A. Bugacov

A. Patel

A. S. Patel

B. Piraux

C. H. Keitel

C. J. Joachain

C. K. Law

J. Grochmalicki

J. R. Vázquez de Aldana

K. C. Kulander

L. Roso

M. Dörr

M. P. de Boer

M. Pont

M. Protopapas

N. J. Kylstra

O. Latinne

P. L. Knight

Q. Su

R. A. Worthington

R. M. A. Vivirito

R. M. Potvliege

T. Katsouleas

U. W. Rathe

V. C. Reed

Physical Review Letters

Crossref

Breakdown of Stabilization of Atoms Interacting with Intense, High-Frequency Laser Pulses

Kylstra, NJ

Worthington, RA

Patel, A

Knight, PL

de Aldana, JRV

Roso, L

Spiral - Imperial College Digital Repository

VOLUME 85, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 28 AUGUST 2000Breakdown of Stabilization of Atoms Interacting with Intense, High-Frequency Laser PulsesN. J. Kylstra,1 R. A. Worthington,1 A. Patel,2 P. L. Knight,2 J. R. Vázquez de Aldana,3 and L. Roso31Department of Physics, Science Laboratories, University of Durham, Durham DH1 3LE, United Kingdom2Optics Section, Blackett Laboratory, Imperial College, London SW7 2BZ, United Kingdom3Departamento de Física Aplicada, Universidad de Salamanca, E-37008 Salamanca, Spain(Received 13 April 2000)An analysis of the influence of the magnetic field of an intense, high-frequency laser pulse on thestabilization of an atomic system is presented. We demonstrate that at relatively modest intensitiesthe magnetic field can significantly alter the dynamics of the system. In particular, a breakdown ofstabilization occurs, thereby restricting the intensity regime in which the atom is relatively stable againstionization. Counterpropagating pulses do not negate the detrimental effects of the magnetic field. Wecompare our quantum mechanical results with classical Monte Carlo simulations.PACS numbers: 32.80.Fb, 32.80.Rm, 42.50.HzTheoretical studies of atoms interacting with high-frequency intense laser pulses have predicted a significantdecrease in the ionization probability with increasinglaser intensity. This phenomenon is referred to as atomicstabilization, and has been extensively studied over thepast decade [1]. Many aspects of this phenomenon can beunderstood by performing a Kramers-Henneberger (KH)transformation to the rest frame of a classical electronin the laser field. In particular, by developing a high-frequency Floquet theory in the KH frame [2], stabilizationcan be seen to have its origin in the rapid quiver motionof the atomic electron in the laser field. This allowsthe electron dynamics to be described by an effectivepotential that, on average, localizes the electron awayfrom the vicinity of the nucleus. Subsequent ab initioFloquet calculations confirmed that ionization ratesdecrease with increasing intensity in a high-frequencyfield [3]. By directly integrating the time-dependentSchrödinger equation numerically, simulations in one [4]and three dimensions [5] demonstrated reductions in theionization probability with increasing laser intensity whenan atom interacts with realistic laser pulses having a finiteduration. Further work has been carried out in orderto elucidate the effects of the pulse shape and duration[6,7]. We also note that evidence of atomic stabilizationof Rydberg states has been observed experimentally [8].In the above-mentioned theoretical studies, the magneticcomponent of the laser pulse was neglected. However, asthe laser intensity increases, relativistic effects that alterthe stabilization dynamics become important. ClassicalMonte Carlo simulations have indicated that the magneticfield pushes the electron in the laser pulse propagationdirection, reducing the degree of stabilization [9]. Rela-tivistic wave equations have also been considered withinthe context of reduced dimensional models [10,11]. How-ever, it is also of interest to study the effects that areneglected in the dipole approximation by using the fullyspace- and time-dependent vector potential in the nonrela-tivistic Schrödinger equation [12]. For atomic hydrogen,0031-90070085(9)1835(4)$15.00 ©this results in the cylindrical symmetry of the system beingbroken, thereby requiring a fully three-dimensional calcu-lation to be carried out for extremely high laser intensities.This is a computationally demanding task. However, atwo-dimensional model atom contains the essential physics[11], allowing studies of laser-atom interactions beyond thedipole approximation to be readily carried out [13]. In thisLetter we investigate the laser-atom dynamics in the stabi-lization regime for a two-dimensional model atom withoutemploying the dipole approximation. We demonstrate thatstabilization breaks down at relatively modest laser inten-sities, i.e., at intensities