Abstract

We suggest and investigate a scheme for non-deterministic noiseless linear amplification of coherent states using successive photon addition, (â)2, where â is the photon creation operator. We compare it with a previous proposal using the photon addition-then-subtraction, ââ, where â is the photon annihilation operator, that works as an appropriate amplifier only for weak light fields. We show that when the amplitude of a coherent state is |α| ≳ 0.91, the (â)2 operation serves as a more efficient amplifier compared to the ââ operation in terms of equivalent input noise. Using ââ and (â)2 as basic building blocks, we compare combinatorial amplifications of coherent states using (ââ)2, â†4, âââ†2, and â†2ââ, and show that (ââ)2, â†2ââ, and â†4 exhibit strongest noiseless properties for |α| ≲ 0.51, 0.51 ≲ |α| ≲ 1.05, and |α| ≳ 1.05, respectively. We further show that the (â)2 operation can be useful for amplifying superpositions of the coherent states. In contrast to previous studies, our work provides efficient schemes to implement a noiseless amplifier for light fields with medium and large amplitudes.

© 2016 Optical Society of America

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2015 (2)

C.-Y. Park and H. Jeong, “Bell-inequality tests using asymmetric entangled coherent states in asymmetric lossy environments,” Phys. Rev. A 91, 042328 (2015).
[Crossref]

J. Kim, J. Lee, S.-W. Ji, H. Nha, P. M. Anisimov, and J. P. Dowling, “Coherent-state optical qudit cluster state generation and teleportation via homodyne detection,” Opt. Commun. 337, 79–82 (2015).
[Crossref]

2014 (2)

D. S. Simon, G. Jaeger, and A. V. Sergienko, “Entangled-coherent-state quantum key distribution with entanglement witnessing,” Phys. Rev. A 89, 012315 (2014).
[Crossref]

B. T. Kirby and J. D. Franson, “Macroscopic state interferometry over large distances using state discrimination,” Phys. Rev. A 89, 033861 (2014).
[Crossref]

2013 (3)

B. T. Kirby and J. D. Franson, “Nonlocal interferometry using macroscopic coherent states and weak nonlinearities,” Phys. Rev. A 87, 053822 (2013).
[Crossref]

G. Torlai, G. McKeown, P. Marek, R. Filip, H. Jeong, M. Paternostro, and G. D. Chiara, “Violation of Bell’s inequalities with preamplified homodyne detection,” Phys. Rev. A 87, 052112 (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, 043832 (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, 043828 (2012).
[Crossref]

M. Cooper, L. J. Wright, C. Soller, and B.J. Smith, “Experimental generation of multiphoton fock states,” Optics Express 21, 5309 (2012).
[Crossref]

R. Blandino, A. Leverrier, M. Barbieri, J. Etesse, P. Grangier, and R. Tualle-Brouri, “Improving the maximum transmission distance of continuous-variable quantum key distribution using a noiseless amplifier,” Phys. Rev. A 86, 012327 (2012).
[Crossref]

M. Mičuda, I. Straka, M. Miková, M. Dušek, N. J. Cerf, J. Fiurášek, and M. Ježek, “Noiseless loss suppression in quantum optical communication,” Phys. Rev. Lett. 109, 180503 (2012).
[Crossref]

Y. Lim, M. Kang, J. Lee, and H. Jeong, “Using macroscopic entanglement to close the detection loophole in Bell-inequality tests,” Phys. Rev. A 85, 062112 (2012).
[Crossref]

J. Park, S.-Y. Lee, H.-W. Lee, and H. Nha, “Enhanced Bell violation by a coherent superposition of photon subtraction and addition,” J. Opt. Soc. Am. B 29, 906–911 (2012).
[Crossref]

2011 (5)

C.-W. Lee and H. Jeong, “Faithful test of nonlocal realism with entangled coherent states,” Phys. Rev. A 83, 022102 (2011).
[Crossref]

T. C. Ralph, “Quantum error correction of continuous-variable states against Gaussian noise,” Phys. Rev. A 84, 022339 (2011).
[Crossref]

A. Zavatta, J. Fiurášek, and M. Bellini, “A high-fidelity noiseless amplifier for quantum light states,” Nature Photon. 5, 52–60 (2011).
[Crossref]

