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Introduction

Magnetic noise sensing is a particular avenue of application for solid-state spin defects in which an external magnetic noise source is detected or measured through changes in the defects longitudinal spin relaxation time, which is conventionally referred to as $T_1$. While there is much which could be discussed regarding the various aspects of $T_1$ measurement, theory, and application this post will focus on the interaction between the spin defect and the magnetic noise source which leads to a reduction in the observed $T_1$, an effect otherwise known as $T_1$ quenching. However, it is this very quenching effect which forms the basis for most noise sensing applications using solid-state spin defects so I think it is a good topic to delve into.

Now while the general theory of how the quenching effect comes about is well understood and heavily reported, there is a rather unsatisfying lack of documentation regarding the derivation of the important equations relevant for modelling the observable changes in $T_1$ from a magnetic noise source, at least within the spin defect community¹. Here I have compiled what I believe to be some of the keys steps in these derivations, framed specifically in the context of solid-state spin defects.

We’ll first start by defining $T_1$ qualitatively and quantitatively by making use of classical rate equations. In this case we’ll consider a three-level system representative of a triplet ground state which is typical of optically addressable spin defects used in quantum sensing². Then we’ll make use of some statistical and quantum mechanics to obtain a formula to calculate the magnetic noise contribution to the overall relaxation rate of the system. Lastly we’ll look at an example where the magnetic noise source is a volume of paramagnetic ions coupled to a single defect embedded in a slab of material.

Defining $T_1$

Figure 1: Defining $T_1$. Phenomenological description of $T_1$ as a cycle between equilibrium and non-equilibrium states. The time taken for relaxation from the non-equilibrium state back to equilibrium is characterised by $T_1$.

Let’s first start with a brief qualitative definition of what the time $T_1$ is. Consider some spin system with at least two-levels where the spins are allowed to transition between the states which is momentarily in a state of static equilibrium. Perturbing the system out of equilibrium by reording the populations in the levels creates a non-equilibrium state. The transitions between the states act to drive the system back towards equilibrium with some characteristic rate and consequently some characteristic decay time which we define as $T_1$. This is a rather loose, phenomenological definition but I think it will suffice for the purposes of this post.

For a quantitative definition we need to be more specific with the system we consider. Here we will use a three-level model, and by using classical rate equations then define $T_1$ for a typical solid-state spin defect system.

Figure 2: Three-level model. The two-way transition rate $k_{01}$ individually couples the $\ket{+1}$ and $\ket{-1}$ states to the $\ket{0}$ state.

As a brief reminder, the population dynamics of a multilevel system with transition rates coupling the levels can be modelled with a matrix rate equation

\begin{equation} \partial_t \vb{\rho}(t) = \Phi \vb{\rho}(t), \end{equation}

where the vector $\vb{\rho}(t)$ describes the population density of the levels within the system at a time $t$ and $\Phi$ is the transition rate matrix which describes the instantaneous rate of transition between the levels of the system. The general solution to the matrix rate equation is dependent on the eigenvalues ($\lambda_n$) and eigenvectors ($\vb{u}_n$) of the transition rate matrix

\begin{equation} \vb{\rho}(t) = \sum_n c_n \vb{u}_n e^{\lambda_n t}, \end{equation}

with coefficients $c_n$ which can be solved for with system specific boundary conditions.

In our three-level system, we’ll couple $\ket{0}$ to $\ket{+1}$ and $\ket{-1}$ with a two-way transition rate $k_{01}$. For simplicity we assume there is no coupling between $\ket{+1}$ and $\ket{-1}$, which is valid if the relaxation dynamics is dominated by magnetic effects rather than electric³⁴ or phonon⁵ processes. I have chosen to use such a simple model in this instance as there isn’t much point in complicating the matter here. For completeness I’ll note this model ignores all the defect dynamics outside of the ground state, which while relevant to the measurement of $T_1$ can for all intents and purposes be ‘absorbed’ into $k_{01}$ (in fact later we’ll make use of this by letting $k_{01} = k_{\text{int}} + k_{\text{ext}}$ to capture both intrinsic ($k_{\text{int}}$) and external ($k_{\text{ext}}$) contributions to the overall rate defining process).

