Optical properties of the NV centre
The first observations of NV centres date back some 50 years  and its isolation on an individual scale was not achieved until the 90s. It was first used as a source of single photons [3,4], and then it was found that this defect also possessed spin properties, which made it possible to use it as a quantum bit or a magnetic field sensor.
Under optical excitation by a green laser, the NV centre emits stable red photoluminescence at room temperature . In the photoluminescence emission spectrum shown in figure opposite, the zero-phonon line at 637 nm can be observed, which corresponds to the optical transition. The coupling to the phonons of the crystal lattice induces a broad satellite band up to 800 nm.
In the following, we will focus on ensembles of NV centres, with a typical density of a few tens of parts per billion (ppb), hosted in bulk diamonds.
Figure 1: Typical photoluminescence emission spectra of NV centres.
Optical detection of the magnetic resonance
The NV centre has a spin S = 1 in its ground state, so its fine structure is composed of the states |ms = 0> and |ms =± 1> [Fig. 2]. Here, ms is the projection of the spin along the quantization axis of the NV centre, formed by the nitrogen atom and the gap, along one of the  crystal orientations of diamond [Fig. 3]. |ms = 0> state is separated from the degenerate |ms =± 1> by D = 2,87 GHz.
The photophysics of the NV centre is governed by spin polarisation and spin state dependent photoluminescence mechanisms . First, under continuous laser excitation, the NV centre will gradually polarise into the |ms = 0>, spin state, thanks to an optical pumping mechanism. Moreover, the |ms = 0>, state emits more photoluminescence than the |ms =± 1>, states, the level of photoluminescence of the NV centre is indicative of its spin state.
These two properties are at the core of the optical detection of the magnetic resonance of the NV centre. It can be all-optically initialized in its spin state |ms = 0>, and the reading of its spin state is done by measuring the photoluminescence signal. By applying a resonant radio frequency excitation with the |ms = 0> ↔ |ms = ±1> transition, a population transfer between these two spin states will lead to a drop in the photoluminescence signal, as observed in figure 4.
In the presence of a magnetic field with a component that follows the quantization axis of the NV centre, the degeneracy of the |ms = ±1> states is lifted, by the Zeeman effect. The separation between the |ms = +1> and |ms = -1> states is proportional to the field component B//, which makes the NV centre a magnetometer at an atomic scale. In figure 5, 8 lines can be observed in the magnetic resonance spectrum, which correspond to the four |ms = 0> ↔ |ms = ±1> transitions in the four  crystal orientations of diamond.
Looking more closely at one of these lines in the magnetic resonance spectrum, the hyperfine structure of the NV centre can be revealed. This structure results from the interaction between the electron spin of the defect and the nuclear spin of the nitrogen atom. Nitrogen 14N has a natural abundance of 99.6% and has a nuclear spin I = 1, which leads to three hyperfine lines [Fig. 6].
Figure 2: Fine structure of the fundamental level of the NV centre.
Figure 3: Diagram of the crystal structure of the NV centre, composed of three carbon atoms, a vacancy and a nitrogen atom.
Figure 4: Magnetic resonance spectrum in zero field. the solid line is a Gaussian fit of the data.
Figure 5: Magnetic resonance spectrum under the application of a bias field of a few mT. The 8 lines correspond to the two transitions between the states |ms = 0> ↔ |ms = -1> and |ms = 0> ↔ |ms = +1> of each of the four crystal orientations.
Figure 6: Typical hyperfine structure of the NV centre.
Longitudinal relaxation time T1
The longitudinal relaxation time, or T1, is the time it takes for the system to return to a thermodynamic balance state once it has been polarised to its spin state |ms = 0>.
The standard T1 measurement sequence is shown in Figure 7. It consists of a first initialisation laser pulse, in order to polarise the system into the spin state |ms = 0>, followed by a free evolution time, in the dark, of variable duration τ. The last step is a readout laser pulse, which extracts the spin state of the NV centre via the photoluminescence signal.
Figure 8 shows the evolution of the photoluminescence signal as a function of the free evolution time τ in the dark. For short durations τ, the system does not have time to return to its thermodynamic balance state. It remains polarised in its spin state |ms = 0>, which results in a high photoluminescence signal. The gradual relaxation of the system towards its thermodynamic balance state results in the fall in photoluminescence for longer times τ.
Fitting the data by an exponential decay allows us to extract the longitudinal relaxation time T1 of 5.4 ± 0.5 ms, which is typical of an NV centre ensemble in a bulk diamond . The T1 time is limited by the interactions between the NV centre and the phonons in the crystal lattice .
