Abstract

This article describes quantum interference in a lambda system driven by two identical pulses that are each sufficiently broadband to drive both dipole-allowed transitions. The first pulse drives the system into a quantum superposition, making the effect of the second pulse depend critically on its optical phase and resulting in Ramsey-like fringes. This method, using just two pulses of the same carrier frequency, is conceptually simpler than in previous Raman–Ramsey studies, which use pump and Stokes pulses in each of two spatially separated regions. The goal here is not efficient population transfer, but to investigate narrow features resulting from quantum interference. I first explore these effects for low-inversion, which illustrates many key features using an easy-to-visualize model. I then use Schrödinger’s equation in a semiclassical model to extend the results to arbitrary inversion. Informative quantum interference features remain when using this simplified scheme.

© 2010 Optical Society of America

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    [Crossref]
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    [Crossref]
  45. N. V. Vitanov and P. L. Knight, “Coherent excitation of a two-state system by a train of short pulses,” Phys. Rev. A 52, 2245–2261 (1995).
    [Crossref]
  46. A. W. Albrecht, J. D. Hybl, S. M. G. Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
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    [Crossref]
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    [Crossref]
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  54. T. H. Yoon, A. Marian, J. L. Hall, and J. Ye, “Phase-coherent multilevel two-photon transitions in cold Rb atoms: Ultrahigh-resolution spectroscopy via frequency-stabilized femtosecond laser,” Phys. Rev. A 63, 011402(R) (2000).
    [Crossref]
  55. A. Marian, M. C. Stowe, J. R. Lawall, D. Felinto, and J. Ye, “United time-frequency spectroscopy for dynamics and global structure,” Science 306, 2063–2068 (2004).
    [Crossref]
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    [Crossref]
  57. B. Dubetsky and P. R. Berman, “Ground-state Ramsey fringes,” Phys. Rev. A 56, R1091–R1094 (1997).
    [Crossref]
  58. M. Weel and A. Kumarakrishnan, “Observation of ground-state Ramsey fringes,” Phys. Rev. A 67, 061602(R) (2003).
    [Crossref]

2010 (1)

2008 (2)

S. T. Müller, D. V. Magalhães, A. Bebeachibuli, T. A. Ortega, M. Ahmed, and V. S. Bagnato, “Free expanding cloud of cold atoms as an atomic standard: Ramsey fringes contrast,” J. Opt. Soc. Am. B 25, 909–914 (2008).
[Crossref]

G. S. Pati, K. Salit, R. Tripathi, and M. S. Shahriar, “Demonstration of Raman-Ramsey fringes using time delayed optical pulses in rubidium vapor,” Opt. Commun. 281, 4676–4680 (2008).
[Crossref]

2006 (4)

J. Fuchs, G. J. Duffy, W. J. Rowlands, and A. M. Akulshin, “Electromagnetically induced transparency in Li6,” J. Phys. B 39, 3479–3489 (2006).
[Crossref]

Y. Xiao, I. Novikova, D. F. Phillips, and R. L. Walsworth, “Diffusion-induced Ramsey narrowing,” Phys. Rev. Lett. 96, 043601 (2006).
[Crossref]

R. Th. Zinkstok, S. Witte, W. Ubachs, W. Hogervorst, and K. S. E. Eikema, “Frequency comb laser spectroscopy in the vacuum-ultraviolet region,” Phys. Rev. A 73, 061801(R) (2006).
[Crossref]

V. Letchumanan, P. Gill, A. G. Sinclair, and E. Riis, “Optical-clock local-oscillator stabilization scheme,” J. Opt. Soc. Am. B 23, 714–717 (2006).
[Crossref]

2005 (6)

R. E. Carley, E. D. Boléat, R. S. Minns, R. Patel, and H. H. Fielding, “Interfering Rydberg wave packets in Na,” J. Phys. B 38, 1907–1922 (2005).
[Crossref]

M. Bellini, S. Cavalieri, C. Corsi, R. Eramo, and M. Materazzi, “Mutually coherent high-order harmonic pulses for XUV Ramsey spectroscopy,” Laser Phys. 15, 324–327 (2005).

S. Witte, R. Th. Zinkstok, W. Ubachs, W. Hogervorst, and K. S. E. Eikema, “Deep-ultraviolet quantum interference metrology with ultrashort laser pulses,” Science 307, 400–403 (2005).
[Crossref] [PubMed]

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

J. Vanier, “Atomic clocks based on coherent population trapping: a review,” Appl. Phys. B 81, 421–442 (2005).
[Crossref]

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref]

2004 (6)

D. Felinto, L. H. Acioli, and S. S. Vianna, “Accumulative effects in the coherence of three-level atoms excited by femtosecond-laser frequency combs,” Phys. Rev. A 70, 043403 (2004).
[Crossref]

S. Bougouffa and A. Kamli, “An analytic approach to a three-level atom interacting with a single-mode quantized field,” J. Opt. B: Quantum Semiclassical Opt. 6, S60–S65 (2004).
[Crossref]

