An analog of photon-assisted tunneling in a periodically modulated waveguide array

We theoretically report an analog of photon-assisted tunneling (PAT) originated from dark Floquet state in a periodically driven lattice array without a static biased potential by studying a three-channel waveguide system in a non-high-frequency regime. This analog of PAT can be achieved by only periodically modulating the top waveguide and adjusting the distance between the bottom and its adjacent waveguide. It is numerically shown that the PAT resonances also exist in the five-channel waveguide system and probably exist in the waveguide arrays with other odd numbers of waveguides, but they will become weak as the number of waveguides increases. With origin different from traditional PAT, this type of PAT found in our work is closely linked to the existence of the zero-energy (dark) Floquet states. It is readily observable under currently accessible experimental conditions and may be useful for controlling light propagation in waveguide arrays.

waveguide system. Our numerical analysis discovers that dark CDT (strong suppression of tunneling) and dark Floquet state still exist in the three-channel waveguide system even in the non-high-frequency regimes where the modulation frequency of the periodically modulated top waveguide is roughly equal to or smaller than the coupling strength between the bottom and its adjacent (middle) waveguide. However, when integer multiples of the modulation frequency approximately equal to the coupling strength between the bottom and its adjacent (middle) waveguide, the light tunneling from the top waveguide to the others is restored as a clear signature of photon-assisted tunneling. In our previous two works (refs 7 and 8), we have addressed a novel extension of coherent destruction of tunneling (CDT) and its application for coherent control. In the current work, however, we report a series of photon-assisted tunneling (PAT) resonances in the considered model by moving the last waveguide (site) closer to its neighbor. Different from the PAT observed in the earlier studies which usually requires a static biased potential to initialize the system in a self-trapped state, this type of PAT is closely linked to a dark Floquet state with zero quasi-energy. Our results are applicable for the five-channel waveguide system and also extendable to waveguide arrays with an odd number of waveguides.

Results
The physical model for periodically modulated waveguide system. As illustrated in Fig. 1, we present an optical implementation of our Hamiltonian in the form of a linear array of tunneling-coupled optical waveguides which is characterized in: (i) that the refractive index of the top boundary waveguide is modulated periodically along the propagation direction; and (ii) that the distance w 2 between the bottom boundary waveguide and its nearest neighbor is different from other identical nearest-neighboring spacings w 1 . Thus, through adjustment of the distance w 2 , the coupling strength between the bottom boundary waveguide and its nearest neighbor can be tuned to be sufficiently large in comparison to the modulation frequency. The role of photon is played by a periodic modulation of the the top boundary waveguide with a certain modulation frequency. With the use of the coupled-mode approach, the optical-field dynamics in such structures are described by the following set of equations 9 , , where a j is field mode amplitude in the j-th waveguide, z the propagation distance, Ω 1 the coupling strength between neighboring waveguides with spacing w 1 , Ω 2 the coupling strength between the bottom boundary waveguide (j = N) and its adjacent waveguide (j = N− 1), and σ(z) the normalized difference between the propagation constants of the top boundary waveguide and the other waveguides of the array. As in ref. 9, instead of modulating an array of waveguide in a uniform fashion, modulating one certain waveguide selectively is implemented here. We consider a harmonic modulation of the linear refractive index of the top boundary waveguide along the propagation direction with σ(z) = A sin (ωz), where A is the relative depth of the harmonic longitudinal modulation, and ω is the spatial modulation frequency. Such a periodic modulation is well within the capacity of current experiments 23,24 . In a different perspective, the above Equation (1) can be regarded as describing the system of a quantum wave in a periodically driven lattice array if z is viewed as time t. As is well known, the periodic time-dependent equation (1)  are periodic with modulation period T = 2π/ω. PAT in three-guide system. We start our consideration for the three-guide system, the minimal one for odd-N-state systems. In this case, the dynamical equations are of the form To study the system's beam dynamics, we solve numerically the coupled-mode equations with the light initially localized in the 1-th waveguide (the top boundary waveguide). With the numerical solution, we compute the intensity of light staying in the initial waveguide by P 1 (z) = |a 1 (z)| 2 and measure the minimum value of P 1 (z) over a long-enough propagation distance. When Min(P 1 ) is not zero, the tunneling is suppressed as the light is not allowed to be fully transferred from the 1-th mode (guide) to the other modes (guides). In Fig. 2(a), we display Min(P 1 ) versus the coupling strength Ω 2 at the fixed parameters A = 6.6, ω = 3, Ω 1 = 1. For Ω 2 = 0, the system is in fact a two-guide system in which the conventional CDT happens only at the isolated degeneracy point of the quasi-energies, and consequently Min(P 1 ) takes a zero value because the ratio of driving amplitude and frequency is set with small deviation from the isolated degeneracy point. When Ω 2 is increased from zero, the value of Min(P 1 ) becomes relatively large except at a series of very sharp dips. In general, periodic modulation of the top boundary waveguide will yield a significant suppression of the light tunneling in the three-guide system even with Ω 1 ≠ Ω 2 , as shown in Fig. 2(a). However, at particular values of the coupling strength Ω 2 , Ω 2 ≈ nω with n being integer, the value of Min(P 1 ) exhibits a series of sharp dips, in analogy to the n-photon-like resonances which destroy the effect of suppression of tunneling. It also can be observed that as the coupling strength Ω 2 is increased, the higher photon-like resonances become very weak and thus are almost not visible.
