Three-level spin system under decoherence-minimizing driving fields: Application to nitrogen-vacancy spin dynamics

Within the framework of a general three-level problem, the dynamics of the nitrogen-vacancy (NV) spin is studied for the case of a special type of external driving consisting of a set of continuous fields with decreasing intensities. Such a set has been proposed for minimizing coherence losses. Each new driving field with smaller intensity is designed to protect against the fluctuations induced by the driving field at the preceding step with larger intensity. We show that indeed this particular type of external driving minimizes the loss of coherence, using purity and entropy as quantifiers for this purpose. As an illustration, we study the coherence loss of an NV spin due to a surrounding spin bath of ${}^{13}\mathrm{C}$ nuclei.

Figure 1

A level diagram for the NV center with time-dependent driving fields.

Figure 2

Level population $|C_1{|}^2$ plotted as functions of time for a fixed driving, a first-order driving field, and a second-order driving field given by Eqs. (6) and (7). The parameters for first- and second-order driving fields are ${\omega }_{+}={\omega }_{-}=0.15,{\Omega }_{\pm }^{\left(1\right)}=0.9,{\Omega }_{\pm }^{\left(2\right)}={\Omega }_{\pm }^{\left(1\right)}/2$, and ${\Delta }_{+}=-1$. Oscillations in level population due to the oscillating driving field can be easily distinguished. Time scale is defined by zero field splitting ${\omega }_0$ and is of the order of a nanosecond.

Figure 3

Level populations $|C_i{|}^2={\left|\langle i|\psi \left(t\right)\rangle \right|}^2,i=1,2,3$ as functions of time. (a) Solid lines correspond to first-order and (b) dashed lines to second-order driving fields. Parameters same as in Fig. 2, ${\omega }_{+}={\omega }_{-}=0.15,{\Omega }_{\pm }^{\left(1\right)}=0.9,{\Omega }_{\pm }^{\left(2\right)}={\Omega }_{\pm }^{\left(1\right)}/2$, and ${\Delta }_{+}=-1$, but results extended to larger time. Green and blue curves of Fig. 2 now in red, solid and dashed, respectively. Time scale is defined by zero-field splitting ${\omega }_0$ and is of the order of a nanosecond.

Figure 4

Purity is plotted as a function of time for constant, first-order, and second-order driving fields. For comparison, the case of no driving field is also shown. As we include higher-order driving fields, the purity of the system improves significantly. In all the cases, $\Gamma =0.05$ and ${\Delta }_{+}=0.9$. For time-dependent driving fields, the parameters are ${\omega }_{+}=1.0,{\omega }_{-}=0.35,{\Omega }_{+}^{\left(1\right)}=1.0,{\Omega }_{-}^{\left(1\right)}=0.8,{\Omega }_{\pm }^{\left(2\right)}={\Omega }_{\pm }^{\left(1\right)}/2$, and ${\Omega }_{\pm }^{\left(3\right)}={\Omega }_{\pm }^{\left(2\right)}/2$. Time scale is defined by zero-field splitting ${\omega }_0$ and is of the order of a nanosecond.

Figure 5

Purity for second-order driving field is plotted as a function of time for $\Gamma =0.05,0.1$, and $0.5$. As in previous figures, $\omega =0.15,{\Omega }_{\pm }^{\left(1\right)}=0.9,{\Omega }_{\pm }^{\left(2\right)}={\Omega }_{\pm }^{\left(1\right)}/2$, and ${\Delta }_{+}=-1$. It can be seen that increasing the decoherence $\Gamma$ reduces the purity. Time scale is defined by zero-field splitting ${\omega }_0$ and is of the order a nanosecond.

Figure 6

The evolution of entropy is plotted for no driving, constant driving, first-order, and second-order drivings. In all the cases, $\Gamma =0.05$ and ${\Delta }_{+}=0.9$. For time-dependent driving fields, the parameters are ${\omega }_{+}=1.0,{\omega }_{-}=0.35,{\Omega }_{+}^{\left(1\right)}=1.0,{\Omega }_{-}^{\left(1\right)}=0.8,{\Omega }_{\pm }^{\left(2\right)}={\Omega }_{\pm }^{\left(1\right)}/2$, and ${\Omega }_{\pm }^{\left(3\right)}={\Omega }_{\pm }^{\left(2\right)}/2$. Time scale is defined by zero-field splitting ${\omega }_0$ and is of the order of a nanosecond.

Figure 7

Purity in the presence of noise of the surrounding heat bath for a constant driving; for first-order driving with noise from the MW source itself; and for the second-order driving case, when the second-order term overcomes the fluctuation in ${\Omega }_{\pm }^{\left(1\right)}$. The inset shows second-order driving for varying noise intensities of the surrounding spin bath with a fixed noise $\left(I_n=0.001\right)$ in the amplitude ${\Omega }_{\pm }^{\left(1\right)}$. Periodic drivings improve the purity in the presence of noise as compared to the constant driving case. Increasing the noise strength lowers the purity, as expected. The various noises for different intensities are also shown. The numerical simulation has been done for 1000 realizations. Parameters are the same as in Fig. 4, ${\omega }_{+}=1.0,{\omega }_{-}=0.35,{\Omega }_{+}^{\left(1\right)}=1.0,{\Omega }_{-}^{\left(1\right)}=0.8,{\Omega }_{\pm }^{\left(2\right)}={\Omega }_{\pm }^{\left(1\right)}/2$, and ${\Delta }_{+}=0.9$. The relaxation parameter is taken as $\Gamma =0.05$. Time scale is defined by zero-field splitting ${\omega }_0$ and is of the order of a nanosecond.

Figure 8

The evolution of entropy is plotted in the presence of noise from the surrounding spin bath for a constant driving; for first-order driving with noise from the MW source itself; and the second-order driving case, when the second-order term overcomes the fluctuation in ${\Omega }_{\pm }^{\left(1\right)}$. The inset shows second-order driving for varying noise intensities of the surrounding spin bath with a fixed noise $\left(I_n=0.001\right)$ in the amplitude ${\Omega }_{\pm }^{\left(1\right)}$. Parameters are the same as in Fig. 6, ${\omega }_{+}=1.0,{\omega }_{-}=0.35,{\Omega }_{+}^{\left(1\right)}=1.0,{\Omega }_{-}^{\left(1\right)}=0.8,{\Omega }_{\pm }^{\left(2\right)}={\Omega }_{\pm }^{\left(1\right)}/2$, and ${\Delta }_{+}=0.9$ and the relaxation parameter $\Gamma =0.05$. Time scale is defined by zero-field splitting ${\omega }_0$ and is of the order of a nanosecond.