It has been demonstrated that using two-mode squeezed vacuum state for phase estimation can break the Heisenberg limit. Our results reveal that the two-mode squeezed vacuum state is also applied to the optical rotation angle measurement. In our scheme, the resolution and sensitivity of the optical rotation angle signal are the same as the case of phase estimation. For the parameter estimation, phase or rotation angle, we discuss the influences of several imperfect factors on the resolution and sensitivity. First, the effect that the upper limit of photon-number resolving has on the maximum amount of available quantum Fisher information has been analyzed. Then, we have also studied the impacts of both the transmission efficiency in the transmission process and the detection efficiency on the detection results. Finally, conditions where all of the above imperfect elements are taken into account at the same time have also been explored. Additionally, other imperfect factors such as squeezing efficiency and dark counts are briefly discussed.
© 2017 Optical Society of America
Quantum metrology [1–3] is the process of employing a quantum system or making use of quantum mechanical properties to estimate several parameters. As an emerging branch of quantum technology, quantum metrology is drawing plenty of attention in the community and showing its potential. The parameter estimation in interferometer has always been a significant problem in quantum metrology. In modern quantum technology, phase coding and polarization coding have been playing extraordinary roles in lots of fields such as quantum communication . Therefore, phase and optical angle are two types of parameters with the most realistic value.
In 1981, Caves pointed out that the classical interferometry could not break through the shot noise limit due to the fact that one of the input ports is a vacuum state, and the vacuum fluctuation limits the measurement accuracy . In order to improve the accuracy of estimation, several non-classical states that can achieve super-resolution and super-sensitivity are theoretically put forward and experimentally prepared [6–9], such as the N00N state and the squeezed state. And it has been studied that the two-mode squeezed vacuum (TMSV) state for phase estimation can break the Heisenberg limit .
In this paper, we demonstrate that the appropriate basis of polarized photons can be utilized to achieve the high-sensitivity estimation of the optical rotation angle under the premise of using the TMSV state. The ultimate estimation result of rotation angle is consistent with that of the phase estimation. This suggests that our scheme is applicable for both phase and optical rotating angle estimation to achieve the same accuracy. On the basis of this scene, we further explore the influence of the finite photon-number resolving on the maximum available quantum Fisher information of the system. We additionally derive an optimal relationship between the average photon number in the incident state and the finite photon-number resolving of the detector. The effects of both transmission efficiency and detection efficiency on estimation accuracy of system are also respectively analyzed. Finally, we simulate and discuss the estimation sensitivity of the system premeditating the above imperfect elements.
2. The principle of system
TMSV state is a dual-mode entangled state with forceful correlation between the two modes. Therefore, it has a major application in the researches of both basic issues in quantum mechanics and the development of the quantum information science and technology. The primary device used in our scheme is a Mach-Zehnder interferometer, which differs from the general phase measurement scheme in that we insert two quarter wave plates at two paths after the first beam splitter (see Fig. 1). The linearly polarized photons passing through the quarter wave plate will be converted to circularly polarized photons, and the right-handed circular polarization may be taken as an example in the following derivation. The two paths in interferometer are respectively placed with phase shifter and optical rotation medium, such as Faraday optical crystal  or Dove prism . The input state can be written as a infinite superposition in two-mode Fock state basis , where |n, n〉 ≡ |n〉a|n〉b, t = N/(N + 2), N = 2sinh2s the average photon number and s is squeezing factor for the input TMSV state. H/V indicates that the mode of the photons is linearly polarized at this time. After the first beam splitter, the TMSV state is naturally decoupled into two single-mode squeezed vacuum states. Note that the mode of photon is circularly polarized at this time. Then two modes go through phase shift and optical rotation, respectively. The phase shift and the optical rotation can be expressed by and under the circularly polarized basis. It is equivalent for circularly polarized photons to withstand optical rotation and phase shift owing to the fact that circularly polarized basis is the eigenstate of the rotational operator. The state after the phase shift and the optical rotation becomes
For the sake of ensuring the high-sensitivity, it is useful to introduce parity detection as detection strategy. Parity detection was originally proposed by Bollinger et al.  and later adopted for interferometry metrology by Gerry et al. [14, 15]. The parity operator at output port A is . So the expectation value of parity detection is
The above derivation used the property of two-mode Fock state  and the summation formula of Legendre polynomial . The details of the beam splitter and other optical components can be found in the work of Yurke et al. .
