Exact theory and numeric results for short pulse ionization of simple model atom in one dimension
aa r X i v : . [ phy s i c s . a t m - c l u s ] M a r Exact theory and numeric results for short pulseionization of simple model atom in one dimension
A.RokhlenkoDepartment of Mathematics, Rutgers UniversityPiscataway, NJ 08854-8019
Abstract
Our exact theory for continuous harmonic perturbation of a one dimen-sional model atom by parametric variations of its potential is generalizedfor the cases when a) the atom is exposed to short pulses of an externalharmonic electric field and b) the forcing is represented by short bursts ofdifferent shape changing the strength of the binding potential. This workis motivated not only by the wide use of laser pulses for atomic ioniza-tion, but also by our earlier study of the same model which successfullydescribed the ionization dynamics in all orders, i.e. the multi-photon pro-cesses, though being treated by the non-relativistic Schr¨odinger equation.In particular, it was shown that the bound atom cannot survive the exci-tation of its potential caused by any non-zero frequency and amplitude ofthe continuous harmonic forcing. Our present analysis found importantlaws of the atomic ionization by short pulses, in particular the efficiencyof ionizing this model system and presumably real ones as well.PACS: 32.80.Fb, 03.65.Ge, 32.80.Rm, 02.30.-f
We study a simple one-dimensional quantum system with the attractivepotential, modeled by the δ -function. This system is assumed to be in thebound state until at some initial time t = 0 a) it becomes exposed to an externalharmonic electric field or b) the strength of the system binding potential getstime dependent. The perturbation after a short interval T is turned off andour objective is to study the time evolution of the bound state on the interval0 < t < T . The pulses of external electric field are modeling the application oflaser beams for atomic ionization.Excitation of the δ -function atom was studied in [1-6] for a simpler case ofharmonic parametric perturbation of its potential when T = ∞ , i.e. of infiniteduration, and the main conclusion was the complete ionization for arbitraryfrequency and amplitude of perturbation. Other results have shown a surprisingsimilarity of main features of the process with observed experimentally andnumerically in spite of simplicity of this model.
1. PROBLEM SET UP
As in [1] we start by considering the one-dimensional stationary system lo-cated at x = 0 with an unperturbed Hamiltonian H = − ~ m d dx − gδ ( x ) , g > , −∞ < x < ∞ , (1)1hich has a single bound state u b ( p, x ) = √ pe − p | x | , p = m ~ g. (2)In the continuous spectrum the eigenfunctions are u ( p, k, x ) = 1 √ π (cid:18) e ikx − pp + i | k | e i | kx | (cid:19) , −∞ < k < ∞ , (3)with energies ~ k / m while the bound state energy is W = − ~ ω = − ~ p / m .Functions u ( p, k, x ) , u b ( p, x ) are normalized to δ ( k − k ′ ) and to unity respectfully.Parameter m represents the mass of the bound charged particle.On the interval 0 ≤ t ≤ T acts a perturbing potential which is described byadding in Eq.(1) a time dependent term V ( x, t ) V ( x, t ) = eExη ( t ) , or V ( x, t ) = Rδ ( x ) η ( t ) , t ∈ [0 , T ] , (4)where η max = 1 and parameters E , R are responsible for the amplitude ofperturbation. Thus we have to solve the time-dependent Schr¨oedinger equation i ~ ∂ψ ( x, t ) ∂t = H ψ ( x, t ) + V ( x, t ) ψ ( x, t ) , t ≥ . (5)After expanding ψ ( x, t ) in the complete set of functions u : ψ ( x, t ) = θ ( t ) u b ( p, x ) e i ( ~ p / m ) t + Z ∞−∞ Θ( k, t ) u ( p, k, x ) e − i ( ~ k / m ) t dk, t ≥ , (6)the survival of the bound state at time t ≤ T can be evaluated by | θ ( t ) | , if weassume the system to be initially in its bound state θ (0) = 1 , Θ( k,
0) = 0 . (7)It is more convenient [1] to proceed in dimensionless units ( ~ = 2 m = g/ u b ( x ) = e −| x | , W b = − , u ( k, x ) = 1 √ π (cid:18) e ikx − e i | kx | i | k | (cid:19) , (8)where W b is the rescaled energy of the bound state. The energies of states u ( k, x ) are W ( k ) = k with multiplicity two for k = 0. These functions arenormalized to δ ( k − k ′ ), while the bound state u b ( x ) - to 1.
