Fig. 1. Schematic diagram of enhanced stochastic phase estimation with an SU(1,1) interferometer. This interferometer consists of two parametric amplifiers (PAs), and the input states are the coherent state and the vacuum state. φ(t) is the stochastic phase to be estimated, and the phase Φ(t) in the other arm is adaptively controlled. r(t) is photocurrent, which is equal to the homodyne measurement results after an added operation. The phase θ(t) of the local oscillator is adaptively controlled simultaneously, and hot is the optimum linear processor of phase tracking.

Fig. 2. Ratio of the two SNRs. The blue surface represents the ratio of the two SNRs. The red surface represents the case in which the two interferometers have equal SNR.

Fig. 3. MSE σf2 of tracking as a function of G2 for MZI (red line) and NLI (blue line). Here we take κ=1.0×104  rad/s, λ=1.0×105  rad/s, and |β|2=1.0×107  s−1.

Fig. 4. MSE ξ as a function of λε for MZI (red line) and NLI (blue line). The horizontal axis is the proportion between ε and the correlation time of φ(t). The proportion equal to 0 represents phase tracking (black dotted line). λε>0 and λε<0 stand for prediction and smoothing, respectively. Here we make κ=1.0×104  rad/s, λ=1.0×105  rad/s, G2=7.4, and |β|2=1.0×107  s−1.

Fig. 5. Optimal smoothing MSE ξ as a function of photon number flux |β|2 for MZI (red line), NLI (blue line), and canonical measurement (black line). Here we take κ=1.0×104  rad/s and λ=1.0×105  rad/s.