Stochastic quantum trajectories demonstrate the quantum Zeno effect in open spin 1/2, spin 1 and spin 3/2 systems

Open quantum systems are ubiquitous since most systems can be found interacting with their environment. The complexity of this interaction as well as uncertainty in the initial state of the environment mean that an element of randomness enters into the description of the environment's influence on a quantum system. This motivates a description of the open quantum system using a randomly evolving (reduced) density matrix.

In order to develop these ideas, consider the commonly used evolution of the density matrix $\rho(t)$ defined through the action of the super-operator S in a time interval $\mathrm dt$. Such a mapping of states of the quantum system must preserve the unit trace and positivity for the quantum state to be considered physically meaningful [58]. The action of the super-operator on ρ can be expressed as

$$\begin{align} S\left[\rho\left(t\right)\right] = & \rho\left(t+dt\right) = \sum_{j}M_{j}\left(\mathrm dt\right)\rho\left(t\right)M_{j}^{\dagger}\left(\mathrm dt\right). \end{align} \tag{ 1 }$$

The Kraus operators $M_{j}(\mathrm dt)$ represent all possible transitions brought about by the system dynamics together with system-environment interactions and can in principle be derived from terms in the Hamiltonian. In order to preserve the trace of ρ they must satisfy the completeness relation $\sum_{j}M_{j}^{\dagger}M_{j} = \mathbb{I}$. The Kraus operators depend on dt and, for continuous evolution of the density matrix, must differ by a small perturbation from the identity operator $\mathbb{I}$. For simplicity we use the set of Kraus operators $M_{l}\equiv M_{k\pm}(\mathrm dt) = \frac{1}{\sqrt{2}}(\mathbb{I}+A_{k\pm})$, where k labels a particular Lindblad channel k, with the ± subscript representing two possible measurement outcomes or transitions associated with the interaction. We employ $A_{k\pm} = -iH_{s}\mathrm dt-\frac{1}{2}L_{k}^{\dagger}L_{k}\mathrm dt\pm L_{k}\sqrt{\mathrm dt}$ where H_s is the system Hamiltonian and L_k the operator associated with the kth Lindblad channel [18, 59–61]. The use of two Kraus operators per Lindblad channel is reminiscent of the approach used by Wiseman, namely Kraus operators $\Omega_{k1} = \sqrt{\mathrm dt}L_{k}$ and $\Omega_{k0} = \mathbb{I}-(iH_{s}+\frac{1}{2}L_{k}^{\dagger}L_{k})\mathrm dt$ [56], such that $\Omega_{k1} = \frac{1}{\sqrt{2}}\left(M_{k+}-M_{k-}\right)$ and $\Omega_{k0} = \frac{1}{\sqrt{2}}\left(M_{k+}+M_{k-}\right)$. Note that the Kraus operators $\Omega_{k0,1}$ and $M_{k\pm}$ yield the same Lindblad master equation but differ in their stochastic realisations of the trajectories, where the former produces discontinuous, 'jump-like' trajectories and the latter diffusive, continuous trajectories.

Such a positive map then results in the ubiquitous Lindblad master equation describing the Markovian evolution of the reduced density matrix ρ averaged over the measurement outcomes:

$$\begin{align} \mathrm d\rho = -i\left[H_{s},\rho\right]\mathrm dt+\sum_{k}\left(L_{k}\rho L_{k}^{\dagger}-\frac{1}{2}\left\{L_{k}^{\dagger}L_{k},\rho\right\}\right)\mathrm dt. \end{align} \tag{ 2 }$$

The density operator ρ may be regarded as representing a statistical ensemble of the pure states that could be adopted by an open quantum system after a measurement has been performed. As such, the Lindblad master equation describes the average evolution of this ensemble under the action of the Kraus operators.

Another possible interpretation of equation (1), when written as

$$\begin{align} \rho\left(t+dt\right) = & \sum_{l}p_{l}\left(\mathrm dt\right)\frac{M_{l}\left(\mathrm dt\right)\rho\left(t\right)M_{l}^{\dagger}\left(\mathrm dt\right)}{Tr\left(M_{l}\left(\mathrm dt\right)\rho\left(t\right)M_{l}^{\dagger}\left(\mathrm dt\right)\right)}, \end{align} \tag{ 3 }$$

is that the dynamics may be represented as stochastic transitions from $\rho(t)$ to $M_{l}(\mathrm dt)\rho(t)M_{l}^{\dagger}(\mathrm dt)/Tr(M_{l}(\mathrm dt)\rho(t)M_{l}^{\dagger}(\mathrm dt))$ that occur with a probability $p_{l}(\mathrm dt) = Tr(M_{l}(\mathrm dt)\rho(t)M_{l}^{\dagger}(\mathrm dt))$ that depends on the density matrix of the system at time t. Equation (1) represents the average evolution of the density matrix as a result of the action of all the Kraus operators on $\rho(t)$. Stochastic quantum trajectories might then be generated through a series of transitions employing a sequence of Kraus operators. This is a strategy that has been used in studies of evolving quantum systems [56, 58].

Our strategy is to generate trajectories reflecting a single possible evolution path of ρ rather than an averaged one, and so we take the stochastic interpretation just mentioned. We are modelling measurement as an interaction between a system and its environment that affects the system in a manner not dissimilar to the effect of a host medium on a Brownian particle. Such an approach can make sense since the state of the environment is unknown, giving rise to an evolving uncertainty in the state of the quantum system. To reflect such an uncertainty it would be natural to consider an ensemble of density matrices at each point in time t, reflecting the multiple possible states of the system as a result of lack of knowledge of the state of the environment. Similarly to Matos et al [59], the average of such an ensemble of density operators may be denoted by the over-bar, $\bar{\rho}$. The evolution of the ensemble averaged density matrix over a time increment dt would then be expressed as [59]

$$\begin{align} \bar{\rho}\left(t+\mathrm dt\right) = \sum_{l}M_{l}\left(\mathrm dt\right)\bar{\rho}\left(t\right)M_{l}^{\dagger}\left(\mathrm dt\right). \end{align} \tag{ 4 }$$

In contrast, the action of a single Kraus operator on a member $\rho(t)$ of such an ensemble represents one of a set of possible system evolutions:

$$\begin{align} \rho\left(t+\mathrm dt\right) = \rho\left(t\right)+\mathrm d\rho = \frac{M_{l}\left(\mathrm dt\right)\rho\left(t\right)M_{l}^{\dagger}\left(\mathrm dt\right)}{Tr\left(M_{l}\left(\mathrm dt\right)\rho\left(t\right)M_{l}^{\dagger}\left(\mathrm dt\right)\right)}, \end{align} \tag{ 5 }$$

and a sequence of such transformations can generate a single, possible evolution path of ρ. Note that the preservation of the trace and positivity of the standard Kraus operator map of equation (1) is well-known [58]. Trace preservation is apparent also in equation (5). However, preservation of positivity in the action of a single Kraus operator of the form $M_{k\pm}(dt) = \frac{1}{\sqrt{2}}(\mathbb{I}+A_{k\pm})$ on one member of an ensemble of density matrices in equation (5) remains to be demonstrated and will be shown in the following.

