Optical interferometers have been widely used to estimated the slight phase changes in quantum metrology. For a Mach–Zehnder interferometer (MZI) with classical-light inputs, the phase sensitivity is bounded by the standard quantum limit (SQL) [1] which scales as of Δφ=1/n̄, with n̄ the mean photon number inside the interferometer. To surpass the SQL in an MZI, one can deploy nonclassical states [2], [3], [4], [5], [6] and photon-number-resolving detectors [7], [8], [9]. The other way of beating the SQL is to deploy an interferometer in which the evolution process is done through the nonlinear elements. A typical interferometer is usually known as an SU(1,1) interferometer introduced by Yurke et al. [10], where two beam splitters in an MZI are replaced by two optical parametric amplifiers. The improvement of the phase sensitivity within an SU(1,1) interferometer has also been shown in both experimental [11], [12], [13] and theoretical studies [14], [15], [16], [17].

For a quantum state in an MZI or SU(1,1) interferometer, the phase sensitivity Δφ with a given detection scheme is bounded by the quantum Cramér–Rao bound (QCRB) defined by the quantum Fisher information [18], [19]. Up to present, three achievable detection schemes, i.e., homodyne detection, intensity detection and parity detection, are widely used to extract phase information from the output states of the interferometer. Parity detection was first proposed by Bollinger et al. for enhanced frequency measurement with an entangled state of trapped ions [20]. Gerry later utilized it to the MZI with the NOON state for achieving phase sensitivity at the Heisenberg limit (HL), Δφ=1/n̄ [21]. In quantum optics, the parity detection is described by the parity operator Πˆ=−1nˆ=eiπnˆ,where nˆ is the single-mode photon number operator. In recent years, it has been proved that the parity detection can achieves the optimal phase sensitivity given by the QCRB for a wide range of input states [3], [4], [7], [8], [9], [22], [23]. For both homodyne detection and intensity detection, one often applies the traditional input–output field operator transformations of the interferometer to derive the signals of the measurement [24], [25], [26], [27], [28], [29]. However, for parity detection, it is usually difficult to obtain the parity signal for general Gaussian or non-Gaussian states by this traditional input–output operator transformation. Noting that the parity operator and the Wigner function Wα are intimately related according to Wα=2Dˆ†αΠˆDˆα/π where Dˆα=expαaˆ†−α∗aˆ is the displacement operator [9], [30]. Then, one can find that Πˆ=πW0/2 with W0 being the origin of the phase space. Therefore, with the help of the transformation of phase space Woutα,β=Winα̃,β̃, many works have investigated the phase sensitivity of the SU(1,1) [16], [17], [31] interferometer and the MZI [32], [33], [34], [35], [36], [37], respectively.

Recently, we consider the whole operation of the SU(1,1) interferometer combined with parity detection as a Hermitian operator by the field operator transformations [38]. Based on this alternative operator method, it is convenient to derive the parity signal for general Gaussian and non-Gaussian states [38], [39]. In this work, we mainly consider the MZI with some concrete detections (such as parity detection and intensity detection). Different from the previous methods, by using of a simple and straightforward way, we introduce a equivalent measurement operator (i.e., a Hermitian operator) which can completely describe the whole operation of the MZI with parity detection and single-mode intensity detection. As a consequence, it can bring convenience to investigate the phase estimations.

The organization of this paper is as follows. In Section 2, we consider the MZI and the parity detection as a whole operation, and propose an alternative operator method to derive the parity signal. The normal ordering form of such equivalent Hermitian operator is derived. At the same time, by the transformations of quantum representations, we can rewrite such Hermitian operator in the coherent state representation. Then, in order to show the advantage of the alternative operator method, we give some applications in Section 3. Last, we conclude with a summary.