Abstract

Photon statistics effects are analyzed for Mach–Zehnder and Michelson interferometers in which a strong coherent state is inserted into one input port and a squeezed vacuum is inserted in the other input port. We study the effects related to entanglement, for the general case, by using Zassenhaus’s perturbation theory as a function of the squeezing parameter. For the special case in which the strong coherent state exits fully through one output port while the squeezed vacuum exits through the other “dark” output port, the Lie group disentangling method is used. For this case we show that the signal can be amplified with sub-Poissonian photon statistics.

© 2008 Optical Society of America

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    [CrossRef] [PubMed]
  4. D. Denot, T. Bschorr, and M. Freyberger, “Adaptive quantum estimation of phase shifts,” Phys. Rev. Lett. 73, 013824 (2006).
  5. M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278-281 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  8. B. Bohmer and U. Leonhardt, “Correlation interferometer for squeezed light,” Opt. Commun. 118, 181-185 (1995).
    [CrossRef]
  9. G. Leuchs, C. Silberhorn, O. Glöckl, C. Marquardt, and N. Korolkova, “Quantum interferometry with intense optical pulses,” Fortschr. Phys. 51, 409-413 (2003).
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  10. M. G. A. Paris, “Increasing the visibility of multiphoton entanglement,” Fortschr. Phys. 48, 511-515 (2000).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  24. O. Assaf and Y. Ben-Aryeh, “Reduction of quantum noise in the Michelson interferometer by use of squeezed vacuum states,” J. Opt. Soc. Am. B 19, 2716-2721 (2002).
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  29. K. McKenzie, D. A. Shaddock, D. E. McClelland, B. C. Buchler, and P. K. Lam, “Experimental demonstration of a squeezing-enhanced power-recycled Michelson interferometer for gravitational wave detection,” Phys. Rev. Lett. 88, 231102 (2002).
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    [CrossRef] [PubMed]
  35. J. Harms, Y. Chen, S. Chelkowski, A. Franzen, H. Vahlbruch, K. Danzmann, and R. Schnabel, “Squeezed-input, optical-spring, signal-recycled gravitational-wave detectors,” Phys. Rev. D 68, 042001 (2003).
    [CrossRef]
  36. H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett. 97, 011101 (2006).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  40. C. Quesne, “Disentangling q-exponentials: A general approach,” Int. J. Theor. Phys. 43, 545-559 (2004).
    [CrossRef]
  41. M. Suzuki, “On the convergence of exponential operators--the Zassenhaus formula, BCH formula and systematic approximants,” Commun. Math. Phys. 57, 193-200 (1977).
    [CrossRef]
  42. W. Witschel, “Ordered operator expansions by comparison,” J. Phys. A 8, 143-154 (1975).
    [CrossRef]
  43. J. Katriel, M. Rasetti, and A. Solomon, “The q-Zassenhause formula,” Lett. Math. Phys. 37, 11-13 (1996).
    [CrossRef]
  44. A. DasGupta, “Disentanglement formulas: An alternative derivation and some applications to squeezed coherent states,” Am. J. Phys. 64, 1422-1427 (1996).
    [CrossRef]
  45. A. Mufti, H. A. Schmitt, and M. Sargent III, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys. 61, 729-733 (1993).
    [CrossRef]
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  47. M. Zahler and Y. Ben Aryeh, “Photon number distribution of detuned squeezed states,” Phys. Rev. A 43, 6368-6378 (1991).
    [CrossRef] [PubMed]
  48. M. Zahler and Y. Ben-Aryeh, “Photon number distribution of detuned two-mode vacuum and excited squeezed states,” Phys. Rev. A 45, 3194-3202 (1992).
    [CrossRef] [PubMed]
  49. F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211-2237 (1972).
    [CrossRef]
  50. A. Jann and Y. Ben-Aryeh, “Phase-sensitive amplification by the use of degenerate squeezed radiation,” J. Opt. Soc. Am. B 12, 840-846 (1995).
    [CrossRef]
  51. K. Zaheer and M. S. Zubairy, “Phase-sensitive amplification in a two-level system,” Opt. Commun. 69, 37-40 (1988).
    [CrossRef]
  52. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).
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    [CrossRef]
  54. R. A. Fisher, M. M. Nieto, and V. D. Sandberg, “Impossibility of naively generalizing squeezed coherent states,” Phys. Rev. D 29, 1107-1110 (1984).
    [CrossRef]

2006

D. Denot, T. Bschorr, and M. Freyberger, “Adaptive quantum estimation of phase shifts,” Phys. Rev. Lett. 73, 013824 (2006).

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

D. Scholz and M. Weyrauch, “A note on the Zassenhaus product formula,” J. Math. Phys. 47, 033505 (2006).
[CrossRef]

2005

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[CrossRef] [PubMed]

2004

C. Quesne, “Disentangling q-exponentials: A general approach,” Int. J. Theor. Phys. 43, 545-559 (2004).
[CrossRef]

T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclassical Opt. 6, 675-683 (2004).
[CrossRef]

2003

J. Harms, Y. Chen, S. Chelkowski, A. Franzen, H. Vahlbruch, K. Danzmann, and R. Schnabel, “Squeezed-input, optical-spring, signal-recycled gravitational-wave detectors,” Phys. Rev. D 68, 042001 (2003).
[CrossRef]

G. Leuchs, C. Silberhorn, O. Glöckl, C. Marquardt, and N. Korolkova, “Quantum interferometry with intense optical pulses,” Fortschr. Phys. 51, 409-413 (2003).
[CrossRef]

2002

P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
[CrossRef]

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

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

K. McKenzie, D. A. Shaddock, D. E. McClelland, B. C. Buchler, and P. K. Lam, “Experimental demonstration of a squeezing-enhanced power-recycled Michelson interferometer for gravitational wave detection,” Phys. Rev. Lett. 88, 231102 (2002).
[CrossRef] [PubMed]

