Photon Propagation in a Discrete Fiber Network: An Interplay of Coherence and Losses

We study light propagation in a photonic system that shows stepwise evolution in a discretized environment. It resembles a discrete-time version of photonic waveguide arrays or quantum walks. By introducing controlled photon losses to our experimental setup, we observe unexpected effects like subexponential energy decay and formation of complex fractal patterns. This demonstrates that the interplay of linear losses, discreteness and energy gradients leads to genuinely new coherent phenomena in classical and quantum optical experiments. Moreover, the influence of decoherence is investigated.

Figure 1

Principal scheme of all-fiber experimental setup (a) without deliberately introduced losses (b) to introduce a controlled loss of photons by measuring all intensity in the $|\uparrow ⟩$ state after every second $50/50$ coupler; SOA: semiconductor optical amplifier; PM: phase modulator; PD: photodiode.

Figure 2

Evolution of wave starting in state $|\downarrow ⟩$ at position $n=0$ in the lossless case [Fig. 1a]; (a)–(d) measurements and simulations of ballistic spreading of intensities $|u_n^m{|}^2$ in the upper and $|v_n^m{|}^2$ in the lower loop without phase modulation; dynamics are the same as in a discrete-time QW (e)–(h) discrete-time Bloch oscillations for linear phase gradient $\alpha =\frac{2\pi }{32}$.

Figure 3

Evolution of wave starting in state $|\downarrow ⟩$ at position $n=0$ in the case with losses [Fig. 1b]; (a),(b) measured and simulated intensities $|a_n^m{|}^2$ in the loop with no phase modulation and (c),(d) intensities $|q_n^m{|}^2$ at the output port; (e),(f) measured and simulated fractal patterns of intensity $|a_n^m{|}^2$ for linear phase gradient $\alpha =\frac{7\pi }{26}$, net gain $g\approx 3.4/\mathrm{step}$; (g),(h) $\alpha =\frac{9\pi }{22}$, $g\approx 3.3$.

Figure 4

Transition to a classical random walk by adding decoherence in the form of random phase shifts; (a) variance of Gaussian fit to measured intensity distribution in the coherent lossy system [Fig. 1b] (blue circles) and to the measured average distribution in the case of decoherence (red crosses); lines: simulation of the coherent system and of a classical random walk; (b) fractional energy loss per step to visualize the low losses of the coherent system and the loss of $1/2$ per propagation step $m$ in the case of decoherence.

Figure 5

Lossy system with linearly increasing phase shift $\alpha$; (a),(b) measured and simulated intensities $|a_n^m{|}^2$ in the loop for $\alpha =\frac{3\pi }{7}$; net gain $g\approx 3.3$ per step; (c),(d) $\alpha =\frac{2\pi }{5}$; $g\approx 3.1$.