Thus, the transition rate matrix $\Phi$ can be constructed by assigning each row and column to one of the states in the system. Off-diagonal elements are then populated with the corresponding transition rates between those states and the diagonal elements are defined so the sum of each row is zero.

\begin{align} \Phi = \begin{bmatrix} -k_{01} & k_{01} & 0\\ k_{01} & -2k_{01} & k_{01}\\ 0 & k_{01} & -k_{01} \end{bmatrix} \end{align}

Solving the matrix rate equation with initial populations $\rho_0(0)$, $\rho_{+1}(0)$, and $\rho_{0}(0)$, gives

\begin{align} \rho_0(t) &= \frac{1}{3} + \bigg[\rho_0(0) - \frac{1}{3} \bigg] e^{-3k_{01}t}\\ \rho_{\pm1}(t) &= \frac{1}{3} - \frac{1}{2}\bigg[\rho_0(0) - \frac{1}{3} \bigg] e^{-3k_{01}t} \pm \frac{1}{2} \bigg[\rho_{+1}(0) - \rho_{-1}(0) \bigg] e^{-k_{01}t}~. \end{align}

By convention, for common solid-state spin defects, $T_1$ is defined as the characteristic decay time out of the $\ket{0}$ state, i.e. the decay in the population of $\rho_0(t)$; thus, $1 / T_1 = 3k_{01}$. Arguably this was an excessive amount of work for an incredibly simple definition but it motivates a choice in the following which would appear rather arbitrary otherwise.

Contribution from external magnetic noise⁶

Let us now turn to deriving an expression for the component of the total relaxation rate attributed to an external magnetic noise source. As I have already alluded to, the transition rate coupling the $\ket{0}$ and $\ket{\pm1}$ states of the spin triplet can be broken down into the sum of intrinsic and extrinsic contributions $k_{01} = k_{\text{int}} + k_{\text{ext}}$. The intrinsic part encapsulates all internal effects (e.g. phonons, electric processes or even internal magnetic noise sources), which in the absence of external magnetic noise is the fundamental relaxation rate defining $T_1$. On the other hand the extrinsic part contains all external contributions which in this case is going to be an external magnetic noise source and is now the subject of our scrutiny.

I’d first like to set the scene by defining our external magnetic noise source and provide a brief qualitative description of its action on the three-level system. Essentially we are going to treat the magnetic noise source as ‘white noise’ meaning it has some fluctuating magnetic field $\vb{B}(t)$ with zero-mean $\langle B(t) \rangle = 0$ and uncorrelated spatial components $i = x,y,z$. As a quick aside, using this definition for the noise source, from the Wiener-Khinchin theorem we can write the spectral density of the noise using the autocorrelation of the fluctuating field,

\begin{equation} S(\omega) = \int B_i(t) B_i(t+\tau) e^{-i \omega \tau} \dd{\tau}. \end{equation}

This is not the most important part of the derivation but this exact integral comes up later. However, it is a useful starting point to understanding where the quenching effect comes from and thus introduce how we can go about determining $T_1$. In other quantum sensing schemes, transitions between the $\ket{0}$ and $\ket{\pm1}$ states are forcibly driven using a (reasonably) well-defined microwave field which can effectively be thought of as a magnetic noise source with a narrow spectral density function (provided the microwave field is on or near resonance with the Rabi frequency). In this sense, the defect can be thought of as acting like a filter for specific frequencies which are contained in the envelope of some spectral density function. Luckily our external magnetic noise source behaves in the same way! It may have a spectral density function which is broader, less well-defined, or weaker, but as long as it contains a frequency component which can drive the $\ket{0} \leftrightarrow \ket{\pm1}$ transition it essentially acts the same as the microwave field.

The analogy between microwave driving and the action of the magnetic noise source sets up the next set of equations nicely as now we know we need to define a mechanism for transitions between states. Let us momentarily abandon the notion of a three-level system and consider instead one with an arbitrary number of levels (something more akin to the system in Fig. 1). For a stationary process $H(t)$ which causes randomly occurring transitions between two states $m$ and $k$, a correlation function can be defined which describes the probability distribution for the likelihood of a transition,

\begin{equation} G_{mk}(\tau) = \big\langle \mel{m}{H(t-\tau)}{k}\mel{k}{H(t)}{m} \big\rangle, \end{equation}

where $\tau$ is some time interval⁷. Note, this is specifically a correlation in time and the transitions are between states are within a closed system of fixed population (much like the three-level model was). Now we want to know how the population of one of the states in the system is evolving through time given there are these randomly occuring transitions. If we consider the state $m$ with population $\rho_m = \mel{m}{\rho}{m}$, then the rate of change in $\rho_m$ is simply the probability of a transition occurring between $m$ and $k$ over any and all time intervals $\tau$,

\begin{align} \dv{\rho_m}{t} &= \frac{1}{\hbar^2} \int_{-\infty}^\infty G_{mk}(\tau) e^{-\frac{i}{\hbar}(E_m - E_k)\tau} \dd{\tau}\\ &= W_{mk}. \end{align}

