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Observation of Bloch oscillations in complex PT-symmetric photonic lattices

Martin Wimmer, Mohammed-Ali Miri, Demetrios Christodoulides, Ulf Peschel

arXiv:1608.08870v1physics.optics

TL;DR

In PT-symmetric photonic lattices, it is unclear whether gain or Bloch oscillations govern light propagation. This paper experimentally studies Bloch oscillations in mesh lattices and reconstructs their complex band structure, observing secondary emissions and resonant restoration of PT symmetry.

Abstract

from arXiv · show

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.

Experimental setup and time multiplexing

Two coupled fiber loops emulate a time-multiplexed lattice: pulse arrival times encode lattice position, and each roundtrip splits pulses into neighboring positions. The setup uses gain and loss controls and supports stable interference over long propagation.

Experimental setup and time multiplexing

  • Each roundtrip maps pulses to earlier or later arrival-time slots, encoding neighboring lattice positions n by ±1.A pulse launched at n=0 splits at the 50/50 coupler into n=−1 and n=1.Source: Experimental setup and time multiplexing · p. 2
  • Two coupled fiber loops of different lengths emulate a 1+1D lattice, with erbium-doped amplifiers compensating losses and modulators set to 50% transmission.Changing modulator transmission generates effective gain or loss; crossover pulses acquire a π/2 phase shift.Source: Experimental setup and time multiplexing · p. 2
  • The model tracks pulse amplitudes in the short and long loops while neglecting pulse shape because dispersive spreading is negligible.The loop directions are labeled u_n^m and v_n^m, respectively.Source: Experimental setup and time multiplexing · p. 3
  • Time multiplexing reuses each experimental element every roundtrip and maintains high-contrast interference for pulses traveling up to 1600 km.The setup suppresses low-frequency noise because all pulses are affected in the same way.Source: Experimental setup and time multiplexing · p. 3

Derivation of the band structure

The mesh lattice is described using Floquet-Bloch modes whose propagation constants and transverse wavenumbers characterize its dispersion. A broad Gaussian excitation can select a narrow region of the Brillouin zone, while a phase gradient shifts the Bloch momentum over time.

Derivation of the band structure

  • The Floquet-Bloch eigenmodes encode the amplitude and phase relation between the short and long loops, and their dispersion has upper and lower bands separated by a gap.The propagation constant θ and transverse wavenumber Q are the mode variables.Source: Derivation of the band structure · p. 3
  • A broad Gaussian distribution, with its short-loop phase tuned to match (U, V)^t, can excite a specific point within the band structure.This allows approximate selection of a Bloch momentum Q0.Source: Derivation of the band structure · p. 4
  • A linear phase modulation φ(m)=mφ0 excites the entire Brillouin zone from a single pulse, whereas a broad Gaussian excitation probes a narrow region.The phase gradient shifts Bloch momentum Q at each time step.Source: Derivation of the band structure · p. 4

Hermitian Bloch oscillations

A phase gradient drives cyclic motion through the Brillouin zone, producing Hermitian Bloch oscillations in real space. Adding balanced, switched gain and loss changes the lattice’s symmetry regime unless periodic phase modulation is also applied.

Hermitian Bloch oscillations

  • Cyclic motion through the Brillouin zone makes the wavepacket undergo Hermitian Bloch oscillations in real space.The phase gradient shifts the Bloch momentum at each time step.Source: Hermitian Bloch oscillations · p. 4
  • Balanced gain and loss switched after every roundtrip create vertical gain and loss channels coupled at discrete time steps.The gain and loss parameters are set equal in magnitude.Source: Hermitian Bloch oscillations · p. 4
  • Without periodic phase modulation, the system is in the broken PT-symmetry regime, with complex bands separated from real bands at exceptional points.Periodic phase modulation is required to establish full PT symmetry.Source: Hermitian Bloch oscillations · p. 4

Bloch oscillations in globally PT-symmetric lattices

In globally PT-symmetric lattices, complex bands shape wavepacket evolution during Bloch oscillations. Exceptional-point crossings produce secondary emissions, while resonant gradients can restore PT symmetry and prevent net power growth over two cycles.

