Experimental snapshot verification of non-Markovianity by quantum probing of convex coefficients

We apply the recently proposed quantum probing protocols with an unknown system-probe coupling to probe the convex coefficients in mixtures of commuting states. By using two reference states instead of one as originally suggested, we are able to probe both the lower and upper bounds for the convex coefficient. We perform extensive analysis for the roles of the parameters characterizing the double peaked Gaussian frequency spectrum in the Markovian-to-non-Markovian transition of the polarization dynamics of a single photon. We apply the probing of the convex coefficient to the transition-inducing frequency parameter and show that the non-Markovianity of the polarization dynamics can be confirmed with a single snapshot measurement of the polarization qubit performed at unknown time. Furthermore, we change the "known"measurement coupling that corresponds to our dynamics of interest to unknown measurement coupling and show that even for unknown measurement couplings the probing method is precise enough to confirm that using the "known coupling"would lead to non-Markovian polarization dynamics. We also show how the protocol can identify Markovian and non-Markovian time intervals in the dynamics. The results are validated with single-photon experiments.