2. DIPOLE FIELD PULSE PERTURBATION
Beginning at t = 0 a perturbing potential Exη ( t ) is applied to the atomand it stops at t = T . Here parameter E represents the electric field of theperturbation whose frequency is ω .For solving Eq.(5) we use Eq.(6) and expand ψ ( x, t ) on the interval (0 , T ) inthe complete set (8) of functions u : ψ ( x, t ) = θ ( t ) u b ( x ) e it + Z ∞−∞ Θ( k, t ) u ( k, x ) e − ik t dk. (9)2hen using their orthonormality and assuming cutoff of the perturbation po-tential for large | x | reduce dimensionless form of Eq.(5) to the following set˙ θ ( t ) = 4 Eη ( t ) √ π Z ∞−∞ Θ( k, t ) e − i ( k +1) t ( k + 1) kdk, (10 a )˙Θ( k, t ) = − Eθ ( t ) e i ( k +1) t sin ωt √ π ( k + 1) k. (10 b )The evolution of Θ( k, t ) is determined in Eq.(10b) by θ ( t ) only. Using Eq.(7),then integrating Eq.(10b) in time, and substituting the result into Eq.(10a) weobtain a single equation which describes the evolution of θ ( t )˙ θ ( t ) = − (4 E ) π η ( t ) Z t θ ( t ′ ) η ( t ′ ) dt ′ Z ∞−∞ e − i ( k +1)( t − t ′ ) ( k + 1) k dk, < t < T. (11)The ionization probability of the bound state is P ( t ) = 1 − | θ ( t ) | , for t ≤ T, (12)and it becomes constant P ( T ) for all later times t ≥ T . The internal integralover k in Eq.(11) can be expressed in terms of Fresnel’s integrals S ( t − t ′ ) = Z ∞−∞ e − i ( k +1) u ( k + 1) k dk = 3 − u A + iu u − iu B, where (13) A = π [1 − Φ( √ iu )] , B = e − iu p π/iu − A, u = t − t ′ ≥ . Numeric realization of ionizing by dipole pulses
We solve Eq.(11) for θ ( t ) by using the following technique, which is impliedby a known method for the Volterra equations. The integral in time is approxi-mated by the summation over discrete equal subintervals defined by equidistantpoints on the interval [0 , t ]: 0 , δ, δ, ..., N δ = t . As it is clear from Eqs.(7) and(11) θ (0) = 1 , ˙ θ (0) = 0, then we have θ ( δ ) ≈ θ (0) + δ ˙ θ (0) = 1. In this way θ (2 δ ) = θ ( δ ) + δ ˙ θ ( δ ) where ˙ θ ( δ ) is evaluated by the integral (11) without involv-ing the end point at t = 2 δ . And so on by consequent computations of θ ( jδ ) viathe terms with numbers 0 , , , ..., j −
1. This technique realizes an approximatequadrature of Eq.(11) and its precision becomes better when subintervals δ getshorter. This approximation replaces Eq.(11) with the sum˙ θ ( nδ ) = − (4 E ) π δη ( nδ ) n X j =0 θ ( jδ ) η ( jδ ) S (( n − j ) δ ) , (14)which allows to find approximately θ (( n + 1) δ ) = θ ( nδ ) + δ ˙ θ ( nδ ). We study inthis work sometimes quite long pulses, they require for an acceptable precisionlong computing time as δ should be very small. This can be helped by improvingthe transition from θ ( nδ ) to θ (( n + 1) δ ) by involving two derivatives of θ : θ (( n + 1) δ ) = θ ( nδ ) + δ ˙ θ ( nδ ) + δ θ ( nδ ) . (15)3ifferentiating Eq.(11) we have¨ θ ( t ) = − (4 E ) π h ˙ η ( t ) Z t θ ( t ′ ) η ( t ′ ) dt ′ Z ∞−∞ e − i ( k +1)( t − t ′ ) ( k + 1) k dk + 5 π θ ( t ) η ( t ) − iη ( t ) Z t θ ( t ′ ) sin ωt ′ dt ′ Z ∞−∞ e − i ( k +1)( t − t ′ ) ( k + 1) k dk i . By expressing the last integral over k in terms of Fresnel functions as V ( u ) = Z ∞−∞ e − iu ( k +1) ( k + 1) k dk = π [1 − Φ( √ iu )] 1 − iu + 4 u e − iu √ πiu iu , (16)Eq.