2001

E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. 69, 79-87 (2001).
[CrossRef]

M. Fiorentino, J. E. Sharping, P. Kumar, D. Levandovsky, and M. Vasilyev, “Soliton squeezing in a Mach-Zehnder fiber interferometer,” Phys. Rev. A 64, 031801 (2001).
[CrossRef]

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

2000

M. G. A. Paris, “Increasing the visibility of multiphoton entanglement,” Fortschr. Phys. 48, 511-515 (2000).
[CrossRef]

S. Scheel, L. Knöll, T. Opatrný, and D. G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
[CrossRef]

1999

M. G. A. Paris, “Entanglement and visibility at the output of a Mach-Zehnder interferometer,” Phys. Rev. A 59, 1615-1621 (1999).
[CrossRef]

1998

1997

S. Inoue and Y. Yamamoto, “Gravitational wave detection using dual input Michelson interferometer,” Phys. Rev. A 236, 183-187 (1997).

1996

K. Sundar, “Amplitude-squeezed quantum states produced by the evolution of a quadrature-squeezed coherent state in a Kerr medium,” Phys. Rev. A 53, 1096-1111 (1996).
[CrossRef] [PubMed]

Z. Y. Ou, “Complementarity and fundamental limit in precision phase measurement,” Phys. Rev. Lett. 77, 2352-2355 (1996).
[CrossRef] [PubMed]

J. Katriel, M. Rasetti, and A. Solomon, “The q-Zassenhause formula,” Lett. Math. Phys. 37, 11-13 (1996).
[CrossRef]

A. DasGupta, “Disentanglement formulas: An alternative derivation and some applications to squeezed coherent states,” Am. J. Phys. 64, 1422-1427 (1996).
[CrossRef]

1995

A. Jann and Y. Ben-Aryeh, “Phase-sensitive amplification by the use of degenerate squeezed radiation,” J. Opt. Soc. Am. B 12, 840-846 (1995).
[CrossRef]

B. Bohmer and U. Leonhardt, “Correlation interferometer for squeezed light,” Opt. Commun. 118, 181-185 (1995).
[CrossRef]

1994

1993

K. Bergman, C. R. Doerr, H. A. Haus, and M. Shirasaki, “Sub-shot-noise measurement with fiber-squeezed optical pulses,” Opt. Lett. 18, 643-645 (1993).
[CrossRef] [PubMed]

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355-1358 (1993).
[CrossRef] [PubMed]

A. Mufti, H. A. Schmitt, and M. Sargent III, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys. 61, 729-733 (1993).
[CrossRef]

1992

M. Zahler and Y. Ben-Aryeh, “Photon number distribution of detuned two-mode vacuum and excited squeezed states,” Phys. Rev. A 45, 3194-3202 (1992).
[CrossRef] [PubMed]

A. Luis and L. L. Sanchez-Soto, “Breaking the standard quantum limit for interferometric measurements,” Opt. Commun. 89, 140-144 (1992).
[CrossRef]

N. C. Wong, “Gravity-wave detection via an optical parametric oscillator,” Phys. Rev. A 45, 3176-3183 (1992).
[CrossRef] [PubMed]

1991

Y. Ben-Aryeh and M. Zahler, “The effects of the beam splitters on the quantum detection properties of the nonlinear Mach-Zehnder interferometer,” Opt. Commun. 85, 132-146 (1991).
[CrossRef]

M. E. Marhic and C. H. Hsia, “Optical amplification and squeezed-light generation in fibre interferometers performing degenerate four-wave mixing,” Quantum Opt. 3, 341-358 (1991).
[CrossRef]

M. Shirasaki, “Squeezing performance of a nonlinear symmetric Mach-Zehnder interferometer using forward degenerate four-wave mixing,” J. Opt. Soc. Am. B 8, 672-680 (1991).
[CrossRef]

M. Zahler and Y. Ben Aryeh, “Photon number distribution of detuned squeezed states,” Phys. Rev. A 43, 6368-6378 (1991).
[CrossRef] [PubMed]

1988

K. Zaheer and M. S. Zubairy, “Phase-sensitive amplification in a two-level system,” Opt. Commun. 69, 37-40 (1988).
[CrossRef]

1987

B. Yurke, P. Grangier, and R. E. Slusher, “Squeezed-state enhanced two-frequency interferometry,” J. Opt. Soc. Am. B 4, 1677-1682 (1987).
[CrossRef]

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278-281 (1987).
[CrossRef] [PubMed]

1986

C. Farina, M. B. Hott, and A. de Souza Dutra, “Zassenhaus formula and the propagator of a particle moving under the action of a constant force,” Am. J. Phys. 54, 377-378 (1986).
[CrossRef]

1984

R. A. Fisher, M. M. Nieto, and V. D. Sandberg, “Impossibility of naively generalizing squeezed coherent states,” Phys. Rev. D 29, 1107-1110 (1984).
[CrossRef]

1981

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

1980

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75-79 (1980).
[CrossRef]

1977

M. Suzuki, “On the convergence of exponential operators--the Zassenhaus formula, BCH formula and systematic approximants,” Commun. Math. Phys. 57, 193-200 (1977).
[CrossRef]

1975

W. Witschel, “Ordered operator expansions by comparison,” J. Phys. A 8, 143-154 (1975).
[CrossRef]

1972

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211-2237 (1972).
[CrossRef]

1963

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10, 84-86 (1963).
[CrossRef]

Am. J. Phys.

E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. 69, 79-87 (2001).
[CrossRef]

C. Farina, M. B. Hott, and A. de Souza Dutra, “Zassenhaus formula and the propagator of a particle moving under the action of a constant force,” Am. J. Phys. 54, 377-378 (1986).
[CrossRef]

A. DasGupta, “Disentanglement formulas: An alternative derivation and some applications to squeezed coherent states,” Am. J. Phys. 64, 1422-1427 (1996).
[CrossRef]

A. Mufti, H. A. Schmitt, and M. Sargent III, “Finite-dimensional matrix representations as calculational tools in quantum optics,” Am. J. Phys. 61, 729-733 (1993).
[CrossRef]

Commun. Math. Phys.

