss in the ring B and C, respectively.The coupled-mode equations are integrated using ran-dom noise of field amplitudes between [0 , .
01] and equi-librium carrier density Z n,m = p A as initial condition.The random noise as initial condition is chosen to trig-ger non-linear behaviour and see whether or not themode is stable. The parameters are chosen similar toRef. [18, 29, 30] and are typical for semiconductor lasers.Figure 6(a) displays the time evolution of the instanta-neous spectrum of the kagome lattice in rhombus geom-etry. It shows that after a transient time, the systemreaches a single laser mode regime. The laser mode isthe topological zero-energy mode with Im( E ) = 0. Fig-ure 6(b) and (c) display the time series of the field ampli-tudes at all the A sites and the adjacent phase differencebetween the A rings. This shows that after a transientregime, only the zero-energy mode survives and reaches asteady state. The amplitudes of all the A sites are equallydistributed over the bulk and have a fixed π -phase differ-ence. The laser system obtained is therefore broad-areaand phase-locked.In addition to this interesting broad-area and phase-locked feature, the laser mode is topologically protected.Figure 6(d) shows the spectrum of the system whenan asymmetric perturbation on the couplings, δt ± , isadded: t e h → ( t + δt ) e h and t e − h → ( t + δt − ) e − h .This asymmetric perturbation accounts for perturbationin the coupling strength as well as for the imaginarygauge field. One can see that the zero-energy mode isstill present. However one can see in Fig. 6(e) and (f) (a)(b)(c) (e)(f)(d) FIG. 6. Time evolution of (a) the instantaneous spectrum of the kagome lattice in rhombus geometry (b) the amplitudesof all the A sites, and (c) the phase differences between the adjacent A sites when there is no disorder. Similarly for (d),(e) and (f) when asymmetric disorders is introduced. The parameters used are: τ p = 40ps, α = 3, τ s = 80ns, p A = 0 . g B τ p = g C τ p = 0 . δ B τ p = δ C τ p = 0 . t τ p = 0 . t τ p = 0 . a slightly different behaviour in the time series of thefield amplitudes and the phase difference between the Arings. They do not reach the same values in amplitudesand phase differences. The amplitudes are not equallydistributed but have slightly different offsets because ofthe different couplings between each sites: a single choicefor the imaginary gauge field h and h (cid:48) cannot satisfy the | r ,i | = 1 conditions between each sites. Similarly, thephase differences are no longer equal to π due to the non-linear Henry factor α . Nevertheless, here, the additionof perturbation in the coupling strengths does not giverise to unstable behaviour in the amplitudes and phasesof the topological extended mode.With the parameters chosen in Fig. 6, the system reaches a stationary state which does not evolve intorandom oscillation in their amplitudes and phase differ-ences. This means that the zero-energy mode does notsuffer from non-linear instabilities. Even though the spa-tial stability of the topological mode is guaranteed by itstopological invariant, it is worth looking at its tempo-ral stability in the parameter space to delimit the regionwhere temporal instabilities arise due to the non-linearterms. In the following, we will refer to the stable regime,the regime of single mode lasing in the topological ex-tended mode. Therefore, we say the system to be unsta-ble (or stable) if oscillations are present (or not) in theiramplitudes or phase differences.Figure 7 shows the stability diagram of the topological (c) (b)(d)(a) FIG. 7. Stability diagrams of the topological extended mode. The color plot corresponds to the probability of the systembeing in the stable regime over 200 realizations in the random initial conditions of the mode amplitudes. The fixed parametersare the same in Fig. 6. extended laser mode for different slices of the param-eter space. These demonstrate stable phase-locking ofthe non-Hermitian gauge laser array in a relatively largeregion of the parameter space. Numerical results showthe stability of the topological extended mode requires aminimum coupling strength (Fig. 7(a)-(b)). One reasonis that the PT -symmetric phase is broken when the cou-plings are too small compared to the gain and loss [11]:the system is no longer in the single lasing mode regime.The second reason is that the non-reciprocal dissipativecouplings need