To clearly observe the behaviors of the signal resolution, in Fig. 2 we plot the curved surface of resolution as functions of φ and θ. Here we fix the average photon number N = 5 in the input state. As shown in Fig. 3, the full width at half maximum (FWHM) of signal is narrower than the curve of the Malus law, that is, our scheme is super-resolution for both phase and rotation angle. According to the definition of visibility , , we calculate the visibility N/(N + 2) by , and Eq. (2). Therefore, the signal visibility can be improved by raising average photon number.
For phase estimation of the TMSV state, it has been verified that parity detection is the optimal strategy by using quantum Fisher information . As for our scheme, the standard deviation of the output signal can be estimated by calculating the error propagation.
The optimal value of above equation shows that the sensitivity of the parameter estimation is preferable to the Heisenberg limit. When the average photon number is large, the above equation is similar to the Heisenberg limit. Additionally, it can be seen that the two parameters have identical consequent in the form of evaluation. Therefore, all conclusions of phase estimation can also be transplanted to our scheme. For the sake of convenience, we uniformly use the parameter δ to indicate phase φ and rotation angle θ in the latter part of discussion.
3. Imperfect elements
The imperfect factor that we first discuss is derived from the finite photon-number resolving of photon counting detector. In the practical measurement process, the photon counting detector must have an upper limit with photon-number resolving, which will inevitably miss part of the measurement information. A copious amount of information in the case of loss will result in the poor measurement or even distortion. For balancing the amount of available information, we need to analyze the quantum Fisher information of the system. Since the input state is an infinite superposition of two-mode Fock state, the amount of the entire quantum Fisher information is equal to the sum of the different two-mode Fock states . The transformation of the Mach-Zehnder interferometer for the estimated parameter can be simplified as
Where is the angular momentum operator of the Schwinger representation, therefore, the quantum Fisher information for pure state can be obtained by the formula [21–23]. Quantum Fisher information of two-mode Fock state is Fn = 2n (n + 1) through the calculation. Further, we can gain the quantum Fisher information of TMSV state FTMSV = N (N + 2). This process depends on the orthogonality of the Fock state vector . In the detection, the upper limit of the photon number will be truncated for this infinity sum, and we simulate the normalized quantum Fisher information of the system with distinct average photon number and upper limit of photon-number resolving. We can find that it can obtain more than 99% of the ideal Fisher information when the maximum photon-number resolving of detector is 6 times greater than average photon number in the input state, the detection signal can restore the real detection information at this time. This conclusion also has guiding and referential significance for numerical simulation.
Transmission efficiency is also a major imperfect factor for the detection results, the transmission efficiency in actual detection is affected by many elements, like the transmittance of optical devices, the absorption of photons in the environment and so on. The expectation value of parity signal can be offered by calculating the Wigner function of the output state. The Wigner function for the two-mode squeezed state is given by
Where the parameter κ is complex phase factor of squeezing parameter ξ, and ξ is generally a complex number to define the two-mode squeezed operator . The modulus of ξ is squeezing factor s (ξ = seiκ). Usually κ = 0 is chosen for simplicity. α and β are parameters characterizing two modes, respectively.