For a deep insight into the tunneling dynamics obtained in Fig. 2(a), we numerically compute the quasi-energies and Floquet states of this system as shown in Fig. 2(b,c). As shown in Fig. 2(b), this three-state system always possesses a Floquet state with zero quasi-energy regardless of the value of Ω 2 . The other two quasi-energies make a set of close approaches to each other as Ω 2 is increased. At the points of close approach, namely, at Ω 2 ≈ nω, the value of Min(P 1 ) displays sharp dips and the tunneling is significantly restored. We also plot the time-averaged population distribution Fig. 2(c). Considering that the dynamic is determined by the Floquet states, self-trapping (suppression of tunneling) of light intensity initially populating at the 1-th mode (guide) will take place if 〈 F 1 〉 > 0.5 holds. As seen in Fig. 2(c), the zero-energy Floquet state has negligible population at the central mode (guide) while the population 〈 F 1 〉 is much larger than 0.5 for all values of Ω 2 except those in the vicinity of Ω 2 ≈ nω. Correspondingly, suppression of tunneling (CDT) occurs for all values of Ω 2 except the locations of photon resonances, as shown in Fig. 2(a). The Floquet state with zero quasi-energy is essentially the dark Floquet state, not only for its zero quasi-energy but also for its negligible population at the central waveguide; the suppression of tunneling (CDT) is of the dark CDT as it is caused by the dark Floquet state rather than level degeneracy (as seen in the inset on the right side of Fig. 2(b)). In fact, the CDT-PAT transition found in Fig. 2(a) is closely related to the sharp localization-delocalization transition of population 〈 F 1 〉 for the zero-energy (dark) Floquet state. Note that the dark Floquet state originally discovered and defined in the high-frequency limit can be reduced to the well-known dark state by means of high-frequency averaging method 7,8 . However, the dark Floquet state and the associated CDT can still exist in the non-high-frequency regimes where the coupling strength Ω 2 is much larger than the modulation frequency and the high-frequency averaging method is invalid. These results will greatly enrich our understanding of dark Floquet state and dark CDT.
To get further study of n-photon-like resonances, we show how the value of Min(P 1 ) varies under conditions that the modulation amplitude is increased, while its frequency is held constant at ω = 3 and the coupling strength held Ω 2 = nω, Ω 1 = 1. At multiphoton resonances, the tunneling is restored in general. However, CDT will occur at certain values of the amplitude of the driving field 10,12 . It means possibility of moving the system between PAT and CDT through variation of the amplitude of the driving field. Figure 3(a) shows the case of n = 1 photon resonance. For A = 0, the system is self-trapped in the 1-th waveguide due to the existence of an imbalanced dark state (− Ω 2 /Ω 1 , 0, 1) T with Ω 2 /Ω 1 > 1, and thus the value of Min(P 1 ) is nonzero. When the periodic driving is applied, as A is increased from zero, the value of Min(P 1 ) rapidly drops to zero, which indicates that the photon resonance destroys the self-trapping effect. When A/ω is increased further, Min(P 1 ) takes extremely low values about zero except at a sequence of very narrow peaks. These peaks are precisely centered at A/ω = 3.83, 7.01, .… , the zeros of J 1 (A/ω). In Fig. 3(b,c), we plot the quasi-energies and the population distributions of the dark Floquet state for the first photon resonance (n = 1). Apparently, the quasi-energies are degenerate when J 1 (A/ω) = 0, and when away from A = 0 the dark Floquet state has averaged population at the 1-th mode (guide) well below the value of 0.5. Therefore it can be concluded that the well-defined quasienergy crossings instead of the dark Floquet state are the origin of the extremely sharp peaks (CDT resonances) seen in Fig. 3(a). The occurrence of CDT resonances centered at the degeneracy points of quasienergies (zeros of Bessel functions) is a feature commonly found in the periodically driven systems 10,12 , even in the case that the conditions of multiphoton resonances are satisfied.