M. Suzuki, “On the convergence of exponential operators--the Zassenhaus formula, BCH formula and systematic approximants,” Commun. Math. Phys. 57, 193-200 (1977).
[CrossRef]

Fortschr. Phys.

G. Leuchs, C. Silberhorn, O. Glöckl, C. Marquardt, and N. Korolkova, “Quantum interferometry with intense optical pulses,” Fortschr. Phys. 51, 409-413 (2003).
[CrossRef]

M. G. A. Paris, “Increasing the visibility of multiphoton entanglement,” Fortschr. Phys. 48, 511-515 (2000).
[CrossRef]

Int. J. Theor. Phys.

C. Quesne, “Disentangling q-exponentials: A general approach,” Int. J. Theor. Phys. 43, 545-559 (2004).
[CrossRef]

J. Math. Phys.

D. Scholz and M. Weyrauch, “A note on the Zassenhaus product formula,” J. Math. Phys. 47, 033505 (2006).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt.

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

T. Corbitt and N. Mavalvala, “Quantum noise in gravitational-wave interferometers,” J. Opt. B: Quantum Semiclassical Opt. 6, 675-683 (2004).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

J. Phys. A

W. Witschel, “Ordered operator expansions by comparison,” J. Phys. A 8, 143-154 (1975).
[CrossRef]

Lett. Math. Phys.

J. Katriel, M. Rasetti, and A. Solomon, “The q-Zassenhause formula,” Lett. Math. Phys. 37, 11-13 (1996).
[CrossRef]

Opt. Commun.

K. Zaheer and M. S. Zubairy, “Phase-sensitive amplification in a two-level system,” Opt. Commun. 69, 37-40 (1988).
[CrossRef]

Y. Ben-Aryeh and M. Zahler, “The effects of the beam splitters on the quantum detection properties of the nonlinear Mach-Zehnder interferometer,” Opt. Commun. 85, 132-146 (1991).
[CrossRef]

B. Bohmer and U. Leonhardt, “Correlation interferometer for squeezed light,” Opt. Commun. 118, 181-185 (1995).
[CrossRef]

A. Luis and L. L. Sanchez-Soto, “Breaking the standard quantum limit for interferometric measurements,” Opt. Commun. 89, 140-144 (1992).
[CrossRef]

Opt. Lett.

Phys. Rev. A

S. Scheel, L. Knöll, T. Opatrný, and D. G. Welsch, “Entanglement transformation at absorbing and amplifying four-port devices,” Phys. Rev. A 62, 043803 (2000).
[CrossRef]

M. Fiorentino, J. E. Sharping, P. Kumar, D. Levandovsky, and M. Vasilyev, “Soliton squeezing in a Mach-Zehnder fiber interferometer,” Phys. Rev. A 64, 031801 (2001).
[CrossRef]

M. G. A. Paris, “Entanglement and visibility at the output of a Mach-Zehnder interferometer,” Phys. Rev. A 59, 1615-1621 (1999).
[CrossRef]

K. Sundar, “Amplitude-squeezed quantum states produced by the evolution of a quadrature-squeezed coherent state in a Kerr medium,” Phys. Rev. A 53, 1096-1111 (1996).
[CrossRef] [PubMed]

N. C. Wong, “Gravity-wave detection via an optical parametric oscillator,” Phys. Rev. A 45, 3176-3183 (1992).
[CrossRef] [PubMed]

S. Inoue and Y. Yamamoto, “Gravitational wave detection using dual input Michelson interferometer,” Phys. Rev. A 236, 183-187 (1997).

M. Zahler and Y. Ben Aryeh, “Photon number distribution of detuned squeezed states,” Phys. Rev. A 43, 6368-6378 (1991).
[CrossRef] [PubMed]

M. Zahler and Y. Ben-Aryeh, “Photon number distribution of detuned two-mode vacuum and excited squeezed states,” Phys. Rev. A 45, 3194-3202 (1992).
[CrossRef] [PubMed]

F. T. Arecchi, E. Courtens, R. Gilmore, and H. Thomas, “Atomic coherent states in quantum optics,” Phys. Rev. A 6, 2211-2237 (1972).
[CrossRef]

Phys. Rev. D

R. A. Fisher, M. M. Nieto, and V. D. Sandberg, “Impossibility of naively generalizing squeezed coherent states,” Phys. Rev. D 29, 1107-1110 (1984).
[CrossRef]

J. Harms, Y. Chen, S. Chelkowski, A. Franzen, H. Vahlbruch, K. Danzmann, and R. Schnabel, “Squeezed-input, optical-spring, signal-recycled gravitational-wave detectors,” Phys. Rev. D 68, 042001 (2003).
[CrossRef]

P. Purdue and Y. Chen, “Practical speed meter designs for quantum nondemolition gravitational-wave interferometers,” Phys. Rev. D 66, 122004 (2002).
[CrossRef]

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

H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin, “Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics,” Phys. Rev. D 65, 022002 (2001).
[CrossRef]

Phys. Rev. Lett.

Z. Y. Ou, “Complementarity and fundamental limit in precision phase measurement,” Phys. Rev. Lett. 77, 2352-2355 (1996).
[CrossRef] [PubMed]

D. Denot, T. Bschorr, and M. Freyberger, “Adaptive quantum estimation of phase shifts,” Phys. Rev. Lett. 73, 013824 (2006).

