Table Of ContentSolution of the time-dependent Dirac equation for describing multiphoton ionization
of highly-charged hydrogenlike ions
Yulian V. Vanne and Alejandro Saenz
AG Moderne Optik, Institut fu¨r Physik, Humboldt-Universita¨t zu Berlin, Newtonstr. 15, D–12489 Berlin, Germany
(Dated: January 21, 2011)
A theoretical study of the intense-field multiphoton ionization of hydrogenlike systems is per-
formed bysolvingthetime-dependentDiracequation withinthedipoleapproximation. Itisshown
that the velocity-gauge results agree to the ones in length gauge only if the negative-energy states
are included in the time propagation. On the other hand, for the considered laser parameters,
no significant difference is found in length gauge if the negative-energy states are included or not.
Within the adopted dipole approximation the main relativistic effect is the shift of the ionization
1 potential. A simple scaling procedure is proposed to account for this effect.
1
0 PACSnumbers: 32.80.Fb,31.30.J-
2
n
a I. INTRODUCTION ing the negative-energy (NE) states was emphasized in
J the case of a treatment beyond the dipole approxima-
0 tion, even for the considered laser parameters where the
Future experiments using an electron-beam ion trap
2 photon energy is insufficient to produce real positron-
(EBIT) at the Linear Coherent Light Source (LCLS) at
electron pairs. Based on general theoretical considera-
] StanfordandX-rayFreeElectronLaser(XFEL)atHam- tions it was conjectured that in length gauge the impor-
h burg are expected to permit the study of photoabsorp-
p tance of the NE states may be reduced. On the other
tionprocessesofhighlychargedatomicionsinthe wave-
- hand, it was concluded on the basis of the numerical re-
t length range of 0.1 to 100nm with a peak intensity up
n sults that within the dipole approximation the inclusion
to 1025W/cm2 or even higher. Also at the GSI (Darm-
a of NE states is not needed, even if the TDDE is solved
u stadt)experimentalinvestigationsofhighly chargedions in velocity gauge. A comparison of the numerically ob-
q exposed to intense laser fields are planned within the
tained ionization rates for various nuclear charges with
[ SPARC project. The analysis of these experiments will
the ones in non-relativistic approximation showed that,
require a relativistic treatment of the ion-laser interac-
1 expectedly,increasingdifferencesarefoundwithincreas-
tion.
v ingcharge. However,itwasconcludedthattheionization
3 Clearly, a full treatment demands to solve the time- rate (shown as a function of the peak value of the laser
9 dependent Dirac equation (TDDE) incorporating also field) obtained within the relativistic TDDE calculation
9
the spatial dependence of the vector potential. Since may be largeror smaller than the non-relativistic result.
3
. such a treatment is very demanding, most earlier treat- The authors found the higher rate easier to understand
1 ments adopted simplifications. Low-dimensional mod- andcouldonlyspeculateonpossiblereasonsforthelower
0
els are especially popular. Starting first with a one- one.
1
1 dimensionaltreatment[1],elaboratetwo-dimensionalcal- A solution of the TDDE within the length gauge was
: culations have been reported, e.g., in [2–4]. However, reportedmorerecentlyin[14]. Adirecttimepropagation
v such models can certainly not provide quantitative pre- on a grid was used. As in [13] the electron-nucleus in-
i
X dictionsanditis,atleastapriori,notevenclearwhether teraction is described by the unmodified non-relativistic
r they are always qualitatively correct. Similarly to the Coulombinteraction. Whilesolelythedipoleapproxima-
a non-relativistic case, also simplified ionization models tionisadopted,notonlyresultsforone-electron,butalso
like the strong-field approximation [5–8], semiclassical for two-electron ions are reported. In the latter case the
tunneling theory [9], or classical models like in [10–12] electron-electroninteractionisalsodescribedbythenon-
have been proposed. However, in order to allow for relativistic Coulomb interaction. No explicit discussion
quantitative predictions or the validation of such sim- of the inclusion or omission of the NE states is given.
