Abstract

We study the phase sensitivity of an SU(1,1) interferometer from two aspects, i.e., the phase estimation determined by the error propagation formula and that by the quantum Cramér-Rao bound (QCRB). The results show that the phase sensitivity by using the intensity detection reaches the sub-shot-noise limit with a coherent state and an m-photon-added squeezed vacuum state (m-PA-SVS) as inputs. The phase sensitivity gradually approaches the Heisenberg limit for increasing m, and the ultimate phase precision improves with the increase of m. In addition, the QCRB can be saturated by the intensity detection with inputting the m-PA-SVS.

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

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2018 (1)

2017 (6)

S. Xu, X. X. Xu, C. J. Liu, H. L. Zhang, and L. Y. Hu, “Hermite polynomial excited squeezed vacuum state: Generation and nonclassical properties,” Optik 144, 664 (2017).
[Crossref]

C. Oh, S. Y. Lee, H. Nha, and H. Jeong, “Practical resources and measurements for lossy optical quantum metrology,” Phys. Rev. A 96, 062304 (2017).
[Crossref]

B. E. Anderson, P. Gupta, B. L. Schmittberger, T. Horrom, C. Hermann-Avigliano, K. M. Jones, and P. D. Lett, “Phase sensing beyond the standard quantum limit with a variation on the SU (1, 1) interferometer,” Optica 4, 752 (2017).
[Crossref]

S. S. Szigeti, R. J. Lewis-Swan, and S. A. Haine, “Pumped-up SU (1, 1) interferometry,” Phys. Rev. Lett. 118, 150401 (2017).
[Crossref]

B. E. Anderson, B. L. Schmittberger, P. Gupta, K. M. Jones, and P. D. Lett, “Optimal phase measurements with bright-and vacuum-seeded SU (1, 1) interferometers,” Phys. Rev. A 95, 063843 (2017).
[Crossref]

Q. K. Gong, X. L. Hu, D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “Intramode-correlation-enhanced phase sensitivities in an SU (1, 1) interferometer,” Phys. Rev. A 96, 033809 (2017).
[Crossref]

2016 (4)

M. V. Chekhova and Z. Y. Ou, “Nonlinear interferometers in quantum optics,” Adv. Opt. Photonics 8, 104 (2016).
[Crossref]

C. Sparaciari, S. Olivares, and M. G. A. Paris, “Gaussian-state interferometry with passive and active elements,” Phys. Rev. A 93, 023810 (2016).
[Crossref]

D. Li, B. T. Gard, Y. Gao, C. H. Yuan, W. P. Zhang, H. Lee, and J. P. Dowling, “Phase sensitivity at the Heisenberg limit in an SU (1, 1) interferometer via parity detection,” Phys. Rev. A 94, 063840 (2016).
[Crossref]

Y. Ouyang, S. Wang, and L. J. Zhang, “Quantum optical interferometry via the photon-added two-mode squeezed vacuum states,” J. Opt. Soc. Am. B 33, 1373 (2016).
[Crossref]

2015 (2)

B. Chen, C. Qiu, S. Y. Chen, J. X. Guo, L. Q. Chen, Z. Y. Ou, and W. P. Zhang, “Atom-Light Hybrid Interferometer,” Phys. Rev. Lett. 115, 043602 (2015).
[Crossref] [PubMed]

S. Y. Liu, Y. Z. Li, L. Y. Hu, J. H. Huang, X. X. Xu, and X. Y. Tao, “Nonclassical properties of Hermite polynomial excitation on squeezed vacuum and its decoherence in phase-sensitive reservoirs,” Laser Phys. Lett. 12, 045201 (2015).
[Crossref]

2014 (6)

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

X. X. Jing, J. Liu, W. Zhong, and X. G. Wang, “Quantum Fisher Information of Entangled Coherent States in a Lossy Mach-Zehnder Interferometer,” Commun. Theor. Phys. 61, 115 (2014).
[Crossref]

R. Birrittella and C. C. Gerry, “Quantum optical interferometry via the mixing of coherent and photon-subtracted squeezed vacuum states of light,” J. Opt. Soc. Am. B 31, 586 (2014).
[Crossref]

