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We propose to take advantage of the very weak coupling of the ground-state helium-3 nuclear spin to its environment to produce very long-lived macroscopic quantum states, here nuclear spin squeezed states, in a gas cell at room temperature. To perform a quantum non-demolition measurement of a transverse component of the previously polarized collective nuclear spin, a discharge is temporarily switched on in the gas, which populates helium-3 metastable state. The collective spin corresponding to the $F=1/2$ metastable level then hybridizes slightly with the one in the ground state by metastability exchange collisions. To access the nuclear spin fluctuations, one continuously measures the light field leaking out of an optical cavity, where it has interacted dispersively with the metastable state collective spin. In a model of three coupled collective spins (nuclear, metastable and Stokes for light) in the Primakoff approximation, and for two measurement schemes, we calculate the moments of the collective nuclear spin squeezed component $ I_z $ conditioned on the optical signal averaged over the observation time $ t $. In the photon counting scheme, we find that the squeezed observable is $ I_z^2 $ rather than $I_z$. In the homodyne detection scheme, we analytically solve the stochastic equation for the state of the system conditioned to the measurement; the conditional expectation value of $I_z $ depends linearly on the signal and the conditional variance of $I_z $ does not depend on it. The conditional variance decreases as $(\Gamma _{\rm sq}t)^{-1}$, where the squeezing rate $ \Gamma _{\rm sq}$, which we calculate explicitly, depends linearly on the light intensity in the cavity at weak atom-field coupling and saturates at strong coupling to the ground state metastability exchange effective rate, proportional to the metastable atom density. Finally, we take into account the de-excitation of metastable atoms at the walls, which induces nuclear spin decoherence with an effective rate $ \gamma _\alpha $. It imposes a limit $\propto (\gamma _\alpha /\Gamma _{\rm sq})^{1/2}$ on the conditional variance reached in a time $ \propto (\gamma _\alpha \Gamma _{\rm sq})^{-1/2}$. A multilingual version is available on the open archive HAL at https://hal.archives-ouvertes.fr/hal-03083577. |
