Phase Measurements And Numerical Simulations Based On Quantum Optical Interferometry

Quantum phase estimation, developed from the combination of quantum mechanics and statistical information, becomes the theoretical basis of high precision quantum measurement, atomic clocks, gravitational wave detection, and other fields. It attracts more and more attention. Essentially, quantum phase measurement is an appropriate data processing, performed on the random outcomes, so as to access a phase estimator. For instance, according to quantum mechanics, detecting a certain quantum state involving the phase information always obtains a series of eigenvalues of observables. The probability of each eigenvalue is the modulus square of the probability amplitude, which can be given by projecting the quantum state to the eigenstate of this observable. The phase estimator can be obtained by performing a data processing on the outcomes and their probabilities. The core of quantum phase estimation is to find appropriate quantum states, to optimize mechanical parameters and to perform data processing, which make the fluctuation of the estimator can be lower than the standard quantum limit.In quantum phase measurement, the measured results are usually continuous variables, such as the measurement of the coordinates x or momentum p Performing special data classification and processing on the measured data artificially, one can realize binary-outcome measurement and then obtain the estimator. For instance, the total momentums can be divided into binary outcomes:p∈ [-α,α] and |p|> α, where a is a controllable parameter.In an optical Mach-Zender interferometer (MZI), a binary-outcome homodyne detection is realized by performing a homodyne detection of the phase quadrature at one of the output ports. According to even/odd and zero/nonzero number of photons detected at the output port, the so-called parity detection and zero/nonzero photon counting measurement are realized successfully in experiments. The above three examples can be considered as the binary-outcome measurement. On the basis of the quantum optical interferometry, we have studied systemically the quantum phase measurement and the numerical simulation. The main research results are as follows.1.The numerical simulation of quantum phase measurement and the final phase precision depend on the high precision of the Wigner-d matrix. The previous researches show that the calculation of the Wigner-d matrix suffers low precision and even divergence due to the presence of large numbers in the original formula; while, the calculation by means of recurrence relations encounters severe numerical instability. Here, we put forward a new method to evaluate the Wigner-d matrix numerically with high accuracy. Specific practices are as follows:(i) The d matrix is expanded into a complex Fourier series with a variable rotation angle, and the expansion coefficients are determined by the eigenvectors of the y component of the angular momentum; (ii) In the eigenbasis of the z component of the angular momentum, the y component of the angular momentum is written in a form of matrix, and all the eigenstates and probabilities are obtained by exactly diagonalizing the matrix numerically. This enables us to avoid the large-number problem. As the main advantage, we show that for given spin and rotation angle, all the elements of Wigner-d matrix and their kth-order derivatives can be obtained by diagonalizing the matrix only once. The absolute error can reach ～10-14 for spin up to 100 and the relative error of ～10-10 within the central region.2. For a general binary-outcome measurement, we show that the sensitivity obtained by the error-propagation formula can saturate the Cramer-Rao bound. For such measurements, the estimator can be obtained by inverting the average output signal. This conclusion is independent of the input states and the presence of noises, such as the photon loss and the phase diffusion and so on.3. Based on the coherent-light Mach-Zehnder interferometer, the validity of above conclusion is demonstrated by two recent experiments. In detail, the binary-outcome homodyne detection, the parity measurement, and the zero/nonzero photon counting measurement are discussed. The resolution of the interference signal and the optimal phase sensitivity are presented in the presence of the phase diffusion. Our results show that the role of phase diffusion is uniquely determined by a product of the injected photon number N and the phase diffusion rate γ. When Nyγ＜＜1, both the resolution and the best sensitivity almost follow the shot-noise scaling; as Ny increases, the scaling undergoes a transition from W-1/2 to N0. The phase diffusion leads to a phase shift of the optimal working point. In addition, we analytically confirm that the zero/nonzero photon counting measurement is better than the parity detection, in agreement with the experiment observations qualitatively.4. The previous researches show that the maximum likelihood estimator normally only can be obtained via the numerical resolution. Its analytic expression can not be given, or even does not exist. For the phase measurement with finite outcomes, we present an analytic expression of the maximum likelihood estimator. Taking the maximal path-entangled state as an example, we simulate the photon counting measurement numerically. As the number of samples and the times of the measurements are both large enough, our results show that the uncertainty of this estimator can saturate the Cramer-Rao bound.