Coupling spin ensembles via superconducting flux qubits

We study a hybrid quantum system consisting of spin ensembles and superconducting flux qubits, where each spin ensemble is realized using the nitrogen-vacancy centers in a diamond crystal and the nearest-neighbor spin ensembles are effectively coupled via a flux qubit. We show that the coupling strengths between flux qubits and spin ensembles can reach the strong and even ultrastrong coupling regimes by either engineering the hybrid structure in advance or tuning the excitation frequencies of spin ensembles via external magnetic fields. When extending the hybrid structure to an array with equal coupling strengths, we find that in the strong-coupling regime, the hybrid array is reduced to a tight-binding model of a one-dimensional bosonic lattice. In the ultrastrong-coupling regime, it exhibits quasiparticle excitations separated from the ground state by an energy gap. Moreover, these quasiparticle excitations and the ground state are stable under a certain condition that is tunable via the external magnetic field. This may provide an experimentally accessible method to probe the instability of the system.

Figure 1

(a) Schematic diagram of the proposed hybrid quantum system. Among the three spin ensembles, every two nearest-neighbor spin ensembles are coupled by a superconducting flux qubit. The qubit loop is on the $y\text{−}z$ plane and perpendicular to the $x$ direction. (b) The Zeeman splitting for the spin-1 sublevels $m_{\mathrm{s}}=0,\pm 1$ of the electronic ground state in a NV center. (c) Hybrid array consisting of spin ensembles and flux qubits, which is a periodic extension of the hybrid quantum system in (a).

Figure 2

The separation ($z_{\mathrm{NV}}$) dependence of the coupling strength $J^{\left(m\right)}$ between the flux qubit and the $m$th spin in the NV-center ensemble, which is located on the symmetric line of the rectangular loop in the $z$ direction. Here $I_p^{\left(i\right)}=0.5\mu \mathrm{A}$, and $a=b=1\mu \mathrm{m}$ for the (blue) dotted curve; $I_p^{\left(i\right)}=0.5\mu \mathrm{A},a=2\mu \mathrm{m}$, and $b=10\mu \mathrm{m}$ for the (green) dashed curve; and $I_p^{\left(i\right)}=0.9\mu \mathrm{A},a=2\mu \mathrm{m}$, and $b=50\mu \mathrm{m}$ for the (red) solid curve.

Figure 3

Dispersion relation for the quasiparticle energy $E_k$ in $\mathrm{Eq}.\left(27\right)$, where $J=0.25\mathrm{GHz}$, ${\omega }_q=6\mathrm{GHz}$, and ${\omega }_s=1\mathrm{GHz}$. Note that an energy gap occurs, which separates the quasiparticle excitations from the ground state.