[Paper Review] Quasiparticle tunneling: P(E) theory

John M. Martinis|ArXiv.org|Apr 14, 2009

Algebraic structures and combinatorial models3 citations

TL;DR

This paper develops a quasiparticle tunneling theory using environmental $P(E)$ formalism to calculate energy decay in superconducting qubits due to non-equilibrium quasiparticles. It extends the $P(E)$ theory to include both electron- and hole-like charge transfer, showing that coherent interference between these channels suppresses dissipation, leading to a $Q$-factor scaling as $\sigma_n^{1/2}$ in thick films and $n_{\textrm{qp}}^{-1}$ in thin films, with key suppression from quasiparticle density and superconducting gap.

ABSTRACT

A calculation of the energy decay rate of a Josephson qubit from non-equilibrium quasiparticles is made using the environmental P(E) theory. For a large-capacitance qubit, we extend the theory to include the tunneling of quasiparticles, which has an electron- and hole-like charge components.

Motivation & Objective

• To model energy decay in Josephson qubits due to non-equilibrium quasiparticles using environmental $P(E)$ theory.
• To extend the standard $P(E)$ formalism to account for both electron-like and hole-like quasiparticle tunneling with coherent charge transfer.
• To quantify the resulting dissipation and quality factor ($Q$) of superconducting resonators and qubits under realistic quasiparticle conditions.
• To derive scaling laws for $Q$-factor in thin-film and thick-film superconducting transmission lines, including dependence on quasiparticle density and material parameters.

Proposed method

• Adapts environmental $P(E)$ theory to include coherent tunneling of quasiparticles with both electron- and hole-like charge components via modified displacement correlators.
• Uses the tunneling Hamiltonian in terms of quasiparticle operators $\gamma$, with $u_k$ and $v_k$ coherence factors from BCS theory.
• Derives the environmental response function $J(t)$ from the impedance $Z(\omega)$ of a resonant LC circuit, incorporating the Josephson junction's capacitance and inductance.
• Computes the probability $P(E)$ for energy absorption/emission by the environment, leading to the photon emission probability $p(\hbar\omega_r) \propto q^2/(2C\hbar\omega_r)$.
• Evaluates the surface impedance $Z_s$ and kinetic inductance contributions in thin- and thick-film limits using $\sigma_1$ and $\sigma_2$ conductivity components.
• Derives the quality factor $1/Q$ as a function of quasiparticle density $n_{\textrm{qp}}$, superconducting gap $\Delta$, and normal-state conductivity $\sigma_n$, with corrections for frequency shifts.

Experimental results

Research questions

• RQ1 How does coherent tunneling of electron- and hole-like quasiparticles affect energy dissipation in superconducting qubits?
• RQ2 What is the correct form of the $P(E)$ function when both electron- and hole-like charge transfer are present in quasiparticle tunneling?
• RQ3 How does the quality factor $Q$ of a superconducting resonator scale with quasiparticle density $n_{\textrm{qp}}$ and material parameters like $\sigma_n$?
• RQ4 What is the role of coherence factors $u_k$ and $v_k$ in suppressing dissipation due to quasiparticle tunneling?
• RQ5 How do the kinetic inductance and surface impedance contributions affect $Q$-factor scaling in thin- and thick-film superconducting transmission lines?

Key findings

• The $P(E)$ function for quasiparticle tunneling includes interference between electron- and hole-like charge transfer, modifying the standard environmental response.
• The probability for photon emission by the environment is $p(\hbar\omega_r) \simeq \frac{q^2}{2C\hbar\omega_r}$, which is small when $q^2/2C \ll \hbar\omega_r$, consistent with weak coupling.
• In the thin-film limit, the inverse quality factor scales as $1/Q \propto \sigma_n^{1/2} \left(\frac{\hbar\omega}{\Delta}\right)^{1/2} \frac{n_{\textrm{qp}}}{D(E_F)\Delta}$, showing suppression by quasiparticle density.
• In the thick-film (dirty) limit, $1/Q \propto \frac{\lambda}{s} \frac{g}{g_m} \gamma \frac{\sqrt{2}}{\pi} \left(\frac{\Delta}{\hbar\omega}\right)^{1/2} \frac{n_{\textrm{qp}}}{D(E_F)\Delta}$, with $\lambda \sim 50\,\textrm{nm}$ for aluminum.
• The fractional frequency shift $\delta\omega/\omega$ due to quasiparticle-induced changes in $\sigma_2$ is equal in magnitude to $1/Q$, indicating a direct link between dissipation and resonance frequency drift.
• The ratio $\sigma_1/\sigma_2 \simeq \pi \frac{\Delta}{\hbar\omega} - 2\frac{\sigma_1}{\sigma_n}$ shows that $\sigma_1 \ll \sigma_2$ for small quasiparticle density, confirming low dissipation in the superconducting state.