M. Xiao, L.-A. Wu, and H. J. Kimble, “Precision measurement beyond the shot-noise limit,” Phys. Rev. Lett. 59, 278-281 (1987).
[CrossRef] [PubMed]

M. J. Holland and K. Burnett, “Interferometric detection of optical phase shifts at the Heisenberg limit,” Phys. Rev. Lett. 71, 1355-1358 (1993).
[CrossRef] [PubMed]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Coherent control of vacuum squeezing in the gravitational-wave detection band,” Phys. Rev. Lett. 97, 011101 (2006).
[CrossRef] [PubMed]

K. McKenzie, D. A. Shaddock, D. E. McClelland, B. C. Buchler, and P. K. Lam, “Experimental demonstration of a squeezing-enhanced power-recycled Michelson interferometer for gravitational wave detection,” Phys. Rev. Lett. 88, 231102 (2002).
[CrossRef] [PubMed]

C. M. Caves, “Quantum-mechanical radiation-pressure fluctuations in an interferometer,” Phys. Rev. Lett. 45, 75-79 (1980).
[CrossRef]

H. Vahlbruch, S. Chelkowski, B. Hage, A. Franzen, K. Danzmann, and R. Schnabel, “Demonstration of a squeezed-light-enhanced power- and signal-recycled Michelson interferometer,” Phys. Rev. Lett. 95, 211102 (2005).
[CrossRef] [PubMed]

R. J. Glauber, “Photon correlations,” Phys. Rev. Lett. 10, 84-86 (1963).
[CrossRef]

Quantum Opt.

M. E. Marhic and C. H. Hsia, “Optical amplification and squeezed-light generation in fibre interferometers performing degenerate four-wave mixing,” Quantum Opt. 3, 341-358 (1991).
[CrossRef]

Other

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, 1994).
[CrossRef]

S. W. Hawking and W. Israel, Three Hundred Years of Gravitation (Cambridge U. Press, 1987).

W. H. Steeb and Y. Hardy, Problems and Solutions in Quantum Computing and Quantum Information, 2nd ed. (World Scientific, 2006).

H. C. Ohanian and R. Ruffini, Gravitation and Spacetime, 2nd ed. (Norton, 1994).

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

R. Gilmore, Lie Groups, Lie Algebras, and Some of Their Applications (Wiley, 1974).

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Figures (5)

Fig. 1
Fig. 1

Probability distribution of the output with input α 2 = 20 input. The effects of the squeezing and optical paths are shown for p ( n 1 , n 2 ) of the interferometer for an input of α 2 = 20 . The contour lines connect all points of equal probability, denoted on them, for different values of ζ and δ. With no squeezing or phase shift the photons are distributed equally (maximum at n 1 = n 2 = 10 ). The squeezing moves the maximum of the distribution to a lower photon number, while the phase shift distributes the photons unevenly between the two arms of the interferometer.

Fig. 2
Fig. 2

Probability distribution of the output with input α 2 = 20 input. The effects of the squeezing and optical paths are shown for one of the arms of the interferometer for an input of α 2 = 20 and γ = π 4 + δ .

Fig. 3
Fig. 3

Entanglement effect on probability distribution with input α 2 = 20 . The effects of the entanglement shown for one of the arms of the interferometer, for an input of α 2 = 20 , γ = π 4 + δ , and ζ = 0 , 0.1 , 0.2 , 0.3 , 0.4 , 0.5 . The output with entanglement (shown as a dashed curve) from Eq. (39) is compared to the outcome without the entanglement part from Eq. (29).

Fig. 4
Fig. 4

Strength [real (left) and imaginary (right) parts] of the coherent parameter of the dark port of the interferometer. The maximum is observed for θ = 2 ϕ .

Fig. 5
Fig. 5

Photon number probability distribution for α δ 2 = 10 and different ζ. The photon number distribution in the dark port for α δ 2 = 10 and different values of ζ of the squeezed state and θ = 2 ϕ in the input ports. An increasing sub-Poisson distribution is achieved as ζ is increased and the maximum point of the photon number distribution is shifted to a higher value of n , as can be seen from the comparison of the graph for ζ = 0.3 , 0.6 , 0.9 with the equivalent Poisson distribution of the dashed graph of a coherent state with α 2 = 24 , 44 , 75 .

Tables (2)

Tables Icon

Table 1 Generators of the SU(1,1) and SU(2) Algebra for the Disentanglement of the Operators E ̂ 12 ( ϕ 1 ζ ) and E ̂ 12 ( ϕ 2 ζ )

Tables Icon

Table 2 Parameters for the Disentanglement of the SU(1,1) and SU(2) Algebra

Equations (104)

