Guangdong Provincial Key Laboratory of Nanophotonic Functional Materials and Devices, School of Information and Optoelectronic Science and Engineering, Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China
† Corresponding author. E-mail:
zhangzhiming@m.scnu.edu.cn yuyafei@m.scnu.edu.cn
Project supported by the National Natural Science Foundation of China (Grant Nos. 11574092, 61775062, 61378012, 91121023, and 60978009), the National Basic Research Program of China (Grant No. 2013CB921804), and the Innovation Project of Graduate School of South China Normal University (Grant No. 2017LKXM088).
1. IntroductionQuantum metrology is a field of exploiting quantum properties to achieve a higher precision in the measurement of a physical parameter.[1,2] Here, quantum state, measurement device, and method of detection contribute to improving the precision. As a typical high-precision measurement device based on optics, the SU(1,1) interferometer (SU(1,1)I)[3–6] consists of two optical parametric amplifiers,[7] two mirrors and a phase shifter, which is similar with Mach–Zehnder interferometer (MZI) but provides a gain factor
over the MZI.[6,8] In addition, researchers have described the structural changes[9–11] and the loss effects of the interferometer,[5,12–14] which has led to the gradual perfection of the phase estimation based on an optical system.
The non-classicality of quantum states, such as entanglement and squeezing, is helpful to improve the accuracy of phase estimation in the SU(1,1)I. For example, in Ref. [15] the thermal state and the squeezed vacuum state as inputs of an SU(1,1)I can approach the Heisenberg limit. There are other combinations of input states beating the standard quantum limit and approaching the Heisenberg Limit, such as coherent state and squeezed vacuum state,[16,17] coherent state and displacement squeezed vacuum state,[8,18] and so on. The accuracy of phase estimation utilizing the SU(1,1)I has also been realized experimentally.[19–23] For instance, an interferometer with the parametric amplifier has proved that the improvement of phase sensitivity[19] is due to the enhanced input intensity. Recently, the absolute sensitivity of SU(1,1)-type interferometer has been investigated and obtained 3-dB improvement[23] over the MZI. The detection method is another factor contributing to achieving better phase sensitivity. Different input states always go with their own optimal measurement methods.[16,24,25] In Ref. [16], for the input combinations, coherent state and squeezed vacuum state, and coherent state and vacuum state, the performance of parity detection is better than intensity detection and homodyne detection, while for the input combination of two coherent states, the parity detection is worse.
The Wigner function is another useful tool to describe a quantum system. In Ref. [26] a new method of coherent mode decomposition is proposed to determine the mode functions and the weights without solving the eigenvalue problem of a large matrix. The phase sensitivity with parity detection is also analyzed by Wigner function in Refs. [8, 17, 27], and [28]. The non-classicality of quantum system can be charactered by the negative Wigner function.[29–31] In addition, quantum interference pattern of Wigner function is used to measure entanglement between two modes of vortex states.[32] Hence, we raise a bold question: could the negativity of Wigner function of quantum state provide certain benefits for the phase estimation of the SU(1,1)I?
In this paper, we propose a combination of even coherent state and squeezed vacuum state as inputs of the SU(1,1)I, which can approach the Heisenberg limit under parity detection when the gain factors of two optical amplifiers are equal. Compared with other combinations of input states, the phase sensitivity is better when even coherent state and squeezed vacuum state are used as input states. In addition, the negativity volume of Wigner function is completely from the input state, which is due to the intrinsic characteristics of the SU(1,1)I. At the same time, the behavior of the phase sensitivity and the negativity volume of Wigner function with the encoding phase is observed in contrast. Its variation with the encoding phase is positively correlated. This relationship may mean the non-classicality indicated by negative Wigner function is a kind of resource that can verify the research results of the phase estimation.
This paper is organized as follows. In Section 2, an SU(1,1)I is proposed to take even coherent state and squeezed vacuum state as inputs, and its phase sensitivity is analyzed by the Wigner function of the output field. The variation between the negativity volume of Winger function and the phase sensitivity is compared in Section 3. Finally, a summary is provided in Section 4.
2. Phase sensitivity of an SU(1,1)I taking even coherent state and squeezed vacuum sate as inputs2.1. Winger function of output fieldWe start with study on the SU(1,1)I, as shown in Fig. 1. After the first OPA, only the path a passes through a phase shift ϕ. In the SU(1,1)I, the relationship between input and output variables of the whole interferometer can be described as
The transformation matrix of the SU(1,1)I is
where
is the transformation of OPA
j,
,
,
gj and
are used to denote the strength and the phase shift in the OPAs (
j = 1, 2).
represents the phase-shift transformation. Here, the balanced configuration corresponds to
and
. The optimal phase sensitivity can be obtained under the balance of the SU(1,1)I, so we focus on the balanced situation.
