Chaotic scattering from hydrogen atoms in a circularly polarized laser field

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1 Chaotic scattering rom hydrogen atoms in a circularly polarized laser ield Elias Okon, William Parker, Will Chism, and Linda E. Reichl Center or Studies in Statistical Mechanics and Complex Systems, The University o Texas at Austin, Austin, Texas Received 31 May 2002; published 15 November 2002 We investigate the classical dynamics o a hydrogen atom in a circularly polarized laser beam with inite radius. The spatial cuto or the laser ield allows us to use scattering processes to examine the laser-atom dynamics. We ind that or certain ield parameters, the delay times, the angular momentum, and the distance o closest approach o the scattered electron exhibit ractal behavior. This ractal behavior is a signature o chaos in the dynamics o the atom-ield system. DOI: /PhysRevA PACS number s : Hz, Rm, Nk I. INTRODUCTION The advent o high-energy lasers has allowed exploration o the dynamics o the laser-atom interaction in parameter regimes where the nonlinear character o the interaction dominates the dynamics. One o the most interesting developments is the stabilization o the atoms in high-energy laser ields. For the case o linearly polarized laser ields, stabilization has been predicted theoretically 1 5 and has been observed experimentally 6,7. The origin o this stabilization appears to be phase-space structures induced by the nonlinear laser-ield interaction. Numerical studies 8 15 have shown that stabilization should also occur or atoms interacting with circularly polarized CP laser ields, although the underlying dynamics is much more complicated. Poincaré suraces o section o the classical phase space show a very complex mixture o chaos and nonlinear resonance structures For hydrogen atoms, interacting with CP laser ields, stabilization appears to be caused by these complex structures which result rom the interaction between the external ield and the nonlinear atomic orces 13. All work done to date on these systems assumes that the laser ield extends spatially to ininity. In this paper, we consider a more realistic spatial dependence or the laser ield. We introduce a cuto on the width o the laser beam. This then allows us to divide the coniguration space into a reaction region interior to the beam, and an asymptotic scattering region exterior to the beam. The dynamics inside the reaction region is nonintegrable and the dynamics in the asymptotic region is integrable. With this more realistic picture o the geometry o the laser beam, we can probe the atom-laser dynamics using techniques o scattering theory, and we can ask dierent questions than those o previous studies. For example, in Res. 11,12, the dynamics o the initially bound electron is studied as the laser pulse is turned on, and it is ound that the ability o the electron to ionize is strongly determined by the position o the electron in the complex phase-space structures mentioned above. By using scattering theory, we can ask the complimentary question: or what initial conditions can an incident electron be captured or a long period o time? As we will see, the cuto does not signiicantly aect the important electron-proton dynamics, so the scattered electron provides a systematic probe o the dynamics in the reaction region. We begin in Sec. II by developing the classical model used to describe the electron-proton system in the presence o a circularly polarized laser beam with inite width. In Sec. III, we discuss the dynamics outside the inluence o the laser beam. In Sec. IV, we look at the eect o the resonance structures and chaos in the reaction region on scattering properties o the electron. And inally, in Sec. V, we make some concluding remarks. II. TWO-DIMENSIONAL CLASSICAL MODEL We investigate the motion o the electron in a hydrogen atom in the limit o an ininitely massive nucleus driven by a CP laser ield which has a inite width. The beam may consist o two linearly polarized laser beams superimposed, so that their Poynting vectors lie along the z axis, and