Observation of topological transport quantization by dissipation in fast Thouless pumps
OObservation of topological transport quantization by dissipation in fast Thoulesspumps
Zlata Fedorova, ∗ Haixin Qiu, † Stefan Linden, ‡ and Johann Kroha § Physikalisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Nussallee 12, 53115 Bonn, Germany. (Dated: August 13, 2020)Quantized dynamics is essential for natural processes and technological applications alike. Thework of Thouless on quantized particle transport in slowly varying potentials (Thouless pumping)has played a key role in understanding that such quantization may be caused not only by discreteeigenvalues of a quantum system, but also by invariants associated with the nontrivial topologyof the Hamiltonian parameter space. Since its discovery, quantized Thouless pumping has beenbelieved to be restricted to the limit of slow driving, a fundamental obstacle for experimentalapplications. Here, we introduce non-Hermitian Floquet engineering as a new concept to overcomethis problem. We predict that a topological band structure and associated quantized transport canbe restored at driving frequencies as large as the system’s band gap. The underlying mechanismis suppression of non-adiabatic transitions by tailored, time-periodic dissipation. We confirm thetheoretical predictions by experiments on topological transport quantization in plasmonic waveguidearrays.
The standard realization for Thouless pumping isthe time-periodic version of the Rice-Mele (RM) model ,which describes a dimerized tight-binding chain whosesystem parameters change cyclically along a closed loopin Hamiltonian parameter space. In the adiabatic regimeand for a completely filled band, the net particle trans-fer per cycle is an integer given alone by the Berryphase associated with the loop or the Chern numberof the band, i.e., a topological invariant robust againsttopology-preserving deformations of the parametric loop.Such nontrivial topology of the Hamiltonian parameterspace or band structure was recognized as the overar-ching concept behind phenomena apparently as diverseas the integer quantum Hall effect , the quantum spinHall effect , topological insulators in solid state andphotonics , quantum spin or charge pumping , Diracor Weyl semimetals , and the electric polarization ofcrystalline solids . Recently, topological or Thoulesspumping was experimentally demonstrated using ultra-cold atoms in dynamically controlled optical lattices or using waveguide arrays .In realistic systems, however, Thouless pumping gener-ically faces two difficulties. First, at nonzero driv-ing frequencies, unavoidable in experiments, the sys-tem becomes topologically trivial. The reason is thatthe nonzero driving frequency defines a Floquet-BlochBrillouin zone (FBBZ) and the dimension of the bandstructure is increased by one compared to the adiabaticcase. The coupling between forward- and backward-propagating states then opens a gap , so that theChern number, or winding number around the FBBZ,of the effectively two-dimensional band becomes trivial,and the particle transport deviates from perfect quantiza-tion. Second, realistic experimental systems are to someextent open and subject to dissipation, so that the quan-tum mechanical time evolution of single-particle statesdeviates from unitarity, which may prevent the closingof the cycle in Hamiltonian parameter space. This moti-vates the interest in non-Hermitian (NH) Hamiltonians. Non-Hermiticity can have profound influence on the sys-tem dynamics. In addition to ubiquitous exponential de-cay, it may cause such peculiar phenomena as dissipation-induced localization in the Caldeira-Legget model , uni-directional robust transport , asymmetric transmissionor reflection , or NH topological edge states associ-ated with exceptional points . Non-Hermiticity hasbeen utilized to probe topological quantities . An-other fascinating example is the so-called non-Hermitianshortcut to adiabaticity , which describes faster evo-lution of a wavefunction in an NH system than in itsHermitian counterpart.Here, we introduce time-periodic modulation of dissi-pation as a new concept to restore topological transportquantization in fast Thouless pumps. Although in many-body systems dissipation would be induced by interac-tions or particle loss, the plasmon polariton dynamicsin our experiments is mathematically identical to thatof a linear, dissipative, periodically driven Schrödingerequation. To analyze systems of this kind theoretically,we utilize the Floquet theory for non-Hermitian, time-periodic systems. Using this formalism, we demonstratefor a driven RM model that time-periodic dissipationcan give rise to a band structure in the two-dimensionalFBBZ with a nontrivial Chern number. Hence, themean displacement of a wave packet per cycle is quan-tized even when the driving frequency is fast, i.e., farfrom adiabaticity. In a real-space picture, this topo-logically quantized transport comes about, because thetime-periodic loss selectively suppresses the hybridizationof a right(left)-moving mode with the counterpropagat-ing one. The theoretical predictions are confirmed byexperiments on arrays of coupled dielectric-loaded sur-face plasmon-polariton waveguides (DLSPPW) . DL-SPPWs are uniquely suited model systems for realiz-ing topological transport with dissipation: The propa-gation of surface plasmon polaritons mathematically re-alizes the single-particle Schrödinger equation on a one-dimensional tight-binding lattice , where the waveg- a r X i v : . [ qu a n t - ph ] A ug uide axis resembles time, and the system parameters, in-cluding losses, can easily be modulated along the waveg-uide axis. Moreover, complete band filling is achievedvia Fourier transform to k-space by pumping a single site(waveguide) of the tight-binding lattice. This is essentialfor probing the band topology which otherwise is possibleonly in fermionic systems at low temperature. RESULTSModel
We consider a periodically driven RM model with additional onsite, periodic dissipation (see Fig. 1), ˆ H ( t ) = ˆ H RM ( t ) − iˆΓ( t ) , ˆ H RM ( t ) = (cid:88) j (cid:16) J ( t )ˆ b † j ˆ a j + J ( t )ˆ a † j +1 ˆ b j + h.c. (cid:17) (1) + (cid:88) j (cid:16) u a ( t )ˆ a † j ˆ a j + u b ( t )ˆ b † j ˆ b j (cid:17) , ˆΓ( t ) = (cid:88) j (cid:16) γ a ( t )ˆ a † j ˆ a j + γ b ( t )ˆ b † j ˆ b j (cid:17) . (2)where j runs over all unit cells, ˆ H RM ( t ) is the Hamilto-nian of the periodically driven, nondissipative RM modeland ˆΓ( t ) describes the losses. ˆ a † j and ˆ b † j ( ˆ a j and ˆ b j ) arecreation (annihilation) operators in unit cell j on sublat-tice A and B , respectively. The inter-/intra-cell hoppingamplitudes, J / ( t ) and the onsite potentials on the twosublattices, u a ( t ) and u b ( t ) , are all real-valued, periodicfunctions of time with frequency Ω = 2 π /T according to u a ( t ) = − u cos(Ω t + ϕ ) , u b ( t ) = u a ( t − T / ,J ( t ) = J e − λ (1 − sin Ω t ) , J ( t ) = J ( t − T / , with u , J , λ > , and ϕ = 0 (unless otherwise speci-fied). The choice of the hopping amplitudes is motivatedby the exponential dependence of the wave-function over-laps on the spacing λ (1 − sin Ω t ) between neighboringsites, as in our experiment below. In our NH modifi-cation of the RM model, the time-periodic decay rates γ a ( t ) ≥ and γ b ( t ) ≥ are nonzero once the onsitepotential exceeds the mean value [ u a ( t ) + u b ( t )] / .This resembles, for instance, a realistic situation whereparticles in a trapping potential are lost from the traponce the trapping potential is not sufficiently deep. Thus,we choose γ a ( t ) = − γ Θ( u a ( t )) cos(Ω t + ϕ ) , γ b ( t ) = γ a ( t − T / . Non-Hermitian Floquet analysis
In the following calculations we use the non-HermitianFloquet formalism discribed in the Methods section be-low. We assume u = J = 1 , λ = 1 . and all energiesare given in units of J . A B 0 P o t en t i a l Decay rateUnit cell a a t/T J J b γ - γ ab u -u a b J - J TT /2 T /4 3 T /4 FIG. 1.
