In this work we analyze two axis twisting in the presence of depolarizing channel dissipation. We find that spin squeezing is only possible if the dissipation is parametrically weaker than the squeezing coupling. Squeezing may be used for meteorologically useful decrease of spin noise but only in the case where the squeezing occurs before measurement, in the case one squeezes as one measures one also squeezes the signal thereby making spin squeezing ineffective for metrological gain. The key mathematical advance made in this work is the observation that TACT in the presence of depolarizing noise is equivalent to TACT with reduced polarization and no noise. We find an exponential gain in signal to noise with the exponent proportional to the ratio between the squeezing strength and the depolarization rate.
Based on the recent twisting-tensor approach [T. Opatrny, ArXiv:1408.3265 (2014)], a specific scenario for fast and deep spin squeezing is proposed. Initially the state is subjected to one-axis twisting under optimum orientation, enabling the maximum squeezing rate allowed by the system nonlinearity. Later on, when for highly squeezed states the one-axis twisting deforms the uncertainty ellipse and deteriorates the squeezing properties, the process is switched to an effective two-axis counter-twisting by a sequence of
π/2 pulses. The squeezing rate then slows to 2/3 of the maximum value, but the process can continue for longer to achieve a very high degree of squeezing.