Consider a density matrix $\rho(t)$ at a time t, specified to be completely positive and hence to have only non-negative eigenvalues. The determinant of such a density matrix is $\det(\rho(t)) = \prod_{i}\lambda_{i}$ where λ_i denote the eigenvalues of $\rho(t)$. The determinant of $\rho(t)$ is thus positive. Now we consider the determinant of the density matrix at time $t+\mathrm dt$, namely $\rho(t+\mathrm dt)$ generated according to equation (5) by a Kraus operator of the form stated above $M = C(\mathbb{I}+A)$, where A is infinitesimal (in the sense that A → 0 as $\mathrm dt\to0$) and C is a constant, such that $\tilde{M} = M/C = \mathbb{I}+A$ differs infinitesimally from the identity $\mathbb{I}$. We write

$$\begin{align} \rho\left(t+\mathrm dt\right) = \frac{\tilde{M}\rho\left(t\right)\tilde{M}^{\dagger}}{Tr\left(\tilde{M}\rho\left(t\right)\tilde{M}^{\dagger}\right)}, \end{align} \tag{ 6 }$$

and employ the identity $\det(\exp A) = \exp(TrA)$ such that $\det(\mathbb{I}+A)\approx1+Tr(A)$ for infinitesimal A, whereby

$$\begin{align} \det\left(\rho\left(t+\mathrm dt\right)\right) & = \frac{\det\left(\tilde{M}\right)\det\left(\rho\left(t\right)\right)\det\left(\tilde{M}^{\dagger}\right)}{Tr\left(\tilde{M}\rho\left(t\right)\tilde{M}^{\dagger}\right)}\nonumber \\ & \approx\frac{\left(1+Tr\left(A\right)\right)\det\left(\rho\left(t\right)\right)\left(1+Tr\left(A^{\dagger}\right)\right)}{Tr\left(\tilde{M}^{\dagger}\tilde{M}\rho\left(t\right)\right)}\nonumber \\ & \approx\frac{\left(1+Tr\left(A+A^{\dagger}\right)\right)\det\left(\rho\left(t\right)\right)}{Tr\left(\left(\mathbb{I}+A+A^{\dagger}\right)\rho\left(t\right)\right)}\nonumber \\ & = \frac{\left(1+Tr(A+A^{\dagger}\right))\det(\rho(t))}{1+Tr((A+A^{\dagger})\rho(t))}\nonumber \\ & \approx\det(\rho(t))\left[1+Tr(A+A^{\dagger}) -Tr((A+A^{\dagger})\rho(t))\right]. \end{align} \tag{ 7 }$$

Next we note that an unacceptable map would produce a density matrix with at least one negative eigenvalue. We can therefore identify unphysical dynamics if a density matrix is generated whose determinant passes through zero at some point in time. This includes cases where an odd number of eigenvalues change sign simultaneously (such that the determinant becomes negative) or where an even number do so (such that the determinant goes through a cusp at zero).

So let us examine the dynamics represented by equation (7). It can be demonstrated that the determinant of $\rho(t+\mathrm dt)$ can never go to zero within a finite time-frame, under certain conditions. We express the change in the determinant of the density matrix $d\det(\rho) = \det(\rho(t+\mathrm dt))-\det(\rho(t))$ as an Itô process: $\mathrm d\det(\rho) = \det(\rho)(a\mathrm dt+b{\mathrm{dW}})$, where ${\mathrm{dW}}$ is a Wiener increment and a and b are specified functions. Letting $y = \ln(\det(\rho)$), the following SDE for y may be obtained:

$$\begin{align} \mathrm dy = & \frac{1}{\det\left(\rho\right)}\mathrm d\det\left(\rho\right)+\frac{1}{2}b^{2}\det\left(\rho\right)^{2}\left(-\frac{1}{\det\left(\rho\right){}^{2}}\right)\mathrm dt = a\mathrm dt+b{\mathrm{dW}}-\frac{1}{2}b^{2}\mathrm dt. \end{align} \tag{ 8 }$$

The unacceptable behaviour $\det(\rho)\to0$ corresponds to $y\to-\infty$ and unless a or b possess singularities it is clear that this can only emerge, if at all, as $t\to\infty$. Thus the dynamics of the single Kraus operator map, equation (5), with $M = C(\mathbb{I}+A)$ and employing a non-singular, infinitesimal A, are physically acceptable.

Employing the set of Kraus operators from before, $M_{k\pm} = \frac{1}{\sqrt{2}}(\mathbb{I}+A_{k\pm})$ and $A_{k\pm} = -iH_{s}\mathrm dt-\frac{1}{2}L_{k}^{\dagger}L_{k}\mathrm dt\pm L_{k}\sqrt{\mathrm dt}$, [58–60] yields the following Lindblad equation:

$$\begin{align} \mathrm d\bar{\rho} = -i\left[H_{s},\bar{\rho}\right]\mathrm dt+\sum_{k}\left(L_{k}\bar{\rho}L_{k}^{\dagger}-\frac{1}{2}\left\{L_{k}^{\dagger}L_{k},\bar{\rho}\right\}\right)\mathrm dt, \end{align} \tag{ 9 }$$

which explicitly describes the evolution of an average over the ensemble of density matrices. We generate stochastic trajectories by unravelling equation (9) in the manner of equation (5) such that an equation concerning a single member of the ensemble of density matrices $\rho(t)$ can be formulated. We next derive such an equation for $\rho(t)$ in the form of an Itô process.

