Floquet-Nambu theory of electron quantum optics with superconductors

TL;DR

This work develops a Floquet-Nambu formalism to analyze time-dependent quantum transport in mesoscopic circuits that couple chiral quantum Hall edge states to superconductors. By combining dynamic scattering theory with a Nambu description of superconductivity, it furnishes the Floquet-Nambu scattering matrix ${\hat{S}}_F$ and the excess correlation function ${\hat{G}}$, enabling precise predictions for the time-dependent current and the quantum-information content of emitted excitations. The authors apply the framework to subgap (leviton) and above-gap driving, show that Lorentzian pulses can yield pure, electron–hole superpositions after Andreev reflection, and quantify purity measures ${\gamma}$ and ${\gamma}_+$ across driving regimes and pairing symmetries (singlet vs triplet). They reveal how the degree of Andreev conversion and the drive frequency relative to the gap govern current, transmitted charge, and information loss due to quasiparticle leakage, outlining how these insights can guide future electron quantum optics experiments with superconductors and paving the way for flying-qubit implementations based on coherent electron–hole superpositions.

Abstract

We present a comprehensive Floquet-Nambu theory to describe the time-dependent quantum transport in mesoscopic circuits involving superconductors. The central object of our framework is the first-order correlation function, which accounts for the excitations that are generated by a time-dependent voltage and their coherent scattering off the interface with a superconductor. We analyze the time-dependent current generated by periodic voltage pulses and how it depends on the excitation energies of the voltage drive compared to the gap of the superconductor. Our general formalism allows us to identify the conditions for the excitations that are scattered off the superconductor to become coherent electron-hole superpositions. To this end, we consider the purity of the outgoing states, which characterizes their ability to carry quantum information. To illustrate our formalism, we apply it to a system composed of chiral quantum Hall edge states connected to a superconductor, and we calculate the current in the outgoing lead and the purity of the outgoing states for Lorentzian and harmonic voltage drives. Our framework paves the way for systematic investigations of time-dependent scattering problems involving superconductivity, and it may help interpret future experiments in electron quantum optics with superconductors.

Figures (9)

• Figure 1: Quantum Hall sample with chiral edge states coupled to a superconductor. (a) Through an external source (circle), voltage pulses are applied to the contact (square box) of a quantum Hall sample with chiral edge states (arrows) that are connected to a superconductor (blue box). The injected electric current ($I_\text{in}(t)$ in blue) is measured in the outputs ($I_\text{out}(t)$ in red) after the incoming charges have been either normal or Andreev scattered at the interface with the superconductor. The operators $\boldsymbol{a}_{e,h}$ describe electrons ($e$) and holes ($h$) before the voltage pulses are applied. The operators $\boldsymbol{b}_{e,h}$ describe particles that have been excited by the pulses, while $\boldsymbol{c}_{e,h}$ correspond to the particles that have reached the outputs. (b) For superconductors of width $W$ and singlet pairing, the spin of an Andreev converted particle is flipped during the scattering process and the particle appears in the other edge channel. For triplet superconductors, the spin is preserved in the scattering process and both normal and Andreev scattered particles remain in the same edge channel.
• Figure 2: Scattering off the superconductor. Probabilities for Andreev conversion, $|S_{he}|^2$, and normal transmission, $|S_{ee}|^2$, in \ref{['eq:NS_SM']} and their sum, $P=|S_{he}|^2+|S_{ee}|^2$, as a function of the energy normalized to the gap of the superconductor. The top and bottom rows show results for singlet and triplet pairings, respectively, with different degrees of conversion $\alpha$.
• Figure 3: Voltage pulses. (a) Lorentzian voltage pulses with period $\mathcal{T}$, half-width $\tau_0$, and amplitude $eV_0=2q\hbar/\tau_0$. (b) Excitation energies compared with the gap, $\Delta$, and the density of states of the superconductor (in blue).
• Figure 4: Electric current. (a) Outgoing current for levitons with charge $q=1$ and width $\tau_0=\hbar/\Delta$ and driving frequency below the gap of the superconductor, $\hbar\Omega=0.1\Delta$. We show results for different values of the electron-hole conversion parameter $\alpha=0$ (full conversion), 0.15, 0.25, 0.3, 0.35, 0.42, 0.45, 0.5, 0.6, 0.7, 0.8, and 1 (normal transmission). (b,c) Similar results for a harmonic drive with $z=eV_0/(\hbar\Omega)=1$ and $q=eV/(\hbar\Omega)=1$ in panel (b) and $q=1/2$ in panel (c). The current is normalized with respect to the maximum of the injected current, $I_0$, which is plotted (with the sign changed) with a black dashed line.
• Figure 5: Influence of driving frequency on the current. (a) Lorentzian drive with $q=1$ and $\tau_0=\hbar/\Delta$ and a superconductor with perfect Andreev conversion, $\alpha=0$. The driving frequencies are $\hbar\Omega/\Delta= 0.1, 0.2, 0.5, 0.66, 1, 2, 10$. (b,c) Similar results for a harmonic drive with $z=eV_0/(\hbar\Omega)=1$ and $q=eV/(\hbar\Omega)=1$ in panel (b) and $q=1/2$ in panel (c). The black dashed line represents the injected current (with the sign changed), and its maximum is $I_0$.