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ψ in = α , ζ ,
= D ̂ 1 ( α ) S ̂ 2 ( ζ ) 0 , 0 ,
D ̂ 1 ( α ) = exp { α a ̂ 1 α * a ̂ 1 } ,
S ̂ 2 ( ζ ) = exp { 1 2 ( ζ * a ̂ 2 2 ζ a ̂ 2 ) 2 } ,
( a ̂ 1 a ̂ 2 ) = ( cos γ sin γ sin γ cos γ ) ( b ̂ 1 b ̂ 2 ) ,
ψ out = exp { α ( cos γ b ̂ 1 + sin γ b ̂ 2 ) α * ( cos γ b ̂ 1 + sin γ b ̂ 2 ) } × exp { 1 2 ζ * ( sin γ b ̂ 1 + cos γ b ̂ 2 ) 2 1 2 ζ ( sin γ b ̂ 1 + cos γ b ̂ 2 ) 2 } 0 , 0 = exp { ( α cos γ b ̂ 1 α * cos γ b ̂ 1 ) + ( α sin γ b ̂ 2 α * sin γ b ̂ 2 ) } exp { 1 2 ( ζ * sin 2 γ b ̂ 1 2 ζ sin 2 γ b ̂ 1 2 ) + 1 2 ( ζ * cos 2 γ b ̂ 2 2 ζ cos 2 γ b ̂ 2 2 ) ( ζ * sin γ cos γ b ̂ 1 b ̂ 2 ζ sin γ cos γ b ̂ 1 b ̂ 2 ) } 0 , 0 = D ̂ 1 ( β 1 ) D ̂ 2 ( β 2 ) exp { 1 2 ( ζ * sin 2 γ b ̂ 1 2 ζ sin 2 γ b ̂ 1 2 ) + 1 2 ( ζ * cos 2 γ b ̂ 2 2 ζ cos 2 γ b ̂ 2 2 ) ( ζ * sin γ cos γ b ̂ 1 b ̂ 2 ζ sin γ cos γ b ̂ 1 b ̂ 2 ) } 0 , 0 ,
β 1 = α cos γ ,
β 2 = α sin γ .
ψ out = exp { 1 2 ( ζ * sin 2 γ b ̂ 1 2 ζ sin 2 γ b ̂ 1 2 ) + 1 2 ( ζ * cos 2 γ b ̂ 2 2 ζ cos 2 γ b ̂ 2 2 ) + sin γ cos γ ( ζ b ̂ 1 b ̂ 2 ζ * b ̂ 1 b ̂ 2 ) } D ̂ 1 ( β 1 ) D ̂ 2 ( β 2 ) 0 , 0 ,
= exp { A ̂ + B ̂ } D ̂ 1 ( β 1 ) D ̂ 2 ( β 2 ) 0 , 0 ,
A ̂ = 1 2 sin 2 γ ( ζ * b ̂ 1 2 ζ b ̂ 1 2 ) + 1 2 cos 2 γ ( ζ * b ̂ 2 2 ζ b ̂ 2 2 ) ,
B ̂ = sin γ cos γ ( ζ b ̂ 1 b ̂ 2 ζ * b ̂ 1 b ̂ 2 ) .
exp { A ̂ + B ̂ } = exp { C ̂ m } exp { C ̂ 2 } exp { B ̂ } exp { A ̂ } ,
e ϵ ( A ̂ + B ̂ ) e ϵ 4 C ̂ 4 e ϵ 3 C ̂ 3 e ϵ 2 C ̂ 2 e ϵ B ̂ e ϵ A ̂ ,
k ϵ k k ! ( A ̂ + B ̂ ) k r A r B r 2 r 3 r 4 ϵ 4 r 4 + 3 r 3 + 2 r 2 + r B + r A r A ! r B ! r 2 ! r 3 ! r 4 ! C ̂ 4 r 4 C ̂ 3 r 3 C ̂ 2 r 2 B ̂ r B A ̂ r A .
ψ out = exp { B ̂ } exp { A ̂ } D ̂ 1 ( β 1 ) D ̂ 2 ( β 2 ) 0 , 0 ,
= E ̂ 12 ( ϕ 1 ζ ) S ̂ 1 ( ψ 1 ζ ) D ̂ 1 ( β 1 ) S ̂ 2 ( ψ 2 ζ ) D ̂ 2 ( β 2 ) 0 , 0 ,
ψ 1 = sin 2 γ ,
ψ 2 = cos 2 γ ,
ϕ 1 = 2 sin γ cos γ ,
E ̂ 12 ( ϕ 1 ζ ) = exp { 1 2 ϕ 1 ( ζ * b ̂ 1 b ̂ 2 ζ b ̂ 1 b ̂ 2 ) } ,
1 2 ! ( A ̂ + B ̂ ) 2 1 2 A ̂ 2 + 1 2 B ̂ 2 + B ̂ A ̂ + C ̂ 2 ,
1 2 ( A ̂ 2 + A ̂ B ̂ + B ̂ A ̂ + B ̂ 2 ) 1 2 A ̂ 2 + 1 2 B ̂ 2 + B ̂ A ̂ + C ̂ 2 .
C 2 ̂ = 1 2 A ̂ B ̂ 1 2 B ̂ A ̂ = 1 2 [ A ̂ , B ̂ ] .
C ̂ 2 = 1 2 sin 3 γ cos γ ( ζ 2 b ̂ 1 b ̂ 2 ζ 2 b ̂ 1 b ̂ 2 ) 1 2 sin γ cos 3 γ ( ζ 2 b ̂ 1 b ̂ 2 ζ 2 b ̂ 1 b ̂ 2 ) = 1 2 sin γ cos γ cos 2 γ ( ζ 2 b ̂ 1 b ̂ 2 ζ 2 b ̂ 1 b ̂ 2 ) .
ψ out = E ̂ 12 ( ϕ 2 ζ ) E ̂ 12 ( ϕ 1 ζ ) S ̂ 1 ( ψ 1 ζ ) D ̂ 1 ( β 1 ) S ̂ 2 ( ψ 2 ζ ) D ̂ 2 ( β 2 ) 0 , 0 ,
ϕ 2 = sin γ cos γ cos 2 γ ,
E ̂ 12 ( ϕ 2 ζ ) = exp { 1 2 ϕ 2 ζ 2 ( b ̂ 1 b ̂ 2 b ̂ 1 b ̂ 2 ) } ,
