Probing topological phases in waveguide superlattices

TL;DR

This paper introduces a general bulk probing method for topological invariants in waveguide superlattices using spatial beam displacement, enabling the detection of topological edge states without relying on Wannier functions.

Contribution

It proposes a novel bulk probing technique for topological properties in superlattices, applicable beyond simple dimeric systems, and introduces a simplified method for quadrimeric superlattices.

Findings

Bulk probing method accurately measures topological invariants.

Method distinguishes topological edge states from non-topological states.

Applicable to complex superlattice structures.

Abstract

One-dimensional superlattices with modulated coupling constants show rich topological properties and tunable edge states. Beyond the dimeric case, probing the topological properties of superlattices is a challenge. Here we suggest a rather general method of bulk probing topological invariants in waveguide superlattices based on spatial displacement of discretized beams. A judiciously tailored initial beam excitation of the lattice, corresponding to superposition of Wannier functions, provides a direct measure of the band gap topological numbers. For a quadrimeric superlattice, a simple bulk probing method, which avoids Wannier states, is proposed to discriminate the existence of zero-energy topological edge states from non-topological ones

Equations

$$\hat{H}=\sum _{n=1}^N\left\{\left(\sum _{l=1}^{M-1}{t}_l{\hat{a}}_{n,l}^{†}{\hat{a}}_{n,l+1}\right)+\tau {\hat{a}}_{n,M}^{†}{\hat{a}}_{n+1,1}+H.c.\right\}$$

$${\mathcal{H}}_{n,m}(k)$$

$${\mathcal{N}}_n={\gamma }_1+{\gamma }_2+...+{\gamma }_n$$

$$a_n(z)=\sum _{l=1}^M{\int }_{-\pi }^{\pi }dk{C}_l(k)u_l(k)\exp [ikn-i{E}_l(k)z]$$

$$C_l(k)=(1/2\pi )\sum _n⟨u_l(k)|a_n(0)⟩\exp (-ikn)$$

$$\mathcal{D}(z)=(1/z){\int }_0^{z}d\xi \sum _{n=-\infty }^{\infty }n⟨a_n(\xi )|a_n(\xi )⟩$$

$${\mathcal{D}}_{as}=2\pi i\sum _{l=1}^M{\int }_{-\pi }^{\pi }dk\left(C_l^{\ast }(k)\frac{d{C}_l}{dk}+|C_l(k){|}^2⟨u_l(k)|\frac{d{u}_l}{dk}⟩\right).$$

$${\mathcal{D}}_{as}=(1/2\pi S)\sum _{l=1}^S{\gamma }_l={\mathcal{N}}_S/(2\pi S)$$

Full text

Introduction. In recent years, topological photonics has emerged as a new paradigm to robustly manipulate light propagation under continuous deformations and to emulate in optics topological phases of matter (see r1 ; r2 and references therein). Topological phases are described by global invariants, such as the Zak phase and Chern numbers. Since topological invariants describe non-local properties of the system, ways to relate topological invariants with measurable quantities is of major relevance both theoretically and experimentally. The bulk-boundary correspondence r3 ; r4 , relating the bulk topological invariants with the number of edge states in finite systems with open boundaries, provides the simplest route to measure topological numbers r4bis ; r5 ; r6 ; r7 ; r7bis ; r8 . However, edge states without topological protection could coexist with topological ones, making edge measurements ambiguous in certain cases. Bulk probing of topological invariants has been proposed and demonstrated in a series of experiments with synthetic matter, such as those based on Bloch oscillations r9 ; r10 , unitary or non-unitary quantum walks (QWs) r11 ; r12 ; r14 ; r15 ; r16 ; r17 ; r18 ; r19 , and topological pumping r20 ; r21 . In one-dimensional (1D) systems the bulk topological properties of the Bloch bands are characterized by the quantized Zak phase r22 ; r23 ; r24 ; r25 ; r26 . In two-band systems, such as in celebrated Su-Schrieffer-Heeger (SSH) model r3 , beam displacement in a photonic QW provides a simple and powerful approach to detect the Zak phase in the bulk r15 ; r16 ; r17 ; r18 ; r19 . Multi band superlattices show richer topological features than two-band models, such as the ability to tune the number of topological edge states by controlling the inter-dimer coupling r26 . In such systems, topological edge states can coexist with non-topological ones r25 , making topological phase characterization of the edge ambiguous. Recently, the displacement method r17 has been extended to measure a global winding number in chiral superlattices r27bis , however an open question is whether photonic QWs can provide a measure of topological numbers in each gap of the superlattice or can be used when the system does not show chiral symmetry.

