Abstract

A new sort of matter wave with the general astigmatism is proposed, whose orientation of the phase front and the density profile are crossed. A generalized ABCD law is developed to treat the evolution of the general astigmatic matter wave. It is revealed that two “cylindrical lens” pulses with oblique orientations can bring general astigmatism to the matter wave. The evolution characteristics of the general astigmatic matter wave in the gravitational field is discussed and illustrated numerically in detail. It is found that the orientations of the density profile and the phase front of the matter wave rotate continuously during evolution.

© 2008 Optical Society of America

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  1. H. J. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer-Verlag New York, 1999).
    [CrossRef]
  2. F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys,  71, 463-512 (1999).
    [CrossRef]
  3. K. Bongs and K. Sengstock, "Physics with coherent matter waves," Rep. Prog. Phys. 67, 907-963 (2004).
    [CrossRef]
  4. Ch. J. Bordé, "Atomic clocks and inertial sensors," Metrologia 39, 435-463 (2002).
    [CrossRef]
  5. M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
    [CrossRef] [PubMed]
  6. G. Roati, M. Zaccanti, C. D. Errico, J. Catani, M. Modugno, A. Simoni, M. Inguscio, and G. Modugno, "39K Bose-Einstein Condensate with Tunerable Interations," Phys. Rev. Lett. 99, 010403 (1)-(4) (2007).
    [CrossRef]
  7. T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
    [CrossRef] [PubMed]
  8. J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt,  8, 1687-1693 (1969).
    [CrossRef] [PubMed]
  9. Q. Lin and Y. Cai, "Tensor ABCD law for partially coherent twisted anisotropic Gaussian-Schell model beams," Opt. Lett. 27, 216-218 (2002).
    [CrossRef]
  10. Ch. Antoine and Ch. J. Bordé, "Exact phase shifts for atom interferometry," Phys. Lett. A 306, 277-284 (2003).
    [CrossRef]
  11. Y. Le. Coq, J. H. Thywissen, S. A. Rangwala, F. Gerbier, S. Richard, G. Delannoy, P. Bouyer, and A. Aspect, "Atom Laser Divergence," Phys. Rev. Lett. 87, 170403 (1)-(4) (2001).
    [CrossRef]
  12. J. -F. Riou, W. Guerin, Y. Le Coq, M. Fauqembergue, V. Josse, P. Bouyer, and A. Aspect, "Beam quality of a nonlineal atom laser," Phys. Rev. Lett. 96, 070404 (1)-(4) (2006).
    [CrossRef]
  13. F. Impens, P. Bouyer, and Ch. J. Bordé, "Matter-wave cavity gravimeter," Appl. Phys. B 84, 603-615 (2006).
    [CrossRef]
  14. Ch. J. Bordé, "Theoretical tools for atom optics and interferometry," C. R. Acad. Sci. Paris, t. 2, Ser. IV, 509-530 (2001).
  15. J. H. Van Vleck, "The correspondence principle in the statistical interpretation of quantum mechanics," Pro. Natl. Acad. Sci., USA 14, 178-188 (1928).
    [CrossRef]
  16. G. Whyte, P. Oöhberg, and J. Courtial, "Transverse laser modes in Bose-Einstein condensates," Phys. Rev. A 69, 053610 (1)-(8) (2004).
    [CrossRef]
  17. D. R. Murry and P. Oöhberg, "Matter wave focusing," J. Phys. B: At. Mol. Opt. Phys. 38, 1227-1234 (2005).
    [CrossRef]

2007 (1)

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

2006 (1)

F. Impens, P. Bouyer, and Ch. J. Bordé, "Matter-wave cavity gravimeter," Appl. Phys. B 84, 603-615 (2006).
[CrossRef]

2005 (1)

D. R. Murry and P. Oöhberg, "Matter wave focusing," J. Phys. B: At. Mol. Opt. Phys. 38, 1227-1234 (2005).
[CrossRef]

2004 (1)

K. Bongs and K. Sengstock, "Physics with coherent matter waves," Rep. Prog. Phys. 67, 907-963 (2004).
[CrossRef]

2003 (1)

Ch. Antoine and Ch. J. Bordé, "Exact phase shifts for atom interferometry," Phys. Lett. A 306, 277-284 (2003).
[CrossRef]

2002 (2)

2001 (1)

Ch. J. Bordé, "Theoretical tools for atom optics and interferometry," C. R. Acad. Sci. Paris, t. 2, Ser. IV, 509-530 (2001).