(15) can be rewritten in the following form θ (( n + 1) δ ) = θ ( nδ ) + 8 δ E π n iδ η ( nδ ) n X j =1 θ ( jδ ) η ( jδ ) V (( n − j ) δ ) (17) − π θ ( nδ ) η ( nδ ) − h η ( nδ ) + δ dηdt ( nδ ) i n X j =1 θ ( jδ ) η ( jδ ) S (( n − j ) δ ) o , convenient for numeric computation of the ionization probability. A. Sin-wave pulses sin( ωt )The ionization of our atom by the sin-wave electric field excitation, whichmodels the laser pulses, means that η ( t ) = sin( ωt ). For illustration everywherein this work the ’short’ harmonic pulses will have only five cycles, N = 5, thiscan be easily realized now by experimental techniques [7-10]. For ω < T >
100 and the useof ¨ θ ( t ) is necessary. The results are exhibited in Fig.1, where time t is measuredin numbers of cycles of perturbation. FIG.1. Ionization probability P ( t ) caused by harmonic pulses ω is doubled in the process,which is suggested in some measure by Eq.(11). When the electric field E = 3the complete ionization occurs practically after the third cycle of the pulse.All computations are done using Maple on the intervals T = 5 πω with various ω and E , therefore T = 78 . , . , . ω = 0 . , . , . δ ≤ .
04, i.e. the sums in Eq.(17) will have up to4000 terms and the computations is time consuming as the sum is of recursivenature.
FIG.2a. E = 1 FIG.2b. E = 0 . ω = 0 . . Fig.2 confirms our observation that when ω is far from the resonance the totalduration of the perturbation pulse can be more important than its frequency,experiments [7-9] confirm this effect in real systems. The same behavior is morevisible in Fig.2b, where the electric field is smaller E = 0 . GV /m . When we rescale | W Cs | = 3 . eV to | W b | = 1 this would correspond in Eq.(14) to E ∼ E = 0 . GV /m , which is a very highelectric field in laser pulses, but reachable for present techniques. The case withthe field strength E = 0 . E ∼ . GV /m for Cs or ∼ . GV /m for Tungsten with | W | = 7 . eV ) was computed too: P ( T ) ≈ .
053 for ω = 0 . .
027 for ω = 0 . P ( t )is similar to plots in Fig.2. These results are only for orientation and clearlyshould be considered as qualitative due to the limitations of our model. It looksthat for smaller E the value of P ( T ) becomes proportional to ∼ E in agreementwith Eq.(17). B. Pulsed dipole forcing
Here we consider electric bursts acting on the model atom: rectangular η ( t ) = 1 and bell-shaped η ( t ) = 4( t/T − t /T ) on the interval 0 ≤ t ≤ T .Though the bell-shaped pulse has the same amplitude as the rectangular one,but its ionization efficiency is much lower in Fig.3 because its total energy issmaller and more importantly it does not have high frequency harmonics. Theionization probability P ( t ) is an oscillating function but there is an important5ifference between plots in Figs.3a and 3b. While the ionization probabilityby a rectangular pulse of duration t is given by P ( t ) in Fig3a, the ionizationevolution for bell-shaped pulses is presented by Fig.3b, but only for the pulseof length 10, i.e. if T = t function P ( t ) in Fig.3b does not represent the finalprobability and the corresponding computation should be performed namely for T = t because δ = δ ( T ). Our calculation with E = 0 . P ( t ) runs very low: its maximum P (4) ∼ .