In Fig. 3(d), we show the values of Min(P 1 ) in the three-guide optical system for the n = 2 resonances. As is expected, the values of Min(P 1 ) exhibit a number of extremely sharp peaks centered on the zeros of J 2 (A/ω) where the quasi-energies will be degenerate; see Fig. 3(e). Similar to the case of n = 1 resonance, the sharp peaks in the n = 2 resonance is also caused by the level degeneracy rather than the dark Floquet state, as it is shown in Fig. 3(f) that the population 〈 F 1 〉 belonging to the dark Floquet state is well below 0.5 at the points of quasi-energy crossings.
In order to observe the n-photon-like resonances from a different angle, we also plot Min(P 1 ) as a function of the modulation frequency ω for two fixed parameters Ω 2 = 2 and Ω 2 = 3 in Fig. 4(a,b) respectively, for the case of the system parameters Ω 1 = 1 and A = 6.6. We can readily observe that the n-photon-like resonances occur at comparatively broad interval around ω = Ω 2 /n. The width of such photon-assisted tunneling resonances is much larger than those of PAT resonances observed in the literature. We now elaborate the physics underlying this photon-assisted tunneling resonances. The actual resonance condition does not refer directly to Ω 2 = nω but rather to the tunneling frequency of the model (2) without periodic modulation. The unmodulated three-guide optical system admits three energy level as ± Ω + Ω 0, 2 . In such a system, the existence of imbalanced dark state with zero energy results in the suppression of tunneling when the periodic modulation is switched off. The periodic modulation effectively creates "photons" that bridge the energy gap between neighboring energy levels. Thus, a photon-assisted tunneling resonance can occur at a modulation frequency which satisfies the resonance condition ω 0 = nω. When Ω 2 is considerably larger than Ω 1 , the energy difference ω 0 of the unmodulated system will become principally characterized by Ω 2 and therefore the resonance condition is approximately given by Ω 2 = nω. As clearly seen in the inset in Fig. 4(c), the position of n-photon-like resonance does slightly shift with increasing Ω 1 due to the dependence of the energy difference (tunneling frequency) ω 0 on Ω 1 .
Scientific RepoRts | 6:35744 | DOI: 10.1038/srep35744 PAT in five-guide system. We now turn to the case of the five-guide optical system and investigate the beam dynamics by direct integration of the time-dependent Schrödinger equation (1) (j = 2, 3 and N = 5) with the light initially localized at the guide 1. In Fig. 5, fixing the parameters A = 6.6, ω = 3, Ω 1 = 1 as before, we show the value of Min(P 1 ) as a function of Ω 2 which exhibits a sequence of PAT resonances with similar behavior as that of a three-guide system. The higher n-photon resonances with n ≥ 3 become very weak, almost invisible, as illustrated in the inset of Fig. 5(a). By comparison of Fig. 2(a) with Fig. 5(a), it is apparent that the same order PAT resonance for the five-guide system is much narrower and weaker than for the three-guide system. Like the case of the three-guide system, this five-guide system also possesses a dark Floquet state with zero quasi-energy and negligible population at all of the even j-th guides (modes), as illustrated in Fig. 5(b,c). Reason for the existence of the analog of PAT resonances in the five-guide system lies in that population distribution 〈 F 1 〉 for the dark Floquet state simultaneously displays a series of sharp dips at the positions of PAT resonances [see Fig. 5(c)].