1999 (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys,  71, 463-512 (1999).
[CrossRef]

1996 (1)

M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
[CrossRef] [PubMed]

1969 (1)

J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt,  8, 1687-1693 (1969).
[CrossRef] [PubMed]

1928 (1)

J. H. Van Vleck, "The correspondence principle in the statistical interpretation of quantum mechanics," Pro. Natl. Acad. Sci., USA 14, 178-188 (1928).
[CrossRef]

Andrews, M. R.

M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
[CrossRef] [PubMed]

Antoine, Ch.

Ch. Antoine and Ch. J. Bordé, "Exact phase shifts for atom interferometry," Phys. Lett. A 306, 277-284 (2003).
[CrossRef]

Arnaud, J. A.

J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt,  8, 1687-1693 (1969).
[CrossRef] [PubMed]

Bongs, K.

K. Bongs and K. Sengstock, "Physics with coherent matter waves," Rep. Prog. Phys. 67, 907-963 (2004).
[CrossRef]

Bordé, Ch. J.

F. Impens, P. Bouyer, and Ch. J. Bordé, "Matter-wave cavity gravimeter," Appl. Phys. B 84, 603-615 (2006).
[CrossRef]

Ch. Antoine and Ch. J. Bordé, "Exact phase shifts for atom interferometry," Phys. Lett. A 306, 277-284 (2003).
[CrossRef]

Ch. J. Bordé, "Atomic clocks and inertial sensors," Metrologia 39, 435-463 (2002).
[CrossRef]

Ch. J. Bordé, "Theoretical tools for atom optics and interferometry," C. R. Acad. Sci. Paris, t. 2, Ser. IV, 509-530 (2001).

Bouyer, P.

F. Impens, P. Bouyer, and Ch. J. Bordé, "Matter-wave cavity gravimeter," Appl. Phys. B 84, 603-615 (2006).
[CrossRef]

Cai, Y.

Dalfovo, F.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys,  71, 463-512 (1999).
[CrossRef]

Durfee, D. S.

M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
[CrossRef] [PubMed]

Fattori, M.

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Fröhlich, B.

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Giorgini, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys,  71, 463-512 (1999).
[CrossRef]

Giovanazzi, S.

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Griesmaier, A.

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Impens, F.

F. Impens, P. Bouyer, and Ch. J. Bordé, "Matter-wave cavity gravimeter," Appl. Phys. B 84, 603-615 (2006).
[CrossRef]

Ketterle, W.

M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
[CrossRef] [PubMed]

Koch, T.

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Kogelnik, H.

J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt,  8, 1687-1693 (1969).
[CrossRef] [PubMed]

Kurn, D. M.

M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
[CrossRef] [PubMed]

Lahaye, T.

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Lin, Q.

Metz, J.

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Mewes, M.-O.

M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
[CrossRef] [PubMed]

Murry, D. R.

D. R. Murry and P. Oöhberg, "Matter wave focusing," J. Phys. B: At. Mol. Opt. Phys. 38, 1227-1234 (2005).
[CrossRef]

Oöhberg, P.

D. R. Murry and P. Oöhberg, "Matter wave focusing," J. Phys. B: At. Mol. Opt. Phys. 38, 1227-1234 (2005).
[CrossRef]

Pfau, T.

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Pitaevskii, L. P.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys,  71, 463-512 (1999).
[CrossRef]

Sengstock, K.

K. Bongs and K. Sengstock, "Physics with coherent matter waves," Rep. Prog. Phys. 67, 907-963 (2004).
[CrossRef]

Stringari, S.

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys,  71, 463-512 (1999).
[CrossRef]

van Druten, N. J.

M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
[CrossRef] [PubMed]

Van Vleck, J. H.

J. H. Van Vleck, "The correspondence principle in the statistical interpretation of quantum mechanics," Pro. Natl. Acad. Sci., USA 14, 178-188 (1928).
[CrossRef]

Appl. Opt (1)

J. A. Arnaud and H. Kogelnik, "Gaussian light beams with general astigmatism," Appl. Opt,  8, 1687-1693 (1969).
[CrossRef] [PubMed]

Appl. Phys. B (1)

F. Impens, P. Bouyer, and Ch. J. Bordé, "Matter-wave cavity gravimeter," Appl. Phys. B 84, 603-615 (2006).
[CrossRef]

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

D. R. Murry and P. Oöhberg, "Matter wave focusing," J. Phys. B: At. Mol. Opt. Phys. 38, 1227-1234 (2005).
[CrossRef]

Metrologia (1)

