1 Introduction Photodetachment of a hydrogen ion H? in an electric eld has been measured and oscillations in the cross section have been observed [1].

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1 A Formula for Laser Pulse Photodetachments of Negative Ions Yongliang Zhao, Mengli Du 1 and Jian-min Mao Department of Mathematics Hong Kong University of Science and Technology Clearwater Bay, Kowloon, Hong Kong Abstract We express the detachment cross section caused by a Gaussian pulsed laser in terms of that caused by a continuous laser. Using this expression and the Poisson's summation formula, we derive a uniform semiclassical formula for the cross section of pulsed-laser photodetachment of a hydrogen ion H? in static parallel electric and magnetic elds. The uniform semiclassical formula does not have the divergence problem at the bifurcation points. Numerical comparisons show an excellent agreement between the uniform semiclassical formula and the quantum formula for the detachment cross section. 1 1 Permanent address: Institute of Theoretical Physics, Academia Sinica, P.O. Box 735, Beijing , China. 1

2 1 Introduction Photodetachment of a hydrogen ion H? in an electric eld has been measured and oscillations in the cross section have been observed [1]. To understand these oscillations, quantum theories [, 3, 4, 5, 6] and the `standard' semiclassical closed-orbit theory [7] have been developed. The standard semiclassical closed-orbit theory has the divergence problem at the bifurcation points of the closed orbits [7, 8] and the problem has been overcome by using a Fresnel integral [9, 10]. In recent years, a laser pulse (rather than a continuous laser) has been used to cause photodetachments of negative ions in a (static) electric eld [11, 1], in a magnetic eld [13] and in parallel electric and magnetic elds [11]. The studies are quantumtheoretical. To the best of our knowledge, neither experiment nor semiclassical theory on the pulsed-laser detachment of ions in external elds is available in the literature [14]. In this article, we derive a formula for the cross sections of the ion photodetachment caused by a laser pulse, expressed in terms of the photodetachment cross sections caused by a continuous laser. Quantum and semiclassical formulas for the latter are readily available in the literature, e.g., in Ref. [3, 9, 5] for parallel electric and magnetic elds, Ref. [7] for crossed elds and Ref. [6] for elds with arbitrary orientation. Using these known results, one can easily obtain formulas for the pulsed-laser cross sections by using the formula. We call the formula `the short-pulse formula'. In this article, we apply the short-pulse formula, and the Poisson's summation formula, to the pulsed-laser detachment of H? in parallel elds to obtain a uniform semiclassical formula for the cross section. The numerical results given by the uniform semiclassical formula agree with the quantum calculations almost perfectly. The uniform semiclassical formula does not have the divergence problem at any value of the laser energy, including those values at which the closed orbits bifurcate. This article is organized as follows. The short-pulse formula is derived in Section. The formula is then applied to the pulsed-laser detachment of H? in parallel elds in Sections 3, mainly to obtain the uniform semiclassical formula for the cross section.

3 (A quantum formula for the same and its approximation are also obtained.) Numerical comparisons between the uniform semiclassical and the quantum formulas are shown in Section 4. The short-pulse formula We consider the case where the electric eld of the laser pulse is Gaussian and is given by, f(t) = Ae?t = cos(!t + ); 1! ; (1) where is the width of the pulse, and A,! and are constants. The cross section of the photodetachment of an ion in an external uniform electric eld F and a magnetic eld B is (E i +!; F; B; ) = 4! Z 1 A 3= c de j g(e? E Df(E) i) j (E? E i ) ; () where Df(E) is the oscillator-strength density and g(e? E i ) is the Fourier transform of f(t). Here the atomic units have been used, c is the speed of light, and E i and E f are the energy of the initial and the nal state of the ion, respectively. Since the oscillator-strength density is proportional to the cross section for the continuous laser (! 1), i.e., (E; F; B; 1) = Df(E), the cross section caused by a pulse can be c written as, Z 1 (E i +!; F; B; ) = A dejg(e? E! i)j (E; F; B; 1) : (3) 3= E? E i In the rotating-wave approximation, the Fourier transform of f(t) is given by g(e? E i ) = q Ae?(E?E i?!) = e?i. This is a Gaussian function peaked at E = E i +! with the width 1=. Hence, the integral in Eq.(3) is eectively limited to an interval centered at E = E i +! and of a few 1= in width, the factor! E?E i in Eq.(3) can be 3

