Abstract

Photon counting measurements are analyzed for obtaining a classical phase parameter in a linear Mach–Zehnder interferometer (MZI) by the use of phase estimation theories. The detailed analysis is made for four cases: (a) coherent states inserted into the interferometer, (b) Fock number state inserted in one input port of the interferometer and the vacuum into the other input port, (c) coherent state inserted into one input port of the interferometer and squeezed-vacuum state into the other input port, and (d) exchanging the first beam splitter of an MZI by a nonlinear system that inserts a NOON (representing a superposition of N particles in the first mode with zero particles in the second mode, and vice versa) state into the interferometer and by using photon counting for parity measurements. The properties of photon counting for obtaining minimal phase uncertainties for the above special cases and for the general case are discussed.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  32. G. Haack, H. Forster, and M. Buttiker, “Parity detection and entanglement with a Mach–Zehnder interferometer,” Phys. Rev. B 82, 155303 (2010).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  39. Y. Ben-Aryeh, “Pancharatnam and Berry phases in three-level photonic systems,” J. Mod. Opt. 50, 2791–2805 (2003).
    [CrossRef]
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    [CrossRef]
  41. P. Kok, H. Lee, and J. P. Dowling, “Creation of large photon-number pathentanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
    [CrossRef]
  42. C. M. Caves, “Quantum-mechanical noise in interferometer,” Phys. Rev. D 23, 1693–1708 (1981).
    [CrossRef]
  43. O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B 4, 49–56 (2002).
    [CrossRef]
  44. O. Assaf and Y. Ben-Aryeh, “Reduction of quantum noise in the Michelson interferometer by the use of squeezed vacuum states,” J. Opt. Soc. Am. B 19, 2716–2721 (2002).
    [CrossRef]
  45. R. Barak and Y. Ben-Aryeh, “Photon statistics and entanglement in coherent-squeezed linear Mach–Zehnder and Michelson interferometers,” J. Opt. Soc. Am. B 25, 361–372 (2008), includes list of references.
    [CrossRef]
  46. Y. Ben-Aryeh, “The use of balanced homodyne and squeezed states for detecting weak optical signals in a Michelson interferometer,” Phys. Lett. A 375, 1300–1303 (2011).
    [CrossRef]
  47. J. P. Dowling, “Quantum optical metrology—the lowdown on high-NOON states,” Contemp. Phys. 49, 125–143 (2008).
    [CrossRef]
  48. S. D. Huver, C. F. Wildfeur, and J. P. Dowling, “Entangled Fock states for robust quantum optical metrology, imaging, and sensing,” Phys. Rev. A 78, 063828 (2008).
    [CrossRef]
  49. I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
    [CrossRef]
  50. M. A. Rubin and S. Kaushik, “Loss-induced limit to phase measurement precision with maximally entangled states,” Phys. Rev. A 75, 053805 (2007).
    [CrossRef]
  51. A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
    [CrossRef]
  52. R. Barak and Y. Ben-Aryeh, “Non-orthogonal positive operator valued measure phase distributions of one- and two-mode electromagnetic fields,” J. Opt. B 7, 123–135 (2005).
    [CrossRef]

2011 (5)

A. Chiruvelli and H. Lee, “Parity measurements in quantum optical metrology,” J. Mod. Opt. 58, 945–953 (2011).
[CrossRef]

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometery with coherent mixed with a squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[CrossRef]

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2011).
[CrossRef]

Y. Ben-Aryeh, “The use of balanced homodyne and squeezed states for detecting weak optical signals in a Michelson interferometer,” Phys. Lett. A 375, 1300–1303 (2011).
[CrossRef]

A. J. Miller, A. E. Lita, B. Calkins, I. Vayshenker, S. M. Gruber, and S. W. Nam, “Compact cryogenic self-aligning fiber-to-detector coupling with losses below one percent,” Opt. Express 19, 9102–9110 (2011).
[CrossRef]

2010 (7)

