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The output phase of the Sagnac interferometer has been measured with quantum balanced homodyne technique when coherent light and squeezed vacuum light are fed into the Sagnac interferometer simultaneously [Chen Kun et al., Acta Phys. Sin. 65 054203(2016)]. Nevertheless, there exist two deficiencies: 1) the phase sensitivity is related to the phase itself; 2) there are strict requirements for the phases of local oscillator light, coherent light and squeezed vacuum light. For overcoming these deficiencies, an adaptive optimal measurement scheme is suggested for the phase estimation. Firstly, we calculate that the quantum Fisher information (QFI) of the squeezed vacuum and coherent state is sinh2r+||2e2r by treating them as a quantum pure state, for they satisfy a condition of the quantum pure state, namely ()=()2. The QFI is related to quantum Cramer-Rao lower bound which can be used to evaluate the performance of the estimator. Secondly, we make an analysis of positive operator-valued measure (POVM) and design a set of the optimal measurement operators for reaching the quantum Cramer-Rao lower bound, whereas the optimal measurement operators depend on the true value of the phase which is what we want to estimate. In order to solve the problem and estimate the parameter effectively, an adaptive method is suggested. We set an initial value of the phase parameter to obtain a set of measurement operators which are not optimal at the first step. And then the initial measurement operators are used for POVM and to obtain a conditional probability function, from which we can obtain a new value of the phase with maximum likelihood estimator. Therefore, the measurement operators and conditional probability function will be updated with the new value. As the measurement operators and conditional probability function are updated step by step, we can estimate the value adaptively. In fact, the results of the maximum likelihood estimator will converge at the true value of the phase parameter gradually, which is then proved with the theoretical analysis. All in all, an adaptive measurement method of estimating the phase parameter of the squeezed vacuum and coherent state in Sagnac interferometer is suggested, and is proved theoretically to be that the scheme will converge at the true value of the phase with a probability of 1 and can reach the quantum Cramer-Rao lower bound.
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Keywords:
- quantum balanced homodyne technique /
- positive operator-valued measure /
- quantum Cramer-Rao lower bound /
- maximum likelihood estimator
[1] Bouyer P 2014 Gyroscopy and Navigation 5 20
[2] Shao L Y, Luo Y, Zhang Z Y, Zou X H, Luo B, Pan W, Yan L S 2015 Opt. Commun. 336 73
[3] Joseph S 2014 Gen. Relativ. Gravit. 46 1710
[4] Trevor L C, Samuel D P, Robert J H, Byungmoon C, David M J 2014 Opt. Lett. 39 513
[5] Kiarash Z A, Michel J F D 2015 J. Opt. Soc. Am. B 32 339
[6] John R E T, Christopher P 2014 Appl. Phys. B 114 333
[7] Chen K, Chen S X, Wu D W, Yang C Y, Wu H 2016 Acta Phys. Sin. 65 054203 (in Chinese) [陈坤, 陈树新, 吴德伟, 杨春燕, 吴昊2016 65 054203]
[8] Kuznetsov A G, Molchanov A V, Chirkin M V, Izmailov E A 2015 Quantum Electron. 45 78
[9] Bertocchi G, Alibart O, Ostrowsky D B 2006 J. Phys. B 39 1011
[10] Kolkiran, Agarwal G S 2007 Opt. Express 15 6798
[11] Scully M O, Zubairy M S 1997 Quantum Optics (Cambridge: Cambridge University Press) pp101-106
[12] Li X, Voss P L, Sharping J E, Kumar P 2005 Phys. Rev. Lett. 94 053601
[13] Yurke B, McCall S L, Klauder J R 1986 Phys. Rev. A 33 4033
[14] Alex M 2006 Phys. Rev. A 73 033821
[15] Luca P, Augusto S 2014 arXiv 1411.5164v1
[16] Takanori S 2015 Phys. Rev. A 91 042126
[17] Barndor-Nielsen O E, Gill R D 2000 J. Phys. A: General Phys. 33 4481
[18] Helstrom C W 1976 Quantum Detection and Estimation Theory (New York: Academic Press) pp266-276
[19] Nagaoka H 2005 Asymptotic Theory of Quantum Statistical Inference (Singapore: World Scientific Press) pp125-132
[20] Okamoto R, Minako I, Satoshi O, Koichi Y, Hiroshi I, Fujiwara A, Shigeki T 2012 Phys. Rev. Lett. 109 130404
[21] Fujiwara A 2006 J. Phys. A: Math. Gen. 39 12489
[22] Fujiwara A 2011 J. Phys. A: Math. Theor. 44 079501
[23] Pezze L, Smerzi A 2008 Phys. Rev. Lett. 100 073601
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[1] Bouyer P 2014 Gyroscopy and Navigation 5 20
[2] Shao L Y, Luo Y, Zhang Z Y, Zou X H, Luo B, Pan W, Yan L S 2015 Opt. Commun. 336 73
[3] Joseph S 2014 Gen. Relativ. Gravit. 46 1710
[4] Trevor L C, Samuel D P, Robert J H, Byungmoon C, David M J 2014 Opt. Lett. 39 513
[5] Kiarash Z A, Michel J F D 2015 J. Opt. Soc. Am. B 32 339
[6] John R E T, Christopher P 2014 Appl. Phys. B 114 333
[7] Chen K, Chen S X, Wu D W, Yang C Y, Wu H 2016 Acta Phys. Sin. 65 054203 (in Chinese) [陈坤, 陈树新, 吴德伟, 杨春燕, 吴昊2016 65 054203]
[8] Kuznetsov A G, Molchanov A V, Chirkin M V, Izmailov E A 2015 Quantum Electron. 45 78
[9] Bertocchi G, Alibart O, Ostrowsky D B 2006 J. Phys. B 39 1011
[10] Kolkiran, Agarwal G S 2007 Opt. Express 15 6798
[11] Scully M O, Zubairy M S 1997 Quantum Optics (Cambridge: Cambridge University Press) pp101-106
[12] Li X, Voss P L, Sharping J E, Kumar P 2005 Phys. Rev. Lett. 94 053601
[13] Yurke B, McCall S L, Klauder J R 1986 Phys. Rev. A 33 4033
[14] Alex M 2006 Phys. Rev. A 73 033821
[15] Luca P, Augusto S 2014 arXiv 1411.5164v1
[16] Takanori S 2015 Phys. Rev. A 91 042126
[17] Barndor-Nielsen O E, Gill R D 2000 J. Phys. A: General Phys. 33 4481
[18] Helstrom C W 1976 Quantum Detection and Estimation Theory (New York: Academic Press) pp266-276
[19] Nagaoka H 2005 Asymptotic Theory of Quantum Statistical Inference (Singapore: World Scientific Press) pp125-132
[20] Okamoto R, Minako I, Satoshi O, Koichi Y, Hiroshi I, Fujiwara A, Shigeki T 2012 Phys. Rev. Lett. 109 130404
[21] Fujiwara A 2006 J. Phys. A: Math. Gen. 39 12489
[22] Fujiwara A 2011 J. Phys. A: Math. Theor. 44 079501
[23] Pezze L, Smerzi A 2008 Phys. Rev. Lett. 100 073601
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