Abstract

We have observed the effect of angled excitation beams on the photon echoes from the 22S1/2–22P1/2 transition of atomic lithium. The degradation of the echo from noncollinear excitations is not so great as had been feared. A calculation with the billiard-ball echo model is in good agreement with our experimental results. In a similar experiment in atomic-sodium vapor we have recorded a decay of echo intensity with increasing pulse separation over nearly 12 orders of magnitude, with the weakest signals coming from atoms that had been more than 23 lifetimes in their excited state.

© 1984 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. O. Scully, M. J. Stephen, and D. C. Burnham, “Photon echo in gaseous media,” Phys. Rev. 171, 213–214 (1968).
    [Crossref]
  2. C. K. N. Patel and R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968).
    [Crossref]
  3. R. Beach, S. R. Hartmann, and R. Friedberg, “Billiard-ball echo model,” Phys. Rev. A 25, 2658–2666 (1982); “Billiard ball echo model,” in Proceedings of the International Conference on Lasers, 1981, Carl B. Collins, ed. (STS, McLean, Va., 1981), pp. 991–996.
    [Crossref]
  4. P. Ye and Y. R. Shen, “Transient four-wave mixing and coherent transient optical phenomena,” Phys. Rev. A 25, 2183–2199 (1982).
    [Crossref]
  5. R. Beach, B. Brody, and S. R. Hartmann, “Elliptical billiard-ball echo model,” Phys. Rev. A 27, 2537–2547 (1983).
    [Crossref]
  6. R. Beach, B. Brody, and S. R. Hartmann, “Photon echoes in lithium vapor with the use of angled excitation beams,” Phys. Rev. A 27, 2925–2929 (1983).
    [Crossref]
  7. I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
    [Crossref]
  8. See, for example, Ref. 9, where a calculation of Doppler damping of photon echoes for noncollinear excitations is made in the appendix. Unfortunately, the starting point of this calculation is Eq. (12) of Ref. 1, which appears incorrectly in Ref. 1 asθ=2τvi·(n-n1/c+…and correctly should beθ=2τvi·(n-n2)/c+….
  9. S. Aoki, “Photon-echo quantum beats on the 7P3/2–6S1/2transition in cesium,” Phys. Rev. A 20, 2013–2021 (1979).
    [Crossref]

1983 (2)

R. Beach, B. Brody, and S. R. Hartmann, “Elliptical billiard-ball echo model,” Phys. Rev. A 27, 2537–2547 (1983).
[Crossref]

R. Beach, B. Brody, and S. R. Hartmann, “Photon echoes in lithium vapor with the use of angled excitation beams,” Phys. Rev. A 27, 2925–2929 (1983).
[Crossref]

1982 (2)

R. Beach, S. R. Hartmann, and R. Friedberg, “Billiard-ball echo model,” Phys. Rev. A 25, 2658–2666 (1982); “Billiard ball echo model,” in Proceedings of the International Conference on Lasers, 1981, Carl B. Collins, ed. (STS, McLean, Va., 1981), pp. 991–996.
[Crossref]

P. Ye and Y. R. Shen, “Transient four-wave mixing and coherent transient optical phenomena,” Phys. Rev. A 25, 2183–2199 (1982).
[Crossref]

1979 (1)

S. Aoki, “Photon-echo quantum beats on the 7P3/2–6S1/2transition in cesium,” Phys. Rev. A 20, 2013–2021 (1979).
[Crossref]

1968 (2)

M. O. Scully, M. J. Stephen, and D. C. Burnham, “Photon echo in gaseous media,” Phys. Rev. 171, 213–214 (1968).
[Crossref]

C. K. N. Patel and R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968).
[Crossref]

1966 (1)

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
[Crossref]

Abella, I. D.

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
[Crossref]

Aoki, S.

S. Aoki, “Photon-echo quantum beats on the 7P3/2–6S1/2transition in cesium,” Phys. Rev. A 20, 2013–2021 (1979).
[Crossref]

Beach, R.

R. Beach, B. Brody, and S. R. Hartmann, “Elliptical billiard-ball echo model,” Phys. Rev. A 27, 2537–2547 (1983).
[Crossref]

R. Beach, B. Brody, and S. R. Hartmann, “Photon echoes in lithium vapor with the use of angled excitation beams,” Phys. Rev. A 27, 2925–2929 (1983).
[Crossref]

R. Beach, S. R. Hartmann, and R. Friedberg, “Billiard-ball echo model,” Phys. Rev. A 25, 2658–2666 (1982); “Billiard ball echo model,” in Proceedings of the International Conference on Lasers, 1981, Carl B. Collins, ed. (STS, McLean, Va., 1981), pp. 991–996.
[Crossref]

Brody, B.