A. Rivas and A. Luis, “Precision quantum metrology and nonclassicality in linear and nonlinear detection schemes,” Phys. Rev. Lett. 105, 010403 (2010).
[CrossRef]

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optics metrology without number-resolving detectors,” New J. Phys. 12, 113025 (2010).
[CrossRef]

C. C. Gerry and J. Mimih, “Heisenberg-limited interferometry with pair coherent states and parity measurements,” Phys. Rev. A 82, 013831 (2010).
[CrossRef]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef]

C. C. Gerry and J. Mimih, “The parity operator in quantum optical metrology,” Contemp. Phys. 51, 497–511 (2010).
[CrossRef]

G. Haack, H. Forster, and M. Buttiker, “Parity detection and entanglement with a Mach–Zehnder interferometer,” Phys. Rev. B 82, 155303 (2010).
[CrossRef]

2009 (4)

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, 040403 (2009).
[CrossRef]

C. F. Wildfeuer, A. J. Pearlman, J. Chen, J. Fan, A. Migdall, and J. P. Dowling, “Resolution and sensitivity of a Fabry–Perot interferometer with a photon-number-resolving detector,” Phys. Rev. A 80, 043822 (2009).
[CrossRef]

M. Rajteri, E. Taralli, C. Portesi, E. Monticone, and J. Beyer, “Photon-number discriminating superconducting transition-edge sensors,” Metrologia 46, S283–S287 (2009).
[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, 013825 (2009).
[CrossRef]

2008 (5)

C. C. Gerry and T. Bui, “Quantum non-demolition measurement of photon number using weak nonlinearities,” Phys. Lett. A 372, 7101–7104 (2008).
[CrossRef]

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

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

R. Barak and Y. Ben-Aryeh, “Photon statistics and entanglement in coherent-squeezed linear Mach–Zehnder and Michelson interferometers,” J. Opt. Soc. Am. B 25, 361–372 (2008), includes list of references.
[CrossRef]

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, 1336–1340 (2008).
[CrossRef]

2007 (3)

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

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wisman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature Lett. 450, 393–396 (2007).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Parity measurements, Heisenberg-limited phase estimation, and beyond,” J. Mod. Opt. 54, 2177–2184 (2007).
[CrossRef]

2006 (3)

R. A. Campos and C. C. Gerry, “Boson permutation and parity operators: Lie algebra and applications,” Phys. Lett. A 356, 286–289 (2006).
[CrossRef]

G. Khoury, H. S. Eisenberg, E. J. S. Fonseca, and D. Bouwmeester, “Nonlinear interferometry via Fock-state projection,” Phys. Rev. Lett. 96, 203601 (2006).
[CrossRef]

J. Dunningham and T. Kim, “Using quantum interferometers to make measurements at the Heisenberg limit,” J. Mod. Opt. 53, 557–571 (2006).
[CrossRef]

2005 (3)

R. Barak and Y. Ben-Aryeh, “Non-orthogonal positive operator valued measure phase distributions of one- and two-mode electromagnetic fields,” J. Opt. B 7, 123–135 (2005).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72, 053818 (2005).
[CrossRef]

R. A. Campos and C. C. Gerry, “Permutation-parity exchange at a beam splitter: application to Heisenberg limited interferometry,” Phys. Rev. A 72, 065803 (2005).
[CrossRef]

2004 (1)

M. W. Mitchell, J. S. Lundeen, and A. M. Steinberg, “Super-resolving phase measurements with a multiphoton entangled state,” Nature 429, 161–164 (2004).
[CrossRef]

2003 (3)

E. Waks, K. Inoue, W. D. Oliver, E. Diamanti, and Y. Yamamoto, “High-efficiency photon-number detection for quantum information processing,” IEEE J. Sel. Top. Quantum Electron. 9, 1502–1511 (2003).
[CrossRef]