4 safely approximated by 1, and Eq.(3) becomes (E i +!; F; B; ) = p Z 1 dee?(e?e i?!) (E; F; B; 1): (4) This is the short-pulse formula that relates the cross section caused by a Gaussian pulsed laser, (E; F; B; ), to that caused by a continuous laser, (E; F; B; 1), under the rotating-wave approximation. This formula implies that the pulsed-laser cross section is convoluted by the energy width of the pulse. In the semiclassical theory, it means suppression of those oscillations in spectra whose returning times of the associated closed orbits are longer than the pulse duration. So, a Gaussian pulse acts essentially as a frequency lter. The short-pulse formula is general in the following senses: it can be used either in a quantum theory or in a uniform semiclassical theory [e.g., if (E; F; B; 1) is semiclassical, then (E; F; B; ) given by Eq.(4) is also semiclassical]; it can be applied to either photodetachment of negative ions or photoabsorption of atoms; and the external eld can be an electric eld, a magnetic eld or any combination of them. The formula gives an easy way to derive formulas for pulsed-laser cross sections since those for continuous laser are widely available in the literature. 3 Applications to Detachments of H? in Parallel Fields We apply the short-pulse formula to the photodetachment of H? in parallel electric and magnetic elds for the case that the laser is polarized in the eld direction. For the continuous-laser detachment of the same system, a quantum theory is given in Ref.[3]. When the parity is -1, the magnetic quantum number m = 0, and the atomic units are used, the continuous-laser detachment cross section in the quantum theory is, (E; F; B; 1) = (E; 0; 0; 1) X n 3 4=3 F 1=3! c [Ai 0 (z n (E))] ; (5) 4

5 where (E; 0; 0; 1) = 0:05408 a 0E 3= =(E b + E) 3 is the eld-free cross section. Here a 0 is the Bohr radius, E b is the binding energy of the hydrogen atom,! c = B=c is the cyclotron frequency, Ai() is the standard Airy function (the prime means derivative with respect to the argument), and z n (E) = [(n + 1=)! c? E]=(F ) = The quantum formula Simply by substituting Eq.(5) into the short-pulse formula Eq.(4), we obtain the following expression for the pulsed-laser detachment cross section for the system, (qm) (E i +!; F; B; ) = 3 p 4=3 F 1=3! c X n Z +1 dee?(e?e i?!) 1 (E; 0; 0; 1) (E) [Ai0 (z n (E))] : (6) 3= This (quantum) formula agrees with Ref.[11] where the result were obtained without using the known results for the continuous-laser detachment [15]. 3. The approximate quantum formula The quantum formula Eq.(6) is approximated, in the Appendix, by using the special properties of the derivative of the Airy function involved in the quantum formula. The resulting formula is (aq) (E i +!; F; B; ) = (E i +!; 0; 0; 1) 1 + 3! c + (E i +!) 3 LX n=0 p 3C0 F (E i +!) 3= n e? F n cos 5= 3F 3 n +! ; (7) where C 0 = R 1 0 dx[a 0 i(x)] is a constant, L is the integer part of ( E i+!! c? 1 ) and n = q?z n (E i +!). 5

