[Paper Review] Signatures of Dynamical Tunneling in the Wave function of a Soft-Walled Open Microwave Billiard

Y. H. Kim, Ulrich Kuhl+2 more|arXiv (Cornell University)|Nov 12, 2004

Nonlinear Dynamics and Pattern Formation5 citations

TL;DR

This paper demonstrates a novel method to detect dynamical tunneling in a soft-walled microwave billiard by analyzing the phase evolution of wave functions across resonances, revealing signatures even when level splittings are unresolved. The key contribution is identifying dynamical tunneling through symmetry changes in wave function phases—transitioning from symmetric to anti-symmetric—as energy increases, providing direct evidence without requiring resolvable spectral splitting.

ABSTRACT

Evidence for dynamical tunneling is observed in studies of the transmission, and wave functions, of a soft-walled microwave cavity resonator. In contrast to previous work, we identify the conditions for dynamical tunneling by monitoring the evolution of the wave function phase as a function of energy, which allows us to detect the tunneling process even under conditions where its expected level splitting remains irresolvable.

Motivation & Objective

• To identify dynamical tunneling in a soft-walled open microwave billiard where traditional spectral splitting detection fails due to broad resonance widths.
• To investigate how quantum states evolve in mixed phase space with KAM islands separated by classically forbidden regions.
• To develop a method for detecting dynamical tunneling using wave function phase symmetry, rather than relying on measurable level splittings.
• To demonstrate that phase asymmetry in eigenstates provides a direct signature of tunneling coupling between degenerate states.

Proposed method

• The experiment uses a 2D microwave cavity with a spatially varying height profile to simulate a soft-walled potential, mimicking quantum dots with surface gates.
• The wave function and phase are measured via S-parameters: |S₃₃| for amplitude and S₃₁ for phase, enabling reconstruction of field patterns and phase maps.
• The phase asymmetry ⟨Δϕ⟩ = ⟨ϕ(x,y) − ϕ(−x,y)⟩ is calculated over the cavity area to detect symmetry transitions from symmetric (⟨Δϕ⟩ ≈ 0) to anti-symmetric (⟨Δϕ⟩ ≈ π).
• The system is probed across resonances, and phase evolution is tracked as a function of frequency (energy), revealing tunneling-induced symmetry changes.
• Theoretical modeling uses the 2D Helmholtz equation with a z-dependent effective potential derived from cavity height variations, approximating the Schrödinger equation for quantum dots.
• The method relies on the fact that dynamical tunneling couples two initially degenerate states into symmetric and anti-symmetric superpositions, detectable via phase asymmetry.

Experimental results

Research questions

• RQ1 Can dynamical tunneling be detected in a microwave cavity when its associated level splitting is below the resolution limit of the measurement system?
• RQ2 How does the phase structure of the wave function evolve across a resonance when dynamical tunneling is active?
• RQ3 Can phase asymmetry serve as a direct, observable signature of dynamical tunneling in systems with unresolved spectral splittings?
• RQ4 What is the role of classical phase space structure—specifically KAM islands and chaotic regions—in mediating tunneling in soft-walled billiards?

Key findings

• The wave function phase asymmetry ⟨Δϕ⟩ transitions from near zero to near π as frequency increases through a resonance, indicating a change from symmetric to anti-symmetric character.
• This phase evolution provides unambiguous evidence of dynamical tunneling, even though the associated level splitting is too small to resolve in the spectrum.
• Cross-like wave function patterns observed in the field intensity do not contribute to classical transport but are shown to be quantum-mechanically active due to tunneling.
• The transition in phase symmetry occurs at frequencies where the wave function exhibits cross-like structures, marked by vertical dashed lines in Fig. 5.
• The phase asymmetry passes through π/2 at each such resonance, confirming the presence of coherent superpositions of symmetric and anti-symmetric states.
• The method successfully detects dynamical tunneling in a system where traditional spectral analysis fails, demonstrating the power of phase-based diagnostics in open quantum systems.