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Interference metrology is a method for achieving high precision detection by phase estimation. The phase sensitivity of a traditional interferometer is subject to the standard quantum limit, while its resolution is constrained by the Rayleigh diffraction limit. The resolution and sensitivity of phase measurement can be enhanced by using quantum metrology. We propose a quantum interference metrology scheme using the entangled squeezed vacuum state, which is obtained using the magic beam splitter, expressed as
Since laser was invented, interference metrology has been extensively applied in the field of precision measurement. However, phase sensitivity is restricted by the standard quantum limit.[1] Simultaneously, the resolution is similarly constrained by the Rayleigh diffraction limit. Quantum metrology can break through the limit, thereby becoming the focus of research. In the study of Lloyd et al., it was shown that by exploiting the quantum states of light, it is probable to overcome the Rayleigh diffraction limit[2,3] (super-resolution) and the standard quantum limit[4,5] (super-sensitivity). Briefly, a series of quantum interference metrology schemes have been proposed.[6–8] Among them, the performance of the N00N state and squeezed vacuum state are relatively outstanding. The research team of Louisiana State University indicated that the sensitivity of the system could be increased from
The ESV state interference measurement device is shown in Fig.
First, the zero–nonzero photon counting at the output port A is analyzed. The counting data {a, b} are grouped into binary outcomes: 0 for a = 0 and ∅ for a ≠ 0.[15,16] We expand the state |ψ2⟩ as
The probability of detecting zero photon on port A is
The reason for this result is that the power of the imaginary number i produces a circle every four times (in = in + 4). The photon probability distribution for a squeezed vacuum state is oscillatory, thereby vanishing for all odd photon numbers. Therefore, the expression of the signal can be written as
However, this is an infinite series summation. Therefore, it is difficult to obtain an analytical solution. We make some rules to truncate the sum. In other words, small probability series items are eliminated. When the total probability of the first m items is more than 99% or the probability of mth is less than 0.1%, the sum operation is stopped. This is analogous to the theory of “random walk.”[17]
To clearly observe the behavior of the signal resolution for this strategy, in Fig.
This result can be perceived, because the counting data of Z detection are grouped into binary outcomes. When the value of 2m is larger, the binomial coefficient of P2m is smaller. That is, the possibility of zero photon appearing on port A is smaller. Therefore, the resolution is poor with the increase of the photon numbers.
According to the definition
We determine that phase sensitivity cannot reach the standard quantum limit, and hence cannot achieve super-sensitivity.
Project detection is proposed by the Louisiana State University research team.[19] Then, this type of method is widely used in the field of quantum metrology. We expand the state |ψ2⟩ as
The phase φ can be determined by measuring the operator
By using the error propagation, the phase sensitivity is given by
The denominator
This strategy selects only one projector, |Am|2 is small; hence,
Parity detection was originally discussed in the context of trapped ions by Bollinger et al.[21] and later adopted for optical interferometry by Gerry and Campos. There are several equivalent approaches of calculating the phase sensitivity.[22–24] The parity operator on output mode A is
Using the conclusions of the N00N state for parity detection,[25] the output signal of the system is
We also use the previously mentioned truncation method to obtain the output signal of the ESV state scheme. The results of the ESV state scheme are compared with the ones of the coherent state scheme when r = 0.9, 1.2, and 1.5 (see Fig.
Here, we discuss two limit problems. The first is the threshold of the signal when achieving super-resolution. With the help of simulation analysis, we found that r = 0.6 is the critical value of our scheme for achieving super-resolution. We can observe from Fig.
The second problem is the maximum visibility of the signal. As shown in Fig.
Particularly, as shown in Fig.
The phase sensitivity can be estimated using the error propagation formula
In Fig.
Our results are also verified for large squeezing parameters through simulation. For example, r = 4, ΔφSQL (4) = 0.03664 and ΔφESV(4 ) = 0.01538. In other words, in the case of large squeezing parameter, the ESV state scheme can still breakthrough the standard quantum limit and achieve super-sensitivity.
Overall, the ESV state scheme can achieve super-resolution and super-sensitivity, the super-resolution range is r > 0.4, the range of visibility reaches 1 when r > 6, and the super-sensitivity is established for r ≥ 1.4.
We present a quantum interference metrology scheme for the ESV state. Moreover, we derive the phase sensitivity and the resolution with Z detection, project detection, and parity detection. The final results reveal that Z detection cannot achieve super-sensitivity and super-resolution. This is because Z detection is to judge whether the output photon number is zero or nonzero. Project detection cannot achieve super-sensitivity, but it can achieve super-resolution; the disadvantage is the visibility is low. The reason for this phenomenon is that project detection selects only one projector. Moreover, parity detection can achieve super-sensitivity and super-resolution. This suggests that parity detection is more suitable for the ESV state scheme. The super-resolution range is achieved when r > 0.4. With the increase of squeezing parameter, the peak of the signal will become narrow, and the visibility of the signal will be enhanced. Therefore, the resolution will be promoted, and the range of visibility reaches 1 when r > 6. Super-sensitivity is established for r ≥ 1.4. When the squeezing parameter increases, the sensitivity of the system will be enhanced. In addition, the squeezing parameter r = 1.45 that corresponds to a squeezing of 12.6 dB was achieved under current experimental conditions.[26] This suggests that super-resolution and super-sensitivity will be obtained. With further perfect experimental conditions, our scheme will achieve better visibility of signal and sensitivity.
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