We unravel the deterministic Lindblad equation by implementing the stochastic evolution of ρ according to the set of available transitions

$$\begin{align} \rho\left(t+dt\right) = \frac{M_{k\pm}\left(\mathrm dt\right)\rho\left(t\right)M_{k\pm}^{\dagger}\left(\mathrm dt\right)}{Tr\left(M_{k\pm}\left(\mathrm dt\right)\rho\left(t\right)M_{k\pm}^{\dagger}\left(\mathrm dt\right)\right)}, \end{align} \tag{ 10 }$$

each adopted with probability $p_{k\pm}(\mathrm dt) = Tr(M_{k\pm}(\mathrm dt)\rho(t)M_{k\pm}^{\dagger}(\mathrm dt))$. The outcome is an Itô process for ρ involving independent Wiener increments ${\mathrm{dW_{k}}}$ (in the case of more than one Lindblad operator) [59]:

$$\begin{align} \mathrm d\rho & = -i\left[H_{s},\rho\right]\mathrm dt+\sum_{k}\left(\left(L_{k}\rho L_{k}^{\dagger}-\frac{1}{2}\left\{L_{k}^{\dagger}L_{k},\rho\right\}\right)\mathrm dt\right.\nonumber\\ & \quad \left. +\ \left(\rho L_{k}^{\dagger}+L_{k}\rho-Tr\left[\rho\left(L_{k}+L_{k}^{\dagger}\right)\right]\rho\right){\mathrm{dW}}_{k}\right). \end{align} \tag{ 11 }$$

Solutions to this equation describe possible stochastic quantum trajectories of the reduced system density matrix ρ, each associated with a particular realisation of the environmental noises $\{{\mathrm{dW}}_{k}\}$ [62], or equivalently, an initial environmental state. The ensemble-averaged density matrix $\bar{\rho}$ then corresponds to taking an average over the noise and hence over all possible trajectories and, as a result, equation (9) is recovered. If the system were closed instead of open, interactions between the system and the environment would vanish and we would be left with the first term on the right-hand side; namely the von Neumann equation. A similar stochastic evolution equation for ρ has been derived by Jacobs by analogy with classical measurement theory [18]. Under this approach, however, the noise is interpreted as a quantum fluctuation stemming from the uncertainty in the measurement: a similar approach is taken by Gambetta et al [27]. In contrast, we consider such a noise to reflect the probabilistic nature of interactions with an underspecified environment causing the quantum system to evolve in a stochastic manner.

We consider a spin system with a Hamiltonian given by

$$\begin{align} H_{s} = \epsilon S_{x}, \end{align} \tag{ 12 }$$

where ε is a positive constant and the operator S_x represents the x-component of the spin: for example in the case of a spin 1/2 system we have $S_{i} = \frac{1}{2}\sigma_{i}$ where σ_i denotes a Pauli spin matrix. The spin operators satisfy the condition $[S_{i},S_{j}] = i\epsilon_{ijk}S_{k}$ where ε_ijk is the Levi–Civita symbol. We shall consider cases of spin 1/2, spin 1 and spin 3/2 systems where the matrix representations of the S_x , S_y and S_z operators accordingly have dimensions 2, 3 and 4, respectively. The Hamiltonian in equation (12) produces Rabi oscillations, as we shall see.

The desired effect of the environment on the system is that it should act as a measuring apparatus. Diffusive evolution of the system's z-component of spin, to be defined shortly, towards one of the eigenstates of the S_z operator, can be achieved using a single Lindblad operator in equation (11) of the form

$$\begin{align} L = \alpha S_{z}, \end{align} \tag{ 13 }$$

where the constant α denotes the degree of coupling between the system and its environment through the operator S_z and can be regarded as the strength of measurement. Figure 1 illustrates the features of the system and its environment. Utilising the above expressions for the Lindblad operator and the system Hamiltonian in equation (11), the following SDE is obtained for the evolution of the reduced density matrix of the system:

$$\begin{align} \mathrm d\rho & = -i\epsilon\left[S_{x},\rho\right]\mathrm dt+\alpha^{2}\left(S_{z}\rho S_{z}-\frac{1}{2}\left(\rho S_{z}^{2}+S_{z}^{2}\rho\right)\right)\mathrm dt +\alpha\left(\rho S_{z}+S_{z}\rho-2\langle S_{z}\rangle\rho\right){\mathrm{dW}}. \end{align} \tag{ 14 }$$

The single Wiener increment ${\mathrm{dW}}$ is a Gaussian noise with a mean of zero and a variance $\mathrm dt$, and $\langle S_{z}\rangle$ is the z-component of spin, defined by $Tr(S_{z}\rho)$. Note that the latter is not to be confused with the usual quantum expectation value of S_z which in our notation is $Tr(S_{z}\bar{\rho})$, where $\bar{\rho}(t)$ is the density matrix at time t averaged over all possible quantum trajectories. The expectation value represents an average of a (projectively) measured system property taking into account the quantum mechanical randomness of measurement outcome as well as the range of quantum states that might be adopted by an open system when coupled to an uncertain environment. By extension, $\langle.\rangle = Tr(.\rho)$ could be interpreted as a conditional average projective measurement outcome given a specific stochastically evolving density matrix. But it could instead be regarded simply as a property of the density matrix, and hence of the physical state, and indeed it is spin components such as $\langle S_{z}\rangle$ that undergo Rabi oscillations.

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Terms in equation (14) that depend on α represent the effects of measurement on the system, brought about by its interaction with the environment. Notice that for ε = 0, the stationary states for the measurement dynamics are $\rho = \vert m_{z}\rangle\langle m_{z}\vert$, where the $\vert m_{z}\rangle$ are eigenstates of S_z satisfying $S_{z}\vert m_{z}\rangle = m_{z}\vert m_{z}\rangle$. Starting from an arbitrary initial state, the coupling to the environment captured by the Lindblad operator in equation (13) evolves the system towards z-spin eigenstates as desired.

The density matrix of the open quantum system is to be parametrised by a set of variables $\{x_{i}\}$ which evolve stochastically according to the general Itô processes [63]:

$$\begin{align} \mathrm dx_{i} = A_{i}\mathrm dt+\sum_{k}B_{ik}{\mathrm{dW}}_{k}, \end{align} \tag{ 15 }$$

where ${\mathrm{dW}}_{k}$ denote the Wiener increments and the A_i and B_ik are designated functions. The number of parameters required to represent the density matrix depends on the system. Three are required for a density matrix describing a spin 1/2 system. For a spin 1 system, eight parameters are needed since this is the number of independent variables required to specify a $3\times3$ Hermitian matrix with unit trace. Through a similar reasoning, fifteen variables are needed to parametrise the density matrix for a spin 3/2 system.