Ch. J. Bordé, "Atomic clocks and inertial sensors," Metrologia 39, 435-463 (2002).
[CrossRef]

Nature (1)

T. Lahaye, T. Koch, B. Fröhlich, M. Fattori, J. Metz, A. Griesmaier, S. Giovanazzi, and T. Pfau, "Strong dipolar effects in a quantum ferrofluid," Nature 448, 672-675 (2007).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Lett. A (1)

Ch. Antoine and Ch. J. Bordé, "Exact phase shifts for atom interferometry," Phys. Lett. A 306, 277-284 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

M.-O. Mewes, M. R. Andrews, N. J. van Druten, D. M. Kurn, D. S. Durfee, and W. Ketterle, "Bose-Einstein condensation in a tightly confining dc magnetic trap," Phys. Rev. Lett. 77, 416-419 (1996).
[CrossRef] [PubMed]

Pro. Natl. Acad. Sci. USA (1)

J. H. Van Vleck, "The correspondence principle in the statistical interpretation of quantum mechanics," Pro. Natl. Acad. Sci., USA 14, 178-188 (1928).
[CrossRef]

Rep. Prog. Phys. (1)

K. Bongs and K. Sengstock, "Physics with coherent matter waves," Rep. Prog. Phys. 67, 907-963 (2004).
[CrossRef]

Rev. Mod. Phys (1)

F. Dalfovo, S. Giorgini, L. P. Pitaevskii, and S. Stringari, "Theory of Bose-Einstein condensation in trapped gases," Rev. Mod. Phys,  71, 463-512 (1999).
[CrossRef]

Ser. (1)

Ch. J. Bordé, "Theoretical tools for atom optics and interferometry," C. R. Acad. Sci. Paris, t. 2, Ser. IV, 509-530 (2001).

Other (5)

G. Whyte, P. Oöhberg, and J. Courtial, "Transverse laser modes in Bose-Einstein condensates," Phys. Rev. A 69, 053610 (1)-(8) (2004).
[CrossRef]

H. J. Metcalf and P. Van der Straten, Laser Cooling and Trapping (Springer-Verlag New York, 1999).
[CrossRef]

G. Roati, M. Zaccanti, C. D. Errico, J. Catani, M. Modugno, A. Simoni, M. Inguscio, and G. Modugno, "39K Bose-Einstein Condensate with Tunerable Interations," Phys. Rev. Lett. 99, 010403 (1)-(4) (2007).
[CrossRef]

Y. Le. Coq, J. H. Thywissen, S. A. Rangwala, F. Gerbier, S. Richard, G. Delannoy, P. Bouyer, and A. Aspect, "Atom Laser Divergence," Phys. Rev. Lett. 87, 170403 (1)-(4) (2001).
[CrossRef]

J. -F. Riou, W. Guerin, Y. Le Coq, M. Fauqembergue, V. Josse, P. Bouyer, and A. Aspect, "Beam quality of a nonlineal atom laser," Phys. Rev. Lett. 96, 070404 (1)-(4) (2006).
[CrossRef]

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Figures (2)

Fig. 1.
Fig. 1.

One EMWpulse with a “focus length” f 1, focuses the matter wave in the x-direction. After an interval time t 1, another EM pulse whose “focus length” is f 2, focuses the matter wave in a direction with an angle α to the x-direction in the x-y plane. The gravitational acceleration g goes in the z-direction.

Fig. 2.
Fig. 2.

The phase front (a)-(d) and the density distribution (e)-(f) of the atom cloud after the interaction with the second pulse: (a) and (e) t 2=0; (b) and (f) t 2=0.06; (c) and (g) t 2=0.2; (d) and (h) t 2=0.3, where t 2 is the evolution time. The horizontal axis of figs. (e)-(f) is the x-coordinate and the vertical axis is the y-coordinate. The origin of the coordinate system is the center of mass of the atom cloud. The initial trapping frequency ω is 100kHz, and the atoms are Rubidium 87.