In Fig. 5(d), we plot Min(P 1 ) obtained in the five-guide system as a function of the modulation parameter A/ω for the 1-photon resonance Ω 2 = 2.8, ω = 3, Ω 2 ≈ ω. As discussed before, we can observe that the values of Min(P 1 ) are peaked at the zeros of J 1 (A/ω), at which CDT occurs, while between the peaks Min(P 1 ) take extremely low values as result of PAT. However, compared with the case of three-guide system, the peaks in Min(P 1 ) are considerably lower and broader. As can be clearly seen from Fig. 5(e,f), the peaks in Min(P 1 ) are indeed centered at the points of closest approach of the quasi-energies where the dark Floquet state has a population 〈 F 1 〉 > 0.5. The numerical results establish again a firm link between PAT and dark Floquet state in our considered systems. Tunneling dynamics in the four-and six-guide optical systems and beyond. Finally, we briefly discuss the case of the four-and six-guide systems. The dynamics for N = 4 and N = 6 are presented in Fig. 6 on the basis of a full numerical analysis of Equation (1) with the light initially populated in the guide 1. It tells the existence of a sharp transition from CDT to complete tunneling for both cases of N = 4 and N = 6 when the coupling strength Ω 2 is increased from zero. A close examination of the tunneling dynamics at Ω 2 = nω shows that the value of Min(P 1 ) displays narrow peaks nearly at zeros of J 0 (A/ω) where a pair of quasi-energies become degenerate. This closely resembles the case of the high-frequency modulation ω ≫ max(Ω 1 , Ω 2 ) where CDT is dominated by the zeros of J 0 (A/ω). As shown in Fig. 6(b,e), the localization centered nearly at zeros of J 0 (A/ω) is fairly smaller for the four-guide system, but still generates high peaks for the six-guide system. Moreover, we have simulated multiwaveguide systems of other numbers of waveguides. The numerical results, which are not displayed here, show that the PAT resonances probably occur in all the odd-N-guide optical systems, while the PAT resonances become weaker with the increase of number of guides. However, all the even-N-guide optical systems exhibit CDT to complete tunneling transition without appearance of n-photon-like resonance when the coupling strength Ω 2 is increased from zero, which is totally different from the case of odd-N-guide system.
Possibility of experimental realization. Now, we discuss the experimental possibility of observing our theoretical predictions based on the coupled-mode equations. A more rigorous dynamics for our system can be simulated by the Schrödinger equation for the dimensionless field amplitude E, which describes the light propagation along the z axis of an array of N waveguides 25,26 Here x and z are the normalized transverse and longitudinal coordinates, p describes the peak contrast (variation) in the refractive index between the unmodulated guiding structure and the substrate. The dimensionless variables x, z and p are related to the corresponding physical quantities x′, z′ and p′ by the scaling  in the experiments 25,26 . The normalized dimensionless power is defined by . After a transformation of dimensionless field intensity to the physical field intensity defined by ρ ′ = ′ E k P P E ( / (0)/ ) 2 2 , we get the physical input power (in W/m) . For our system, the refractive index of the first waveguide is harmonically modulated along the propagation direction, while all other N− 1 waveguides are unmodulated. The corresponding refractive index distribution of this kind of waveguide system is given by 1 with the position of each waveguide being x j , the channel width w x , the longitudinal modulation amplitude μ, and the modulation frequency ω. Therein the super-Gaussian function −x w exp( / ) x 6 6 describes the profile of a single waveguide with width w x . In our discussion, all the waveguide spacings [x j − x j−1 ] are identical except that the spacing between the bottom boundary waveguide and its neighbor is variable.
In what follows, we will illustrate our main results with a triplet waveguide system (N = 3) by directly integrating the field propagation Equation (3) with realistic experimental parameters. We set w x = 0.3, p = 2.78, Figure 5. PAT in five-guide optical system. The left column: (a) the minimum value of population distribution at guide 1, Min(P 1 ), versus Ω 2 with A = 6.6, ω = 3, Ω 1 = 1; (b) the corresponding quasi-energy ε versus Ω 2 and (c)the time-averaged population 〈 F j 〉 belonging to the zero-energy (dark) Floquet state versus Ω 2 . The right column: (d) Min(P 1 ) versus A/ω for the 1-photon resonance ω = 3, Ω 2 = 2.8, Ω 1 = 1; (e) the corresponding quasi-energy ε versus A/ω and (f) the time-averaged population 〈 F j 〉 belonging to the zero-energy (dark) Floquet state versus A/ω. Scientific RepoRts | 6:35744 | DOI: 10.1038/srep35744 μ = 0.2 and ω = 3.45 × (π/100). We characterize two distinct waveguide spacings as w 1 and w 2 respectively, where w 1 = x 1 − x 0 stands for the separation between the top waveguide and the middle waveguide and w 2 = x 0 − x −1 the separation between the bottom waveguide and the middle waveguide. Further we set w 1 = 3.2 and choose different values of w 2 to observe PAT resonance. As in the experiments 25,26 , w x and x j are in units of 10 μm, and p = 2.78 corresponds to a real refractive index of 3.1 × 10 −4 (weakly guided). The small change of index between the guide (core) and the substrate, which can be fabricated by using a femtosecond-writing method 27 , allows the weak-guidance approximation to be employed for obtaining the scalar wave equation (3). In all simulations we excited the top channel at z = 0, using the fundamental linear mode of the isolated waveguide. It is instructive to normalize the modulation frequency to the beating frequency of the unmodulated linear dual-core coupler with spacing w 1 , Ω b = 2Ω 1 = 2π /Z b , where Z b is a beating period representing the shortest distance for the light returning to the input waveguide. For our set of parameters one has Z b = 100 and thus ω = 3.45 Ω 1 .
