Abstract

We investigate the performance of the super-resolving quantum lidar with the entangled coherent states of light in the presence of loss and noise, especially in the noisy case. An exact analytical expression of the output signal has been derived with the binary-outcome photon counting measurements. Numerical results show that the resolution of our scheme with parity detection is √N (N) times enhanced relative to that of the coherent-state strategy with the same (intensity) detection in the lossless and noiseless cases. The influences of phase diffusion on resolution and sensitivity have been analyzed and discussed. It is found that the super-resolution emerges in the whole diffusion rate regions, whereas the super-sensitivity just exists in the high and low diffusion rate regimes. Comparisons are made with the well known N00N states, the results show that the entangled coherent states performs better resolution and sensitivity than those of the N00N scheme in the whole diffusion regimes. In addition, the effects of photon loss on resolution and sensitivity have also been studied. The phase sensitivity can beat the shot noise limit and the resolution is much better than the Rayleigh diffraction limit in the whole loss regions. Finally, the zero-nonzero photon counting measurement gives much worse sensitivity than that of the parity detection, which is just opposite from the case as demonstrated in a recent coherent-light Mach-Zehnder experiment.

© 2016 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  31. L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22(10), 11945–11953 (2014).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
  42. J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
    [Crossref] [PubMed]
  43. C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
    [Crossref]
  44. J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
    [Crossref]
  45. J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
    [Crossref] [PubMed]
  46. A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312(5770), 83–86 (2006).
    [Crossref] [PubMed]
  47. A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448(7155), 784–786 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]

2016 (1)

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

2015 (1)

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

2014 (3)

X. Jing, J. Liu, W. Zhong, and X. Wang, “Quantum Fisher information of entangled coherent states in a Lossy Mach-Zehnder interferometer,” Commum. Theor. Phys. 61(1), 115–120 (2014).
[Crossref]

L. Cohen, D. Istrati, L. Dovrat, and H. S. Eisenberg, “Super-resolved phase measurements at the shot noise limit by parity measurement,” Opt. Express 22(10), 11945–11953 (2014).
[Crossref] [PubMed]

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

2013 (5)

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111(3), 033603 (2013).
[Crossref] [PubMed]

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: Coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114(19), 193102 (2013).
[Crossref]

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88(4), 043832 (2013).
[Crossref]

W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, “Fisher information under decoherence in Bloch representation,” Phys. Rev. A 87(2), 022337 (2013).
[Crossref]

B. Roy Bardhan, K. Jiang, and J. P. Dowling, “Effects of phase fluctuations on phase sensitivity and visibility of path-entangled photon Fock states,” Phys. Rev. A 88(2), 023857 (2013).
[Crossref]

2012 (6)

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
[Crossref]

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, “Quantum metrological limits via a variational approach,” Phys. Rev. Lett. 109(19), 190404 (2012).
[Crossref] [PubMed]

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

M. Jarzyna and R. Demkowicz-Dobrzanski, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
[Crossref]

J. J. Cooper, D. W. Hallwood, J. A. Dunningham, and J. Brand, “Robust quantum enhanced phase estimation in a multimode interferometer [corrected],” Phys. Rev. Lett. 108(13), 130402 (2012).
[Crossref] [PubMed]

2011 (5)

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
[Crossref]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7(5), 406–411 (2011).
[Crossref]

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106(15), 153603 (2011).
[Crossref] [PubMed]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

2010 (6)

Z. Dutton, J. H. Shapiro, and S. Guha, “LADAR resolution improvement using receivers enhanced with squeezed-vacuum injection and phase-sensitive amplification,” J. Opt. Soc. Am. B 27(6), A63–A72 (2010).
[Crossref]

B. Teklu, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Phase estimation in the presence of phase diffusion: the qubit case,” Phys. Scr. T140, 014062 (2010).
[Crossref]

Y. C. Liu, G. R. Jin, and L. You, “Quantum-limited metrology in the presence of collisional dephasing,” Phys. Rev. A 82(4), 045601 (2010).
[Crossref]

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27(6), A170–A174 (2010).
[Crossref]

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81(4), 043624 (2010).
[Crossref]

2009 (4)

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

A. Al-Qasimi and D. F. V. James, “Nonexistence of entanglement sudden death in dephasing of high NOON states,” Opt. Lett. 34(3), 268–270 (2009).
[Crossref] [PubMed]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

2008 (4)