R. Beach, B. Brody, and S. R. Hartmann, “Photon echoes in lithium vapor with the use of angled excitation beams,” Phys. Rev. A 27, 2925–2929 (1983).
[Crossref]

R. Beach, B. Brody, and S. R. Hartmann, “Elliptical billiard-ball echo model,” Phys. Rev. A 27, 2537–2547 (1983).
[Crossref]

Burnham, D. C.

M. O. Scully, M. J. Stephen, and D. C. Burnham, “Photon echo in gaseous media,” Phys. Rev. 171, 213–214 (1968).
[Crossref]

Friedberg, R.

R. Beach, S. R. Hartmann, and R. Friedberg, “Billiard-ball echo model,” Phys. Rev. A 25, 2658–2666 (1982); “Billiard ball echo model,” in Proceedings of the International Conference on Lasers, 1981, Carl B. Collins, ed. (STS, McLean, Va., 1981), pp. 991–996.
[Crossref]

Hartmann, S. R.

R. Beach, B. Brody, and S. R. Hartmann, “Elliptical billiard-ball echo model,” Phys. Rev. A 27, 2537–2547 (1983).
[Crossref]

R. Beach, B. Brody, and S. R. Hartmann, “Photon echoes in lithium vapor with the use of angled excitation beams,” Phys. Rev. A 27, 2925–2929 (1983).
[Crossref]

R. Beach, S. R. Hartmann, and R. Friedberg, “Billiard-ball echo model,” Phys. Rev. A 25, 2658–2666 (1982); “Billiard ball echo model,” in Proceedings of the International Conference on Lasers, 1981, Carl B. Collins, ed. (STS, McLean, Va., 1981), pp. 991–996.
[Crossref]

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
[Crossref]

Kurnit, N. A.

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
[Crossref]

Patel, C. K. N.

C. K. N. Patel and R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968).
[Crossref]

Scully, M. O.

M. O. Scully, M. J. Stephen, and D. C. Burnham, “Photon echo in gaseous media,” Phys. Rev. 171, 213–214 (1968).
[Crossref]

Shen, Y. R.

P. Ye and Y. R. Shen, “Transient four-wave mixing and coherent transient optical phenomena,” Phys. Rev. A 25, 2183–2199 (1982).
[Crossref]

Slusher, R. E.

C. K. N. Patel and R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968).
[Crossref]

Stephen, M. J.

M. O. Scully, M. J. Stephen, and D. C. Burnham, “Photon echo in gaseous media,” Phys. Rev. 171, 213–214 (1968).
[Crossref]

Ye, P.

P. Ye and Y. R. Shen, “Transient four-wave mixing and coherent transient optical phenomena,” Phys. Rev. A 25, 2183–2199 (1982).
[Crossref]

Phys. Rev. (2)

M. O. Scully, M. J. Stephen, and D. C. Burnham, “Photon echo in gaseous media,” Phys. Rev. 171, 213–214 (1968).
[Crossref]

I. D. Abella, N. A. Kurnit, and S. R. Hartmann, “Photon echoes,” Phys. Rev. 141, 391–406 (1966).
[Crossref]

Phys. Rev. A (5)

R. Beach, S. R. Hartmann, and R. Friedberg, “Billiard-ball echo model,” Phys. Rev. A 25, 2658–2666 (1982); “Billiard ball echo model,” in Proceedings of the International Conference on Lasers, 1981, Carl B. Collins, ed. (STS, McLean, Va., 1981), pp. 991–996.
[Crossref]

P. Ye and Y. R. Shen, “Transient four-wave mixing and coherent transient optical phenomena,” Phys. Rev. A 25, 2183–2199 (1982).
[Crossref]

R. Beach, B. Brody, and S. R. Hartmann, “Elliptical billiard-ball echo model,” Phys. Rev. A 27, 2537–2547 (1983).
[Crossref]

R. Beach, B. Brody, and S. R. Hartmann, “Photon echoes in lithium vapor with the use of angled excitation beams,” Phys. Rev. A 27, 2925–2929 (1983).
[Crossref]

S. Aoki, “Photon-echo quantum beats on the 7P3/2–6S1/2transition in cesium,” Phys. Rev. A 20, 2013–2021 (1979).
[Crossref]

Phys. Rev. Lett. (1)

C. K. N. Patel and R. E. Slusher, “Photon echoes in gases,” Phys. Rev. Lett. 20, 1087–1089 (1968).
[Crossref]

Other (1)

See, for example, Ref. 9, where a calculation of Doppler damping of photon echoes for noncollinear excitations is made in the appendix. Unfortunately, the starting point of this calculation is Eq. (12) of Ref. 1, which appears incorrectly in Ref. 1 asθ=2τvi·(n-n1/c+…and correctly should beθ=2τvi·(n-n2)/c+….

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

A π pulse has a 100% probability of promoting a ground-state wave packet (solid-line contour) to an excited-state wave packet (clashed-line contour). Because the excited-state wave packet has absorbed a photon having momentum ħk, it recoils with velocity vrecoil = ħk/m, where m is the atomic mass.

