Influence of a stray magnetic field on the measurement of long-range spin-spin interaction

In the measurement [13] it was assumed that in the system there is only an applied magnetic field $\vec{B}={\vec{B}}_{\mathrm{tot}}$, which is directed along the line connecting the two ions, cf figure 1(a). Here, we consider the presence of an additional (stray) magnetic field ${\vec{B}}_{\perp }$ in the system. Without loss of generality one can assume that it acts along the perpendicular direction to the line connecting the two ions, since the other component of the stray magnetic field can be added to the applied field. Hence, the total magnetic field ${\vec{B}}_{\mathrm{tot}}$ can be expressed as follows

$$\begin{eqnarray}&&{\vec{B}}_{\mathrm{tot}}={B}_{\perp }\hat{n}+B\hat{p},\end{eqnarray} \tag{ 1 }$$

where $\hat{n}$ and $\hat{p}$ represent the two unit vectors, one is perpendicular and another is parallel to the line connecting the two ions, cf figure 1(b). We, particularly, consider these two unit vectors laying in the xz plane are perpendicular to each other. In order to illustrate the possible role of the stray magnetic field in the measurement of the spin-spin interaction, we have drawn the schematics of the two-electron system in figure 1, where the initially prepared state $| \uparrow \downarrow \rangle$ is sketched for both of the cases. Despite the spin-spin interaction does not depend on an applied magnetic field, the measurement outcome depend on spin basis functions employed for its detection. While the spin basis functions are the spin states with defined projection of the spin momentum on a direction of total magnetic field. Thus, if the perpendicular field component ${\vec{B}}_{\perp }$ is acting in the system, the total magnetic field ${\vec{B}}_{\mathrm{tot}}$ is the vector sum of ${\vec{B}}_{\perp }$ and $\vec{B}$ with ${B}_{\mathrm{tot}}=\sqrt{{B}^{2}+{B}_{\perp }^{2}}$. Therefore, an angle θ is created between ${\vec{B}}_{\mathrm{tot}}$ and the line connecting two ions as shown in figure 1(b). In present work we aim to investigate the effects due to the presence of a stray magnetic field in the measurement of long-range spin-spin interaction strength.

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In order to understand the effect of the perpendicular field component on the long-range spin-spin interaction measurement, we start with the spin-part of the two-electron Hamiltonian. The spin-spin interaction operator of two electrons can be expressed with the help of the Breit operator [1], as

$$\begin{eqnarray}&&{V}_{{\rm{SS}}}=\displaystyle \frac{{\mu }_{0}}{4\pi }{\mu }_{B}^{2}\left\{\displaystyle \frac{{\hat{\sigma }}_{1}\cdot {\hat{\sigma }}_{2}}{| {\vec{r}}_{12}{| }^{3}}-\displaystyle \frac{3({\hat{\sigma }}_{1}\cdot {\vec{r}}_{12})({\hat{\sigma }}_{2}\cdot {\vec{r}}_{12})}{| {\vec{r}}_{12}{| }^{5}}\right\}.\end{eqnarray} \tag{ 2 }$$

Here, μ₀ is the vacuum permeability, μ_B is the Bohr magneton (${\mu }_{B}=e{\rm{\hslash }}/2m$), and σ_i is the Pauli spin matrix acting on i^th electron. In equation (2), ${\vec{r}}_{12}$ describes the separation distance between two electrons. The electrons are considered to be bound in two separate ions, which are located at well-defined distance d from each other. Since the distance between ions is much larger than a typical atomic size, i.e., d ≫ a₀, where a₀ is the Bohr radius, one can use the point-dipole approximation, $| {\vec{r}}_{12}| \approx d$. Thus, the spin-spin interaction operator of two electrons is approximated by

$$\begin{eqnarray}&&{V}_{\mathrm{SS}}^{\mathrm{dip}}=\displaystyle \frac{{\mu }_{0}}{4\pi }{\mu }_{B}^{2}\left\{\displaystyle \frac{{\hat{\sigma }}_{1}\cdot {\hat{\sigma }}_{2}}{{d}^{3}}-\displaystyle \frac{3({\hat{\sigma }}_{1}\cdot \vec{d})({\hat{\sigma }}_{2}\cdot \vec{d})}{{d}^{5}}\right\}.\end{eqnarray} \tag{ 3 }$$