Y. Ben-Aryeh, “Pancharatnam and Berry phases in three-level photonic systems,” J. Mod. Opt. 50, 2791–2805 (2003).
[CrossRef]

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
[CrossRef]

2002 (5)

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource of maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[CrossRef]

H. Lee, P. Kok, and J. P. Dowling, “A quantum Rosetta Stone for interferometry,” J. Mod. Opt. 49, 2325–2338 (2002).
[CrossRef]

O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B 4, 49–56 (2002).
[CrossRef]

P. Kok, H. Lee, and J. P. Dowling, “Creation of large photon-number pathentanglement conditioned on photodetection,” Phys. Rev. A 65, 052104 (2002).
[CrossRef]

O. Assaf and Y. Ben-Aryeh, “Reduction of quantum noise in the Michelson interferometer by the use of squeezed vacuum states,” J. Opt. Soc. Am. B 19, 2716–2721 (2002).
[CrossRef]

2000 (1)

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

1998 (1)

B. Cabrera, R. M. Clarke, P. Colling, A. J. Miller, S. Nam, and R. W. Romani, “Detection of single infrared, optical, and ultraviolet photons using superconducting transition edge sensors,” Appl. Phys. Lett. 73, 735–737 (1998).
[CrossRef]

1997 (1)

Z. Y. Ou, “Fundamental quantum limit in precision phase measurement,” Phys. Rev. A 55, 2598–2609 (1997).
[CrossRef]

1996 (3)

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

S. L. Braunstein, C. M. Caves, and G. J. Milburn, “Generalized uncertainty relations: theory, examples, and Lorentz invariance,” Ann. Phys. 247, 135–173 (1996).
[CrossRef]

C. Brif and Y. Ben-Aryeh, “Improvement of measurement accuracy in SU(1,1) interferometers,” Quantum Semiclass. Opt. 8, 1–5 (1996).
[CrossRef]

1994 (1)

S. L. Braunstein and C. M. Caves, “Statistical distance and the geometry of quantum states,” Phys. Rev. Lett. 72, 3439–3443 (1994).
[CrossRef]

1992 (1)

J. W. Noh, A. Fougeres, and L. Mandel, “Operational approach to the phase of a quantum field,” Phys. Rev. A 45, 424–442 (1992).
[CrossRef]

1991 (1)

J. W. Noh, A. Fougeres, and L. Mandel, “Measurement of the quantum phase by photon counting,” Phys. Rev. Lett. 67, 1426–1429 (1991).
[CrossRef]

1981 (1)

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

1974 (1)

C. W. Helstrom, “Estimation of a displacement parameter of a quantum system,” Int. J. Theor. Phys. 11, 357–378 (1974).
[CrossRef]

Afek, I.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

Agarwal, G. S.

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optics metrology without number-resolving detectors,” New J. Phys. 12, 113025 (2010).
[CrossRef]

Ambar, O.

I. Afek, O. Ambar, and Y. Silberberg, “High-NOON states by mixing quantum and classical light,” Science 328, 879–881 (2010).
[CrossRef]

Anisimov, P. M.

K. P. Seshadreesan, P. M. Anisimov, H. Lee, and J. P. Dowling, “Parity detection achieves the Heisenberg limit in interferometery with coherent mixed with a squeezed vacuum light,” New J. Phys. 13, 083026 (2011).
[CrossRef]

W. N. Plick, P. M. Anisimov, J. P. Dowling, H. Lee, and G. S. Agarwal, “Parity detection in quantum optics metrology without number-resolving detectors,” New J. Phys. 12, 113025 (2010).
[CrossRef]

P. M. Anisimov, G. M. Raterman, A. Chiruvelli, W. N. Plick, S. D. Huver, H. Lee, and J. P. Dowling, “Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit,” Phys. Rev. Lett. 104, 103602 (2010).
[CrossRef]

Assaf, O.