G. Gilbert, M. Hamrick, and Y. S. Weinstein, “Use of maximally entangled N-photon states for practical quantum interferometry,” J. Opt. Soc. Am. B 25(8), 1336–1340 (2008).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78(6), 063828 (2008).
[Crossref]

J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
[Crossref]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

2007 (2)

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448(7155), 784–786 (2007).
[Crossref] [PubMed]

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75(5), 053805 (2007).
[Crossref]

2006 (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312(5770), 83–86 (2006).
[Crossref] [PubMed]

2005 (1)

S. J. van Enk, “Decoherence of multidimensional entangled coherent states,” Phys. Rev. A 72(2), 022308 (2005).
[Crossref]

2004 (1)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

2000 (2)

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000).
[Crossref] [PubMed]

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
[Crossref]

1996 (1)

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref] [PubMed]

1995 (1)

B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75(16), 2944–2947 (1995).
[Crossref] [PubMed]

1989 (1)

B. C. Sanders, “Quantum dynamics of the nonlinear rotator and the effects of continual spin measurement,” Phys. Rev. A 40(5), 2417–2427 (1989).
[Crossref] [PubMed]

1981 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D Part. Fields 23(8), 1693–1708 (1981).
[Crossref]

Abrams, D. S.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000).
[Crossref] [PubMed]

Al-Qasimi, A.

Andersen, U. L.

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111(3), 033603 (2013).
[Crossref] [PubMed]

Anisimov, P. M.

Banaszek, K.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

Bollinger, J. J.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref] [PubMed]

Boto, A. N.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000).
[Crossref] [PubMed]

Brand, J.

J. J. Cooper, D. W. Hallwood, J. A. Dunningham, and J. Brand, “Robust quantum enhanced phase estimation in a multimode interferometer [corrected],” Phys. Rev. Lett. 108(13), 130402 (2012).
[Crossref] [PubMed]

Braunstein, S. L.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000).
[Crossref] [PubMed]

Brivio, D.

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D Part. Fields 23(8), 1693–1708 (1981).
[Crossref]

Cialdi, S.

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

Cipriani, D.

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

Cohen, L.

Cooper, J. J.

J. J. Cooper, D. W. Hallwood, J. A. Dunningham, and J. Brand, “Robust quantum enhanced phase estimation in a multimode interferometer [corrected],” Phys. Rev. Lett. 108(13), 130402 (2012).
[Crossref] [PubMed]

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81(4), 043624 (2010).
[Crossref]

Davidovich, L.

B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, “Quantum metrological limits via a variational approach,” Phys. Rev. Lett. 109(19), 190404 (2012).
[Crossref] [PubMed]

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7(5), 406–411 (2011).
[Crossref]

de Matos Filho, R. L.

B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, “Quantum metrological limits via a variational approach,” Phys. Rev. Lett. 109(19), 190404 (2012).
[Crossref] [PubMed]

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7(5), 406–411 (2011).
[Crossref]

Demkowicz-Dobrzanski, R.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

M. Jarzyna and R. Demkowicz-Dobrzanski, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
[Crossref]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

Distante, E.

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111(3), 033603 (2013).
[Crossref] [PubMed]

Dorner, U.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

Dovrat, L.

Dowling, J. P.

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: Coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114(19), 193102 (2013).
[Crossref]

B. Roy Bardhan, K. Jiang, and J. P. Dowling, “Effects of phase fluctuations on phase sensitivity and visibility of path-entangled photon Fock states,” Phys. Rev. A 88(2), 023857 (2013).
[Crossref]

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27(6), A170–A174 (2010).
[Crossref]

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
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J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78(6), 063828 (2008).
[Crossref]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000).
[Crossref] [PubMed]

Dunningham, J. A.

J. J. Cooper, D. W. Hallwood, J. A. Dunningham, and J. Brand, “Robust quantum enhanced phase estimation in a multimode interferometer [corrected],” Phys. Rev. Lett. 108(13), 130402 (2012).
[Crossref] [PubMed]

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81(4), 043624 (2010).
[Crossref]

Durkin, G. A.

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
[Crossref]

Dutton, Z.

Eisenberg, H. S.

Escher, B. M.

B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, “Quantum metrological limits via a variational approach,” Phys. Rev. Lett. 109(19), 190404 (2012).
[Crossref] [PubMed]

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7(5), 406–411 (2011).
[Crossref]

Feng, X.

