Phase Squeezed Spin State

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Phase squeezed spin states (PSS) are collective spin states.

In the canonical measurement scheme it is sensible, for cyclic variables, to define uncertainty is in terms of the sharpness

$S = \langle e^{i(\varphi-\phi)}\rangle= \int d\phi P(\phi)e^{i(\varphi-\phi)}$ $= \sum_{\mu= -J}^{\mu= J} {}_z\langle J,\mu+1|\psi\rangle\langle\psi|J,\mu\rangle_z.$

As long as $|\psi\rangle$ is oriented in the $x$ direction, $S$ is real and positive, and it is always less than 1. Unlike the variance, the sharpness respects the periodicity of the phase $\phi$. A state with a $S$ close to 1 has a low phase uncertainty, and vice versa. For such states, the variance of $P(\phi), \int_{\phi_0}^{\phi_0+2\pi} P(\phi)\phi^2 d\phi$ , is given by the approximate formula $V \simeq 2(1-S)$ , provided that $φ 0$ is not near the peak of $P (φ)$ . (Another variance can be defined using $S$ , the Holevo variance $S - 2 - 1$ but for our purposes it is simpler to take a quantity linear in $S$ .)

Now that we have a measure for phase uncertainty, we can define a new figure of merit, appropriate to a single-shot phase estimate with no prior information:

ζ 2 = 4 J (1 - S).

This is the Phase Squeezing parameter for spin states. This measure has, $\zeta^2 \approx 1$ for coherent states (c.f. with Spin Squeezed State), and phase squeezing implies entanglement