Bloch oscillations in globally PT-symmetric lattices

  • Wavepackets are strongly amplified in regions where eigenvalues have nonzero imaginary parts, and exceptional-point crossings generate secondary emissions.An emission appears each time the wavepacket passes through an exceptional point.Source: Bloch oscillations in globally… · p. 4
  • At resonant Bloch gradients, balanced passage along positive- and negative-imaginary eigenvalue branches keeps average power constant over two Bloch oscillations.The wavepacket’s power increases and decreases periodically, and reciprocal resonant gradients are equidistantly spaced according to the amplification value.Source: Bloch oscillations in globally… · p. 4

Local PT symmetry

In locally PT-symmetric mesh lattices, Bloch oscillations persist despite gain and loss, with cyclic amplification and attenuation yielding stable average power. Monitoring wavepacket motion and intensity also enables reconstruction of the complex band structure.

Local PT symmetry

  • Bloch oscillations proceed conventionally despite gain and loss, with cyclic amplification and attenuation leaving average power constant.The lattice amplifies and attenuates the field during the cycle, but its average power stays constant.Source: Local PT symmetry · p. 5
  • Integrating the imaginary dispersion across Q = −π to π gives zero, explaining the absence of net power increase after one cycle.The phase potential therefore supports pseudo-Hermitian propagation.Source: Local PT symmetry · p. 5
  • Monitoring the wavepacket’s center of mass and intensity at each time step, then taking temporal derivatives, provides group velocity and power change.The measurements use a Gaussian fit.Source: Local PT symmetry · p. 5
  • The phase potential shifts Bloch momentum by φ0/2 per step; integrating group velocity determines the real dispersion up to an unknown constant.The imaginary dispersion is evaluated from the wavepacket’s growth or decay rate.Source: Local PT symmetry · pp. 5–6
  • Evaluating the reconstruction equations yields the full complex band structure, in excellent agreement with the theoretical dispersion relation.The experimentally obtained reconstruction is compared with the theoretically calculated band structure.Source: Local PT symmetry · p. 6

Conclusions

The experiments demonstrate optical Bloch oscillations in global and local PT-symmetric mesh lattices. They show stable propagation under different conditions in each arrangement and use Bloch oscillations to reconstruct the complex band structure.

Conclusions

  • Optical Bloch oscillations were experimentally demonstrated in both global and local PT-symmetric mesh lattices.The conclusion identifies both arrangements as experimental demonstrations.Source: Conclusions · p. 6
  • In global PT-symmetric systems, stable propagation occurs at specific resonant Bloch gradients; other values produce secondary emissions during growing revivals.In locally PT-symmetric arrangements, propagation is always stable through cyclic amplification and attenuation.Source: Conclusions · p. 6
  • Bloch oscillations can be used to reconstruct the whole band structure, including complex eigenvalues.The authors suggest the results may be pertinent to other areas involving non-Hermiticity in periodic environments.Source: Conclusions · p. 6

Figures

The figures move from the time-multiplexed fiber-loop implementation to Bloch oscillations and PT-symmetric band behavior. They also illustrate resonant restoration and reconstruction of a complex band structure.

Figures

  • Two fiber loops joined by a 50/50 coupler implement the mesh lattice; their roundtrips map to different lattice directions.The phase modulator creates the Bloch gradient, while acousto-optical modulators set effective gain and loss.Source: Figure 1 · p. 9
  • The conservative mesh lattice has two bands separated by a gap, while a temporal Bloch gradient increases momentum from Q0 ≈ 0.A broad Gaussian packet undergoes oscillatory motion as its state shifts through the Brillouin zone.Source: Figure 2 · p. 9, Figures · p. 9
  • Alternating gain and loss without symmetric phase modulation breaks PT symmetry, producing exceptional points between conservative bands and complex-eigenvalue regions.Source: Figure 3 · p. 10
  • For an upper-band Gaussian packet at G = 1.1, exceptional-point crossings produce secondary branches; resonant gradients restore pseudo-Hermitian propagation.The caption gives resonant φ0 values of approximately π/31.2 experimentally and π/31.9 in simulation.Source: Figure 4 · p. 10
  • Blue and red encode the real and imaginary band components; compare the resulting band in panel b with the reconstruction in panel f.The imaginary part in panel f is magnified threefold; panel d maps the horizontal axis to Bloch momentum Q.Source: Figure 5 · p. 11
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