Observation of Bloch oscillations in complex PT-symmetric photonic lattices

Light propagation in periodic environments is often associated with a number of interesting and potentially useful processes. If a crystalline optical potential is also linearly ramped, light can undergo periodic Bloch oscillations, a direct outcome of localized Wannier-Stark states and their equidistant eigenvalue spectrum. Even though these effects have been extensively explored in conservative settings, this is by no means the case in non-Hermitian photonic lattices encompassing both amplification and attenuation. Quite recently, Bloch oscillations have been predicted in parity-time-symmetric structures involving gain and loss in a balanced fashion. While in a complex bulk medium, one intuitively expects that light will typically follow the path of highest amplification, in a periodic system this behavior can be substantially altered by the underlying band structure. Here, we report the first experimental observation of Bloch oscillations in parity-time-symmetric mesh lattices. We show that these revivals exhibit unusual properties like secondary emissions and resonant restoration of PT symmetry. In addition, we present a versatile method for reconstructing the real and imaginary components of the band structure by directly monitoring the light evolution during a cycle of these oscillations.

In a system encompassing amplifying and attenuating regions, one may anticipate that light will follow the direction of gain. Yet, as we will see, Bloch oscillations in the presence of both gain and loss provide a counter example refuting this perception. Thus, the question arises, as to which process will ultimately dominate the other. Will light still follow the direction of amplification, or will Bloch oscillations instead force the wavepacket to enter the loss region and hence experience attenuation?
In this Report, we address this intriguing question, by considering the role of the complex band structure on Bloch oscillations when taking place in non-Hermitian PT-symmetric mesh lattices. We show that by controlling the period of Bloch oscillations, one can establish pseudo-Hermitian wavepacket propagation that continuously undergoes through successive cycles of amplification and attenuation. Along these lines, we observe a resonant restoration of PT symmetry as well as secondary emissions. This special case of PT-symmetric Bloch oscillations is then generalized to arbitrary Bloch gradients in locally PT-symmetric systems. In this case, the wavepacket trajectories in this mesh lattice can be analyzed in order to reconstruct both the real and imaginary parts of the band diagram associated with this system.

Experimental setup and time multiplexing
In what follows, we provide a brief introduction into time multiplexed systems based on coupled fiber loops and their relation to mesh lattices. In our experiments, we use two coupled fiber loops of different lengths, to emulate a 1+ 1D lattice (see Fig. 1a) 12 . Erbium doped fiber amplifiers compensate losses inside the fiber loops and additionally, acousto-optical modulators are inserted and set to a transmission of 50%. By increasing or decreasing this transmission ratio, an effective gain or loss is generated. A detailed explanation of the experiment is reported in 21 . Depending on whether pulses travel through the longer or shorter loop, they either propagate on the equivalent lattice from "North West" to "South East" or vice versa (see Fig. 1b). Thus, their position on the lattice is encoded by the arrival times after m roundtrips 22 . Note that in each roundtrip, a pulse can jump either to an earlier or later arrival time slot, thus decreasing or increasing its position n on the equivalent lattice by ±1. In the experiment, we start with a single pulse at time = m 0 and at position = n 0. After passing through the 50/50 coupler the pulse splits up into two pulses at time = m 1 and at position = − n 1 and = n 1 with half of the intensity. Furthermore, the fiber coupler induces a phase shift of π/2 for the case of a crossover pulse.
To mathematically describe this process, pulses passing through the short/long loop and therefore arriving earlier/later or traveling to the West/East in the equivalent spatial scheme (see Fig. 1b) are labeled as u n m /v n m , respectively. As dispersive spreading of the pulses is negligible only their amplitudes are of interest, while the pulse shape itself is neglected. In this setting, the evolution of an initial distribution of pulse amplitudes u n 0 and v n 0 can be described by iteratively applying the evolution equations 12 Step m PosiƟon n where G u and G v are the gain and loss parameters for pulses inside the short (long) loop which travel on the lattice toward the East (West). In addition, a phase modulator is inserted into the shorter loop, which allows an arbitrary modulation of the phases of the pulses by inducing a time dependent shift ϕ ( ) m . This technique, which is referred to as time multiplexing 22 , does not only drastically reduce the number of components, since each element in the experiment is involved in every single round trip, but it also provides an extremely stable measurement platform. Although pulses from a non-temperature-stabilized DFB laser diode propagate for up to 1600 km corresponding to 400 roundtrips, they can still interfere with each other with high contrast. This provides an additional advantage given that in the two coupled fiber loops, only pulses which have traveled for the same time through the longer and shorter loop can interfere. Therefore, any low frequency noise is not involved in our measurements since all pulses are affected in exactly the same way. In principle, the setup is equivalent to a self-adjusting interferometer.