their electric ields are polarized along the x axis and y axis, respectively. Outside some radius r in the x-y plane, the electric ield drops to zero. A realistic model or this cuto is a Gaussian unction o the radial distance r along with a parameter c that determines the rate o decay o the ield strength. We restrict our model to the two-dimensional plane o polarization o the laser the x-y plane. This simpliication should capture the most important eatures o the dynamics, i.e., ionization or appearance o stable orbits which are expected to occur in that plane. In polar coordinates (p r,r,p, ) and in atomic units a.u., the planar Hamiltonian is given by H 1 2 p 2 r p 2 r r 2 Fre cr 2 cos t, where F is the ield strength, is the driving requency, and is a smoothing parameter. We set 0.8 a.u., so the ionization potential is the same as that o hydrogen 10. A suitable value or the cuto parameter c is determined by experimentally obtainable sizes o ocused laser beam spots. The size o the smallest possible laser beam spot is determined by the diraction limit, and cannot be smaller than the wavelength o the laser. For visible light, which is the one employed or the calculations throughout this paper the parameter c must be in the order o 10 8 a.u. or smaller. However, because o computational considerations, we used c 10 5 a.u. This value o c is large enough to permit practi /2002/66 5 / /$ The American Physical Society

2 OKON et al. FIG. 1. Zero-velocity surace or F a.u., a.u., and c 10 5 a.u. cal calculations but small enough to retain most o the interesting structures in the system. The time dependence o the Hamiltonian can be eliminated by perorming a time-dependent canonical transormation, with generating unction F p,,p r,,t p t p r 2 relating lab coordinates (p r,r,p, ) to a coordinate rame (P,,P, ) that rotates with the electric ield at a constant angular velocity. This leads to the Hamiltonian H 1 2 P 2 P F 2 2 e c cos P 2, where the quasienergy is a conserved energy in this rotating rame. I we introduce Cartesian coordinates in the rotating rame, x cos( ), y sin( ), the electric ield always lies along the x axis. Note that the Hamiltonian depends on, so the angular momentum P is not conserved. In the subsequent discussion, we choose unless otherwise stated a.u., which corresponds to 400 nm laser excitation, and F a.u. (I W/cm 2 ) which is a moderate intensity associated with stabilization seen in numerical studies. In the rotating rame, noninertial orces appear in the system, and lead to a mixing o position and momentum coordinates. This precludes the construction o a potential-energy surace in the usual sense. However, as shown in Res. 11,12, it is possible to construct a zero-velocity surace ZVS, which separates physical rom unphysical regions o the phase space. I we recast the Hamiltonian in terms o the 3 FIG. 2. Poincaré SOS at a.u. or a c 0 and b c 10 5 a.u. velocities P and P / 2, and then set 0 and 0, we obtain the equation or the ZVS, 1 ZVS F e c 2 cos. 4 The irst two terms on the right-hand side o Eq. 4 result rom the atomic orce and centriugal orce, respectively, and have no angular dependence. They orm a Coulomb well at small and a quadratic allo at large. The angledependent term, due to the laser ield, breaks the rotational symmetry. Just as was ound in Res. 11,12 or the case c 0, the ZVS can be depicted as a volcano with a conined caldera region see Fig. 1. The ZVS helps to identiy exclusion regions in phase space because a particle with nonzero velocity cannot intersect the ZVS. The ZVS possesses two critical points, denoted by x and x, that lie along the x axis on opposite sides o the proton and have quasienergies, zvs (x ) and zvs (x ), respectively. For any nonzero ield,