In this Letter it is shown that photonic QWs of judiciously-tailored discretized optical beams in waveguide lattices, corresponding to a superposition of Wannier functions, provide a general route to measure the Zak phase of various superlattice bands. In case of a quadrimeric superlattice, a simple setup is suggested to detect particle-hole zero-energy modes r27 and to discriminate them from non-topological edge states.

Topological properties of superlattices and edge states. We consider a 1D waveguide superlattice with open boundary conditions comprising $N$ unit cells, each unit cell hosting $M$ equal waveguides, with inter-cell coupling constant $\tau$ and intra-cell couplings $t_{1},t_{2},...,t_{M-1}$ (Fig.1). The different coupling constants can be readily controlled by adjusting waveguide spacing (see, for instance, r28 ; r29 ). The Hamiltonian of the system reads

[TABLE]

where $\hat{a}^{{\dagger}}_{n,l}$ is the photon creation operator at the $l-th$ waveguide of the lattice in the $n-th$ unit cell. For $M=2$, this model reproduces the celebrated two-band SSH model. In bulk momentum space representation, the Hamiltonian is described by the $M\times M$ matrix $\mathcal{H}(k)$ defined by

[TABLE]

where $k$ is the Bloch wave number, which varies in the interval $(-\pi,\pi)$. The $M$ energy bands $E_{l}(k)$ and corresponding Bloch functions $u_{l}(k)$ of the superlattice $(l=1,2,..,M$) are defined by the eigenvalue equation $\mathcal{H}(k)u_{l}(k)=E_{l}(k)u_{l}(k)$, with the normalization condition $\langle u_{l}(k)|u_{n}(k)\rangle=\delta_{n,l}$. Owing to the particle-hole symmetry r26 ; r27 , the energy spectrum appears to be mirror symmetric with respect to the zero energy $E=0$. The topological properties of the $(M-1)$ band gaps are characterized by the sum of Zak phases of all the isolated bands below the corresponding band gap r22 ; r23 ; r25 ; r26 , i.e. by the numbers

[TABLE]

where $n=1,2,..,M-1$ is the band gap number (ordered for increasing values of energy) and $\gamma_{l}\equiv i\int_{-\pi}^{\pi}dk\langle u_{l}(k)|(du_{l}/dk)\rangle$ is the Zak phase of the $l-th$ band. As shown in previous works r22 ; r23 ; r26 , the Zak phase and thus the gap numbers $\mathcal{N}_{l}$ are defined apart from integer multiplies than $2\pi$, depend on the choice of the origin in the unit cell, and they are not quantized rather generally. However, when the system possesses inversion symmetry, i.e. when $t_{M-l}=t_{l}$, the Zak phase is quantized and can take two possible values: [math] or $\pi$ r23 ; r26 . Correspondingly, the gap numbers $\mathcal{N}_{l}$ are topological and quantized, taking the two possible values [math] or $\pi$ (mod $2\pi$). Interestingly, in the gaps with topological number $\pi$, nearly-degenerate topological edge states, localized one on the left (L) and the other on the right (R) edges of the lattice, do exist r26 .

The eigenenergies and domain of existence of edge states can be determined, in the most general case, by extending the procedure earlier introduced in r29 for surface bound states in the continuum. Indicating by $\mathcal{H}_{L}$ the $(M-1)\times(M-1)$ matrix defined by $(\mathcal{H}_{L})_{n,m}=t_{n}\delta_{n,m-1}+t_{m}\delta_{n,m+1}$ and by $E_{L,\sigma}$, $v^{(L,\sigma)}$ the eigenvalues and corresponding eigenvectors of $\mathcal{H}_{L}$, i.e. $\mathcal{H}_{L}v^{(L,\sigma)}=E_{L,\sigma}v^{(L,\sigma)}$, it can be shown that there are up to $(M-1)$ left edge states with energies $E_{L,\sigma}$; moreover, the left edge state with energy $E_{L,\sigma}$ does exist provided that $\tau>t_{M-1}|v_{M-1}^{(L,\sigma)}/v_{1}^{(L,\sigma)}|$. Likewise, let us indicate by $\mathcal{H}_{R}$ the $(M-1)\times(M-1)$ matrix defined by $(\mathcal{H}_{R})_{n,m}=t_{M-n}\delta_{n,m-1}+t_{M-m}\delta_{n,m+1}$, and by $E_{R,\sigma}$, $v^{(R,\sigma)}$ the eigenvalues and corresponding eigenvectors of $\mathcal{H}_{R}$, i.e. $\mathcal{H}_{R}v^{(R,\sigma)}=E_{R,\sigma}v^{(R,\sigma)}$. Then, it can be shown that there are up to $(M-1)$ right edge states with energies $E_{R,\sigma}$; moreover, the right edge state with energy $E_{R,\sigma}$ does exist provided that $\tau>t_{1}|v_{M-1}^{(R,\sigma)}/v_{1}^{(R,\sigma)}|$. Note that, in the presence of inversion symmetry, one has $\mathcal{H}_{L}=\mathcal{H}_{R}$ and thus the energies of left and right edge states become degenerate: this mean that in superlattices with inversion symmetry the edge states appear in pairs at both left and right edges with the same energy.

