Abstract

Mach-Zehnder interferometer is a common device in quantum phase estimation and the photon losses in it are an important issue for achieving a high phase accuracy. Here we thoroughly discuss the precision limit of the phase in the Mach-Zehnder interferometer with a coherent state and a superposition of coherent states as input states. By providing a general analytical expression of quantum Fisher information, the phase-matching condition and optimal initial parity are given. Especially, in the photon loss scenario, the sensitivity behaviors are analyzed and specific strategies are provided to restore the phase accuracies for symmetric and asymmetric losses.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2017 (6)

J. Liu and H. Yuan, “Quantum parameter estimation with optimal control,” Phys. Rev. A 96, 012117 (2017).
[Crossref]

J. Liu and H. Yuan, “Control-enhanced multiparameter quantum estimation,” Phys. Rev. A 96, 042114 (2017).
[Crossref]

H. Yuan and C.-H. F. Fung, “Quantum parameter estimation with general dynamics,” Npj:Quantum Information 3, 14 (2017).

P. Liu, P. Wang, W. Yang, G. R. Jin, and C. P. Sun, “Fisher information of a squeezed-state interferometer with a finite photon-number resolution,” Phys. Rev. A 95, 023824 (2017).
[Crossref]

P. Liu and G. R. Jin, “Ultimate phase estimation in a squeezed-state interferometer using photon counters with a finite number resolution,” J. Phys. A: Math. Theor. 50, 405303 (2017).
[Crossref]

M. Takeoka, K. P. Seshadreesan, C. You, S. Izumi, and J. P. Dowling, “Fundamental precision limit of a Mach-Zehnder interferometric sensor when one of the inputs is the vacuum,” Phys. Rev. A 96, 052118 (2017).
[Crossref]

2016 (7)

J. Liu, J. Chen, X.-X. Jing, and X. Wang, “Quantum Fisher information and symmetric logarithmic derivative via anti-commutators,” J. Phys. A: Math. Theor. 49, 275302 (2016).
[Crossref]

M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
[Crossref]

B. P. Abbott and LIGO Scientific Collaboration and the Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

S. Vitale, “Multiband gravitational-wave astronomy: parameter estimation and tests of general relativity with space-and ground-based detectors,” Phys. Rev. Lett. 117, 051102 (2016).
[Crossref]

J. Luo, L.-S. Chen, H.-Z. Duan, Y.-G. Gong, S. Hu, J. Ji, Q. Liu, J. Mei, V. Milyukov, M. Sazhin, C.-G. Shao, V. T. Toth, H.-B. Tu, Y. Wang, Y. Wang, H.-C. Yeh, M.-S. Zhan, Y. Zhang, V. Zharov, and Z.-B. Zhou, “TianQin: a space-borne gravitational wave detector,” Class. Quant. Grav. 33, 035010 (2016).
[Crossref]

J. Liu, X.-M. Lu, Z. Sun, and X. Wang, “Quantum multiparameter metrology with generalized entangled coherent state,” J. Phys. A: Math. Theor. 49, 115302 (2016).
[Crossref]

H. Yuan, “Sequential feedback scheme outperforms the parallel scheme for Hamiltonian parameter estimation,” Phys. Rev. Lett. 117, 160801 (2016).
[Crossref] [PubMed]

2015 (4)

H. Yuan and C.-H. F. Fung, “Optimal feedback scheme and universal time scaling for Hamiltonian parameter estimation,” Phys. Rev. Lett. 115, 110401 (2015).
[Crossref] [PubMed]

J. Liu, X.-X. Jing, and X. Wang, “Quantum metrology with unitary parametrization processes,” Sci. Rep. 5, 8565 (2015).
[Crossref] [PubMed]

M. Jarzyna and R. Demkowicz-Dobrzanski, “True precision limits in quantum metrology,” New J. Phys. 17, 013010 (2015).
[Crossref]

