Optical metrology relies on light interferometry as its primary tool for phase estimation.^[1,2] A general protocol to estimate an unknown parameter ϕ corresponding to a quantum process in quantum metrology is comprised of three steps.^[3,4] Firstly, a quantum state ρ, which plays an important role in the measurement scheme, is prepared that will serve as the probe. Secondly, this probe state interacts with a system of interest such that ρ evolves to a state ρ_ϕ which depends on the parameter ϕ to be estimated. In the third and last step, an observable O of the probe state is measured in order to gain information about the value of ϕ. Therefore, the phase sensitivity δϕ of an interferometer, followed by a detection scheme described by an operator O, can be characterized using the error propagation formula^[5]

The usual tool to estimate phase is the physical process of quantum interference by using a Mach–Zenhder interferometer (MZI).^[7] Enormous efforts have been devoted to improve the sensitivity of the interferometers by using a different input light field. In 1981, Caves pointed out that by using coherent light together with a squeezed vacuum one could beat SNL δϕ < δϕ_SNL (supersensitivity).^[8] Similar works had been also studied by Pezzé et al.^[9] and Seshadreesan et al.,^[10] which showed that the phase sensitivity can reach the Heisenberg limit. In the work of Boto et al.,^[11] it is possible to beat SNL by exploiting special states of light, such as the N00N state.^[12] Hu et al. investigated quantum interference with Fock and conventional cat states,^[13] where the phase sensitivity approaches to the HL with the increasing values of Fock number and the total average photon number, and Lee et al. demonstrated the more enhanced performance in phase estimation by using a four-headed cat state rather than the conventional cat state.^[14] In addition, some works show that entanglement is important in order to achieve supersensitivity.^[15]

Recently, both Anisimov et al.^[16] and Zhang et al.^[17] studied the sensitivity of phase measurement in an MZI with a two-mode squeezed vacuum state (TMSVS). The former scheme showed that sub-Heisenberg sensitivity was obtained with parity detection, the latter one presented that phase sensitivity was obtained with the measurement, which may be useful to estimate a small phase shift at high precision. As an extension of that work, we shall study the similar work with a two-mode squeezed thermal state (TMSTS) as the input resource. Therefore, the TMSVS is only a special case of our present work. It is emphasized that we shall tackle such input–output issues by using the Wigner-function (WF) method,^[18] which can be viewed as a quantum analogy to the operational formulation of classical mechanics.^[19,20] Very recently, our group carried out some similar works on the phase estimation with the phase space method.^[21,22]

In this paper, we shall focus on demonstrating the utility of the WF method in studying the parity detection and the phase sensitivity of MZI with TMSTS as the input resource. The paper is organized as follows. In Section 2 we make a brief review of the two-mode squeezed thermal state (TMSTS) and derive its WF after obtaining its normal ordering form. In Section 3, we simply obtain the output WF after employing the input–output transformation. In Section 4, the analytical expression of the parity is obtained. Then the analytical expression of the phase sensitivity is obtained in Section 5. Finally, our results are summarized in Section 6.

In this section, we give a brief description of the two-mode squeezed thermal states and derive its normal ordering form and Wigner function.

2.3. Wigner function of TMSTS

Knowing the normal ordering form of the density operator, we can easily calculate the Wigner function of the two-mode squeezed thermal state. For a two-mode quantum state ρ, the Wigner function in the coherent state representation is given by the following equation^[28]

where |z₁,z₂〉 = | z₁〉_a |z₂〉_b is the two-mode coherent state and , . Substituting Eq. (13) into Eq. (15), we obtain the WF of TMSTS as follows:

where we have set

The Gaussian–Wigner function in Eq. (16) shows that TMSTS is a two-mode Gaussian state.

In this section, based on the phase space method, we derive the output WF of MZI with the TMSTS as an input resource.

Fig. 1.

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	Fig. 1. A lossless Mach–Zehner interferometer with TMSTS as inputs. The modes are labeled by the annihilation operators ai, bi for the input port and af, bf for the final output ports, respectively. BS1 and BS2 are 50:50 beam splitters. At the output of the MZI, a parity measurement on mode af is implemented.

An MZI is composed of optical elements such as beam splitters, mirrors, and phase shifts. As shown in Fig. 1, two input optical modes merge at a 50/50 beam splitter, propagate along two paths of different lengths to accumulate an unknown relative phase shift ϕ, and then merge at a second 50/50 beam splitter. Thus an MZI can be considered as a losses and passive four-port device. Propagating through MZI, two input optical fields (a_i, b_i mode) evolute into two output fields (a_f, b_f mode) yielding

Considering the TMSTS as the input resource, the output WF can be obtained by using Eqs. (16) and (21) as well as Eq. (22). Therefore, the state of light at the output of the MZI is described by the following output WF

Parity detection should be a better alternative to the detection scheme since parity measurement can determine the evenness or oddness of the photon number in a given beam. Photon number states are assigned a parity of +1 if their photon number is even and a parity of −1 if odd. In experiment, using photon-number-resolving detectors or using optical non-linearities without any photon counting, the parity measurement can be obtained in the low photon number regime. Parity detection was adopted for optical interferometry by Gerry’s group.^[29]

Here we consider a detector that is placed at one of the output beams in the a_f mode. The parity operator for this mode is

Performing this operation in calculating the WF in Eq. (23), we obtain the parity for the a_f arm

Let us compare the signal outcomes of the TMSTS with different parameters (n̄₁, n̄₂,r). Specially, when n̄₁ = n̄₂ = 0, and r = 0, TMSTS is just TMSVS and 〈Π_af〉|_{n̄1=n̄2=r=0} = 1, i.e., it always remains one.

