Measuring a magnetic field with NV spins

Introduction

In the previous section, we saw that we can initialize and readout the spin state of NV centers optically. This makes it possible to observe the Zeeman effect even at room temperature. In order to eventually use NV centers as magnetometer, we need to know quantitatively the behavior of these spin states when an external magnetic field is applied. Therefore in this section we'll discuss how to determine the amplitude and direction of the external magnetic field from the electron spin resonance (ESR) frequencies of NV centers, and how to experimentally measure these frequencies.

Zeeman Splitting of NV Spin States

In the absence of external fields, the $m_S = \pm 1$ states of NV center are "degenerate" i.e. they have the same energy. When a magnetic field $\mathbf{B}$ is applied, this degeneracy will be lifted, and $m_S = \pm 1$ states will have different energies $E_u$ and $E_l$ (subscripts refer to "upper" and "lower"), or in terms of transition frequencies ("ESR frequencies"), $f_u$ and $f_l$ (linked to the energies by relation $E = hf$). The exact values of these frequencies are functions of both amplitude and direction of the magnetic field, specifically

$$f_i = f_i(B,\cos{\theta}),\quad i = u,l$$

where $B$ is the amplitude of external field $\mathbf{B}$, and $\theta$ is the angle between $\mathbf{B}$ and the NV axis. Importantly, the splitting of spin states is determined by the projection of external field along the NV direction, indicated by the $\cos\theta$ dependence.

As a result, if we can measure $f_u$ and $f_l$ experimentally, we will be able to solve for $B$ and $\cos{\theta}$, thus determine the amplitude and orientation with respect to the NV center of the external field. This relation is given by

$$\gamma B(f_u,f_l) = \frac{1}{\sqrt{3}}\sqrt{f_u^2+f_l^2-f_u f_l-D^2}$$

$$\cos^2\theta(f_u,f_l) = \frac{-(f_u+f_l)^3+3f_u^3+3f_l^3}{27D(\gamma B)^2}+\frac{2D^2}{27(\gamma B)^2}+\frac{1}{3}$$

where $\gamma = 28\mathrm{GHz/T}$ is the gyromagnetic ratio of electron, and $D = 2.87\mathrm{GHz}$ is the zero-field splitting of NV ground state (i.e. the splitting between $m_S = 0$ and $m_S = \pm 1$ at zero external field). The derivation for these equations (involves some quantum physics methodology) is included in the optional contents.

As a special case, when $\mathbf{B}$ is aligned with the NV axis i.e. $\theta = 0$, the above relations will be reduced to a simple linear form:

$$f_u = D+\gamma B,\quad f_l = D-\gamma B$$

Below is an illustration showing the $B$ dependence of ESR frequencies at different values of $\theta$.

Measuring the ESR Frequencies

In order to experimentally determine $f_u$ and $f_l$, we apply a microwave to the NV center after the initialization process discussed in previous section. We sweep the frequency of the microwave across the expected ESR frequency while monitering the NV photoluminescence. Since the $m_S = \pm 1$ states have lower PL intensity, we will notice a drop in measured PL intensity once we hit the ESR frequencies and drive the transition to $m_S = \pm 1$. Therefore, we can determine $f_u$ and $f_l$ by measuring the PL intensity versus microwave frequency (the "ESR spectrum"), and readout the location of the dips in that curve, as shown in the figure below.

If we're measuring with only one orientation of NV centers, we would then expect two dips in our ESR spectrum corresponding to $f_u$ and $f_l$. In reality, as discussed before, there are 4 possible orientations of NV centers and in our experiment all of them are contributing to the measured photoluminescence. Therefore there could be 4 different sets of $\{ f_u,f_l\}$ due to the different angles between NV centers and external field, resulting in a maximum of 8 dips in the spectrum. This can in principle allow us to exactly reconstruct the orientation of the magnetic field if we assign the dips to the correct NV orientation. See the optional contents for details on this.

There are many cases, however, where less dips are visible in the spectrum. This is because the ESR frequencies are determined by the projection of magnetic field on NV axis, which can be the same for a subset of the 4 possible orientations when the field direction fulfills specific symmetry. When this happens, the dips in the spectrum will also overlap, thus resulting in a reduced number of visible dips.

Exercise 3 - Analyze the behavior of ESR spectrum in different configurations

In the lab sessions of this practicum, you will be mostly working in one of the following configurations. For each of them, calculate the projection of external field on the 4 NV orientations and determine how many dips will there be:

1. There is no external field
2. The external field is in the $\langle 100 \rangle$ direction of the diamond
3. The external field is in the $\langle 110 \rangle$ direction of the diamond
4. The external field lies in the $\langle 100 \rangle$ plane with an arbitrary rotation (i.e. in the direction of $(0,y,z)$)
5. The external field is in the $\langle 111 \rangle$ direction of the diamond (i.e. aligned with one orientation of NV centers)