O. Assaf and Y. Ben-Aryeh, “Reduction of quantum noise in the Michelson interferometer by the use of squeezed vacuum states,” J. Opt. Soc. Am. B 19, 2716–2721 (2002).
[CrossRef]

O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B 4, 49–56 (2002).
[CrossRef]

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, 040403 (2009).
[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, 013825 (2009).
[CrossRef]

Barak, R.

R. Barak and Y. Ben-Aryeh, “Photon statistics and entanglement in coherent-squeezed linear Mach–Zehnder and Michelson interferometers,” J. Opt. Soc. Am. B 25, 361–372 (2008), includes list of references.
[CrossRef]

R. Barak and Y. Ben-Aryeh, “Non-orthogonal positive operator valued measure phase distributions of one- and two-mode electromagnetic fields,” J. Opt. B 7, 123–135 (2005).
[CrossRef]

Bartlett, S. D.

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wisman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature Lett. 450, 393–396 (2007).
[CrossRef]

Ben-Aryeh, Y.

Y. Ben-Aryeh, “The use of balanced homodyne and squeezed states for detecting weak optical signals in a Michelson interferometer,” Phys. Lett. A 375, 1300–1303 (2011).
[CrossRef]

R. Barak and Y. Ben-Aryeh, “Photon statistics and entanglement in coherent-squeezed linear Mach–Zehnder and Michelson interferometers,” J. Opt. Soc. Am. B 25, 361–372 (2008), includes list of references.
[CrossRef]

R. Barak and Y. Ben-Aryeh, “Non-orthogonal positive operator valued measure phase distributions of one- and two-mode electromagnetic fields,” J. Opt. B 7, 123–135 (2005).
[CrossRef]

Y. Ben-Aryeh, “Pancharatnam and Berry phases in three-level photonic systems,” J. Mod. Opt. 50, 2791–2805 (2003).
[CrossRef]

O. Assaf and Y. Ben-Aryeh, “Quantum mechanical noise in coherent-state and squeezed-state Michelson interferometers,” J. Opt. B 4, 49–56 (2002).
[CrossRef]

O. Assaf and Y. Ben-Aryeh, “Reduction of quantum noise in the Michelson interferometer by the use of squeezed vacuum states,” J. Opt. Soc. Am. B 19, 2716–2721 (2002).
[CrossRef]

C. Brif and Y. Ben-Aryeh, “Improvement of measurement accuracy in SU(1,1) interferometers,” Quantum Semiclass. Opt. 8, 1–5 (1996).
[CrossRef]

Benmoussa, A.

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Parity measurements, Heisenberg-limited phase estimation, and beyond,” J. Mod. Opt. 54, 2177–2184 (2007).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Quantum nondemolition measurement of parity and generation of parity eigenstates in optical fields,” Phys. Rev. A 72, 053818 (2005).
[CrossRef]

R. A. Campos, C. C. Gerry, and A. Benmoussa, “Optical interferometry at the Heisenberg limit with twin Fock states and parity measurements,” Phys. Rev. A 68, 023810 (2003).
[CrossRef]

C. C. Gerry, A. Benmoussa, and R. A. Campos, “Nonlinear interferometer as a resource of maximally entangled photonic states: application to interferometry,” Phys. Rev. A 66, 013804 (2002).
[CrossRef]

Berry, D. W.

G. Y. Xiang, B. L. Higgins, D. W. Berry, H. M. Wiseman, and G. J. Pryde, “Entanglement-enhanced measurement of a completely unknown optical phase,” Nat. Photonics 5, 43–47 (2011).
[CrossRef]

B. L. Higgins, D. W. Berry, S. D. Bartlett, H. M. Wisman, and G. J. Pryde, “Entanglement-free Heisenberg-limited phase estimation,” Nature Lett. 450, 393–396 (2007).
[CrossRef]

Beyer, J.