R. Demkowicz-Dobrzanski, M. Jarzyna, and J. Kolodynski, “Chapter four-Quantum limits in optical interferometry,” Progress in Optics,  60, 345–435 (2015).
[Crossref]

2014 (8)

J. Liu, X. Jing, W. Zhong, and X. Wang, “Quantum Fisher information for density matrices with arbitrary ranks,” Commun. Theor. Phys. 61, 45–50 (2014).
[Crossref]

J. Liu, H.-N. Xiong, F. Song, and X. Wang, “Fidelity susceptibility and quantum Fisher information for density operators with arbitrary ranks,” Physica A 410, 167–173 (2014).
[Crossref]

S. Pang and T. A. Brun, “Quantum metrology for a general Hamiltonian parameter,” Phys. Rev. A 90, 022117 (2014).
[Crossref]

Z. Jiang, “Quantum Fisher information for states in exponential form,” Phys. Rev. A 89, 032128 (2014).
[Crossref]

P. A. Knott, T. J. Proctor, Kae Nemoto, J. A. Dunningham, and W. J. Munro, “Effect of multimode entanglement on lossy optical quantum metrology,” Phys. Rev. A 90, 033846 (2014).
[Crossref]

R. Demkowicz-Dobrzanski and L. Maccone, “Using entanglement against noise in quantum metrology,” Phys. Rev. Lett. 113, 250801 (2014).
[Crossref]

Y. Yao, L. Ge, X. Xiao, X. Wang, and C. P. Sun, “Multiple phase estimation for arbitrary pure states under white noise,” Phys. Rev. A 90, 062113 (2014).
[Crossref]

G. Toth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
[Crossref]

2013 (5)

M. Tsang, “Quantum metrology with open dynamical systems,” New J. Phys. 15, 073005 (2013).
[Crossref]

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

L. Pezze and A. Smerzi, “Ultrasensitive two-mode interferometry with single-mode number squeezing,” Phys. Rev. Lett. 110, 163604 (2013).
[Crossref] [PubMed]

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, 043832 (2013).
[Crossref]

J. Liu, X. Jing, and X. Wang, “Phase-matching condition for enhancement of phase sensitivity in quantum metrology,” Phys. Rev. A 88, 042316 (2013).
[Crossref]

2012 (4)

N. Gkortsilas, J. J. Cooper, and J. A. Dunningham, “Measuring a completely unknown phase with sub-shot-noise precision in the presence of loss,” Phys. Rev. A 85, 063827 (2012).
[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, 043828 (2012).
[Crossref]

R. W. Boyd and J. P. Dowling, “Quantum lithography: Status of the field,” Quantum Inf. Process. 11, 891 (2012).
[Crossref]

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

2011 (3)

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, 406 (2011).
[Crossref]

V. Giovannetti, S. Lloyd, and L. Maccone, “Advances in quantum metrology,” Nat. Photonics 5, 222 (2011).
[Crossref]

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

2010 (3)

R. Schnabel, N. Mavalvala, D. E. McClelland, and P. K. Lam, “Quantum metrology for gravitational wave astronomy,” Nat. Commun. 1, 121 (2010).
[Crossref] [PubMed]

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

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

2009 (2)

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]

M. G. A. Paris, “Quantum estimation for quantum technology,” Int. J. Quantum Inform. 7, 125 (2009).
[Crossref]

2008 (3)

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, 063828 (2008).
[Crossref]

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[Crossref]

L. Pezze and A. Smerzi, “Mach-Zehnder interferometry at the Heisenberg limit with coherent and squeezed-vacuum light,” Phys. Rev. Lett. 100, 073601 (2008).
[Crossref] [PubMed]

2007 (1)

Y. H. Shih, “Quantum imaging,” IEEE J. Sel. Top. Quantum Electron. 13, 1016 (2007).
[Crossref]

2006 (1)

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

2004 (1)

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

1999 (1)