Here, $\vec{d}=d(\hat{z}\cos \theta -\hat{x}\sin \theta )$ represents the inter-ion distance in a chosen coordinate system. The vector $\vec{d}$ forms an angle $\theta ,\theta ={\tan }^{-1}({B}_{\perp }/B)$ with the spin-quantization axis z. Since within the employed approximation the spin-spin interaction operator does not depend on spatial coordinates, the electron spin dynamics can be described by a reduced spin Hamiltonian (H_spin), which acts only on spin coordinates of the electron wave functions. In addition to the spin-spin interaction one has to include the interaction of the spins with the total magnetic field ${\vec{B}}_{\mathrm{tot}}$. As a result the spin Hamiltonian takes a form:

$$\begin{eqnarray}{H}_{\mathrm{spin}} & = & {\mu }_{B}{B}_{\mathrm{tot}}({\sigma }_{z,1}+{\sigma }_{z,2})+{\rm{\hslash }}\xi (2{\sigma }_{z,1}{\sigma }_{z,2}-{\sigma }_{x,1}{\sigma }_{x,2}-{\sigma }_{y,1}{\sigma }_{y,2})\\ & & -{\rm{\hslash }}\delta {\xi }_{1}({\sigma }_{z,1}{\sigma }_{z,2}-{\sigma }_{x,1}{\sigma }_{x,2})-{\rm{\hslash }}\delta {\xi }_{2}({\sigma }_{z,1}{\sigma }_{x,2}-{\sigma }_{x,1}{\sigma }_{z,2}),\end{eqnarray} \tag{ 4 }$$

where B_tot represents the magnitude of the total magnetic field, which direction is chosen as z axis, $\xi ={\mu }_{0}{\mu }_{B}^{2}/(4\pi {\rm{\hslash }})\times 1/{d}^{3}$ defines the spin-spin interaction strength between the electrons, while δξ₁ and δξ₂ describe the corrections originated due to the misalignment of the quantization axis and the line connecting two ions, $\delta {\xi }_{1}=3\xi {\sin }^{2}\theta$ and $\delta {\xi }_{2}=3\xi \sin \theta \cos \theta$. The first term in equation (4) describes the sum of Zeeman energies of electrons placed in the magnetic field, while the rest terms correspond to the spin-spin interaction. In the case of absence of any stray magnetic field (θ = 0), δξ₁ and δξ₂ are zero and the last two terms vanish, thus, the Hamiltonian used in [13] is restored. The terms σ_z,1σ_z,2 can be understood as a shift of the spin-flip energy of one of the electron conditioned on the spin-state of another electron. The terms σ_x,1σ_x,2 and σ_y,1σ_y,2 lead to the collective spin-flips in which a spin excitation is interchanged, while the terms proportional to δξ₂ result in the single spin-flips. Here, we note that single spin-flips are only allowed when $\theta \ne 0$. The spin Hamiltonian can be also represented in the matrix form, e.g., in the basis of spin functions $| \uparrow \uparrow \rangle ,| \uparrow \downarrow \rangle$, $| \downarrow \uparrow \rangle ,| \downarrow \downarrow \rangle$ it is given by:

$$\begin{eqnarray}{H}_{\mathrm{spin}}={\rm{\hslash }}\,\left(\begin{array}{cccc}2\xi -\delta {\xi }_{1}+{\omega }_{z} & -\delta {\xi }_{2} & -\delta {\xi }_{2} & 0\\ -\delta {\xi }_{2} & -2\xi +\delta {\xi }_{1} & -2\xi +\delta {\xi }_{1} & \delta {\xi }_{2}\\ -\delta {\xi }_{2} & -2\xi +\delta {\xi }_{1} & -2\xi +\delta {\xi }_{1} & \delta {\xi }_{2}\\ 0 & \delta {\xi }_{2} & \delta {\xi }_{2} & 2\xi -\delta {\xi }_{1}-{\omega }_{z}\end{array}\right),\end{eqnarray} \tag{ 5 }$$