A. Godone, S. Micalizio, and F. Levi, “Pulsed optically pumped frequency standard,” Phys. Rev. A 70, 023409 (2004).
[Crossref]

V. Letchumanan, P. Gill, E. Riis, and A. G. Sinclair, “Optical Ramsey spectroscopy of a single trapped Sr+88 ion,” Phys. Rev. A 70, 033419 (2004).
[Crossref]

A. Shelkovnikov, C. Grain, R. J. Butcher, A. Amy-Klein, A. Goncharov, and C. Chardonnet, “Two-Photon Ramsey Fringes at 30 THz Referenced to an H Maser/Cs Fountain via an Optical-Frequency Comb at the 1-Hz Level,” IEEE J. Quantum Electron. 40, 1023–1029 (2004).
[Crossref]

A. Marian, M. C. Stowe, J. R. Lawall, D. Felinto, and J. Ye, “United time-frequency spectroscopy for dynamics and global structure,” Science 306, 2063–2068 (2004).
[Crossref]

2003 (4)

S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003).
[Crossref]

M. Weel and A. Kumarakrishnan, “Observation of ground-state Ramsey fringes,” Phys. Rev. A 67, 061602(R) (2003).
[Crossref]

E. A. Curtis, C. W. Oates, and L. Hollberg, “Quenched narrow-line second- and third-stage laser cooling of Ca40,” J. Opt. Soc. Am. B 20, 977–984 (2003).
[Crossref]

R. Netz, A. Nazarkin, and R. Sauerbrey, “Observation of selectivity of coherent population transfer induced by optical interference,” Phys. Rev. Lett. 90, 063001 (2003).
[Crossref]

2001 (2)

A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A 65, 013814 (2001).
[Crossref]

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A 64, 063413 (2001).
[Crossref]

2000 (2)

L. Marmet and A. A. Madej, “Optical Ramsey spectroscopy and coherence measurements of the clock transition in a single trapped Sr ion,” Can. J. Phys. 78, 495–507 (2000).
[Crossref]

T. H. Yoon, A. Marian, J. L. Hall, and J. Ye, “Phase-coherent multilevel two-photon transitions in cold Rb atoms: Ultrahigh-resolution spectroscopy via frequency-stabilized femtosecond laser,” Phys. Rev. A 63, 011402(R) (2000).
[Crossref]

1999 (1)

A. W. Albrecht, J. D. Hybl, S. M. G. Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[Crossref]

1998 (2)

B. Gross, A. Huber, M. Niering, M. Weitz, and T. W. Hänsch, “Optical Ramsey spectroscopy of atomic hydrogen,” Europhys. Lett. 44, 186–191 (1998).
[Crossref]

Walter F. Buell, “Laser-pumped and laser-cooled atomic clocks for space applications,” Laser Part. Beams 16, 627–639 (1998).
[Crossref]

1997 (2)

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee, and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

B. Dubetsky and P. R. Berman, “Ground-state Ramsey fringes,” Phys. Rev. A 56, R1091–R1094 (1997).
[Crossref]

1995 (1)

N. V. Vitanov and P. L. Knight, “Coherent excitation of a two-state system by a train of short pulses,” Phys. Rev. A 52, 2245–2261 (1995).
[Crossref]

1993 (1)

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]

1991 (2)

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, “Atomic velocity selection using stimulated Raman transitions,” Phys. Rev. Lett. 66, 2297–2300 (1991).
[Crossref]

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry: Time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

1990 (1)

N. F. Ramsey, “Experiments with separated oscillatory fields and hydrogen masers,” Rev. Mod. Phys. 62, 541–552 (1990).
[Crossref]

1989 (1)

1988 (1)

1986 (2)

B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459–472 (1986).
[Crossref]

P. R. Hemmer, G. P. Ontai, and S. Ezekiel, “Precision studies of stimulated-resonance Raman interactions in an atomic beam,” J. Opt. Soc. Am. B 3, 219–230 (1986).
[Crossref]

1985 (2)

L. S. Vasilenko, I. D. Matveyenko, and N. N. Rubtsova, “Study of narrow resonances of coherent radiation in time separated fields in SF6,” Opt. Commun. 53, 371–374 (1985).
[Crossref]

H.-I. Yoo and J. H. Eberly, “Dynamical theory of an atom with two or three levels interacting with quantized cavity fields,” Phys. Rep. 118, 239–337 (1985).
[Crossref]

1982 (1)

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Observation of Ramsey fringes using a stimulated, resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867–870 (1982).
[Crossref]

1981 (1)

R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference technique applied to high precision spectroscopy,” J. Phys. (Paris) 42, C8–45 (1981).
[Crossref]

1978 (1)

M. M. Salour, “Quantum interference effects in two-photon spectroscopy,” Rev. Mod. Phys. 50, 667–681 (1978).
[Crossref]

1977 (2)

M. M. Salour and C. Cohen-Tannoudji, “Observation of Ramsey’s interference fringes in the profile of Doppler-free two-photon resonances,” Phys. Rev. Lett. 38, 757–760 (1977).
[Crossref]

R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764 (1977).
[Crossref]

1976 (2)

Ye. V. Baklanov, V. P. Chebotayev, and B. Ya Dubetsky, “The resonance of two-photon absorption in separated optical fields,” Appl. Phys. 11, 201–202 (1976).
[Crossref]

Ye. V. Baklanov, B. Ya. Dubetsky, and V. P. Chebotayev, “Non-linear Ramsey resonance in the optical region,” Appl. Phys. 9, 171–173 (1976).
[Crossref]

Abgrall, M.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Acioli, L. H.