A. Peters, K. Y. Chung, and S. Chu, “Measurement of gravitational acceleration by dropping atoms,” Nature 400, 849–852 (1999).
[Crossref]

1998 (1)

M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, “Measurement of the Earth’s gravity gradient with an atom interferometer-based gravity gradiometer,” Phys. Rev. Lett. 81, 971 (1998).
[Crossref]

1997 (1)

C. C. Gerry, “Generation of Schrödinger cats and entangled coherent states in the motion of a trapped ion by a dispersive interaction,” Phys. Rev. A 55, 2478 (1997).
[Crossref]

1994 (2)

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A 50, 67 (1994).
[Crossref] [PubMed]

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

1992 (1)

B. C. Sanders, “Entangled coherent states,” Phys. Rev. A 45, 6811 (1992).
[Crossref] [PubMed]

1986 (1)

B. Yurke, S. L. McCall, and J. R. Klauder, “SU(2) and SU(1,1) interferometers,” Phys. Rev. A 33, 4033 (1986).
[Crossref]

1981 (1)

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

Abbott, B. P.

B. P. Abbott and LIGO Scientific Collaboration and the Virgo Collaboration, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116, 061102 (2016).
[Crossref] [PubMed]

Apellaniz, I.

G. Toth and I. Apellaniz, “Quantum metrology from a quantum information science perspective,” J. Phys. A: Math. Theor. 47, 424006 (2014).
[Crossref]

Banaszek, K.

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]

Barbieri, M.

P. C. Humphreys, M. Barbieri, A. Datta, and I. A. Walmsley, “Quantum enhanced multiple phase estimation,” Phys. Rev. Lett. 111, 070403 (2013).
[Crossref] [PubMed]

Barrett, M. D.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

Bollinger, J. J.

D. J. Wineland, J. J. Bollinger, W. M. Itano, and D. J. Heinzen, “Squeezed atomic states and projection noise in spectroscopy,” Phys. Rev. A 50, 67 (1994).
[Crossref] [PubMed]

Bouyer, P.

M. J. Snadden, J. M. McGuirk, P. Bouyer, K. G. Haritos, and M. A. Kasevich, “Measurement of the Earth’s gravity gradient with an atom interferometer-based gravity gradiometer,” Phys. Rev. Lett. 81, 971 (1998).
[Crossref]

Bowen, W. P.

M. A. Taylor and W. P. Bowen, “Quantum metrology and its application in biology,” Phys. Rep. 615, 1–59 (2016).
[Crossref]

Boyd, R. W.

R. W. Boyd and J. P. Dowling, “Quantum lithography: Status of the field,” Quantum Inf. Process. 11, 891 (2012).
[Crossref]

Braunstein, S. L.

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

Britton, J.

D. Leibfried, M. D. Barrett, T. Schaetz, J. Britton, J. Chiaverini, W. M. Itano, J. D. Jost, C. Langer, and D. J. Wineland, “Toward Heisenberg-limited spectroscopy with multiparticle entangled States,” Science 304, 1476 (2004).
[Crossref] [PubMed]

Brun, T. A.

S. Pang and T. A. Brun, “Quantum metrology for a general Hamiltonian parameter,” Phys. Rev. A 90, 022117 (2014).
[Crossref]

Cable, H.

R. T. Glasser, H. Cable, J. P. Dowling, F. De Martini, F. Sciarrino, and C. Vitelli, “Entanglement-seeded, dual, optical parametric amplification: Applications to quantum imaging and metrology,” Phys. Rev. A 78, 012339 (2008).
[Crossref]

Caves, C. M.

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

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

Chen, J.

