Abstract

High-precision inertial sensing demonstrations with light pulse atom interferometry have typically used Raman pulses having durations orders of magnitude shorter than the dwell time between interferometer pulses. Environmentally robust sensors operating at high-bandwidth will be required to operate at short (millisecond scale) dwell times between Raman pulses. In such an operational mode, the Raman pulse duration becomes an appreciable fraction of the dwell time between pulses. In addition, high-precision inertial sensing applications have typically been demonstrated in mildly dynamic or nondynamic environments having low rate of change of inertial input, ensuring that applied Raman pulses satisfy the Raman resonance condition. Application of nonresonant pulses will be inevitable in sensors registering time-varying inertial input. We present a diagrammatic technique for calculation of atomic output state populations for multipulse atom optics manipulations that explicitly account for the effects of finite pulse duration and finite Raman detuning effects on the laser-induced atomic phase. We analyze several atom interferometer sequences. We report accelerometer and gyroscope phase evolution for fixed Raman laser frequency difference incorporating corrections in powers of the ratio of pulse duration to time interval between interferometer pulses. Our accelerometer result agrees with other published results.

© 2011 Optical Society of America

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References

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  1. T. L. Gustavson, P. Bouyer, and M. A. Kasevich, “Precision rotation measurements with an atom interferometer gyroscope,” Phys. Rev. Lett. 78, 2046–2049 (1997).
    [CrossRef]
  2. 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, 971–974 (1998).
    [CrossRef]
  3. T. L. Gustavson, A. Landragin, and M. A. Kasevich, “Rotation sensing with a dual atom interferometer Sagnac gyroscope,” Class. Quant. Gravity 17, 2385–2398 (2000).
    [CrossRef]
  4. A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38, 25–61(2001).
    [CrossRef]
  5. J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65, 033608 (2002).
    [CrossRef]
  6. D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
    [CrossRef]
  7. H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
    [CrossRef] [PubMed]
  8. T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
    [CrossRef]
  9. Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
    [CrossRef]
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    [CrossRef] [PubMed]
  12. D. S. Weiss, B. C. Young, and S. Chu, “Precision measurement of ℏ/mCs based on photon recoil using laser-cooled atoms and atomic interferometry,” Appl. Phys. B 59, 217–253 (1994).
    [CrossRef]
  13. B. Young, M. Kasevich, and S. Chu, “Precision atom interferometry with light pulses,” Atom Interferometry, P.Berman, ed. (Academic, 1997), pp. 363–406.
    [CrossRef]
  14. A. Peters, “High precision gravity measurements using atom interferometry,” Ph.D. thesis (Stanford University, 1998).
  15. P. Storey and C. Cohen-Tannoudji, “The Feynman path integral approach to atomic interferometry. A tutorial,” J. Phys. II (France) 4, 1999–2027 (1994).
    [CrossRef]
  16. D. A. Steck, “Quantum and atom optics,” http://steck.us/teaching (2006).
  17. J. K. Stockton, “Continuous quantum measurement of cold alkali-atom spins,” Ph.D. thesis (California Institute of Technology, 2007).
  18. G. Baym, Lectures on Quantum Mechanics (Benjamin/Cummings Publication, 1969).
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    [CrossRef]
  20. E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
    [CrossRef]
  21. C. Antoine, “Matter wave beam splitters in gravito-inertial and trapping potentials: generalized ttt scheme for atom interferometry,” Appl. Phys. B 84, 585–597 (2006).
    [CrossRef]
  22. C. Antoine, “Rotating matter-wave beam splitters and consequences for atom gyrometers,” Phys. Rev. A 76, 033609 (2007).
    [CrossRef]

2011 (1)

2010 (2)

Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
[CrossRef]

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef] [PubMed]

2009 (1)

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

2007 (1)

C. Antoine, “Rotating matter-wave beam splitters and consequences for atom gyrometers,” Phys. Rev. A 76, 033609 (2007).
[CrossRef]

2006 (2)

C. Antoine, “Matter wave beam splitters in gravito-inertial and trapping potentials: generalized ttt scheme for atom interferometry,” Appl. Phys. B 84, 585–597 (2006).
[CrossRef]

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

2002 (1)

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65, 033608 (2002).
[CrossRef]

2001 (1)

A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38, 25–61(2001).
[CrossRef]

2000 (1)

T. L. Gustavson, A. Landragin, and M. A. Kasevich, “Rotation sensing with a dual atom interferometer Sagnac gyroscope,” Class. Quant. Gravity 17, 2385–2398 (2000).
[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, 971–974 (1998).
[CrossRef]

1997 (1)

T. L. Gustavson, P. Bouyer, and M. A. Kasevich, “Precision rotation measurements with an atom interferometer gyroscope,” Phys. Rev. Lett. 78, 2046–2049 (1997).
[CrossRef]

1994 (2)

D. S. Weiss, B. C. Young, and S. Chu, “Precision measurement of ℏ/mCs based on photon recoil using laser-cooled atoms and atomic interferometry,” Appl. Phys. B 59, 217–253 (1994).
[CrossRef]

P. Storey and C. Cohen-Tannoudji, “The Feynman path integral approach to atomic interferometry. A tutorial,” J. Phys. II (France) 4, 1999–2027 (1994).
[CrossRef]

1992 (1)

K. Moler, D. S. Weiss, M. Kasevich, and S. Chu, “Theoretical analysis of velocity-selective Raman transitions,” Phys. Rev. A 45, 342–348 (1992).
[CrossRef] [PubMed]

1950 (1)

E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
[CrossRef]

Antoine, C.