The beam dynamics of a three-guide optical system are visualized in Fig. 7 for three values of w 2 , which firmly verifies the predictions from the coupled-mode Equation (2). In Fig. 7. the left column shows the refractive index distribution R(x, z) and the right column shows the evolution of light intensity |E(x, z)| 2 along the propagation direction. It can be readily observed from Fig. 7(a,b) that the light tunneling is almost completely suppressed, as the three-channel waveguide system has equal channel spacing w 1 = w 2 = 3.2. At w 2 = 2.22, the light coupling between the waveguide channels is restored [see Fig. 7(c,d)]. The revival of light tunneling is a signature of PAT resonance predicted by the coupled-mode theory. In fact, our numerical simulation (not shown here) reveals that the beating period of an unmodulated linear dual-core coupler with a channel spacing 2.22 is about 100/3.45. As such, we have Ω 2 ≈ 3.45Ω 1 and Ω 2 ≈ ω, which is in fact the position of the first photon resonance. As the channel spacing w 2 is reduced further, it is expectable to observe again the strong suppression of light tunneling [see Fig. 7(e,f)]. These results are in good qualitative agreement with those in Fig. 2(a) based on the coupled-mode equation.

Discussion
We have theoretically reported an analog of PAT in a three-channel waveguide system, in which the space separation between the bottom and the middle waveguides is adjustable and the refractive index of the top waveguide is modulated periodically along the light propagation direction. With the standard coupled-mode theory, the system can be described by a driven three-state discrete model with two distinct coupling strengths Ω 1 and Ω 2 , where Ω 1 stands for the coupling strength between states 1 and 2, and Ω 2 between states 2 and 3. In studying the three-state discrete model, we have found that (i) a strong suppression (CDT) associated with the zero-energy (dark) Floquet state persists even in the non-high-frequency modulation regimes where ω ≤ max(Ω 1 , Ω 2 ) except at a series of resonance positions; (ii) at particular values of the coupling strength Ω 2 , Ω 2 ≈ nω with n being integer, the tunneling dynamics is (partly) restored, analogous to the n-photon-like resonances which overcome the effect of suppression of tunneling. The numerical calculations illustrate that the PAT resonances exist in the five-state system and also probably exist in the systems with arbitrary odd number of coupled states. In particular, the PAT resonances will become weaker with the increase of number of states (modes). This type of PAT found in our work has a different origin from traditional PAT. It is closely related to the existence of the dark Floquet state. The main results are demonstrated by the direct numerical simulations of propagation dynamics based on the full continuous model with realistic experimental parameters, which indicates that the PAT found in our work can be readily tested in the current experimental setup. Because of the equivalence between the Schrödinger equation and the optical wave equation, we select the engineered photonic lattice as our model system due to its robust and feasibility in the realistic experiment and, obviously, our results can also be applied to the quantum systems such as cold atom or trapped ion in optical lattices and electron transport in quantum dot chains.

Methods
To calculate the Floquet states and corresponding quasienergies of the coupled-mode Equation (1), we carry out a procedure that runs in principle as follows. We first numerically integrate Eq. (1) from z = 0 to T with the initial conditions a j (0) = δ jm , m = 1, 2, 3, … N. The time evolution operator over one modulation period U(T, 0) is given by a N × N propagator matrix, where the matrix element U j,m is the value a j (T) obtained by solving Eq. (1) with the initial condition a l (0) = δ jm . Given that the Floquet states are eigenstates of U(T, 0) with eigenvalues exp(− iεT), the quasienergies and corresponding Floquet eigenmodes are numerically computed by direct diagonalization of U(T, 0).
In our numerical simulations, the coupled-mode equation is numerically solved with Runge-Kutta method and the simulations of propagation dynamics based on the continuous wave equation are implemented with the use of the split-step Fourier method. In the meantime, imaginary-time evolution method is used in finding the fundamental mode of the isolated top waveguide which is taken as input beam for numerical demonstrations of PAT in the continuous model.