Non-Hermitian driven Rice-Mele model (a) Schematic of the periodically driven, NH RM lattice forfour equidistant times during a pumping cycle. Lossy sites aredepicted by red color, large (small) hopping amplitudes J , by short (long) distances between sites. (b) Pumping cycle inthe parameter space ( J − J , u a − u b , γ a − γ b ) . In view of the experimental setup discussed below, wealso consider the time evolution of states | Ψ A ( t ) (cid:105) and | Ψ B ( t ) (cid:105) which have been initialized (“injected”) at time t = 0 with nonzero amplitude only at a single site of the A or B sublattice, respectively. For the parametric cycleconfiguration shown in Fig. 1 (b) the chosen initial timemoment leads to an asymmetric amplitude distributionof the counter-propagating Floquet states with respectto the two sublattices. As seen in Fig. 2, such initialconditions populate, by Fourier expansion, almost ho-mogeneously an entire right- or left-moving band. Thus,it is a way to create the topologically important completeband filling, which would otherwise be possible only infermionic systems. In the Hermitian case ( γ = 0 ), wesee from Fig. 2 (a) that the counterpropagating bands hy-bridize, accompanied by avoided crossings and gaps withwidth G opening at the Floquet Brillouin zone bound-aries, so that the bands become topologically trivial. Asa result, the charge pumped per period deviates fromthe quantized value. This marks the generic breakdownof quantized Thouless pumping at any finite pumpingfrequncy Ω , as also noted in . Note that computingthe gap size G , as visible in Fig. 2 (a), involves diago-nalization of the entire Floquet Hamiltonian matrix. Inleading order perturbation theory, G would be given bythe Fourier amplitude of the periodic drive, i.e., for thefirst FBBZ by J , which strongly differs from the exactvalue.We now consider the NH RM model driven with γ =0 . J (see Figs. 2 (b-e)). Adding losses leads to severalprofound effects. First, the quasienergies become com-plex, whereby the right- and left-moving bands acquireconsiderably different dampings shown in Fig. 2 (e) andseen as different broadenings of the spectral band occu-pation in Fig. 2 (b), (c). Second, the two inputs are nolonger equivalent in respect to the relative populationsof the two bands. In particular, for the input A we al-most exclusively excite right-moving states, while for theinput B in addition to the lossy left-moving states, wepartially populate right moving-states. Third, and mostimportantly, the gap G closes and, hence, the bands windaround the entire 2D FBBZ as illustrated in Fig. 2 (d). k E b I m ( ε ) k Input BInput A0- π π π π π π k k ae -γ π - π J ] G G [ Ω ] γ =0 γ =0.05 γ =0.3 γ =0.6 γ =0.9 f c right moving r. m. l. m. r. m. l. m. - - | Ψ ( E , k ) | ~ k E d FIG. 2.
Floquet analysis of the driven NH RM modelin the non-adiabatic regime.
Calculated band structuresof the RM model for driving frequency
Ω = 1 . J . Thin lines:left- and right-moving Floquet quasienergy bands (real parts).(a) Band structure of the Hermitian RM model ( γ = 0 ). Theband gaps at E = ± Ω / indicate a topologically trivial bandstructure, i.e., the breakdown of transport quantization. colorcode: normalized spectral occupation density of a state | Ψ( t ) (cid:105) injected at time t = 0 on a single site of the sublattice A , cal-culated from Equation (13). It is seen that this injection al-most homogeneously populates the right-moving bands, andalmost no mixing of different Floquet modes occurs, as de-scribed by Equation (14). (b) Same as in (a) but for the NHRM model with γ = 0 . J when the system is excited at asingle site of the initially nonlossy sublattice A . As in (a),almost no mixing of Floquet modes occurs. The gap at theFBBZ boundary is closed, restoring transport quantization.(c) Same as in (b), but for a state injected at a site of theinitially lossy sublattice B . Although the band gaps remainclosed by the dissipation, this predominantly populates theleft-moving band with a broad distribution, and the lossesare high. (d) The first FBBZ which evolves into a 2D torusdue to the periodicity along the E axis (coincidence of dashed-dotted lines) as well as the k axis (coinciding dashed lines).The magenta and green lines are the forward- and backward-propagating dispersions analogous to (b) and (c). They windaround the torus with winding numbers Z = ± (c.f. Meth-ods). (e) Imaginary part of the quasienergy bands presentedin (b, c), showing low dissipatiion in the right-moving band.(f) The size of the band gap G in dependence on the drivingfrequency at different loss amplitudes γ . The black dashedline shows Ω = 1 . J . In the Methods section it is shown that this restores thequantized transport (see Equation (17)). Note, that theseeffects only occur once γ is larger than some thresh-old value. In order to study this threshold behaviour γ =0 γ =0.05 γ =0.1 γ =0.3 ad. Input A Input B < x > t[T] t[T] < x > a b FIG. 3.
The center-of-mass shift in the NH RM model.