Now, let us consider even coherent state and squeezed vacuum state,
, as inputs in the SU(1,1)I. The corresponding Winger function can be written as[33]
respectively, where
with
,
. When even coherent state and squeezed vacuum state pass through the SU(1,1)I, the Winger function of output field is
Combined with Eqs. (
1)–(
5), the Winger function of output field of SU(1,1)I is given by
with
2.2. Phase sensitivity with parity detectionThe important character of the accuracy of phase estimation in optical interferometer is phase sensitivity
, which can be characterized using the error propagation formula
The operator
is observable for detection. Now, we adopt parity detection, where the parity operator on the output mode
a can be defined as
. Owing to the integration property of the characteristic function for the parity operator, the average value
can be written as
[8]By substituting Eq. (
6) into Eq. (
10), we can get
with
When we take the average value
into Eq. (9) and simplify the expression through a bunch of tedious simplifications, the phase sensitivity is found to be optimal point at
and is given by
with
From Eq. (
19), the phase sensitivity is related to not only the intensity
and the phase angle
of input state but also the phase shift
of OPA
1. After some simple simulation calculations, the phase sensitivity is optimized with phase angle
,
.
Then, in Fig. 2, the optimal phase sensitivity is compared with the Heisenberg limit (HL) and the shot noise limit (SNL), which are related to the total photon number and displayed as
where
is the average photon number in the mode
a of the input states. The phase sensitivity becomes better gradually and remains below the shot noise limit with the gain coefficient
g increasing. When
with
,
r = 1, the optimal phase sensitivity goes close to the Heisenberg limit.
To analyze the effects of different combinations of input states on the accuracy of phase estimation, the phase sensitivities with parity detection are compared among different combinations of input states, as shown in Table 1. According to Table 1, the phase sensitivities with parity detection are expressed as the presence of a constant Nc, which is caused by the amplification process of OPAs. It is possible from the comparison that the phase sensitivity of the input combination of squeezed vacuum state and even coherent state is better than those of other combinations. To clarify this point, the phase sensitivities of the input combinations including squeezed vacuum states is chosen to be plotted with gain coefficient g in Fig. 3. It is true that the phase sensitivity of
as inputs is better than that for other input combinations, even the combination of coherent state and squeezed vacuum state
. And the phase sensitivity of the input combination
is the worst. In brief, even coherent state can replace coherent state when
as input of the SU(1,1)I to improve the accuracy of phase estimation when the gain coefficient g is not large enough.
Table 1.
Table 1.
 | Table 1.
Different combinations of input states and their phase sensitivities with parity detection.
. |
3. Comparison between the phase sensitivity and negativity volume of the Winger functionAn improvement in the accuracy of phase estimation is usually attributed to the non-classicality in quantum states, such as the negative Wigner function. In the following, we will analyze comparatively the phase sensitivity and the negativity volume of output Wigner function. The negativity volume of Wigner function[30] is defined as
where
. By combining Eq. (
6) and Eq. (
25), we plot the negativity volume of Wigner function
versus the encoding phase in Fig.
4. As can be seen from Fig.
4, the negativity volume of Wigner function maximizes at the point of
ϕ = 0 and varies periodically with the encoding phase. In Fig.
4(a), it is shown that the increasing of the intensity of even coherent state as input can enable the maximum negativity volume of Wigner function at the point of
ϕ = 0 rising. Compared with Fig.
4(b) and Fig.
4(c), the maximum of the negativity volume of Wigner function remains unchanged for different squeezing parameters and gain coefficients when the average photon number of the input states is fixed, which displays that the negativity of the output field completely comes from the input state.
The negativity of the output field varies periodicity with the encoding phase for squeezing parameters and gain coefficients, which is determined by the intrinsic properties of the SU(1,1)I. Interestingly, the researchers have also found that the phase sensitivity varies periodically with the encoding phase. To see more clearly that the relation between the phase sensitivity and the negativity of Wigner function with certain input state, we will now analyze them comparatively. Due to periodicity and symmetry of the encoding phase, we perform numerical simulation to obtain Fig. 5 in the range of ϕ = 0 to ϕ = π/4 based on Eq. (11) and Eq. (25). Here, one can see clearly in Fig. 5(a) and Fig. 5(b) that the phase sensitivity is positively correlated with the negativity volume of the Winger function. This positive correlation with different squeezing parameters and gain coefficients indicates that the negativity volume of the Wigner function can be used as a verification for the variation of phase sensitivity with parameters.
4. ConclusionsIn this work, we have investigated the phase sensitivity and the negativity of Winger function in the SU(1,1)I with even coherent state and squeezed vacuum state as inputs under parity detection. Compared with other combinations of input states, the phase sensitivity for even coherent state and squeezed vacuum state is optimal, by which the Heisenberg limit can be approached under appropriate parameters. The negativity volume of Winger function in the detection mode comes entirely from the input state and varies periodically with the encoding phase. Through numerical simulation, it can be found that the phase sensitivity is positively correlated with the negativity volume of the Winger function, which provides a supplementary verification of phase estimation.