where the laser-atom interactioncan be described by the nonrelativistic Schrödinger equa-tion. In particular, we show that the dipole approximationleads to survival probabilities which are too large becausethe magnetic field, neglected in the dipole approximation,strongly influences the stabilization dynamics. By compar-ing our results with those obtained from classical MonteCarlo simulations, we confirm the validity of classical ap-proaches in describing atoms interacting with short, high-intensity, and high-frequency laser pulses.Our results have been obtained using a two-dimensionalmodel atom which is described by the time-dependentSchrödinger equationi≠≠tCr, t ∑12µp 11cAh∂21 V r∏Cr, t ,(1)with r  x, z. The spin of the electron is neglectedand we use atomic units (a.u.) throughout. The laser fieldis described classically by the fully spatially dependentvector potentialAh E0cvx̂fh sinh . (2)The function fh is the laser pulse envelope, which istaken to be trapezoidal, and h  vt 2 zc so that thelaser pulse travels in the positive z direction and c isthe speed of light. We use a smoothed two-dimensional2000 The American Physical Society 1835VOLUME 85, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 28 AUGUST 2000Coulomb potentialV r  21pr2 1 a2. (3)The smoothing parameter a is used to avoid problems intwo dimensions associated with the singularity at r  0.We have set a  0.80 so that the ground state has the samebinding energy as hydrogen, i.e., 20.50. The potentialsupports an infinite number of Rydberg series of boundstates. Equation (1) is numerically integrated on a uniformgrid using either a split operator or a Crank-Nicholsonmethod. Both widely used approaches are easily modifiedto account for the spatially dependent vector potential.The probability that the atom does not ionize during thelaser pulse is determined by projecting the wave functionat the end of the laser pulse onto the bound field-free statesof the atom,Psurvival Xn,mjcnm jCtj2, (4)where t is the pulse duration. The field-free states jcnmare labeled by their principal quantum number n andazimuthal quantum number m. In cylindrical coordinates,cnmr  2p12Rnmr exp2imf so that the radialfunctions are solutions of the differential equation∑212r≠≠rµr≠≠r∂1m22r21 V r 2 En,m∏Rnmr  0 .(5)To understand the influence of the magnetic field, we alsoperformed calculations in the dipole approximation by set-ting h  vt in Eq. (2) for the vector potential.In the upper plots of Fig. 1, the ionization probability isshown as a function of the peak electric field strength ofthe laser, E0, for angular frequencies of v  1 (left) andv  2 (right). As indicated in the plots, the dashed linesare nondipole (NDP) results and solid lines are the resultsof the dipole (DP) approximation. A trapezoidal pulsewith three-cycle turn-on (turn-off) and six-cycle constantintensity was used. From the results, it is evident that thedipole approximation breaks down when the electric fieldE0 * 10 for v  1. However, when v  2 it remainsa good approximation up to E0  20, the maximum fieldstrength considered here.In the lower plots of Fig. 1, the populations in theground state and in the even parity bound states are shown.For a twelve-cycle pulse and v  1, nearly all of the sur-viving population returns to the ground state for E0 & 5.This is an indication of the so-called “adiabatic stabiliza-tion” of the atom. When E0 * 10, the evolution of thesystem is highly nonadiabatic, and in the dipole approxi-mation there is significant population transfer (“shake up”)to the excited states. Note that, in this “dynamical sta-bilization” regime, only even parity states are populated,i.e., the evolution of the system can be described to a goodapproximation by an effective time-averaged potential in1836FIG. 1. The upper plot shows the survival probability as afunction of the maximum electric field strength, E0, of thelaser. On the left, results are shown for an angular frequencyof v  1 while, on the right, v  2. A trapezoidal pulse witha three-cycle turn-on (turn-off ) and six-cycle constant intensitywas used. In the lower two plots, the populations in the groundstate and in the even parity excited states are shown. The dipole(DP) results are given by the solid curves, nondipole (NDP) re-sults are given by the dashed curves while the dotted curvescorrespond to results for two counterpropagating (CP) pulses.the KH frame [7]. For the nondipole case, the surviv-ing population moves progressively to the odd parity statesas the dipole approximation breaks down. When v  2,adiabaticity is maintained up to E0  10. We have per-formed additional calculations for a