C. R. Myers and T. C. Ralph, “Coherent state topological cluster state production,” New J. Phys. 13, 115015 (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 (5)

M. Bellini and A. Zavatta, “Manipulating Light States by Single-Photon Addition and Subtraction,” Prog. Optics 55, 41–83 (2010).
[Crossref]

F. Ferreyrol, M. Barbieri, R. Blandino, S. Fossier, R. Tualle-Brouri, and P. Grangier, “Implementation of a nondeterministic optical noiseless amplifier,” Phys. Rev. Lett. 104, 123603 (2010).
[Crossref] [PubMed]

G. Y. Xiang, T. C. Ralph, A. P. Lund, N. Walk, and G. J. Pryde, “Heralded noiseless linear amplification and distillation of entanglement,” Nature Photon. 4, 316–319 (2010).
[Crossref]

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nature Phys. 6, 767–771 (2010).
[Crossref]

M. Paternostro and H. Jeong, “Testing nonlocal realism with entangled coherent states,” Phys. Rev. A 81, 032115 (2010).
[Crossref]

2009 (7)

C.-W. Lee and H. Jeong, “Effects of squeezing on quantum nonlocality of superpositions of coherent states,” Phys. Rev. A 80, 052105 (2009).
[Crossref]

C. C. Gerry, J. Mimih, and A. Benmoussa, “Maximally entangled coherent states and strong violations of Bell-type inequalities,” Phys. Rev. A 80, 022111 (2009).
[Crossref]

H. Jeong and T. C. Ralph, “Failure of Local Realism Revealed by Extremely-Coarse-Grained Measurements,” Phys. Rev. Lett. 102, 060403 (2009).
[Crossref] [PubMed]

J. Fiurášek, “Engineering quantum operations on traveling light beams by multiple photon addition and subtraction,” Phys. Rev. A 80, 053822 (2009).
[Crossref]

S. Fossier, E. Diamanti, T. Debuisschert, R. Tualle-Brouri, and P. Grangier, “Improvement of continuous-variable quantum key distribution systems by using optical preamplifiers,” J. Phys. B: At. Mol. Opt. Phys. 42, 114014 (2009).
[Crossref]

A. Ourjoumtsev, F. Ferreyrol, R. Tualle-Brouri, and P. Grangier, “Preparation of non-local superpositions of quasi-classical light states,” Nature Phys. 5, 189–192 (2009).
[Crossref]

S. R. Huisman, N. Jain, S. A. Babichev, Frank Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Optics Letters 34, 2739 (2009).
[Crossref] [PubMed]

2008 (3)

P. Marek, H. Jeong, and M. S. Kim, “Generating “squeezed” superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[Crossref]

A. P. Lund, T. C. Ralph, and H. L. Haselgrove, “Fault-tolerant linear optical quantum computing with small-amplitude coherent states,” Phys. Rev. Lett. 100, 030503 (2008).
[Crossref] [PubMed]

H. Jeong, “Testing Bell inequalities with photon-subtracted Gaussian states,” Phys. Rev. A 78, 042101 (2008).
[Crossref]

2007 (2)

H. Jeong and T. C. Ralph, “Violation of Bell’s inequality using classical measurements and nonlinear local operations,” Phys. Rev. A 75, 052105 (2007).
[Crossref]

V. Parigi, A. Zavatta, M. Kim, and M. Bellini, “Probing Quantum Commutation Rules by Addition and Subtraction of Single Photons to/from a Light Field,” Science 317, 1890 (2007).
[Crossref] [PubMed]

2006 (1)

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

2005 (1)

H. Jeong, A. P. Lund, and T. C. Ralph, “Production of superpositions of coherent states in traveling optical fields with inefficient photon detection,” Phys. Rev. A 72, 013801 (2005).
[Crossref]

2004 (3)

A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, “Conditional production of superpositions of coherent states with inefficient photon detection,” Phys. Rev. A 70, 020101 (2004).
[Crossref]

J. Wenger, R. Tualle-Brouri, and P. Grangier, “Non-Gaussian statistics from individual pulses of squeezed light,” Phys. Rev. Lett. 92, 153601 (2004).
[Crossref] [PubMed]