S ̂ 1 ( ψ 1 ζ ) D ̂ 1 ( β 1 ) S ̂ 2 ( ψ 2 ζ ) D ̂ 2 ( β 2 ) 0 , 0 = n 1 , n 2 = 0 ( e i θ tanh r 1 ) n 1 2 ( e i θ tanh r 2 ) n 2 2 2 [ ( n 1 + n 2 ) 2 ] n 1 ! n 2 ! cosh r 1 cosh r 2 × exp [ 1 2 ( β 1 2 + β 2 2 e i θ β 1 2 tanh r 1 e i θ β 2 2 tanh r 2 ) ] H n 1 ( β 1 e i θ 2 2 cosh r 1 sinh r 1 ) × H n 2 ( β 2 e i θ 2 2 cosh r 2 sinh r 2 ) n 1 , n 2 = n 1 , n 2 = 0 F n 1 n 2 ( β 1 , β 2 , ζ 1 , ζ 2 ) n 1 , n 2 ,
[ J ̂ 0 , J ̂ ± ] = ± J ̂ ± ,
[ J ̂ + , J ̂ ] = { 2 J ̂ 0 for SU ( 2 ) 2 J ̂ 0 for SU ( 1 , 1 ) } .
P ̂ = ξ J ̂ + ξ * J ̂ ,
e λ J ̂ + e μ J ̂ 0 e ν J ̂ ,
E ̂ 12 ( ϕ 1 ζ ) = exp { b ̂ 1 b ̂ 2 e i θ tanh r sin 2 γ 2 } exp { ( 1 + n ̂ 1 + n ̂ 2 ) ln ( cosh r sin 2 γ 2 ) } × exp { b ̂ 1 b ̂ 2 e i θ tanh r sin 2 γ 2 } ,
E ̂ 12 ( ϕ 1 ζ ) = exp { b ̂ 1 b ̂ 2 e i θ tan ξ } exp { ( n ̂ 1 + n ̂ 2 ) ln ( cos ξ ) } exp { b ̂ 1 b ̂ 2 e i θ tan ξ } .
ψ out = n 1 , n 2 = 0 exp [ ( 1 + n 1 + n 2 ) ln ( cosh r sin 2 γ 2 ) ] F n 1 n 2 ( β 1 , β 2 , ζ 1 , ζ 2 ) × { n 1 , n 2 + m 2 = 1 ( n 1 + m 2 ) ( n 2 + m 2 ) m 2 ! ( e i θ tanh r sin 2 γ 2 ) m 2 n 1 + m 2 , n 2 + m 2 + m 1 = 1 n 12 exp [ 2 m 1 ln ( cosh r sin 2 γ 2 ) ] ( n 1 m 1 + 1 ) ( n 2 m 1 + 1 ) m 1 ! ( e i θ tanh r sin 2 γ 2 ) m 1 × [ m 2 = 1 ( n 1 m 1 + m 2 ) ( n 2 m 1 + m 2 ) m 2 ! ( e i θ tanh r sin 2 γ 2 ) m 2 n 1 m 1 + m 2 , n 2 m 1 + m 2 + n 1 m 1 , n 2 m 1 ] } ,
sin ( π 4 + δ ) 1 2 ( 1 + δ ) ,
cos ( π 4 + δ ) 1 2 ( 1 δ ) .
p ( n 1 , n 2 ) = exp [ ( 1 + n 1 + n 2 ) ln ( cosh r 2 ) ] { F n 1 n 2 ( β 1 , β 2 , ζ 1 , ζ 2 ) + m 2 = 1 n 12 1 n 1 n 2 m 2 ! exp [ 2 m 2 ln ( cosh r 2 ) ] ( tanh r 2 ) m 2 F n 1 m 2 , n 2 m 2 ( β 1 , β 2 , ζ 1 , ζ 2 ) + m 1 = 1 ( n 1 + 1 ) ( n 2 + 1 ) m 1 ! ( tanh r 2 ) m 1 F n 1 + m 1 , n 2 + m 1 ( β 1 , β 2 , ζ 1 , ζ 2 ) + m 1 = 1 m 2 = 1 n 12 + m 1 1 n 1 n 2 ( n 1 m 2 + 1 ) ( n 2 m 2 + 1 ) m 1 ! m 2 ! ( 1 ) m 1 ( tanh r 2 ) m 1 + m 2 exp [ 2 m 2 ln ( cosh r 2 ) ] × F n 1 + m 1 m 2 , n 2 + m 1 m 2 ( β 1 , β 2 , ζ 1 , ζ 2 ) } 2 .
E ̂ 12 ( ϕ 1 ζ ) exp { i 2 ϕ 1 ( r α 2 cos γ sin γ sin ( θ 2 ϕ ) ) } .
J ̃ + = ( 0 1 0 0 ) , J ̃ = ( 0 0 1 0 ) , J ̃ z = ( 1 2 0 0 1 2 ) ,
J ̃ x = ( 0 1 2 1 2 0 ) , J ̃ y = ( 0 1 2 i 1 2 i 0 ) ,
A ̃ 1 ( I ̂ + 2 J ̃ z ) 2 ,
A ̃ 2 ( I ̂ 2 J ̃ z ) 2 ,
B ̃ 2 J ̃ x = J ̃ + + J ̃ ,
C 2 ̃ 2 i J ̃ y = J ̃ + J ̃ .
A ̃ 1 = ζ * b ̂ 1 2 ζ b ̂ 1 2 2 ζ ( 1 0 0 0 ) ,
A ̃ 2 = ζ b ̂ 2 2 ζ * b ̂ 2 2 2 ζ ( 0 0 0 1 ) ,
B ̃ = ζ b ̂ 1 b ̂ 2 ζ * b ̂ 1 b ̂ 2 ζ ( 0 1 1 0 ) ,
C 2 ̃ = b ̂ 1 b ̂ 2 b ̂ 1 b ̂ 2 ( 0 1 1 0 ) .
A ̂ = ζ ( sin 2 γ A ̃ 1 cos 2 γ A ̃ 2 ) ,