D. Felinto, L. H. Acioli, and S. S. Vianna, “Accumulative effects in the coherence of three-level atoms excited by femtosecond-laser frequency combs,” Phys. Rev. A 70, 043403 (2004).
[Crossref]

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A 64, 063413 (2001).
[Crossref]

Ahmed, M.

Akulshin, A. M.

J. Fuchs, G. J. Duffy, W. J. Rowlands, and A. M. Akulshin, “Electromagnetically induced transparency in Li6,” J. Phys. B 39, 3479–3489 (2006).
[Crossref]

Albrecht, A. W.

A. W. Albrecht, J. D. Hybl, S. M. G. Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[Crossref]

Amy-Klein, A.

A. Shelkovnikov, C. Grain, R. J. Butcher, A. Amy-Klein, A. Goncharov, and C. Chardonnet, “Two-Photon Ramsey Fringes at 30 THz Referenced to an H Maser/Cs Fountain via an Optical-Frequency Comb at the 1-Hz Level,” IEEE J. Quantum Electron. 40, 1023–1029 (2004).
[Crossref]

Audoin, C.

J. Vanier and C. Audoin, The Quantum Physics of Atomic Frequency Standards (IOP Publishing, 1989).
[Crossref]

Bagnato, V. S.

Baklanov, Ye. V.

Ye. V. Baklanov, B. Ya. Dubetsky, and V. P. Chebotayev, “Non-linear Ramsey resonance in the optical region,” Appl. Phys. 9, 171–173 (1976).
[Crossref]

Ye. V. Baklanov, V. P. Chebotayev, and B. Ya Dubetsky, “The resonance of two-photon absorption in separated optical fields,” Appl. Phys. 11, 201–202 (1976).
[Crossref]

Bebeachibuli, A.

Bellini, M.

M. Bellini, S. Cavalieri, C. Corsi, R. Eramo, and M. Materazzi, “Mutually coherent high-order harmonic pulses for XUV Ramsey spectroscopy,” Laser Phys. 15, 324–327 (2005).

Berman, P. R.

B. Dubetsky and P. R. Berman, “Ground-state Ramsey fringes,” Phys. Rev. A 56, R1091–R1094 (1997).
[Crossref]

Bernacki, B. E.

Bize, S.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Boléat, E. D.

R. E. Carley, E. D. Boléat, R. S. Minns, R. Patel, and H. H. Fielding, “Interfering Rydberg wave packets in Na,” J. Phys. B 38, 1907–1922 (2005).
[Crossref]

Bosco, C. A. C.

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A 64, 063413 (2001).
[Crossref]

Bougouffa, S.

S. Bougouffa and A. Kamli, “An analytic approach to a three-level atom interacting with a single-mode quantized field,” J. Opt. B: Quantum Semiclassical Opt. 6, S60–S65 (2004).
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[Crossref]

Butcher, R. J.

A. Shelkovnikov, C. Grain, R. J. Butcher, A. Amy-Klein, A. Goncharov, and C. Chardonnet, “Two-Photon Ramsey Fringes at 30 THz Referenced to an H Maser/Cs Fountain via an Optical-Frequency Comb at the 1-Hz Level,” IEEE J. Quantum Electron. 40, 1023–1029 (2004).
[Crossref]

Cacciapuoti, L.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Carley, R. E.

R. E. Carley, E. D. Boléat, R. S. Minns, R. Patel, and H. H. Fielding, “Interfering Rydberg wave packets in Na,” J. Phys. B 38, 1907–1922 (2005).
[Crossref]

Carlson, R. J.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry: Time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Cavalieri, S.

M. Bellini, S. Cavalieri, C. Corsi, R. Eramo, and M. Materazzi, “Mutually coherent high-order harmonic pulses for XUV Ramsey spectroscopy,” Laser Phys. 15, 324–327 (2005).

Chardonnet, C.

A. Shelkovnikov, C. Grain, R. J. Butcher, A. Amy-Klein, A. Goncharov, and C. Chardonnet, “Two-Photon Ramsey Fringes at 30 THz Referenced to an H Maser/Cs Fountain via an Optical-Frequency Comb at the 1-Hz Level,” IEEE J. Quantum Electron. 40, 1023–1029 (2004).
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Chebotayev, V. P.