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Equations (101)

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J^x=(a^b^+b^a^)/2,J^y=(a^b^b^a^)/2i,J^z=(a^a^b^b^)/2,
J^2=J^x2+J^y2+J^z2=(N^/2)(N^/2+1),
|ψ=U^BS1|ψin,
|ψ=j=0m=jjCj,m|j,m,
|ψ=n(1),n(2)=0Cn(1),n(2)|n(1),n(2)
n(1)+n(2)2=j,n(1)n(2)2=m.
|ψ˜=exp(iJ^zφ)|ψ=j=0m=jjCj,mexp(imφ)|j,m.
|ψout=U^BS2|ψ˜,
O^=ψ|outO^|ψout=ψ˜|U^BS2O^U^BS2|ψ˜=ψ˜|O^in|ψ˜,
O^in=U^BS2O^U^BS2.
δφ=ΔO^|/φO^|,ΔO^=O^2O^2.
O^=ψ˜|O^in|ψ˜,O^2=ψ˜|O^in2|ψ˜.
O^=J^z=(a^a^b^b^)out,O^2=J^z2=(a^a^b^b^)out2,
U^BS2=exp(iπ2J^y).
O^in=J^x,O^in2=J^x2,
O^in=ψ˜|J^x|ψ˜,O^in2=ψ˜|J^x2|ψ˜.
|ψout=l(1),l(2)=0Cl(1),l(2)(out)|l(1),l(2),
l^(1)=(a^a^)out,l^(2)=(b^b^)out.
ψout|P^|ψout=ψout|(1)a^a^|ψout=l(1),l(2)=0|Cl(1),l(2)(out)|2(1)l(1).
dξE^(ξ)=1(unit operator).
p(ξ|X)=Tr(E^(ξ)ρ^(X)).
(δφ)21FQ,
FQ=Tr[ρ^(φ)A^2].
ρ^(φ)φ=12[A^ρ^(φ)+ρ^(φ)A^].
(A)i,j=2pi+pj[ρ^(φ)]i,j,
FQ=4[ψ(φ)|ψ(φ)|ψ(φ)|ψ(φ)|2].
dL2=1|ψ(s)|ψ(s)|2,
(dLds)2=dψ(s)ds|dψ(s)dsdψ(s)ds|ψ(s)ψ(s)|dψ(s)ds.
{dψ(s)ds|dψ(s)ds|ψ(s)ψ(s)|}{|dψ(s)dsψ(s)|dψ(s)ds|ψ(s)}=2(dLds)2,
|ψ=|α1|β2=exp(|α|22)n(1)=0αn(1)n(1)!|n(1exp(|β|22)n(2)=0βn(2)n(2)!|n(2,
a^|α1=α|α1,b^|β2=β|β2
|β=||β|exp(iθ2)|β=||β|exp[i(θ2+φ)],|ψ=|α1|β2|ψ˜=|α1|β2,
O^in=ψ˜|J^x|ψ˜=α|β|a^b^+b^a^2|α|β=α*β+β*α2=|α||β|cos(φ+θ2θ1),
O^in2=ψ˜|J^x2|ψ˜=α|β|(a^b^+b^a^2)2|α|β=α*2β2+β*2α2+2α*αβ*β+α*α+β*β4=|α|2|β|2[cos[2(φ+θ2θ1)]+1]2+|α|2+|β|24=|α|2|β|2cos2[(φ+θ2θ1)]+|α|2+|β|24.
ΔO^=O^in2O^in2=|α|2+|β|24,|φO^in|=|α||β||sin(φ+θ2θ1)|,δφ=ΔO^|/φO^|=|α|2+|β|22|α||β||sin(φ+θ2θ1)|.
|ψ˜=|α1|β2=|α1n(2)=0Cn(2)|n(2),Cn(2)=exp(|β|22)[|β|exp[i(θ2+φ)]n(2)n(2)!,
