Abstract

We consider coherent control of the shape of an atomic electron’s wavefunction using a train of short transform-limited laser pulses. This type of control is experimentally demonstrated by exciting with a train of three pulses and measuring the resulting quantum state distribution. We also present a general theory for control with a train of N pulses in the weak field limit and discuss the extension of this theory to the strong field limit.

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References

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  1. Paul Brumer and Moshe Shapiro, "Coherence Chemistry: Controlling Chemical Reactions with Lasers," Acc. Chem. Res. 22, 407 (1989).
    [CrossRef]
  2. Y.-Y. Yin, C. Chen, D. S. Elliott, and A. V. Smith, "Asymmetric photoelectron angular distributions from interfering photoionization processes," Phys. Rev. Lett. 69, 2353 (1992).
    [CrossRef] [PubMed]
  3. E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, "Phase-controlled currents in semiconductors," Phys. Rev. Lett. 74, 3596 (1995).
    [CrossRef] [PubMed]
  4. W. S. Warren, H. Rabitz, and M. Dahleh, "Coherent control of quantum dynamics: The dream is alive," Science 259, 1581 (1993).
    [CrossRef] [PubMed]
  5. D. Neuhauser and H. Rabitz, "Paradigms and algorithms for controlling molecular motion," Acc. Chem. Res. 26, 496 (1993).
    [CrossRef]
  6. C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, "Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses," Opt. Lett. 19, 737 (1994).
    [CrossRef] [PubMed]
  7. Marc M. Wefers and Keith A. Nelson, "Generation of high-fidelity programmable ultrafast optical waveforms," Opt. Lett. 20, 1047 (1995).
    [CrossRef] [PubMed]
  8. Marc M. Wefers, Keith A. Nelson, and Andrew M. Weiner, "Multidimensional shaping of ultrafast optical waveforms," Opt. Lett. 21, 746 (1996).
    [CrossRef] [PubMed]
  9. J. L. Krause, R. M. Whitnell, K. R. Wilson, Y. Yan, and S. Mukamel, "Optical control of molecular dynamics: Molecular cannons, reflectrons, and wave-packet focusers," J. Chem. Phys. 99, 6562 (1993).
    [CrossRef]
  10. B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, and K. R. Wilson, "Quantum control of wave packet evolution with tailored femtosecond pulses," Phys. Rev. Lett. 74, 3360 (1995).
    [CrossRef] [PubMed]
  11. D. W. Schumacher, J. H. Hoogenraad, D. Pinkos, and P. H. Bucksbaum, "Programmable cesium Rydberg wavepackets," Phys. Rev. A 52, 4719 (1995).
    [CrossRef] [PubMed]
  12. A. M. Weiner, "Enhancement of coherent charge oscillations in coupled quantum wells by femtosecond pulse shaping," J. Opt. Soc. Am. B 11, 2480 (1994).
    [CrossRef]
  13. J. Parker and C. R. Stroud, Jr., "Coherence and decay of Rydberg wave packets," Phys. Rev. Lett. 56, 716 (1986).
    [CrossRef] [PubMed]
  14. J. Parker and C. R. Stroud, Jr., "Rydberg wave packets and the classical limit," Phys. Scr. T12, 70 (1986).
    [CrossRef]
  15. W. Schleich, M. Pernigo, and Fam Le Kien, "Nonclassical state from two pseudoclassical states," Phys. Rev. A 44, 2172 (1991).
    [CrossRef] [PubMed]
  16. V. Buzek, A. Vidiella-Barranco, and P. L. Knight, "Superpositions of coherent states: Squeezing and dissipation," Phys. Rev. A 45, 6570 (1992).
    [CrossRef]
  17. Michael W. Noel and C. R. Stroud, Jr., "Excitation of an Atomic Electron to a Coherent Superposition of Macroscopically Distinct States," Phys. Rev. Lett. 77, 1913 (1996).
    [CrossRef] [PubMed]

Other

Paul Brumer and Moshe Shapiro, "Coherence Chemistry: Controlling Chemical Reactions with Lasers," Acc. Chem. Res. 22, 407 (1989).
[CrossRef]

Y.-Y. Yin, C. Chen, D. S. Elliott, and A. V. Smith, "Asymmetric photoelectron angular distributions from interfering photoionization processes," Phys. Rev. Lett. 69, 2353 (1992).
[CrossRef] [PubMed]

E. Dupont, P. B. Corkum, H. C. Liu, M. Buchanan, and Z. R. Wasilewski, "Phase-controlled currents in semiconductors," Phys. Rev. Lett. 74, 3596 (1995).
[CrossRef] [PubMed]

W. S. Warren, H. Rabitz, and M. Dahleh, "Coherent control of quantum dynamics: The dream is alive," Science 259, 1581 (1993).
[CrossRef] [PubMed]

D. Neuhauser and H. Rabitz, "Paradigms and algorithms for controlling molecular motion," Acc. Chem. Res. 26, 496 (1993).
[CrossRef]

C. W. Hillegas, J. X. Tull, D. Goswami, D. Strickland, and W. S. Warren, "Femtosecond laser pulse shaping by use of microsecond radio-frequency pulses," Opt. Lett. 19, 737 (1994).
[CrossRef] [PubMed]

Marc M. Wefers and Keith A. Nelson, "Generation of high-fidelity programmable ultrafast optical waveforms," Opt. Lett. 20, 1047 (1995).
[CrossRef] [PubMed]