C. Antoine, “Rotating matter-wave beam splitters and consequences for atom gyrometers,” Phys. Rev. A 76, 033609 (2007).
[CrossRef]

C. Antoine, “Matter wave beam splitters in gravito-inertial and trapping potentials: generalized ttt scheme for atom interferometry,” Appl. Phys. B 84, 585–597 (2006).
[CrossRef]

Audoin, C.

J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards, Vol.  II (Adam Hilger, 1989).
[CrossRef]

Baym, G.

G. Baym, Lectures on Quantum Mechanics (Benjamin/Cummings Publication, 1969).

Berg, P.

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

Bodart, Q.

Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
[CrossRef]

Bouyer, P.

Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
[CrossRef]

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, 971–974 (1998).
[CrossRef]

T. L. Gustavson, P. Bouyer, and M. A. Kasevich, “Precision rotation measurements with an atom interferometer gyroscope,” Phys. Rev. Lett. 78, 2046–2049 (1997).
[CrossRef]

Butts, D. L.

Chu, S.

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef] [PubMed]

A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38, 25–61(2001).
[CrossRef]

D. S. Weiss, B. C. Young, and S. Chu, “Precision measurement of ℏ/mCs based on photon recoil using laser-cooled atoms and atomic interferometry,” Appl. Phys. B 59, 217–253 (1994).
[CrossRef]

K. Moler, D. S. Weiss, M. Kasevich, and S. Chu, “Theoretical analysis of velocity-selective Raman transitions,” Phys. Rev. A 45, 342–348 (1992).
[CrossRef] [PubMed]

B. Young, M. Kasevich, and S. Chu, “Precision atom interferometry with light pulses,” Atom Interferometry, P.Berman, ed. (Academic, 1997), pp. 363–406.
[CrossRef]

Chung, K. Y.

A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38, 25–61(2001).
[CrossRef]

Cohen-Tannoudji, C.

P. Storey and C. Cohen-Tannoudji, “The Feynman path integral approach to atomic interferometry. A tutorial,” J. Phys. II (France) 4, 1999–2027 (1994).
[CrossRef]

Durfee, D. S.

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

Ertmer, W.

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

Fixler, J. B.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65, 033608 (2002).
[CrossRef]

Foster, G. T.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65, 033608 (2002).
[CrossRef]

Gilowski, M.

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

Gustavson, T. L.

T. L. Gustavson, A. Landragin, and M. A. Kasevich, “Rotation sensing with a dual atom interferometer Sagnac gyroscope,” Class. Quant. Gravity 17, 2385–2398 (2000).
[CrossRef]

T. L. Gustavson, P. Bouyer, and M. A. Kasevich, “Precision rotation measurements with an atom interferometer gyroscope,” Phys. Rev. Lett. 78, 2046–2049 (1997).
[CrossRef]

Hahn, E. L.

E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
[CrossRef]

Haritos, K. G.

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, 971–974 (1998).
[CrossRef]

Kasevich, M.

K. Moler, D. S. Weiss, M. Kasevich, and S. Chu, “Theoretical analysis of velocity-selective Raman transitions,” Phys. Rev. A 45, 342–348 (1992).
[CrossRef] [PubMed]

B. Young, M. Kasevich, and S. Chu, “Precision atom interferometry with light pulses,” Atom Interferometry, P.Berman, ed. (Academic, 1997), pp. 363–406.
[CrossRef]

Kasevich, M. A.

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65, 033608 (2002).
[CrossRef]

T. L. Gustavson, A. Landragin, and M. A. Kasevich, “Rotation sensing with a dual atom interferometer Sagnac gyroscope,” Class. Quant. Gravity 17, 2385–2398 (2000).
[CrossRef]

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, 971–974 (1998).
[CrossRef]

T. L. Gustavson, P. Bouyer, and M. A. Kasevich, “Precision rotation measurements with an atom interferometer gyroscope,” Phys. Rev. Lett. 78, 2046–2049 (1997).
[CrossRef]

Kinast, J. M.

Landragin, A.

Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
[CrossRef]

T. L. Gustavson, A. Landragin, and M. A. Kasevich, “Rotation sensing with a dual atom interferometer Sagnac gyroscope,” Class. Quant. Gravity 17, 2385–2398 (2000).
[CrossRef]

Malossi, M.

Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
[CrossRef]

McGuirk, J. M.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65, 033608 (2002).
[CrossRef]

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, 971–974 (1998).
[CrossRef]

Merlet, S.

Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
[CrossRef]

Moler, K.

K. Moler, D. S. Weiss, M. Kasevich, and S. Chu, “Theoretical analysis of velocity-selective Raman transitions,” Phys. Rev. A 45, 342–348 (1992).
[CrossRef] [PubMed]

Müller, H.

