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1、arXiv:quant-ph/0007004v1 3 Jul 2000 Phase-Control of Photoabsorption in Optically Dense Media David Petrosyan1,2and P. Lambropoulos1,3,4 1 Institute of Electronic Structure neither the fundamental, nor the harmonic experience any remarkable distortion of their shapes or total energy Sj(z) R dt|Ej(z,
2、t)|2, j = f,h, over distances of propagation z as large as 50 cm. The accumulated over this distance change of the relative phase is only 103 rad, which is due to the fi eld independent phase shift of the fundamental, given by the term in parentheses of Eq. (5a). Consider next the case (0,t) = 0, i.
3、e., at the entrance to the cell the two fi elds interfere constructively. The results corresponding to the parameters of Fig. 1 with Imax f = 81010W/cm2are collected in Figs. 2 and 3. One can see in Fig. 2 that, in the course of propagation, the relative phase (taken at the dynamic pulse maximum tma
4、x+z/c) grows rapidly and over a distance of the order of 1 cm reaches the value , at which the initial constructive interference of the two fi elds turns to destructive. At the same time, the total energy of the harmonic pulse, after a small reduction over a short interval of z, begins to increase a
5、s a result of the energy transfer from the strong fundamental fi eld, in the parametric conversion process. This small reduction of the harmonic takes place only at the beginning of the propagation, when the relative phase is still close to 0 and the two fi elds interfere constructively, in the proc
6、ess of excitation of atoms from the ground state |1i to the state |2i, while the generated part of the harmonic fi eld is out of phase with the fundamental approximately by and continues to build up with a slight oscillation around the value of the phase. It is important to mention that throughout t
7、he propagation, the amplitude and the phase of the fundamental fi eld do not change signifi cantly. This is because the number of photons contained in that pulse exceeds by many ( 6) orders of magnitude the number of atoms the pulse interacts with over the distance of z 20 cm. Comparing the three gr
8、aphs of Fig. 2, one can see that with increasing of and Sh , the ionization probability fi rst also grows, which is consistent with the previous discussion related to that intensity of the fundamental fi led. But as approaches , the ion yield drops almost exponentially until Q 10%. This residual ion
9、ization that is present even at (and tends to 0 rather slowly) is caused by the fact that, because of the signifi cant increase of the total energy of harmonic fi eld, conditions (6) are not completely satisfi ed and the upper atomic level |2i acquires population due to that fraction of the generate
10、d fi eld which exceeds the initial. Since the temporal widths of the pulses are less than the (radiative) relaxation time of the atomic coherence 21(1 2 ns), a signifi cant fraction of the harmonic pulse amplitude is generated behind the fundamental (Fig. 3). That part of the amplitude is then atten
11、uated due to the atomic relaxation. Thus the total energy of the harmonic, after passing a maximum at z 5 7 cm, decays then slowly back. Under these conditions, the leading part of the harmonic pulse that falls under the temporal shape of the fundamental is by out of phase with the latter and theref
12、ore the ionization vanishes, while the generated tail is continuously scattered by the atoms in the process of radiative decay. The oscillations of the relative phase around are also slowly damped and the propagation reaches a “dynamic equilibrium” We note fi nally that a similar behavior of the sys
13、tem is obtained for a range of intensities we have explored. The main diff erence is that for weaker fi elds (Imax f = 3 1010and 1 1010W/cm2) the ion yield does not exhibit a maximum other than at z = 0 and drops to zero much faster as z increases, which is consistent with the discussion above. We h
14、ave examined the problem of propagation in the simplest context of phase control, namely the excitation of a bound state, which has been used as a prototype in much of the initial work 1,2. As discussed here, the problem bears resemblance to earlier works 710 on cancellation in third harmonic genera