3 CHAOTIC SCATTERING FROM HYDROGEN ATOMS IN A... FIG. 3. A SOS in the neighborhood o the hyperbolic ixed point induced by the spatial cuto o the laser ield. The initial conditions lie in the interval 64 a.u. P 66 a.u., with 0.50 a.u. and c 10 5 a.u.. Thus, the quasienergy will all in one o the three regions: 1, 2, or 3. When a.u. and F a.u., we have a.u. and a.u. We will use these values in the remainder o this paper. I, the interior region inside the caldera cannot be reached by an electron lying outside the ZVS but still inside the reaction region. However, i, then some orbits can travel between the exterior region and the caldera and visit the vicinity o the proton. The system whose dynamics is given by Eq. 3 has two degrees o reedom. Thereore, it is possible to construct a Poincaré surace o section SOS. We will plot the radial momentum P and radial coordinate each time 0 and (mod ). This can be thought o as a plot o P x vs x, each time y 0 regardless o the sign o P y. It is also useul to plot x vs y each time P 0 to obtain inormation about regions o coniguration space accessible to the electron. In the suraces o section shown in the subsequent sections, our initial conditions generally consist o a range o points with P 0 and 0. The value o is chosen randomly between 0 and a given max and the initial value o P is determined by the quasienergy. Figure 2 shows a comparison o suraces o section with and without the cuto or a.u. We note that the SOS with c 10 5 in Fig. 2 b, retains the signiicant nonlinear resonance structures and surrounding chaotic sea that can be seen in Fig. 2 a and which has been seen by previous authors or the case when the laser ield extends to ininity. The large external resonance still exists 14 and the location o its central ixed point does not change signiicantly. However, the size o the external resonance and o the chaotic sea surrounding it is reduced. In addition, in Fig. 2 b there is a pair o hyperbolic ixed points not present in the system without the cuto. A plot with 0.50 a.u. and initial conditions, 64 a.u. P 66 a.u., conined to the neighborhood o the cuto induced hyperbolic point see FIG. 4. All graphs in this igure correspond to a.u. and c 10 5 a.u. a Surace o section, y vs x in a.u. or initial conditions p x 0, 70 a.u. x 70 a.u. b Surace o section, p x vs x in a.u. or the same initial conditions as in a. Fig. 3 shows clearly the contracting and expanding directions o the phase space in the neighborhood o the hyperbolic ixed point. By extending the initial conditions urther out rom the nucleus, but still along 0, we ound yet another pair o elliptic and hyperbolic points in each side o the nucleus. III. ASYMPTOTIC SCATTERING REGION When is large enough so that the ield term can be neglected, the Hamiltonian reduces to an asymptotic Hamiltonian H asym given by H asym 1 2 P 2 P P. 5 This Hamiltonian is equivalent to that o the Kepler problem particle in a central attractive inverse square-law orce, as

4 OKON et al. FIG. 5. Scattering data or a.u. and initial angular momentum P i a.u. a Delay time T D in a.u. vs 0 or b T D vs 0 or c Final angular momentum P in a.u. vs 0 or d Radius o closest approach in a.u. vs 0 or seen in a rame rotating with requency. The equations o motion o this system can be integrated analytically yielding closed-orm solutions to the orbits in the asymptotic region. The radial component o the motion in the rotating rame is identical to that o the nonrotating rame. However, in the rotating rame the actual trajectories cease to be conic sections due to Coriolis orces which are present in the rotating rame. Let us consider the dynamics generated by the Hamiltonian H asym. The orbits o H asym can be speciied uniquely in terms o three quantities: the two conserved quantities, and P which are both conserved by H asym ); and by a third quantity 0, which is the initial angle o the orbit. The total energy o an orbit in the lab rame, E asym P, determines i that orbit is closed bounded or open unbounded. I E 0, the motion is bounded. I E 0, the motion is unbounded and the trajectory comes in rom ininity up to a point o closest approach,, then returns to ininity. For E 0, the point o closest approach is a(1 e), where a 1/2E and e 1 2P 2 E is the eccentricity. The value o is determined by the total energy E and the total angular momentum P. I we draw a circle o radius R centered at (x 0, y 0), we can determine the time required or an orbit to traverse the interior o the circle ater it irst enters it at R. Let be the time to travel rom R to.itis straightorward to solve the equation o motion and obtain an analytic expression or. We ind a R R aln 2 R R R 2a aln 2e a. Because o the symmetry o the orbit, the total time the orbit spends inside the circle is 2 when F 0. IV. FRACTAL BEHAVIOR IN THE SCATTERING PROCESS We can now use the inormation about the dynamics when F 0 to probe the dynamics o the system when F 0. The

