Spectroscopic Evidence of the Aharonov-Casher Effect in a Cooper Pair Box

We observe the effect of the Aharonov-Casher (AC) interference on the spectrum of a superconducting system containing a symmetric Cooper pair box (CPB) and a large inductance. By varying the charge $n_g$ induced on the CPB island, we observe oscillations of the device spectrum with the period $\mathrm{\Delta }{n}_g=2e$. These oscillations are attributed to the charge-controlled AC interference between the fluxon tunneling processes in the CPB Josephson junctions. The measured phase and charge dependences of the frequencies of the $|0⟩\to |1⟩$ and $|0⟩\to |2⟩$ transitions are in good agreement with our numerical simulations. Almost complete suppression of the single fluxon tunneling due to destructive interference is observed for the charge $n_g=e(2n+1)$. The CPB in this regime enables fluxon pairing, which can be used for the development of parity-protected superconducting qubits.

Figure 1

(a) The schematics of the circuit containing the device and the read-out lumped-element resonator. The CPB Josephson junctions are shown as crosses. (b) The layout of the device, the read-out resonator, and the MW transmission line. The superinductor consists of 36 coupled cells, each cell represented a small superconducting loop interrupted by three larger and one smaller Josephson junctions [14].

Figure 2

(a) The transmitted microwave power $|S_{21}{|}^2$ at the first-tone frequency $f_1$ as a function of the second-tone frequency $f_2$ and the gate voltage $V_g$ measured at a fixed value of ${\mathrm{\Phi }}_L=0.5{\mathrm{\Phi }}_0$. The power maxima correspond to the resonance excitations of the device ($f_2=f_{01}$), the superinductor ($f_L$), and the read-out resonator ($f_R$). Note that the resonance measurements could not be extended below $\sim 1\mathrm{GHz}$ because of a high-pass filter in the second-tone feed line. (b) The frequency dependence of the transmitted microwave power measured at $V_g=0V$ and ${\mathrm{\Phi }}_{SL}=0.5{\mathrm{\Phi }}_0$.

Figure 3

(a) The flux dependence of the device energy levels obtained by numerical diagonalization of the Hamiltonian (see the Supplemental Material [15] for details; the fitting parameters are listed below). The solid curves correspond to $n_g=0.5$, the dashed curves to $n_g=0$ (the blue curves correspond to the ground state $|0⟩$, the yellow curves to the state $|1⟩$, and the green curves to the state $|2⟩$). For comparison, we also plotted the dotted curves that correspond to the fully suppressed fluxon tunneling; in this case, there are no avoided crossings between the parabolas that represent the superinductor energies $E_L(m,\mathrm{\Phi })=\frac{1}{2}{E}_L[m-(\mathrm{\Phi }/{\mathrm{\Phi }}_0){]}^2$ plotted for different $m$. (b) The dependences of the resonance frequencies $f_{01}$ (red dots, $n_g=0$; red squares, $n_g=0.5$) and $f_{02}$ (blue down triangles, $n_g=0$; blue up triangles $n_g=0.5$) on the flux in the device loop. The theoretical fits (solid curves, $n_g=0.5$; dashed curves, $n_g=0$) were calculated with the following parameters: $E_J=6.25\mathrm{GHz}$, the asymmetry between the CPB junctions $△E_J=0.5\mathrm{GHz}$, $E_C=6.7\mathrm{GHz}$, $E_L=0.4\mathrm{GHz}$ [$L=({\mathrm{\Phi }}_0/2\pi {)}^2/E_L\eqsim 0.4\mu \mathrm{H}$], $E_{CL}=5\mathrm{GHz}$.