range of pulse dura-tions. These results, which will be presented elsewhere,show that the degree of adiabaticity of the laser-atom in-teraction does not modify significantly the breakdown ofthe dipole approximation.The fact that the dipole approximation breaks down atrelatively moderate intensities, and that the dipole approxi-mation improves as the frequency increases, can be un-derstood by using the Lorentz force equations. To lowestorder in 1c for a free electron initially at rest, the forceexerted on the electron and the velocity of the electron inthe laser pulse areyxt AxhcÇz0, (6)Fzt  2yxtc≠Axh≠zÇz012c3≠A2xh≠tÇz0, (7)yzt 12c3A2xvt . (8)VOLUME 85, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 28 AUGUST 2000Note that the magnetic field induces a velocity in the propa-gation direction which is never negative. This velocity iscomposed of a drift term and an oscillating term. In ad-dition, the displacement of the electron in the propagationdirection increases as the pulse duration increases. By in-tegrating Eq. (8), this simple model implies that the dipoleapproximation is expected to fail when the the displace-ment of the electron arising from its “figure eight” mo-tion due to the magnetic field becomes comparable to thesize of the initial wave packet, i.e., E20cv3  1. This isconfirmed by our calculations. When this occurs, the to-tal displacement of a classical electron in the laser fieldis proportional to tE20cv2, where t is taken to be ap-proximately the duration of the flat part of the pulse. Weemphasize that for E0 fixed, decreasing v will result inan eventual breakdown of the dipole approximation in thishigh-intensity regime.One might imagine that the detrimental effect of themagnetic field can be eliminated by having the atom in-teract with two counterpropagating laser pulses. Taking avector potential of the formAh, h0 cE02vx̂sinh0 1 sinh , (9)with h0  vt 1 zc, will result in fixed points in spacewhere the magnetic field vanishes, while the electric fielddoes not. In particular, at z  0 the electric field oscillatesbetween 2E0 and E0, but the magnetic field vanishes forall times. It therefore seems reasonable that for a standingwave the effect of the magnetic field would be minimizedand stabilization maintained. Near z  0 the magneticfield, of course, does not vanish.We have calculated the time evolution of the initialground state of our two-dimensional model atom interact-ing with two linearly polarized counterpropagating laserpulses described by the vector potentialAh, h0 cE02vx̂ fh0 sinh0 1 fh sinh	 . (10)In Fig. 2, plots of the wave function probability density areshown after 3, 6, and 9 cycles of the counterpropagatingpulses (right). Results for the dipole approximation arealso shown (left). Here E0  15, v  1 and again fhdescribes a pulse with a three-cycle linear turn-on (turn-off) and six cycles of constant intensity. Clearly magneticfield effects strongly influence the stabilization dynamicseven for counterpropagating pulses. From Fig. 1, it is seenthat in fact the counterpropagating pulses result in survivalprobabilities that are smaller than when a single pulse isused. During the laser turn-on, the magnetic fields do notexactly cancel to lowest order in 1c. This leads to adrift velocity that is negative for z negative and positivefor z positive. In addition, after the turn-on and to lowestorder in zc, the force along the z direction on the electron− 15015Z− 30− 15015X− 15015Z− 30− 15015X− 15015Z− 30− 15015X− 15015Z− 30− 15015X− 15015Z− 30− 15015X− 15015Z− 30− 15015XFIG. 2. Plots of the probability density of the wave func-tion for linearly polarized (along the x axis), counterpropa-gating pulses (right). Dipole approximation results are alsoshown (left). These snapshots are taken after three (top), six(middle) and nine (bottom) cycles. The pulse shapes are thesame as in Fig. 1, the maximum electric field strength of thepulses is E0  15 and the angular frequency is v  1.in the laser pulse (and in the absence of the nucleus) isgiven byFzt, z yxtc3z∑≠2Axh, h0≠t2∏z0, (11)yx Axh, h0cÇz0. (12)From the above equations, we can see that the force willalways act such that it drives the electron away from theorigin. This is also detrimental to stabilization. Note that,while the force Fzt, z is a factor zvc smaller thanFzt defined in Eq. (7), it is nevertheless comparable tothe force of the time-averaged potential in the KH framethat binds the electron (in the dipole approximation). Byconsidering different laser pulse turn-ons, we have verifiedthat the force given by Eq. (11) is large enough to desta-bilize the atom.Finally, we compare our results with