3.2.1. Spin 1/2.

The density matrix of a spin 1/2 system can be expressed using the Bloch sphere formalism, namely $\rho = \frac{1}{2}\left(\mathbb{I}+\boldsymbol{r}\cdot\boldsymbol{\sigma}\right)$ with coherence (or Bloch) vector $\boldsymbol{r} = (x,y,z) = Tr(\rho\boldsymbol{\sigma}) = \langle\boldsymbol{\sigma}\rangle$ [64]. SDEs in the form of equation (15) can be derived starting from

$$\begin{align} \mathrm dr_{i} = Tr\left(\sigma_{i}\mathrm d\rho\right), \end{align} \tag{ 16 }$$

with the insertion of equation (14). For clarity of presentation, however, we shall employ a one parameter representation for the special case of a pure state with $Tr\rho^{2} = 1$ or $\vert\boldsymbol{r}\vert = 1$, with coherence vector confined to the (y, z) plane, i.e.

$$\begin{align} \rho = \frac{1}{2}\left(\mathbb{I}+\cos\phi\,\sigma_{z}-\sin\phi\,\sigma_{y}\right), \end{align} \tag{ 17 }$$

namely x = 0, $y = -\sin\phi$ and $z = \cos\phi$. The SDE for z arising from $\mathrm dz = Tr(\sigma_{z}\mathrm d\rho)$ and equation (14) is

$$\begin{align} \mathrm dz = -\epsilon\sin\phi \mathrm dt+\alpha\sin^{2}\phi\,{\mathrm{dW}}. \end{align} \tag{ 18 }$$

The Rabi angle φ represents the angle of rotation of the coherence vector about the x axis. In the absence of environmental coupling, the action of the system Hamiltonian $\frac{1}{2}\epsilon\sigma_{x}$ is to increase φ linearly in time. In the presence of such coupling the evolution of the Rabi angle represents the effect of measurement on the unitary dynamics of the system. The SDE for $\phi = \cos^{-1}z$ can be found through use of Itô's lemma [63]:

$$\begin{align} \mathrm d\phi = \frac{\mathrm d\phi}{\mathrm dz}\mathrm dz+\frac{1}{2}\beta^{2}\frac{\mathrm d^{2}\phi}{\mathrm dz^{2}}\mathrm dt, \end{align} \tag{ 19 }$$

where $\beta = \alpha\sin^{2}\phi$. Inserting equation (18) produces

$$\begin{align} \mathrm d\phi = \left(\epsilon-\frac{1}{4}\alpha^{2}\sin2\phi\right)\mathrm dt-\alpha\sin\phi\,{\mathrm{dW}}. \end{align} \tag{ 20 }$$

When α = 0 the Rabi angle increases at a constant rate, while for $\alpha\ne0$ the Wiener noise ${\mathrm{dW}}$ disturbs its evolution. The average rate of change of φ is given by

$$\begin{align} \frac{\mathrm d\langle\phi\rangle}{\mathrm dt} = \epsilon-\frac{1}{4}\alpha^{2}\langle\sin2\phi\rangle, \end{align} \tag{ 21 }$$

where the brackets again represent an ensemble average. The QZE will emerge if the term proportional to α² represents an average retardation of the rotation. In order to resolve this matter we consider the Fokker–Planck equation for $p(\phi,t)$, the probability distribution function (PDF) of the Rabi angle φ:

$$\begin{align} \frac{\partial p}{\partial t} = -\frac{\partial\left(\left(\epsilon-\frac{1}{4}\alpha^{2}\sin2\phi\right)p\right)}{\partial\phi}+\frac{1}{2}\alpha^{2}\frac{\partial^{2}\left(\sin^{2}\phi\,p\right)}{\partial\phi^{2}}. \end{align} \tag{ 22 }$$

For small α, an approximate stationary PDF may be obtained:

$$\begin{align} p_{{st}}\left(\phi\right)\propto1+\frac{3\alpha^{2}}{4\epsilon}\sin2\phi, \end{align} \tag{ 23 }$$

which shows that the PDF is disturbed from uniformity, in this regime, by increasing the environmental coupling α, or by reducing ε and hence slowing the rate of Rabi oscillation for an isolated system. We shall investigate the shape of the stationary PDF numerically for a range of α in section 4. For now, let us notice that using this approximation the average needed in equation (21) is proportional to

$$\begin{align} \int_{0}^{2\pi}\mathrm d\phi\left(1+\frac{3\alpha^{2}}{4\epsilon}\sin2\phi\right)\sin2\phi\propto\int_{0}^{2\pi}\mathrm d\phi\sin^{2}2\phi\gt 0, \end{align} \tag{ 24 }$$

showing that the effect of measurement at small α is indeed a mean retardation of the Rabi oscillations, consistent with the quantum Zeno effect.

3.2.2. Spin 1.

The generalised Bloch sphere formalism is used to represent the density matrix of a spin 1 system. This permits any $3\times3$ density matrix to be written in terms of the Gell–Mann matrices λ_i through

$$\begin{align} \rho = \frac{1}{3}\left(\mathbb{I}+\sqrt{3}\boldsymbol{R}\cdot\boldsymbol{\lambda}\right), \end{align} \tag{ 25 }$$

where $\boldsymbol{R} = (s,m,u,v,k,x,y,z)$ is an eight dimensional coherence (or Bloch) vector and the Gell–Mann matrices (see appendix A) form the elements of the vector $\boldsymbol{\lambda} = (\lambda_{1},\lambda_{2},\lambda_{3},\lambda_{4},\lambda_{5},\lambda_{6},\lambda_{7},\lambda_{8})$ [65]. The following density matrix emerges for the spin 1 system:

$$\begin{align} \rho = \frac{1}{3}\begin{pmatrix}1+\sqrt{3}u+z & -i\sqrt{3}m+\sqrt{3}s & \sqrt{3}v-i\sqrt{3}k\\ i\sqrt{3}m+\sqrt{3}s & 1-\sqrt{3}u+z & \sqrt{3}x-i\sqrt{3}y\\ \sqrt{3}v+i\sqrt{3}k & \sqrt{3}x+i\sqrt{3}y & 1-2z \end{pmatrix}. \end{align} \tag{ 26 }$$

Stochastic quantum trajectories can be produced from the equations of motion for the eight variables parametrising ρ. As before, these are generated from

$$\begin{align} \mathrm dR_{i} = \frac{\sqrt{3}}{2}Tr\left(\mathrm d\rho\lambda_{i}\right), \end{align} \tag{ 27 }$$

using the properties of the Gell–Mann matrices, and are specified in appendix B. The components of the coherence vector R are denoted R_i .