15、tion experiments, and so does the whole issue of phase control. The relevance and possible impact of propagation has been recognized by Chen and Elliott 6 who presented data and an interpretation in terms of rate equations 10. Their study showed evidence of non-linear coupling, such as those discuss
16、ed above, and called for “more rigorous techniques” in the approach to this basic problem. In the limit of validity of rate equations, our results do indeed recapture the equations employed in their analysis. It will be interesting to explore this issue under more general conditions, such as the exc
17、itation of states embedded in continua, on which we expect to report elsewhere. The basic features of our analysis should, 3 however, remain valid. In summary, we have shown that the propagation of a bichromatic fi eld with a preselected initial relative phase, has a profound eff ect. Over a rather
18、short scaled distance and independent of its initial value, the relative phase settles to a value that makes the medium transparent to the radiation, precluding thus further excitation and consequently control. The scaled distance zN does of course involve the density of the species and the cross-se
19、ction of the laser beam, which suggests some fl exibility on the choice of these parameters. In any case, however, the actual length of the interaction region over which control can be active will be defi ned and limited by the combination of the above parameters, as well as by the geometry of the f
20、ocused or unfocused laser beam. Briefl y, for not very low atomic densities (N 1012cm3 ), the harmonic fi eld settles to the steady-state value within a thin layer where a focused beam is well approximated by a plane wave. In the presence of large ac Stark shifts, however, a detailed analysis includ
21、ing specifi c experimental parameters is mandatory. 1 M. Shapiro, J. W. Hepburn, and P. Brumer, Chem. Phys. Lett. 149, 451 (1988); C. K. Chan, P. Brumer, and M. Shapiro, J. Chem. Phys. 94, 2688 (1991). 2 C. Chen, Y. Yin, and D. S. Elliott, Phys. Rev. Lett. 64, 507 (1990); C. Chen, and D. S. Elliott,
22、 ibid. 65, 1737 (1990). 3 L. Zhu, K. Suto, J. A. Fiss, R. Wada, T. Seideman, and R. J. Gordon, Phys. Rev. Lett. 79, 4108 (1997); L. Zhu, V. Kleiman, X. Li, S. Lu, K. Trentelman, and R. J. Gordon, Science, 270, 77 (1995) and references therein. 4 T. Nakajima and P. Lambropoulos, Phys. Rev. Lett. 70,
23、1081 (1993); P. Lambropoulos and T. Nakajima, ibid. 82, 2266 (1999); T. Nakajima and P. Lambropoulos, Phys. Rev. A 50, 595 (1994); T. Nakajima, J. Zhang, and P. Lambropoulos, J. Phys. B 30, 1077 (1997). 5 J. C. Camparo and P. Lambropoulos, Phys. Rev. A 55, 552 (1997); 59, 2515 (1999). 6 Ce Chen and
24、D. S. Elliott, Phys. Rev. A 53, 272 (1996). 7 R. N. Compton, J. C. Miller, A. E. Carter, and P. Kruit, Chem. Phys. Lett. 71, 87 (1980); J. C. Miller, R. N. Compton, M. G. Payne, and W. W. Garret, Phys. Rev. Lett. 45, 114 (1980). 8 D. J. Jackson and J. J. Wynne, Phys. Rev. Lett. 49, 543 (1982); J. J.
25、 Wynne, ibid. 52, 751 (1984); D. J. Jackson, J. J. Wynne, and P. H. Kes, Phys. Rev. A 28, 781 (1983). 9 D. Charalambidis, X. Xing, J. Petrakis, and C. Fotakis, Phys. Rev. A 44, R24 (1991). 10 M. Elk, P. Lambropoulos, and X. Tang, Phys. Rev. A 46, 465 (1992). 0.00.51.01.52.0 Relative phase ( rad) 0.0
26、 0.2 0.4 0.6 0.8 Ion yield FIG. 1. Ion yield Q = (11122)t 111 (t ) versus relative phase for three diff erent peak intensities of the fundamental: Imax f = 1 1010W/cm2(dashed line), Imax f = 3 1010W/cm2(dot-dashed line), Imax f = 8 1010W/cm2(solid line). 4 05101520 Propagation length z (cm) 0.0 0.2
27、0.4 0.6 0.8 Ion yield 0.0 1.0 2.0 3.0 Normalized energy 0.0 0.5 1.0 1.5 Relative phase ( rad) 00.250.5 0.9 1.0 1.1 (a) (b) (c) FIG. 2. Relative phase (z,t = tmax+ z/c) (a), normalized energy Sh(z)/Sh (0) of harmonic fi eld (b), and ion yield Q(z) (c) versus propagation length z for the case Imax f = 8 1010W/cm2. 024 Time (ns) 0 2 4 6 8 0 2 4 6 Harmonic field amplitude (10 5 V/m) 0 1 2 3 024 Time (ns) 0 2 4 6 0 2 4 6 0 2 4 6 8 z=0 z=1cm z=5cm z=10cm z=15cm z=20cm FIG. 3. Temporal profi le of the amplitude Eh of harmonic fi eld at diff erent z. All parameters are as in Fig. 2. 5