Where $W_{mk}$ has been defined as the rate of a transition from $m$ to $k$. Some quantum mechanics and numerous lines of algebra have been skipped here in relation to density matrices but essentially the exponential term comes from solutions to the quantum mechanical form of ⁸. Note, you can also get to this equation through some time dependent perturbation theory. Regardless, the expression for $W_{mk}$ looks like a Fourier transform so lets define some relationships based on properties of Fourie transforms,

\begin{align} G_{mk}(\tau) &= \frac{1}{2\pi} \int_{-\infty}^\infty J_{mk}(\omega) e^{-\frac{i}{\hbar}\omega\tau} \dd{\omega}\\ J_{mk}(\omega) &= \int_{-\infty}^\infty G_{mk}(\tau) e^{-\frac{i}{\hbar}\omega\tau} \dd{\tau}. \end{align}

Here we have introduced $J_{mk}$ as the inverse Fourier transform of $G_{mk}$⁹. Note $J_{mk}$ is essentially a spectral density for the interaction described by $G_{mk}$ (not really relevant but it is the source of the spectral density expression coming up later). Thus we can neatly write,

\begin{equation} W_{mk} = \frac{J_{mk}(\omega_m-\omega_k)}{\hbar^2}, \end{equation}

where, for convenience I have taken $\omega_a = E_a / \hbar$.

Now we have all the parts for describing the action of the system, but we are missing the physical mechanisms for driving it. Hence, lets find a way to include the magnetic field. We introduce the magnetic field as a part of the stationary process $H(t)$, as the field is acting on some spin (in this case a solid state spin defect) we take the vector projection of the field along a spin quantization axis, i.e.

\begin{align} H(t) &= \gamma \hbar \vb{S} \vdot \vb{B}\\ &= \gamma \hbar \big[ B_x S_x + B_y S_y + B_z S_z \big]\\ &= \gamma \hbar \sum_i B_i S_i \end{align} where $\gamma$ is the gyromagnetic ratio relevant to the spin defect and the time dependence is absorbed into the components of the magnetic field (we’ll show it explicitly in the next line)¹⁰. Now we simply substitute this expression into the initial expression for $G_{mk}(\tau)$ giving,

\begin{equation} G_{mk}(\tau) = \gamma^2 \hbar^2 \sum_{i,j} \expval{B_i(t)B_j(t+\tau)} \mel{m}{S_i}{k} \mel{k}{S_j}{m}. \end{equation}

As we want the components $i,j = x,y,z$ of the field to be uncorrelated in accordance with our initial assumption about its fluctuating nature we can let $i=j$. From the Fourier relation previously defined,

\begin{equation} J_{mk} = \gamma^2 \hbar^2 \sum_i \abs{\mel{m}{S_i}{k}}^2 \int \expval{B_i(t)B_i(t+\tau)} e^{-i \omega \tau} \dd{\tau}. \end{equation}

We have found the expression for the spectral density of the the fluctuating field! Its appearance is hardly an accident though, the Wiener-Khinchin theorem simply offers it up, reassuring us we are on the right track and close to our desired expression for $T_1$. Apparently there are only a few cases where this integral can be solved. A physically relevant case is when the fields fluctuate between two values (e.g. positive and negative) at a rate independent of the time between transitions, $\expval{B_i(t)B_i(t+\tau)} = \expval{B_i^2} \exp(-\abs{\tau} / \tau_c )$ where $\tau_c$ is the correlation time which defines the width of the spectral density.