3.2.3. Spin 3/2.

In order to construct the density matrix of the spin 3/2 system, an obvious representation would be to utilise SU(4) generators in the generalised Bloch sphere representation:

$$\begin{align} \rho = \frac{1}{4}\left(\mathbb{I}+\boldsymbol{s}\cdot\boldsymbol{\Sigma}\right). \end{align} \tag{ 28 }$$

The 15 component coherence vector is $\boldsymbol{s} = (v,e,f,g,h,j,k,l,m,n,o,p,q,s,u)$ and the vector of generators Σ has components $\Sigma_{k}$ with $k = 1,15$. However, Aerts and Sassoli de Bianchi discuss how such a representation beyond the SU(2) generators could potentially allow the system to explore unphysical states [64]. Instead, we use equation (28) with generators $\Sigma_{k}$ constructed using the tensor product of two Pauli matrices: $\Sigma_{k} = \sigma_{i}\otimes\sigma_{j}$: full details are given in appendix C. The following density matrix is then obtained for the spin 3/2 system:

$$\begin{align} \rho = \frac{1}{4}\begin{pmatrix}1+f+p+u & -ie+q-is+v & g+k-il-io & h-ij-im-n\\ ie+q+is+v & 1-f+p-u & h+ij-im+n & g-k-il+io\\ g+k+il+io & h-ij+im+n & 1+f-p-u & -ie-q+is+v\\ h+ij+im-n & g-k+il-io & ie-q-is+v & 1-f-p+u \end{pmatrix}. \end{align} \tag{ 29 }$$

As before, stochastic quantum trajectories can be produced from the equations of motion of the 15 variables parametrising the density matrix. SDEs for components of the coherence vector s_k are obtained through

$$\begin{align} \mathrm ds_{k} = Tr\left(\mathrm d\rho\Sigma_{k}\right), \end{align} \tag{ 30 }$$

and the details can be found in appendix D.

The stochastic trajectories of the variables parametrising the density matrices introduced for the spin quantum systems enable us to describe the system dynamics exactly, but do not offer much direct physical insight into the QZE. A more useful way of visualising the behaviour of such systems is to consider the evolution of the time dependent quantities $\langle S_{x}\rangle$, $\langle S_{y}\rangle$ and $\langle S_{z}\rangle$. Recall that these could be interpreted as conditional average values under projective measurements, hence a set of statistics of the dynamics, though we prefer to regard them as actual physical properties of the current quantum state. It is these quantities that perform noisy Rabi oscillations and we shall refer to them simply as spin components. Two methods exist to determine their evolution.

The first method simply employs $\langle S_{i}\rangle = Tr(S_{i}\rho)$ such that the spin components can be written in terms of the variables parametrising the density matrices in equations (17), (26) and (29). Alternatively, by considering $\mathrm d\langle X\rangle = Tr(X\mathrm d\rho)$, where operators X are various functions of the S_i , the SDEs that govern the evolution of the spin components can be found using equation (14) and solved. We consider these approaches for the three spin systems in turn.

3.3.1. Spin 1/2.

For the special case density matrix in equation (17) the spin components $\langle S_{y}\rangle$ and $\langle S_{z}\rangle$ can be written in terms of the Rabi angle $\phi(t)$ as follows:

$$\begin{align} \langle S_{y}\rangle & = -\frac{1}{2}\sin\phi,\qquad\langle S_{z}\rangle = \frac{1}{2}\cos\phi, \end{align} \tag{ 31 }$$

so the phase space explored by $\langle S_{y}\rangle$ and $\langle S_{z}\rangle$ is a circle of radius $1/2$. We expect a typical stochastic trajectory to dwell increasingly in the vicinity of the z-spin eigenstates at φ = 0 and π as the measurement strength increases, and will demonstrate this in section 4. We also expect the mean rate of passage between the two eigenstates to reduce as the measurement strength increases.

The following three SDEs describe the spin components:

$$\begin{align} & \mathrm d\langle S_{z}\rangle = \epsilon\langle S_{y}\rangle \mathrm dt+2\alpha\left(\frac{1}{4}-\langle S_{z}\rangle^{2}\right){\mathrm{dW}}\nonumber \\ & \mathrm d\langle S_{x}\rangle = -\frac{\alpha^{2}}{2}\langle S_{x}\rangle \mathrm dt-2\alpha\langle S_{z}\rangle\langle S_{x}\rangle {\mathrm{dW}}\nonumber \\ & \mathrm d\langle S_{y}\rangle = -\epsilon\langle S_{z}\rangle \mathrm dt-\frac{\alpha^{2}}{2}\langle S_{y}\rangle \mathrm dt-2\alpha\langle S_{z}\rangle\langle S_{y}\rangle {\mathrm{dW}}. \end{align} \tag{ 32 }$$

For α = 0 we clearly see the emergence of Rabi oscillations in $\langle S_{y}\rangle$ and $\langle S_{z}\rangle$. Similarly, for ε = 0 we identify stationary states at $\langle S_{x}\rangle = \langle S_{y}\rangle = 0$ and $\langle S_{z}\rangle = \pm1/2$, which correspond to the measurement eigenstates.

It may be shown that the purity of the system defined by $P = Tr(\rho^{2})$ satisfies

$$\begin{align} \mathrm dP = \alpha^{2}\left(1-r_{z}^{2}\right)\left(1-P\right)\mathrm dt+2\alpha r_{z}\left(1-P\right){\mathrm{dW}}, \end{align} \tag{ 33 }$$

such that purity moves towards P = 1 and stays there under the given dynamics. Furthermore, $\langle S_{x}\rangle = 0$ is a fixed point of the dynamics of equation (32) so the special case considered in equation (17) describes a more general asymptotic behaviour.

