Decoherence in Superconducting Quantum Bits by Phonon Radiation

We discuss a fundamental limitation for the coherent operation of superconducting quantum bits originating from phonon radiation generated in the Josephson junctions of the device. The time dependent superconducting phase across the junction produces an electric field that couples to the underlying crystal lattice via the piezoelectric effect. We determine the radiation resistance of the junction due to phonon emission and derive substantial decoherence rates for the quantum bits, which are compatible with quality factors measured in recent experiments.

Figure 1

Phonon radiation from a Josephson junction (schematic): The time dependent phase $\phi (t)$ induces a voltage $V=\hslash \dot{\phi }/2e\propto \cos ({\omega }_{01}t)$ across the junction generating charges $\pm Q=CV$. The electric field $\mathbf{E}=V{\mathbf{e}}_z/d$ polarizes the insulator (dielectric screening) and induces displacements $\mathbf{u}$ via the piezoelectric effect. In addition, displaced metallic ions in the superconductor contribute ionic screening of the surface charges $\pm Q$. The oscillating displacement field $\mathbf{u}\propto \cos ({\omega }_{01}t)$ produces phonon radiation into the substrate, which contributes to the energy relaxation or decoherence of the qubit.