The nature of the Lamb shift in weakly-anharmonic atoms: from normal mode splitting to quantum fluctuations

TL;DR

This paper distinguishes between classical normal mode splitting and quantum vacuum fluctuation contributions to frequency shifts in weakly-anharmonic systems like transmon qubits, clarifying their respective roles in observed Lamb shifts.

Contribution

It introduces a framework separating normal mode splitting from quantum vacuum fluctuation effects in weakly-anharmonic systems, clarifying the quantum origin of frequency shifts.

Findings

Normal mode splitting can mimic Lamb shifts without quantum fluctuations.

Quantum vacuum fluctuations contribute significantly to frequency shifts in weakly-anharmonic systems.

The framework helps interpret spectroscopic data in circuit QED experiments.

Abstract

When a two level system (TLS) is coupled to an electromagnetic resonator, its transition frequency changes in response to the quantum vacuum fluctuations of the electromagnetic field, a phenomenon known as the Lamb shift. Remarkably, by replacing the TLS by a harmonic oscillator, normal mode splitting leads to a quantitatively similar shift, without taking quantum fluctuations into account. In a weakly-anharmonic system, lying in between the harmonic oscillator and a TLS, the origins of such shifts can be unclear. An example of this is the dispersive shift of a transmon qubit in circuit quantum electrodynamics (QED). Although often referred to as a Lamb shift, the dispersive shift observed in spectroscopy in circuit QED could contain a significant contribution from normal-mode splitting that is not driven by quantum fluctuations, raising the question: how much of this shift is quantum in origin? Here, we treat normal-mode splitting separately from shifts induced by quantum vacuum fluctuations in the Hamiltonian of a weakly-anharmonic system, providing a framework for understanding the extent to which observed frequency shifts can be attributed to quantum fluctuations.