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef] [PubMed]

Müller, T.

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

Peters, A.

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef] [PubMed]

A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38, 25–61(2001).
[CrossRef]

A. Peters, “High precision gravity measurements using atom interferometry,” Ph.D. thesis (Stanford University, 1998).

Rasel, E. M.

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

Santos, F. Pereira Dos

Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
[CrossRef]

Schubert, C.

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

Shaham, Y. K.

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

Snadden, M. J.

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65, 033608 (2002).
[CrossRef]

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, 971–974 (1998).
[CrossRef]

Steck, D. A.

D. A. Steck, “Quantum and atom optics,” http://steck.us/teaching (2006).

Stockton, J. K.

J. K. Stockton, “Continuous quantum measurement of cold alkali-atom spins,” Ph.D. thesis (California Institute of Technology, 2007).

Stoner, R. E.

Storey, P.

P. Storey and C. Cohen-Tannoudji, “The Feynman path integral approach to atomic interferometry. A tutorial,” J. Phys. II (France) 4, 1999–2027 (1994).
[CrossRef]

Timmons, B. P.

Vanier, J.

J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards, Vol.  II (Adam Hilger, 1989).
[CrossRef]

Weiss, D. S.

D. S. Weiss, B. C. Young, and S. Chu, “Precision measurement of ℏ/mCs based on photon recoil using laser-cooled atoms and atomic interferometry,” Appl. Phys. B 59, 217–253 (1994).
[CrossRef]

K. Moler, D. S. Weiss, M. Kasevich, and S. Chu, “Theoretical analysis of velocity-selective Raman transitions,” Phys. Rev. A 45, 342–348 (1992).
[CrossRef] [PubMed]

Wendrich, T.

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

Young, B.

B. Young, M. Kasevich, and S. Chu, “Precision atom interferometry with light pulses,” Atom Interferometry, P.Berman, ed. (Academic, 1997), pp. 363–406.
[CrossRef]

Young, B. C.

D. S. Weiss, B. C. Young, and S. Chu, “Precision measurement of ℏ/mCs based on photon recoil using laser-cooled atoms and atomic interferometry,” Appl. Phys. B 59, 217–253 (1994).
[CrossRef]

Zaiser, M.

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

Appl. Phys. B (2)

D. S. Weiss, B. C. Young, and S. Chu, “Precision measurement of ℏ/mCs based on photon recoil using laser-cooled atoms and atomic interferometry,” Appl. Phys. B 59, 217–253 (1994).
[CrossRef]

C. Antoine, “Matter wave beam splitters in gravito-inertial and trapping potentials: generalized ttt scheme for atom interferometry,” Appl. Phys. B 84, 585–597 (2006).
[CrossRef]

Appl. Phys. Lett. (1)

Q. Bodart, S. Merlet, M. Malossi, F. Pereira Dos Santos, P. Bouyer, and A. Landragin, “A cold atom pyramidal gravimeter with a single laser beam,” Appl. Phys. Lett. 96, 134101 (2010).
[CrossRef]

Class. Quant. Gravity (1)

T. L. Gustavson, A. Landragin, and M. A. Kasevich, “Rotation sensing with a dual atom interferometer Sagnac gyroscope,” Class. Quant. Gravity 17, 2385–2398 (2000).
[CrossRef]

Eur. Phys. J. D (1)

T. Müller, M. Gilowski, M. Zaiser, P. Berg, C. Schubert, T. Wendrich, W. Ertmer, and E. M. Rasel, “A compact dual atom interferometer gyroscope based on laser-cooled rubidium,” Eur. Phys. J. D 53, 273–281 (2009).
[CrossRef]

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

J. Phys. II (France) (1)

P. Storey and C. Cohen-Tannoudji, “The Feynman path integral approach to atomic interferometry. A tutorial,” J. Phys. II (France) 4, 1999–2027 (1994).
[CrossRef]

Metrologia (1)

A. Peters, K. Y. Chung, and S. Chu, “High-precision gravity measurements using atom interferometry,” Metrologia 38, 25–61(2001).
[CrossRef]

Nature (1)

H. Müller, A. Peters, and S. Chu, “A precision measurement of the gravitational redshift by the interference of matter waves,” Nature 463, 926–929 (2010).
[CrossRef] [PubMed]

Phys. Rev. (1)

E. L. Hahn, “Spin echoes,” Phys. Rev. 80, 580–594 (1950).
[CrossRef]

Phys. Rev. A (3)

C. Antoine, “Rotating matter-wave beam splitters and consequences for atom gyrometers,” Phys. Rev. A 76, 033609 (2007).
[CrossRef]

J. M. McGuirk, G. T. Foster, J. B. Fixler, M. J. Snadden, and M. A. Kasevich, “Sensitive absolute-gravity gradiometry using atom interferometry,” Phys. Rev. A 65, 033608 (2002).
[CrossRef]

K. Moler, D. S. Weiss, M. Kasevich, and S. Chu, “Theoretical analysis of velocity-selective Raman transitions,” Phys. Rev. A 45, 342–348 (1992).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