A. Zavatta, S. Viciani, and M. Bellini, “Quantum-to-classical transition with single-photon-added coherent states of light,” Science 306, 660–662 (2004).
[Crossref] [PubMed]

2003 (4)

J. Wenger, M. Hafezi, F. Grosshans, R. Tualle-Brouri, and P. Grangier, “Maximal violation of Bell inequalities using continuous-variable measurements,” Phys. Rev. A 67, 012105 (2003).
[Crossref]

H. Jeong, W. Son, and M. S. Kim, “Quantum nonlocality test for continuous-variable states with dichotomic observables,” Phys. Rev. A 67, 012106 (2003).
[Crossref]

T. C. Ralph, A. Gilchrist, G. J. Milburn, W. J. Munro, and S. Glancy, “Quantum computation with optical coherent states,” Phys. Rev. A 68, 042319 (2003).
[Crossref]

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

T. C. Ralph, “Coherent superposition states as quantum rulers,” Phys. Rev. A 65, 042313 (2002).
[Crossref]

W. J. Munro, K. Nemoto, G. J. Milburn, and S. Braunstein, “Weak-force detection with superposed coherent states,” Phys. Rev. A 66, 023819 (2002).
[Crossref]

H. Jeong and M. S. Kim, “Efficient quantum computation using coherent states,” Phys. Rev. A 65, 042305 (2002).
[Crossref]

D. Wilson, H. Jeong, and M. S. Kim, “Quantum nonlocality for a mixed entangled coherent state,” J. Mod. Opt. 49, 851–864 (2002).
[Crossref]

2001 (5)

R. Filip, J. Rehácek, and M. Dusek, “Entanglement of coherent states and decoherence,” J. Opt. B: Quantum Semiclass. Opt. 3, 341–345 (2001).
[Crossref]

L. M. Duan, M. D. Lukin, J. I. Cirac, and P. Zoller, “Long-distance quantum communication with atomic ensembles and linear optics,” Nature Photon. 414, 413–418 (2001).

S. J. van Enk and O. Hirota, “Entangled coherent states: teleportation and decoherence,” Phys. Rev. A 64, 022313 (2001).
[Crossref]

H. Jeong, M. S. Kim, and J. Lee, “Quantum-information processing for a coherent superposition state via a mixedentangled coherent channel,” Phys. Rev. A 64, 052308 (2001).
[Crossref]

C. Gerry and R. Campos, “Generation of maximally entangled photonic states with a quantum-optical Fredkin gate,” Phys. Rev. A 64, 063814 (2001).
[Crossref]

1999 (1)

P. T. Cochrane, G. J. Milburn, and W. J. Munro, “Macroscopically distinct quantum-superposition states as a bosonic code for amplitude damping,” Phys. Rev. A 59, 2631–2634 (1999).
[Crossref]

1998 (1)

D. T. Pegg, L. S. Phillips, and S. M. Barnett, “Optical state truncation by projection synthesis,” Phys. Rev. Lett. 81, 1604–1606 (1998).
[Crossref]

1996 (1)

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]

1995 (1)

A. Mann, B. C. Sanders, and W. J. Munro, “Bell’s inequality for an entanglement of nonorthogonal states,” Phys. Rev. A 51, 989–991 (1995).
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[Crossref] [PubMed]

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P. Marek, H. Jeong, and M. S. Kim, “Generating “squeezed” superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
[Crossref]

A. P. Lund, H. Jeong, T. C. Ralph, and M. S. Kim, “Conditional production of superpositions of coherent states with inefficient photon detection,” Phys. Rev. A 70, 020101 (2004).
[Crossref]

H. Jeong, W. Son, and M. S. Kim, “Quantum nonlocality test for continuous-variable states with dichotomic observables,” Phys. Rev. A 67, 012106 (2003).
[Crossref]

D. Wilson, H. Jeong, and M. S. Kim, “Quantum nonlocality for a mixed entangled coherent state,” J. Mod. Opt. 49, 851–864 (2002).
[Crossref]

H. Jeong and M. S. Kim, “Efficient quantum computation using coherent states,” Phys. Rev. A 65, 042305 (2002).
[Crossref]