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

Furusawa, A.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

Gao, Y.

Gebrehiwot, B. T.

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

Genoni, M. G.

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106(15), 153603 (2011).
[Crossref] [PubMed]

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

B. Teklu, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Phase estimation in the presence of phase diffusion: the qubit case,” Phys. Scr. T140, 014062 (2010).
[Crossref]

Gerry, C. C.

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: Coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114(19), 193102 (2013).
[Crossref]

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
[Crossref]

Gilbert, G.

Giovannetti, V.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

Grangier, P.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448(7155), 784–786 (2007).
[Crossref] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312(5770), 83–86 (2006).
[Crossref] [PubMed]

Guha, S.

Guta, M.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

Hallwood, D. W.

J. J. Cooper, D. W. Hallwood, J. A. Dunningham, and J. Brand, “Robust quantum enhanced phase estimation in a multimode interferometer [corrected],” Phys. Rev. Lett. 108(13), 130402 (2012).
[Crossref] [PubMed]

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81(4), 043624 (2010).
[Crossref]

Hamrick, M.

Hao, L.

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

Hayasaka, K.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

Heinzen, D. J.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
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Huver, S. D.

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78(6), 063828 (2008).
[Crossref]

Istrati, D.

Itano, W. M.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref] [PubMed]

James, D. F. V.

Jarzyna, M.

M. Jarzyna and R. Demkowicz-Dobrzanski, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
[Crossref]

Jeong, H.

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
[Crossref]

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448(7155), 784–786 (2007).
[Crossref] [PubMed]

Ježek, M.

E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111(3), 033603 (2013).
[Crossref] [PubMed]

Jiang, K.

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: Coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114(19), 193102 (2013).
[Crossref]

B. Roy Bardhan, K. Jiang, and J. P. Dowling, “Effects of phase fluctuations on phase sensitivity and visibility of path-entangled photon Fock states,” Phys. Rev. A 88(2), 023857 (2013).
[Crossref]

Jin, G.

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

Jin, G. R.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88(4), 043832 (2013).
[Crossref]

Y. C. Liu, G. R. Jin, and L. You, “Quantum-limited metrology in the presence of collisional dephasing,” Phys. Rev. A 82(4), 045601 (2010).
[Crossref]

Jing, X.

X. Jing, J. Liu, W. Zhong, and X. Wang, “Quantum Fisher information of entangled coherent states in a Lossy Mach-Zehnder interferometer,” Commum. Theor. Phys. 61(1), 115–120 (2014).
[Crossref]

Joo, J.

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
[Crossref]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

Kaplan, L.

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

Kaushik, S.

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75(5), 053805 (2007).
[Crossref]

Knysh, S.

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
[Crossref]

Kok, P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000).
[Crossref] [PubMed]

Kolodynski, J.

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
[Crossref] [PubMed]

Laurat, J.

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312(5770), 83–86 (2006).
[Crossref] [PubMed]

Lee, H.

K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: Coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114(19), 193102 (2013).
[Crossref]

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27(6), A170–A174 (2010).
[Crossref]

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

Lee, T. W.

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

Li, X. W.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88(4), 043832 (2013).
[Crossref]

Liu, J.

X. Jing, J. Liu, W. Zhong, and X. Wang, “Quantum Fisher information of entangled coherent states in a Lossy Mach-Zehnder interferometer,” Commum. Theor. Phys. 61(1), 115–120 (2014).
[Crossref]

Liu, Y. C.

Y. C. Liu, G. R. Jin, and L. You, “Quantum-limited metrology in the presence of collisional dephasing,” Phys. Rev. A 82(4), 045601 (2010).
[Crossref]

Lloyd, S.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

Luine, J.

Lundeen, J. S.

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

Ma, J.

W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, “Fisher information under decoherence in Bloch representation,” Phys. Rev. A 87(2), 022337 (2013).
[Crossref]

Maccone, L.

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
[Crossref] [PubMed]

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

McCracken, S. B.

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

Milburn, G. J.

B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75(16), 2944–2947 (1995).
[Crossref] [PubMed]

Min, C. J.

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

Munro, W. J.

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
[Crossref]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

Nemoto, K.

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
[Crossref]

Nori, F.

W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, “Fisher information under decoherence in Bloch representation,” Phys. Rev. A 87(2), 022337 (2013).
[Crossref]

Olivares, S.