Marc M. Wefers, Keith A. Nelson, and Andrew M. Weiner, "Multidimensional shaping of ultrafast optical waveforms," Opt. Lett. 21, 746 (1996).
[CrossRef] [PubMed]

J. L. Krause, R. M. Whitnell, K. R. Wilson, Y. Yan, and S. Mukamel, "Optical control of molecular dynamics: Molecular cannons, reflectrons, and wave-packet focusers," J. Chem. Phys. 99, 6562 (1993).
[CrossRef]

B. Kohler, V. V. Yakovlev, J. Che, J. L. Krause, M. Messina, and K. R. Wilson, "Quantum control of wave packet evolution with tailored femtosecond pulses," Phys. Rev. Lett. 74, 3360 (1995).
[CrossRef] [PubMed]

D. W. Schumacher, J. H. Hoogenraad, D. Pinkos, and P. H. Bucksbaum, "Programmable cesium Rydberg wavepackets," Phys. Rev. A 52, 4719 (1995).
[CrossRef] [PubMed]

A. M. Weiner, "Enhancement of coherent charge oscillations in coupled quantum wells by femtosecond pulse shaping," J. Opt. Soc. Am. B 11, 2480 (1994).
[CrossRef]

J. Parker and C. R. Stroud, Jr., "Coherence and decay of Rydberg wave packets," Phys. Rev. Lett. 56, 716 (1986).
[CrossRef] [PubMed]

J. Parker and C. R. Stroud, Jr., "Rydberg wave packets and the classical limit," Phys. Scr. T12, 70 (1986).
[CrossRef]

W. Schleich, M. Pernigo, and Fam Le Kien, "Nonclassical state from two pseudoclassical states," Phys. Rev. A 44, 2172 (1991).
[CrossRef] [PubMed]

V. Buzek, A. Vidiella-Barranco, and P. L. Knight, "Superpositions of coherent states: Squeezing and dissipation," Phys. Rev. A 45, 6570 (1992).
[CrossRef]

Michael W. Noel and C. R. Stroud, Jr., "Excitation of an Atomic Electron to a Coherent Superposition of Macroscopically Distinct States," Phys. Rev. Lett. 77, 1913 (1996).
[CrossRef] [PubMed]

Supplementary Material (1)

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Figures (4)

Fig. 1:
Fig. 1:

(QuickTime animation 2.2 Mb) Phase evolution of a circular orbit wave packet. The probability distribution in the x - y plane is plotted as the height out of this plane and the phase distribution is represented using different colors for the different phases of the wavefunction. The color wheel to the right of the figure shows the continuous mapping of the phase onto color. This wave packet is made up of a Gaussian distribution (full width at half maximum of 8) of circular states centered near principle quantum number n 0 = 180. Time is in units of the Kepler period, TK , and the frame shown above is at the time where the one-third fractional revival appears. The diameter of the orbit is slightly over 3 μm.

Fig. 2:
Fig. 2:

Experimental setup used to produce a train of phase locked pulses. The beam splitters (BS) split a single laser pulse into three. The mirrors (M1, M2, and M3) are adjusted to recombine these three pulses so that they exactly spatially overlap, but are delayed in time. Two HeNe laser beams are used to actively stabilize the relative phase between pairs of pulses. This is done interferometrically by sending the HeNe beam in one of the delay arms through a glass wedge (W) to produce tilt fringes in the recombined HeNe beams. The position of these fringes and thus the delay between pulses can be locked with a simple servo system through feedback to the piezo electric translator (PZT) on which the associated mirror is mounted. By translating the fringe detector in the servo system the delay between pulses can be varied with an accuracy of 1/100th of the optical period.

Fig. 3:
Fig. 3:

Control with a pair of pulses separated by one-third the Kepler period, TK . The top state distribution, (a), was taken with only one pulse present and is used as a reference to identify the locations of the different states in the superposition. The next three state distributions, (b)–(d), are for three choices of phase between the two pulses. The phases were chosen to have the effect of nearly eliminating population from one of the three central states in the superposition.

Fig. 4:
Fig. 4:

Here the effect of using three pulses in the control train was investigated. The top state distribution, (a), is again a reference trace taken with only one pulse present. Next, in (b), the second pulse was turned on and its phase adjusted to equalize the population in n = 65 and 66. Finally, the third pulse was turned on, and its phase adjusted to form the distribution shown in (c), which is dominated by every third state.

Equations (10)

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E ( t ) = j = 1 N ε j f ( t T j ) [ e ( t T j ) + e ( t T j ) ] z ̂ ,
ψ ˙ n = i j d n ħ ε j f ( t T j ) e i Δ n t e T j ψ g .
ψ n = i d n ħ j = 1 N F j t f ( t T j ) e i Δ n t dt ,
ψ n = i d n ħ j = 1 N F j e i Δ n j T K N F ( Δ n ) ,
F ( Δ n ) = f ( X ) e i Δ n X dX .
Ψ n = j = 1 N F j e i Δ n j T K N .
Ψ = 𝗠 F ,
F = 𝗠 1 Ψ .
ψ n = i d n ħ j = 1 N F j t f ( t T j ) e i Δ n t ψ g ( t ) dt .
ψ g = exp [ γ j j F j * F j 0 t f ( t T j ) f ( t T j ) dt ] .

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