D. S. Durfee, Y. K. Shaham, and M. A. Kasevich, “Long-term stability of an area-reversible atom-interferometer sagnac gyroscope,” Phys. Rev. Lett. 97, 240801 (2006).
[CrossRef]

T. L. Gustavson, P. Bouyer, and M. A. Kasevich, “Precision rotation measurements with an atom interferometer gyroscope,” Phys. Rev. Lett. 78, 2046–2049 (1997).
[CrossRef]

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, 971–974 (1998).
[CrossRef]

Other (6)

B. Young, M. Kasevich, and S. Chu, “Precision atom interferometry with light pulses,” Atom Interferometry, P.Berman, ed. (Academic, 1997), pp. 363–406.
[CrossRef]

A. Peters, “High precision gravity measurements using atom interferometry,” Ph.D. thesis (Stanford University, 1998).

D. A. Steck, “Quantum and atom optics,” http://steck.us/teaching (2006).

J. K. Stockton, “Continuous quantum measurement of cold alkali-atom spins,” Ph.D. thesis (California Institute of Technology, 2007).

G. Baym, Lectures on Quantum Mechanics (Benjamin/Cummings Publication, 1969).

J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards, Vol.  II (Adam Hilger, 1989).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Energy level structure for a simplified three-level atom undergoing a stimulated Raman transition. The magnitude of the detunings Δ and δ are exaggerated for clarity.

Fig. 2
Fig. 2

Diagrammatic representation of a Raman pulse operator applied at time t = t 1 . An initial state purely in the lower ground state is shown on the left; a pure state beginning in the upper ground state is shown on the right.

Fig. 3
Fig. 3

Diagrammatic illustration of the action of a second Raman pulse on the output amplitudes of a first pulse as shown in Fig. 2. The lower ground initial state of Fig. 2 is assumed. Input to each node is scaled as per the rules of Fig. 2 with the resultant output amplitudes shown. The sum of amplitudes associated with the respective ground states yields the net probability amplitudes.

Fig. 4
Fig. 4

Diagrammatic illustration of the action of a third Raman pulse on the output amplitudes of a second pulse as shown in Fig. 3. The output amplitudes shown are the output probability amplitudes for the three-pulse interferometer.

Fig. 5
Fig. 5

Timing diagrams for (a) two and (b) three Raman pulse sequences.

Fig. 6
Fig. 6

Rotating interferometer (not to scale): atoms (black dot) are launched on a linear trajectory at t = 0 and propagate with mean velocity v l . The instrument is taken to be rotating about the point of atom launch. The rotation induces relative motion of the Raman beams with respect to the atom trajectory. L is the distance between Raman beam pairs as shown, d is the diameter of the Raman beam pairs, Ω o is the rate of rotation. The vector direction of rotation is along + y (points out of the page). The initial velocity of launch is in the + x direction. k eff is directed from top to bottom in the figure (along z).

Tables (1)

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Table 1 Definitions of Parameters and Variables

Equations (74)

