Andreev molecules at distance

TL;DR

This paper demonstrates that Andreev Molecules can be realized at long junction separations by mediating coupling through an embedding circuit, enabling hybridization of excited Andreev bound states (ALQs) without direct electron transfer. It develops a Lindblad framework for driven ALQs in a general linear circuit, derives the effective Hamiltonian with a tunable splitting λ, and analyzes steady-state populations and inverse inductance signatures across multiple frequency regimes. Key contributions include explicit predictions for resonance- and flux-tuned features, phase-dependent interference effects, and a detailed two-tone spectroscopy scheme (potentially using separate readout oscillators) to resolve AM states. The findings offer a route to non-local Josephson coupling and state readout in superconducting circuits, with practical implications for circuit QED and hybrid qubit architectures.

Abstract

Andreev molecule states arise from hybridization of Andreev bound states in different Josephson Junctions. Extensive theoretical and experimental research concentrates on direct coherent electron coupling between the junctions: this implies the distance between the junctions is of the order of superconducting coherence length, that is, short. We propose and discuss the possibility to create Andreev molecules at long (in principle, arbitrary long) distance between the junctions. In this case, the hybridized states are excited quasi-particle singlets and the coupling is provided by an embedding electric circuit. To achieve a strong hybridization, one aligns the energies of the Andreev bound states with associated phase differences. In fact, a recent experiment realizes such setup. With circuit theory we derive the hybridization level splitting and estimate the scale of the effect. Since the phenomenon encompasses excited states, we derive and solve the associated Lindblad equation under condition of persistent resonant excitation. By analyzing the resulting dissipative dynamics we identify relevant regimes where the hybridization and resonant excitation peaks are most pronounced. The low-frequency mutual inductance of the Josephson junctions is an important signature of the molecular state and associated non-local Josephson effect. We demonstrate the peak structures for both mutual and self-inductance, and compute them in various frequency regimes. In an interesting common case the embedding circuit includes an oscillator, which can be used both to enhance hybridization and for state readout with two-tone spectroscopy. We derive and solve Lindblad equations for the conditions of two-tone spectroscopy to demonstrate the the readout of molecular states.

Figures (10)

• Figure 1: Left: Andreev molecule at short distances. The energies of quasiparticle states (with spin up) versus flux. If the ABS energies at two different junctions (blue and orange) are aligned, weak electron transfer between the junctions hybridizes the states resulting in level repulsion and delocalized superpositions. Right: Andreev molecule at long distances. The energies of excited singlet states versus flux. Since the junctions are separated by a large distance, the electron transfer is impossible. However, the junctions are coupled via a shared electric circuit. This also results in hybridization, level repulsion, and delocalized superpositions.
• Figure 2: Andreev molecule at long distance. Two JJs with admittances $Y_{1, 2}$ are embedded in a linear electric circuit characterized by the impedance matrix $Z_{\alpha \beta}$, where $\alpha$ and $\beta$ are used to label the ports. The circuit provides the coupling and hybridization between the excited states in the two junctions. An oscillator (admittance $Y_{\rm osc}$) can also be included to the circuit. The system can be excited with an AC current source connected at port "0".
• Figure 3: (a) Convenient "coordinates" $f$ and $\varphi$. The red and blue lines give the half-splitting of the first and second ALQ, while the green line is at half the oscillator frequency. The black dashed line gives the average qubit half-splitting $(E_1{+}E_2)/2$. All lines are plotted versus the superconducting phase difference across the second junction junction in the vicinity of the degeneracy point $E_1=E_2$. The "coordinates" $\varphi$ and $f$ are defined as shown. (b) The eigenvalues of $H_{{\rm eff}}$ in the large drive limit at $A_1 = 0.5 f$, $A_2 = 0.8f$, and $\theta = 0$ versus $\varphi$. The black dashed lines correspond to $\lambda=0$ while the magenta lines give the spectrum at $\lambda = 0.2 f$ where the degeneracy is lifted. (c) The eigenvalues of $H_{{\rm eff}}$ in the small drive limit at $\varphi = 0$ and and $\theta=\pi{/}3$ versus $f$. The black dashed lines correspond to $A_1=A_2=0$, while the magenta lines correspond to $A_{1} = A_2 = 0.1\lambda$. Finite gaps linear in $A_{1}$ and $A_{2}$ open at $f = \pm\sqrt{\varphi^2 + \lambda^2/2}$. Note the asymmetry of positive and negative energies around the gaps while the limiting spectrum is mirror-symmetric. The degeneracy of $|ee\rangle$ and $|gg\rangle$ at $f=0$ is lifted with a much smaller gap quadratic in the drive.
• Figure 4: Degenerate ALQ. The excitation probabilities at $\varphi=0$ and $\theta=1.0$ versus $f$. Only the first ALQ is directly driven, $A_2=0$. The panes (a-c) and (d-f) show $\langle\sigma_1^z \rangle$ and $\langle\sigma_2^z \rangle$, respectively, and different $A_1$ increasing as the curves extends upwards. In the panes (a) and (d) $A_1$ increases linearly from curve to curve in the range $0.5 \gamma$ to $5 \gamma$. In the pane (b) and (e) $A_1$ increases linearly from curve to curve in the range $0.16 \sqrt{\gamma\lambda}$ to $3.2 \sqrt{\gamma\lambda}$. In the panes (c) and (f), $A_1$ increases linearly from $0.1\lambda$ to $1.5 \lambda$. The vertical dashed lines and corresponding numbers indicate the changing horizontal scale. The pane (g) shows $\langle \sigma_1^z \rangle$ versus $f$ for $A_1{=}A_2{=}0.2\lambda$ and various values of their relative phase $\theta$. The different colored curves correspond to equally spaced $\theta$ in the range from $0.2$ to $0.2{+} 2\pi$. They are offset for clarity, so the distance between each dashed line corresponds to $\langle \sigma_1^z \rangle \in [-1, 0]$. In all plots, $\gamma_1{=}\gamma_2{=} 10^{-3} \lambda$.
• Figure 5: Tuning resonances with flux. The ALQ excitation probabilities for a fixed $f{=}10 \lambda$ versus $\varphi$. Only the first ALQ is driven directly, $A_2 =0$, and $\theta=1.0$. The panes(a-c) and (d-f) show $\langle\sigma_1^z \rangle$ and $\langle\sigma_2^z \rangle$ respectively for varying $A_1$. In the panes (a) and (d), $A_1$ increases linearly in the range from $0.5 \gamma$ to $5 \gamma$ from lower to upper curves. In the panes (b) and (e), $A_1$ increases linearly from $0.5 {\gamma (f/\lambda)}$ to $10 {\gamma (f/\lambda)}$. In the panes (c) and (f), $A_1$ increases linearly from $0.1f$ to $3 f$ from the darkest to the lightest curve. For all plots, $\gamma_1{=}\gamma_2{=}10^{-3}f$. The horizontal scale in (a), (b), (d), (e) is not uniform as indicated by the vertical dashed lines and corresponding numbers.