One of the many quantum sensors widely used by scientists today is Nitrogen-Vacancy Centers, known as the NV centers. NV centers are defects where one carbon atom lying next to a vacancy (a “missing atom”) is replaced by one nitrogen atom in a diamond crystal or thin film, resulting in a mobile electron and a nearby vacancy [1] . This is an NV center (see also Figure 1). They can be natural, or created artificially and controllably by a variety of methods. One such method is irradiation [2] [3] providing a nitrogen atom some energy (at high temperatures) so it can break the covalent bonds between carbon atoms and replace a carbon atom. Since this NV center gains no electrons from its external environment (i.e., the lattice structure surrounding it) and only has 1 unpaired electron, it is called an NV⁰ [1] [3] .

However, in quantum sensing, NV¹⁻, where the defect site gains an electron and has 2 unpaired electrons at the vacancy, is of particular interest. Having a quantum state with two electrons yields a total spin S = 1, which then splits into three quantum sublevels indexed by the quantum number m = −1, 0, +1. The acquired electron comes from the isolated Nitrogen atom in the lattice structure such that the net charge of the overall system remains the same [4] .

NV centers are artificial atoms, whose energy levels are discrete, like those of actual atoms. The discovery of the energy level quantization in atoms was one of the seminal discoveries of the first quantum revolution in the early 1900’s. Electrons occupy a region outside the atom’s nucleus at certain special locations called “orbitals” (Figure 2). Each orbit corresponds to a certain amount of

energy called the energy level, which means energy levels are discrete. This is the model that was originally developed by Danish physicist Niels Bohr [5] . The different energy levels are enumerated with an integer label n, known as the principal quantum number. In the case of hydrogen, according to Bohr’s model, if its electron is at the n = 1 energy level, the atom is at its ground state. If the electron has energy levels greater than n = 1, the atom is in an excited state.

We can identify two main sets of quantum states in an NV center: ³A, triplet ground state, and ³E, triplet excited state. Each triplet state has 3 different spin states: |0>, |−1>, |+1>. Initially, these 3 spin states are “degenerate”—they have the same energy level. However, because of the intrinsic magnetic environment provided by the lattice structure, the |0> state splits with |±1> states. This is called the zero-field splitting (ZFS) and it is due to spin-spin interactions which are influenced by the lattice [6] [7] . These formerly degenerate levels now have different energies. In addition to these two main sets of states, there is an intermediate singlet state between the ground and excited energy states [8] . The latter plays an important role in the radiative processes that connect the ³A and ³E states. There is no external magnetic field applied at this point. An illustration of the energy levels is shown in Figure 3.

To excite the spins coherently, namely so the NV center can be in a quantum superposition of |0> or |±1>, an electromagnetic wave is used to excite the NV center. In this case, a microwave pulse is required [8] because the energy splitting between the levels is in the range of electromagnetic radiation with frequency in the GHz frequency range. For instance, to reach the ³A |±1> states from ³A |0> state (without an external magnetic field applied), a microwave pulse at a frequency of 2.87 GHz needs to be directed at the NV center (Figure 3) [9] .

If there is a magnetic field applied to this system, the Zeeman effect would take

place, where the |±1> states further split into |−1> and |+1> states, with the |+1> state at higher energy shown in Figure 4 [7] . The Zeeman splitting occurs because spins pointing in different directions (such as those of the m = −1 and +1 levels) gain or lose energy when they are parallel or antiparallel to the external magnetic field.

If a beam of green laser is directed to the NV center, the electrons would be optically pumped from the ground state to the high energy ³E states. This is illustrated by Figure 5, they will be pumped to the ³E |±1> states from ³A |±1> if they

were initially at ³A |±1>; they will be pumped to the ³E |0> states from ³A |0> if they were initially at ³A |0>. These transitions are selective on the value of m because the spin is conserved upon photoexcitation [10] .

After being pumped to the higher energy states, the electrons would decay back to the ground state. The decay can be through 2 processes: 1) spin-conserving photon emission; or 2) non-radiative relaxation [8] . Spin-conserving photon emission means that electrons will emit photons, that is, light, hence there will be a fluorescence observed. Non-radiative relaxation means that electrons would decay back via the intermediate singlet state, in the process releasing heat (vibrations of the diamond lattice) and emitting no light. When the NV center is excited into the ³E |0> spin state, the electrons in the NV center would decay by mainly the spin-conserving photon emission back to the ³A |0> spin state, emitting a red fluorescence and some heat. When the NV center is at ³E |±1> spin state, the electrons relax to the ³A |±1> state non-radiatively with higher probability [8] , emitting less fluorescence compared to when the NV centers decays from ³E |0>. This is shown in Figure 5. The difference in the amount of fluorescence provides the optical contrast necessary to perform the so-called Optically Detected Magnetic Resonance (ODMR).

The emitted fluorescence is around 30% higher for the decay from ³E |0> to ³A |0> state than for the ³E |±1> to ³A |±1> states [7] [11] . Hence, there is quite a strong contrast that can be observed with regards to the fluorescence intensity.