Fig. 2.

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Fig. 2. The parity signal as a function of the shift phase with different input TMSTS. (a) r = 0.3, n̄1 = 0 (solid), 0.2 (dashed), 0.5 (dotted), 0.8 (dot–dashed), n̄2 = 0; (b) r = 1, n̄1 = 0 (solid), 0.2 (dashed), 0.5 (dotted), 0.8 (dot–dashed), n̄2 = 0; (c) r = 0.2 (solid), 0.5 (dashed), 1 (dotted), 2 (dot–dashed), n̄1 = 0, n̄2 = 0; (d) r = 0.2 (solid), 0.5 (dashed), 1 (dotted), 2 (dot–dashed), n̄1 = 0.2, n̄2 = 0.2.

In Fig. 2 we plot the parity signal 〈Π_af〉 as a function of the shift phase ϕ with different input TMSTS according to Eq. (26). The signal of the parity detection scheme is periodic with period 2π. For the symmetrical TMSTS, it is shown from Figs. 2(c) and 2(d) that two peaks of the signal are always found at ϕ = ±π/2 and the width of each peak becomes narrower with the increasing r. When n̄₁ = n̄₂ = 0, it attains its maximum value of one at ϕ = ±π/2, while this maximum value decreases with the increasing n̄₁ = n̄₂, which is independent of the squeezing parameter r. Moreover, for the case of asymmetrical TMSTS with n̄₁ ≠ 0 and n̄₂ = 0 (see Figs. 2(a) and 2(b)), the corresponding maximum value not only becomes smaller but also drifts away from ϕ = ±π/2 as n̄₁ increases. The case of n̄₁ = 0 and n̄₂ ≠ 0 is the same as that of n̄₁ ≠ 0 and n̄₂ = 0.

The phase sensitivity is quantified by an average value of how much the measured phase could differ from the actual value. It is an important aspect of optical interferometry. Based on the outcome of the parity measurement (O = Π_af), knowing the signal suffices for the sensitivity calculation since , the variance of the phase estimation could be estimated as the following formula

Fig. 3.

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	Fig. 3. Phase sensitivity δϕ with parity detection as a function of the shift phase ϕ with different input TMSTS. (a) r = 0.4, n̄1 = 0 (solid), 2 (dashed), n̄2 = 0; (b) r = 1, n̄1 = 0 (solid), 2 (dashed), n̄2 = 0; (c) r = 0.2 solid), 2 (dashed), n̄1 = 0, n̄2 = 0; (d) r = 0.2 (solid), 2 (dashed), n̄1 = 0.2, n̄2 = 0.2.

To see clearly the behaviors of the phase sensitivity for the different TMSTS, in Fig. 3 we plot phase sensitivity δϕ with parity detection as a function of the shift phase ϕ with different input TMSTS according to Eq. (29). In principle, the smaller the value of δϕ, the higher the phase sensitivity. A numerical calculation of the phase sensitivity for the symmetrical case shows the highest phase sensitivity δϕ at the optimal working point being close to π/2. In addition, the minimum value for δϕ is 0.000331577 at the point of r → 4.07803, n̄ → 0.00270621, and ϕ → π/2. In addition, for the asymmetrical case, the optimal working point deviates from π/2, as shown in Figs. 3(a) and 3(b).

By numerical simulation, we find that one minimum of phase sensitivity is 0.00582434 with r → 4.39585, n̄₁ → 2.20112, n̄₂ → 5.39877, ϕ → π/2. In the vicinity of this parameter interval, we plot the phase sensitivity δϕ as a function of the total photon number N̄ by fixing n̄₁ → 2.20112, n̄₂ → 5.39877, ϕ → π/2, and modulating r between 3.8 and 4.8 (black solid line), SNL (blue dashed line), and HL (red dot–dashed line) in Fig. 4. The curve of phase sensitivity is interchanged with the SNL curve at the threshold value with N̄ = 4 × 10⁴ and is always above the HL line. The figure demonstrates that δϕ > δϕ_SNL > δϕ_HL in the interval of N̄ ≤ 4 × 10⁴ and δϕ_SNL > δϕ > δϕ_HL in the interval of N̄ ≥ 4 × 10⁴. The result shows that only the TMSTS with a large number of photons can beat supersensitivity.

Fig. 4.

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	Fig. 4. Phase sensitivity δϕ as a function of the total photon number N̄ by fixing n̄1 → 2.20112, n̄2 → 5.39877, ϕ → π/2, and modulating r between 3.8 and 4.8 (black solid line), SNL (blue dashed line), and HL (red dot–dashed line).

To verify our result, we also reduce our work to a similar work with the condition n̄₁ = n̄₂ = 0 studied in Ref. [16]. Thus equation (27) is simplified as

In Fig. 3(c), we can clearly see that phase sensitivity obtained by the parity measurement at ϕ = π/2 saturates the lower bound. Thus by the parity measurement at φ = 0, the phase sensitivity is

Fig. 5.

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	Fig. 5. Phase sensitivity δϕ as a function of the total photon number N̄ at φ = 0 for input TMSVS (black solid line), SNL (blue dashed line), and HL (red dotdashed line).

In summary, we have studied the application of parity detection for phase estimation in Mach–Zehner interferometry with a two-mode squeezed thermal state based on the Wigner-function method rather than the traditional operator method. The detected parity is easily obtained from the output WF. The results from the numerical calculation show that the supersensitivity can be reached only if the input TMSTS have a large number photons. Our result can be reduced to the case of the two-mode squeezed vacuum state (TMSVS) in Ref. [16]. On the other hand, the present paper further demonstrates the power of the Wigner-function method in studying quantum optics and quantum metrology.