Thus we have

\begin{align} J_{mk} &= \gamma^2 \hbar^2 \sum_i \abs{\mel{m}{S_i}{k}}^2 \expval{B_i^2} \frac{2 \tau_c}{1 + \omega^2 \tau_c^2}\\ \implies W_{mk} &= \gamma^2 \frac{2 \tau_c}{1 + \omega_0^2 \tau_c^2} \sum_i \abs{\mel{m}{S_i}{k}}^2 \expval{B_i^2} \end{align}

where $\omega_0 = \omega_m-\omega_k$ is the resonant frequency of the transition between the states $m$ and $k$. The summation is a necessity because we are considering contributions from all components of the fluctuating field. All we have left to do now is to determine the matrix elements $\mel{m}{S_i}{k}$. For the relevant solid state spin defects we are considering a spin-$1$ system. Therefore using the following spin matrices,

\begin{align} \mel{m}{S_x}{k} &= \frac{1}{2} (\delta_{m,k+1} + \delta_{m+1,k}) \sqrt{S(S+1) - mk}\\ \mel{m}{S_y}{k} &= \frac{1}{2} (\delta_{m,k+1} - \delta_{m+1,k}) \sqrt{S(S+1) - mk}\\ \mel{m}{S_z}{k} &= m\delta_{m,k} \end{align}

we can calculate the appropriate matix elements¹¹. First off we’ll just consider one possible transition between $m \rightarrow k = \ket{1} \rightarrow \ket{0}$. This gives,

\begin{equation} \sum_i \abs{\mel{m}{S_i}{k}}^2 \expval{B_i^2} = \expval{B_x^2} + \expval{B_y^2} = B_\perp^2 \end{equation}

where I have introduced $B_\perp^2 = \expval{B_x^2} + \expval{B_y^2}$ to simplify the expression. Now hopefully it is fairly obvious this is also the case for $m \rightarrow k = \ket{-1} \rightarrow \ket{0}$ (for completeness $m \rightarrow k = \ket{-1} \rightarrow \ket{1}$ gives only zeros which makes sense as they are not magnetically coupled). Hence we have,

\begin{equation} W_{0,+1} = W_{0,-1} = \gamma^2 B_\perp^2 \frac{2 \tau_c}{1 + \omega_0^2 \tau_c^2}. \end{equation}

If we assume a transition from $\ket{0}$ to $\ket{+1}~\textbf{OR}~\ket{-1}$ are equally likely then the probability rate of transition to both states is $W_{0,\pm 1} = \frac{1}{2}W_{0,+1}$ (i.e. $\ket{0}$ to $\ket{+1}~\textbf{AND}~\ket{-1}$). We have made it! We have an equation which describes the rate of transition between the states of the three-level system driven by an external fluctuating magnetic noise source! Thus by making the substitution $k_{\text{ext}} = W_{0,\pm 1}$ in to $k_{01} = k_{\text{int}} + k_{\text{ext}}$ an overall rate equation for the system is obtained,

\begin{equation} k_{01} = k_{\text{int}} + \gamma^2 B_\perp^2 \frac{\tau_c}{1 + \omega_0^2 \tau_c^2}. \end{equation}

Finally to re-express in terms of $T_1$ we can use the pathetic relation we derived at the very start $1 / T_1 = 3k_{01}$ to get,

\begin{equation}\label{total_T1} \frac{1}{T_1} = \frac{1}{T_1^{\text{int}}} + 3 \gamma^2 B_\perp^2 \frac{\tau_c}{1 + \omega_0^2 \tau_c^2} \end{equation}

where $1 / T_1^{\text{int}} = 3k_{\text{int}}$ in accordance with the original definition of $T_1$. So that’s it! Here is the equation which tells us the measured value for $T_1$ will be reduced if a solid-state spin defect is in the presence of a fluctuating magnetic field. The strength of the effect is scaled by the variance in the magnetic field perpendicular to the quantization axis of the spin defect ($B_\perp^2$) and the correlation time of the magnetic noise. Both of these are dependent on the nature of the noise source but the variance in particular is also dependent on the geometry of the system as we shall see in the following example.

Example: paramagnetic ions

Figure 3: Example $T_1$ sesning scenario. Schematic representation of a solid-state spin defect embedded in a material at a depth $h$. Parmagnetic ions fill the space external to the material.

Detecting paramagnetic ions is a basic test case for the $T_1$ quenching effect. We will use this scenario as an example for calculating the magnetic field variance from a particular magnetic noise source. It is important to remember this is just one example of calculating the variance and if you are considering a different noise source it is very likely you will need to use the relevant expressions for the magnetic field.