Ye. V. Baklanov, B. Ya. Dubetsky, and V. P. Chebotayev, “Non-linear Ramsey resonance in the optical region,” Appl. Phys. 9, 171–173 (1976).
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Ye. V. Baklanov, V. P. Chebotayev, and B. Ya Dubetsky, “The resonance of two-photon absorption in separated optical fields,” Appl. Phys. 11, 201–202 (1976).
[Crossref]

Chu, S.

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]

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, “Atomic velocity selection using stimulated Raman transitions,” Phys. Rev. Lett. 66, 2297–2300 (1991).
[Crossref]

Cina, J. A.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry: Time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Clairon, A.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref]

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
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Cohen-Tannoudji, C.

M. M. Salour and C. Cohen-Tannoudji, “Observation of Ramsey’s interference fringes in the profile of Doppler-free two-photon resonances,” Phys. Rev. Lett. 38, 757–760 (1977).
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Corsi, C.

M. Bellini, S. Cavalieri, C. Corsi, R. Eramo, and M. Materazzi, “Mutually coherent high-order harmonic pulses for XUV Ramsey spectroscopy,” Laser Phys. 15, 324–327 (2005).

Cundiff, S. T.

S. T. Cundiff and J. Ye, “Colloquium: Femtosecond optical frequency combs,” Rev. Mod. Phys. 75, 325–342 (2003).
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Curtis, E. A.

Dalton, B. J.

B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459–472 (1986).
[Crossref]

de Clercq, E.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref]

Dimarcq, N.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref]

Du, M.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry: Time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Dubetsky, B.

B. Dubetsky and P. R. Berman, “Ground-state Ramsey fringes,” Phys. Rev. A 56, R1091–R1094 (1997).
[Crossref]

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J. Fuchs, G. J. Duffy, W. J. Rowlands, and A. M. Akulshin, “Electromagnetically induced transparency in Li6,” J. Phys. B 39, 3479–3489 (2006).
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Eckstein, J.

R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764 (1977).
[Crossref]

Eikema, K. S. E.

R. Th. Zinkstok, S. Witte, W. Ubachs, W. Hogervorst, and K. S. E. Eikema, “Frequency comb laser spectroscopy in the vacuum-ultraviolet region,” Phys. Rev. A 73, 061801(R) (2006).
[Crossref]

S. Witte, R. Th. Zinkstok, W. Ubachs, W. Hogervorst, and K. S. E. Eikema, “Deep-ultraviolet quantum interference metrology with ultrashort laser pulses,” Science 307, 400–403 (2005).
[Crossref] [PubMed]

Eramo, R.

M. Bellini, S. Cavalieri, C. Corsi, R. Eramo, and M. Materazzi, “Mutually coherent high-order harmonic pulses for XUV Ramsey spectroscopy,” Laser Phys. 15, 324–327 (2005).

Ezekiel, S.

P. R. Hemmer, M. S. Shahriar, H. Lamela-Rivera, S. P. Smith, B. E. Bernacki, and S. Ezekiel, “Semiconductor laser excitation of Ramsey fringes by using a Raman transition in a cesium atomic beam,” J. Opt. Soc. Am. B 10, 1326–1329 (1993).
[Crossref]

P. R. Hemmer, M. S. Shahriar, V. D. Natoli, and S. Ezekiel, “Ac Stark shifts in a two-zone Raman interaction,” J. Opt. Soc. Am. B 6, 1519–1528 (1989).
[Crossref]

P. R. Hemmer, G. P. Ontai, and S. Ezekiel, “Precision studies of stimulated-resonance Raman interactions in an atomic beam,” J. Opt. Soc. Am. B 3, 219–230 (1986).
[Crossref]

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Observation of Ramsey fringes using a stimulated, resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867–870 (1982).
[Crossref]

R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference technique applied to high precision spectroscopy,” J. Phys. (Paris) 42, C8–45 (1981).
[Crossref]

Faeder, S. M. G.

A. W. Albrecht, J. D. Hybl, S. M. G. Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[Crossref]

Felinto, D.

D. Felinto, L. H. Acioli, and S. S. Vianna, “Accumulative effects in the coherence of three-level atoms excited by femtosecond-laser frequency combs,” Phys. Rev. A 70, 043403 (2004).
[Crossref]

A. Marian, M. C. Stowe, J. R. Lawall, D. Felinto, and J. Ye, “United time-frequency spectroscopy for dynamics and global structure,” Science 306, 2063–2068 (2004).
[Crossref]

D. Felinto, C. A. C. Bosco, L. H. Acioli, and S. S. Vianna, “Accumulative effects in temporal coherent control,” Phys. Rev. A 64, 063413 (2001).
[Crossref]

Fielding, H. H.

R. E. Carley, E. D. Boléat, R. S. Minns, R. Patel, and H. H. Fielding, “Interfering Rydberg wave packets in Na,” J. Phys. B 38, 1907–1922 (2005).
[Crossref]

Fleming, G. R.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry: Time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
[Crossref]

Fuchs, J.

