$$ \require{bbox} \require{upgreek} \require{physics} \def\du{\mathrm{d}} \newcommand{\vect}[1]{\boldsymbol{#1}} \def\eu{\mathrm{e}} \def\upMu{{\large \upmu}} \def\iu{\mathrm{i}} $$

Introduction

A bias field perturbs our defect system. This fact is obvious, however it is easy to overlook the effect (detrimental or otherwise) this perturbation has on our measurements. Any bias field that is not exactly aligned with the anisotropy (if present) of our spin defect will cause spin-mixing: the electron eigenstates will no longer be eigenstates of the spin operators. In this post I will repeat the simple model introduced in Tetienne et al. 2012¹, then elaborate by applying it to a few key situations metrologists will hopefully find useful. I will primarily consider NVs, but the model should be general to any colour centre. The model is basic undergraduate quantum mechanics mixed with a little photodynamics. For case studies I will address the change to the PL intensity, ODMR (optically detected ESR) contrast, $T_1$ time, and Rabi frequency, as compared with an ideally-aligned bias field (I will leave level anti-crossing dynamics for another post). These examples should provide simple examples to cover the main applications of MW free, DC and AC sensing.

Introduction and Assumptions

We begin by assuming we have a defect with uniaxial anisotropy, that is that there is some singular preferred axis to the defect Hamiltonian. For example, the NV-centre has such an anisotropy caused by spin-spin interactions between its two electrons, which is shares symmetry with the geometric N-V (111) axis in the diamond crystal. The basic form of the NV ground state electronic Hamiltonian is \begin{equation} \mathcal{H} = DS_z^2 + \gamma_e \vect{B} \cdot \vect{S} \end{equation} where $\gamma_e = g {\large \upmu}_{\rm B} = 28$ GHz/T and $D=2.87$ GHz is the temperature-dependent zero-field splitting parameter. If the bias field is along the defect axis ($z$) then the only operator in $\mathcal{H}$ is $S_z$, thus our stationary states are also eigenstates of $S_z$, and $m_s$ is a good quantum number. The important thing to note here is that this means some state $\ket{m_s=0}$ is conserved as a function of magnetic field strength² and so it can continue to be labelled correctly as $\ket{m_s=0}$.

Instead, if we have some off-axis bias field, then we assume that the eigenstates of the system ($\ket{i}$) are linear combinations of the zero-field eigenstates ($\ket{j}$, which form a complete set): \begin{equation}\label{eq:lincomp} \ket{i} = \sum_{j} \alpha_{ij}(\vect{B})\ket{j} \end{equation} and we won’t worry about how we identify the eigenstates $\ket{i}, \ket{j}$ just yet. The $\alpha_{ij}$ coefficients transform from the old to new basis and can be determined by diagonalising $\mathcal{H}$ in the $S_z$ basis. For example, we might set up our system numerically in the zero-field basis (i.e. our basis is formed by the eigenstates of our defect anisotropy; $S_z$). Each element of a column vector would represent the $m_s = 1, 0, -1$ states, and so we would have the regular Pauli matrices (or higher spin analogues³) for our spin operators, and so on. Then when we diagonalize our full Hamiltonian $\mathcal{H}$ in this basis, we get the energy levels and the transform $\alpha_{ij}$ from the eigenvalues and eigenvectors, respectively.

We usually measure our defects optically, so we need to consider what happens to the defect transition rates. A (spontaneous emission) transition rate can be calculated from Fermi’s golden rule: \begin{equation} k_{ab} \propto | \bra{a} \vect{r} \ket{b} |^2 \end{equation} where the matrix element is over the electric dipole. We don’t really want to evaluate all that exactly, so instead we’ll try and express $k_{ij}$ in terms of the zero field rates $k^0_{pq}$. That transform is pretty easy, just plug Eq. \ref{eq:lincomp} into each side of the matrix element \begin{align} k_{ab} &\propto | \Big (\sum_c \alpha_{ac}^\dagger(B)\bra{c} \Big ) \ \vect{r}\ \Big ( \sum_d \alpha_{bd}(B)\ket{d} \Big ) |^2 \\ &= \sum_{c} \sum_d |\alpha_{ac}|^2 | \alpha_{bd}|^2 | \bra{c} \vect{r} \ket{d} | ^2 \\ &= \sum_{c} \sum_d |\alpha_{ac}|^2 | \alpha_{bd}|^2 k_{cd}^0~, \end{align} which is exactly Eq. 3 from Tetienne et al. 2012¹. This immediately gives us the change in the $T_1$ time, which scales as something like $1 / 3k_{0,1}$⁴. Note that the real bias induced change to $T_1$ will dependent on the specifics of any environmental magnetic noise, which needs to be treated separately from the intrinsic transition rates, and in general will have its own directionality. I haven’t thought about this in detail yet, so I don’t want to post some half-baked analysis here.