D. Li, C. H. Yuan, Z. Y. Ou, and W. P. Zhang, “The phase sensitivity of an SU (1, 1) interferometer with coherent and squeezed-vacuum light,” New J. Phys. 16, 073020 (2014).
[Crossref]

F. Hudelist, J. Kong, C. J. Liu, J. T. Jing, Z. Y. Ou, and W. P. Zhang, “Quantum metrology with parametric amplifier-based photon correlation interferometers,” Nat. Commun. 5, 3049 (2014).
[Crossref] [PubMed]

Q. S. Tan, J. Q. Liao, X. G. Wang, and F. Nori, “Enhanced interferometry using squeezed thermal states and even or odd states,” Phys. Rev. A 89, 053822 (2014).
[Crossref]

2013 (2)

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. Kong, F. Hudelist, Z. Y. Ou, and W. P. Zhang, “Cancellation of internal quantum noise of an amplifier by quantum correlation,” Phys. Rev. Lett. 111, 033608 (2013).
[Crossref] [PubMed]

2012 (6)

R. Carranza and C. C. Gerry, “Photon-subtracted two-mode squeezed vacuum states and applications to quantum optical interferometry,” J. Opt. Soc. Am. B 29, 2581 (2012).
[Crossref]

A. M. Marino, N. V. Corzo Trejo, and P. D. Lett, “Effect of losses on the performance of an SU (1, 1) interferometer,” Phys. Rev. A 86, 023844 (2012).
[Crossref]

Z. Y. Ou, “Enhancement of the phase-measurement sensitivity beyond the standard quantum limit by a nonlinear interferometer,” Phys. Rev. A 85, 023815 (2012).
[Crossref]

H. Yonezawa, D. Nakane, T. A. Wheatley, K. Iwasawa, S. Takeda, H. Arao, K. Ohki, K. Tsumura, D. W. Berry, and T. C. Ralph, “Quantum-enhanced optical-phase tracking,” Science 337, 1514 (2012).
[Crossref] [PubMed]

J. W. Pan, Z. B. Chen, C. Y. Lu, H. Weinfurter, A. Zeilinger, and M. Żukowski, “Multiphoton entanglement and interferometry,” Rev. Mod. Phys. 84, 777 (2012).
[Crossref]

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

2011 (3)

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

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

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

2010 (6)

C. Gross, T. Zibold, E. Nicklas, J. Estève, and M. K. Oberthaler, “Nonlinear atom interferometer surpasses classical precision limit,” Nature 464, 1165 (2010).
[Crossref] [PubMed]

W. N. Plick, J. P. Dowling, and G. S. Agarwal, “Coherent-light-boosted, sub-shot noise, quantum interferometry,” New J. Phys. 12, 083014 (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]

M. Zwierz, C. A. Pérez-Delgado, and P. Kok, “General optimality of the Heisenberg limit for quantum metrology,” Phys. Rev. Lett. 105, 180402 (2010).
[Crossref]

L. Y. Hu and H. Y. Fan, “Nonclassicality of photon-added squeezed vacuum and its decoherence in thermal environment,” J. Mod. Optic 57, 1344 (2010).
[Crossref]

K. Si, X. H. Ji, and H. Y. Jia, “Nonclassicality of photon-added squeezed vacuum states,” Chin. Phys. B 19, 064205 (2010).
[Crossref]

2008 (4)

S. Boixo, A. Datta, M. J. Davis, S. T. Flammia, A. Shaji, and C. M. Caves, “Quantum metrology: dynamics versus entanglement,” Phys. Rev. Lett. 101, 040403 (2008).
[Crossref] [PubMed]

S. M. Roy and S. L. Braunstein, “Exponentially enhanced quantum metrology,” Phys. Rev. Lett. 100, 220501 (2008).
[Crossref] [PubMed]

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]

L. Pezzé 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)

A. Zavatta, V. Parigi, and M. Bellini, “Experimental nonclassicality of single-photon-added thermal light states,” Phys. Rev. A 75, 052106 (2007).
[Crossref]