The center-of-mass position of the injected wavepacket afterup to 5 full pumping cycles (
Ω = 1 . J ) at different loss apli-tudes γ for a single-site input on (a) sublattice A and (b)sublattice B . we numerically evaluated the gap size G at various driv-ing frequencies and loss amplitudes (see Fig. 2 (f)). Inthe Hermitian case ( γ = 0 ) the gap size has a complexoscillatory behaviour as a function of the driving fre-quency. Our analysis shows that a larger gap size requiresstronger damping in order to close it. For instance, at thepreviously analyzed driving frequency Ω = 1 . J the lossamplitude γ should be larger than . J to close the gap.Next, we investigate the position of the cen-ter of mass (CoM) of the wave-packet, (cid:104) x (cid:105) ( t ) = (cid:104) Ψ( t ) | x | Ψ( t ) (cid:105) / (cid:104) Ψ( t ) | Ψ( t ) (cid:105) , after up to 5 completed driv-ing cycles at various losses and fixed driving frequency Ω = 1 . J for different initial conditions input A or B (see Fig. 3 (a,b)). In the adiabatic case the mean dis-placement is almost +1 ( − ) unit cell per cycle for delta-like excitations on sublattice A ( B ). Small deviationsfrom unity result from slight inhomogeneity of the bandpopulation. At the driving frequency Ω = 1 . J the dis-placement per cycle is considerably smaller in the Hermi-tian case ( γ = 0 ) indicating deviation from the quantizedtransport. With increasing losses this deviation becomessmaller and smaller for input A and for γ ≥ . the dis-placement can not be distinguished from the adiabaticcase. Surprisingly, for the input B we observe that theCoM position switches direction with time. This is a sig-nature of the chirality of the Floquet bands and is due tothe fact that the propagation of even poorly populatedlow-loss states in positive x -direction starts to dominateafter the first few periods, while the states propagatingin negative x -direction are quickly damped due to thephase relation of the periodic losses with respect to thehopping amplitude. Experiments
In order to test our theoretical predictions we per-formed experiments based on DLSPPWs. The exper-imental realization of the model described by Equa-tions (1), (2) is based on the mathematical equivalencebetween the time-dependent Schrödinger equation intight-binding approximation and the paraxial Helmholtzequation which describes propagation of light in coupledwaveguide arrays . Figure 4 shows a scheme of a DL- z ( t i m e ) x y G o l d PMMA G l a ss d ( z ) h ( z ) a h ( z ) b d ( z ) T AB U n i t c e ll a w ( z ) a w ( z ) b t =05 μ m b z μ m c x μ m] a FIG. 4.
Plasmonic implementation of the NH RMmodel. (a) Sketch of the plasmonic implementation of theNH RM model. (b) Scanning electron micrograph of a typ-ical sample corresponding to J = 0 . µ m − , Ω = 1 . J , u = 1 . J , γ = 0 . J . The red dotted box highlights thegrating coupler deposited onto the input waveguide A. (c)AFM scan of the same sample as shown in (b). SPPW array (a) as well as a scanning electron micro-graph (b) and an AFM scan (c) of a typical sample. Thesample fabrication process and the typical geometricalparameters of the arrays are described in the Methodssection. The waveguide array represents a dimerized 1Dlattice, where each unit cell contains two waveguides, A and B . Here, the propagation direction z plays the role oftime. Periodic modulation of the effective hopping am-plitudes is reached by sinusoidally varying the spacingbetween the adjacent waveguides d , ( z ) while the on-site potential variation is realized by changing the waveg-uides’ cross-sections (heights h a,b ( z ) and widths w a,b ( z ) ).In addition, the variation of the waveguides’ cross-sectionaffects the instantaneous losses γ a,b ( z ) . When the cross-section decreases, the confinement of the guided modeweakens. As a result, the modes can couple to free-propagating surface plasmon polaritons (SPPs) and scat-ter out from the array. We employ this effect to introducetime-dependent losses γ a,b ( z ) .We first consider a pumping cycle that encloses thecritical point. For this purpose we choose the geometricalparameters of the DLSPPW array such that u = 1 . J and Ω = 1 . J . By comparing the real-space inten-sity distribution to numerical calculations we estimatethe loss amplitude to be γ = 0 . J . The real-space SPPintensity distribution I ( x, z ) recorded by leakage radia-tion microscopy (see Methods) for single site excitationat site A is shown in Fig. 5 (a). According to the afore-mentioned quantum optical analogy this corresponds tothe probability density I ( x, t ) = | Ψ( x, t ) | . We observefor all z a strongly localized wave packet, whose CoM istransported in positive x -direction in a quantized man- N o r m . I n t. -0.5 -1 -1.5 x / a a z (time) [T]0 1 32 4 (d) z (time) [T]0 1 32 40-4 Δ u Δ J φ =042-2 x / a c N o r m . I n t. -0.5 -1 -1.5 N o r m . I n t. -1 -2 k ( qua s i ene r g y ) z -4 -2 0 2 k (momentum) [ π / a ] x 0 -3 -1 31free SPPs1st BZ Ω{ b φ=π/2 Δ J Δ u N o r m . I n t. -1 -2 k ( qua s i ene r g y ) z -4 -2 0 2 k (momentum) [ π / a ] x 0 -3 -1 31free SPPs1st BZ d FIG. 5.