B ̂ = ζ sin γ cos γ B ̃ ,
C ̂ 2 = 1 2 ζ 2 cos γ sin γ cos 2 γ C 2 ̃ ,
D ̂ = ζ 2 cos 2 γ sin 2 γ cos 2 γ [ A ̃ 1 + A ̃ 2 ] .
[ A ̂ , B ̂ ] = 2 C ̂ 2 ,
[ A ̂ , C ̂ 2 ] = 1 2 ζ 2 cos 2 2 γ B ̂ ,
[ B ̂ , C ̂ 2 ] = D ̂ = ζ 3 cos 2 γ sin 2 γ cos 2 γ ( A ̃ 1 + A ̃ 2 ) .
[ A ̃ 1 , A ̃ 2 ] = 0 ,
[ A ̃ 1 , B ̃ ] = [ A ̃ 2 , B ̃ ] = C ̃ 2 ,
[ A ̃ 1 , C ̃ 2 ] = [ A ̃ 2 , C ̃ 2 ] = B ̃ ,
[ B ̃ , C ̃ 2 ] = 2 ( A ̃ 1 + A ̃ 2 ) .
ψ out = exp { A ̂ + B ̂ } D ̂ 1 ( β 1 ) D ̂ 2 ( β 2 ) 0 , 0 ,
= exp { η J ̃ } exp { η + J + ̃ } exp { η 2 A ̃ 2 } exp { η 1 A ̃ 1 } D ̂ 1 ( β 1 ) D ̂ 2 ( β 2 ) 0 , 0 .
exp { A ̂ + B ̂ } = exp { ζ ( sin 2 γ cos γ sin γ cos γ sin γ cos 2 γ ) } exp { ζ M ̃ } .
exp { ζ M ̃ } = n = 0 ( ζ M ̃ ) n n ! = I + n = 1 M ̃ ζ n n ! = I + ( e ζ 1 ) M ̃ .
exp { η J ̃ } exp { η + J + ̃ } exp { η 2 A ̃ 2 } exp { η 1 A ̃ 1 } = ( 1 0 η 1 ) ( 1 η + 0 1 ) ( 1 0 0 e η 2 ) ( e η 1 0 0 1 ) = ( e η 1 η + e η 2 η e η 1 e η 2 ( 1 + η η + ) ) .
η 1 = ln [ e ζ sin 2 γ + cos 2 γ ] ,
η 2 = ln [ e ζ e ζ sin 2 γ + cos 2 γ ] ,
η = ( e ζ 1 ) sin γ cos γ e ζ sin 2 γ + cos 2 γ ,
η + = ( e ζ 1 ) sin γ cos γ [ e ζ sin 2 γ + cos 2 γ ] e ζ .
ψ out = exp { η B ̃ C ̃ 2 2 } exp { η + B ̃ + C ̃ 2 2 } exp { η 2 A ̃ 2 } exp { η 1 A ̃ 1 } D ̂ 1 ( β 1 ) D ̂ 2 ( β 2 ) 0 , 0 .
sin ( π 2 + δ ) 1 ,
cos ( π 2 + δ ) δ ,
η 1 ζ ,
η 2 0 ,
η ( e ζ 1 ) δ ,
η + ( 1 e ζ ) δ ,
ψ out = exp { δ ( 1 cosh ζ ) C ̃ 2 } exp { δ sinh ζ B ̃ } S ̂ 1 ( ζ ) D ̂ 1 ( α δ ) D ̂ 2 ( α ) 0 , 0 .
ψ out 1 = D ̂ ( α δ ( 1 cosh r ) ) D ̂ ( α * δ e i θ sinh r ) S ̂ ( ζ ) D ̂ ( α δ ) 0 1 .
D ̂ ( α 2 ) D ̂ ( α 1 ) = D ̂ ( α 1 + α 2 ) exp { 1 2 ( α 2 α 1 * α 2 * α 1 ) } ,
D ̂ ( α ) S ̂ ( ζ ) = S ̂ ( ζ ) D ̂ ( α cosh r + α * e i θ sinh r ) ,
ψ out 1 = G 1 G 2 S ̂ ( ζ ) D ̂ ( κ ) 0 1 ,
G 1 = exp { i p δ sinh r sin ( θ 2 ϕ ) ( 1 cosh r ) } ,
G 2 = exp { i α 2 δ 2 sinh r [ sin ( θ 2 ϕ ) 2 cosh r sin ( θ 2 ϕ ) ] } ,
κ = α δ [ ( cosh r cosh 2 r + e i ( θ 2 ϕ ) sinh r sinh 2 r 2 e i ( θ 2 ϕ ) sinh r cosh r 1 ) ] .
n ̂ 1 = 0 D ̂ ( κ ) S ̂ ( ζ ) b ̂ 1 b ̂ 1 S ̂ ( ζ ) D ̂ ( κ ) 0 = 0 D ̂ ( κ ) ( b ̂ 1 cosh r b ̂ 1 e i θ sinh r ) ( b ̂ 1 cosh r b ̂ 1 e i θ sinh r ) D ̂ ( κ ) 0 = κ 2 cosh 2 r κ 2 e i θ sinh r cosh r κ * 2 e i θ sinh r cosh r + κ 2 sinh 2 r 0 = κ 2 [ cosh 2 r cos ( θ 2 ϕ ) sinh 2 r ] + sinh 2 r .
Δ n ̂ 1 2 = κ 2 [ cosh 4 r cos ( θ 2 ϕ ) sinh 4 r ] + 2 sinh 2 r cosh 2 r .
η 1 = ln [ ( 1 + ζ + ) sin 2 γ + cos 2 γ ] ln [ 1 + ζ sin 2 γ ] ζ sin 2 γ .
η 2 = ln [ sin 2 γ + cos 2 γ ( 1 + ζ + ) ] 1 ln [ ζ cos 2 γ 1 ] 1 ζ cos 2 γ .
