Magnetic flux controlled current phase relationship in double Quantum Dot Josephson junction

TL;DR

This work tackles how magnetic flux controls the current-phase relation in a Josephson junction with parallel double quantum dots. It introduces a surrogate three-level discretization of the superconducting leads and couples it to the DQD system, enabling exact diagonalization and spectral access; a complementary low-energy effective Hamiltonian provides physical interpretation. Across interaction regimes, the study uncovers flux-tunable ground-state transitions among singlet, doublet, and triplet states, with Andreev bound states (ABS) driving subgap current and giving rise to complex phase boundaries, including a triple point in the ($\phi$, $\phi_B$) space. The findings offer a computationally efficient framework to map phase diagrams of multi-dot superconducting devices and illuminate how flux control can tailor the Josephson response in nano-scale circuits.

Abstract

In this work, we study a Josephson junction with parallel-connected quantum dots (QDs) threaded by a magnetic flux in the central region. We discretize the superconducting (SC) electrode into three discrete energy levels and modify the tunneling coefficients to construct a finite-dimensional surrogate Hamiltonian. By directly diagonalizing this Hamiltonian, we compute the physical quantities of the system. Additionally, we employ a low-energy effective model to gain deeper physical insight. Our findings reveal that when only one QD exhibits Coulomb interaction, the system undergoes a phase transition between singlet and doublet states. The magnetic flux has a minor influence on the singlet state but significantly affects the doublet state. When both QDs have interactions, the system undergoes two phase transitions as the SC phase difference increases: the ground state evolves from a doublet to a singlet and finally into a triplet state at $φ= π$. Increasing the magnetic flux suppresses the doublet and triplet phases, eventually stabilizing the singlet state. In this regime, enhancing the interaction strength does not induce a singlet-doublet transition but instead drives a transition between upper and lower singlet states, leading to a critical current peak as $U$ increases. Finally, we examine the case where the tunneling coefficient $Γ$ exceeds the SC pairing potential $Δ$. Here, doublet states dominate, and the system only exhibits a phase transition between doublet and triplet states when $φ_B = 0$. In the presence of a magnetic flux, the three states converge, resulting in a triple point in the ($φ$, $φ_B$) parameter space.

Figures (8)

• Figure 1: (Color online) Schematic diagram of two QD parallel connected to two superconductor.
• Figure 2: (Color online) (a) Current suppressed with the different $\phi_{B}$. $\varepsilon_1=\varepsilon_2=-1$ and $\Gamma=0.25$. Blank square curves are currents calculated by surrogate Hamiltonian. (b) Current is non-zero at $\phi_{B}=\pi$. $\varepsilon_1=0, \varepsilon_2=-1$ and $\Gamma=0.25$. Blank square curves are current calculated by surrogate Hamiltonian. (c) Continuum current at different $\phi_{B}$. (d) Sub-gap current at different $\phi_{B}$
• Figure 3: (Color online) Phase diagram as the function of $\phi$ and $\phi_{B}$. The curves denotes the phase boundary. Parameters are set as follows: $\varepsilon_1=-\frac{U_1}{2},\varepsilon_2=0$ and $\Gamma=0.25$.
• Figure 4: (Color online) (a) Josephson current calculated by surrogate Hamiltonian and by low energy effective model.$\phi_{B}$ is taken to be $0.4\pi$ and $U_1=2,\varepsilon_1=-\frac{U_1}{2},\varepsilon_2=0,\Gamma=0.25$. (b) Entropy calculated by surrogate Hamiltonian. Parameters are set as Fig .4(a) .(c) Energy spectrum of the system, highlighting the lowest three eigenstates. The spectrum is calculate with low energy effective Hamiltonian. (d) Energy spectrum of the system, highlighting the lowest three eigenstates. The spectrum is calculate with surrogate Hamiltonian. Panels (e)--(h) correspond to (a)--(d), with the parameter $\phi_{B}=\pi$
• Figure 5: (Color online) (a) Current phase relationship for different U. Parameters are set as follows: $\phi_{B}=0$ and $\Gamma=0.25$. (b)Entropy with $\phi$ at $U_1=U_2=0.3$.