013 and P (10) is less than 0 . FIG.3a. Rectangular pulse FIG.3b. Bell-shaped pulseFig.3. Ionization probability caused by rectangular and bell-shaped electric bursts
Our approach is modified below for studying the ionization caused by theparametric modulations of the binding potential. Though this is hardly achiev-able in practice but, as we already mentioned, it exhibits some illuminatingfeatures of the ionization by short pulses which are quite universal.
3. PULSED MODULATION OF BINDING POTENTIAL.
Here we consider short pulse bursts of the potential strength which has tobe studied by a different computations technique. Eq.(5) now has the followingform i ∂ψ ( x, t ) ∂t = H ψ ( x, t )+ Rδ ( x ) η ( t ) ψ ( x, t ) , η ( t ) = 0 when t / ∈ [0 , T ] , t ≥ , (18)with the initial conditions given by Eq.(7). Using the expansion (6) in terms ofthe stationary eigen-functions and the methods developed in [1] yields a simpleequation for the function θ ( t ) θ ( t ) = 1 + 2 i Z t Y ( t ′ ) dt ′ , (19)where Y ( t ) is defined by the following integral equation Y ( t ) = Rη ( t ) (cid:26) Z t [2 i + M ( t − t ′ )] Y ( t ′ ) dt ′ (cid:27) . (20)The function M ( s ) in (20), see [1], is M ( s ) = 2 iπ Z ∞ u e − is (1+ u ) u du = − i + r iπs e − is + i Φ( √ is ) . (21)6t behaves as p i/πs − i when s → s − / e − is when s → ∞ . A. Case of rectangular pulse η ( t ) = 1 . For approximate evaluation of the function θ ( t ) in this set up we have tosolve numerically the integral equation (20) that can be done if there is aneffective way of computing M ( s ) for not very large values of s as our pulses arenot long. Using [11] equation for the integral in Eq.(21) can be written in termsof gamma functions Z ∞ √ xe − isx x dx = e is Γ (cid:18) (cid:19) Γ (cid:18) − , is (cid:19) , and then one can apply the power series expansion for the confluent hyper-geometric function and present M ( s ) in a rapidly convergent form M ( s ) = − i − r isπ X n =0 ( − is ) n (2 n − n ! . (22)For a given pulse duration T the upper limit in the sum (22) can be only slightlylarger than eT to give a good precision for M ( s ).Our next step is express this function by a power series using the behaviorof Y ( t ) near zero, Y ( t ) = X m =0 c m t m/ , < t < T, (23)and substitute Eqs.(22) and (23) into the integral equation (20). The resultreads R − X m =0 c m t m/ = 1 + i X m =0 c m t m/ m/ X k,n =0 B (cid:18) k , n + 12 (cid:19) c k a n t n + k +12 , where B ( x, y ) = Γ( x )Γ( y ) / Γ( x + y ) is the β -function [11]. Thus we obtainequations for finding coefficients c ∗ : c = R, c = B (cid:18) , (cid:19) Ra c = 2 R r iπ , (24) c m = 2 iRm c m − + R ⌊ m − ⌋ X n =0 B (cid:18) m + 12 − n, n + 12 (cid:19) a n c m − n − , m ≥ . The symbol ⌊ x ⌋ as usual means the integer part of x , and coefficients c m in (24)can be calculated consequently as they are expressed in terms of c ∗ evaluatedearlier. Clearly we will keep in (22) and (23) only finite numbers of terms.By combining Eqs.(19), (23), and (12) we find the ionization probability asa function of the pulse duration TP ( T ) = 1 − (cid:12)(cid:12)(cid:12) i X m =0 c m m + 2 T m/ (cid:12)(cid:12)(cid:12) . (25)The results of computation by Eqs.(22-25), where we keep about 80 terms insums, are presented in Figs.4 which show this probability for different values7f pulse amplitude R and T ≤
10. Left Fig.4 shows also that a very shortattractive pulse with R = − . , T ≈ . FIG.4 Plots of ionization probability P ( T ) for rectangular pulses In agreement with the common sense when the external pulse decreases thebinding energy,
R > T ≤ .