H. Jeong, M. S. Kim, and J. Lee, “Quantum-information processing for a coherent superposition state via a mixedentangled coherent channel,” Phys. Rev. A 64, 052308 (2001).
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B. T. Kirby and J. D. Franson, “Nonlocal interferometry using macroscopic coherent states and weak nonlinearities,” Phys. Rev. A 87, 053822 (2013).
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A. Ourjoumtsev, R. Tualle-Brouri, J. Laurat, and P. Grangier, “Generating optical Schrödinger kittens for quantum information processing,” Science 312, 83–86 (2006).
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C.-W. Lee and H. Jeong, “Faithful test of nonlocal realism with entangled coherent states,” Phys. Rev. A 83, 022102 (2011).
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C.-W. Lee and H. Jeong, “Effects of squeezing on quantum nonlocality of superpositions of coherent states,” Phys. Rev. A 80, 052105 (2009).
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J. Kim, J. Lee, S.-W. Ji, H. Nha, P. M. Anisimov, and J. P. Dowling, “Coherent-state optical qudit cluster state generation and teleportation via homodyne detection,” Opt. Commun. 337, 79–82 (2015).
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Y. Lim, M. Kang, J. Lee, and H. Jeong, “Using macroscopic entanglement to close the detection loophole in Bell-inequality tests,” Phys. Rev. A 85, 062112 (2012).
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H. Jeong, M. S. Kim, and J. Lee, “Quantum-information processing for a coherent superposition state via a mixedentangled coherent channel,” Phys. Rev. A 64, 052308 (2001).
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Leonhardt, U.

U. Leonhardt and H. Paul, “High-accuracy optical homodyne detection with low-efficiency detectors: ‘preamplification’ from antisqueezing,” Phys. Rev. Lett. 72, 4086–4089 (1994).
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Ann. Phys. (1)

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J. Opt. Soc. Am. B (1)

J. Phys. B: At. Mol. Opt. Phys. (1)

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Nature Phys. (2)

M. A. Usuga, C. R. Müller, C. Wittmann, P. Marek, R. Filip, C. Marquardt, G. Leuchs, and U. L. Andersen, “Noise-powered probabilistic concentration of phase information,” Nature Phys. 6, 767–771 (2010).
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C. R. Myers and T. C. Ralph, “Coherent state topological cluster state production,” New J. Phys. 13, 115015 (2011).
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Opt. Commun. (2)

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Optics Express (1)

M. Cooper, L. J. Wright, C. Soller, and B.J. Smith, “Experimental generation of multiphoton fock states,” Optics Express 21, 5309 (2012).
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Optics Letters (1)

S. R. Huisman, N. Jain, S. A. Babichev, Frank Vewinger, A. N. Zhang, S. H. Youn, and A. I. Lvovsky, “Instant single-photon Fock state tomography,” Optics Letters 34, 2739 (2009).
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Phys. Rev. (1)

E. Wigner, “On the quantum correction for thermodynamic equilibrium,” Phys. Rev. 40, 749 (1932).
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Phys. Rev. A (33)

P. Marek, H. Jeong, and M. S. Kim, “Generating “squeezed” superpositions of coherent states using photon addition and subtraction,” Phys. Rev. A 78, 063811 (2008).
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Figures (5)

Fig. 1
Fig. 1

(a) Maximum fidelities, (b) amplitude gains, and (c) EINs when the amplification methods ââ (solid curve) and â†2 (dashed curve) are applied to the coherent state of initial amplitude αi. (a) The fidelities of the ââ-amplified coherent states are higher than Fmax > 0.98, which are close to 1 for small and large αi. The fidelity of the â†2-amplified coherent state approaches 1 for large αi. (b) The higher amplitude gain is obtained when the amplification is performed by â†2 rather than ââ. The gains from ââ and â†2 approach 2 and 3 as αi → 0, and are dropped to 1 as αi → ∞. (c) The upper solid and dashed curves represent average EINs, i.e. EINs averaged over all values of λ, while the lower solid and dashed curves correspond to EINs with λ = 0, which gives the lowest EINs. The â†2-amplification exhibits lower average EINs than the ââ-amplification with large amplitude αi ≳ 0.91, while the opposite is true for αi ≲ 0.91. As αi → 0, the average EINs approach −3/8 for ââ, and −2/9 for â†2. The average EINs approach zero as αi increases for both the cases.

