Strong Coupling of a Spin Ensemble to a Superconducting Resonator

We report the realization of a quantum circuit in which an ensemble of electronic spins is coupled to a frequency tunable superconducting resonator. The spins are nitrogen-vacancy centers in a diamond crystal. The achievement of strong coupling is manifested by the appearance of a vacuum Rabi splitting in the transmission spectrum of the resonator when its frequency is tuned through the nitrogen-vacancy center electron spin resonance.

Figure 1

Scheme of the experiment. (a)–(c) Photograph, top and side view of the sample: a diamond crystal is glued on top and in the middle of a superconducting coplanar resonator with its (001) surface facing the chip. A SQUID array (d) is inserted in the central conductor to tune the resonator frequency ${\omega }_r(\Phi )$ with an on-chip wire producing a flux $\Phi$ in the SQUIDs. A magnetic field ${\mathbf{B}}_{\mathrm{N}V}$ is applied parallel to the [100] direction. (e) Sketch of a $\mathrm{N}\mathrm{\text{−}}V$ center with its vacancy ($V$) and nitrogen atom (N), as well as three neighboring carbon atoms. Four equivalent $⟨111⟩$ directions exist for the $\mathrm{N}\mathrm{\text{−}}V$ axis, all making the same angle $\theta =55°$ with $[100]$, and thus with ${\mathbf{B}}_{\mathrm{N}V}$. (f) Energy diagram of the $\mathrm{N}\mathrm{\text{−}}V$ center, with the $|m_S=0⟩$ ground state separated by ${\omega }_{\pm }$ from the excited states $|\pm ⟩$, which are linear combinations of the pure spin states $|m_S=+1⟩$ and $|m_S=-1⟩$.

Figure 2

(a) Resonance line of the coplanar resonator with no diamond on top, measured at 40 mK. (b) Measured (dots) and fitted (dashed line) central frequency ${\omega }_r$ as a function of the applied flux $\Phi$, yielding a SQUID critical current $2I_{C0}=2.2\mu \mathrm{A}$ and a bare resonator frequency ${\omega }_{r0}/2\pi =3.012\mathrm{GHz}$. (c) Optically detected ESR of the $\mathrm{N}\mathrm{\text{−}}V$ centers in the sample, in zero magnetic field and at room temperature. The ${\omega }_{\pm }$ lines are fitted by a sum of two 6 MHz wide Lorentzians separated by 10.4 MHz. (d) Measured (dots) and fitted (dotted lines) dependence of ${\omega }_{\pm }$ on $B_{\mathrm{N}V}$, yielding a zero-field splitting $D/2\pi =2.873\mathrm{GHz}$ and a strain-induced splitting $E/2\pi =4.3\mathrm{MHz}$.

Figure 3

Transmission $|S_{21}|$ of the resonator with diamond on top (at 40 mK) as a function of $\Phi$ for $B_{\mathrm{N}V}=0$, 0.99, and 1.98 mT. Two anticrossings are observed symmetrically around 2.88 GHz, when the resonator frequency is resonant with the $\mathrm{N}\mathrm{\text{−}}V$ transitions $|0⟩\to |+⟩$ and $|0⟩\to |-⟩$. Red solid (yellow dashed) lines are fits to the eigenfrequencies of the coupled (uncoupled) resonator-spin system as described in the text. A transmission spectrum (white overlay) is also shown in linear units in the middle of one of the two anticrossings, for each $B_{\mathrm{N}V}$ value. Right-hand panel: Experimental frequencies ${\tilde{\omega }}_{\pm }$ (dots) resulting from the fits of $|S_{21}|(\omega ,\Phi )$ for several $B_{\mathrm{N}V}$ values. The data points are then fitted (solid lines) to the eigenfrequencies of Hamiltonian $H_{\mathrm{N}V}$ yielding $D/2\pi =2.878\mathrm{GHz}$ and $E/2\pi =7.2\mathrm{MHz}$. For comparison, the ESR frequencies ${\omega }_{\pm }(B_{\mathrm{N}V})$ measured at room temperature (see Fig. 2) are also shown as dashed lines.