Improving the lifetime of the nitrogen-vacancy-center ensemble coupled with a superconducting flux qubit by applying magnetic fields

One of the promising systems to realize quantum computation is a hybrid system where a superconducting flux qubit plays the role of a quantum processor and the nitrogen-vacancy- $\left({\mathrm{NV}}^{-}\right)$ center ensemble is used as a quantum memory. We have theoretically and experimentally studied the effect of magnetic fields on this hybrid system, and found that the lifetime of the vacuum Rabi oscillation is improved by applying a few mT magnetic field to the ${\mathrm{NV}}^{-}$ center ensemble. Here, we construct a theoretical model to reproduce the vacuum Rabi oscillations with and without magnetic fields applied to the ${\mathrm{NV}}^{-}$ centers, and we determine the reason why magnetic fields can affect the coherent properties of the ${\mathrm{NV}}^{-}$ center ensemble. From our theoretical analysis, we quantitatively show that the magnetic fields actually suppress the inhomogeneous broadening from the strain in the ${\mathrm{NV}}^{-}$ centers.

Figure 1

Illustration of the hybrid system composed of a superconducting flux qubit and an ensemble of ${\mathrm{NV}}^{-}$ centers. The flux qubit has four Josephson junctions that form a two-level system. There are two control lines for the flux qubit. We use one of the control lines to change the energy bias $\epsilon$, and use the other line to change the energy gap $\Delta$. Diamond crystal is glued on top of a flux qubit, and this diamond contains ${\mathrm{NV}}^{-}$ centers. The electron spins trapped in the ${\mathrm{NV}}^{-}$ center form a three-level system, and so we have a V-type energy level structure for the ${\mathrm{NV}}^{-}$ center.

Figure 2

Vacuum Rabi oscillations between the FQ and ${\mathrm{NV}}^{-}$ centers without an applied external magnetic field. Red dots show the experimental results, while the blue line shows the results of a numerical model where we use $N=1200,\epsilon =0,{\Gamma }_c/2\pi =0.3\mathrm{MHz},{\Gamma }_b/2\pi ={\Gamma }_d/2\pi =0.44\mathrm{MHz},\delta D_k/2\pi =0.08\mathrm{MHz}$ (FWHM), $\delta \left(g{\mu }_B{B}_z^{\left(k\right)}\right)/2\pi =3.1\mathrm{MHz}$ (FWHM), $\delta E_{1,k}/2\pi =\delta E_{2,k}/2\pi =4.4\mathrm{MHz}$ (FWHM), $A_{\text{HF}}=2.3\mathrm{MHz}$, and $\sqrt{N}g/2\pi =13\mathrm{MHz}.$

Figure 3

Vacuum Rabi oscillations between the FQ and ${\mathrm{NV}}^{-}$ centers with an applied external magnetic field of $2.6$ mT along the [100] crystalline axis. Except for the magnetic field, we use the same parameters as those in Fig. 2.

Figure 4

Numerical simulations of vacuum Rabi oscillations with the applied magnetic field for several values of the strains. Except for the values of the strains, we use the same parameters as those in Fig. 3.