Figure 7: Standard T1 relaxation time measurement sequence, consiting of two laser pulses.
Figure 8: Measuring the time T1 on a set of NV centres hosted in a bulk diamond. The solid line is a fit by an exponential decay.
Coherent control of spin state: Rabi oscillations
In the following, a bias magnetic field of a few mT will be applied along the quantization axis of the NV centre, in order to lift the degeneracy between the |ms = ±1> states. We will only consider a two-level system, composed of the states |ms = 0> and |ms = +1>.
Coherent control of the spin state of an NV centre can be achieved by performing Rabi oscillations between the states |ms = 0> and |ms = +1> . The sequence used is shown in Figure 9.
It consists of a first initialization laser pulse, which polarizes the NV centre into its spin state |ms = 0>. This is followed by a radio-frequency pulse of variable duration τ, resonant with the transition
|ms = 0> ↔ |ms = +1> : this is the manipulation step. The spin state after this step is a superposition of state α|ms = 0> + β|ms = +1>. The spin state of the system is then extracted by a final readout laser pulse. The Bloch sphere representation of each step is shown in Figure 10.
A typical Rabi oscillation signal is shown in Figure 11. For very short microwave pulse durations, the populations do not have time to switch from the |ms = 0> state to the |ms = +1> state, the photoluminescence signal remains maximum. For a radio frequency pulse duration Tπ, a population maximum of the |ms = 0> state has switched to the |ms = +1> state, so the photoluminescence signal is minimum. The coherent oscillations between these two states are represented by the oscillations of the photoluminescence signal. These Rabi oscillations allow us to define the times Tπ and Tπ/2, which are at the heart of the spin dynamics sequences.
Figure 9: Typical Rabi oscillation measurement sequence.
Figure 10: Representation on the Bloch sphere of the spin of the system ( red arrow) at each stage.
Figure 11: Typical Rabi oscillations measured on an NV centres ensembles hosted in a bulk diamond. The solid line is a fit by a sinusoidal function with an exponential decay envelope.
Transverse coherence time T2*
Decoherence is the loss of quantum information of a system initially placed in a state of quantum superposition,
This decoherence is characterised by the transverse coherence time T2*.
This T2*decoherence time can be measured by the Ramsey fringe method. The sequence is shown in Figure 13. The manipulation step consists of a first radio frequency pulse of duration Tπ/2 to place the system in a superposition state, as described by the equation above.
The system then evolves freely for a time τ and will perform a free precession. The coherences will accumulate a phase, which depends on the environment of the NV centre and in particular on the magnetic fluctuations of the spin bath. Finally, a second radio-frequency pulse of duration Tπ/2, followed by the readout laser pulse allows the spin state of the system to be read out. This spin state manipulation step is shown on the Bloch sphere in Figure 14.
A typical Ramsey fringe measurement is plotted in Figure 15. The decay of the oscillations corresponds to the transverse coherence time T2*, which can be extracted by fitting these data [8, 9], to give a T2* time of 456 ± 45 ns. The decoherence of the NV centre is governed by the interactions with the spin bath, composed of :
- Nuclear spins, such as 13C atoms , with a natural abundance of 1.1% and a nuclear spin
I = 1.
- Electronic spins, such as nitrogen impurities .
- Electron spins on the surface of diamond .
This coherence time can be extended by so-called dynamic decoupling sequences, which isolate the system from the magnetic noise generated by the surrounding spin bath.
Figure 13: Ramsey fringe measurement sequence.
Figure 14: Representation on the Bloch sphere of the spin state during the manipulation step.
Figure 15: Ramsey fringe measurement on a set of NV centres, the T2* coherence time is extracted by a fit described in reference 
Dynamic decoupling: coherence time T2echo
The simplest dynamic decoupling sequence to implement is the Hahn echo sequence . As schematised in Figure 16, the handling phase is similar to that presented above for the T2* time measurement, except that a radio frequency pulse of duration Tπ is applied in the middle of the free precession time. The phase shift accumulated during the time τ/2 is thus compensated by the phase accumulated during the second free precession time τ/2, as shown on the Bloch sphere in Figure 17. This method avoids the slow fluctuations of the spin bath, thus extending the coherence time.
A typical T2echo time measurement is plotted in Figure 18. Fitting the data with an exponential decay to the power n  allows a T2echo lifetime of 3.8 ± 0.2 µs to be extracted.
Figure 16: Hanh echo sequence.
Figure 17: Representation on the Bloch sphere of the spin state during the manipulation step.
Figure 18: Measurement of T2echo coherence time. The data is fitted by an exponential decay to the power of 2.
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