Abstract

The mechanical action of light on atoms is currently a tool used ubiquitously in cold atom physics. In the semiclassical regime, where atomic motion is treated classically, the computation of the mean force acting on a two-level atom requires numerical approaches in the most general case. Here we show that this problem can be tackled in a purely analytical way. We provide an analytical yet simple expression of the mean force that holds in the most general case, where the atom is simultaneously exposed to an arbitrary number of lasers with arbitrary intensities, wave vectors, and phases. This yields a novel tool for engineering the mechanical action of light on single atoms.

© 2017 Optical Society of America

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References

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  1. R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
    [Crossref]
  2. P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
    [Crossref]
  3. S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
    [Crossref]
  4. M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
    [Crossref]
  5. R. J. Cook, “Atomic motion in resonant radiation: an application of Ehrenfest’s theorem,” Phys. Rev. A 20, 224–228 (1979).
    [Crossref]
  6. J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
    [Crossref]
  7. M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
    [Crossref]
  8. M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
    [Crossref]
  9. S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
    [Crossref]
  10. C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
    [Crossref]
  11. C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
    [Crossref]
  12. X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
    [Crossref]
  13. D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
    [Crossref]
  14. V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, 1987).
  15. V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
    [Crossref]
  16. G. S. Agarwal, “Quantum statistical theories of spontaneous emission and their relation to other approaches,” in Springer Tracts in Modern Physics (Springer, 1974), Vol. 70, pp. 1–129.
  17. B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).
  18. All nonzero mj integers are setwise coprime (the greatest common divisor of these integers is 1) but not necessarily pairwise coprime (each pair of nonzero mj is not necessarily setwise coprime).
  19. L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, Vol. 146 of Translations of Mathematical Monographs (American Mathematical Society, 1995).
  20. The monodromy matrix C of a set of equations x˙=A(t)x+b(t), where A(t) and b(t) are T-periodic (comprising the case b(t) constant) is the matriciant XI(t) of the set of equations evaluated at time t=t0+T: C=XI(t0+T). The matriciant is the unique matrix solution X(t) to the matricial equation X˙=A(t)X subscribed to the initial conditions X(t0)=I, with I the identity matrix and where X has the same dimension as A(t). The matriciant and monodromy matrices are both invertible. A logarithm of an invertible matrix B is a matrix L such that eL=B. Every invertible matrix admits a logarithm, which is not unique but with invariant real parts of the eigenvalues.
  21. V. A. Iakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Halsted, 1975), Vol. 1.
  22. F. Riesz, Les Systèmes d’Équations Linéaires à une Infinité d’Inconnues (Gauthier-Villars, 1913).
  23. As a consequence of the linear system [Eq. (12)] that connects w0 to wmlj, ∀l,j, then to wmlj+ml′j′, ∀l′,j′, and so on, the convergence in Eq. (13) is highly optimized by reordering the vector of unknowns w (and accordingly the lines and columns of I+W) so as to have w0 symmetrically surrounded by w±|mlj| for all distinct nonzero mlj, then by w±|mlj+ml′j′| for all additional distinct mlj+ml′j′, and so on.
  24. C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).
  25. H. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer, 1999).
  26. The specific links between the unknowns wn in the linear system [Eq. (12)] [23] can make the ratios qn+mlj and hence the Fourier components Fj,n not necessarily significant for the first n>0 values. In this case, the forces Fj(t) could be better described in the periodic regime by an almost periodic behavior with an almost period smaller than Tc depending on each specific case.
  27. V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).
  28. M. Cashen and H. Metcalf, “Optical forces on atoms in nonmonochromatic light,” J. Opt. Soc. Am. B 20, 915–924 (2003).
    [Crossref]
  29. L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
    [Crossref]
  30. G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).

2016 (1)

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

2015 (2)

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

2013 (1)

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

2011 (1)

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

2004 (1)

L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
[Crossref]

2003 (1)

2000 (1)

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

1999 (1)

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

1997 (1)

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

1995 (1)

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

1988 (1)

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

1986 (1)

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

1979 (2)

R. J. Cook, “Atomic motion in resonant radiation: an application of Ehrenfest’s theorem,” Phys. Rev. A 20, 224–228 (1979).
[Crossref]

V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
[Crossref]

1973 (1)

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

1972 (1)

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

Adrianova, L. Y.

L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, Vol. 146 of Translations of Mathematical Monographs (American Mathematical Society, 1995).

Agarwal, G. S.

G. S. Agarwal, “Quantum statistical theories of spontaneous emission and their relation to other approaches,” in Springer Tracts in Modern Physics (Springer, 1974), Vol. 70, pp. 1–129.

Aldridge, L.

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

Anderson, M. H.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Anisimov, P. M.

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

Arnold, B.

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

Ashkin, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Berger, R. L.

G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).

Bjorkholm, J. E.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Bouyer, Ph.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Cable, A.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Casella, G.

G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).

Cashen, M.

Cashen, M. T.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

Chi, F.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

Chu, S.