3.3.2. Spin 1.

Similarly, the spin components $\langle S_{x}\rangle$, $\langle S_{y}\rangle$ and $\langle S_{z}\rangle$ can be written in terms of the variables parametrising the density matrix in equation (26):

$$\begin{align} \langle S_{x}\rangle & = \sqrt{\frac{2}{3}}\left(x+s\right)\nonumber \\ \langle S_{y}\rangle & = \sqrt{\frac{2}{3}}\left(m+y\right)\nonumber \\ \langle S_{z}\rangle & = \frac{u}{\sqrt{3}}+z. \end{align} \tag{ 34 }$$

The purity P of the system can be written

$$\begin{align} P = Tr\left(\rho^{2}\right) = \frac{2}{3}\left(s^{2}+m^{2}+u^{2}+v^{2}+k^{2}+x^{2}+y^{2}+z^{2}\right)+\frac{1}{3}, \end{align} \tag{ 35 }$$

and this evolves asymptotically to unity. The dynamics in terms of spin components and related quantities take the form of eight coupled SDEs:

$$\begin{align} \mathrm d\langle S_{z}\rangle & = \epsilon\langle S_{y}\rangle \mathrm dt+2\alpha\left(\langle S{}_{z}^{2}\rangle-\langle S_{z}\rangle^{2}\right){\mathrm{dW}}\nonumber \\ \mathrm d\langle S_{x}\rangle & = -\frac{\alpha^{2}}{2}\langle S_{x}\rangle \mathrm dt+\alpha\left(\langle S_{z}S_{x}\rangle+\langle S_{x}S_{z}\rangle-2\langle S_{z}\rangle\langle S_{x}\rangle\right){\mathrm{dW}}\nonumber \\ \mathrm d\langle S_{y}\rangle & = -\epsilon\langle S_{z}\rangle \mathrm dt-\frac{\alpha^{2}}{2}\langle S_{y}\rangle \mathrm dt+\alpha\left(\langle S_{z}S_{y}\rangle+\langle S_{y}S_{z}\rangle-2\langle S_{z}\rangle\langle S_{y}\rangle\right){\mathrm{dW}}\nonumber \\ \mathrm d\langle S{}_{y}^{2}\rangle & = -\epsilon\left(\langle S_{y}S_{z}\rangle+\langle S_{z}S_{y}\rangle\right)\mathrm dt-\alpha^{2}\left(i\langle S_{x}S_{y}S_{z}\rangle+\langle S_{z}^{2}\rangle+\langle S_{y}^{2}\rangle-1\right)dt\nonumber\\ & \quad +\alpha\langle S_{z}\rangle\left(1-2\langle S_{y}^{2}\rangle\right){\mathrm{dW}}\nonumber \\ \mathrm d\langle S_{z}^{2}\rangle & = \epsilon\left(\langle S_{y}S_{z}\rangle+\langle S_{z}S_{y}\rangle\right)\mathrm dt+2\alpha\langle S_{z}\rangle\left(1-\langle S_{z}^{2}\rangle\right)\mathrm {\mathrm{dW}}\nonumber \\ \mathrm d\left(\langle S_{y}S_{z}\rangle+\langle S_{z}S_{y}\rangle\right) & = 2\epsilon\left(\langle S_{y}^{2}\rangle-\langle S_{z}^{2}\rangle\right)\mathrm dt-\frac{\alpha^{2}}{2}\left(\langle S_{y}S_{z}\rangle+\langle S_{z}S_{y}\rangle\right)\mathrm dt\nonumber\\ & \quad +\alpha\left(\langle S_{y}\rangle-2\langle S_{z}\rangle\left(\langle S_{y}S_{z}\rangle+\langle S_{z}S_{y}\rangle\right)\right){\mathrm{dW}}\nonumber \\ \mathrm d\left(\langle S_{z}S_{x}\rangle+\langle S_{x}S_{z}\rangle\right) & = -\frac{\alpha^{2}}{2}\left(\langle S_{x}S_{z}\rangle+\langle S_{z}S_{x}\rangle\right)\mathrm dt+\alpha\left(\langle S_{x}\rangle-2\langle S_{z}\rangle\left(\langle S_{x}S_{z}\rangle+\langle S_{z}S_{x}\rangle\right)\right){\mathrm{dW}}\nonumber \\ \mathrm d\langle S_{x}S_{y}S_{z}\rangle & = i\epsilon\left(\langle S_{y}S_{z}\rangle+\langle S_{z}S_{y}\rangle\right)\mathrm dt+i\alpha^{2}\left(\langle S_{z}^{2}\rangle+2i\langle S_{x}S_{y}S_{z}\rangle\right)\mathrm dt\nonumber\\ & \quad +i\alpha\langle S_{z}\rangle\left(1+2i\langle S_{x}S_{y}S_{z}\rangle\right){\mathrm{dW}}. \end{align} \tag{ 36 }$$

Again, the regular Rabi dynamics around a circle in the phase space spanned by $\langle S_{y}\rangle$ and $\langle S_{z}\rangle$ is apparent when α = 0. When $\alpha\ne0$ the trajectories are more complicated. For ε = 0 the eigenstates of the z-spin are stationary: the relevant spin components being $\langle S_{x}\rangle = \langle S_{y}\rangle = 0$ and $\langle S_{z}\rangle = \pm1,0$.

3.3.3. Spin 3/2.

Once again, the spin components can be written in terms of the variables parametrising the density matrix for the spin 3/2 system in equation (29) as follows:

$$\begin{align} \langle S_{x}\rangle & = \frac{1}{2}\left(h+n+\sqrt{3}v\right)\nonumber \\ \langle S_{y}\rangle & = \frac{1}{2}\left(\sqrt{3}e-j+m\right)\nonumber \\ \langle S_{z}\rangle & = \frac{1}{2}f+p. \end{align} \tag{ 37 }$$

The purity of the system P may be written

$$\begin{align} P & = \frac{1}{4}\left(1+e^{2}+f^{2}+g^{2}+h^{2}+j^{2}+k^{2}+l^{2} +m^{2}+n^{2}+o^{2}+p^{2}+q^{2}+s^{2}+u^{2}+v^{2}\right). \end{align} \tag{ 38 }$$

For the spin 3/2 system a closed set of fifteen stochastic differential equations for the spin components and related quantities could not be found. Thus equation (37) and the SDEs in appendix D are the only way to solve the dynamics for this spin system.

The equations of motion for the variables parametrising the density matrices of the spin 1/2, spin 1 and spin 3/2 systems, given by equations (20), (27) and (30), were solved numerically using the Euler-Maruyama update method [66]. From the evolution of these variables, stochastic quantum trajectories in the phase space of the z-spin component $\langle S_{z}\rangle$ and the y-spin component $\langle S_{y}\rangle$ were produced. For spin 1/2 we also generated stochastic trajectories of the Rabi angle φ. We chose the x-spin component to be zero initially, where it remains.

The evolution of the Rabi angle with time for the spin 1/2 system for various strengths of measurement is shown in figure 2. Figure 2(a) shows that the mean rate of change of the Rabi angle, namely the mean frequency of Rabi oscillations, decreases with increasing measurement coupling constant, as was suggested by equation (21). Such a decrease demonstrates the slowing down of the unitary dynamics as a result of stronger, or equivalently, more frequent measurement as is expected for the QZE. Figure 2(b) illustrates the dynamics on a finer scale. Behaviour at high measurement strength resembles the quantum jumps conventionally considered to occur between eigenstates of a monitored observable, brought about by an interaction present in the Hamiltonian.