5. One can seethe resonances and quite a high probability 0 .
32 when T = 0 .
55 because thepulse fronts have infinite slopes and thus introduce very high harmonics thoughthe total pulse energy is limited. Note also that R should be compared with g = 2 in our units, | R | = 1 is the largest considered here. B. Bell-shaped pulse η ( t ) = 4( t/T − t /T )On the interval 0 < t < T we consider a pulse, symmetric about t = T / R , see Eq.(4). In this case the same approachas before converges slower and one needs more terms in Eqs.(22) and (23), weused up to 400-500 of them.A straightforward analysis of Eq.(20) shows that the power series expansionfor Y ( t ) has the following form Y ( t ) = X m =0 c m t m/ , < t < T. (26)By substituting (26) into Eq.(20) and using Eq.(22) we solve Eq.(26) and findthe coefficients which define Y ( t ) T c R = 1 , c = 0 , T c R = − T , T c R = a c B (cid:18) , (cid:19) , T c R = i c a c B (cid:18) , (cid:19) ,T c R = i c a c B (cid:18) , (cid:19) + a c B (cid:18) , (cid:19) − T a c B (cid:18) , (cid:19) , (27)8 c m R = 2 ic m − m − ic m − T ( m −
2) + a c m − B (cid:18) , m + 12 (cid:19) + ⌊ m − ⌋ X n =0 c m − n − (cid:18) n + 1 m a n +1 − a n T (cid:19) B (cid:18) n + 12 , m − − n (cid:19) , m ≥ . One can see again that as in Eqs.(24) each coefficient c ∗ is defined via corre-sponding ones found earlier. FIG.5a FIG.5b
Ionization probability by bell-shaped Time evolution of ionizationpulses of different duration T when T = 2 . R = ± . t = T or inFig.5a. But for the present bell-shaped pulse the meaning of plots in Fig.5b isdifferent, see Part 2B, and also the ionization probability is found using Eqs.(26),(27) and (25) in the slightly modified form P ( T ) = 1 − | θ ( T ) | = 1 − (cid:12)(cid:12)(cid:12) i X m =0 c m m + 4 T m/ (cid:12)(cid:12)(cid:12) , (28)where all c k , k = 0 , , ... depend on T . P ( T ) in Fig.5a is the ionization probability at the end of the correspondingpulse with R = − . T , but the plot for maximum M ax ( P ) whichoccurs at an intermediate time 0 < t < T and for longer pulses it is muchlarger than P ( T ). Fig.5b shows P ( t ) behavior in the case T = 2 . R = ± .
5. Positive pulses are more effective as before: P ( T ) = 0 .
13 is close to
M ax ( P ) when T = 2 . P ( T ) and M ax ( P ) are0 .
139 and 0 .