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H [ Ψ e Ψ i Ψ g ] = i t [ Ψ e Ψ i Ψ g ] ,
H 0 = [ ω o / 2 0 0 0 ω i 0 0 0 ω o / 2 ] ,
H 1 = [ 0 H 1 B 0 H 1 B * 0 H 1 A * 0 H 1 A 0 ] ,
H 1 A = 2 | Ω A | exp [ i ϕ ˜ A ( t ) ] ,
H 1 B = 2 | Ω B | exp [ i ϕ ˜ B ( t ) ] .
ϕ ˜ A ( t ) = 0 t [ ω A ( t ) k A · z ˙ ( t ) e ^ z ] d t + ϕ A ,
ϕ ˜ B ( t ) = 0 t [ ω B ( t ) k B · k eff m k B · z ˙ ( t ) e ^ z ] d t + ϕ B .
[ Ψ e ( t ) Ψ i ( t ) Ψ g ( t ) ] = [ b e ( t ) exp { i 2 [ ϕ ˜ A ( t ) ϕ ˜ B ( t ) ( ϕ A ϕ B ) ] } b i ( t ) exp { i 2 [ ϕ ˜ A ( t ) + ϕ ˜ B ( t ) ( ϕ A + ϕ B ) ] } b g ( t ) exp { i 2 [ ϕ ˜ A ( t ) ϕ ˜ B ( t ) ( ϕ A ϕ B ) ] } ] .
[ b e ( t ) b g ( t ) ] = exp [ i Ω e AC + Ω g AC 2 ( t t j ) ] [ c e ( t ) c g ( t ) ] ,
d d t [ c e ( t ) c g ( t ) ] = i 2 Ω ( t ) · σ [ c e ( t ) c g ( t ) ] .
δ ( t ) = [ ω A ( t ) ω B ( t ) ] [ ω o + k eff 2 2 m 2 m ( k A 2 k B 2 ) + k eff · e ^ z z ˙ ( t ) ] ,
[ c e ( t j + τ j ) c g ( t j + τ j ) ] ( 0 ) = exp ( i τ j 2 Ω · σ ) [ c e ( t j ) c g ( t j ) ] .
Ω j ( t ) = Ω j ( 0 ) δ ˙ | t = M j ( t M j ) e ^ z 1 2 δ ¨ | t = M j ( t M j ) 2 e ^ z + ,
[ c e ( t ) c g ( t ) ] = [ c e ( t ) c g ( t ) ] ( 0 ) + [ c e ( t ) c g ( t ) ] ( 1 ) + [ c e ( t ) c g ( t ) ] ( 2 ) +
d d t { [ c e ( t ) c g ( t ) ] ( 0 ) + [ c e ( t ) c g ( t ) ] ( 1 ) + [ c e ( t ) c g ( t ) ] ( 2 ) + } = i 2 [ Ω j ( 0 ) δ ˙ | t = M j ( t M j ) e ^ z 1 2 δ ¨ | t = M j ( t M j ) 2 e ^ z + ] · σ × { [ c e ( t ) c g ( t ) ] ( 0 ) + [ c e ( t ) c g ( t ) ] ( 1 ) + [ c e ( t ) c g ( t ) ] ( 2 ) + } .
d d t [ c e ( t ) c g ( t ) ] ( 0 ) = i 2 Ω j ( 0 ) · σ [ c e ( t ) c g ( t ) ] ( 0 ) , d d t [ c e ( t ) c g ( t ) ] ( 1 ) = i 2 Ω j ( 0 ) · σ [ c e ( t ) c g ( t ) ] ( 1 ) + i 2 δ ˙ | t = M j ( t M j ) e ^ z · σ [ c e ( t ) c g ( t ) ] ( 0 ) , d d t [ c e ( t ) c g ( t ) ] ( 2 ) = i 2 Ω j ( 0 ) · σ [ c e ( t ) c g ( t ) ] ( 2 ) + i 2 δ ˙ | t = M j ( t M j ) e ^ z · σ [ c e ( t ) c g ( t ) ] ( 1 ) + i 4 δ ¨ | t = M j ( t M j ) 2 e ^ z · σ [ c e ( t ) c g ( t ) ] ( 0 ) , etc. ,
[ c e ( t = t j ) c g ( t = t j ) ] ( 0 ) = [ c e ( t = t j ) c g ( t = t j ) ] , [ c e ( t = t j ) c g ( t = t j ) ] ( 1 , 2 , 3 , ) = [ 0 0 ] .
[ c e ( t ) c g ( t ) ] ( 1 ) = t j t d t { exp [ i 2 ( t t ) Ω j ( 0 ) · σ ] · i 2 δ ˙ ( M j ) e ^ z ( t M j ) · σ · exp [ i 2 ( t t j ) Ω j ( 0 ) · σ ] } [ c e ( t j ) c g ( t j ) ] ,
[ c e ( t j + τ j ) c g ( t j + τ j ) ] [ c e ( t j + τ j ) c g ( t j + τ j ) ] ( 0 ) + [ c e ( t j + τ j ) c g ( t j + τ j ) ] ( 1 ) = exp ( i τ j 2 Ω j ( 0 ) · σ ) { 1 i 2 [ Ω j ( 0 ) ] 2 δ ˙ | t = M j × [ σ · ( Ω ^ j ( 0 ) × e ^ z ) { sin ( Ω j ( 0 ) τ j ) Ω j ( 0 ) τ j 2 [ cos ( Ω j ( 0 ) τ j ) 1 ] } + σ · [ Ω ^ j ( 0 ) ( Ω ^ j ( 0 ) · e ^ z ) e ^ z ] [ Ω j ( 0 ) τ j 2 sin ( Ω j ( 0 ) τ j ) + cos ( Ω j ( 0 ) τ j ) 1 ] ] } [ c e ( t j ) c g ( t j ) ] ,