Fig. 2
Fig. 2

Wigner functions of (a) a coherent state of amplitude αi = 2 and the states after photonic operations, (b) â, (c) ââ, and (d) (â)2. The amplitudes of the amplified states, i, are noted for comparison.

Fig. 3
Fig. 3

(a) Maximum fidelities, (b) amplitude gains, and (c) average EINs after two-cycle amplifications (ââ)2 (solid curve), â†4 (dashed curve), âââ†2 (dot-dashed curve), and â†2ââ (dotted curve). (a) The maximum fidelity of the (ââ)2-amplified coherent state approaches 1 for small and large αi’s, which is the highest among the two-cycle amplifications. All the maximum fidelities become perfect as the initial amplitude αi increases. (b) The higher gain is obtained by the two-cycle amplifications compared to the one-cycle amplifications. The gains from (ââ)2, â†4, â†2ââ, and âââ†2 become 4, 5, 4, and 6 as αi → 0, respectively, and drop to 1 as αi increases. (c) In the regions of αi ≲ 0.51, 0.51 ≲ αi ≲ 1.05, and αi ≳ 1.05, (ââ)2, â†4, and â†2ââ show the lowest EIN, respectively, which are all lower than EINs obtained by the one-cycle amplifications.

Fig. 4
Fig. 4

(a) Maximum fidelities, (b) amplitude gains, and (c) optimal phase uncertainties when the amplification methods ââ (solid curve) and â†2 (dashed curve) are applied to the even SCS of initial amplitude αi. The same functions for the case of the odd SCS are plotted in panels (d), (e) and (f). The dot-dashed curves in panels (c) and (f) represent optimal phase uncertainties of the even and odd SCSs, respectively. The amplitude gains using ââ approach 3 and 2 for even and odd SCSs, respectively, as the initial amplitude αi → 0. The amplitude gain using either of the amplification methods approaches 1 as αi → ∞.

Fig. 5
Fig. 5

(a) Maximum fidelities and (b) required levels of squeezing (dB) when the amplification methods ââ (solid curve) and â†2 (dashed curve) are applied to the squeezed vacuum state to approximate the even SCS of amplitude αf. The same functions for the cases of the squeezed single-photon state for approximating the odd SCS are plotted in panels (c) and (d). The dot-dashed curves represent the cases with the squeezed vacuum and single-photon states, respectively, without the amplification methods. In (a) and (c), higher fidelities are obtained for approximating the even and odd SCSs of large amplitudes using the amplification methods ââ and â†2 (αf ≳ 1.47 and αf ≳ 2.04, respectively), compared to the cases without the amplification methods. The amplification method â†2 achieves the fidelities to even and odd SCSs up to F + S a ^ 2 max 0.943 and F S a ^ 2 max 0.959 around αf ≃ 2.12 and αf ≃ 2.59, respectively.

Equations (47)