Observation of fast Thouless pumping in aDLSPPW array. (a) Real-space SPP intensity distributionfor u = 1 . J , γ = 0 . J , and ϕ = 0 . Plot on the rightshows the projection of the corresponding pumping cycle ontothe plane ( J − J , u a − u b ) . (b) Fourier-space SPP intensitydistribution corresponding to (a). (c) Real-space SPP inten-sity distribution for u = 1 . J , γ = 0 . J , and ϕ = π / .Plot on the right again shows the corresponding cycle in pa-rameter space. (d) Fourier-space SPP intensity distributioncorresponding to (c). ner, i.e., by one unit cell per driving period (see dottedlines), even though the driving frequency Ω is larger thanthe modulation amplitudes of all relevant parameters.The corresponding momentum resolved spectrum I ( k x , k z ) is obtained by Fourier-space leakage radiationmicroscopy and is shown in Fig. 5 (b). This intensitydistribution is analogous to the spectral energy densitypresented in Fig. 2. We note that this technique providesthe full decomposition in momentum components in thehigher Brillouin zones . The main feature of the spec-trum is a continuous band with average slope a /T . Theabsence of gaps in the band indicates that the band windsaround the 2D FBBZ {− Ω / ≤ k z < Ω / − π /a ≤ k x < π /a } . This is a hallmark of a quantized pumping andconfirms our theoretical predictions (see Fig. 2 (b)).As a reference measurement, we consider the paramet-ric cycle, where all parameters are changing with thesame amplitudes as in the previous case but the phaseis chosen as ϕ = π / . Under these conditions the Hamil-tonian is symmetric under space and time inversion. InFig. 5 (c) we present the real-space SPP intensity distri-bution for this parametric cycle. In contrast to the pre-vious case the wave packet is spreading and we do not z (time) [ μ m] x / a x / a x / a μ m] b z [ T ]0 1 32 401234-1 < x > / a Ω =1.45 J Ω=0.7 J Ω=1.1 J u =1.1 J γ =0.8 J u =0 γ =0 N o r m . I n t. - . - u =1.1 J γ =0.8 J u =0 γ =0 Ω =1.45 J Ω =1.1 J Ω =0.7 J Ω =1.45 J Ω =1.1 J Ω =0.7 J a FIG. 6.
Influence of driving frequency on the trans-port. (a) Real-space SPP intensity distributions for differentdriving frequencies and single-site excitation at waveguide A .The left and right column correspond to arrays with cross-section modulation ( u = 1 . J , γ = 0 . ) and without cross-section modulation ( u = 0 , γ = 0 ), respectively. (b) TheCoM position of the SPP intensity in dependence on propaga-tion distance z calculated from the experimental results shownin (a). Note that the z -axis here is normalized to the period T . Red markers correspond to arrays with cross-section mod-ulation and blue markers correspond to no modulation. Theblack dashed line shows the anticipated adiabatic behavior. observe CoM transport in x -direction. The correspond-ing momentum resolved spectrum shows a complicatedband structure with multiple band gaps (see Fig. 5 (d)).Obviously, none of the bands winds around the 2D FBBZ.Directional transport of light in periodically curvedwaveguides can be in principle also achieved by usinga simple combination of directional couplers with con-stant effective mode index, i.e., constant waveguide cross-section . However, due to periodic exchange of powerbetween two coupled waveguides this effect has a res-onant character and the period of modulation plays inthis case a crucial role. In order to demonstrate that thedirectional transport in our system has a different origin,we repeat the experiment shown in Fig. 5 (a) for three dif-ferent driving frequencies Ω ( . J , . J , . J ). More-over we prepare two sets of samples, one with modula-tion of the waveguide cross-section as before ( u = 1 . J , γ = 0 . J ) and the second with constant cross-section( u = 0 , γ = 0 ). The measured real space intensitydistributions are depicted in Fig. 6 (a). We extract fromthis data the CoM position after up to 4 complete periodsas displayed in Fig. 6 (b). In the case with cross-sectionmodulation (red markers) the CoM is shifted by one unitcell per period T at all chosen driving frequencies. Wenote that the somewhat lower than unit slope of the CoMplots in Figs. 6 (b), 7 (b) during the first pumping cy- z (time) [ T ] x / a x / a x / a T ] b z [ T ]0 1 32 4 u =1.1 J γ =0.8 J < x > / a u =1.5 J γ =1.1 J u =0.3 J γ =0.1 J N o r m . I n t. - . - Input A Input B u =1.1 J γ =0.8 J u =1.1 J γ =0.8 J u =0.3 J γ =0.1 J u =0.3 J γ =0.1 J u =1.5 J γ =1.1 J u =1.5 J γ =1.1 J a FIG. 7.
Influence of cross-section modulation and in-put conditions on the transport. (a) Real-space SPPintensity distributions for the arrays with different strengthsof the cross-section modulation ( u = 0 . J , γ = 0 . J ),( u = 1 . J , γ = 0 . J ), and ( u = 1 . J , γ = 1 . J ). Mea-surements on the left-hand side show the SPP propagation af-ter excitation at sublattice A (low-loss input). Measurementson the right-hand side show the SPP propagation after exci-tation at sublattice B (high-loss input) (b) The CoM positionof the SPP intensity in dependence on propagation distance z calculated from the experimental results shown in (a). Notethat the z -axis here is normalized to the period T . The blackdashed line shows the anticipated adiabatic behavior. cle is an artifact which arises from non-ideal excitationconditions, such as weak excitation of the neighboringwaveguides. The deviations at large distance are statis-tical and result from increasing measurement errors dueto camera noise and decaying signal intensity. In Fourierspace changing the modulation frequency influences pri-marily the width of the Floquet BZ: the lower is the fre-quency, the smaller is the distance between the neighbor-ing bands and the smaller is the tilt of these bands whichreflects the wavepacket group velocity in absolute values(see Supplementary Figure 1).Without cross-section modulation (blue markers inFig. 