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P. A. Knott, T. J. Proctor, Kae Nemoto, J. A. Dunningham, and W. J. Munro, “Effect of multimode entanglement on lossy optical quantum metrology,” Phys. Rev. A 90, 033846 (2014).
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Figures (3)

Fig. 1
Fig. 1 Schematic of an Mach-Zehnder Interferometer. The input ports are labeled as A and B. The photon losses in the interferometer are simulated with two fictitious beam splitters, with corresponding operators B AC T 1 and B BD T 2. Here C and D are fictitious ports and T1, T2 are transmission rates. No photon losses exist for T1 = T2 = 1.
Fig. 2
Fig. 2 The values of max Θ F m as a function of T and |α|. The areas below the solid black line, above the dashed black line and between these lines represent the regimes that N ex < ( 2 + 2 e 2 | α | 2 ) 1 (optimal Θ is 0), N ex < ( 2 2 e 2 | α | 2 ) 1 (optimal Θ is π) and N ex [ ( 2 + 2 e 2 | α | 2 ) 1 , ( 2 2 e 2 | α | 2 ) 1 ]. The PMC here is Φ = 0, π. max Θ F m takes the logarithmic values in the figure.
Fig. 3
Fig. 3 The difference between Eq. (15) and T |α|2 under the PMC Φ = 0. The expression T |α|2 is a good approximation from |α| ≈ 3 for the coefficients values in the figure. A larger transmission rate T requires a larger |α| for this approximation.

Equations (39)