Figure 2 shows an example of numerically-computed energy spectrum (left panels) and corresponding gap numbers $\mathcal{N}_{n}$ (right panels) versus the inter-cell coupling $\tau$ in two superlattices comprising $M=4$ waveguides in each unit cell. In Fig.2(a) there is inversion symmetry ($t_{3}=t_{1}$), and the gap numbers $\mathcal{N}_{n}$ are quantized for all gaps. Note that degenerate left and right edge states appear in the middle gap (energy $E=0$) and in the upper/lower gaps (energies $\pm\sqrt{t_{1}^{2}+t_{2}^{2}}$) as $\tau$ is increased above the two values $\tau_{1}=t_{1}^{2}/t_{2}$ and $\tau_{2}=t_{2}$, respectively. The topological nature of the edge states, signaled by the quantization of the gap numbers, can be also revealed by the fact that, as $\tau$ is varied, the edge states appear and disappear as the band gaps close and reopen. Figure 2(b) corresponds to a superlattice without inversion symmetry. In this case the gap numbers in the upper and lower band gaps are not quantized and vary continuously as the inter-cell coupling $\tau$ is varied. Correspondingly, L and R edge states observed by increasing $\tau$ are not topological (they emanate from the bands without gap closing) and are not degenerate in energy. Interestingly, for the middle gap the number $\mathcal{N}_{2}$ is quantized and topological zero-energy degenerate L and R edge states are observed by increasing $\tau$ after band gap closing and reopening at $\tau=\tau_{0}\equiv t_{1}t_{3}/t_{2}$ [Fig.2(b)]. The born of such edge states corresponds to a sudden jump of the gap number $\mathcal{N}_{2}$ from trivial (0) to nontrivial ($\pi$) values. This means that in a superlattice with broken inversion symmetry topological and non-topological edge states can coexist.

Wannier function excitation, beam displacement and gap numbers. Recent works have shown that, in a two-band lattice such as in the SSH model, the average beam displacement in a photonic QW with single-site initial excitation provides a direct measure in the bulk of Zak phase r15 ; r17 ; r18 ; r19 . A main open question is whether photonic QSs in a multi-gap superlattice can be related to the Zak phases and gap numbers of the various bands/gaps. The propagation of the field amplitudes in the various waveguides can be written as a superposition of Bloch modes

[TABLE]

where we have set $a_{n}\equiv(a_{n,1},a_{n,2},...,a_{n,M})^{T}$ and where the spectral amplitudes $C_{l}(k)$ are determined by the initial excitation condition $a_{n}(0)$ of the array according to

[TABLE]

with the normalization condition $2\pi\sum_{l}\int_{-\pi}^{\pi}dk|C_{l}(k)|^{2}=1$. We are interested to calculate the integral mean displacement of the discretized beam along the propagation distance $z$, defined by

[TABLE]

which is a measurable quantity. Substitution of Eq.(4) into Eq.(6), after some calculations it can be shown that, in the large $z$ limit, $\mathcal{D}(z)$ reaches the asymptotic value

[TABLE]

Let us assume that $C_{l}(k)$ is independent of $k$, i.e. let us assume uniform band filling. For $C_{l}(k)=(1/2\pi)\delta_{l,n}$, i.e. when the $n-th$ Bloch band solely is uniformly excited at input plane, the initial beam excitation $a_{n}(0)=(1/2\pi)\int_{-\pi}^{\pi}dku_{n}(k)\exp(ikn)$ reproduces the Wannier function of the $n-th$ band r30 and correspondingly one has $\mathcal{D}_{as}=\gamma_{n}/(2\pi)$: this is a well-known result, that relates Zak phase to the Wannier function displacement in a crystal r30 . The most interesting case is attained when we excite uniformly the lowest $S$-bands of the superlattice. By assuming $C_{l}(k)=1/(2\sqrt{S}\pi)$, i.e. $a_{n}(0)=(1/2\sqrt{S}\pi)\sum_{l=1}^{S}\int_{-\pi}^{\pi}dku_{l}(k)\exp(ikn)\equiv A_{n}^{(S)}$, from Eq.(7) it follows that