6 (b)) the CoM displacement per period at thesefrequencies is much smaller than in the quantized caseand depends on the driving frequency. These measure-ments confirm that the observed directional transport inour system is not a resonant directional coupler effect.Up to now we only considered experiments with exci-tation at sub-lattice A (low loss input). The numericalcalculations predict that the transport in the oppositedirection for single site excitation at sub-lattice B isstrongly suppressed by the time-periodic losses. To testthis, we perform additional experiments to study how thetransport properties depend on the initial conditions fordifferent strengths of cross-section modulation. In doingso we tune the amplitude of the on-site potential u andsimultaneously the loss amplitude γ . Fig. 7 (a) showsthe real space intensity distributions for the excitation atthe waveguides A (left column) and B (right column) forthree different cross-section modulations and the drivingfrequency Ω = 1 . J . The CoM displacement derivedfrom this data is depicted in Fig. 7 (b) (waveguideA: circles, waveguide B: triangles). In case of smallmodulation strength ( u = 0 . J , γ = 0 . J , redmarkers) SPPs excited at site A and B are transportedin + x and - x directions, respectively. However, for bothinputs the mean displacement of the CoM is less than1 unit cell per period. For the modulation strength( u = 1 . J , γ = 0 . J , blue markers) input A showsquantized displacement of the CoM while the sign of themean displacement for input B switches from + to -.This effect becomes even stronger at higher modulationstrength u = 1 . J , γ = 1 . J (green) – as predictedby theory (compare with Fig. 3 (a-b)). In Fourier spaceincreasing the modulation strength results in a strongband broadening caused by a growing damping rate.This effect is more pronounced for the input B (seeSupplementary Figure 2). DISCUSSION
In this work, we introduced the concept of time-periodic dissipation in Floquet topological systems. Thetheoretical analysis required the generalization of Flo-quet theory to quantum mechanics with non-Hermitian,time periodic Hamiltonians. Such quantum systems canbe simulated experimentally in dielectric-loaded surface-plasmon polariton waveguide (DLSPPW) arrays. Specif-ically, we considered a non-Hermitian extension of theperiodically driven Rice-Mele model. While fast drivingof dissipationless systems always destructs the quanti-zation of Thouless pumping, we predicted theoreticallythat time- and space-periodic dissipation can lead to therestoration of quantized transport for nonadiabatic driv-ing conditions. This finding results from the fact thatperiodic loss can modify the Floquet-Bloch band struc-ture in such a way that the band gaps present in thenon-lossy Floquet-driven system close. In this way, achiral Floquet band is established that winds around thetwo-dimensional Floquet-Bloch Brillouin zone, and whichthus carries quantized transport given by the Chern num-ber. We emphasize that this is not merely due to adissipation-induced band smearing, but a true renormal-ization of the real part of the energy eigenvalues, in-duced by the nonlinearity of the eigenvalue equation.In a real-space picture, the phenomenon of gap closingcan be understood as selective suppression of one of thecounter-propagating states. In order to examine the the-oretical predictions, we used evanescently coupled plas-monic waveguide arrays to implement the model. Com-bining real- and Fourier-space imaging, we demonstrated fast, quantized transport in the waveguide arrays. Inreal space, the center of mass of the excited surface-plasmon polariton wave packet was shifted by one unitcell per driving cycle. In Fourier space quantized pump-ing is seen as a chiral Floquet band that winds aroundthe quasienergy Brillouin zone. Additional experimentsshowed that, first, unlike in a simple combination of di-rectional couplers, the SPP transport in our system isindependent on the driving frequency. Second, the trans-port in the opposite direction is strongly suppressed. Ourexperimental results agree well with the theoretical pre-dictions based on Floquet theory.Our findings may open a new line of research usingdissipative Floquet engineering to control periodicallydriven quantum systems. Specifically, it will be interest-ing to see whether in a conserving system time-periodicimaginary parts in an effective single-particle equationof motion can be induced not only by losses but ratherby interactions, and if they can be controlled so as toestablish topologically nontrivial, effective band struc-tures. The present plasmonic waveguide setup consti-tutes a model for man-body systems with particle loss toan external bath. The latter are often described by theLindblad formalism, if the bath is Markovian. The topo-logical structure of a periodically driven system, however,becomes visible in Floquet space only. In a many-bodydescription, this would call for a combination of the Flo-quet and the Lindblad techniques, which is a combinationis a fundamental, unresolved problem . Note added inproof: After submission of the final manuscript, a studyon a very similar subject appeared . METHODSNon-Hermitian Floquet theory
In momentum space the Hamiltonian of the drivenRice-Mele model with periodic dissipation reads, ˆ H k ( t ) = ( J + J ) cos ka σ x + ( J − J ) sin ka σ y + ( u a − i γ a )(1+ σ z ) / u b − i γ b )(1 − σ z ) / , (3)where the coefficients have the above time dependence, σ x , σ y , σ z are the Pauli matrices acting in ( A, B ) sub-lattice space, and k and a denote momentum and thelattice constant, respectively.We now develop the Floquet formalism for non-Hermitian, periodic Hamiltonians. Due to time periodic-ity, the eigenstates of ˆ H k obey the Floquet theorem , | Ψ kα ( t ) (cid:105) = e − i ε kα t | φ kα ( t ) (cid:105) , (4)where a Greek index α ∈ { , } denotes the band quan-tum number originating from the two sublattices, and | φ kα ( t ) (cid:105) = | φ kα ( t + T ) (cid:105) are time-periodic states which,by construction, obey the Floquet equation H k ( t ) | φ kα ( t ) (cid:105) = ε kα | φ kα ( t ) (cid:105) , (5)with ˆ H k ( t ) := [ ˆ H k ( t ) − i ∂ t ] . The non-Hermiticity is ac-counted for by complex quasienergies ε kα . Note thatfor nonlinear or interacting, dissipative systems the Flo-quet theorem would generally not hold due to non-periodic, decaying density terms in the Hamiltonian. Ex-panding the | φ kα ( t ) (cid:105) in the basis of time-periodic func-tions, | φ kα ( t ) (cid:105) = (cid:80) n e − i n Ω t | u