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Numerically generated stationary PDFs for the Rabi angle are shown in figure 3 for an isolated system (α = 0) and for three nonzero measurement strengths. Initial states were selected from a uniform probability density over φ. Angles φ = 0 and π correspond to the $|\frac{1}{2}\rangle$ and $|\!-\!\frac{1}{2}\rangle$ eigenstates of S_z , respectively. Values of φ from the trajectories are mapped into the range 0 to 2π. It can be seen that as the measurement strength is increased, the stationary PDFs become narrower since the measurement dynamics are better able to localise the system in the vicinity of the eigenstates of S_z , resisting the pull of the unitary dynamics induced by the Hamiltonian, which seek to drive the system into Rabi oscillations. In accordance with the Born rule, for α > 0 the stationary PDFs contain two approximately equivalent peaks around the $|\!-\!\frac{1}{2}\rangle$ and $|\frac{1}{2}\rangle$ eigenstates, since the system should have an equal probability of localising at either. The PDF in figure 3(a) is uniform over φ since the trajectories represent the dynamics without measurement, namely Rabi oscillations. Note that the stationary PDF in figure 3(b) is roughly sinusoidal, as suggested by the PDF for small α given in equation (23). Indeed, that approximate result suggested peaks at $\phi = \pi/4$ and $3\pi/4$: a consequence of the competition between measurement-induced dwell near the eigenstates and the rotational pull of the Hamiltonian. As measurement strength increases, the peaks are drawn closer to φ = 0 and π.

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The evolution of $\langle S_{z}\rangle$ and $\langle S_{y}\rangle$ for a spin 1 system, for different values of the measurement strength α, is illustrated in figure 4. The system was initialised in the $|\!-\!1\rangle$ spin eigenstate of the S_z operator. Figure 4(a) illustrates the dynamics when there is no measurement: the Hamiltonian drives Rabi oscillations in $\langle S_{z}\rangle$ and $\langle S_{y}\rangle$ corresponding to the precession of the spin vector around the orthogonal $\langle S_{x}\rangle$ axis. The $\vert\!\pm\!1\rangle$ and $\vert0\rangle$ eigenstates of S_z lie at $\langle S_{x}\rangle = 0$, $\langle S_{y}\rangle = 0$ and $\langle S_{z}\rangle = \pm1$ and 0, respectively. If the initial state of the system had been the $|0\rangle$ eigenstate of S_z , then the spin vector would lie on the $\langle S_{x}\rangle$ axis and would be unable to precess around it, and furthermore, $\langle S_{z}\rangle$ and $\langle S_{y}\rangle$ would be zero.

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Disturbance of the circular trajectories can be seen in figure 4(b). As a result of the non-zero measurement strength the system can now pass near the $|0\rangle$ eigenstate of S_z , located at the origin. As the α is increased further in figures 4(c), (d), a figure-of-eight pathway forms and the system dwells more often at the $|0\rangle$ eigenstate until, in figures 4(e), (f), visits to this state always occur in between visits to the $\vert\!-\!1\rangle$ and $\vert1\rangle$ eigenstates.

Note that the spin 1 system is not moving at a constant speed around these figure-of-eight pathways. Instead, the system makes rapid jumps between the eigenstates, typically in a counter-clockwise direction, separating periods of dwell in the vicinity of the eigenstates. Movies of examples of the spin 1 [67] and spin 3/2 [68] system dynamics are available. The behaviour is illustrated in plots of the relative probability of occupation of patches of the phase space by the system, shown in figure 5. The behaviour is similar to that of the spin 1/2 system: a greater tendency to localise near the eigenstates as α is increased, while the pull of the Hamiltonian produces a counterclockwise displacement.

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Figure 6 further illustrates how increasing the measurement coupling constant causes the system to dwell for longer at the three eigenstates of S_z and transit more abruptly between them. The behaviour at high measurement strength increasingly resembles the textbook behaviour of a system making jumps between eigenstates.

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In order to quantify the QZE, we consider the α dependence of two quantities derived from the trajectories in figure 4. First, the residence probabilities, which characterise the fraction of time spent in the vicinity of each eigenstate of S_z over a long simulation. The system is defined to occupy an eigenstate if the $\langle S_{z}\rangle$ coordinate lies within ± 0.1 of the appropriate eigenvalue. The second quantity considered is the mean return time. This is the average period from the moment the system leaves a particular eigenstate (with occupation defined as above) to the moment it returns to it having visited another eigenstate in between.

Figure 7(a) illustrates how residence probabilities are small for low values of α, which is a reflection of the only slightly disturbed oscillatory behaviour of $\langle S_{z}\rangle$ in figure 6(a). But as the coupling strength is raised, so do the residence probabilities, and this can similarly be understood by considering the trajectory shown in figure 6(b). Little time is spent in regions far away from the eigenstates. Visits to the $\langle S_{z}\rangle = 0$ eigenstate are less frequent than to the ±1 eigenstates for small α, which is a consequence of having started the trajectory at $\langle S_{z}\rangle = -1$, but they occur with approximately equal probability at high values of α.

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Figure 7(b) shows how the system takes longer times to return to an eigenstate for stronger measurement. Again, this is consistent with the behaviour shown in figure 6(b). The mean return time for a given eigenstate is dominated by the period of dwell at the eigenstate (or eigenstates) to which it moves. The mean return time to the $\langle S_{z}\rangle = 0$ state initially decreases with α but this is an artefact of the initial condition for the motion, which makes visits to this eigenstate rare when α is small (as is apparent in figure 4). The mean return time for the $\langle S_{z}\rangle = 0$ eigenstate is typically less than the return times for $\langle S_{z}\rangle = \pm1$ (roughly half), possibly because direct transitions between the $\langle S_{z}\rangle = 1$ and −1 states are unlikely. Return to the central eigenstate is characterised by just one period of dwell at one of the two outer eigenstates, while return to an outer eigenstate might require waiting while the system hops between the other two states. The mean return times in figure 6(b) can be compared to the average time between two jumps τ found from the analytical expression of the jump rates in Bauer et al [34]. Specifically, for $\alpha = 7,$ the average time between two jumps for the $|\pm1\rangle$ eigenstates is $\tau_{|\pm1\rangle} = 49$ and for the $|0\rangle$ eigenstate is $\tau_{|0\rangle} = 24.5$, showing good agreement with our results.