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106(15), 153603 (2011).
[Crossref] [PubMed]

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

B. Teklu, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Phase estimation in the presence of phase diffusion: the qubit case,” Phys. Scr. T140, 014062 (2010).
[Crossref]

Ourjoumtsev, A.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448(7155), 784–786 (2007).
[Crossref] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312(5770), 83–86 (2006).
[Crossref] [PubMed]

Paris, M. G. A.

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106(15), 153603 (2011).
[Crossref] [PubMed]

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

B. Teklu, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Phase estimation in the presence of phase diffusion: the qubit case,” Phys. Scr. T140, 014062 (2010).
[Crossref]

Park, K.

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
[Crossref]

Qiao, T.

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

Roy Bardhan, B.

B. Roy Bardhan, K. Jiang, and J. P. Dowling, “Effects of phase fluctuations on phase sensitivity and visibility of path-entangled photon Fock states,” Phys. Rev. A 88(2), 023857 (2013).
[Crossref]

Rubin, M. A.

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75(5), 053805 (2007).
[Crossref]

Sanders, B. C.

B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75(16), 2944–2947 (1995).
[Crossref] [PubMed]

B. C. Sanders, “Quantum dynamics of the nonlinear rotator and the effects of continual spin measurement,” Phys. Rev. A 40(5), 2417–2427 (1989).
[Crossref] [PubMed]

Santamato, A.

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

Sasaki, M.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

Shapiro, J. H.

Smelyanskiy, V. N.

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
[Crossref]

Smith, B. J.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

Spiller, T. P.

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
[Crossref]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

Sun, Z.

W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, “Fisher information under decoherence in Bloch representation,” Phys. Rev. A 87(2), 022337 (2013).
[Crossref]

Suzuki, S.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

Takahashi, H.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

Takeoka, M.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

Teklu, B.

B. Teklu, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Phase estimation in the presence of phase diffusion: the qubit case,” Phys. Scr. T140, 014062 (2010).
[Crossref]

Tualle-Brouri, R.

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448(7155), 784–786 (2007).
[Crossref] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312(5770), 83–86 (2006).
[Crossref] [PubMed]

Uskov, D. B.

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

van Enk, S. J.

S. J. van Enk, “Decoherence of multidimensional entangled coherent states,” Phys. Rev. A 72(2), 022308 (2005).
[Crossref]

Veronis, G.

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

Vezzoli, S.

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

Wakui, K.

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

Walmsley, I. A.

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

Wang, Q.

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

Wang, X.

X. Jing, J. Liu, W. Zhong, and X. Wang, “Quantum Fisher information of entangled coherent states in a Lossy Mach-Zehnder interferometer,” Commum. Theor. Phys. 61(1), 115–120 (2014).
[Crossref]

W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, “Fisher information under decoherence in Bloch representation,” Phys. Rev. A 87(2), 022337 (2013).
[Crossref]

Wasilewski, W.

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

Weinstein, Y. S.

Wildfeuer, C. F.

Y. Gao, P. M. Anisimov, C. F. Wildfeuer, J. Luine, H. Lee, and J. P. Dowling, “Super-resolution at the shot-noise limit with coherent states and photon-number-resolving detectors,” J. Opt. Soc. Am. B 27(6), A170–A174 (2010).
[Crossref]

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78(6), 063828 (2008).
[Crossref]

Williams, C. P.

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000).
[Crossref] [PubMed]

Wineland, D. J.

J. J. Bollinger, W. M. Itano, D. J. Wineland, and D. J. Heinzen, “Optimal frequency measurements with maximally correlated states,” Phys. Rev. A 54(6), R4649–R4652 (1996).
[Crossref] [PubMed]

Xu, L.

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

Xu, Y.

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

Yang, C.

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

Yang, W.

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88(4), 043832 (2013).
[Crossref]

Yang, X.

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

You, L.

Y. C. Liu, G. R. Jin, and L. You, “Quantum-limited metrology in the presence of collisional dephasing,” Phys. Rev. A 82(4), 045601 (2010).
[Crossref]

Zagury, N.

B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, “Quantum metrological limits via a variational approach,” Phys. Rev. Lett. 109(19), 190404 (2012).
[Crossref] [PubMed]

Zhang, Y.

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

Zhang, Y. M.

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88(4), 043832 (2013).
[Crossref]

Zhang, Z.

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

Zhao, Y.

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

Zhong, W.