Equations (39)

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( r 1 v 1 ) = ( ξ ξ ˙ ) + M ( r 0 v 0 ) ,
M = ( A B C D ) .
S = m ξ ˙ ˜ ( r 1 ξ ) + t 0 t 1 L ( t ) d t t 0 t 1 V ( t ) d t
+ m 2 [ ( r ˜ 1 ξ ˜ ) D B 1 ( r 1 ξ ) 2 ( r ˜ 1 ξ ˜ ) ( B 1 ) T r 0 + r ˜ 0 B 1 A r 0 ] ,
S ( r 1 , t 1 , r 0 , t 0 ) = m ξ ˙ ˜ ( r 1 ξ ) + S 0 + m 2 ( r 0 r 1 ξ ) T U ( r 0 r 1 ξ ) ,
S 0 = t 0 t 1 L ( t ) d t t 0 t 1 V ( t ) d t
U = [ B 1 A B 1 ( C D B 1 A ) D B 1 ] .
{ ( B 1 A ) T = B 1 A , ( B 1 ) T = C D B 1 A , ( D B 1 ) T = D B 1 .
ψ ( r 1 , t 1 ) = K ( q 1 , t 1 , r 0 , t 0 ) ψ ( r 0 , t 0 ) d r 0 .
K ( r 1 , t 1 , r 0 , t 0 ) = ( m 2 π i h ¯ ) 3 2 det B 1 2 exp [ iS h ¯ ] ,
ψ ( r 0 , t 0 ) = exp [ im 2 h ¯ ( r 0 r 0 c ) T Q 0 1 ( r 0 r 0 c ) ] exp [ im v ˜ 0 ( r 0 r 0 c ) h ¯ ] ,
Q 0 1 = R 0 1 + 2 i h ¯ m W 0 2 ,
ψ ( r 1 , t 1 )
= ( m 2 π i h ¯ ) 3 2 det B 1 2 exp [ iS 0 h ¯ ] exp [ im ξ ˙ ˜ ( r 1 ξ ) h ]
exp { im 2 h ¯ [ r 0 r 1 ξ T U r 0 r 1 ξ + r ˜ 0 Q 0 1 r 0 + 2 v ˜ 0 r 0 ] } d r 0
= ( m 2 π i h ¯ ) 3 2 det B 1 2 exp [ iS c h ¯ ]
exp [ im 2 h ¯ ( r 1 r c ) T Q 1 1 ( r 1 r c ) ] exp [ im v ˜ c ( r 1 r c ) h ¯ ]
exp [ im 2 h ¯ ( B 1 A + Q 0 1 ) 1 2 r 0 ( B 1 A + Q 0 1 ) 1 2 B 1 ( r 1 r c ) 2 ] d r 0
= A + BQ 0 1 1 2 exp [ iS c h ¯ ]
exp [ im 2 h ¯ ( r 1 r c ) T Q 1 1 ( r 1 r c ) ] exp [ im v ˜ c ( r 1 r c ) h ¯ ]
Q 1 1 = ( C + D Q 0 1 ) ( A + B Q 0 1 ) 1 ,
S c = m 2 v ˜ 0 DBv 0 + m ξ ˙ ˜ Bv 0 + m 2 t 0 t 1 ( ξ ˙ 2 + ξ ˜ γ ξ + 2 g ˜ ξ ) dt t 0 t 1 V ( t ) dt .
r c = Bv 0 + ξ , v c = Dv 0 + ξ ˙ .
E ( x ) = E 0 ( 1 x 2 w x 2 ) u ,
Ω ac ( x ) = Ω ac 0 ( 1 2 x 2 w x 2 ) .
ψ ( t 0 + τ ) = ρ 1 exp [ i 2 Ω ac 0 τ w x 2 x 2 ] ψ ( t 0 ; v 0 + h ¯ k eff m , Q 0 1 ) .
ψ ( t 0 + τ ) = ρ 1 ψ ( t 0 ; v 0 + h ¯ k eff m , Q 1 1 ) ,
M = ( A B C D ) = ( I 0 F I ) ,
F = 1 f ( 1 0 0 0 0 0 0 0 0 ) ,
f = mw x 2 4 h ¯ Ω ac 0 τ .
M = M 2 M t 1 M 1 .
M 1 , 2 = ( I 0 F 1 , 2 I ) ,
F 1 = 1 f 1 ( 1 0 0 0 0 0 0 0 0 ) ,
F 2 = 1 f 2 ( cos 2 α sin α cos α 0 sin α cos α sin 2 α 0 0 0 0 ) .
M t 1 = ( I t 1 I 0 I ) .
Q 0 1 = ( q 1 0 0 0 q 1 0 0 0 q 1 ) ,
Q 2 1 = ( q 1 1 f 1 1 t 1 f 1 + t 1 q 1 cos 2 α f 2 sin α cos α f 2 0 sin α cos α f 2 q 1 1 + t 1 q 1 sin 2 α f 2 0 0 0 q 1 1 + t 1 q 1 ) .
M t = M t 2 M ,
M t 2 = ( I t 2 I 0 I )

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