|ψ˜|ψ˜φ=|α1n(2)=0Cn(2)φ|n(2)=|α1n(2)=0in(2)Cn(2)|n(2).
ψ˜|ψ˜=n(2)=0|Cn(2)|2n(2)2=n(2)2,|ψ˜|ψ˜|2=[n(2)=0|Cn(2)|2n(2)]2=n(2)2.
FQ=4[n(2)2n(2)2]=4n(2)=4|β|2,δφ12|β|.
|ψin=a^inN|0/N!|0a|ψ=(a^+b^)N(2)NN!|0a|0b,
|ψ˜N=(a^+b^eiφ)N(2)NN!|0a|0b,
O^=ψ˜|Na^b^+b^a^2|ψ˜N.
[a^,(a^+b^eiφ)N]|0a|0b=N(a^+b^eiφ)N1|0a|0b,[b^,(a^+b^eiφ)N]|0a|0b=Neiφ(a^+b^eiφ)N1|0a|0b,
O^=0|b0|a(a^+b^eiφ)N1(2)NN!(a^+b^eiφ)N1(2)NN!N2cosφ|0a|0b=ψ˜|ψ˜N1N1Ncosφ2=Ncosφ2.
O^2=ψ˜|J^x2|ψ˜NN=ψ˜|N(a^b^+b^a^2)2|ψ˜N=ψ˜|N(a^2b^2+b^2a^2+2a^b^a^b^+a^a^+b^b^)|ψ˜N.
[a^2,(a^+b^eiφ)N]|0a|0b=N(N1)(a^+b^eiφ)N2|0a|0b,[b^2,(a^+b^eiφ)N]|0a|0b=N(N1)ei2φ(a^+b^eiφ)N2|0a|0b,
O^2=ψ˜|ψ˜N2N2(N(N1)(cos2φ+1)8)+ψ˜|ψ˜N1N1N4=(N(N1)cos2φ+N4).
O^2O^2=N(N1)cos2φ+N4N2cos2φ4=Nsin2φ4.
δφ=O^2O^2|/φ(Ncosφ/2)|=Nsin(φ)/2Nsin(φ)/2=1N,
ψ˜|Nψ˜|Nφ=iN2eiφψ˜|N1b^,|ψ˜N|ψ˜Nφ=ieiφN2b^|ψ˜N1,
b^|ψ˜N=N2eiφ|ψ˜N1,ψ˜|Nb=N2eiφψ˜|N1.
|ψ˜|N|ψ˜N|2=|ψ˜|Nφ|ψ˜N|2=N24,
ψ˜|N|ψ˜Nψ˜|Nφ|ψ˜Nφ=N2ψ˜|N1b^b^|ψ˜N1=N2ψ˜|N1|ψ˜N1+N2ψ˜|N1b^b^|ψ˜N1=N2+N(N1)4=N2+N4.
FQ=4[ψ(φ)|ψ(φ)|ψ(φ)|ψ(φ)|2]=4[N4+N24N24]=N.
(δφ)1N.
(a^+b^)N(a^b^)N=(a^2b^2)N.
|ψ0=D^a(α)|0aS^b(ς)|0b,
D^a(α)=exp(αa^α*a^),S^b(ς)=exp{12(ς*b^2ςb^2)},
ς=riθ(r,θreal).
a^a^+b^2,b^a^b^2,
|ψin=exp[α(a^+b^)2α*(a^+b^)2]exp{12[ς*(a^b^)22ς(a^b^)22]}|0a|0b.
b^b^eiφ,b^b^eiφ,
|ψ˜=exp[α(a^+b^eiφ)2α*(a^+b^eiφ)2]exp{12[ς*(a^b^eiφ)22ς(a^b^eiφ)22]}|0a|0b.
|ψ˜=n(1),n(2)=0Cn(1),n(2)|n(1)a|n(2)b,
c^=a^+b^eiφ2,d^=a^b^eiφ2.
|ψ˜=D^c(α)S^d(ς)|0c|0d,
a^=c^+d^2,b^=eiφ(c^d^2),
J^x=(a^b^+b^a^)/2=cosφ[c^c^d^d^]+isinφ[d^c^c^d^].
D^c1(α)c^D^c(α)=c^+α;D^c1(α)c^D^c(α)=c^+α*,S^d1(ς)d^S^d(ς)=d^cosh(r)d^eiθsinh(r),S^d1(ς)d^S^d(ς)=d^cosh(r)d^eiθsinh(r).