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F = i = 1 M ( ϕ p i ) 2 p i + i = 1 M 4 p i ϕ ψ i | ϕ ψ i i , j = 1 M 8 p i p j p i + p j | ψ i | ϕ ψ j | 2 .
F = i = 1 M 4 p i ψ i | H 2 | ψ i i , j = 1 M 8 p i p j p i + p j | ψ i | H | ψ j | 2 .
F = 4 ( ψ ϕ | H 2 | ψ ϕ | ψ ϕ | H | ψ ϕ | 2 ) .
U MZ = B x ( π 2 ) U ( ϕ ) B x ( π 2 ) = exp ( i ϕ J y AB ) .
F = ( 2 n ¯ A n ¯ B + n ¯ A + n ¯ B 2 | a | 2 | b | 2 ) 2 Re ( a 2 b 2 a 2 b 2 ) ,
F = 2 n ¯ A ( n ¯ B | b | 2 ) + n ¯ A + n ¯ B 2 Re [ β 2 ( b 2 b 2 ) ] .
| Arg ( β 2 ) Arg ( b 2 b 2 ) | = π ,
F m = 2 n ¯ A ( n ¯ B | b | 2 + | b 2 b 2 | ) + n ¯ A + n ¯ B .
| ψ = N α ( | α + e i Θ | α ) ,
| Φ A Φ B | = π 2 ,
F m = n ¯ A ( 2 | α | 2 + 1 ) + n ¯ B ( 2 n ¯ A + 1 ) .
F m n ^ 2 ,
N α | α e i ( Φ π 2 ) A ( | α B + e i Θ | α B ) .
ρ 1 = N α 2 ( 1 + p t 2 + 2 e 2 | α | 2 cos Θ 1 p t 2 ( p t + p r e i Θ ) e i | α | 2 δ T sin Φ 1 p t 2 ( p t + p r e i Θ ) e i | α | 2 δ T sin Φ 1 p t 2 ) ,
F = i = ± 4 λ i λ i | ( J z AB ) 2 | λ i i , j = ± 8 λ i λ j λ i + λ j | λ i | J z AB | λ j | 2 .
F = 2 ( δ T ) 2 N α 6 | α | 4 p t Δ ( 1 p t 2 ) [ 4 p r ( p r + p t cos Θ ) ( p t + p r cos Θ ) Δ N α 4 p t ] 16 ( δ T ) 2 N α 8 Δ | α | 4 ( 1 p r 2 ) e 4 | α | 2 sin 2 Θ 2 δ T N α 2 | α | 2 e 2 | α | 2 ( 4 T N α 2 | α | 2 1 ) sin Θ sin Φ + 2 T 2 | α | 4 N α 2 [ 1 2 N α 2 ( 1 p r 2 ) ( Δ 2 N α 2 + 2 N α 2 e 4 | α | 2 sin 2 Θ ) sin 2 Φ ] + 2 T N α 2 | α | 2 .
F = 2 T 2 | α | 4 N α 2 [ 1 2 N α 2 ( 1 p r 2 ) ( Δ 2 N α 2 + 2 N α 2 e 4 | α | 2 sin 2 Θ ) sin 2 Φ ] + 2 T N α 2 | α | 2 .
F m = 2 T N α 2 | α | 2 { 1 + T | α | 2 [ 1 2 N α 2 ( 1 p r 2 ) ] } .
N α 2 = N ex : = 1 + T | α | 2 4 T | α | 2 ( 1 p r 2 )
N α 2 [ 1 2 ( 1 + e 2 | α | 2 ) , 1 2 ( 1 e 2 | α | 2 ) ] ,
max Θ F m = T | α | 2 1 e 2 | α | 2 [ 1 + T | α | 2 ( 1 1 e 2 | α | 2 R 1 e 2 | α | 2 ) ] .
max Θ F m = T | α | 2 1 + e 2 | α | 2 [ 1 + T | α | 2 ( 1 1 e 2 | α | 2 R 1 + e 2 | α | 2 ) ] .
max Θ F m = ( 1 + T | α | 2 ) 2 4 ( 1 e 2 | α | 2 R ) .
cos Θ = 2 T | α | 2 e 2 | α | 2 ( 1 T ) + ( T | α | 2 1 ) e 2 | α | 2 1 + T | α | 2 .
F sin Φ = 2 δ T N α 2 | α | 2 e 2 | α | 2 sin Θ ( 1 4 T N α 2 | α | 2 ) 2 T 2 | α | 4 ( Δ + 4 N α 4 e 4 | α | 2 sin 2 Θ ) sin Φ .
sin Φ = N ex : = N α 2 ( 4 T N α 2 | α | 2 1 ) sin Θ T 2 | α | 2 ( Δ e 2 | α | 2 + 4 N α 4 e 2 | α | 2 sin 2 Θ ) δ T .
F m T | α | 2 ,
n ¯ = 2 | α | 2 1 + e 2 | α | 2 cos Θ .
| ψ in = | α e i ( Φ + π 2 ) N α ( | α + e i Θ | α ) .
B x T | α A | β A = | α T + i β 1 T A | β T + i α 1 T A ,
| ψ 0 = N α ( | i f + A | f B + e i Θ | i f A | f + B ) ,
| ψ 1 = N α ( | A | f + R 1 C | i f R 2 D + e i Θ | | f R 1 C | i f + R 2 D ) ,
ρ 1 = Tr CD ( | ψ 1 ψ 1 | ) = N α 2 [ | A A | + | | + p r e i ( | α | 2 δ T sin Φ Θ ) | A | + p r e i ( | α | 2 δ T sin Φ Θ ) | A | ] ,
| A = 1 1 p t 2 ( | p t e i | α | 2 δ T sin Φ | A ) .
ρ 1 = N α 2 ( 1 + p t 2 + 2 e 2 | α | 2 cos Θ 1 p t 2 ( p t + p r e i Θ ) e i | α | 2 δ T sin Φ 1 p t 2 ( p t + p r e i Θ ) e i | α | 2 δ T sin Φ 1 p t 2 ) ,
| λ ± = ( v ± p t + p r e i Θ p t 2 + p r 2 + 2 e 2 | α | 2 cos Θ p t v 1 p t 2 ) | A ± v e i | α | 2 δ T sin Φ 1 p t 2 | ,
v ± = 1 2 1 ± p t 2 + e 2 | α | 2 cos Θ Δ ( 1 + e 2 | α | 2 cos Θ ) ,
= 1 Δ Δ 2 ± Δ N α 2 ( p t 2 + e 2 | α | 2 cos Θ ) ,
Δ = 1 4 det ρ 1 .

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