147 respectively for T = 3. This probably means that change ofsign of the external force plays an important role and might suggest that morerealistic perturbation (a harmonic one with some envelope) can be more efficient.One can see that the system can be ionized before the end of perturbation pulse,i.e. the value of M ax ( P ) is interesting. C. Short sin-wave pulses η ( t ) = sin( ωt )9ur pulse exists on the interval 0 ≤ t ≤ T and as in Part 2 its lengthhas always an integer number N = 5 of cycles, T = 2 πN/ω . We assumein computations that the frequency of principal harmonic of Y ( t ) is ω andchoose an integer K of harmonics sufficient for modeling Y ( t ). Then using theGalerkin [12] method for solving Eq.(20) the function Y ( t ) is approximated bythe following sum Y ( t ) = K X k = − K a k f k ( t ) , where f k ( t ) = e iωkt . (29)The solution method requires the discrepancy of using the approximation (29)in Eq.(20) be orthogonal to all functions f k . This procedure creates the linearalgebraic system for coefficients a kK X k = − K a k Z T dt ¯ f m ( t ) ( f k ( t ) − Rη ( t ) Z t [2 i + M ( t ′ )] f k ( t − t ′ ) dt ′ ) = (30) R Z T η ( t ) ¯ f m ( t ) dt, − K ≤ m ≤ K. By substituting Eq.(29) into (30) this system can be rewritten in the stan-dard form K X k = − K C k,m a k = B m , (31)where all coefficients with k = m ± C k,m = A k − m + R ω (cid:18) M − k − M − m k − m + 1 − M − k − M − − m k − m − (cid:19) . (32)For k = m − k = m + 1 we have respectively C m − ,m = R ω [(1 + 2 iωT ) M − m − M − − m − iωM − m ] , (33) C m +1 ,m = R ω [(1 − iωT ) M − − m − M − m + 2 iωM − − m ] . Other terms in Eqs.(31-33) are A k − m = ( , k = m,T, k = m, B m = R i [ A − m − A − − m ] , (34) M n = Z T [2 i + M ( t )] e inωt dt, M n = Z T [2 i + M ( t )] te inωt dt. Our computation by Eqs.(31-34) for the cases when T = 5, R = 0 . , . ω = 2 πN/T ≈ . > ω ) areshown in Fig.6: 10 IG.6. Ionization probability P ( T ) caused by sin-wave pulse One can see that with pulse amplitude 0 . K = 5, in computations but checked the precision bytaking K = 10 which produced the curves P ( t ) almost identical to ones in Fig.4.The roughly linear time dependence of the ionization probability is caused bythe fact that ω is significantly larger than ω , i.e. the energy of ionizing photonsin Fig.4 exceeds the binding energy of the model atom and thus we observe thefirst order effect. In reality this process might be compared with the soft X-rayionization.For solving Eq.(20) this method of approximation Y ( t ) by Eq.(29) and theroutine defined by Eqs.(31-34) becomes inefficient when ω is much smaller be-cause the interval T is too long in this case and oscillations of M ( s ) are not wellmodeled by harmonics of η ( t ). We use the special properties of the VolterraEq.(20) to model function Y ( t ) by a set of its discrete points. When the ioniza-tion is provided by at least two photons and ω = 0 .
