Spin Squeezed State

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Spin squeezed states (SSS) are collective spin states defined as follows. First consider an individual spin $i$ with the vector spin operator

$\vec{f}^{(i)} = [ f^{(i)}_x, f^{(i)}_y, f^{(i)}_z]$

The individual spin operators have the commutation relations:

$\left[ f_x , f_y \right] = i \hbar f_z$
$\left[ f_z , f_x \right] = i \hbar f_y$
$\left[ f_y , f_z \right] = i \hbar f_x$

For two operators $A$ and $B$ , we have the general uncertainty relation:

$\langle \Delta A ^2 \rangle \langle \Delta B ^2 \rangle \geq \frac{1}{4} |\langle [ A , B ] \rangle |^2 + \frac{1}{4}|\langle \{ \Delta A , \Delta B \}\rangle |^2$

Now consider the collective spin vector operators

$\vec{F} = \sum_{i}^{N} \vec{f}^{(i)} = [ F_x, F_y, F_z ]$

Using the definition of the collective spin components and the general uncertainty relation, we get the collective spin uncertainty relations

$\langle \Delta F_x ^2 \rangle \langle \Delta F_y ^2 \rangle \geq \frac{\hbar^2}{4} |\langle F_z \rangle |^2 + \frac{1}{4}|\langle \{ \Delta F_x , \Delta F_y \}\rangle |^2$
$\langle \Delta F_z ^2 \rangle \langle \Delta F_x ^2 \rangle \geq \frac{\hbar^2}{4} |\langle F_y \rangle |^2 + \frac{1}{4}|\langle \{ \Delta F_z , \Delta F_x \}\rangle |^2$
$\langle \Delta F_y ^2 \rangle \langle \Delta F_z ^2 \rangle \geq \frac{\hbar^2}{4} |\langle F_x \rangle |^2 + \frac{1}{4}|\langle \{ \Delta F_y , \Delta F_z \}\rangle |^2$

Now consider a spin state with all spins aligned along the x-axis. This separable state is referred to as a coherent spin state (CSS) and has $\langle F_x \rangle = \hbar F$ , where $F = N f$ is unitless. Despite the fact that all spins are aligned as well as possible, there remains uncertainty in the transverse directions (y and z) due to the final uncertainty relation above (in which the last term is zero by symmetry). For the CSS, these uncertainties are equal $\langle \Delta F_y ^2 \rangle = \langle \Delta F_y ^2 \rangle = \hbar^2 F/2$ .

For a spin squeezed state, one of these uncertainties is decreased, while the other necessarily increases, e.g. $\langle \Delta F_z ^2 \rangle < \hbar^2 F/2$ while $\langle \Delta F_y ^2 \rangle > \hbar^2 F/2$ .

The spin squeezing parameter (for a state aligned along x and squeezed along z) is defined as

$\xi^2 = \frac{ 2 F \langle \Delta F_z^2 \rangle }{\langle F_x \rangle^2}$

For a CSS, $ξ 2 = 1$ . In contrast, a spin squeezed state is defined as a state with $\langle F_y \rangle =\langle F_z \rangle =0$ and $ξ 2 < 1$ . By this definition, it has been shown that spin squeezed states are necessarily entangled.