X. Jing, J. Liu, W. Zhong, and X. Wang, “Quantum Fisher information of entangled coherent states in a Lossy Mach-Zehnder interferometer,” Commum. Theor. Phys. 61(1), 115–120 (2014).
[Crossref]

W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, “Fisher information under decoherence in Bloch representation,” Phys. Rev. A 87(2), 022337 (2013).
[Crossref]

Commum. Theor. Phys. (1)

X. Jing, J. Liu, W. Zhong, and X. Wang, “Quantum Fisher information of entangled coherent states in a Lossy Mach-Zehnder interferometer,” Commum. Theor. Phys. 61(1), 115–120 (2014).
[Crossref]

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J. P. Dowling, “Quantum optical metrology–the lowdown on high-N00N states,” Contemp. Phys. 49(2), 125–143 (2008).
[Crossref]

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K. Jiang, H. Lee, C. C. Gerry, and J. P. Dowling, “Super-resolving quantum radar: Coherent-state sources with homodyne detection suffice to beat the diffraction limit,” J. Appl. Phys. 114(19), 193102 (2013).
[Crossref]

Q. Wang, L. Hao, Y. Zhang, C. Yang, X. Yang, L. Xu, and Y. Zhao, “Optimal detection strategy for super-resolving quantum lidar,” J. Appl. Phys. 119(2), 023109 (2016).
[Crossref]

J. Opt. Soc. Am. B (3)

Nat. Commun. (1)

R. Demkowicz-Dobrzański, J. Kołodyński, and M. Guţă, “The elusive Heisenberg limit in quantum-enhanced metrology,” Nat. Commun. 3, 1063 (2012).
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Nat. Phys. (1)

B. M. Escher, R. L. de Matos Filho, and L. Davidovich, “General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology,” Nat. Phys. 7(5), 406–411 (2011).
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Nature (1)

A. Ourjoumtsev, H. Jeong, R. Tualle-Brouri, and P. Grangier, “Generation of optical ‘Schrödinger cats’ from photon number states,” Nature 448(7155), 784–786 (2007).
[Crossref] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Optik (Stuttg.) (1)

Q. Wang, Y. Zhang, Y. Xu, L. Hao, Z. Zhang, T. Qiao, and Y. Zhao, “Pseudorandom modulation quantum secured lidar,” Optik (Stuttg.) 126(22), 3344–3348 (2015).
[Crossref]

Phys. Rev. A (19)

M. A. Rubin and S. Kaushik, “Loss-induced limits to phase measurement precision with maximally entangled states,” Phys. Rev. A 75(5), 053805 (2007).
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R. Demkowicz-Dobrzanski, U. Dorner, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Quantum phase estimation with lossy interferometers,” Phys. Rev. A 80(1), 013825 (2009).
[Crossref]

S. D. Huver, C. F. Wildfeuer, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78(6), 063828 (2008).
[Crossref]

B. C. Sanders, “Quantum dynamics of the nonlinear rotator and the effects of continual spin measurement,” Phys. Rev. A 40(5), 2417–2427 (1989).
[Crossref] [PubMed]

Y. C. Liu, G. R. Jin, and L. You, “Quantum-limited metrology in the presence of collisional dephasing,” Phys. Rev. A 82(4), 045601 (2010).
[Crossref]

D. Brivio, S. Cialdi, S. Vezzoli, B. T. Gebrehiwot, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Experimental estimation of one-parameter qubit gates in the presence of phase diffusion,” Phys. Rev. A 81(1), 012305 (2010).
[Crossref]

X. Feng, G. Jin, and W. Yang, “Quantum interferometry with binary-outcome measurements in the presence of phase diffusion,” Phys. Rev. A 90(1), 013807 (2014).
[Crossref]

S. J. van Enk, “Decoherence of multidimensional entangled coherent states,” Phys. Rev. A 72(2), 022308 (2005).
[Crossref]

T. W. Lee, S. D. Huver, H. Lee, L. Kaplan, S. B. McCracken, C. J. Min, D. B. Uskov, C. F. Wildfeuer, G. Veronis, and J. P. Dowling, “Optimization of quantum interferometric metrological sensors in the presence of photon loss,” Phys. Rev. A 80(6), 063803 (2009).
[Crossref]

J. J. Cooper, D. W. Hallwood, and J. A. Dunningham, “Entanglement-enhanced atomic gyroscope,” Phys. Rev. A 81(4), 043624 (2010).
[Crossref]