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Cohen-Tannoudji, C.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).

Cook, R. J.

R. J. Cook, “Atomic motion in resonant radiation: an application of Ehrenfest’s theorem,” Phys. Rev. A 20, 224–228 (1979).
[Crossref]

Corder, C.

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

Cornell, E. A.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Danileiko, M. V.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Dupont-Roc, J.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).

Elgin, J.

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

Enshner, J. R.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Eyler, E. E.

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

Galica, S. E.

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

Grimm, R.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Grynberg, G.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).

Hua, X.

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

Iakubovich, V. A.

V. A. Iakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Halsted, 1975), Vol. 1.

Jacquinot, P.

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Letokhov, V. S.

V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, 1987).

Liberman, S.

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Matthews, M. R.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Metcalf, H.

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

C. Corder, B. Arnold, X. Hua, and H. Metcalf, “Laser cooling without spontaneous emission using the bichromatic force,” J. Opt. Soc. Am. B 32, B75–B83 (2015).
[Crossref]

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
[Crossref]

M. Cashen and H. Metcalf, “Optical forces on atoms in nonmonochromatic light,” J. Opt. Soc. Am. B 20, 915–924 (2003).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

H. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer, 1999).

Minogin, V. G.

V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
[Crossref]

V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, 1987).

Negriiko, A. M.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Ovchinnikov, Yu. B.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Picqué, J. L.

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Pinard, J.

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Riesz, F.

F. Riesz, Les Systèmes d’Équations Linéaires à une Infinité d’Inconnues (Gauthier-Villars, 1913).

Romanenko, V. I.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Salomon, Ch.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Schieder, R.

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

Serimaa, O. T.

V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
[Crossref]

Shore, B. W.

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).

Söding, J.

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

Stack, D.

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

Starzhinskii, V. M.

V. A. Iakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Halsted, 1975), Vol. 1.

Van der Straten, P.

H. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer, 1999).

Voitsekhovich, V. S.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Walther, H.

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

Wieman, C. E.

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Williams, M. R.

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

Wöste, L.

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

Yatsenko, L.

L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
[Crossref]

Yatsenko, L. P.

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

J. Opt. Soc. Am. B (2)

Opt. Commun. (3)

V. G. Minogin and O. T. Serimaa, “Resonant light pressure forces in a strong standing laser wave,” Opt. Commun. 30, 373–379 (1979).
[Crossref]

R. Schieder, H. Walther, and L. Wöste, “Atomic beam deflection by the light of a tunable dye laser,” Opt. Commun. 5, 337–340 (1972).
[Crossref]

P. Jacquinot, S. Liberman, J. L. Picqué, and J. Pinard, “High resolution spectroscopic application of atomic beam deflection by resonant light,” Opt. Commun. 8, 163–165 (1973).
[Crossref]

Phys. Rev. A (7)

X. Hua, C. Corder, and H. Metcalf, “Simulation of laser cooling by the bichromatic force,” Phys. Rev. A 93, 063410 (2016).
[Crossref]

D. Stack, J. Elgin, P. M. Anisimov, and H. Metcalf, “Numerical studies of optical forces from adiabatic rapid passage,” Phys. Rev. A 84, 013420 (2011).
[Crossref]

R. J. Cook, “Atomic motion in resonant radiation: an application of Ehrenfest’s theorem,” Phys. Rev. A 20, 224–228 (1979).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Measurement of the bichromatic optical force on Rb atoms,” Phys. Rev. A 60, R1763–R1766 (1999).
[Crossref]

M. R. Williams, F. Chi, M. T. Cashen, and H. Metcalf, “Bichromatic force measurements using atomic beam deflections,” Phys. Rev. A 61, 023408 (2000).
[Crossref]

S. E. Galica, L. Aldridge, and E. E. Eyler, “Four-color stimulated optical forces for atomic and molecular slowing,” Phys. Rev. A 88, 043418 (2013).
[Crossref]

L. Yatsenko and H. Metcalf, “Dressed-atom description of the bichromatic force,” Phys. Rev. A 70, 063402 (2004).
[Crossref]

Phys. Rev. Lett. (3)

J. Söding, R. Grimm, Yu. B. Ovchinnikov, Ph. Bouyer, and Ch. Salomon, “Short-distance atomic beam deceleration with a stimulated light force,” Phys. Rev. Lett. 78, 1420–1423 (1997).
[Crossref]

C. Corder, B. Arnold, and H. Metcalf, “Laser cooling without spontaneous emission,” Phys. Rev. Lett. 114, 043002 (2015).
[Crossref]

S. Chu, J. E. Bjorkholm, A. Ashkin, and A. Cable, “Experimental observation of optically trapped atoms,” Phys. Rev. Lett. 57, 314–317 (1986).
[Crossref]

Science (1)

M. H. Anderson, J. R. Enshner, M. R. Matthews, C. E. Wieman, and E. A. Cornell, “Observation of Bose-Einstein condensation in a dilute atomic vapor,” Science 269, 198–201 (1995).
[Crossref]

Sov. Phys. Tech. Phys. (1)

V. S. Voitsekhovich, M. V. Danileiko, A. M. Negriiko, V. I. Romanenko, and L. P. Yatsenko, “Light pressure on atoms in counterpropagating amplitude-modulated waves,” Sov. Phys. Tech. Phys. 33, 690–691 (1988).