Spin contrast

Determining the spin contrast will dependent on the specifics of the defect in question. Again, we shall consider spin contrast measured optically, so we need to model the PL signal $\mathcal{R}$. We already have one piece of information, which is the lifetime ($\tau_i$) of each eigenstate, simply the reciprocal of the relevant transition rates. Taking $\eta$ as the collection efficiency, the time-dependent PL signal will look something like (again stealing from Tetienne et al. 2012¹) \begin{equation} \mathcal{R}(t, \vect{B}) = \eta \sum_a \sum_b k_{ab} n_a(0) \eu^{-t / \tau_a}. \end{equation} The transition rates are given above, but the populations much be determined from the classical rate equations of the system \begin{equation} \frac{\du{n_i}}{\du t} = \sum_j (k_{ji} n_j - k_{ij} n_i), \end{equation} under suitable boundary conditions. This equation simply states that the change in population ($n$) of the state $i$ is the sum of the rates (the $k$’s are the probability of a single transition per unit time, so we need to scale by the population in the ‘old’ state) entering $i$, minus the sum of rates leaving $i$.

Last thing we need is the mean PL rate, after adding in some (CW) microwave coupling \begin{equation} \overline{\mathcal{R}}(\vect{B}) = \eta \sum_{\rm e} \sum_{\rm g} \overline{n}_{\rm e} k_{\rm eg}, \end{equation} where $e,g$ represent the excited and ground states of the optical transitions, respectively. Now we can define the ESR contrast for a transition $p \to q$ as \begin{equation} \mathcal{C}(\vect{B}) = \frac{\overline{\mathcal{R}}(k_{pq}=0) - \overline{\mathcal{R}}(k_{pq})}{\overline{\mathcal{R}}(k_{pq}=0)}~, \end{equation} i.e. the normalised difference in PL between MW on and MW off, under continuous laser excitation. Most of the details above were rushed over, but the key elements were the spin-mixed transition rates, after that it was just rate equations. I have some rough code available here.

Figure 1: ESR Contrast. (a) ODMR (optically detected ESR) spectra for a single NV at various magnetic fields, fit with Lorentzians. (b) ODMR frequencies as a function of field strength, used to extract the angular misalignment (inset). (c) ODMR contrast as a function of field strength. Reproduced from Ref. 1, Copyright 2012 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft, licensed under CC BY-NC-SA 3.0.

There are some nice plots in Tetienne et al. 2012¹, if you want to see what sort of trends you get.

Rabi frequency

Alrighty, onto the AC domain, let’s look at the Rabi frequency. In general, for some periodic perturbation Hamiltonian term $\mathcal{H}_2(t) = \mathcal{H}_2\eu^{-\iu \omega t}$, the Rabi frequency is \begin{equation} \hbar \Omega = \bra{f} \mathcal{H}_2 \ket{i}~, \end{equation} between some unperturbed $i$nitial and $f$inal states. Our full Hamiltonian will look something like \begin{align} \mathcal{H} &= DS_z^2 + \gamma \vect{S} \cdot \vect{B}_1 + \gamma \vect{S} \cdot \vect{B}_2(t) \\ &= \mathcal{H}_0 + \mathcal{H}_1 + \mathcal{H}_2(t)~. \end{align} Where the MW field is of the form \begin{equation} \vect{B}_2(t) = \vect{B}_2 \eu^{-\iu \omega t} \end{equation} We are going to consider a few cases: ${\rm \textbf{I}}$, bias perfectly aligned with $z$ and mw with $y$; ${\rm \textbf{II}}$, bias perfectly aligned with $z$ but mw direction varied and ${\rm \textbf{III}}$, bias and mw of arbitrary direction.

