Four-spin cross relaxation in a hybrid quantum device

Understanding the spin relaxation in superconducting quantum circuit and solid-state spin hybrid systems is of great importance especially for quantum storage purposes. We have studied the longitudinal relaxation for electron spins of substitutional nitrogen (P1) centers in a hybrid quantum device containing a diamond and a superconducting coplanar waveguide resonator. From a series of pump-probe experiments we conclude that the dominated spin relaxation mechanism is a cross-relaxation process induced by a four-spin interaction (four-spin cross relaxation) among different hyperfine split spin transitions. Some features of the four-spin process are discussed based on a set of rate equations. This work provides interesting perspectives in understanding the coherence properties of the hyperfine split spin ensembles in a hybrid quantum system.

Figure 1

(a) Illustration of the hybrid device. The magnetic field is applied along the diamond's $\langle 001\rangle$ direction to achieve the same Zeeman splitting for all the spin transitions along the $\langle 111\rangle$ direction. (b) Energy levels of electron spins of P1 centers considering Zeeman splitting and hyperfine splitting. (c) Intensity plot of the $S_{21}$ spectra under varied magnetic fields. Three clear avoided crossings ambiguously manifest the strong coupling between the three electron spin transitions and the resonator mode. The white markers and lines give the fitted peak positions in each of the vertical cuts. The inset in panel (c) is a vertical cut of panel (c) at $B_{\parallel }=154.4$ mT [indicated by the white dashed line in panel (c)]. It shows the vacuum Rabi split transmission peaks with a splitting value of $2g_{\mathrm{ens}}/2\pi =70.7$ MHz.

Figure 2

(a) Fast longitudinal relaxation process for $I_1$ spins. Panel (a) shows the evolution of the ground-state population after pumping the $I_1$ spins. (b) Spin cross relaxation from $I=1$ to $I=0$. After pumping the $I_1$ spins, the GS population for $I_0$ spins shows a fast decay, which indicates an excitation happens to the $I_0$ spins after the pump pulse. The error bar gives a standard deviation of the measured data. The red (gray) arrows label the calculation results based on the four-spin model. The red (gray) line is an exponential fitting of the data points. The insets in panels (a) and (b) show illustrations of the corresponding pump-probe schemes.

Figure 3

The evolution of the GS population of $I_1$ spins after pumping at $I_0$ spins (a) or $I_{-1}$ spins (b). The red (gray) arrows label the calculation results based on the four-spin model. The insets in panels (a) and (b) give illustrations of the corresponding pump-probe schemes.

Figure 4

Four-spin-process-mediated spin-relaxation process. A pumping pulse is firstly applied at $I_1$ spins. After 100 ms waiting, the system reaches a steady state and there is about 14% of spin excitation left in $I_1$ spins. Then we pump $I_{-1}$ spins and probe the GS population of $I_1$ spins as a function of the pump-probe delay. It can be seen that the $I_1$ spins start relaxing to the ground state mediated by the relaxation of $I_{-1}$ spins, which is consistent with the description of the four-spin relaxation process. The red (gray) arrows label the calculation results based on the four-spin model. The inset gives an illustration of the pump-probe scheme.

Figure 5

Calibration of the measured transmission intensity to the frequency shift of the VRS. Panel (a) shows one of the VRS peaks at different pump powers. The pump tone diminished the ground-state spin population and the VRS peaks shift correspondingly. As shown in panel (b), for a given frequency of $f=4.401$ GHz [indicated by a red dashed line in panel (a)] we could calibrate the strength of the transmission signal to the frequency shift. The error bars give standard deviations of the measured data points.

Figure 6

The slow spin diffusion process. The residual spin excitation after the four-spin relaxation process will be slowly relaxed with a time constant of about hundreds of seconds. Such a process is commonly known as spin diffusion out of the resonator mode region. The error bars give the standard deviations of the data points.