J. Fuchs, G. J. Duffy, W. J. Rowlands, and A. M. Akulshin, “Electromagnetically induced transparency in Li6,” J. Phys. B 39, 3479–3489 (2006).
[Crossref]

Gill, P.

V. Letchumanan, P. Gill, A. G. Sinclair, and E. Riis, “Optical-clock local-oscillator stabilization scheme,” J. Opt. Soc. Am. B 23, 714–717 (2006).
[Crossref]

V. Letchumanan, P. Gill, E. Riis, and A. G. Sinclair, “Optical Ramsey spectroscopy of a single trapped Sr+88 ion,” Phys. Rev. A 70, 033419 (2004).
[Crossref]

Godone, A.

A. Godone, S. Micalizio, and F. Levi, “Pulsed optically pumped frequency standard,” Phys. Rev. A 70, 023409 (2004).
[Crossref]

Goncharov, A.

A. Shelkovnikov, C. Grain, R. J. Butcher, A. Amy-Klein, A. Goncharov, and C. Chardonnet, “Two-Photon Ramsey Fringes at 30 THz Referenced to an H Maser/Cs Fountain via an Optical-Frequency Comb at the 1-Hz Level,” IEEE J. Quantum Electron. 40, 1023–1029 (2004).
[Crossref]

Grain, C.

A. Shelkovnikov, C. Grain, R. J. Butcher, A. Amy-Klein, A. Goncharov, and C. Chardonnet, “Two-Photon Ramsey Fringes at 30 THz Referenced to an H Maser/Cs Fountain via an Optical-Frequency Comb at the 1-Hz Level,” IEEE J. Quantum Electron. 40, 1023–1029 (2004).
[Crossref]

Gross, B.

B. Gross, A. Huber, M. Niering, M. Weitz, and T. W. Hänsch, “Optical Ramsey spectroscopy of atomic hydrogen,” Europhys. Lett. 44, 186–191 (1998).
[Crossref]

Grünert, J.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Guerandel, S.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref]

Hall, J. L.

T. H. Yoon, A. Marian, J. L. Hall, and J. Ye, “Phase-coherent multilevel two-photon transitions in cold Rb atoms: Ultrahigh-resolution spectroscopy via frequency-stabilized femtosecond laser,” Phys. Rev. A 63, 011402(R) (2000).
[Crossref]

Hänsch, T. W.

B. Gross, A. Huber, M. Niering, M. Weitz, and T. W. Hänsch, “Optical Ramsey spectroscopy of atomic hydrogen,” Europhys. Lett. 44, 186–191 (1998).
[Crossref]

R. Teets, J. Eckstein, and T. W. Hänsch, “Coherent two-photon excitation by multiple light pulses,” Phys. Rev. Lett. 38, 760–764 (1977).
[Crossref]

Hemmer, P. R.

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee, and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

P. R. Hemmer, M. S. Shahriar, H. Lamela-Rivera, S. P. Smith, B. E. Bernacki, and S. Ezekiel, “Semiconductor laser excitation of Ramsey fringes by using a Raman transition in a cesium atomic beam,” J. Opt. Soc. Am. B 10, 1326–1329 (1993).
[Crossref]

P. R. Hemmer, M. S. Shahriar, V. D. Natoli, and S. Ezekiel, “Ac Stark shifts in a two-zone Raman interaction,” J. Opt. Soc. Am. B 6, 1519–1528 (1989).
[Crossref]

P. R. Hemmer and M. G. Prentiss, “Coupled-pendulum model of the stimulated resonance Raman effect,” J. Opt. Soc. Am. B 5, 1613–1623 (1988).
[Crossref]

P. R. Hemmer, G. P. Ontai, and S. Ezekiel, “Precision studies of stimulated-resonance Raman interactions in an atomic beam,” J. Opt. Soc. Am. B 3, 219–230 (1986).
[Crossref]

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Observation of Ramsey fringes using a stimulated, resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867–870 (1982).
[Crossref]

R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference technique applied to high precision spectroscopy,” J. Phys. (Paris) 42, C8–45 (1981).
[Crossref]

Hogervorst, W.

R. Th. Zinkstok, S. Witte, W. Ubachs, W. Hogervorst, and K. S. E. Eikema, “Frequency comb laser spectroscopy in the vacuum-ultraviolet region,” Phys. Rev. A 73, 061801(R) (2006).
[Crossref]

S. Witte, R. Th. Zinkstok, W. Ubachs, W. Hogervorst, and K. S. E. Eikema, “Deep-ultraviolet quantum interference metrology with ultrashort laser pulses,” Science 307, 400–403 (2005).
[Crossref] [PubMed]

Hollberg, L.

Holleville, D.

T. Zanon, S. Guerandel, E. de Clercq, D. Holleville, N. Dimarcq, and A. Clairon, “High contrast Ramsey fringes with coherent-population-trapping pulses in a double lambda atomic system,” Phys. Rev. Lett. 94, 193002 (2005).
[Crossref]

Huber, A.