A note on ladder operators

To simplify the working, we’ll quickly remind readers of the ladder operators. \begin{align} J_+ &= J_x + \iu J_y \\ J_- &= J_x - \iu J_y~, \end{align} or rearranged \begin{align} J_x &= \frac{1}{2}(J_+ + J_-) \\ J_y &= \frac{1}{2\iu}(J_+ - J_-)~, \end{align} with eigenvalues \begin{align} J_+ \ket{j, m} &= \hbar \sqrt{(j - m)(j+m+1)}\ket{j, m+1} \\ J_- \ket{j, m} &= \hbar \sqrt{(j + m)(j-m+1)}\ket{j, m-1}~. \end{align}

I. Aligned bias and MW driving

The simplest case to consider is \begin{align} \vect{B}_1 &= B_1 \vect{\hat{z}} \\ \vect{B}_2 &= B_2 \vect{\hat{x}}~. \end{align} As the bias field is aligned with the anisotropy ($\mathcal{H}_0$), the electron states are simple eigenstates of $S_z$, \begin{equation} \ket{-S}, \ket{-S + 1}, \ldots \ket{S}~. \end{equation} The Rabi frequency is thus simple to write down, for example \begin{equation} \hbar \Omega_{\rm I} = \gamma B_2 \bra{g} S_x \ket{e}~. \end{equation} To proceed further we need to pick our states and spin number. We will proceed with the calculation for the ‘highest’ transition. \begin{align} \hbar \Omega_{I} &= \gamma \bra{S-1} \vect{S} \cdot \vect{B} \ket{S} \\ &= \gamma B_2 \bra{S-1} S_x B_x \ket{S} \\ &= \frac{\gamma B_2}{2} \bra{S-1} S_+ + S_- \ket{S} \\ &= \frac{\sqrt{2S} \gamma \hbar B_2}{2} \end{align}

II. Aligned bias and general MW driving

Now we will consider the case of a MW field at an angle to the bias ($\theta$ from $\hat{\bf z}$, $\phi$ from $\hat{\bf x}$) \begin{align} \vect{B}_1 &= B_1 \hat{\bf z}\\ \vect{B}_2 &= B_2 \sin(\theta)\cos(\phi)\hat{\bf x} + B_2 \sin(\theta)\sin(\phi)\hat{\bf y} + B_2 \cos(\theta)\hat{\bf z}~. \end{align} Let’s again consider the highest transition, noting that any $z$ component will cancel: \begin{align} \hbar \Omega_{\rm II}^{S} &= \gamma \bra{S - 1} \vect{S} \cdot \vect{B} \ket{S}\\ &= \gamma B_2 \bra{S - 1} S_x \sin(\theta) \cos(\phi) + S_y \sin(\theta) \sin(\phi) \ket{S} \\ &= \frac{\sqrt{2S} \gamma \hbar B_2}{2} \big ( \sin(\theta) \cos(\phi) - \sin(\theta) \sin(\phi) \big )~. \end{align}

III. Misaligned bias and general MW driving

Now we need to take into consideration the spin mixing from our misaligned bias. This process is probably possible analytically, but it’s a whole lot easier numerically as we did earlier. Remember here that you can phase shift your driving field however you like, but the Rabi frequency needs to be real at the end - or just take the absolute value of your result and you’ll be happy. When you go to numerics, it will be useful to have expressions for the higher spin operators, so consult the footnotes³. Aside from taking into consideration the spin mixing instead of working analytically with $\bra{S-1}$ and $\ket{S}$, the process is identical for the two cases above, so I won’t work through everything in detail here - but I have some rough code available. Note that it’s probably easier at this point to simulate the time evolution of the density matrix in general (no perturbation approximation), and then fit/Fourier transform the population curves to get the flopping frequency. However this approach is more computationally complex, and I like the above modelling as it makes the spin-mixing more tangible.

Conclusion

I hope this rough introduction to the effects of bias field induced spin mixing on metrology measurements. These effects usually result in simply a reduced sensitivity - for example from the drop in contrast in an ODMR measurement. On the other hand, if for example you are comparing two $T_1$ measurements, it is important to either take controls for each measurement (with/without noise source), or ensure that the bias field is exactly the same in each. I would be remiss if I did not point out a few papers that have used these effects as a tool, for example to align the bias field⁵, or as an sensing protocol in and of itself⁶.

References

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