It is approximately equal for the optical path of two modes in the practical application of the interferometry metrology. So we presume that the two paths share the same transmission coefficient T, by making the substitutes
It can be found that the visibility of the signal expectation is still N/(N + 2), that is, the transmission efficiency does not affect the visibility of the system. In accordance with the error propagation, the sensitivity can be written as
It is liable to check that Eq. (3) is equivalent to Eq. (8) when T = 1, and we conduct a simulation analysis with fixing N = 5 and T = 0.8, 0.9, 1, the results are demonstrated in Fig. 4(a). In the case of transmission efficiency less than 1, large transmission efficiency can still break through the shot noise limit and achieve super-sensitivity. We also investigate the relationship between the average photon number and the minimum transmission efficiency under the premise of achieving super-sensitivity. As can be seen from Fig. 4(b), smaller minimum transmission efficiency is required to achieve super-sensitivity when the average photon number is larger. In other words, it is more robust to resist losses for large average photon number. In particular, when N = 680, 10% of transmission efficiency is required to achieve the shot noise limit, from another point of view, at this time the system can tolerate 90% of the loss. This phenomenon is not bothersome to understand from the physical level. Because of the amount of information required for the shot noise limit is proportional to the N, and the Heisenberg limit information is proportional to N2. Transmission efficiency T is a linear coefficient, the greater the N, the smaller the T that satisfies N2T greater than N, super-sensitivity can be achieved under this circumstance. Of course, here the relationship (N2T greater than the N) is a qualitative analysis based on the magnitude, not a quantitative calculation.
Detection efficiency of the detector is also an extremely critical defect ingredient. For the parity detection, the implementation of the detector is more demanding. Superconducting detectors can only achieve the detection of a few photons, which are obviously not conducive for practical applications. Therefore, most of the experiments to achieve the parity detection are based on the APD array [25–28], APD is a typical binary detector, which only response to the the presence of photons, even if more than one photon reach the APD can only count once a time. So the core idea of the APD array detector is to allocate an average photon number less than one in each APD by fiber distribution as possible.
This scheme takes advantage of readily available assemblies such as APD and time-multiplexed device . In this kind of proposal, the importing photons are splitted into numerous spatially or temporally separate bins to make the presence of more than one photon per bin impossible. Subsequently, each bin is detected with single APD and the resolution results of photon number are obtained by inversely reckoning the number of outcomes from all of the bins.
This type of photon-number-resolving detector is characterized by positive operate valued measure (POVM) formalism  in which all photon-counting operations equivalent to POVM elements .
Where λn,m = [CW · L (η)]n,m, n, m and W denote the detection pattern, photon number and the number of bins involved in time-multiplexed process, respectively. The other details about working principle of the detector can refer to the work of Achilles et al. , where CW and L (η) represent weight and detection efficiency of detector individually. According to the above analysis, we simulate the influence of the detection efficiency on the detection results in the case of transmission efficiency is 1. Figures 5(a) and 5(b) are the results of resolution and sensitivity, respectively.
It can be viewed in Fig. 5(a) that the visibility of the detection signal increases with the increase of the detection efficiency. But when the detection efficiency is 1, the maximum detection expectation value cannot reach 1 due to the loss of optical fiber in the process of photon assignation. And this loss is inherent in the detection model. This also affects the sensitivity curve to an enormous extent. Moreover, through the calculation we can note that the effect of detection efficiency on the signal FWHM is extremely asthenic. This reveals that the super-resolution characteristic is favorable maintained.
As for the sensitivity, the results in Fig. 5(b) display that the optimal sensitivity will deteriorate as the detection efficiency decreases. The above result is built on the average photon number N = 5. It will increase the visibility and sensitivity if increasing the average photon number, however, the change trend with the detection efficiency will not change.
The squeezing efficiency of preparation stage is also an important imperfect factor. The cause of imperfect squeezing efficiency has diverse possibilities. For simplicity, an analytical scenario is to regard the input state as a mixed state containing TMVS state and minor coherent state (minor pump light that is not converted and filtered for some reasons). A well-known conclusion is that the estimation sensitivity of the coherent state is limited by the shot noise limit. Therefore, the mixture of coherent state pulls down the overall estimation sensitivity. This also means that the sensitivity decreases as the squeezing efficiency decreases. As for the resolution, the peak position of coherent state and that of TMSV state are respectively 0 and ±π/2 in the case of parity detection. Hence, the impact for visibility is not significant except for a bulge appears between the two signal peaks.