B. Gross, A. Huber, M. Niering, M. Weitz, and T. W. Hänsch, “Optical Ramsey spectroscopy of atomic hydrogen,” Europhys. Lett. 44, 186–191 (1998).
[Crossref]

Hybl, J. D.

A. W. Albrecht, J. D. Hybl, S. M. G. Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[Crossref]

Jonas, D. M.

A. W. Albrecht, J. D. Hybl, S. M. G. Faeder, and D. M. Jonas, “Experimental distinction between phase shifts and time delays: Implications for femtosecond spectroscopy and coherent control of chemical reactions,” J. Chem. Phys. 111, 10934–10956 (1999).
[Crossref]

Kamli, A.

S. Bougouffa and A. Kamli, “An analytic approach to a three-level atom interacting with a single-mode quantized field,” J. Opt. B: Quantum Semiclassical Opt. 6, S60–S65 (2004).
[Crossref]

Kasapi, S.

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, “Atomic velocity selection using stimulated Raman transitions,” Phys. Rev. Lett. 66, 2297–2300 (1991).
[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]

M. Kasevich, D. S. Weiss, E. Riis, K. Moler, S. Kasapi, and S. Chu, “Atomic velocity selection using stimulated Raman transitions,” Phys. Rev. Lett. 66, 2297–2300 (1991).
[Crossref]

Katz, D. P.

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee, and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

Kieu, T. D.

B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459–472 (1986).
[Crossref]

Knight, P. L.

N. V. Vitanov and P. L. Knight, “Coherent excitation of a two-state system by a train of short pulses,” Phys. Rev. A 52, 2245–2261 (1995).
[Crossref]

B. J. Dalton, T. D. Kieu, and P. L. Knight, “Theory of ultra-high-resolution optical Raman Ramsey spectroscopy,” Opt. Acta 33, 459–472 (1986).
[Crossref]

Kumarakrishnan, A.

M. Weel and A. Kumarakrishnan, “Observation of ground-state Ramsey fringes,” Phys. Rev. A 67, 061602(R) (2003).
[Crossref]

Lamb, W. E.

M. Sargent, M. O. Scully, and W. E. Lamb, Laser Physics (Westview, 1974).

Lamela-Rivera, H.

Laurent, P.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Lawall, J. R.

A. Marian, M. C. Stowe, J. R. Lawall, D. Felinto, and J. Ye, “United time-frequency spectroscopy for dynamics and global structure,” Science 306, 2063–2068 (2004).
[Crossref]

Lee, A.

M. S. Shahriar, P. R. Hemmer, D. P. Katz, A. Lee, and M. G. Prentiss, “Dark-state-based three-element vector model for the stimulated Raman interaction,” Phys. Rev. A 55, 2272–2282 (1997).
[Crossref]

Leiby, C. C.

J. E. Thomas, P. R. Hemmer, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Observation of Ramsey fringes using a stimulated, resonance Raman transition in a sodium atomic beam,” Phys. Rev. Lett. 48, 867–870 (1982).
[Crossref]

R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference technique applied to high precision spectroscopy,” J. Phys. (Paris) 42, C8–45 (1981).
[Crossref]

Lemonde, P.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Letchumanan, V.

V. Letchumanan, P. Gill, A. G. Sinclair, and E. Riis, “Optical-clock local-oscillator stabilization scheme,” J. Opt. Soc. Am. B 23, 714–717 (2006).
[Crossref]

V. Letchumanan, P. Gill, E. Riis, and A. G. Sinclair, “Optical Ramsey spectroscopy of a single trapped Sr+88 ion,” Phys. Rev. A 70, 033419 (2004).
[Crossref]

Levi, F.

A. Godone, S. Micalizio, and F. Levi, “Pulsed optically pumped frequency standard,” Phys. Rev. A 70, 023409 (2004).
[Crossref]

Loudon, R.

R. Loudon, The Quantum Theory of Light, 3rd ed. (Oxford Univ. Press, 2000).

Luiten, A.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Luiten, A. N.

Madej, A. A.

L. Marmet and A. A. Madej, “Optical Ramsey spectroscopy and coherence measurements of the clock transition in a single trapped Sr ion,” Can. J. Phys. 78, 495–507 (2000).
[Crossref]

Magalhães, D. V.

Major, F. G.

F. G. Major, The Quantum Beat—The Physical Principles of Atomic Clocks (Springer, 1998).

Maksimovic, I.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Marian, A.

A. Marian, M. C. Stowe, J. R. Lawall, D. Felinto, and J. Ye, “United time-frequency spectroscopy for dynamics and global structure,” Science 306, 2063–2068 (2004).
[Crossref]

T. H. Yoon, A. Marian, J. L. Hall, and J. Ye, “Phase-coherent multilevel two-photon transitions in cold Rb atoms: Ultrahigh-resolution spectroscopy via frequency-stabilized femtosecond laser,” Phys. Rev. A 63, 011402(R) (2000).
[Crossref]

Marion, H.