6 3.3 The uniform semiclassical formula The uniform semiclassical expression for the pulsed-laser detachment cross sections is obtained by using the short-pulse formula and the Poisson's summation formula, P 1n= f(n) = P 1 R 1 m= f(x)e i mx dx. Applying the Poisson's summation formula to the summation in Eq.(5), we have p 3F (E; F; B; 1) = (E; 0; 0; 1) p E cos 4 E 3= 3= 3F! + 1X j=1 T j (E) ; (8) where T j (E) = 3F j E! c 3= Re Z e i uj (E) j (E) due iu! : (9) Here j (E) = je=! c? 3 j 3 F =(3! 3 c)? j + =, u j = (E? E r j )=(F ( j! c ) 1= ), and E r j = j F =(! c) +! c =, see Ref. [10]. The integral in Eq.(9) is a modied Fresnel integral [16]. It decreases when u j decreases, and is negligible if u j is less then -. Therefore only a nite number (M) of terms in the summation is eective. Setting u M q (E + 5=4 F 1= E 1=4 ), where equal to?, M can be estimated by M = Int Int() means the integer part of the argument. To get a formula for the pulsed-laser cross section, we substitute Eq.(8) into the short-pulse formula, Eq.(4). In the resulting equation, there are three integrals over E, corresponding to the three terms in Eq.(8). They are in the form of! c F Z +1 dee?(e?e i?!) 1 l (E; 0; 0; 1) h(e); (10) E 3= where l and h(e) for the three integrals are dierent: l = 0 and h(e) = 1 for the rst integral, l = 1 and h(e) = cos[4 p E 3= =(3F )] for the second integral, and l = 1 and h(e) = Re(e i j (E) R u j (E) due iu ) for the third integral. Because the integrand in Eq.(10) 6

7 has the factor e?(e?e i?!), we approximate E by E i +! in the terms (E; 0; 0; 1) and 1=E 3=. So, Eq.(10) is approximated by " (E i +!; 0; 0; 1) 1 # l Z +1 (E i +!) 3= dee?(e?e i?!) h(e); (11) Hence, Eq.(4) becomes (E i +!; F; B; ) = p p (E i +!; 0; 0; fty) 3F Z = (E i +!) 3= dee?(e?e i?!) cos 4 p 3F E3=! + MX j=1 3F j! 3= Z +1 (E i +!)! c dee?(e?e i?!) Re Z e i uj (E) j (E)! due : iu (1) The rst integral over E in this equation can, under some further approximations, be carried out, Z +1 dee?(e?e i?!) cos 4 p 3F E3=! p " p # (E i +!) 4 e? F cos 3F (E i +!) 3= : (13) Here we have made a Taylor expansion for the factor E 3= at E = E i +! and have kept only the rst two terms, i.e., E 3= (E i +!) 3= + (E i +!) 1= (E? E i?!). The second integral over E in Eq.(1) can also be carried out, Z +1 dee?(e?e i?!) Re = p e? j! c Re e i j (E i +!) Z e i uj (E) j (E) Z uj (E i +!) due iu! due iu! : (14) This is because j (E) = j! c (E? E i?!) + j (E i +!), R u j (E) e iu du = 1= 3= (1 + i) + R uj (E) 0 cos u du + i R u j (E) 0 sin u du, and R 1 0 e?ax cos(bx)dx = 1 q a e?b =4a. Substi- 7

8 tuting Eqs.(13) and (14) into Eqs.(1), we have (E i +!; F; B; ) = (E i +!; 0; 0; 1) + 3F p (E i +!) e? F cos 3= (E i +!) MX j=1 4 p! 3F (E i + omega) 3 e? j! c T j (E i +!) ; (15) where T j () is dened in Eq.(9). We write the integral in Eq.(9) as a sum of two integrals, R uj (E) = R +1? R 1 u j (E). Consequently, T j(e) becomes a sum of two terms, T j (E) = C j (E) + R j (E); (16) where j 3= C j (E) = 3F Re e i j (E) E! c R j (E) =?3F j E! c 3= Re Z +1 Z +1 e i j (E) u j (E) due iu ; (17) due iu! : (18) Carrying out the integral, Eq.(17) becomes C j (E) =? 3j3= F sin S j? j (E) 3=! c? ; (19) 4 where S j = (je? 3 j 3 F =3)=! c and j (E) = j? 1 can be regarded as the action and the Maslov index for the jth closed orbit, respectively. Hence, C j (E) may be considered as the contribution of the j-th closed orbit, and R j (E) as the quantum correction, to the semiclassical cross section. Finally, Eq.(15) becomes (sc) (E i +!; F; B; ) = (E i +!; 0; 0; 1) 1 + 3F 4 p e? 3= (E i +!) (E i +!) F cos 4p 3F (E i +!) 3 8!

