[Paper Review] Zitterbewegung, Bloch Oscillations and Landau-Zener Tunneling in a Quantum Walk

Alois Regensburger, Christoph Bersch+5 more|arXiv (Cornell University)|Apr 1, 2011

Quantum Information and Cryptography3 citations

TL;DR

This paper experimentally demonstrates Zitterbewegung, Bloch oscillations, and Landau-Zener tunneling in a discrete-time quantum walk using coupled fiber loops with tunable phase gradients. By applying position-dependent phase shifts, the system realizes analogs of solid-state phenomena: moderate gradients induce periodic localization and recovery (Bloch oscillations), while strong gradients trigger coherent tunneling across lattice sites (Landau-Zener tunneling), all within a 70-step photonic quantum walk at 1545 nm.

ABSTRACT

We experimentally investigate a discrete time quantum walk in a system of coupled fiber loops and observe typical phenomena known from the wave propagation in periodic structures as ballistic spreading or an oscillation between two internal quantum states similar to Zitterbewegung (trembling motion). If a position-dependent phase gradient is applied we find localization and Bloch oscillations of the field for moderate as well as Landau-Zener tunneling for strong phase gradients.

Motivation & Objective

• To explore the emergence of wave-mechanical phenomena—Zitterbewegung, Bloch oscillations, and Landau-Zener tunneling—within a discrete-time quantum walk.
• To investigate how position-dependent phase gradients in a photonic quantum walk system mimic external forces found in solid-state physics.
• To demonstrate the experimental feasibility of simulating complex quantum dynamics using classical optical systems with coherent light in fiber loop networks.
• To validate theoretical predictions of band structure effects and coherent transport in quantum walks via measurable intensity patterns and interference.

Proposed method

• A discrete-time quantum walk is implemented using two 540 m coupled single-mode fiber loops with a 11.4 m length difference, enabling temporal separation of internal states.
• A 50/50 coupler performs the unbiased quantum coin operation $ C = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & i \\ i & 1 \end{pmatrix} $, inducing superposition and stepwise evolution in position space.
• Semiconductor optical amplifiers maintain signal coherence over 70 steps, simulating a single quantum particle’s evolution with minimal noise.
• A phase modulator introduces a position-dependent phase shift $ \exp(in\alpha) $ to the $ |{\downarrow}\rangle $ state, creating a constant force analog in momentum space.
• Intensity distributions in the upper and lower loops are measured via fast photodiodes to reconstruct the walker’s state $ |\Psi(m)\rangle = \sum_n \left( u_n^m |{\uparrow}\rangle + v_n^m |{\downarrow}\rangle \right) \otimes |n\rangle $.
• The system’s band structure is analyzed via the dispersion relation $ \beta(\kappa) $, revealing flat bands at $ \kappa = \pm \pi/2 $, corresponding to ballistic propagation.

Experimental results

Research questions

• RQ1 Can Zitterbewegung—trembling motion due to interference between positive and negative energy components—be observed in a photonic quantum walk?
• RQ2 Does a position-dependent phase gradient induce Bloch oscillations analogous to those in electronic systems under a constant electric field?
• RQ3 At what phase gradient strength does Landau-Zener tunneling emerge, and how does it disrupt periodic localization in the quantum walk?
• RQ4 How does the discrete, stepwise evolution of the quantum walk affect the completeness of recovery and the onset of tunneling compared to continuous models?
• RQ5 To what extent can classical optical systems simulate complex quantum phenomena like coherent tunneling and band structure effects in a scalable, experimentally accessible setup?

Key findings

• Zitterbewegung is observed as hyperbolic interference patterns in the intensity distribution, arising from counter-propagating waves with opposite diffraction signs.
• For a moderate phase gradient $ \alpha = 2\pi/32 $, the system exhibits periodic localization and recovery of the wave packet after $ m = 64 $ steps, confirming Bloch oscillations.
• At strong phase gradients $ \alpha = 2\pi/5 $, the wave packet begins to tunnel coherently to distant lattice sites, with recovery lost due to Landau-Zener tunneling across band gaps.
• The tunneling occurs between sites separated by approximately $ p = 5 $ steps, corresponding to phase-matching conditions in the $ 2\pi $-periodic band structure.
• Ballistic spreading is initially observed but decays as $ 2^{-m} $, with incomplete recovery when $ p $ is not an integer, due to the discrete nature of the stepwise evolution.
• The system’s band structure, with flat bands at $ \kappa = \pm \pi/2 $, explains the absence of first-order diffraction and enables ballistic propagation, a hallmark of quantum walks.