6, 5 cycles of sin wave make
T >
60, and thus one needs about N = 10 points for a decent approximationof Y ( t ). Denoting temporarily F ( s ) = 2 i + M ( s ) the integral equation (20) isreplaced by the following one, where we keep the integral term in the intervalof divergent behavior of M ( s ) R − Y ( t n ) = η ( t n ) + η ( t n ) " ∆ n − X m =1 F ( t n − t m ) Y ( t m ) + Z ∆0 F ( s ) Y ( t n − s ) ds . (35)Here t n = n ∆, n runs from 1 to N and ∆ = T /N . The integral in Eq.(35) isapproximated using Eq.(22) for M ( s ) when s is small and only the linear termof Y ( t )-dependence on the interval ( t n − , t n ): F ( s ) ≈ i + r iπs (cid:20) is + s − i s (cid:21) , Y ( t n − s ) ≈ Y ( t n ) + Y ( t n − ) − Y ( t n )∆ s. (36)Near small s we keep more terms in singular F ( s ) than in Y ( t n − s ) because Y ( t )has a regular behavior there. By evaluating the integral in Eq.(35) we come tothe recurrent equation for computing the set Y ( t n ) consequently starting from11 = 2. Y ( t n ) = Rη ( t n )1 − B − i ∆ / " n − X m =1 F ( t n − t m ) Y ( t m ) + ( A + i ∆ / Y ( t n − ) , (37) A = r i ∆ π (cid:18)
23 + 2 i ∆5 + ∆ − i ∆ (cid:19) , B = r i ∆ π (cid:18)
43 + 4 i ∆15 + 2∆ − i ∆ (cid:19) . Here we will neglect terms of the order ∆ and higher.It is clear that Y (0) = 0 and for approximating Y ( t ) by a polynomial withthe same precision we substitute Y (∆ − s ) into Eq.(20) and using Eqs.(36,37)obtain the following relation X k =0 c k ∆ k/ = R (cid:18) ω ∆ − ω ∆
3! + ω ∆ (cid:19) ( X n =0 c n " i ∆ n/ n/ r iπ H n (∆) , where H n (∆) = (38)∆ n (cid:20) B (cid:18) n , (cid:19) + i ∆ B (cid:18) n , (cid:19) + ∆ B (cid:18) n , (cid:19) − i ∆ B (cid:18) n , (cid:19)(cid:21) . These equations are sufficient to evaluate c k , k = 0 , ...,
10 and find Y ( t ) = Y (∆). We applied Eqs.(35-38) to compute the ionization by the five cycle pulsesof lower than in Fig.6 frequency ω = 0 . R = 0 . , R = 0 . FIG.7. Ionization initiated by sin-wave pulse with ω = 0 . The dynamics of the process is shown in Fig.7. One can see that the ionizationprobability is higher than in the case of ω ≈ R ) roughly the same. The sin-wave pulsesappeared to be quite efficient for ionization and P ( T ) seems to be proportional to R for smaller R like for the laser pulse perturbation. The time of perturbationin Fig.5 is measured in the number of harmonic cycles, the pulse ends when N = 5 and T ≈ . . eV and 3 . eV respectively. It is easy to see that ω = 0 . nm for W and 530 nm for Cs in our cases. In experiments often are used ∼ −
10 fsec laser pulses of λ = 800 − nm , [7-10], and having in mind a qualitative application of ourtheory we perform a somewhat less precise computation for smaller ω = 0 . . T is longer. FIG.8. Ionization by short pulse harmonic waves of ω = 0 . The results presented in Figs.8 and 9 confirm the importance of total pulseduration. As before our pulses have only 5 cycles, the time of their action is T = 157 dimensionless units when ω = 0 . ω = 0 . R = 0 . . λ ∼ − nm . The amplitude R = 0 . λ ∼ nm ). FIG.9. Short pulse ionization when atomic binding energy is larger
The ionization level is lower than in case of Cs though the wave length is twice13horter. This clearly agrees with greater binding energy in W atoms.As the parametric perturbation acts directly on the binding energy it ismore efficient in Figs.7-9 than the more realistic perturbation by the externalharmonic electric field in Part 2. These results show that the ionization of ourmodel atom in some measure describes qualitative behavior of real systems.
4. SUMMARY
The atomic ionization by short pulses of external forces is studied on a sim-ple one-dimensional model which allows to construct an exact theory of theprocess and realize its conclusions by several methods of numerical computa-tions. This creates a basis for comparison with approximate solutions of morerealistic models, simulations, and experiments. Our main results include theobservation that for external frequencies, much lower than the resonance ones,the total duration of the pulse is more important for effective ionization than itsfrequency. When the ionization level is substantially far from the complete oneit is increasing approximately linearly in time and has resonances as a functionof pulse duration. For ionization caused by the dipole electric field the frequencyof these resonances is twice larger than the frequency of external forcing.
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