S. Bize, P. Laurent, M. Abgrall, H. Marion, I. Maksimovic, L. Cacciapuoti, J. Grünert, C. Vian, F. Pereira dos Santos, P. Rosenbusch, P. Lemonde, G. Santarelli, P. Wolf, A. Clairon, A. Luiten, M. Tobar, and C. Salomon, “Cold atom clocks and applications,” J. Phys. B 38, S449–S468 (2005).
[Crossref]

Marmet, L.

L. Marmet and A. A. Madej, “Optical Ramsey spectroscopy and coherence measurements of the clock transition in a single trapped Sr ion,” Can. J. Phys. 78, 495–507 (2000).
[Crossref]

Materazzi, M.

M. Bellini, S. Cavalieri, C. Corsi, R. Eramo, and M. Materazzi, “Mutually coherent high-order harmonic pulses for XUV Ramsey spectroscopy,” Laser Phys. 15, 324–327 (2005).

Matro, A.

N. F. Scherer, R. J. Carlson, A. Matro, M. Du, A. J. Ruggiero, V. Romero-Rochin, J. A. Cina, G. R. Fleming, and S. A. Rice, “Fluorescence-detected wave packet interferometry: Time resolved molecular spectroscopy with sequences of femtosecond phase-locked pulses,” J. Chem. Phys. 95, 1487–1511 (1991).
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R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference technique applied to high precision spectroscopy,” J. Phys. (Paris) 42, C8–45 (1981).
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A. S. Zibrov and A. B. Matsko, “Optical Ramsey fringes induced by Zeeman coherence,” Phys. Rev. A 65, 013814 (2001).
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R. Th. Zinkstok, S. Witte, W. Ubachs, W. Hogervorst, and K. S. E. Eikema, “Frequency comb laser spectroscopy in the vacuum-ultraviolet region,” Phys. Rev. A 73, 061801(R) (2006).
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S. Witte, R. Th. Zinkstok, W. Ubachs, W. Hogervorst, and K. S. E. Eikema, “Deep-ultraviolet quantum interference metrology with ultrashort laser pulses,” Science 307, 400–403 (2005).
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Appl. Phys. (2)

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Appl. Phys. B (1)

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Europhys. Lett. (1)

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IEEE J. Quantum Electron. (1)

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P. R. Hemmer, M. S. Shahriar, H. Lamela-Rivera, S. P. Smith, B. E. Bernacki, and S. Ezekiel, “Semiconductor laser excitation of Ramsey fringes by using a Raman transition in a cesium atomic beam,” J. Opt. Soc. Am. B 10, 1326–1329 (1993).
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J. Phys. (Paris) (1)

R. E. Tench, B. W. Peuse, P. R. Hemmer, J. E. Thomas, S. Ezekiel, C. C. Leiby, R. H. Picard, and C. R. Willis, “Two laser Raman difference technique applied to high precision spectroscopy,” J. Phys. (Paris) 42, C8–45 (1981).
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Figures (6)

Fig. 1
Fig. 1 “Resonant” frequency ω o is defined as the energy difference (in frequency units) measured from the upper level to midway between the lower levels. The detuning is Δ ω o ω . The figure is drawn with Δ > 0 .
Fig. 2
Fig. 2 (a) The two pulses are different parts of the same sinusoid. This could, for example, be obtained by pulsed amplification of a stable cw laser. (b) The second pulse is a time-delayed copy of the first pulse. Unless T is an integer multiple of the optical period, the second pulse will not be in phase with a mathematical continuation (shown dotted) of the first pulse. In both (a) and (b), τ is the duration of each pulse, and T is the time from the beginning of the first pulse to the beginning of the second pulse.
Fig. 3
Fig. 3 Final population in state | 3 as a function of detuning Δ. All curves have Ω = 10 and τ = 0.05 in units chosen so that δ = 1 . The heavy solid curve has T = 4 π ; the dashed curve has T = 12 π ; the light solid curve has T = 5.3 π (to illustrate a non-integer multiple of π). As is typical with Ramsey fringes, the fringe width is of order 1 T . In contrast, a single pulse of duration τ = 0.05 would result in a peak much broader than the entire figure.
Fig. 4
Fig. 4 The only change among the three graphs is the velocity averaging interval. All three graphs have Ω = 5 and τ = 0.2 in units chosen so that δ = 1 . For comparison, the dashed curves show the non-velocity-averaged population in state | 3 as a function of detuning for a pulse delay of T = 4 π . The solid curves show the result when averaged using Eq. (36). (a) Averages over T = 4 π ± 1 . (b) Averages over T = 4 π ± 4 . (c) Averages from T = 4 π 10 to 4 π + 40 . The purpose of the asymmetric averaging interval in (c) is simply to allow a very large range of T without going below τ, which would be unphysical. The striking qualitative change is discussed in the text.
Fig. 5
Fig. 5 (a) Population in state | 3 versus detuning Δ and pulse delay time T. Units are chosen so that δ = 1 . The other parameters are τ = 0.2 , Ω = 7 , and γ = 0 . The maximum population is 0.49 and is indicated by white; black indicates no population in | 3 . (b) White indicates places where one would have expected high inversion from the extremely naïve guess based on Eqs. (18, 19). The agreement is quite good, considering that the naïve guess should become less and less reliable as inversion increases.
Fig. 6
Fig. 6 The dashed curve is identical to those in Fig. 4, again having T = 4 π , Ω = 5 , and τ = 0.2 in units chosen so that δ = 1 . The heavy solid curve has τ = 4 , giving the pulse a narrower bandwidth and hence identifying the central peak. The pulse area is the same in both cases, with Ω τ = 1 . The lighter solid curve confirms that decay ( γ = 0.2 ) simply lowers the overall signal. There is no velocity averaging in this figure.