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N A ^ ( α i ) A ^ | α i ,
N a ^ a ^ ( α i ) = ( α i 4 + 3 α i 2 + 1 ) 1 2 ,
N a ^ 2 ( α i ) = ( α i 4 + 4 α i 2 + 1 ) 1 2 ,
F A ^ = { N A ^ ( α i ) } 2 | α f | A ^ | α i | 2 .
F a ^ a ^ = { N a ^ a ^ ( α i ) } 2 e ( α f α i ) 2 ( α f α i + 1 ) 2 ,
F a ^ 2 = { N a ^ 2 ( α i ) } 2 α f 4 e ( α f α i ) 2 .
F max A ^ ( α i ) = max α f F A ^
g λ A ^ = | N A ^ ( α i ) | 2 | α i | A ^ x ^ λ A ^ | α i | | α i | x ^ λ | α i | ,
g λ a ^ a ^ = { N a ^ a ^ ( α i ) } 2 ( α i 4 + 4 α i 2 + 2 ) ,
g λ a ^ 2 = { N a ^ 2 ( α i ) } 2 ( α i 4 + 6 α i 2 + 6 ) .
E λ A ^ = Δ x λ out 2 ( g λ A ^ ) 2 Δ x λ in 2 ,
E λ a ^ a ^ ( α i ) = { 2 α i 6 + 11 α i 4 + 11 α i 2 + 2 ( α i 4 + 5 α i 2 + 3 ) α i 2 cos ( 2 λ ) + 1 } 2 { N a ^ a ^ ( α i ) } 2 ( α i 4 + 4 α i 2 + 2 ) 2 2 α i 2 cos 2 λ 1 2 E λ a ^ 2 ( α i ) = { 2 N a ^ 2 ( α i ) } 2 ( α 2 + 2 ) { α 4 + ( α 2 + 6 ) α 2 cos ( 2 λ ) + 6 α 2 + 2 } + 1 { 2 N a ^ 2 ( α i ) } 4 ( α 4 + 6 α 2 + 6 ) 2 2 α 2 cos 2 λ 1 2
F max ( a ^ a ^ ) 2 ( α i ) > F max a ^ a ^ a ^ 2 ( α i ) > F max a ^ 2 a ^ a ^ ( α i ) > F max a ^ 4 ( α i ) .
g λ a ^ 4 ( α i ) < g λ a ^ 2 a ^ a ^ ( α i ) > g λ a ^ a ^ a ^ 2 ( α i ) > g λ ( a ^ a ^ ) 2 ( α i ) .
p add | λ g | 2 ( n ¯ + 1 ) ,
p sub R n ¯ .
| ± α = 1 2 ( 1 ± e 2 | α | 2 ) ( | α ± | α ) ,
| ± α i A ^ = N ± A ^ ( α i ) A ^ | ± α i ,
N ± a ^ a ^ ( α i ) = [ 2 { ± e 2 α i 2 ( α i 4 3 α i 2 + 1 ) + ( α i 4 + 3 α i 2 + 1 ) } ] 1 2 , N ± a ^ 2 ( α i ) = [ 2 { ± e 2 α i 2 ( α i 4 4 α i 2 + 2 ) + ( α i 4 + 4 α i 2 + 2 ) } ] 1 2
F ± A ^ max ( α i ) = max α f | ± α f | ± α i A ^ | 2 ,
F ± a ^ a ^ = 2 { N ± a ^ a ^ ( α i ) } 2 e ( α f + α i ) 2 { e 2 α f α i ( α f α i + 1 ) ( α f α i 1 ) } 2 ± e 2 α f 2 + 1 , F ± a ^ 2 = 2 { N ± a ^ 2 ( α i ) } 2 α f 4 { ± e 1 2 ( α i + α f ) 2 + e 1 2 ( α i α f ) 2 } 2 ± e 2 α f 2 + 1 .
g ± A ^ ( α i ) = | α f | | α i | ,
δ ϕ ± = 1 ± ,
± = 4 ( Δ n ^ ) 2
S ^ ( r ) | 0 = n = 0 ( tanh r ) n ( cosh r ) 1 / 2 ( 2 n ) ! 2 n n ! | 2 n ,
S ^ ( r ) | 1 = n = 0 ( tanh r ) n ( cosh r ) 3 / 2 ( 2 n + 1 ) ! 2 n n ! | 2 n + 1 ,
F + S 1 ^ max ( α f ) = max r | + α f | S ^ ( r ) | 0 | 2 ,
F S 1 ^ max ( α f ) = max r | α f | S ^ ( r ) | 1 | 2 .