5 CHAOTIC SCATTERING FROM HYDROGEN ATOMS IN A... ollowing method is employed. For the case when F 0, an electron with ixed and P and with a range o values or 0 is launched inwards rom a point ( R, 0 ). We choose R large enough so that it lies in the asymptotic region. The trajectory enters the circle at time t 0, interacts with the the laser ield and atomic orces, and ater a inite time T R, it leaves the circle. The unctional dependence on 0 o the excursion time T R, the angular momentum P, and the distance o closest approach to the nucleus,, are analyzed. Speciically we numerically generate plots o the delay time, T D T R 2, the inal angular momentum P, and the distance o closest approach,, as a unction o 0. The delay time T D is the actual time the particle spends inside the circle, minus the time 2 calculated or the trajectory with the asymptotic Hamiltonian alone. Below we explore the dynamics in the quasienergy regimes and. In subsequent plots, we choose the radius o the circle to be R a.u., or which the laser-ield term in the Hamiltonian can be saely neglected. The values o 0 are evenly distributed in the interval 0,2 and dierent values o 0 are chosen. Below we irst consider the regime, and then the regime. A. The quasienergy regime Ë À We begin with quasienergies in the regime, where the ZVS divides the phase space into two unconnected areas, the central caldera and the exterior region. For a.u., the SOS exhibits a large exterior resonance, as shown in Figs. 4 a and 4 b. The initial conditions or both Figs. 4 a and 4 b include a line o points at p x 0 ranging rom x 70 a.u. to x 70 a.u. Although both these plots show orbits in the interior and exterior regions, this only happens because some initial points span those two regions. There are no orbits that transit between these regions. In Fig. 5, we show plots o the delay time T D, the inal angular momentum P, and the radius o closest approach, i, or initial conditions, a.u., P a.u., and These plots show ractal behavior or initial angles in the interval In Fig. 5 b, we ocus on a small interval 0.70 a.u a.u. rom the ractal segment in Fig. 5 a. The ractal behavior continues to repeat itsel on smaller scales. In Fig. 5 c, we show the inal angular momentum P or initial angles in the interval This shows the same ractal behavior as the delay time. In Fig. 5 d, we show the radius o closest approach,, or the range o initial conditions This also shows ractal behavior in the interval Note that the electron never gets close to the proton. As discussed in Res , ractal behavior in the scattering properties is an indicator that the electron has traversed a network o heteroclinic tangles chaotic structures as it passes through the reaction region. To see this more clearly, in Fig. 6, we examine some typical orbits rom the ractal region in Fig. 5. Figure 6 a shows a surace o section o p x vs x or a set o orbits with initial angular momentum P i a.u. and initial angles in the interval FIG. 6. Some typical orbits rom the ractal regime or a.u. and P i a.u. a Surace o section o p x vs x in a.u. or b A single trajectory in the x-y plane or P i a.u. and c P vs time t in a.u. or trajectories in the interval In the SOS, orbits are plotted or 0 and. These orbits clearly lie along the contracting and expanding maniolds associated with the exterior resonance. Figure 6 b shows the path in the x-y plane not a SOS o a single orbit with initial angular momentum P i a.u. and initial angle It is clearly trapped in the exterior part o the reaction region or a long period o time. Figure 6 c shows the variation o the angular momentum o the orbits in Fig. 6 a as a unction o time