The definition of the vicinity of an eigenstate is, of course, open to debate. The choice we make is simple but is sufficient to reveal the effects that characterise the QZE. It is natural that the value of $\langle S_{z}\rangle$ should feature prominently in the definition, but there are other characteristics of the eigenstates that could be taken into account. An eigenstate of the S_z operator corresponds to a point in a multidimensional parameter space, and its vicinity could be defined by putting conditions on a variety of parameters. We have investigated a more elaborate scheme along these lines but the resulting residence probabilities and mean return times are broadly similar to those shown in figure 7. In the interests of simplicity we therefore focus on the value of $\langle S_{z}\rangle$ alone, and the chosen range of ±0.1 about eigenvalues.

The effects of measurement on a spin 3/2 system, viewed in terms of the evolution of spin components $\langle S_{z}\rangle$ and $\langle S_{y}\rangle$, is illustrated in figure 8 for different values of the measurement strength α. The system was initialised in the $|\!-\!\frac{3}{2}\rangle$ eigenstate of S_z . When α = 0 there is no measurement and, as with the spin 1 system, the Hamiltonian drives circular trajectories, as depicted in figure 8(a), passing through the $|\frac{3}{2}\rangle$ and $|\!-\!\frac{3}{2}\rangle$ eigenstates of S_z at $\langle S_{y}\rangle = 0$ and $\langle S_{z}\rangle = \pm3/2$. The spin vector processes around the $\langle S_{x}\rangle$ axis. As α is increased (figures 8(b)–(f)), the circular trajectory is disturbed such that the system passes through further regions of phase space, including the vicinities of the $|\frac{1}{2}\rangle$ and $|\!-\!\frac{1}{2}\rangle$ eigenstates of S_z at $\langle S_{y}\rangle = 0$ and $\langle S_{z}\rangle = \pm1/2$. When the measurement strength is sufficiently high, a triple figure-of-eight trajectory emerges, such that the $|\frac{1}{2}\rangle$ and $|\!-\!\frac{1}{2}\rangle$ eigenstates are encountered between visits to the $|\frac{3}{2}\rangle$ eigenstate and the $|\!-\!\frac{3}{2}\rangle$ eigenstate, and visa-versa. The system dwells in the vicinities of all four eigenstates of S_z for high enough α.

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Similarly to the spin 1 case, figure 11 reveals how increasing α leads to longer dwell at the eigenstates of S_z and more 'jump-like' behaviour in between.

Figure 9(a) illustrates how the residence probabilities for the spin 3/2 system depend on α. As with the spin 1 case, the system is defined to occupy an eigenstate if the $\langle S_{z}\rangle$ coordinate lies within ±0.1 of the appropriate eigenvalue. At low values of α, the Hamiltonian term dominates and therefore occupation of the $|\frac{1}{2}\rangle$ and $|\!-\!\frac{1}{2}\rangle$ eigenstates of S_z is lower than for the $|\frac{3}{2}\rangle$ and $|\!-\!\frac{3}{2}\rangle$ eigenstates. For higher values of α, the residence probabilities for all four eigenstates increase and become roughly equal, in concord with the Born rule, and a triple figure-of-eight trajectory is followed. For large α, the probability of occupying each eigenstate lies around 0.25, implying that the system is unlikely to occupy a point in the phase space outside the vicinity of the eigenstates. Note that increasing the value of α amplifies the noise term in equation (14), introducing greater statistical uncertainty into the simulation.

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Figure 9(b) shows that as the measurement coupling constant is raised the mean return times increase, hence demonstrating the QZE. Notably, the mean return times for the $|\frac{1}{2}\rangle$ and $|\!-\!\frac{1}{2}\rangle$ eigenstates of S_z are approximately half those of the $|\frac{3}{2}\rangle$ and $|\!-\!\frac{3}{2}\rangle$ eigenstates. When the system resides at the $|\frac{1}{2}\rangle$ and $|\!-\!\frac{1}{2}\rangle$ eigenstates, and α is high enough, there are two states to which it can transfer, as opposed to one when it resides at the $|\frac{3}{2}\rangle$ and $|\!-\!\frac{3}{2}\rangle$ eigenstates. The $|\!\pm\!\frac{1}{2}\rangle$ lie within the ladder of eigenstates while the $|\!\pm\!\frac{3}{2}\rangle$ are its termini. Twice the number of paths for a return to $|\frac{1}{2}\rangle$ compared with $|\frac{3}{2}\rangle$ suggests half the mean return time. The analytical average time between two jumps τ, found from the analytical jump rates in Bauer et al, also reflect this trend. Namely, for $\alpha = 7,$ the average time between two jumps for the $|\pm\frac{3}{2}\rangle$ eigenstates is $\tau_{|\pm\frac{3}{2}\rangle}\approx32.7$ and for the $|\pm\frac{1}{2}\rangle$ eigenstates is $\tau_{|\pm\frac{1}{2}\rangle} = 14$ [34].

We speculate that higher spin systems will continue this pattern of behaviour. Systems with large measurement strength will follow stochastic transitions along a multiple figure-of-eight pathway in the phase space of $\langle S_{z}\rangle$ and $\langle S_{y}\rangle$. The behaviour will evolve from regular Rabi oscillations towards a situation where the system (effectively) jumps stochastically between eigenstates of the monitored observable.

Finally, the evolution of the purity of the spin 3/2 system under measurement was studied to ensure it remained within the expected range of $\frac{1}{4}\unicode{x2A7D} P\unicode{x2A7D}1$, starting with the system in the fully mixed state $\rho = \frac{1}{4}\left(|\!-\!\frac{3}{2}\rangle\langle\!-\frac{3}{2}|+|\!-\!\frac{1}{2}\rangle\langle\!-\frac{1}{2}|+|\frac{1}{2}\rangle\langle\frac{1}{2}|+|\frac{3}{2}\rangle\langle\frac{3}{2}|\right)$. In figure 10 it can be seen that the effect of measurement is to cause the system to purify. The purification occurs in a continuous but stochastic manner, taking approximately 0.4 time units for the parameters chosen. The approximation of instantaneous projective measurements and apparent jumps in purity emerges only in the limit $\alpha\rightarrow\infty$.

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It is worth noting that the dynamical framework we use has the effect that interaction brings about a disentanglement of the system from its environment, namely an increase in purity, which is the opposite of what is often supposed. However, this is an appropriate outcome for measurement, where the idea is to convey the system into a (pure) eigenstate of the appropriate observable. More general system-environment interactions might change the system purity in different ways, but to take this into account would require a more explicit representation of environmental degrees of freedom than that which we have employed.