9 + MX j=1 e? j! c? 3j 3= F sin S j? j ((E i +!)) 3=! c? + R j (E i +!) ; (0) 4 where R j () is dened in Eq.(18). This is the uniform semiclassical formula for the pulsedlaser detachment cross section of H? in parallel electric and magnetic elds. When pulse width tends to innity, this equation becomes Eq.(8), the formula for continuous-laser detachment, as it should be. The uniform semiclassical formula agrees with the quantum formula very well, as shown by the numerical comparisons in the next section. This formula does not have divergence problem at any value of the energy. 4 Numerical Results We numerically compare the cross sections calculated by using the uniform semiclassical formula Eq.(0), the quantum formula Eq.(6) and its approximation Eq.(7). The comparison is shown in Fig.1 for the electric eld F =30 V/cm, the magnetic eld B=1 Tesla and the pulse width =60 ps. The three formulas agree with each other very well. The discrepancy is so tiny that it is hard to distinguish them in the gure. But there is discrepancy and they can be understood as follows. For lower energy, the approximate quantum formula is better than the uniform semiclassical one since the quantum eect is relatively strong for lower energy. For higher energy, on the other hand, the quantum eect is relatively weak, and therefore the uniform semiclassical formula gives more accurate results than the approximate quantum formula does. Though not presented here, we have also attained excellent agreements between the three formulas for F ranged from 5 V/cm to 90 V/cm, from 8 ps to 60 ps, and B xed at 1 Tesla. The fact that the uniform semiclassical formula agrees with the quantum formula almost perfectly con- rms that the uniform semiclassical theory is a good one, and that the approximations we made in the derivation of the uniform semiclassical formula are justied. The eects of bifurcations can also be observed in Fig.1. When E < 7cm, there exists only one oscillation, which is associated with the parallel orbit. The orbit bifurcates 9

10 at E 7cm and a new close orbit is created. The cross section for 7cm < E < 18cm is a superposition of two oscillations associated with the parallel orbit and the new orbit. At E 18cm, the parallel orbit bifurcates again, and so on. At these bifurcation points, the uniform semiclassical formula also agrees with the quantum formula almost perfectly. (As well known, the standard uniform semiclassical closed-orbit theory gives an innitely large amplitude of the oscillation at these bifurcation points.) The occurrences of the bifurcations can be seen more clearly if one varies the pulse width, as shown in Fig.. In Fig.(a), all oscillations are suppressed by the short laser pulse. When is increased to 0 ps, as shown in Fig.(b), the oscillation associated with the parallel orbit is survived from the suppression imposed by the laser pulse. In Fig.(c), the second oscillation can be seen when E > 7 cm, as the result of the rst bifurcation of the parallel orbit. In Figs.(d) and (e), the third oscillation can be seen for E > 18 cm, as the result of the second bifurcation of the parallel orbit. The calculations are done by using the uniform semiclassical formula. But the results will be almost the same if the quantum formula or the approximate quantum formula is used because the three formulas almost perfectly agree with each other, as shown in Fig.1. Conclusions We have derived the short-pulse formula, Eq.(4), relating the cross section of the photodetachment caused by a Gaussian laser pulse and that caused by a continuous laser under the rotating-wave approximation. Applying this formula and the Poisson's summation formula to the laser pulse photodetachment of H? in parallel electric and magnetic elds, we have obtained the uniform semiclassical formula for the detachment cross section, Eq.(0). It gives an excellent agreement with the quantum calculation at any value of the laser energy including those values for the bifurcations of the closed orbits. 10

11 Acknowledgments This work was partially supported by the Research Grant Committee of Hong Kong under grant nos. HKUST606/95P and DAG94/95.SC01. Appendix: The Approximate Quantum Formula Here we approximate the quantum formula Eq.(6) by using the special properties of the derivative of the Airy function in the quantum formula. Ai 0 (z n ) for z n > 0 and that for z n < 0 behave very dierently [17]. For z n > 0, it is always negative and is almost zero for z n >. For z n < 0, it oscillates with z n. Hence we write the cross section in the quantum formula as = (>) + (<) ; (1) where (>) ( (<) ) is the contributions of the terms for z n > 0 (z n < 0) to the cross section. Due to the Gaussian function in the integral in Eq.(6), (>) can be approximated by (>) p 3C0 F (E i +!) 3= (E i +!; 0; 0; 1); () where C 0 = R 1 0 dx[a 0 i(x)] is a constant. For z n < 0 and jz n j large, we have [17] Therefore, [A 0 i(z n )] 1 q jz n j (<) (E i +!; 0; 0; 1) cos 3 jz nj 3= + ; z n < 0: (3) 3! c (E i +!) 3 where n = q?z n (E i +!), or explicitly, n = X z n<0 n e? F n cos 5= 3F 3 n +! ; (4) r E i +!? n + 1!c. Finally, we add (>) to (<) and obtain Eq.(7) as the approximation to the quantum formula Eq.(6). 11