where ${\omega }_{z}=2{\mu }_{B}{B}_{\mathrm{tot}}/{\rm{\hslash }}$ is the Larmor frequency of electron spin precession in the magnetic field ${\vec{B}}_{\mathrm{tot}}$. As it can be clearly seen from (5) the matrix elements proportional to δξ₂ mix the states which differ by the spin state of one of the electron. In order to investigate the spin evolution driven by the spin-dependent Hamiltonian matrix (5), we solve the time dependent Schrödinger equation of the two electron system

$$\begin{eqnarray}&&i{\rm{\hslash }}\displaystyle \frac{\partial }{\partial t}\psi (t)={H}_{\mathrm{spin}}\psi (t),\end{eqnarray} \tag{ 6 }$$

where ψ(t) describes the spin wave function of the two electron system, which can be decomposed as follows

$$\begin{eqnarray}&&\psi (t)=A(t)| \uparrow \uparrow \rangle +B(t)| \uparrow \downarrow \rangle +C(t)| \downarrow \uparrow \rangle +D(t)| \downarrow \downarrow \rangle .\end{eqnarray} \tag{ 7 }$$

In turn solving the time dependent Schrödinger equation with initial conditions at t = 0: B(0) = 1 and A(0) = C(0) = D(0) = 0, we get

$$\begin{eqnarray}A(t) & = & (i/2)\sin (2\delta {\xi }_{2}t){e}^{-i{\omega }_{z}t},\\ B(t) & = & {e}^{i(2\xi -\delta {\xi }_{1})t}[{\cos }^{2}(\delta {\xi }_{2}t)\cos ((2\xi -\delta {\xi }_{1})t)-i{\sin }^{2}(\delta {\xi }_{2}t)\sin ((2\xi -\delta {\xi }_{1})t)],\\ C(t) & = & {e}^{i(2\xi -\delta {\xi }_{1})t}[-{\sin }^{2}(\delta {\xi }_{2}t)\cos ((2\xi -\delta {\xi }_{1})t)+i{\cos }^{2}(\delta {\xi }_{2}t)\sin ((2\xi -\delta {\xi }_{1})t)],\\ D(t) & = & (-i/2)\sin (2\delta {\xi }_{2}t){e}^{i{\omega }_{z}t}.\end{eqnarray} \tag{ 8 }$$

In figure 2 we display the population probabilities P of spin-state of two electrons as the projections on the basis functions $| \uparrow \uparrow \rangle ,| \uparrow \downarrow \rangle ,| \downarrow \uparrow \rangle$, and $| \downarrow \downarrow \rangle$ after an interaction time t = 15 s and for the inter-ion distance d = 2.41 μm. As one can see from the figure, only in the case of θ = 0° the dynamics of the spin-states of two electrons is limited to the $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ subspace. But, if a stray magnetic field is presented in the system, there is a nonzero probability that the spin-states of electrons can escape to the $| \uparrow \uparrow \rangle$ and $| \downarrow \downarrow \rangle$ subspace. This implies that the probability of having the spin-states of two electrons in the $\{| \uparrow \downarrow \rangle ,| \downarrow \uparrow \rangle \}$ subspace is leaked to the subspace spanned by the $| \uparrow \uparrow \rangle$ and $| \downarrow \downarrow \rangle$ states. Therefore, the dynamics of the spin-states within the $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ subspace becomes affected due to an angular misalignment between the magnetic field and the line joining two ions. Hence, it is expected that the measurement of the interaction strength, which is based on the analysis of dynamics of the spin-states in the $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ subspace, will be influenced by the presence of the stray magnetic field.

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In figure 3 we show the population probabilities P of spin-state of two electrons as a function of interaction time (t) on the basis functions $| \uparrow \uparrow \rangle ,| \uparrow \downarrow \rangle ,| \downarrow \uparrow \rangle$, and $| \downarrow \downarrow \rangle$ for the case of θ = 10° and θ = 0° and for the inter-ion distance d = 2.41 μm. From figure 3, it can be seen that only in the case of θ = 0° the population probabilities P of spin-state of two electrons are limited to the $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ subspace at different t. This implies that the probability of having the spin-states of two electrons in the $\{| \uparrow \downarrow \rangle ,| \downarrow \uparrow \rangle \}$ subspace for the case of θ = 10° is diverged for larger evolution time (t) from the same at θ = 0°. Therefore, it is foreseen that the measurement of the interaction strength, which is based on the analysis of dynamics of the spin-states in the $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ subspace, will be more affected at larger t by the influence of the stray magnetic field.