[ Ψ e ( t j + τ j ) Ψ g ( t j + τ j ) ] = exp { i σ z 2 0 t j + τ j [ δ ( t ) + ω o ] d t } exp ( i Ω e AC + Ω g AC 2 τ j ) × exp ( i τ j 2 Ω j ( 0 ) · σ ) exp { i σ z 2 0 t j [ δ ( t ) + ω o ] d t } [ Ψ e ( t j ) Ψ g ( t j ) ] .
δ ( t ) 2 σ z [ b e ( t ) b g ( t ) ] = i d d t [ b e ( t ) b g ( t ) ] .
[ b e ( t d + τ d ) b g ( t d + τ d ) ] = exp [ i σ z 2 t d t d + τ d d t δ ( t ) ] [ b e ( t d ) b g ( t d ) ] .
[ Ψ e ( t d + τ d ) Ψ g ( t d + τ d ) ] DARK = exp ( i σ z 2 ω o τ d ) [ Ψ e ( t d ) Ψ g ( t d ) ] .
R j exp ( i τ j 2 Ω j ( 0 ) · σ ) ,
Φ j j + 1 0 t j + τ j d t [ δ j + 1 ( t ) δ j ( t ) ] + t j + τ j t j + 1 d t δ j + 1 ( t ) ,
Φ j j + 1 t j + τ j t j + 1 d t δ ( t ) .
[ Ψ e ( t 1 + τ 1 ) Ψ g ( t 1 + τ 1 ) ] = exp { i σ z 2 0 t 1 + τ 1 d t [ δ 1 ( t ) + ω o ] } R 1 × exp { i σ z 2 0 t 1 d t [ δ 1 ( t ) + ω o ] } [ Ψ e ( t 1 ) Ψ g ( t 1 ) ] .
[ Ψ e ( t 2 ) Ψ g ( t 2 ) ] = exp { i σ z 2 ω o [ t 2 ( t 1 + τ 1 ) ] } [ Ψ e ( t 1 + τ 1 ) Ψ g ( t 1 + τ 1 ) ] .
[ Ψ e ( t 2 + τ 2 ) Ψ g ( t 2 + τ 2 ) ] = exp { i σ z 2 0 t 2 + τ 2 d t [ δ 2 ( t ) + ω o ] } R 2 × exp { i σ z 2 0 t 2 d t [ δ 2 ( t ) + ω o ] } [ Ψ e ( t 2 ) Ψ g ( t 2 ) ] ,
[ Ψ e ( t 3 ) Ψ g ( t 3 ) ] = exp { i σ z 2 ω o [ t 3 ( t 2 + τ 2 ) ] } [ Ψ e ( t 2 + τ 2 ) Ψ g ( t 2 + τ 2 ) ] .
[ Ψ e ( t 3 + τ 3 ) Ψ g ( t 3 + τ 3 ) ] = exp { i σ z 2 0 t 3 + τ 3 d t [ δ 3 ( t ) + ω o ] } R 3 × exp { i σ z 2 0 t 3 d t [ δ 3 ( t ) + ω o ] } [ Ψ e ( t 3 ) Ψ g ( t 3 ) ] .
[ Ψ e ( t 3 + τ 3 ) Ψ g ( t 3 + τ 3 ) ] = exp { i σ z 2 0 t 3 + τ 3 d t [ δ 3 ( t ) + ω o ] } R 3 exp ( i σ z 2 Φ 2 3 ) R 2 × exp ( i σ z 2 Φ 1 2 ) R 1 exp { i σ z 2 0 t 1 d t [ δ 1 ( t ) + ω o ] } [ Ψ e ( t 1 ) Ψ g ( t 1 ) ] .
[ Ψ e ( t 2 + τ 2 ) Ψ g ( t 2 + τ 2 ) ] = exp { i σ z 2 0 t 2 + τ 2 d t [ δ 2 ( t ) + ω o ] } R 2 exp ( i σ z 2 Φ 1 2 ) R 1 × exp { i σ z 2 0 t 1 d t [ δ 1 ( t ) + ω o ] } [ Ψ e ( t 1 ) Ψ g ( t 1 ) ] .
[ Ψ e ( t 3 + τ 3 ) Ψ g ( t 3 + τ 3 ) ] = R 3 exp ( i σ z 2 Φ 2 3 ) R 2 exp ( i σ z 2 Φ 1 2 ) R 1 [ Ψ e ( t 1 ) Ψ g ( t 1 ) ] PURE ,
[ Ψ e ( t 1 ) Ψ g ( t 1 ) ] PURE = { [ 1 0 ] , [ 0 1 ] } .
[ Ψ e ( t 2 + τ 2 ) Ψ g ( t 2 + τ 2 ) ] = R 2 exp ( i σ z 2 Φ 1 2 ) R 1 [ Ψ e ( t 1 ) Ψ g ( t 1 ) ] PURE .
R j [ 1 0 ] = C j * [ 1 0 ] i S j [ 0 1 ] ,
R j [ 0 1 ] = C j [ 0 1 ] i S j * [ 1 0 ] ,
Ψ e ( t 2 + τ 2 ) | 2 - pulse = i C 1 S 2 * exp ( i 2 Φ 1 2 ) i S 1 * C 2 * exp ( i 2 Φ 1 2 ) ,
Ψ g ( t 2 + τ 2 ) | 2 - pulse = C 1 C 2 exp ( i 2 Φ 1 2 ) S 1 * S 2 exp ( i 2 Φ 1 2 ) ,