S. Knysh, V. N. Smelyanskiy, and G. A. Durkin, “Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state,” Phys. Rev. A 83(2), 021804 (2011).
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M. Jarzyna and R. Demkowicz-Dobrzanski, “Quantum interferometry with and without an external phase reference,” Phys. Rev. A 85(1), 011801 (2012).
[Crossref]

Y. M. Zhang, X. W. Li, W. Yang, and G. R. Jin, “Quantum Fisher information of entangled coherent states in the presence of photon loss,” Phys. Rev. A 88(4), 043832 (2013).
[Crossref]

M. G. Genoni, S. Olivares, D. Brivio, S. Cialdi, D. Cipriani, A. Santamato, S. Vezzoli, and M. G. A. Paris, “Optical interferometry in the presence of large phase diffusion,” Phys. Rev. A 85(4), 043817 (2012).
[Crossref]

W. Zhong, Z. Sun, J. Ma, X. Wang, and F. Nori, “Fisher information under decoherence in Bloch representation,” Phys. Rev. A 87(2), 022337 (2013).
[Crossref]

B. Roy Bardhan, K. Jiang, and J. P. Dowling, “Effects of phase fluctuations on phase sensitivity and visibility of path-entangled photon Fock states,” Phys. Rev. A 88(2), 023857 (2013).
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[Crossref] [PubMed]

C. C. Gerry, “Heisenberg-limit interferometry with four-wave mixers operating in a nonlinear regime,” Phys. Rev. A 61(4), 043811 (2000).
[Crossref]

J. Joo, K. Park, H. Jeong, W. J. Munro, K. Nemoto, and T. P. Spiller, “Quantum metrology for nonlinear phase shifts with entangled coherent states,” Phys. Rev. A 86(4), 043828 (2012).
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Phys. Rev. Lett. (11)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum metrology,” Phys. Rev. Lett. 96(1), 010401 (2006).
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B. C. Sanders and G. J. Milburn, “Optimal quantum measurements for phase estimation,” Phys. Rev. Lett. 75(16), 2944–2947 (1995).
[Crossref] [PubMed]

A. N. Boto, P. Kok, D. S. Abrams, S. L. Braunstein, C. P. Williams, and J. P. Dowling, “Quantum interferometric optical lithography: exploiting entanglement to beat the diffraction limit,” Phys. Rev. Lett. 85(13), 2733–2736 (2000).
[Crossref] [PubMed]

U. Dorner, R. Demkowicz-Dobrzanski, B. J. Smith, J. S. Lundeen, W. Wasilewski, K. Banaszek, and I. A. Walmsley, “Optimal quantum phase estimation,” Phys. Rev. Lett. 102(4), 040403 (2009).
[Crossref] [PubMed]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

J. J. Cooper, D. W. Hallwood, J. A. Dunningham, and J. Brand, “Robust quantum enhanced phase estimation in a multimode interferometer [corrected],” Phys. Rev. Lett. 108(13), 130402 (2012).
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E. Distante, M. Ježek, and U. L. Andersen, “Deterministic superresolution with coherent states at the shot noise limit,” Phys. Rev. Lett. 111(3), 033603 (2013).
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M. G. Genoni, S. Olivares, and M. G. A. Paris, “Optical phase estimation in the presence of phase diffusion,” Phys. Rev. Lett. 106(15), 153603 (2011).
[Crossref] [PubMed]

J. Joo, W. J. Munro, and T. P. Spiller, “Quantum metrology with entangled coherent states,” Phys. Rev. Lett. 107(8), 083601 (2011).
[Crossref] [PubMed]

B. M. Escher, L. Davidovich, N. Zagury, and R. L. de Matos Filho, “Quantum metrological limits via a variational approach,” Phys. Rev. Lett. 109(19), 190404 (2012).
[Crossref] [PubMed]

H. Takahashi, K. Wakui, S. Suzuki, M. Takeoka, K. Hayasaka, A. Furusawa, and M. Sasaki, “Generation of large-amplitude coherent-state superposition via ancilla-assisted photon subtraction,” Phys. Rev. Lett. 101(23), 233605 (2008).
[Crossref] [PubMed]

Phys. Scr. (1)

B. Teklu, M. G. Genoni, S. Olivares, and M. G. A. Paris, “Phase estimation in the presence of phase diffusion: the qubit case,” Phys. Scr. T140, 014062 (2010).
[Crossref]

Science (2)

V. Giovannetti, S. Lloyd, and L. Maccone, “Quantum-enhanced measurements: beating the standard quantum limit,” Science 306(5700), 1330–1336 (2004).
[Crossref] [PubMed]

A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312(5770), 83–86 (2006).
[Crossref] [PubMed]

Other (1)

M. Lanzagorta, Quantum Radar (Morgan and Claypool, 2012).

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Figures (6)

Fig. 1
Fig. 1

Scheme of a two-mode interferometric quantum lidar.