The numerical simulations that take all imperfect elements into account are discussed. We still fix the N =5 and the resolution limit of detector is 6N, which is to ensure the adequate amount of information. The simulation results are individually illustrated in Figs. 6(a) and 6(b). Figure 6(a) suggests that the effect of the two kinds of efficiencies on the visibility of the signal resolution. It is not exertive to find that the resolution is not affected by the transmission efficiency, which confirms our previous conclusion. In addition, the decrease in detection efficiency brings about resolution visibility decline. As to the sensitivity, Fig. 6(b) shows that the optimum sensitivity deteriorates with either the transmission efficiency or the detection efficiency. In contrast, this diversification is more conspicuous as the detection efficiency changes. The above results reveal that the entire detection upshot is more affected by the detection efficiency of the detector.
In addition, response time delay and dark counts of detector are momentous imperfect factors. Suppose that the rate of dark counts is r, the probability of m dark counts follows the Poisson distribution . The parity is estimated just from the parity of the photon number at output port. Ideally, the sum of the photon number detected by two detectors is always even due to the property of TMSV state and the probability of even counts is . Therefore, the count is invalidated if the total number of detected photon is odd. The probability of even counts under such a protocol can be rewritten as
Where DA/B (e/o) indicates that the probability of even or odd dark counts generated by detector A or detector B. The situation, odd dark counts of one detector and even dark counts of the other detector, is invalid owing to violating the protocol. Similarly, we obtain . The expectation value of the output signal existing dark counts is
An intuitive phenomenon is that output signal remains the visibility of N/N + 2. The system sensitivity can be calculated and we plot Fig. 7 for observing the effect of dark counts on the system sensitivity. Under the current technology, the range of r is generally between 10−8 to 10−3. The response time delay forces the sampling detection gates to increase the width and introduces added dark counts. The role of the response time delay can be seen as a negative impact, which is to raise the dark counts. Take N=5 as an example with r = 10−2 (simulating the case of taking into account the response time delay and dark counts) and r = 10−3 (simulating the situation of only considering dark counts). It can be clearly observed that the effects of both response time delay and dark counts on sensitivity are slight and the sensitivity is still lower than the Heisenberg limit.
In general, response time delay and dark counts do not interfere the visibility of output signal, while the property of super-resolution is not changed. A relaxing phenomenon is that the influences of two imperfect factors on sensitivity are also feeble under the circumstance of actual detector parameter.
In conclusion, we manifest a quantum metrology scheme of rotating angle measurement based on the TMSV state. It can be used to estimate the rotation angle and obtain the same resolution and sensitivity compared to the phase estimation. On the basis of this theory, we also discuss the relationship between the upper limit Nup of photon-number resolving and the average photon number N in the incident state from the perspective of quantum information theory. The results indicate that the detector can obtain 99% of information about the estimate parameter when the condition Nup ≥ 6N is satisfied. Subsequently, the effects of transmission efficiency on sensitivity and resolution are also discussed. The outcomes demonstrate that the transmission efficiency will not affect the visibility of the system, but the sensitivity will decrease as the transmission efficiency decreases. In addition, increasing the average photon number will increase the ability of the system to achieve super-sensitivity in the presence of transmission loss. The effect of the detection efficiency is also considered, and decrease in detection efficiency will result in reduction in resolution and sensitivity. Additionally, all the imperfections are analyzed synthetically. Numerical results establish that only the detection efficiency has an influence on the system resolution, while the transmission efficiency and detection efficiency will contribute to the deterioration of the system sensitivity. The influence of the detection efficiency on the sensitivity is more serious. Finally, the impacts of multitudinous imperfect elements on the detection results have been studied, and the effect of response time delay and that of dark counts are inconspicuous in accordance with theoretical analysis. Overall, the function of TMSV state has also been expanded, which is of realistic connotation to the application of TMSV state.
The Scientific Research Fund of Heilongjiang Provincial Education Department (12541839).
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