S^d1(ς)D^c1(α)J^xD^c(α)S^d(ς)=cosφ{(c^+α*)(c^+α)(d^cosh(r)d^eiθsinh(r))(d^cosh(r)d^eiθsinh(r))}+isinφ{(d^cosh(r)d^eiθsinh(r))(c^+α)(c^+α*)(d^cosh(r)d^eiθsinh(r))}.
0|d0|cS^d1(ς)D^c1(α)J^xD^c(α)S^d(ς)|0c|0d=cosφ[|α|2sinh2r],
|φJ^x|sinφ=1=|α|2sinh2(r)|α|2.
0|d0|c{S^c1(ς)D^d1(α)J^x2D^d(α)S^c(ς)}|0c|0d|α|2[cosh2(r)+sinh2(r)]+α2sinh(r)cosh(r)eiθ+α*2sinh(r)cosh(r)eiθ.
J^x2|α|2[cosh(r)sinh(r)]2=|α|2e2r.
ΔO^=O^2O^2=[J^x2J^x2]cosφ=0,2fθ=π|α|2e2r,|φJ^x|sinφ=1|α|2.
δφ|α|er|α|2=er|α|=erN.
|ψin=(|Na|0b+|0a|Nb)/2.
|ψin=(|2a|0b+|0a|2b)/2.
|ψin=(|j=1,m=1+|j=1,m=1)/2.
|ψ˜=(|1,1eiφ+|1,1eiφ)/2.
|ψin=(|j,j+|j,j)/2.
|ψ˜=(|j,jeijφ+|j,jeijφ)/2.
ψ|outP^|ψout=ψ˜|U^BS2(1)jJ^zU^BS2|ψ˜,
U^BS2=exp[(iπ/2)J^x],
Q^=U^BS2(1)jJ^zU^BS2=exp[(iπ/2)J^x](1)jJ^zexp[(iπ/2)J^x]=(1)jexp(iπJ^y)=μ,ν=jj(1)jdν,μj(π)|j,νj,μ|,
dν,μj(π)=(1)2νδ(ν,μ),
ψ|outO^2|ψoutψ|outP^2|ψout=ψ˜|Q^2|ψ˜=1.
ψ|outO^|ψoutψ|outP^|ψout=ψ˜|Q^|ψ˜={eijφj,j|+eijφj,j|}2{|j,jj,j|+|j,jj,j|}{eijφ|j,j+eijφ|j,j}2=cos(2jφ).
ΔO^=O^2O^2ψ|outP^2|ψout(ψ|outP^|ψout)2=1cos2(2jφ)=sin(2jφ),
|O^φ|=|ψ|outP^|ψoutφ|=|φ[cos(2jφ)]|=|2jsin(2jφ)|.
δφ=12j=1N,2j=n1+n2=N.
ψ˜(φ)|ψ˜(φ)={ijeijφj,j|+(ij)eijφj,j|}2{ijeijφ|j,j+ijeijφ|j,j}2=j2,
ψ˜(φ)|ψ˜(φ)={ijeijφj,j|+(ij)eijφj,j|}2{eijφ|j,j+eijφ|j,j}2=0.
FQ=4j2,
(δφ)214j2.
(δφ)min=12j=1N,
FQ=4[{j=0m,m=jjCj,m*Cj,m(mm)2}{j=0m,m=jjCj,m*Cj,m(mm)}2].
FQ=4[ψout|J^z2|ψout]ψout|J^z|ψout2].
δφ1FQ=12[ψout|J^z2|ψout]ψout|J^z|ψout2]=12[(mm)2mm2],
δφmin2Δm=1,
δφmin=12j=1n1+n2=1Ntot.

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