12 References [1] H.C. Bryant, A. Mohagheghi, J.E. Stewart, J.B. Donaue, C.R. Quick, R.A. Reeder, V. Yuan, C.R. Hummer, W.W. Smith, S. Cohen, W.P. Reinhardt, and L. Overman, Phys. Rev. Lett. 58, 41 (1987). [] A.R.P. Rau and H.-Y. Wong, Phys. Rev. A37, 63 (1988); H.-Y. Wong, A.R.P. Rau and C.H. Greene, ibid, 37, 393 (1988). [3] M.L. Du, Phys. Rev. A40, 1330 (1989). [4] I.I. Fabrikant, Phys. Rev. A43, 58 (1991). [5] Q.L. Wang and A.F. Starace, Phys. Rev. A55, 815 (1997). [6] Z.Y. Liu and D.H. Wang, Phys. Rev. A55, 4605 (1997). [7] A.D. Peters and J.B. Delos, Phys. Rev. A47, 300 (1993); ibid, A47, 3036 (1993). [8] J.M. Mao and J.B. Delos, Phys. Rev. A45, 1746 (199); J.M. Mao, J. Shaw and J.B. Delos, J. Stat. Phys. 68, 51 (199). [9] A.D. Peters, C. Jae and J.B. Delos, Phys. Rev. Lett. 73, 85 (1994); ibid., Phys. Rev. A56, 331 (1997); A.D. Peters, C. Jae, J. Gao and J.B. Delos, Phys. Rev. A56, 345 (1997). [10] G.C. Yang and M.L. Du, Chinese Phys. Lett. 13, 817 (1996). [11] Q.L. Wang and A.F. Starace, Phys. Rev. A48, R1741 (1993); ibid, A51, 160 (1995). [1] M.L. Du, Phys. Rev. A5, 1143 (1995). [13] I.Y. Kiyan and D.J. Larson, Phys. Rev. Lett. 73, 943 (1994). 1

13 [14] Eorts have been make in Ref.[11] to understand the results of their quantum theory in terms of classical or semiclassical concepts. [15] Eq.(6) is equivalent to Eq.(4) in the second paper of Ref.[11]. To see this, we note that, for the laser polarized in the eld direction (which is what we study in this article), the term with Ai(?) in their equation does not contribute [3]. We also note that the Gaussian function in our equation is the `quasi--function' in their equation. Because of the the Gaussian function, the term (E; 0; 0; 1)=E 3= (E b + E)?3 in our equation can be replaced by!?3 (and then placed outside of the integral) if! is large enough. [16] The Fresnel integral has also been employed in Ref.[9] for the contribution to the photodetachment cross section arising from the combined eects of the parallel orbit and the nth new orbit at energies close to the nthe bifurcation. [17] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1964). 13

14 Figure captions Fig.1 Comparison between the three formulas for the pulsed-laser detachment cross section of H? in parallel elds: the quantum formula Eq.(6) (bold curves), the approximate quantum formula Eq.(7) (dotted curves), and the uniform semiclassical formula Eq.(0) (thin curves). The calculations are for the electric eld F =30 V/cm, the magnetic eld B=1 Tesla and the pulse width =60 ps. The three formulas agree with each other so well that it is really hard to see three dierent curves here. Fig. Pulsed-laser detachment cross sections given by the uniform semiclassical formula when the elds are xed at F =30 v/cm and B=1 Tesla, and varied. When increases, more oscillations are seen as the results of the bifurcations. The oscillation amplitudes at the bifurcation points (i.e., E 7 cm ; 18 cm ) are nite. 14