| Ψ e ( t 2 + τ 2 ) | 2 - pulse | 2 = | C 1 | 2 | S 2 | 2 + | S 1 | 2 | C 2 | 2 + 2 Re [ C 1 * S 1 * C 2 * S 2 exp ( i Φ 1 2 ) ] ,
| Ψ g ( t 2 + τ 2 ) | 2 - pulse | 2 = | C 1 | 2 | C 2 | 2 + | S 1 | 2 | S 2 | 2 2 Re [ C 1 * S 1 * C 2 * S 2 exp ( i Φ 1 2 ) ] .
Ψ e ( t 3 + τ 3 ) | 3 - pulse = ( Ψ e ) int + ( Ψ e ) loss ,
( Ψ e ) int = i C 1 S 2 * C 3 * exp ( i 2 Φ 1 2 ) exp ( i 2 Φ 2 3 ) + i S 1 * S 2 S 3 * exp ( i 2 Φ 1 2 ) exp ( i 2 Φ 2 3 ) ,
( Ψ e ) loss = i S 1 * C 2 * C 3 * exp ( i 2 Φ 1 2 ) exp ( i 2 Φ 2 3 ) i C 1 C 2 S 3 * exp ( i 2 Φ 1 2 ) exp ( i 2 Φ 2 3 ) ,
Ψ g ( t 3 + τ 3 ) | 3 - pulse = ( Ψ g ) int + ( Ψ g ) loss ,
( Ψ g ) int = C 1 S 2 * S 3 exp ( i 2 Φ 1 2 ) exp ( i 2 Φ 2 3 ) S 1 * S 2 C 3 exp ( i 2 Φ 1 2 ) exp ( i 2 Φ 2 3 ) ,
( Ψ g ) loss = S 1 * C 2 * S 3 exp ( i 2 Φ 1 2 ) exp ( i 2 Φ 2 3 ) + C 1 C 2 C 3 exp ( i 2 Φ 1 2 ) exp ( i 2 Φ 2 3 ) .
| Ψ e ( t 3 + τ 3 ) | 3 - pulse | 2 = | S 1 | 2 | S 2 | 2 | S 3 | 2 + | C 1 | 2 | S 2 | 2 | C 3 | 2 2 Re [ exp ( i ϕ int ) C 1 S 1 ( S 2 * ) 2 C 3 * S 3 ] + | S 1 | 2 | C 2 | 2 | C 3 | 2 + | C 1 | 2 | C 2 | 2 | S 3 | 2 + 2 Re [ exp ( i ϕ loss ) C 1 * S 1 * ( C 2 * ) 2 C 3 * S 3 ] + 2 Re [ exp ( i ϕ int / 2 ) exp ( i ϕ loss / 2 ) C 1 S 1 C 2 S 2 * ( | C 3 | 2 | S 3 | 2 ) ] + 2 Re [ exp ( i ϕ int / 2 ) exp ( i ϕ loss / 2 ) C 2 * S 2 * C 3 * S 3 ( | C 1 | 2 | S 1 | 2 ) ] ,
| Ψ g ( t 3 + τ 3 ) | 3 - pulse | 2 = | C 1 | 2 | S 2 | 2 | S 3 | 2 + | S 1 | 2 | S 2 | 2 | C 3 | 2 + 2 Re [ exp ( i ϕ int ) C 1 S 1 ( S 2 * ) 2 C 3 * S 3 ] + | S 1 | 2 | C 2 | 2 | S 3 | 2 + | C 1 | 2 | C 2 | 2 | C 3 | 2 2 Re [ exp ( i ϕ loss ) C 1 * S 1 * ( C 2 * ) 2 C 3 * S 3 ] 2 Re [ exp ( i ϕ int / 2 ) exp ( i ϕ loss / 2 ) C 1 S 1 C 2 S 2 * ( | C 3 | 2 | S 3 | 2 ) ] 2 Re [ exp ( i ϕ int / 2 ) exp ( i ϕ loss / 2 ) C 2 * S 2 * C 3 * S 3 ( | C 1 | 2 | S 1 | 2 ) ] .
Φ 1 2 = t 1 + τ 1 t 2 d t δ ( t ) = t 1 + τ 1 t 2 { [ ω A ( t ) ω B ( t ) ] ω o } d t .
C j * 1 2 [ 1 + i δ ( M j ) Ω ( M j ) ] 1 2 exp [ i δ ( M j ) τ j Ω ( M j ) τ j ] 1 2 exp [ i 2 π δ ( M j ) τ j ] ,
C 1 * C 2 * = 1 2 exp { i 2 π [ δ ( M 1 ) τ 1 + δ ( M 2 ) τ 2 ] } for     Ω ( M j ) τ j = π 2 .
C 1 * C 2 * 1 2 exp ( i 4 π δ τ ) for     τ 1 = τ 2 = τ ,
δ T ( 1 + 4 τ π T ) ,
| Ψ e ( t 2 + τ 2 ) | 2 - pulse | 2 = | C 1 | 2 | S 2 | 2 + | S 1 | 2 | C 2 | 2 + 2 | C 1 * S 1 * C 2 * S 2 | cos [ δ T ( 1 + 4 τ π T ) ] ,
| Ψ g ( t 2 + τ 2 ) | 2 - pulse | 2 = | C 1 | 2 | C 2 | 2 + | S 1 | 2 | S 2 | 2 2 | C 1 * S 1 * C 2 * S 2 | cos [ δ T ( 1 + 4 τ π T ) ] .