Equations (38)

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δ ω 3 ω 1 = ω 3 .
Δ ω 0 ω ,
ω 0 ω 2 ω 1 + ω 3 2 = ω 2 ω 3 2 ;
Δ 1 ω 2 ω = Δ + δ 2 .
Δ 3 ω 2 ω 3 ω = Δ δ 2 .
E ( t ) = f ( t ) cos ( ω t ) ,
f ( t ) = { E 0 0 t τ and T t T + τ 0 otherwise } .
H ̂ 0 | j = ϵ j | j ,
| Ψ = C 1 ( t ) e i Δ 1 t | 1 + C 2 ( t ) e i ω 2 t | 2 + C 3 ( t ) e i Δ 1 t | 3 .
d d t | Ψ = i [ H ̂ 0 + V ̂ ( t ) ] | Ψ ,
V i j = q e i | x ̂ | j f ( t ) cos ( ω t ) = Ω i j ( t ) cos ( ω t ) .
( C ̇ 1 i Δ 1 C 1 ) e i Δ 1 t | 1 + ( C ̇ 2 i ω 2 C 2 ) e i ω 2 t | 2 + ( C ̇ 3 i Δ 1 C 3 ) e i Δ 1 t | 3 = i [ V C 1 e i Δ 1 t | 1 + ( ϵ 2 + V ) C 2 e i ω 2 t | 2 + ( ϵ 3 + V ) C 3 e i Δ 1 t | 3 ] .
C ̇ 1 = i 2 ( 2 Δ 1 C 1 Ω 1 C 2 ) ;
C ̇ 2 = i 2 ( Ω 1 * C 1 + Ω 3 C 3 ) ;
C ̇ 3 = i 2 ( Ω 3 * C 2 Δ 3 C 3 ) .
d d t C = i 2 M C ,
M [ 2 Δ 1 Ω 1 ( t ) 0 Ω 1 * ( t ) i γ Ω 3 ( t ) 0 Ω 3 * ( t ) 2 Δ 3 ] .
Δ 1 T = 2 n π
Δ 3 T = 2 m π ,
( ω 2 ω 1 ) T ( ω 2 ω 3 ) T = ω 3 T δ T = 2 p π .
Ω 1 ( t ) = Ω 3 ( t ) = { Ω , pulse on 0 , pulse off } .
| C 1 | 1 and | C 3 | | C 2 | 1 .
C 1 = e i Δ 1 t .
C ̇ 2 { i 2 Ω e i Δ 1 t , pulse on 0 , pulse off } .
C ̇ 3 { i 2 Ω C 2 , pulse on i Δ 3 C 3 , pulse off } .
C 2 { i Ω 2 t , 0 t τ i Ω 2 τ , τ t T i Ω 2 [ τ + ( t T ) e i Δ 1 T ] , T t T + τ } ;
C 3 { Ω 2 8 t 2 , 0 t τ Ω 2 8 τ 2 e i Δ 3 t , τ t T Ω 2 8 [ 2 τ ( t T ) + ( t T ) 2 e i Δ 1 T + τ 2 e i Δ 3 T ] , T t T + τ } .
τ 1 Δ i ( in this section ) ,
C 3 f Ω 2 8 τ 2 ( 2 + e i Δ 1 T + e i Δ 3 T ) .
P 3 = | C 3 f | 2 Ω 4 64 τ 4 ( 2 + e i Δ 1 T + e i Δ 3 T ) ( 2 + e i Δ 1 T + e i Δ 3 T ) = Ω 4 τ 4 32 ( 3 + 2 cos Δ 1 T + 2 cos Δ 3 T + cos δ T ) .
P 3 = Ω 4 τ 4 32 ( 3 + cos δ T + 4 cos Δ T cos δ T 2 ) .
C ( t ) = e i 2 M t C ( 0 ) .
C 1 ( T ) = e i Δ 1 ( T τ ) C 1 ( τ ) ,
C 2 ( T ) = e γ 2 ( T τ ) C 2 ( τ ) ,
C 3 ( T ) = e i Δ 3 ( T τ ) C 3 ( τ ) .
C f = e i 2 M τ C ( T ) .
P 3 = | C 3 f | 2 .
P 3 avg ( Δ , T max , T min ) = 1 T max T min T min T max P 3 ( Δ , T ) d T .

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