Figure 6 shows the graph of intensity of fluorescence emitted as a function of the gigahertz supplied to the ³E |0> state electrons, without and with an external magnetic field.

As seen from Figure 6(a), when the microwave frequency applied to the NV center is 2.87 GHz, the intensity of the fluorescence detected dipped. This shows that the NV center decays from ³E |±1> to ³A |±1>, which means that the NV center is pumped from ³A |0> to ³A |±1> by 2.87 GHz.

If there is a magnetic field applied to this system, the Zeeman effect would take place [7] , where the |±1> states further split into |−1> and |+1> states, with the |+1> state at higher energy. To increase the NV center from ³A |0> spin state to ³A |+1> spin state would now require a higher microwave frequency of around 2.88 GHz with reference to Figure 6(b).

With 3.6 Gauss of external magnetic field applied to the setup in Figure 6(b), the first dip in fluorescence is at about 2.86 GHz. When the microwave is applied at 2.85 GHz, the NV center would be pumped from the ³A |0> state to the ³A |−1> state. The green laser then sends the NV center up to the ³E |−1> state, which

then decays back to the ³E |−1> state via mainly the non-radiative relaxation method, causing the intensity of the fluorescence detected to dip.

The second dip in fluorescence is at about 2.88 GHz. When the microwave is applied at 2.88 GHz, the NV center would be pumped from the ³A |0> state to the ³A |+1> state. The green laser then sends the NV center up to the ³E |+1> state, and then decays back to the ³A |+1> state via mainly the non-radiative relaxation method, causing again the fluorescence detected to dip. The red arrow indicates the energy difference between the ³A |−1> and ³A |+1> states.

The larger the frequency difference indicated by the arrow in Figure 6(b), the larger the energy difference between spin |−1> and spin |+1> states, hence the stronger the Zeeman effect and greater the external magnetic field applied. The magnetic field strength is directly proportional to the energy difference between the ³A |−1> and ³A |+1> states [10] , providing a mechanism to spectroscopically and quantitatively measure the magnitude of the magnetic field, shown in Figure 7.

The spin coherence time is of significance for the detection of magnetic field strength. The spin states |0>, |−1> and |+1> are pure states. If the electron spins are not coherent, the NV center will not be in a pure spin state, but it would exist in multiple states at the same time (mixed state), so the signals detected would mix up, so sensitive and precise detection would not be possible. The spin coherence time of NV centers exceeds 400 μs at room temperature with dynamical decoupling [12] , which is a relatively long coherence time compared to other quantum sensors. Therefore, using an NV center instead of other quantum sensors is advantageous. Other than that, its high spatial resolution and high sensitivity

will be helpful in understanding material properties [13] , to discover new materials or enhance biomedical techniques including single-molecule detection.

NV centers can be used to carry out NMR from ten-picolitre volumes, which is from about 10⁻¹¹ liters [14] , to better understand and characterize organic and biological molecules and macromolecules (proteins, enzymes, etc.) including several of crucial importance for life science and biotechnology applications [15] . The high spatial resolution of NV centers [15] enables them to image biological samples with much higher resolution than MRI scanners. They can also reveal changes in chemical compositions from the micro to nano scale.

A thin diamond chip that is a few millimeters thick with a thin surface layer of NV ensembles, can be used to detect the magnetic fields produced by individual cells, enabling their identification. Since NV centers are highly sensitive to external magnetic fields but otherwise occupy a very small area, the image they produce has a high spatial resolution when they are scanned on an area of interest [15] . This is a new technology to diagnose cancer cells from normal cells. The problem with this method is that the cells may not always align to the N-V axis, and it is difficult to make them align to the axis. Hence, researchers insert nanodiamonds containing NV centers into the interior of cells or bacteria to better detect its magnetic properties and discover properties of cells or certain cancer biomarkers such as the presence of certain proteins [15] .

NV centers are highly sensitive to magnetic noises at the NV spin transition frequency—it can be used to detect magnetic ions such as Gd³⁺ [15] . Gd³⁺ can improve the clarity of the MRI image, by improving the visibility of inflammation, tumors and blood supply for some organs [16] .

NV centers are highly sensitive to free radicals in tissues since free radicals have a large spin noise, due to the presence of an unpaired electron [17] . Hence, NV centers can be used to detect free radicals in the human body, which is highly related to cardiovascular diseases, neurological disorders, and the immune system.

NV centers can further be used to study materials under high pressure [1] , since under high pressure, some materials become superconducting, which makes them perfect diamagnets. This perfect diamagnetism (also known as the Meissner effect) produces a characteristic magnetic response to external fields. NV centers embedded in the high-pressure cells (which use diamond anvils [18] ) can hence study the transition of a material from normal to superconducting state. In Figure 8, the 2 diamond anvils are exerting pressure on the material to transit it into its superconducting state.

NV centers can be inserted into the culet of the Diamond Anvils to detect the changes in magnetic properties of the material. This is to ensure that the distance between the NV center and the sample is as close as possible, to ensure the NV center can detect the magnetic field of the sample with maximum sensitivity (see also Figure 9). The “boxes” in Figure 9 placed at the culets of the Anvils

show where the NV centers are supposed to be placed at.