6 OKON et al. FIG. 7. Scattering data or a.u. and P i 5.0 a.u. a Delay time T D in a.u. vs 0 or b Final angular momentum P in a.u. vs 0 or Not all initial conditions allow the trajectory to enter the exterior resonance region. In Figs. 7 a and 7 b, we show plots o the delay time T D, and the asymptotic angular momentum P, respectively, as a unction o 0 or a.u., P i 5.0 a.u., and In both cases, we obtain simple smooth curves. Note that T D has a large range o values, 1000 a.u. T D 1500 a.u., and that large T D values are associated with small values o P. On the other hand, the negative values o T D correspond to orbits that receive a large increase in the angular momentum P, in the reaction region, and leave the reaction region much aster due to the presence o the laser ield. B. The quasienergy regime À Ï Ï FIG. 8. Graphs correspond to a.u. and c 10 5 a.u. a Surace o section y vs x or initial conditions p x 0 and 70 a.u. x 70 a.u. b Surace o section p x vs x in a.u. or the same initial conditions as in a. Let us now consider the regime, in which the caldera and the exterior region are connected by a passageway. For some initial conditions, the electron can enter the caldera and come near the proton. In Fig. 8 a, we show a surace o section o x vs y or a.u. and plotted each time p x 0. The initial condition includes a line o points, 70 a.u. x 70 a.u. In Fig. 8 b, we show a SOS o p x vs x or the same initial conditions used in Fig. 8 a. We see that the exterior resonance is still intact but the area o the regular island is much smaller than in Figs. 4 a and 4 b, and it has moved closer to the origin. Also, the area aected by the heteroclinic tangles has increased considerably. The inward shit o the location o the elliptic ixed point o the resonance with the increase o causes more orbits to collide with the ZVS, leading to their eventual chaotic ionization. In Figs. 9 a and 9 c, we show plots o the delay time T D and the inal angular momentum P, as a unction o 0 in the interval All plots in Fig. 9 are or quasienergy a.u. and initial angular momentum P i 5.0 a.u. We again see a range o ractal behavior in each

7 CHAOTIC SCATTERING FROM HYDROGEN ATOMS IN A... FIG. 9. Scattering data or a.u. and P i 5.0 a.u. a Delay time T D in a.u. vs 0 or b T D in a.u. vs 0 or c Final angular momentum P in a.u. vs 0 or d P in a.u. vs 0 or plot. In Fig. 9 a, we show the delay time T D or the entire range o initial angles, 0 0 2, and in Fig. 9 b we ocus on a small interval and magniy the horizontal scale o one o the unresolved regions. We can see that the unction is still not resolved. This clearly suggests a complex structure at even smaller scales, and we ind that to be the case. In Figs. 9 c and 9 d, we show plots or the angular momentum P o the scattered electron or the same two ranges o the initial angle as shown in Figs. 9 a and 9 b. The angular momentum o the scattered particle can take on a huge range o values ater passing through the resonance region. For a small range o initial angles, the electron does enter the interior region and comes relatively close to the proton. In order to conirm that the ractal behavior in the T D plots is related to the chaotic structures in the phase space, it is useul to construct a SOS or the orbits that generate the ractal structures in Fig. 9. In Fig. 10 a we show a SOS o p x vs x or initial condidtions, a.u., P i 5.0 a.u., and Only those orbits with ẏ 0 are shown. These orbits get trapped or long time in the heteroclinic tangles associated with the exterior region and do not enter the caldera. In Fig. 10 b we show a single orbit, with P i 5.0 a.u. and This orbit does not enter the caldera, but does get delayed or a signiicant period o time in the reaction region. In Fig. 10 c, we show the angular momentum as a unction o time or the orbits in Fig. 10 a. The angular momentum appears to decrease while the orbit is trapped in the exterior resonance region. In Fig. 11 a, we show a single orbit which enters the caldera or the case a.u., taken rom the ractal region with P i 5.0 a.u. and a.u. This orbit gets trapped or a long time inside the caldera. In Fig. 11 b, we show the angular momentum as a unction o time or the same orbit as in Fig. 11 a. It is interesting that during the