to be high enough in order to reach a(soft) synchronization [31]. On the other hand, insta-bilities arise when the detunings δ b and δ c are too high(Fig. 7(c)). The major critical case is with positive de-tunings (Fig. 7(c)-(d)). However, stability is retrieved ifthe dissipation is large enough to compensate the detun-ings in the rings B and C.Extended lasing modes present an important advan-tage in getting a better slope efficiency, compared to thecompact lasing mode. Figure 8 plots the total intensity, I = (cid:80) m,n | ψ n,m | against the pump amplitudes p A forthe compact and extended lasing mode where the colorplot corresponds to the system being in the stable or un-stable regime. The numerical results show that the com-pact mode has lower slope efficiency compared to the ex- FIG. 8. Total intensity against pump amplitudes p A for thelocalized and delocalized topological mode with N s = 11. Thetriangles (circles) correspond to the topological edge (bulk)laser. The colors plot indicate if the system in the unstable(blue) or stable (red) regime. The inset shows the dependencyof the slope efficiency with respect to N s . The parameters arethe same as in Fig. 6. tended mode. The remarkable difference is the scaling ofthe slope efficiency with the size of the system for the de-localized mode while it remains constant for the compactmode, as shown in the inset of Fig. 8. This is because ofthe extended nature of the mode whose contribution tothe lasing intensity increases with the size of the system.Clearly, the imaginary gauge field helps to stabilise thesystem in the zero-energy lasing mode. The extendednature of the topological mode over the bulk allows thezero-energy mode to carry all the gain of the system whilesuppressing all the other bulk modes. The color plot inFig. 8 indicates that the compact mode reaches an un-stable regime for relatively low values of pump intensitywhereas the extended mode is unstable for higher pumpintensities. The stability diagrams for the compact modeare shown in Fig. 9(a)-(c). Compared to the extendedmode in Fig. 9(b)-(d), these diagrams demonstrate thatextended modes have, indeed, larger regions of stabilityin the parameter space than compact modes.Figure 10 demonstrates the slope efficiency is robustagainst asymmetric disorder in the couplings, for boththe compact and extended mode, as expected the topo-logical nature of the lasing mode. The compact modehas a constant slope efficiency with increasing disorderstrength. While the extended mode gives varying slopeefficiency because of varying field distribution of the ex-tended mode in the bulk as explain before, its slope ef-ficiency remains higher compared to the compact mode.However, for high values of disorder strength in the cou-plings, the slope efficiency of the extended mode startsto decrease. This is explained because the delocalizationof the topological mode, originating from non-reciprocalcouplings, is highly perturbed by the asymmetric pertur-bation δt ± in the couplings: the topological mode maynot be completely delocalized anymore. V. CONCLUSION
To summarize, we have shown a procedure to get topo-logical modes extended over a d -dimensional bulk usingan imaginary gauge field. In particular, we have demon-strated the existence of a topological extended mode inthe kagome lattice in the rhombus geometry. This topo-logical extended mode in the kagome lattice has beenstudied in the context of a lasing system where the laserrate equation is included to take into account non-lineareffect. Investigating its temporal stability, we proved thatstable topological broad-area phase-locked laser opera-tion is possible in a large region in the parameter space.Furthermore, it has been shown that the topologicalextended mode presents clear advantages over the topo-logical localized mode. The extended nature of the for-mer topological mode over the bulk enhances its temporalstability, and yields higher slope efficiency that scale withthe size of the system. In terms of footprint, a higherdimensional extended mode is also more advantageouscompared to their lower dimensional one since at equiv-alent slope efficiency, for example, the system occupies asmaller region in real space when increasing the dimen-sionality of the system. This can lead to applicationswhere transport of high energy density is possible. ACKNOWLEDGMENTS
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