nk,α (cid:105) (Floquet representa-tion), Equation (5) takes the form of a discrete matrixFloquet-Schrödinger equation, (cid:88) l,γ ( H k ) nlβγ u lmk,γα = ε kα u nmk,βα , (6)where ( H k ) nlβγ = [( H k ) nlβγ − n Ω δ nl δ βγ ] is the time-independent Floquet Hamiltonian and ( H k ) nlβγ the repre-sentation of ˆ H k ( t ) in the basis of time-periodic functions, { e − i n Ω t | n ∈ Z } . Equation (6) determines the eigenval-ues ε kα and the eigenvector components u nmk,βα ∈ C forthe above Floquet expansion. Since there are as manyeigenvectors as the dimension of the Floquet Hamilto-nian, these components not only carry a RM sublatticeindex β and a Floquet expansion index n , but also aband index α and a Floquet index m to label the differ-ent eigenvectors. Thus, ( u nmk,βα ) is the matrix comprisedof column eigenvectors of Equation (6). Note that Equa-tion (6) cannot be diagonalized separately in the sublat-tice space ( βα ) and in the Floquet space ( nm ) becauseof the entanglement of both spaces.In quantum mechanics, expectation values of an ob-servable ˆ A are calculated as overlap matrix elements like (cid:104) Ψ | ˆ A | Ψ (cid:105) , which defines the standard scalar product inHilbert space. However, the eigenstates of a NH Hamil-tonian ˆ H k are generally not simultaneously eigenstatesof ˆ H † k . As a consequence, they do not constitutean orthonormal basis with respect to the standard scalarproduct of quantum mechanics. This hampers the expan-sion of a quantum state | Ψ( t ) (cid:105) , prepared with a given ini-tial condition | Ψ( t = 0) (cid:105) as in the experiments, in termsof Hamiltonian eigenstates. For the sake of orthonormalbasis expansions, a scalar product in Hilbert space can bedefined by constructing the dual (bra) states (cid:104) (cid:101) Ψ kα | cor-responding to the ket states | Ψ kα (cid:105) in the following way.When | u mkα (cid:105) is an eigenstate of Equation (6), it is clearthat there exists an, in general different, adjoint state | (cid:101) u mk,α (cid:105) such that H † k | (cid:101) u mk,α (cid:105) = ε ∗ kα | (cid:101) u mk,α (cid:105) . (7)The dual state is then obtained as (cid:104) (cid:101) u mk,α | = | (cid:101) u mk,α (cid:105) † , defin-ing the scalar product as (cid:104) (cid:101) u mk,α | u nk (cid:48) ,β (cid:105) . Using Equation (6)and the Hermitian conjugate of Equation (7), it is easyto show that the Floquet states fulfill the biorthonormal-ity (and corresponding completeness) relation (for non-degenerate ε kα (cid:54) = ε k (cid:48) β ) (cid:104) (cid:101) u mkα | u nk (cid:48) β (cid:105) = (cid:88) l,γ ( (cid:101) u mlk,αγ ) ∗ u lnk (cid:48) ,γβ = δ kk (cid:48) δ αβ δ mn . (8) The retarded Green’s function to the NH Hamiltonian H k is then the causal part of the time evolution operatorin Floquet representation, G nmk,βα ( t − t (cid:48) ) = − iΘ( t − t (cid:48) ) (cid:104) (cid:101) u nkβ | e − i H k ( t − t (cid:48) ) | u mkα (cid:105) , (9)which yields the spectral representation G nmk,βα ( E ) = (cid:88) l,γ ( (cid:101) u nlk,βγ ) ∗ u lmk,γα E − ε γ − l Ω + i0 . (10)Note that the lossy dynamics ( Im ε kα ≤ ) ensures theconvergence of the Fourier integral.An arbitrary state | Ψ( t ) (cid:105) can now be expanded in thebasis of Floquet states as | Ψ( t ) (cid:105) = (cid:88) k,α,n C nkα e − i( ε kα + n Ω) t | u nk,α (cid:105) , C nkα = (cid:104) (cid:101) u nk,α | Ψ(0) (cid:105) , (11)where the time-independent expansion coefficients C nkα are calculated at the initial time t = 0 using thebiorthonormality relation (8) and, thus, incorportate theinitial conditions on | Ψ( t ) (cid:105) .Using the expansion (11), physical expectation val-ues for time-evolving states can now be calculated in astraight-forward way and decay exponentially in time dueto the lossy dynamics of the system. For instance, thedensity of a driven-dissipative Floquet state reads, (cid:104) Ψ kα ( t ) | Ψ kα ( t ) (cid:105) = e − Γ kα t , (12)with the decay rate Γ kα = − ε kα > . In ourDLSPPW experiments below it is possible to directlymeasure the momentum- and energy-resolved populationdensity, i.e., intensity of the Fourier transform | Ψ k ( E ) (cid:105) ,which reads, I ( E, k ) = (cid:104) Ψ k ( E ) | Ψ k ( E ) (cid:105) (13) = (cid:88) n,m,αβ (cid:88) l,γ C l ∗ kβ C lkα ( u nlkβγ ) ∗ u lmk,γα ( E − ε ∗ kβ − l Ω − i0)( E − ε kα − l Ω + i0) . It is seen that, in general, this expression involves themixing of the RM bands ( α, β ), leading to a broad spec-tral distribution in the FBBZ. A distribution of this typeis shown in Fig. 2 (c). It is also possible to effectivelypopulate only one RM band α by populating at the ini-tial time T = 0 one single site of the initially nonlossysublattice, see Fig. 2 (a) and (b). In this case, the α − β cross terms vanish, and Equation (13) simplifies to I α ( E, k ) = (cid:104) Ψ nkα ( E ) | Ψ nkα ( E ) (cid:105) = (cid:88) n,m,l,γ C l ∗ kα C lkα ( u nlkαγ ) ∗ u lmk,γα | E − ε kα − l Ω | . (14)This is similar, albeit not identical, to the spectral densityobtained from the imaginary part of the Green’s functionin Equation (9). Thus, measurements of the populationdensity I ( E, k ) of a wave function initialized at t = 0 provide detailed information about the stationary spec-tral function. Dissipative transport quantization
For an adiabatic Thouless pump the number of par-ticles transported by one lattice constant per cycle isgiven by the Berry phase, i.e., the Berry flux penetratinga closed loop in Hamiltonian parameter space. There-fore it is quantized and time plays no role . For fastdriving, considering the states localized on single sitesas approximate eigenstates for small hopping amplitude,the driving-induced hopping to neighboring sites can beviewed as Landau-Zener tunneling in real space. How-ever, the topological nature of the process is better an-alyzed by working in momentum and frequency space.Namely, any nonzero driving frequency Ω turns the prob-lem into an effectively two-dimensional (2D) one dueto the periodicity in space and time. In this case, theHermitian RM model possesses two counterpropagatingchiral Floquet bands in the 2D FBBZ {− Ω / ≤ ε < Ω / − π /a ≤ k < π /a } , as depicted in Fig. 2 (a).Quantized tranport in a Floquet band is controlled