| + S r A ^ = M + A ^ ( r ) A ^ S ^ ( r ) | 0 ,
| S r A ^ = M A ^ ( r ) A ^ S ^ ( r ) | 1 ,
M ± a ^ a ^ ( r ) = ( sech [ r ] ) 2 [ { ( 1 1 ) tanh 4 r + ( 12 8 ) tanh 2 r + 5 3 } 2 ] 1 2 , M ± a ^ 2 ( r ) = ( sech [ r ] ) 2 [ ( 5 4 ) tanh 2 r + 4 2 ] 1 2 .
F ± S a ^ a ^ = | ± α f | ± S r a ^ a ^ | 2 = 2 { M ± a ^ a ^ ( r ) } 2 α f 1 1 e α f 2 ( tanh r 1 ) ( α f 2 tanh r 3 1 2 ) 2 ( ± 1 + e 2 α f 2 ) cosh 2 1 r ,
F ± S a ^ 2 = | ± α f | ± S r a ^ 2 | 2 = 2 { M ± a ^ 2 ( r ) } 2 α f 5 1 e α f 2 ( tanh r + 1 ) ( 1 + e 2 α f 2 ) cosh 2 1 r ,
F ± S A ^ max ( α f ) = max r | ± α f | ± S r A ^ | 2 .
W ( x , y ) = 2 π Tr [ D ^ ( x + i y ) ρ ^ D ^ ( x + i y ) ( 1 ) n ^ ] ,
D ^ ( x + i y ) = exp [ ( x + i y ) a ^ ( x i y ) a ^ ]
W 1 ^ ( x , y ) = 2 π e 2 ( x α i ) 2 2 y 2 .
W a ^ ( x , y ) = 2 π e 2 ( x α i ) 2 2 y 2 ( α i 2 x ) 2 + 4 y 2 1 1 + α i 2 .
W a ^ a ^ ( x , y ) = 2 π e 2 ( x α i ) 2 2 y 2 α i 4 + α i 2 ( 4 x 2 + 4 y 2 3 ) 4 α i 3 x + 4 α i x + 1 α i 4 + 3 α i 2 + 1 .
W a ^ a ^ ( x , y ) = 2 π e 2 ( x α i ) 2 2 y 2 × 16 y 4 + 8 y 2 { ( α i 2 x ) 2 2 } + ( α i 2 x ) 2 { ( α i 2 x ) 2 4 } + 2 α i 4 + 4 α i 2 + 2 .
N ( a ^ a ^ ) 2 ( α i ) = ( α i 8 + 10 α i 6 + 25 α i 4 + 15 α i 2 + 1 ) 1 2 , N a ^ 4 ( α i ) = ( α i 8 + 16 α i 6 + 72 α i 4 + 96 α i 2 + 24 ) 1 2 , N a ^ a ^ a ^ 2 ( α i ) = ( α i 8 + 15 α i 6 + 63 α i 4 + 78 α i 2 + 18 ) 1 2 , N a ^ 2 a ^ a ^ ( α i ) = ( α i 8 + 11 α i 6 + 31 α i 4 + 22 α i 2 + 2 ) 1 2 .
F ( a ^ a ^ ) 2 = { N ( a ^ a ^ ) 2 ( α i ) } 2 e ( α i α f ) 2 ( α i 2 α f 2 + 3 α i α f + 1 ) 2 , F a ^ 4 = { N a ^ 4 ( α i ) } 2 α i 8 e ( α i α f ) 2 , F a ^ a ^ a ^ 2 = { N a ^ a ^ a ^ 2 ( α i ) } 2 α f 4 e ( α i α f ) 2 ( α i α f + 3 ) 2 , F a ^ 2 a ^ a ^ = { N a ^ 2 a ^ a ^ ( α i ) } 2 α f 4 e ( α i α f ) 2 ( α i α f + 1 ) 2 .
g λ ( a ^ a ^ ) 2 = { N ( a ^ a ^ ) 2 ( α i ) } 2 ( α i 8 + 12 α i 6 + 38 α i 4 + 32 α i 2 + 4 ) , g λ a ^ 4 = { N a ^ 4 ( α i ) } 2 ( α i 8 + 20 α i 6 + 120 α i 4 + 240 α i 2 + 120 ) , g λ a ^ a ^ a ^ 2 = { N a ^ a ^ a ^ 2 ( α i ) } 2 ( α i 8 + 18 α i 6 + 96 α i 4 + 168 α i 2 + 72 ) , g λ a ^ 2 a ^ a ^ = { N a ^ 2 a ^ a ^ ( α i ) } 2 ( α i 8 + 14 α i 6 + 54 α i 4 + 60 α i 2 + 12 ) ,