Fig. 2
Fig. 2

A π/2 pulse splits the atomic wave packet into a 50–50 superposition state. The wave-packet amplitudes describing the ground state (solid-line contour) and the excited state (dashed-line contour) then separate with time because the excited state recoils with the momentum of the absorbed photon.

Fig. 3
Fig. 3

Sequence of events leading to a photon echo at time 2τ after the first excitation pulse. The ground and excited states are denoted by solid and dashed lines, respectively.

Fig. 4
Fig. 4

Recoil diagram for the π/2, π pulse sequence that leads to a photon echo at time 2τ. Ground- and excited-state trajectories are denoted by solid and dashed lines, respectively. The wavy lines denote photons that are either emitted or absorbed, as indicated by the arrowheads.

Fig. 5
Fig. 5

The effect of a long laser pulse in the small-pulse-area limit. The contours are drawn where the wave-packet intensities are down by a factor of 1/e from their maxima. Ground and excited states are denoted by solid and dashed lines, respectively.

Fig. 6
Fig. 6

Schematic diagram of our experimental setup with angled beams. Here BS stands for beam splitter, BC for beam combiner, PH for pinhole, AOM for acousto-optic modulator, GP for Glan prism, PC for Pockels cell shutter, and PMT for photomultiplier tube (an RCA C31034). The second-harmonic pump beams from two Quanta-Ray DCR-1A YAG lasers are drawn as heavier lines.

Fig. 7
Fig. 7

Number of photons in the echo versus time between pulse 1 and the echo in units of the natural lifetime of the 22P1/2 state of lithium (27 nsec). The angles between the excitation pulses are 0.70 ± 0.06 and 1.35 ± 0.06 for the filled and open circles, respectively. These experiments were performed at a temperature of 609 K, as measured by thermocouples strapped to the outside of the sample cell.

Fig. 8
Fig. 8

Sequence of events leading to an echo in an angled-beam experiment with sub-Doppler excitation pulses. Solid- and dashed-line contours denote ground- and excited-state wave packets, respectively.

Fig. 9
Fig. 9

The ratio of the echo signals for the two angles plotted separately in Fig. 7 (circles). The solid line is the prediction of the billiard-ball model, as given by Eq. (18), for this ratio.

Fig. 10
Fig. 10

Echo intensity versus time between the first excitation pulse and the echo in a two-pulse photon-echo experiment performed on the sodium 32S1/2–32P3/2 transition with an angle of 0.64 ± 0.07 mrad between the excitation beams. Time is indicated in units of the 15.9-nsec natural lifetime of the 32P3/2 state.

Equations (21)

Equations on this page are rendered with MathJax. Learn more.

E = k 2 d r ( n ^ × p ) × n ^ exp ( i k R - r ) R - r ,
v Doppler τ dephasing = ƛ optical .
v recoil τ splitting = ƛ wave packet ,
ƛ wave packet = m v Doppler
m v recoil = / ƛ optical .
v recoil ƛ optical = v Doppler ƛ wave packet .
τ dephasing = τ splitting ,
E ( t ) = E 0 exp [ - 1 2 ( t τ pulse ) 2 ] cos Ω t ,
f 1 ( r ) = ( q 0 2 π ) 3 / 4 exp ( - 1 2 q 0 2 r 2 ) ,
q 0 = ƛ wave packet = ( 2 m k B T ) 1 / 2 ,
ψ ( r , t < 0 ) - f 1 ( r ) 1 ,
ψ ( r , t > 0 ) = f 1 ( r ) 1 - i Θ 2 f 2 ( r - k m t ) × exp [ - i ( Ω t - k · r ) ] 2
f 2 ( r ) = ( q 0 2 π ) 1 / 2 ( q 1 / e 4 / q 0 2 π ) 1 / 4 × exp [ - 1 2 ( q 0 2 r 2 + q 1 / e 2 r k 2 ) ] ,
q 1 / e = 1 k m τ pulse = 1 v recoil τ pulse
L / W = q 0 / q 1 / e = v recoil τ pulse ƛ wave packet .
I echo ( θ , τ ) = I 0 ( θ ) exp [ - 2 ( q 0 v recoil θ τ ) 2 ] × exp ( - 2 τ / T 1 ) .
R ( τ ) = I echo ( θ 1 = 1.35 mrad , τ ) I echo ( θ 2 = 0.70 mrad , τ ) = R 0 exp [ - 2 ( q 0 v recoil τ ) 2 ( θ 1 2 - θ 2 2 ) ]
R ( τ ) = R 0 exp [ - ( τ τ eff ) 2 ]
τ eff = 55 nsec .
θ=2τvi·(n-n1/c+
θ=2τvi·(n-n2)/c+.

Metrics