bythe winding or Chern number of the band around theFBBZ . Here we investigate transport quantization ina general, fast pumped, dissipative situation. The ve-locity operator reads ˆ v = Re d ˆ H k /dk ( (cid:126) = 1 ) , i.e., foreach k -state | Ψ kα ( t ) (cid:105) , its eigenvalue is the group velocity d Re ε kα /dk . Thus, the spatial displacement of the parti-cle number during one pumping cycle carried by a singleFloquet state | Ψ kα ( t ) (cid:105) with a loss rate Γ kα is given by (cid:90) T dt (cid:104) Ψ kα ( t ) | ˆ v | Ψ kα ( t ) (cid:105)(cid:104) Ψ kα ( t ) | Ψ kα ( t ) (cid:105) = d Re ε kα dk T. (15)Note that the velocity expectation value is normalizedby the exponentially decaying probability density, Equa-tion (12), such that in Equation (15) the exponential de-cay factor exp( − Γ kα t ) drops out. The shift per cyclecarried by a band α with population density I α ( E, k ) (c.f. Equation (13)) is obtained by integrating over theFBBZ, that is, over the energy E and all k states, andreads, L α = (cid:90) FBBZ dE Ω (cid:90) π /a − π /a dk π /a I α ( E, k ) d Re ε kα dk T. (16)For a homogenously filled band, (cid:82) dE I α ( E, k ) / Ω = 1 ,this reduces to L α a = (cid:90) π /a − π /a dk π d Re ε kα dk T = Z . (17)For a periodically driven system, the dispersion Re ε k α is not only a periodic function of k , but its values arealso periodic with period Ω . That is, ε kα is a mappingfrom the 1D circle onto the 1D circle, Re ε α : S → S ,and wraps around the 2D torus of the FBBZ as shownin Fig. 2 (d). Equation (17) is the definition of thewinding number around the circle. It is seen that it as-sumes nonzero, integer values if the dispersion contin-uously covers the entire FBBZ in the frequency direc-tion, i.e., if it is gapless, (cid:82) dkd Re ε kα /dk = Z Ω = Z π T , since ε π /a , α = ε − π /a , α . This proves the second equalityin Equation (17) for a gapless dispersion and indicatestransport quantization. Samples
The DLSPPW arrays are fabricated by negative-tone gray-scale electron beam lithography (EBL) .The waveguides consist of poly(methyl methacrylate)(PMMA) ridges deposited on top of a 60 nm thick goldfilm evaporated on a glass substrate. The mean center-tocenter distance between the ridges is . µ m and the max-imum deflection from the center is . µ m . The resultingvariation of coupling constants is J ( z ) = J e − λ (1 − sin Ω z ) , J ( z ) = J ( z − T / with J = 0 . µ m − and λ = 1 . .The cross-section of each waveguide is controlled by theapplied electron dose during the lithographic process. Byvarying the electron dose along the z-axis we modulatethe waveguides’ cross-sections and hence the propagationconstants as β a ( z ) ≈ ¯ β − u cos (Ω z + ϕ ) − i γ a ( z ) , β b ( z ) = β a ( z − T / , where ¯ β = 6 .
62 + i0 . µ m − correspondsto the mean height
100 nm and the mean width
250 nm ofa waveguide and γ a ( t ) ≈ − γ Θ( u a ( z )) cos(Ω z + ϕ ) is theperiodic loss rate induced by coupling to free SPPs. Thechoice of such geometrical parameters is motivated bythe fact that strong losses due to coupling to continuumof free propagating SPPs occur when the height and thewidth of a waveguide are smaller than the correspondingmean values, i.e., β j ( z ) < ¯ β . Other sources of losses canbe assumed to be independent of z because their variationis negligibly small in comparison to this effect. Leakage radiation microscopy
SPPs are excited by focusing a TM-polarized laserbeam with free space wavelength λ =980 nm (NA ofthe focusing objective is 0.4) onto the grating couplerdeposited on top of the central waveguide (either sublat-tice A or B ). The propagation of SPPs in an array ismonitored by real- and Fourier-space leakage radiationmicroscopy . For this purpose, we use an oil immer-sion objective (63× magnification, NA=1.4) to collectthe leakage radiation. Real space intensity distributionsare recorded by imaging the sample plane onto a CMOScamera. The corresponding Fourier images are obtainedby imaging the back-focal plane of the objective onto thecamera. The directly transmitted laser beam is blockedby Fourier filtering. We note that we work in the single-mode waveguide regime for all cross sections used in theexperiments at the design wavelength. DATA AVAILABILITY
The data that support the findings of this study areavailable from the corresponding author upon reasonablerequest. All these data are directly shown in the corre- sponding figures without further processing. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] D. J. Thouless, “Quantization of particle transport,” Phys-ical Review B , 6083 (1983). Q. Niu and D. J. Thouless, “Quantised adiabatic chargetransport in the presence of substrate disorder and many-body interaction,” Journal of Physics A , 2453 (1984). M. J. Rice and E. J. Mele, “Elementary excitations of a lin-early conjugated diatomic polymer,” Physical Review Let-ters , 1455 (1982). D. J. Thouless, M. Kohmoto, M. P. Nightingale, andM. den Nijs, “Quantized hall conductance in a two-Dimensional periodic potential,” Physical Review Letters , 405 (1982). C. L. Kane and E. J. Mele, “Z2 topological order and thequantum spin Hall effect,” Physical Review Letters ,146802 (2005), arXiv:cond-mat/0506581. M. Franz and L. Molenkamp, eds.,
Topological Insulators ,Vol. 6 (Elsevier, 2013). M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer,D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Sza-meit, “Photonic Floquet topological insulators,” Nature , 196 (2013), arXiv:1212.3146. M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. M. Tay-lor, “Imaging topological edge states in silicon photonics,”Nature Photonics , 1001 (2013), arXiv:1302.2153. L. Fu and C. L. Kane, “Time reversal polarization and aZ2 adiabatic spin pump,” Physical Review B , 195312(2006), arXiv:cond-mat/0606336. N. P. Armitage, E. J. Mele, and A. Vishwanath, “Weyl andDirac semimetals in three-dimensional solids,” Reviews ofModern Physics , 15001 (2018), arXiv:1705.01111. R. D. King-Smith and D. Vanderbilt, “Theory of polar-ization of crystalline solids,” Physical Review B , 1651(1993). M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger,and I. Bloch, “A Thouless quantum pump with ultracoldbosonic atoms in an optical superlattice,” Nature Physics , 350 (2016), arXiv:1507.02225. S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa,L. Wang, M. Troyer, and Y. Takahashi, “TopologicalThouless pumping of ultracold fermions,” Nature Physics , 296 (2016), arXiv:1507.02223. Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zil-berberg, “Topological states and adiabatic pumping in qua-sicrystals,” Physical Review Letters , 106402 (2012),arXiv:1109.5983. P. Titum, E. Berg, M. S. Rudner, G. Refael, and N. H.Lindner, “Anomalous Floquet-Anderson Insulator as anonadiabatic quantized charge pump,” Physical Review X , 21013 (2016), arXiv:1506.00650. N. H. Lindner, E. Berg, and M. S. Rudner, “Uni-versal chiral quasisteady states in periodically drivenmany-body systems,” Physical Review X , 11018 (2017),arXiv:1603.03053. L. Privitera, A. Russomanno, R. Citro, and G. E. Santoro,“Nonadiabatic Breaking of Topological Pumping,” Physi-cal Review Letters , 106601 (2018), arXiv:1709.08457. A. O. Caldeira and A. J. Leggett, “Path integral approachto quantum Brownian motion,” Physica A , 587 (1983). S. Longhi, D. Gatti, and G. Della Valle, “Robust lighttransport in non-hermitian photonic lattices,” Scientific re-ports , 13376 (2015). T. Kottos, H. Ramezani, Z. Lin, T. Eichelkraut, H. Cao,and D. N. Christodoulides, “Unidirectional invisibility ofphotonic periodic structures induced by PT-symmetricarrangements,” Optics InfoBase Conference Papers ,213901 (2011). L. Feng, Y. L. Xu, W. S. Fegadolli, M. H. Lu, J. E. Oliveira,V. R. Almeida, Y. F. Chen, and A. Scherer, “Experimentaldemonstration of a unidirectional reflectionless parity-timemetamaterial at optical frequencies,” Nature Materials ,108 (2013). T. E. Lee, “Anomalous Edge State in a Non-HermitianLattice,” Physical Review Letters , 133903 (2016),arXiv:1610.06275. D. Leykam, K. Y. Bliokh, C. Huang, Y. D. Chong,and F. Nori, “Edge Modes, Degeneracies, and Topologi-cal Numbers in Non-Hermitian Systems,” Physical ReviewLetters , 40401 (2017), arXiv:1610.04029. S. Malzard, C. Poli, and H. Schomerus, “Topologi-cally Protected Defect States in Open Photonic Systemswith Non-Hermitian Charge-Conjugation and Parity-TimeSymmetry,” Physical Review Letters , 200402 (2015),arXiv:1508.03985. M. S. Rudner and L. S. Levitov, “Topological transition ina non-hermitian quantum walk,” Physical Review Letters , 65703 (2009), arXiv:0807.2048. T. Ozawa and I. Carusotto, “Anomalous and quantum halleffects in lossy photonic lattices,” Physical review letters , 133902 (2014). C. M. Bender, D. C. Brody, H. F. Jones, and B. K. Meis-ter, “Faster than Hermitian quantum mechanics,” PhysicalReview Letters , 40403 (2007), arXiv:quant-ph:0609032. S. Ibáñez, S. Martínez-Garaot, X. Chen, E. Torrontegui,and J. G. Muga, “Shortcuts to adiabaticity for non-Hermitian systems,” Physical Review A , 23415 (2011),arXiv:1106.2776. B. T. Torosov, G. Della Valle, and S. Longhi, “Non-Hermitian shortcut to adiabaticity,” Physical Review A ,52502 (2013), arXiv:1306.0698. F. Bleckmann, Z. Cherpakova, S. Linden, and A. Alberti,“Spectral imaging of topological edge states in plasmonicwaveguide arrays,” Physical Review B , 45417 (2017). D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Dis-cretizing light behaviour in linear and nonlinear waveguidelattices,” Nature , 817 (2003). J. K. Asboth, L. Oroszlany, and A. Palyi,
A Short Courseon Topological Insulators , Lecture Notes in Physics, Vol.919 (Springer, 2015) arXiv:1509.02295. J. E. Avron and Z. Kons, “Quantum response at finite fieldsand breakdown of Chern numbers,” Journal of Physics A , 6097 (1999), arXiv:math-ph/9901018. Z. Fedorova (Cherpakova), C. Jörg, C. Dauer, F. Letscher,M. Fleischhauer, S. Eggert, S. Linden, and G. von Frey-mann, “Limits of topological protection under local peri-odic driving,” Light: Science and Applications , 63 (2019),arXiv:1807.02321. S. Longhi, “Rectification of light refraction in curvedwaveguide arrays,” Optics Letters , 458 (2009),arXiv:1001.0992. F. Dreisow, Y. V. Kartashov, M. Heinrich, V. A. Vys-loukh, A. Tünnermann, S. Nolte, L. Torner, S. Longhi,and A. Szameit, “Spatial light rectification in an opticalwaveguide lattice,” Europhysics Letters , 44002 (2013). A. Schnell, A. Eckardt, and S. Denisov, “Is there a floquetlindbladian?” Physical Review B , 100301 (2020). B. Höckendorf, A. Alvermann, and H. Fehske, “Topologi-cal origin of quantized transport in non-Hermitian Floquetchains,” Phys. Rev. Research , 023235 (2020). A. Eckardt, “Colloquium: Atomic quantum gases in pe-riodically driven optical lattices,” Reviews of ModernPhysics (2017), arXiv:1606.08041. Y. Chen and H. Zhai, “Hall conductance of a non-Hermitian Chern insulator,” Physical Review B , 245130(2018), arXiv:1806.06566. A. Gómez-León and G. Platero, “Floquet-Bloch theory andtopology in periodically driven lattices,” Physical ReviewLetters , 200403 (2013), arXiv:1303.4369. D. C. Brody, “Biorthogonal quantum mechanics,” Journalof Physics A , 35305 (2014), arXiv:1308.2609. T. Kitagawa, E. Berg, M. Rudner, and E. Demler,“Topological characterization of periodically driven quan-tum systems,” Physical Review B , 235114 (2010),arXiv:1010.6126. A. Block, C. Etrich, T. Limboeck, F. Bleckmann, E. So-ergel, C. Rockstuhl, and S. Linden, “Bloch oscillations in plasmonic waveguide arrays,” Nature Communications ,3843 (2014), arXiv:1612.01850. A. Drezet, A. Hohenau, D. Koller, A. Stepanov, H. Ditl-bacher, B. Steinberger, F. R. Aussenegg, A. Leitner, andJ. R. Krenn, “Leakage radiation microscopy of surface plas-mon polaritons,” Materials Science and Engineering B ,220 (2008), arXiv:1002.0725. J. Petráček and V. Kuzmiak, “Transverse Anderson local-ization of channel plasmon polaritons,” Physical Review A , 023806 (2018). ACKNOWLEDGMENTS
This work was supported by the Deutsche Forschungs-gemeinschaft (DFG) within SFB/TR 185 (277625399)and the Cluster of Excellence ML4Q (390534769).
AUTHOR CONTRIBUTIONS
ZF fabricated the samples, conducted the experiments,and performed the numerical calculations with the help ofHQ. SL and JK conceived the project and supervised ZFand HQ, respectively. All authors contributed to the dis-cussion and interpretation of the results as well as writingthe manuscript.