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When the applied magnetic field ($\vec{B}$) is perfectly directed along the line connecting the two ions, the dynamics of the spin-states of two electrons within the $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ subspace depends only on the spin-spin interaction strength ξ between the electrons. In particular, the initially prepared state $| \uparrow \downarrow \rangle$ oscillates in time between $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ states with an oscillation rate 2ξ. Thus, the spin-spin interaction strength can be accessed by measuring the visibility of such an oscillation:

$$\begin{eqnarray}&&{ \mathcal V }=| {B}^{* }(t)C(t)| ,\end{eqnarray} \tag{ 9 }$$

where B(t) and C(t) are time-dependent coefficients in the expansion of spin wave function given in equation (6). In order to measure the visibility the following experimental scheme was proposed and realized in [13]. First, the spin-state of two electrons initialized to $| \uparrow \downarrow \rangle$ was allowed to evolve under the spin-spin interaction up to particular time (t), what creates the mixture of $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ states. Further, a controlled magnetic field gradient was applied for a fixed duration to accumulate the relative phase between these spin-states. The next step was a global π/2 spin rotation, which was achieved by applying an external magnetic field resonantly modulated along the direction perpendicular to the applied (constant) magnetic field $\vec{B}$. After making the collective π/2 rotation, the projective measurements were performed in order to obtain the ions population at states $| \uparrow \uparrow \rangle$, $| \downarrow \downarrow \rangle$, and $[| \uparrow \downarrow \rangle +| \downarrow \uparrow \rangle ]$ by the state selective fluorescence [13]. As a result, the measured visibility in the case of absence of any perpendicular magnetic field reads

$$\begin{eqnarray}&&{{ \mathcal V }}_{\mathrm{meas}}(\theta =0)=\sin 4\xi t.\end{eqnarray} \tag{ 10 }$$

In the case of the presence of the stray magnetic field, which is characterized by the angle θ, we repeat the experimental scheme realized in [13] for the measurement of the visibility and arrive to

$$\begin{eqnarray}&&{{ \mathcal V }}_{\mathrm{meas}}(\theta )=\sin (2(2\xi -\delta {\xi }_{1})t)\pm \displaystyle \frac{1}{2}{\sin }^{2}(2\delta {\xi }_{2}t)\sin (2{\omega }_{z}t),\end{eqnarray} \tag{ 11 }$$

which exactly coincides with equation (10) for the case when δξ₁ = δξ₂ = 0. Here, ± sign comes from the uncertainties in the geometry of π/2 rotation, since the resonantly modulated external magnetic field might be applied either parallel or perpendicular to the stray magnetic field ${B}_{\perp }$. Thus, the obtained results will differ exactly by the sign of the second term. Further, we will surmise that in the experiment some additional stray magnetic field is present and what is measured is the visibility given by equation (11). However, the spin-spin interaction strength is deduced by the dependence given by equation (10). Therefore, if we assume that there is a stray magnetic field, the experimentally deduced spin-spin interaction strength ξ_exp is given by the expression

$$\begin{eqnarray}&&{\xi }_{\exp }=\displaystyle \frac{1}{4t}{\sin }^{-1}\left[\sin (2(2\xi -\delta {\xi }_{1})t)\pm \displaystyle \frac{1}{2}{\sin }^{2}(2\delta {\xi }_{2}t)\sin (2{\omega }_{z}t)\right],\end{eqnarray} \tag{ 12 }$$

where the first term corresponds to the change in the interaction strength due to the rotation of the magnetic field created by one of the electron, while the second term describes the leakage of probability to the $\{| \uparrow \uparrow \rangle ,| \downarrow \downarrow \rangle \}$ subspace. In equation (12), ξ_exp is only an apparent interaction strength, which was measured from the oscillation rate between the $| \uparrow \downarrow \rangle$ and $| \downarrow \uparrow \rangle$ states [13]. This differs from the physical interaction strength, since that oscillation rate could depend on interaction time (t) by the presence of stray magnetic field as pointed out in figure 3. Equation (12) allows us to investigate how the experimentally deduced spin-spin interaction strength depends on the presence of a stray magnetic field (${B}_{\perp }$) at different inter-ion distances (d).