2006 (2)

H. Y. Fan, X. G. Meng, and J. S. Wang, “New form of Legendre polynomials obtained by virtue of excited squeezed state and IWOP technique in quantum optics,” Commun. Theor. Phys. 46, 845 (2006).
[Crossref]

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

2005 (1)

A. Zavatta, S. Viciani, and M. Bellini, “Single-photon excitation of a coherent state: Catching the elementary step of stimulated light emission,” Phys. Rev. A 72, 023820 (2005).
[Crossref]

2004 (1)

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660 (2004).
[Crossref] [PubMed]

2000 (1)

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, 2733 (2000).
[Crossref] [PubMed]

1998 (1)

M. Dakna, L. Knöll, and D. G. Welsch, “Photon-added state preparation via conditional measurement on a beam splitter,” Opt. Commun. 145, 309 (1998).
[Crossref]

1994 (2)

G. M. D’Ariano and M. G. A. Paris, “Lower bounds on phase sensitivity in ideal and feasible measurements,” Phys. Rev. A 49, 3022 (1994).
[Crossref]

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

1993 (2)

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355 (1993).
[Crossref] [PubMed]

M. Hillery and L. Mlodinow, “Interferometers and minimum-uncertainty states,” Phys. Rev. A 48, 1548 (1993).
[Crossref] [PubMed]

1992 (1)

Z. Zhang and H. Fan, “Properties of states generated by excitations on a squeezed vacuum state,” Phys. Lett. A 165, 14 (1992).
[Crossref]

1987 (2)

M. Xiao, L. A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278 (1987).
[Crossref] [PubMed]

P. Grangier, R. E. Slusher, B. Yurke, and A. LaPorta, “Squeezed-light-enhanced polarization interferometer,” Phys. Rev. Lett. 59, 2153 (1987).
[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]

1969 (1)

C. W. Helstrom, “Quantum detection and estimation theory,” J. Stat. Phys 1, 231 (1969).
[Crossref]

Abadie, J.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Abbott, B. P.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
[Crossref]

Abbott, R.

J. Abadie, B. P. Abbott, R. Abbott, T. D. Abbott, M. Abernathy, C. Adams, R. Adhikari, C. Affeldt, B. Allen, and G. S. Allen, “A gravitational wave observatory operating beyond the quantum shot-noise limit,” Nat. Phys. 7, 962 (2011).
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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of an SU(1,1) interferometer. The two input beams are in a coherent state and a photon-added squeezed vacuum state, respectively. OPA : optical parametric amplifier; Da and Db: detectors.
Fig. 2
Fig. 2 Phase sensitivity based on the intensity detection as a function of ϕ with g=1, r =1, and α=1. The solid lines correspond to the detection method I, while the dashed lines are for the detection method II.
Fig. 3
Fig. 3 Phase sensitivity based on the intensity detection as a function of (a) g with α=1 and r =1, (b) r with g=1 and α=1. The solid lines correspond to the detection method I, while the dashed lines are for the detection method II.
Fig. 4
Fig. 4 The total average photon number NT as a function of (a) g with α=1 and r =1. (b) r with α=1 and g=1.
Fig. 5
Fig. 5 (a) Quantum Fisher information Fm versus gain factor g for α=1 and r =1. Phase sensitivity ΔϕF as a function of (b) g with α=1 and r =1.(c) r with α=1 and g=1.(d) α with r =1 and g=1.
Fig. 6
Fig. 6 Pase sensitivity based on the intensity detection against g for different m, (a) m=0, (b) m=1, (c) m=2, (d) m=3. The subscript 1(2) of m corresponds to the detection method I(II). The orange dashed line is for the shot-noise limit, the purple dashed line is for the QCRB, the cyan dashed line is for the Heisenberg limit, where α=1 and r =1.
Fig. 7
Fig. 7 The same as Fig. 6 but α = 0.
Fig. 8
Fig. 8 The phase sensitivity Δϕ as a function of g with (a) α = (tanh 2g)er/2 and r =0.2, (b) α=0 and r =1, (c) α = (tanh 2g)er/2 and r =0, (d) α=1 and r =0. The subscript 1(2) of m corresponds to the intensity detection method I (II).