Other (13)

V. G. Minogin and V. S. Letokhov, Laser Light Pressure on Atoms (Gordon & Breach, 1987).

G. S. Agarwal, “Quantum statistical theories of spontaneous emission and their relation to other approaches,” in Springer Tracts in Modern Physics (Springer, 1974), Vol. 70, pp. 1–129.

B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley, 1990).

All nonzero mj integers are setwise coprime (the greatest common divisor of these integers is 1) but not necessarily pairwise coprime (each pair of nonzero mj is not necessarily setwise coprime).

L. Y. Adrianova, Introduction to Linear Systems of Differential Equations, Vol. 146 of Translations of Mathematical Monographs (American Mathematical Society, 1995).

The monodromy matrix C of a set of equations x˙=A(t)x+b(t), where A(t) and b(t) are T-periodic (comprising the case b(t) constant) is the matriciant XI(t) of the set of equations evaluated at time t=t0+T: C=XI(t0+T). The matriciant is the unique matrix solution X(t) to the matricial equation X˙=A(t)X subscribed to the initial conditions X(t0)=I, with I the identity matrix and where X has the same dimension as A(t). The matriciant and monodromy matrices are both invertible. A logarithm of an invertible matrix B is a matrix L such that eL=B. Every invertible matrix admits a logarithm, which is not unique but with invariant real parts of the eigenvalues.

V. A. Iakubovich and V. M. Starzhinskii, Linear Differential Equations with Periodic Coefficients (Halsted, 1975), Vol. 1.

F. Riesz, Les Systèmes d’Équations Linéaires à une Infinité d’Inconnues (Gauthier-Villars, 1913).

As a consequence of the linear system [Eq. (12)] that connects w0 to wmlj, ∀l,j, then to wmlj+ml′j′, ∀l′,j′, and so on, the convergence in Eq. (13) is highly optimized by reordering the vector of unknowns w (and accordingly the lines and columns of I+W) so as to have w0 symmetrically surrounded by w±|mlj| for all distinct nonzero mlj, then by w±|mlj+ml′j′| for all additional distinct mlj+ml′j′, and so on.

C. Cohen-Tannoudji, J. Dupont-Roc, and G. Grynberg, Atom-Photon Interactions, Basic Processes and Applications (Wiley, 1992).

H. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer, 1999).

The specific links between the unknowns wn in the linear system [Eq. (12)] [23] can make the ratios qn+mlj and hence the Fourier components Fj,n not necessarily significant for the first n>0 values. In this case, the forces Fj(t) could be better described in the periodic regime by an almost periodic behavior with an almost period smaller than Tc depending on each specific case.

G. Casella and R. L. Berger, Statistical Inference, 2nd ed. (Duxbury, 2002).

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Figures (1)

Fig. 1.
Fig. 1.

Stimulated bichromatic force F as a function of the atomic velocity v computed via our formalism for a detuning δ=10Γ, a Rabi frequency of 3/2δ, and a phase shift of π/2 for one laser.

Equations (21)

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F=Γ2s1+sk.
Fj=Γ2sj1+seffkj
ddtρ^(t)=1i[H^(t),ρ^(t)]+D(ρ^(t))
x˙(t)=A(t)x(t)+b
A(t)=(Γ/2δ¯Im(Ω(t))δ¯Γ/2Re(Ω(t))Im(Ω(t))Re(Ω(t))Γ),
Ωj=Dge·Ejei(kj·r+ϕj)/ΩR,jeiϕj,
x(t)=PI(t)eR(tt0)(x0xp(t0))+xp(t),
x(t)=n=+xneinωct,
un=i(τn+j=1NΩjwnmjτnj=1NΩj*wn+mj),vn=(τn+j=1NΩjwnmj+τnj=1NΩj*wn+mj),
wn+mM0Wn,mwn+m=12(1+s˜)δn,0.
s˜j=Re[ΩjΓ/2iδjl=1l:δl=δjNΩl*Γ].
(I+W)w=c,
qn=limkΔk(n)Δk(0),
w0=1211+seff,
sj=Re[ΩjΓ/2iδjl=1NΩl*Γqmlj].
Rj(t)=Re[Ωj(v(t)+iu(t))ei(ωjω¯)t].
Rj,n=Γ2sj,n1+seff
σj,n=ΩjΓ/2+i(nωcδj)l=1NΩl*Γqn+mlj.
F¯j=Γ2sj1+seffkj,
sj|Ωj|2/2δj2+Γ2/4.
wn+Wn,nswn+ns+Wn,nswnns=12(1+s˜)δn,0.

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