[Paper Review] Selection rules in a strongly coupled qubit-resonator system

Thomas M. Niemczyk, Frank Deppe+12 more|arXiv (Cornell University)|Jul 5, 2011

Semiconductor Lasers and Optical Devices3 citations

TL;DR

This paper experimentally demonstrates controlled manipulation of selection rules in a strongly coupled superconducting qubit-resonator system by tuning the qubit's flux bias, enabling coexistence of one- and two-photon transitions. The vacuum coupling strength is directly extracted from two-photon spectra with minimal cavity excitation, confirming strong coupling and validating theoretical predictions in circuit QED.

ABSTRACT

Superconducting qubits acting as artificial two-level atoms allow for controlled variation of the symmetry properties which govern the selection rules for single and multiphoton excitation. We spectroscopically analyze a superconducting qubit-resonator system in the strong coupling regime under one- and two-photon driving. Our results provide clear experimental evidence for the controlled transition from an operating point governed by dipolar selection rules to a regime where one- and two-photon excitations of the artificial atom coexist. We find that the vacuum coupling between qubit and resonator can be straightforwardly extracted from the two-photon spectra where the detuned two-photon drive does not populate the relevant resonator mode significantly.

Motivation & Objective

• To investigate how symmetry breaking in a superconducting qubit-resonator system alters selection rules for single- and multi-photon transitions.
• To demonstrate the transition from dipole-selection-rule-dominated behavior to a regime where two-photon excitations become accessible.
• To extract the vacuum coupling strength $ g $ from two-photon spectroscopy data with minimal cavity population.
• To validate the applicability of the Jaynes-Cummings Hamiltonian in describing the strongly coupled system under multi-photon driving.
• To establish a method for measuring coupling rates in the dispersive regime using off-resonant two-photon drives.

Proposed method

• Employed a flux-tunable superconducting qubit coupled to a coplanar waveguide (CPW) resonator in the strong coupling regime.
• Performed one- and two-photon spectroscopy using microwave drives detuned from the resonator mode to minimize cavity excitation.
• Used dispersive two-photon spectroscopy to extract the qubit-cavity coupling rate $ g $, relying on the fact that the drive does not significantly populate the resonator mode.
• Analyzed transmission spectra to identify vacuum Rabi splitting and anticrossings, confirming strong coupling.
• Applied a flux bias $ ilde{ heta} $ to tune the qubit's energy level splitting and break symmetry, enabling two-photon transitions.
• Fitted the measured energy level spectra to the Jaynes-Cummings Hamiltonian to extract $ g $, $ \gamma $, and $ \kappa $.

Experimental results

Research questions

• RQ1 How does tuning the flux bias of a superconducting qubit affect the selection rules for one- and two-photon transitions?
• RQ2 Can two-photon excitations be coherently accessed in a strongly coupled qubit-resonator system, and under what symmetry conditions?
• RQ3 Is the vacuum coupling strength $ g $ reliably extractable from two-photon spectroscopy when the resonator is not significantly populated?
• RQ4 How does the system's behavior transition from dipole-selection-rule dominance to a regime with mixed one- and two-photon excitation pathways?
• RQ5 What is the quantitative agreement between coupling rates extracted from two-photon spectroscopy and those from one-photon transmission and spectroscopy?

Key findings

• The system exhibits a clear transition from dipole-selection-rule-dominated behavior to a regime where both one- and two-photon excitations coexist, controlled by flux tuning.
• The vacuum Rabi splitting was measured as $ \tilde{g}_3 / 2\pi = 77.4\,\text{MHz} $, yielding a coupling strength $ g_3 / 2\pi \approx 87.0\,\text{MHz} $ after accounting for the mixing angle $ \sin\theta \approx 0.89 $.
• The qubit decay rate $ \gamma / 2\pi \approx 43\,\text{MHz} $ was extracted from the FWHM of the spectroscopy signal, consistent with one-photon spectroscopy results.
• Two-photon spectroscopy enabled clean extraction of $ g $ because the detuned drives did not significantly populate the resonator mode, minimizing cavity-induced broadening.
• The absence of flux-independent features at $ \omega_s = \omega_3 / 2 $ in the harmonic limit confirms the suppression of multi-photon transitions in the symmetric case.
• The observed anticrossings in the two-photon spectrum are direct evidence of strong coupling, with resolved vacuum Rabi splitting confirming coherent qubit-resonator exchange.