plified models a full-dimensional solution of the TDDE Comparing for Ne9+, Ne8+, U91+, and U90+ the pho-
is needed. toionizationcrosssectionsobtainedbysolvingtheTDDE
Very recently, a three-dimensional solution of the for 10-cycle pulses (using a single pulse with one set of
TDDE for hydrogenlike systems has been reported in laserparametersforeveryion)withthe onesofrelativis-
[13]. The radial solutions were expanded on a grid and tic perturbation theory, a good quantitative agreement
the TDDE was either solved by a direct propagation on is found. This result is, of course, expected, if the laser
the grid or using a spectral expansion in field-free eigen- parametersarechoseninawaythatperturbationtheory
states. The spatial dependence of the vector potential is applicable, as was the case in [14] where relatively low
of the carrier part of the laser pulse was also consid- intensities (in relation to the ionization potentials and
ered, while the one in the envelope was ignored. The photon frequencies) were considered.
velocity gauge was used and the importance of includ- In this work a further approachfor solving the TDDE
2
is presented. It is based on the spectral expansion in and the interaction with the electromagnetic field V(t)
field-free eigenstates where the radial wavefunctions are can be presented within the dipole approximationeither
expressed in a B-spline basis. It may be seen as an in the velocity (V) or length (L) gauge,
extension of the corresponding approaches for solving
thenon-relativistictime-dependentSchr¨odingerequation VV(t)=cα·A(t)=cαzA(t),
(5)
(TDSE) for one- and two-electron atoms in, e.g., [15– VL(t)=r·F(t)=zF(t).
18], for molecular hydrogen [19–23], and for in princi-
ple arbitrary molecules within the single-active-electron Here,β andthecomponentsofthevectorαaretheDirac
approximation [24, 25]. While [13] and [14] considered matrices, p = −i∇ is the momentum operator, c ≈ 137
mostly (orsolely)one-photonionization,anextensionto isthespeedoflight,andF(t)=−dA(t)/dtistheelectric
multiphoton ionization is presented. Clearly, such cal- field.
culations are very demanding as they require larger ex- Asiswellknown(see,e.g.,[26]),theeigenstatesofHD
0
pansions, because more angular momenta are involved. can be presented as four component spinors
Within the dipole approximationa systematic investiga-
tion of the importance of NE states as well as the con- Φ (r)= 1 Pκ(r)χκ,m(ˆr) (6)
vergence behavior within either length or velocity gauge κm r iQκ(r)χ−κ,m(ˆr)
(cid:18) (cid:19)
is performed. A simple scaling relation is proposed that
allows to relate relativistic TDDE solutions to the ones where χκ,m(ˆr) is an ls coupled spherical spinor, κ is the
obtainedwiththenon-relativisticTDSE,ifbothareper- relativistic quantum number of angular momentum, re-
formed within the dipole approximation. lated to the orbital and total angular momenta, l and j,
In this work atomic units (e = m = ~ = 1) are used as
e
unless specified otherwise.
−(j+1/2)=−(l+1), for j =l+1/2,
κ= (7)
( j+1/2 = l, for j =l−1/2;
II. THEORY
andtheradialfunctionsP (r)andQ (r)aresolutionsof
κ κ
The dynamics of a highly charged atomic ion exposed the coupled equations
to an external electromagnetic field is considered by
κ d
means of both a relativistic and a non-relativistic treat- [U(r)+c2−E]P (r)+c − Q (r)=0
κ κ
r dr
mentwithinthedipoleapproximation. Thevectorpoten- (cid:20) (cid:21) (8)
κ d
tialA(t) ischoseninthe formofanN-cyclecos2-shaped [U(r)−c2−E]Qκ(r)+c + Pκ(r)=0.