Derivation of the band structure
To better understand our system, we will first focus our discussion on the equivalent spatial mesh lattice, depicted in Fig. 1b. In this arrangement, the Floquet-Bloch mode eigenstates can be described according to: where θ is the longitudinal propagation constants and Q is the transverse wavenumber, also referred to as Bloch momentum. In our system, the eigenmodes ( , ) U V t implicitly contain information as to the amplitude and phase relation between the longer and shorter loop. The band structure or dispersion relation of this system is given by 23 (derived by inserting Eq. 2 into Eq. 1). Evidently, it resembles a set of coupled two level systems, with an upper and a lower band separated by a gap (see Fig. 2a). A specific point inside the band structure can

Hermitian Bloch oscillations
In our platform, applying a constant phase gradient ϕ ( , ) m n either along the temporal (vertical) or the spatial (horizontal) direction is similar to a constant electric field applied to a crystal (see also Supplementary Note 1). Thus, if a single pulse is injected into the fiber system in presence of a phase modulation ϕ ϕ ( ) = m m 0 , the entire Brillouin zone is excited, thus resulting in a Bloch oscillation of the wavepacket (see Fig. 2b). In contrast, spectrally resolved measurements require the use of broad initial distributions, which only excite a narrow region within the Brillouin zone. Hence, starting with a Gaussian profile, one can approximately select a specific value of the Bloch momentum Q 0 . In the presence of a phase gradient, at each time step, the Bloch momentum Q is shifted by a small step, i.e.: The cyclic motion through the Brillouin zone makes the wavepacket undergo Hermitian Bloch oscillations in real space (see Fig. 2c).

Bloch oscillations in globally PT-symmetric lattices
Introducing amplification and attenuation into the system in a balanced way = = − G G G u v 1 and by switching the gain and loss after every roundtrip leads to an equivalent photonic lattice with vertical gain and loss channels that are coupled at discrete time steps m (see Fig. 3a). In order to establish full PT symmetry, one would also have to apply a periodic phase modulation to the system 12 . In the absence of a periodic phase modulation, however, the system enters the broken PT symmetry regime and as a result, a part of the band structure becomes complex, with exceptional points marking the transition between the real and complex band regions (see Fig. 3b). As in the Hermitian case, a ladder of eigenvalues corresponding to Wannier-Stark states exists (see Supplementary Figs 1 and 2 and Supplementary Note 2).
The idiosyncrasies of this lattice are also reflected in our measurements, where we again start with a broad Gaussian distribution in order to probe the spectral features of the band structure. In this regard, we find that the wave packet is strongly amplified in the regions of the band structure, where the eigenvalues have a non-vanishing imaginary part. Furthermore, the exceptional points lead to the formation of secondary emissions, which emerge each time the wave packet passes through the exceptional point (see Fig. 4). At this point, a degeneracy of the band structure occurs, which causes a redistribution of the initial excitation between both bands. However, as the group velocity of the two bands has an opposite sign, both excitations of the upper and lower band move into opposite directions.
While in general, the existence of complex eigenvalues introduces exponential growth during Bloch oscillations, there are specific resonant values of the Bloch gradient 6 , which result in a pseudo-Hermitian motion with a vanishing net increase in power. In this case, a balanced passage along branches of the band structure with positive and negative imaginary eigenvalues causes the energy of the wave packet to periodically increase and decrease while the average total power remains constant over two Bloch oscillations (see Fig. 4). In other words, PT symmetry is restored at these resonant values. This process is elucidated in Fig. 4. As predicted in ref. 6 we found the reciprocal values of these resonant Bloch gradients to have an equidistant spacing defined by the amplification value.

Local PT symmetry
Another interesting non-Hermitian platform where Bloch oscillations can be observed and studied is that of local PT-symmetric mesh lattices 21 . Such system can be readily implemented by time multiplexing techniques where all pulses inside the long loop are amplified, while at the same time the pulses inside the short loop are attenuated (see Fig. 5a) 21 . Figure 5a depicts the spatial analogue of this lattice that happens to be transversely PT-symmetric at every longitudinal step = m m 0 (locally). Using the same ansatz as in Eq. 2, one can directly find the band structure of this new arrangement, which is given by (see also Supplementary Note 3): PosiƟon n φ 0 =π/30.4

Measurement
Step in the experiment and π/ . 31 9 in our simulation), a pseudo-Hermitian propagation is restored, where the total energy stays constant over many time steps. Here, the gain/loss coefficient is = .
G 1 1. where γ = ( ) G log . In this case, the propagation constant θ is complex for all values of the Bloch momentum Q, except at the center and the edges of the Brillouin zone ( = Q 0 and π = ± Q ) (see Fig. 5b). Hence, light propagation in this lattice is by nature non-Hermitian and following a naive interpretation one would always expect exponential growth.
Still, one may ask the question: Will light follow the direction of gain by always propagating in the gain loop or will it perform Bloch oscillations, if a linear phase gradient ϕ 0 is applied? Interestingly, the band structure still dictates the dynamics and as a result, Bloch oscillations take place in real space in the conventional way, thus disregarding the presence of gain and loss channels (see Fig. 5c,d and the Supplementary Figure). Because light follows one of the bands and as there are no exceptional points to switch between the bands, the field distribution is periodically amplified and attenuated during these Bloch oscillations (see Fig. 5e and Supplementary Fig. 3) but the average power stays constant.
Integrating the imaginary part of the dispersion relation (see Fig. 5b) from π = − Q to π = Q yields a vanishing imaginary component, which explains why after one cycle of Bloch oscillations no overall a) Step m I 0 e) PosiƟon n Step m