[TABLE]

i.e. a measure of the mean beam displacement yields the topological number $\mathcal{N}_{S}$ of the $S-th$ band gap, scaled by the factor $2\pi S$. Figure 3 shows examples of numerically-computed evolution of integral beam displacement in a quadrimeric superlattice with [Fig.3(a,b)] and without [Fig.3(c)] inversion symmetry, corroborating the theoretical prediction of Eq.(8). In the figure, the amplitudes of the excitation field at the input plane $z=0$ is shown in the insets, while propagation distancee $z$ is normalized to the inverse of the inter-cell coupling constant $\tau$. In all cases the initial field distribution turns out to be real and requires to excite some waveguides ($\sim 10-30$) with tailored intensity and phase ([math] or $\pi$ depending on the sign of $a_{n}$). Intensity control can be provided by proper beam shaping methods (for example using a spatial light modulator r31 or a binary phase plate r32 and a Fourier filter), while $\pi$ phase shifts could be realized by waveguide segmentation r33 . Note that the asymptotic value $\mathcal{D}_{as}$ is reached after a propagation length $z\sim 40/\tau$. For example, for case of Fig.3(a) and for a typical value $\tau\sim 3\;{\rm cm}^{-1}$ in waveguide lattices manufactured by fs-laser writing r29 ; r31 , such a distance corresponds to a sample length $L\sim 13$ cm, which is feasible with current technology. It should be noted that perturbations of the excitation condition from the target one $A_{n}$ and/or disorder in the system will introduce some deviations of $\mathcal{D}_{s}$ from the predicted value (8). As an example, the right panels in Fig.3 show the statistical distribution of $\mathcal{D}_{s}$ obtained when the array is excited with the perturbed amplitudes $a_{n}(0)=A_{n}^{(S)}(1+f_{n})$, where $f_{n}$ is taken from a Gaussian distribution with zero mean and $\sigma=0.1$ standard deviation.

Bulk probing of zero-energy topological edge mode. The Wannier function excitation protocol discussed above provides a rather general and powerful tool for topological spectroscopy of 1D superlattices. However, it requires the knowledge of Wannier functions, its practical implementation requires special beam shaping methods, and it is sensitive to perturbation of the excitation condition. It is thus worth exploring whether simpler excitation configurations can provide some useful information about the topological properties of the superlattice, without the knowledge of Wannier functions. The most interesting case is the one of a chiral superlattice ($M$ even), that can sustain a zero-energy mode protected by the particle-hole symmetry r27 . As shown in Fig.2(b), in a superlattice with broken inversion symmetry the topological zero-energy mode can coexist with other non-topological edge states in the other bang gaps. Clearly, edge excitation of the array – like e.g. in the recent experiment r8 –can not discriminate in a simple way the existence of the topological zero-energy mode from other non-topological edge states, since in both cases some light remains trapped at the edge and oscillatory behavior is observed, as shown in Fig.4. For $M=4$, we found the following remarkable result. Let us consider single-cell excitation in the bulk corresponding to the initial condition $a_{n}=(1/\sqrt{2})(1,0,\pm i,0)^{T}\delta_{n,0}$. This means that only two waveguides of the array, separated apart by two sites, are equally excited at the input plane with a $\pm\pi/2$ phase shift; a possible excitation scheme is depicted in Fig.4(a). The spectral amplitudes $C_{l}$ corresponding to such an excitation condition are obtained from Eq.(5), and the corresponding asymptotic value of mean beam displacement $\mathcal{D}_{as}$ can be then calculated using Eq.(7). The final result is very simple: $\mathcal{D}_{as}=0$ for $\tau<\tau_{0}$ and $\mathcal{D}_{as}=-1/4$ for $\tau>\tau_{0}$, where $\tau_{0}\equiv t_{1}t_{3}/t_{2}$ is the gap closing point above which the topological zero-energy mode does appear [Fig.2(b)]. This means that the bulk parameter $\mathcal{D}_{as}$ for the excitation $a_{n}(0)=(1/\sqrt{2})(1,0,\pm i,0)^{T}\delta_{n,0}$ provides a topological number that signals the existence (or not) of the topological zero-energy mode; see Figs.4(b) and (c). This topological number is analogous to the global winding number of the SSH4 model recently introduced in r27bis , however in our setup it is not required to trace the chiral (sublattice) average displacements over a basis of states of the unit cell, taken as different initial excitation conditions.

Conclusions. Identifying ways to relate global topological invariants with measurable quantities is a quest of major relevance in topological physics. Several methods have been suggested and demonstrated using synthetic topological matter. Here a general method of topological spectroscopy of 1D photonic superlattices has been suggested, which is based on bulk waveguide array excitation by a superposition of Wannier functions. In case of a quadrimeric superlattice, we also suggested a simple method to detect topological zero-energy edge states when they coexist with non-topological ones. Our results are feasible for a description of diverse 1D topological systems beyond optics, such as electronic, polaritonic, mechanical, and acoustic superlattices. Future research directions include extension of the Wannier function method to other systems, such as to topological non-Hermitian lattices, and improvement of the robustness against lattice disorder and perturbations.