classical MonteCarlo simulations. Previous classical simulations have in-dicated a breakdown of stabilization due to the magneticfield component [9]. However, the applicability of ap-proaches based on classical mechanics has remained anopen question. Our results are shown in Fig. 3. Theensemble of initial conditions was obtained in a manneranalogous to the microcanonical distribution used for theKepler problem [9]. Typically, 500 trajectories were cal-culated for each value of E0. It is seen that the results of the1837VOLUME 85, NUMBER 9 P H Y S I C A L R E V I E W L E T T E R S 28 AUGUST 2000FIG. 3. Quantum (solid curves) and classical Monte Carlo(dotted curves) survival probabilities as a function of the maxi-mum electric field strength, E0, of the laser. On the left, resultsare shown for an angular frequency of v  1 while, on theright, v  2. The remaining pulse parameters are the same asin Fig. 1. Dipole (DP), nondipole (NDP), and counterpropagat-ing pulses (CP) results are shown. Error estimates are given forthe Monte Carlo results.quantum and classical calculations demonstrate a remark-able agreement for each of the cases considered. Evidently,aspects of the wave packet dynamics of the system in thehigh-frequency, high-intensity regime can be described toa good approximation by using classical mechanics.We have presented the results of numerical simulationsinvolving a two-dimensional model atom interacting withan intense laser pulse that includes the magnetic field of thelaser pulse. We have clearly shown that the magnetic fieldinduces a drift along the propagation direction that is suffi-cient, even for moderate intensities, to disrupt the electrondynamics that are required for atomic stabilization. Byconsidering the Lorentz force on the electron in the propa-gation direction of the laser pulse, it can be seen that evena small drift velocity, integrated over the duration of thelaser pulse, is sufficient to modify the simple oscillatingmotion along the polarization direction, thereby leading toa breakdown of stabilization. It should be stressed that,for the laser frequencies and intensities considered here,the dynamics of the electron in the laser field is essentiallynonrelativistic. Indeed, for the highest field strength con-sidered, E0  20, the ratio of the ponderomotive energy1838of the electron in the laser field to its rest mass energyis small: E204v2c2  5 3 1023. In addition, for thelaser field parameters considered, the laser coupling of theelectron spin degrees of freedom is expected to have a neg-ligible influence on the stabilization dynamics.In conclusion, our results indicate that important revi-sions are needed of existing stabilization models so thatthe effect of the magnetic field is properly accounted for.This work has been supported by the UK Engineeringand Physical Sciences Research Council (EPSRC) and theSpanish Dirección General de Enseñanza Superior e Inves-tigación Científica (Grant No. PB98-0268).[1] For reviews, see C. J. Joachain, M. Dörr, and N. J. Kylstra,Adv. At. Mol. Opt. Phys. 42, 225 (1999); M. Protopapas,C. H. Keitel, and P. L. Knight, Rep. Prog. Phys. 60, 389(1997).[2] M. Pont, N. R. Walet, M. Gavrila, and C. W. McCurdy,Phys. Rev. Lett. 61, 939 (1988); M. Pont, N. R. Walet, andM. Gavrila, Phys. Rev. A 41, 477 (1990); M. Pont andM. Gavrila, Phys. Rev. Lett. 65, 2362 (1990).[3] M. Dörr, R. M. Potvliege, D. Proulx, and R. Shakeshaft,Phys. Rev. A 43, 3729 (1991).[4] Q. Su, J. H. Eberly, and J. Javanainen, Phys. Rev. Lett.64, 862 (1990); C. K. Law, Q. Su, and J. H. Eberly, Phys.Rev. A 44, 7844 (1991).[5] K. C. Kulander, K. J. Schafer, and J. L. Krause, Phys. Rev.Lett. 66, 2601 (1991).[6] V. C. Reed, P. L. Knight, and K. Burnett, Phys. Rev.Lett. 67, 1415 (1991); R. M. A. Vivirito and P. L. Knight,J. Phys. B 28, 4357 (1995).[7] A. S. Patel, N. J. Kylstra, and P. L. Knight, J. Phys. B 32,5759 (1999).[8] M. P. de Boer, J. H. Hoogenraad, R. B. Vrijen, L. D. Noor-dam, and H. G. Muller, Phys. Rev. Lett. 71, 3263 (1993);see also B. Piraux and R. M. Potvliege, Phys. Rev. A 57,5009 (1998).[9] J. Grochmalicki, M. Lewenstein, and K. Rza̧żewski, Phys.Rev. Lett. 66, 1038 (1991); C. H. Keitel and P. L. Knight,Phys. Rev. A 51, 1420 (1995); see also T. Katsouleas andW. B. Mori, Phys. Rev. Lett. 70, 1561 (1993).[10] M. Protopapas, C. H. Keitel, and P. L. Knight, J. Phys. B29, L591 (1996); N. J. Kylstra, A. 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Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses

Breakdown of stabilization of atoms interacting with intense, high-frequency laser pulses

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