Energy levels and decoherence properties of single electron and nuclear spins in a defect center in diamond

The coherent behavior of the single electron and single nuclear spins of a defect center in diamond and a ${}^{13}\mathrm{C}$ nucleus in its vicinity, respectively, are investigated. The energy levels associated with the hyperfine coupling of the electron spin of the defect center to the ${}^{13}\mathrm{C}$ nuclear spin are analyzed. Methods of magnetic resonance, together with the optical readout of single defect centers, have been applied in order to observe the coherent dynamics of the electron and nuclear spins. Long coherence times, in the order of microseconds for electron spins and tens of microseconds for nuclear spins, recommend the studied system as a good experimental approach for implementing a two-qubit gate.

Figure 1

(a) The experimental cw ODMR spectrum recorded for an N-V center coupled to ${}^{13}\mathrm{C}$. (b) The stem plot shows the calculated transitions strengths and the frequencies at which they occur; the transitions strengths are equal, without considering any effects related to optical readout. (c) The energy-level scheme of the N-V center coupled to a ${}^{13}\mathrm{C}$ nucleus, calculated with the Hamiltonian presented in the text; the spin functions are of the form $\mid m_s{m}_I⟩$. Transitions are indicated with arrows and identified correspondingly in the cw spectrum in (b).

Figure 2

(a) The FID on the electron spin. The inset to (a) shows the microwave pulse sequence employed. There is no visible decay of the oscillations in the time range considered. For short interpulse delays, a modulation due to ${}^{14}\mathrm{N}$ can be observed. The FID measurement was recorded for a different magnetic field magnitude than that was for recording the cw spectrum in Fig. 1b. (b) Fourier transform of the FID data; the line at $12\mathrm{MHz}$ corresponds to the splitting between the levels 1 and 2 [see Fig. 1a], while the less intense lines are due to the ${}^{14}\mathrm{N}$ coupling. The Fourier transform was performed on the data for interpulse delays up to $1.5\mu \mathrm{s}$. (c) The Fourier transform on the data corresponding to interpulse delay higher than $1.5\mu \mathrm{s}$. The satellite lines in the previous time range are not present anymore due to the fast decay of the ${}^{14}\mathrm{N}$-induced modulation of the FID.

Figure 3

Hahn echo decay on a single electron spin for different values of the external magnetic field. The Hahn echo corresponding to zero field decays faster due to cross relaxation.

Figure 4

Hahn echo performed on the ${}^{13}\mathrm{C}$ nucleus. The pulse sequence employed, modified accordingly for the optical readout scheme, is shown in the inset to the figure. For times up to $30\mu \mathrm{s}$ the amplitude of the echoes show no decay.