Systematic Schrieffer-Wolff-transformation approach to Josephson junctions: quasiparticle effects and Josephson harmonics

TL;DR

The paper develops a systematic Schrieffer-Wolff transformation to derive low-energy Hamiltonians for Josephson junctions from charge-conserving BCS leads. It reproduces the conventional $-E_J\cos\hat{\varphi}$ term at second order and extends the framework to include quasiparticles, revealing correlated quasiparticle–Cooper-pair dynamics and intralead scattering. At fourth order, it uncovers a second Josephson harmonic with $\cos(2\hat{\varphi})$ and analyzes how multi-channel tunneling can tune or suppress the first harmonic. The approach provides operator-level expressions and a clear pathway to engineer junction properties via microscopic parameters, with potential implications for qubit spectra and performance.

Abstract

We use the Schrieffer-Wolff transformation (SWT) to analyze Josephson junctions between superconducting leads described by the charge-conserving BCS theory. Starting from the single-electron tunneling terms, we directly recover the conventional effective Hamiltonian, $-E_J\cos\hat{\varphi}$, with an operator-valued phase bias $\hat{\varphi}$. The SWT approach has the advantage that it can be systematically extended to more complex scenarios. We show that if a Bogoliubov quasiparticle is present its motion couples to that of Cooper pairs, introducing correlated dynamics that reshape the energy spectrum of the junction. Furthermore, higher-order terms in the SWT naturally describe Josephson harmonics, whose amplitudes are directly related to the microscopic properties of the superconducting leads and the junction. We derive expressions that could facilitate tuning the ratio between the different harmonics in a controlled way.

Figures (2)

• Figure 1: Spectrum of the Josephson junction with a single quasiparticle. The parameters were set to $T=0.5$ and $h=t=0.1$ in units of $E_J$. The dashed curve shows the spectrum of the quasiparticle-free case. In reality, the quasiparticle-free and one-quasiparticle branches are separated by $\Delta$ which is not displayed in the plot since it greatly exceeds all other energy scales.
• Figure 2: The fourth-order contributions to the first and second Josephson harmonics as a function of $\mathcal{D}\Delta$. The parameters $T_{CP}^{(4)}$ and $D^{(4)}$ are measured in units of $T^4\mathcal{D}^2/\Delta$. For the plot, constant density of states and momentum-independent tunneling amplitude is assumed. Furthermore, the bandwidth is assumed to be $W = \sqrt{(1/\mathcal{D})^2 + \Delta^2}$, where $\mathcal{D}$ is the normal-state density of states. Inset: the ratio of the two contributions indicating that larger $\mathcal{D}\Delta$ generates more significant second harmonic.