exp { η 2 A ̃ 2 } exp { η 1 A ̃ 1 } exp { ζ ( sin 2 γ A ̃ 1 cos 2 γ A ̃ 2 ) } exp { A ̂ } .
η + η ζ cos γ sin γ ,
exp { η B ̃ C ̃ 2 2 } exp { η + B ̃ + C ̃ 2 2 } exp { B ̃ ζ cos γ sin γ } .
L ̂ = 1 3 ! ( A ̂ + B ̂ ) 3 = 1 6 ( A ̂ 3 + A ̂ B ̂ A ̂ + B ̂ A ̂ 2 + B ̂ 2 A ̂ + A ̂ 2 B ̂ + A ̂ B ̂ 2 + B ̂ A ̂ B ̂ + B ̂ 3 ) ,
R ̂ = 1 6 A ̂ 3 + 1 6 B ̂ 3 + C ̂ 3 + 1 2 B ̂ A ̂ 2 + 1 2 B ̂ 2 A ̂ + C ̂ 2 B ̂ + C ̂ 2 A ̂ = 1 6 ( A ̂ 3 + B ̂ 3 + 6 C ̂ 3 + 3 B ̂ A ̂ 2 + 3 B ̂ 2 A ̂ + 3 A ̂ B ̂ 2 3 B ̂ A ̂ B ̂ + 3 A ̂ B ̂ A ̂ 3 B ̂ A ̂ 2 ) .
C ̂ 3 = 1 6 ( A ̂ 2 B ̂ + B ̂ A ̂ 2 2 B ̂ 2 A ̂ 2 A ̂ B ̂ 2 + 4 B A ̂ B ̂ 2 A ̂ B ̂ A ̂ ) = 1 6 ( 2 [ B ̂ , [ A ̂ , B ̂ ] ] + [ A ̂ , [ A ̂ , B ̂ ] ] ) .
C ̂ 3 = 1 3 [ A ̂ , C ̂ 2 ] + 2 3 [ B ̂ , C ̂ 2 ] ,
1 3 [ A ̂ , C ̂ 2 ] = 1 6 cos γ sin γ cos 2 2 γ ζ 2 ( ζ b ̂ 1 b ̂ 2 ζ * b ̂ 1 b ̂ 2 ) ,
2 3 [ B ̂ , C ̂ 2 ] = 1 3 cos 2 γ sin 2 γ cos 2 γ ζ 2 ( ( ζ * b ̂ 1 2 ζ b ̂ 1 2 ) ( ζ * b ̂ 2 2 ζ b ̂ 2 2 ) ) .
ψ out = E ̂ 12 ( ϕ 3 ζ ) S ̂ 1 ( ψ 3 ζ ) S ̂ 2 ( ψ 3 ζ ) E ̂ 12 ( ϕ 2 ) E ̂ 12 ( ϕ 1 ζ ) × S ̂ 1 ( ψ 1 ζ ) D ̂ 1 ( β 1 ) S ̂ 2 ( ψ 2 ζ ) D ̂ 2 ( β 2 ) 0 , 0 ,
ϕ 3 = 1 3 sin γ cos γ cos 2 2 γ ζ 2 ,
ψ 3 = 2 3 sin 2 γ cos 2 γ cos 2 γ ζ 2 ,
L ̂ = 1 4 ! ( A ̂ + B ̂ ) 4 = 1 24 ( A ̂ 4 + A ̂ B ̂ A ̂ 2 + B ̂ A ̂ 3 + B ̂ 2 A ̂ 2 + A ̂ 2 B ̂ A ̂ + A ̂ B ̂ 2 A ̂ + B ̂ A ̂ B ̂ A ̂ + B ̂ 3 A ̂ + A ̂ 3 B ̂ + A ̂ B ̂ A ̂ B ̂ + B ̂ A ̂ 2 B ̂ + B ̂ 2 A ̂ B ̂ + A ̂ 2 B ̂ 2 + A ̂ B ̂ 3 + B ̂ A ̂ B ̂ 2 + B ̂ 4 ) ,
R ̂ = 1 24 A ̂ 4 + 1 24 B ̂ 4 + 1 6 B ̂ A ̂ 3 + 1 6 B ̂ 3 A ̂ + 1 4 B ̂ 2 A ̂ 2 + 1 2 C ̂ 2 B ̂ 2 + 1 2 C ̂ 2 A ̂ 2 + C ̂ 2 B ̂ A ̂ + 1 2 C ̂ 2 2 + C ̂ 3 B ̂ + C ̂ 3 A ̂ + C ̂ 4 = 1 24 A ̂ 4 + 1 24 B ̂ 4 + 1 6 B ̂ A ̂ 3 + 1 6 B ̂ 3 A ̂ + 1 4 B ̂ 2 A ̂ 2 + 1 4 A ̂ B ̂ 3 1 4 B ̂ A ̂ B ̂ 2 + 1 4 A ̂ B ̂ A ̂ 2 1 4 B ̂ A ̂ 3 + 1 2 A ̂ B ̂ 2 A ̂ 1 2 B ̂ A ̂ B ̂ A ̂ + 1 8 A ̂ B ̂ A ̂ B ̂ 1 8 B ̂ A ̂ 2 B ̂ 1 8 A ̂ B ̂ 2 A ̂ + 1 8 B ̂ A ̂ B ̂ A ̂ + 1 6 A ̂ 2 B ̂ 2 + 1 6 B ̂ A ̂ 2 B ̂ 1 3 B ̂ 2 A ̂ B ̂ 1 3 A ̂ B ̂ 3 + 2 3 B ̂ A ̂ B ̂ 2 1 3 A ̂ B ̂ A ̂ B ̂ + 1 6 A ̂ 2 B ̂ A ̂ + 1 6 B ̂ A ̂ 3 1 3 B ̂ 2 A ̂ 2 1 3 A ̂ B ̂ 2 A ̂ + 2 3 B ̂ A ̂ B ̂ A ̂ 1 3 A ̂ B ̂ A ̂ 2 + C ̂ 4 .
C ̂ 4 = 1 24 ( 3 A ̂ B ̂ A ̂ 2 B ̂ A ̂ 3 + 3 B ̂ 2 A ̂ 2 3 A ̂ 2 B ̂ A ̂ 6 B ̂ A ̂ B ̂ A ̂ 3 B ̂ 3 A ̂ + A ̂ 3 B ̂ + 6 A ̂ B ̂ A ̂ B ̂ + 9 B ̂ 2 A ̂ B ̂ 3 A ̂ 2 B ̂ 2 + 3 A ̂ B ̂ 3 9 B ̂ A ̂ B ̂ 2 ) = 1 24 ( [ A ̂ , [ A ̂ , [ A ̂ , B ̂ ] ] ] + 3 [ B ̂ , [ B ̂ , [ A ̂ , B ̂ ] ] ] + 3 [ B ̂ , [ A ̂ , [ A ̂ , B ̂ ] ] ] ) ,

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