The system we will consider is a volume of paramagnetic ions situatued over a slab of material containing a defect embedded at a destance $d$ from the surface. Unpaired electrons give the ions a net magnetic moment which rapidly fluctuate under ambient conditions. We can treat the these ions as magnetic dipoles with fields coupling to the defect. The magnetic field vector $\mathbf{B}_n$ from a dipole $n$ with spin $\mathbf{S}_n$ is given by,

\begin{equation} \mathbf{B}_n = \frac{\mu_0 \gamma_e \hbar}{4\pi r_n^3} \big[\mathbf{S}_n - 3(\mathbf{S}_n \cdot \mathbf{u}_n)\mathbf{u}_n \big], \end{equation}

where $\mathbf{u}_n$ defines the position of the dipole relative to the defect and $r_n$ is the distance between them¹².

In free space (under ambient conditions and in the absence of any static fields) we assume all projections of the spin are equally probable so by taking the trace over a purely mixed state described by the density matrix $\rho = [1 /(2S+1)]\mathbf{1}_{2S+1}$ we can write the variance of the field from a single dipole as,

\begin{equation} B_{\perp,n}^2 = \langle B_{x,n}^2 \rangle + \langle B_{y,n}^2 \rangle = \text{Tr} [\rho(B_{x,n}^2 + B_{y,n}^2)] = \bigg( \frac{\mu_0 \gamma_e \hbar}{4\pi} \bigg)^2 C_S \frac{2 + \sin^2{\theta_n}}{r_n^6} \end{equation}

where we have used¹³

\begin{equation} \rm{Tr}[\rho S _{i,n}^2] = \frac{1}{2S+1} \sum_{m=-S}^S m^2 = \frac{S(S+1)}{3} = C_S, \end{equation}

with $i=x,y,z$. In this instance $\theta_n$ is the angle between the quantization axis of the spin defect, which we will take to lie along the $z$-axis for simplicity, and $\mathbf{u}_n$. Summing over all possible position for the paramagnetic ions (i.e. integrating over the space they occupy) gives us the total variance for the field acting on the spin defect.

\begin{align} B_\perp^2 &= \sum_n B_{\perp,n}^2 = \bigg( \frac{\mu_0 \gamma_e \hbar}{4\pi} \bigg)^2 \rho_{\text{ext}} C_S \int_0^{2\pi} {\rm d}\phi \int_0^{\frac{\pi}{2}} {\rm d}\theta \sin{\theta} \int_{\frac{d}{\cos{\theta}}}^{\infty} {\rm d}r \frac{2 + \sin^2{\theta}}{r^4}\\ &= \bigg( \frac{\mu_0 \gamma_e \hbar}{4\pi} \bigg)^2 \frac{\pi \rho_{\text{ext}} C_S}{3 d^3}. \end{align}

where $\rho_{\text{ext}}$ is density of the paramagnetic ions which we have also taken to be occupying all space external to the material. For this particular example we see the variance is proportional to the ion density $\rho_{\text{ext}}$, and inversely proportional to $d^3$ indicating the sensitivity of the defect to magnetic noise is strongly dependent on its depth.

Inputing physically relevant parameters and constants here and into Eq. $\ref{total_T1}$ gives an estimation for the expected measurement value for $T_1$ in the presence of paramagnetic ions (reminder: you also need to include the measured $T_1$ value without any paramagnetic ions present). I have not personally had the opportunity to apply similar methods to calculate the variance for other noise sources but if anyone has the maths on hand please reach out! Otherwise, I imagine at the very least the process is loosely similar so hopefully this example can be a useful guide for tackling such problems.

Conclusion

The main purpose of this post was to detail some of the steps and reasoning behind deriving the equation for the contribution to $T_1$ from an external magnetic noise source. While there were some detours and shortcuts along the way, my intention was to give enough context while skipping the boring bits to motivate the choices surrounding assumptions about the system and modelling. At the end of the day it is a rather simple derivation with somewhat lacklustre final equations, but hopefully it saves some from unnecessarily pouring through various SIs and textbooks because they were similarly disappointed by these equations (or similar) appearing all over the place without much explanation. Minor rantings aside I hope this answers some questions or scratches at a niggling curiosity itch. There are some intentional (and probably unintentional) gaps in the story which can hopefully be addressed in later posts, I am particularly interested in seeing some articles regarding the intricacies of $T_1$ measurement.

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