E λ ( a ^ a ^ ) 2 ( α i ) = { 4 α i 14 + 62 α i 12 + 382 α i 10 + 1101 α i 8 + 1554 α i 6 + 955 α i 4 + 226 α i 2 + 2 ( 2 α i 12 + 26 α i 10 + 125 α i 8 + 254 α i 6 + 225 α i 4 + 66 α i 2 + 7 ) α i 2 cos ( 2 λ ) + 15 } / { 2 ( α i 8 + 12 α i 6 + 38 α i 4 + 32 α i 2 + 4 ) 2 } , E λ a ^ 4 ( α i ) = 4 { α i 14 + 26 α i 12 + 276 α i 10 + 1488 α i 8 + 4344 α i 6 + 6624 α i 4 + 4896 α i 2 + ( α i 12 + 24 α i 10 + 228 α i 8 + 1056 α i 6 + 2520 α i 4 + 2880 α i 2 + 1440 ) α i 2 cos ( 2 λ ) + 1152 } / { ( α i 8 + 20 α i 6 + 120 α i 4 + 240 α i 2 + 120 ) 2 } ,
E λ a ^ a ^ a ^ 2 ( α i ) = 3 { 2 α i 14 + 49 α i 12 + 486 α i 10 + 2421 α i 8 + 6432 α i 6 + 8784 α i 4 + 5688 α i 2 + 2 ( α i 12 + 22 α i 10 + 189 α i 8 + 780 α i 6 + 1626 α i 4 + 1584 α i 2 + 648 ) α i 2 cos ( 2 λ ) + 1188 } / { 2 ( α i 8 + 18 α i 6 + 96 α i 4 + 168 α i 2 + 72 ) 2 } , E λ a ^ 2 a ^ a ^ ( α i ) = { 6 α i 14 + 103 α i 12 + 714 α i 10 + 2395 α i 8 + 4128 α i 6 + 3344 α i 4 + 1192 α i 2 + 2 ( 3 α i 12 + 46 α i 10 + 271 α i 8 + 716 α i 6 + 886 α i 4 + 400 α i 2 + 72 ) α i 2 cos ( 2 λ ) + 124 } / { 2 ( α i 8 + 14 α i 6 + 54 α i 4 + 60 α i 2 + 12 ) 2 } .
+ 1 ^ ( α i ) = 4 α i 2 ( 4 e 2 α i 2 α i 2 + e 4 α i 2 1 ) ( e 2 α i 2 + 1 ) 2 , + a ^ a ^ ( α i ) = 16 ( N + a ^ a ^ ) 4 e 2 α i 2 α i 2 [ ( N + a ^ a ^ ) 2 e α i 2 { 4 ( 2 α i 4 + 1 ) sinh α i 2 + ( α i 4 + 14 ) α i 2 cosh α i 2 } 4 { ( α i 4 + 4 ) α i sinh α i 2 + 5 α i 3 cosh α i 2 } 2 ] , + a ^ 2 ( α i ) = 16 ( N + a ^ 2 ) 4 e 2 α i 2 [ e α i 2 ( N + a ^ 2 ) 2 { ( 13 α i 4 + 46 ) α i 2 sinh α i 2 + ( α i 8 + 46 α i 4 + 8 ) cosh α i 2 } 4 { ( α i 4 + 14 ) α i 2 sinh α i 2 + 4 ( 2 α i 4 + 1 ) cosh α i 2 } 2 ]
1 ^ ( α i ) = 4 α i 2 ( 4 e 2 α i 2 α i 2 + e 4 α i 2 1 ) ( e 2 α i 2 1 ) 2 , a ^ a ^ ( α i ) = 16 ( N a ^ a ^ 4 ) 4 e 2 α i 2 [ ( N a ^ a ^ ) 2 e α i 2 { ( α i 4 + 14 ) α i 2 sinh α i 2 + 4 ( 2 α i 4 + 1 ) cosh α i 2 } 4 { ( α i 4 + 4 ) α i cosh α i 2 + 5 α i 3 sinh α i 2 } 2 ] , a ^ 2 ( α i ) = 16 ( N a ^ 2 ) 4 e 2 α i 2 [ e α i 2 ( N a ^ 2 ) 2 { ( 13 α i 4 + 46 ) α i 2 cosh α i 2 + ( α i 8 + 46 α i 4 + 8 ) sinh α i 2 } 4 { 4 ( 2 α i 4 + 1 ) sinh α i 2 + ( α i 4 + 14 ) α i 2 cosh α i 2 } 2 ] ,

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