ϕ loss = k eff · e ^ z z ˙ ( t = 0 ) [ t 2 ( t 1 + τ 1 ) + t 3 ( t 2 + τ 2 ) ] + t 2 + τ 2 t 3 [ ω A ( t ) ω B ( t ) ( ω o + k eff 2 2 m ) ] d t + t 1 + τ 1 t 2 [ ω A ( t ) ω B ( t ) ( ω o + k eff 2 2 m ) ] d t t 1 + τ 1 t 2 { [ 0 t k eff · e ^ z z ¨ ( t ) d t ] } d t t 2 + τ 2 t 3 { [ 0 t k eff · e ^ z z ¨ ( t ) d t ] } d t ,
| Ψ e ( t 3 + τ 3 ) | 3 - pulse , DS | 2 = | S 1 | 2 | S 2 | 2 | S 3 | 2 + | C 1 | 2 | S 2 | 2 | C 3 | 2 + | S 1 | 2 | C 2 | 2 | C 3 | 3 + | C 1 | 2 | C 2 | 2 | S 3 | 2 [ exp ( i ϕ int ) C 1 S 1 ( S 2 * ) 2 C 3 * S 3 + c c ] ,
| Ψ g ( t 3 + τ 3 ) | 3 - pulse , DS | 2 = | C 1 | 2 | S 2 | 2 | S 3 | 2 + | S 1 | 2 | S 2 | 2 | C 3 | 2 + | S 1 | 2 | C 2 | 2 | S 3 | 3 + | C 1 | 2 | C 2 | 2 | C 3 | 2 + [ exp ( i ϕ int ) C 1 S 1 ( S 2 * ) 2 C 3 * S 3 + c c ] ,
exp ( i ϕ int ) C 1 S 1 ( S 2 * ) 2 C 3 * S 3 + c c = exp ( i ϕ int ) exp [ i ( ϕ 1 2 ϕ 2 + ϕ 3 ) ] | S 1 | | S 2 | 2 | S 3 | C 1 C 3 * + c c .
ϕ 1 2 ϕ 2 + ϕ 3 = ( ϕ A , o ϕ B , o ) 1 k eff · e ^ z z ( t = 0 ) 2 [ ( ϕ A , o ϕ B , o ) 2 k eff · e ^ z z ( t = 0 ) ] + ( ϕ A , o ϕ B , o ) 3 k eff · e ^ z z ( t = 0 ) = ( ϕ A , o ϕ B , o ) 1 2 ( ϕ A , o ϕ B , o ) 2 + ( ϕ A , o ϕ B , o ) 3 .
C 1 C 3 * = 1 2 exp { i 2 π [ δ ( M 3 ) τ 3 δ ( M 1 ) τ 1 ] } for     Ω ( M j ) τ j = π 2 .
δ ( M 3 ) δ ( M 1 ) = ( ω A ω B ) ( ω o + k eff 2 2 m ) k eff · e ^ z z ˙ ( M 3 ) ( ω A ω B ) + ( ω o + k eff 2 2 m ) + k eff · e ^ z z ˙ ( M 1 ) = k eff a o ( τ + 2 T ) .
C 1 C 3 * = 1 2 exp { i 2 π [ k eff a o ( τ + 2 T ) ] τ } .
ϕ int = k eff [ t 2 + τ 2 t 3 z ˙ ( t ) d t t 1 + τ 1 t 2 z ˙ ( t ) d t ] = k eff a o [ T 2 τ 2 2 4 ] .
2 Re [ exp ( i ϕ int ) C 1 S 1 ( S 2 * ) 2 C 3 * S 3 ] + c c = 2 Re { exp ( i ϕ int ) exp [ i ( ϕ 1 2 ϕ 2 + ϕ 3 ) ] | S 1 | | S 2 | 2 | S 3 | C 1 C 3 * } = 1 2 cos { k eff a o T 2 [ 1 + 4 τ π T + τ 2 T 2 ( 2 π τ 2 2 4 τ 2 ) ] } 1 2 cos { k eff a o T 2 [ 1 + 4 τ π T + τ 2 T 2 ( 2 π 1 ) ] } , for     τ 2 = 2 τ ,
cos ϕ Peters = cos [ k eff a o ( T p + 2 τ ) T p ] 2 k eff a o ( T p + 2 τ ) Ω eff sin [ k eff a o ( T p + 2 τ ) T p ] cos [ k eff a o ( T p + 2 τ ) ( T p + 4 τ π ) ] ϕ Peters = k eff a o T 2 [ 1 + 4 τ π T + τ 2 T 2 ( 4 π 1 ) ] ,
v j = Ω o ( v o · e ^ x ) M j e ^ z ,
δ j ( t ) = [ ω A ( t ) ω B ( t ) ] j ω o k eff 2 2 m + 2 m ( k A 2 k B 2 ) k eff Ω o ( v o · e ^ x ) M j .
ϕ int = Φ 2 3 Φ 1 2 = 2 k eff Ω o ( v l · e ^ x ) ( T + τ 2 ) T .
( ω A ω B ) 1 = ω o + k eff 2 2 m 2 m ( k A 2 k B 2 ) + k eff Ω o v l M 1 e ^ z , for     δ ( M 1 ) = 0.
C 1 C 3 * = 1 2 exp { i 2 π τ [ δ ( M 3 ) δ ( M 1 ) ] } = 1 2 exp [ i 2 π k eff Ω o v l ( 2 T + τ ) τ ]
ϕ int + 2 π τ [ δ ( M 3 ) δ ( M 1 ) ] 2 k eff Ω o v l T 2 [ 1 + τ 2 T + 2 τ π T + 1 π ( τ T ) 2 ] .

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