A Dayem Loop Qubit Based on Interfering Superconducting Nanowires

Abstract

We propose a qubit design based on two parallel superconducting nanowires (i.e., a "Dayem loop qubit"). The inclusion of two nanowires instead of one leads to the Little-Parks effect, which provides an oscillator behavior for the qubit frequency as well as anharmonicity. Our key result is that even if the nanowires have an increasingly linear CPR at low supercurrents, the quantum interference between two condensates, induced by a magnetic field, leads to a restoration of cubic nonlinearity, which is predicted to be sufficient to create a functional transmon qubit based on thin superconducting wires. We consider both generic (cubic) current-phase relationships (CPR) as well as more realistic microscopic CPR, having higher-order nonlinearities. For higher-order CPRs, we propose a simple power-law phenomenological approximation valid at very low temperatures, at which superconducting qubits normally operate.

Figures (7)

• Figure 1: Schematic of a Dayem loop qubit. It involved two superconducting thin-film electrodes connected by two parallel nanowires, which are assumed identical. The electrodes form a coplanar capacitor $C$. The magnetic field $B$ is applied perpendicular to the device, as shown by the circles with "x" symbols in them.
• Figure 2: Evolution of the total supercurrent versus magnetic field for $n_v=0$ and $n_v=1$. For sufficiently high values of $b-n_v$, the local maxima and minima are suppressed towards zero supercurrent.
• Figure 3: (a) $I_c(b)$ curve for $\phi_0 = 10\pi$. (b) Plots of $(\phi, b)$ with $\phi_0 = 10\pi$, where the inductance diverges (blue), and the potential reaches its maximum (red). Note, they are not the same curve.
• Figure 4: Increasing anharmonicity of a Dayem loop qubit with a cubic order CPR via magnetic-field tuning. Shown is the potential energy together with the lowest three eigenstates: ground (blue), first excited (red), and second excited (green). Solid curves represent the eigenfunctions, while dashed horizontal lines indicate the corresponding eigenenergies. The vertical axis shows the normalized energy $2eE/(\hbar I_0)$, with $n_v=0$, $I_0 = 1~\mu\mathrm{A}$, corresponding to an energy normalization factor $\hbar I_0/2e=497~\mathrm{GHz}$. The qubit parameters are $E_c/h \approx 0.3~\mathrm{GHz}$ and $\phi_0 = 10\pi$. The qubit excitation energy depends on the magnetic field. Note, all the presented energies below have units of $\hbar I_0/(2e)$. (a) Zero magnetic field ($b = 0$). The maximum of the potential is $U_{max}=15.7$. The energy levels are $E_0$, $E_1$, and $E_2$. The ground state is $E_0 = 3.566 \times 10^{-3}$. The qubit excitation energy is $E_{01}=E_1-E_0 = 7.13\times10^{-3}$. Thus, the qubit excitation energy is $E_{01}=3.54$GHz. The second excitation energy gap is $E_{12}=E_2-E_1=E_{01}+\alpha$, where $\alpha=-2.427\times 10^{-6}$ is the absolute anharmonicity. This yields a small relative anharmonicity $\alpha_r =(E_{12}-E_{01})/E_{01}=-1+(E_{01}+\alpha)/E_{01}=\alpha/E_{01}=-0.00034$, which is $-0.034\%$. The small anharmonicity arises because the eigenfunctions are localized far below the potential maximum, where the potential is nearly harmonic, resulting in an almost linear energy spectrum. (b) Finite magnetic field ($b = 5.7$). The applied magnetic field pushes down the maximum of the potential, and the wave functions expand and become closer to the maximum. Thus, the effect of the non-linear terms of the potential is enhanced. The maximum of the potential energy is $U_{max}=1\times 10^{-2}$. The ground state energy is $E_0 = 5.6145 \times 10^{-4}$, the first excitation energy $E_{01} = E_1 - E_0 = 1.111\times 10^{-3}$, and the second one is $E_{12} = E_2 - E_1 = 1.093\times 10^{-3}$. Therefore, the absolute anharmonicity is $\alpha = E_{12} - E_{01} = -1.8\times 10^{-5}$. This results in a significantly increased relative anharmonicity of $\alpha_r = (E_{12} - E_{01})/E_{01} = -0.016$ which is $-1.6\%$.
• Figure 5: Plots of log (base 10) of normalized inductance of a symmetric 2 nanowire SQUID Here, we consider a device with $\phi_0 = 10\pi$ at magnetic fields of $b=0$ (blue), $b=3$ (red), $b=5.7$ (green).