Fig. 2
Fig. 2

Output signals for binary-outcome parity photon counting measurement with various number of photons N = 9, 16, 25, 36, 49, 64, 81, 100 (blue thin dotted lines). The green solid lines denote the parity signals for the coherent-state scheme and the red dot-dashed lines represent the traditional intensity signals. In the upper left picture, δφ represents the resolution which is the phase difference between the adjacent wave troughs.

Fig. 3
Fig. 3

(a) The resolution and (b) the optimal sensitivity determined by QFI for ECS as a function of the transmittance T with a fixed number of photons N=25 .

Fig. 4
Fig. 4

(a) The output signals of binary-outcome parity photon counting detection with κ=0 (green thick dashed line), κ=5× 10 4 (red solid line), and κ=5× 10 3 (black dot-dashed line) for N=9 . (b) The resolution as a function of the diffusion rate κ for the ECS (red dot-dashed line), the N00N states (black solid line) and the CS (blue dotted line) schemes with the parity detection, and for the ECS scheme (green dashed line) with the Z-detection (which has been specified in Sec. 4).

Fig. 5
Fig. 5

(a) The sensitivity with κ=0 (red solid line), κ= 10 4 (blue dotted line), κ= 10 3 (green dashed line), and κ= 10 2 (black dot-dashed line) for N=9 . (b) The best sensitivity as a function of the diffusion rate κ for the ECS (red dot-dashed line), the N00N (black solid line) and the CS (blue dotted line) schemes with the parity photon counting measurement. The green dashed line in (b) denotes the best sensitivity for the ECS with the Z-detection.

Fig. 6
Fig. 6

The output signal and phase sensitivity of the Z-detection for N=9 under the lossless and noiseless cases. (a) The output signal, (b) the phase sensitivity, the red dot-dashed line: the parity detection, the green dashed line: the Z-detection.

Equations (10)

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ρ ^ out =Tr( M ^ ρ ^ in M ^ )= | N α | 2 { | α a | i α b b i α | a α |+ | i α e iφ a | α e iφ b b α e iφ | a i α e iφ | + e R | α | 2 ( | α a | i α b b α e iφ | a i α e iφ |+H.c. ) },
P( n,m|T,φ )= | N α | 2 { 2 e 2 | α | 2 | α | 2( n+m ) / ( n!m! ) + e | α | 2 [ ( i | α | 2 e iφ ) n ( i | α | 2 e iφ ) m / ( n!m! ) +H.c. ] },
P( n|T,φ )= | N α | 2 { 2 e | α | 2 | α | 2n / n! + e | α | 2 [ e i | α | 2 e iφ ( i | α | 2 e iφ ) n / n! +H.c. ] },
P( ±|T,φ )= n odd or even n P( n|T,φ ) = 1 2 ± | N α | 2 [ e T | α | 2 + e | α | 2 ( 1Tsinφ ) cos( T | α | 2 cosφ ) ],
^ ( T,φ ) =P( + )P( )=2 | N α | 2 [ e T | α | 2 + e | α | 2 ( 1Tsinφ ) cos( T | α | 2 cosφ ) ],
ρ ^ κ,out ( φ )= R { exp[ ( xφ ) 2 / 4κ ] ρ ^ out ( φ ) / 4πκ }dx ,
^ κ = R { exp[ ( xφ ) 2 / ( 4κ ) ] ^ / 4πκ }dx ,
Δ φ = 1 ^ 2 / | ^ / φ | .
P( 0|T,φ )= m P( n=0,m|T,φ ) =2 | N α | 2 [ e T | α | 2 /2 + e | α | 2 ( 1 Tsinφ /2 ) cos( T | α | 2 cosφ /2 ) ],
Δ φ Z = Z ^ Z ^ 2 / | Z ^ / φ | .

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