8 OKON et al. FIG. 11. A typical orbit rom the ractal regime or a.u. and initial angular momentum P i 5.0 a.u. a A single trajectory in the x-y plane notasos or P i 5.0 a.u. and b P vs time t in a.u. or the same orbit as in a. FIG. 10. Some typical orbits which get trapped in the exterior region or a.u. a p x vs x in a.u. or P i 5.0 a.u. and b x vs y notasos or a single orbit with P i 5.0 a.u. and c P vs time t or P i 5.0 a.u. and in a.u.. time the orbit spends inside the caldera, it has a very low angular momentum as one might expect since the radius remains small. V. CONCLUSIONS We have examined the classical dynamics o an electronproton system the hydrogen atom when they are bound which interacts with an intense circularly polarized laser beam o inite radius. We have explored dynamics o the system by launching electrons into the reaction region and plotting the delay time, the exiting angular momentum, and the distance o closest approach as a unction o the incident angle. The dynamics has two distinct regimes depending on the value o the quasienergy o the incident particle. Below a certain cuto value o the quasienergy, no initial condition will allow the incident electron to come close to the proton. Above that cuto value quasienergy, a inite range which grows with increasing value o quasienergy can come arbitrarily close to the proton this separation o regimes was also noted in Res We have ound that incident orbits can exhibit ractal behavior in their delay time, the exiting angular momentum, and the distance o closest approach as a unction o the incident angle ater they exit the reaction region. The appearance o ractal behavior in the scattering plots is closely related to the trapping o incoming orbits in the chaotic structures surrounding the stable islands in the phase space. Using scattering theory to probe the electron-proton dynamics gives a systematic way o inding the chaotic regions

9 CHAOTIC SCATTERING FROM HYDROGEN ATOMS IN A... o the phase space and regions that can trap the electron or long periods. Use o scattering theory can also give some inormation about ionizing orbits, since or every incident orbit, there will be a time-reversed ionizing orbit. Our results show a strong directional dependence o initial states that can be delayed or long periods in the reaction region. It might be possible to observe this directional dependence in the scattering o electrons o a proton beam as it passes through a circularly polarized laser beam. ACKNOWLEDGMENTS The authors wish to thank the Welch Foundation, Grant No. F-1051, NSF Grant No. INT , and DOE Contract No. DE-FG03-94ER14405 or partial support o this work. We also thank the University o Texas at Austin High Perormance Computing Center or use o their computer acilities, and we thank Christo Jung or very useul conversations. 1 J.H. Eberly and K.C. Kulander, Science 262, Q. Su and J.H. Eberly, J. Opt. Soc. Am. B 7, Q. Su, J.H. Eberly, and J. Javanainen, Phys. Rev. Lett. 64, R.V. Jensen and B. Sundaram, Phys. Rev. Lett. 65, B. Sundaram and R.V. Jensen, Phys. Rev. A 47, M.P. de Boer et al., Phys. Rev. Lett. 71, ; Phys. Rev. A 50, N.J. Druten, R.C. Constantinescu, J.M. Schins, H. Nieuwenhuize, and H.G. Muller, Phys. Rev. A 55, J. Zakrzewski and D. Delande, J. Phys. B 28, L M. Pont and M. Gavrila, Phys. Rev. Lett. 65, ; M. Pont, Phys. Rev. A 40, A. Patel, M. Protopapas, D.G. Lappas, and P.L. Knight, Phys. Rev. A 58, R ; A. Patel, N.J. Kylstra, and P.L. Knight, Opt. Express 4, A.F. Brunello, T. Uzer, and D. Farrelly, Phys. Rev. A 55, D. Farrelly and T. Uzer, Phys. Rev. Lett. 74, Will Chism, Dae-Il Choi, and L.E. Reichl, Phys. Rev. A 61, W. Chism and L.E. Reichl, Phys. Rev. A 65, 21404R Iwo Bialymicki-Birula, Maciej Kalinski, and J.H. Eberly, Phys. Rev. Lett. 73, E. Ott and T. Tl, Chaos 3, E. Ott, Chaos in Dynamical Systems Cambridge University Press, Cambridge, England, C. Jung and T.H. Seligman, Phys. Rep. 285,