Equations (21)

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( a 2 b 2 + ) = S ( a 0 b 0 + ) ,
S = S O P A 2 S ϕ S O P A 1 ,
S O P A 1 = ( cosh g 1 e i θ 1 sinh g 1 e i θ 1 sinh g 1 cosh g 1 ) ,
S ϕ = ( e i ϕ 0 0 1 ) ,
S O P A 2 = ( cosh g 2 e i θ 2 sinh g 2 e i θ 2 sinh g 2 cosh g 2 ) ,
a 2 = ( e i ϕ cosh 2 g sinh 2 g ) a 0 [ e i θ 1 ( e i ϕ 1 ) sinh g cosh g ] b 0 + ,
b 2 + = [ e i θ 1 ( e i ϕ 1 ) sinh g cosh g ] a 0 + ( cosh 2 g e i ϕ sinh 2 g ) b 0 + .
| r , m b = N m b + m | r , 0 b ,
N m 2 = m ! cosh m r P m ( cosh r ) ,
N = A a N a + A b N b ,
Δ 2 ϕ = N 2 N 2 | ϕ N | 2 .
( Δ 2 ϕ ) I = 1 8 ( 1 + n ¯ m + α 2 ) 2 { 8 [ 1 + B m + 2 n ¯ m n ¯ m 2 + 2 ( 1 + A m + n ¯ m ) α 2 + 4 A m α 2 csch 2 2 g ] + csch 4 2 g [ 8 α 2 csc 2 ϕ 2 + [ 3 B m 1 + 2 n ¯ m 3 n ¯ m 2 2 ( n ¯ m + A m 1 ) α 2 + 4 ( α 2 B m n ¯ m + n ¯ m 2 ) cosh 4 g + ( 1 + B m + 2 n ¯ m n ¯ m 2 + 2 ( 1 + A m + n ¯ m ) α 2 ) cosh 8 g ] sec 2 ϕ 2 ] } ,
( Δ 2 ϕ ) I I = 1 8 sinh 4 2 g ( e 2 i ϕ 1 ) 2 ( 1 + n ¯ m + α 2 ) 2 { 2 e 2 i ϕ [ 9 B m 7 + 2 n ¯ m 9 n ¯ m 2 + 2 ( 1 + A m 7 n ¯ m ) α 2 + ( 3 + 3 B m + 6 n ¯ m 3 n ¯ m 2 + ( 6 10 A m + 6 n ¯ m ) α 2 ) cos 2 ϕ ] 4 ( e 2 i ϕ 1 ) 2 [ 1 + B m + 2 n ¯ m n ¯ m 2 + 2 ( n ¯ m + 1 A m ) α 2 ] cosh 4 g + [ 1 + B m + 2 n ¯ m n ¯ m 2 + 2 ( 1 + n ¯ m + A m ) α 2 ] [ 8 e 2 i ϕ cos ϕ + ( e i ϕ 1 ) 4 cosh 8 g ] } ,
n ¯ m = r , m | b + b | r , m b = N m 2 N m + 1 2 1 ,
A m = r , m | b 2 | r , m b = N m 2 coth r [ N m + 1 2 ( m + 1 ) N m 2 ] ,
B m = r , m | b + 2 b 2 | r , m b = N m 2 ( N m + 2 2 4 N m + 1 2 ) + 2 .
N T , m = cosh 2 g ( 1 + α 2 + n ¯ m ) 1 .
F = 4 [ ψ ϕ | ψ ϕ | ψ ϕ | ψ ϕ | 2 ] ,
F = 4 Δ 2 n a ,
F m = 4 [ α 2 cosh 4 g + ( B m + n ¯ m n ¯ m 2 ) sinh 4 g ] + [ 1 + n ¯ m + ( 2 A m + 1 + 2 n ¯ m ) α 2 ] sinh 2 2 g .
Δ ϕ F 1 v F m ,

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