laser pulse that is linear-polarized along the z axis, r dr
(cid:20) (cid:21)
e A cos2 πt sin(ωt), t∈(−T,T); For two spinor states, Φi and Φf, character-
A(t)= z 0 T 2 2 (1) ized by the quantum numbers {ni,κi,ji,li,mi} and
(cid:18) (cid:19) {n ,κ ,j ,l ,m },respectively,thetime-dependentma-
0, otherwise, f f f f f
trix element V (t)=hΦ |V(t)|Φ i can be written as
fi f i
where ω isthe radiation frequency and T = 2πN/ω is
V (t)=δ δ W M (t) (9)
the pulse duration. Ignoring finite nuclear size effects, fi mf,mi |lf−li|,1 fi fi
the interaction of an electron with the nucleus may be
where
approximated by the Coulomb potential
Z W =(−1)jf−mf(−1)ji+1/2 (2j +1)(2j +1)
U(r)=− , (2) fi f i
r q (10)
j 1 j j 1 j
× f i f i
where Z is the nuclear charge. In all calculations pre- −m 0 m −1/2 0 1/2
f i
sented in this work the hydrogenlike system is initially (cid:18) (cid:19)(cid:18) (cid:19)
prepared in its ground state. and
ML(t)=F(t) drr P (r)P (r)+Q (r)Q (r) , (11)
fi i f i f
A. Relativistic calculations
Z h i
The relativistic dynamics of the quantum system is
MV(t)=−icA(t) dr (1+∆ )P (r)Q (r)
governedby the TDDE fi fi i f
Z h (12)
∂Ψ(t) −(1−∆ )Q (r)P (r)
i = HD+V(t) Ψ(t) (3) fi i f
∂t 0
i
where HD is the field-fre(cid:2)e Dirac Ham(cid:3)iltonian HD, with ∆fi =(−1)jf−ji(κf −κi).
0 0 Inordertodescribebothboundandcontinuumstates,
HD =cα·p+c2β+U(r), (4) the atom is confined within a spherical box boundary of
0
3
radius R. This leads to the discretization of the contin- and transforming the radial Schr¨odinger equation into
uumspectrum,whereastheboundstatesremainunmod- the n ×n generalized banded eigenvalue problem with
ified, if R is chosen sufficiently large and not too highly respect to a coefficient vector
excited Rydberg states are considered.
Introducing in a region [0,R] a B-spline set consisting C=(ρ1,ρ2,...,ρn), (19)
ofn+2splinefunctionsB (r)oforderk,theradialfunc-
i which yields n solutions for every orbital quantum num-
tions P(r) andQ(r) canbe expandedin aB-spline basis
ber l.
as
For two states, Φ and Φ , characterized by the
i f
n n quantum numbers {n ,l ,m } and {n ,l ,m }, respec-
i i i f f f
P(r)= piBi+1(r) Q(r)= qiBi+1(r) (13) tively, the time-dependent matrix element Vfi(t) =
i=1 i=1 hΦ |V(t)|Φ i can also be written in form of Eq. 9, but
X X f i
with
where the first and the last spline are removed from the
expansion to ensure the boundary conditions P(0) =
W =(−1)lf−mf(−1)lf (2l +1)(2l +1)
Q(0) = P(R) = Q(R) = 0. Defining the coefficient vec- fi f i
tor l 1 l lq 1 l (20)
× f i f i
−m 0 m 0 0 0
C=(p1,q1,p2,q2,...,pn,qn), (14) (cid:18) f i(cid:19)(cid:18) (cid:19)
and
thesystemofequations(8)istransformedintoa2n×2n
generalized banded eigenvalue problem which is effi-
ML(t)=F(t) drrR (r)R (r), (21)
ciently solved using LAPACK routine DSBGVX. This fi i f
Z
procedure yields for every value of κ exactly n negative
energy solutions and n positive energy solutions.
l
As has beendiscussedin literature since the firstsolu- MV(t)=−iA(t) drR (r) R (r)+R′(r) . (22)
fi f r i i
tionofthetime-independent(stationary)Diracequation
Z h i
using B splines [27], there is the problem of the occur-
rence of spurious states [28] that are non-physical. Dif-
C. Time propagation
ferent procedures were proposed to avoid this problem
[29–31]. While there can be a problem with their iden-
Within the spectral approach, the integration of the
tification in the case of many-electron systems, it is the
TDDE(3)andtheTDSE(15)isperformedbyexpanding
lowest positive energy state for κ > 0 that represents a
the function Ψ(t) describing the dynamics of the system
spurious state in the present case.
in the basis of field-free eigenstates Φ ,
K
Ψ(r,t)= C (t)e−iEKtΦ (r) (23)
B. Non-relativistic calculations K K
K
X
The non-relativistic TDSE where the compound index K represents the full set of
quantum numbers and the coefficients C (t = −T/2)
K
aresettozeroforallstatesexceptfortheinitialstatefor
∂Ψ(t)
i =[H +V(t)]Ψ(t) (15) which its value is set to 1. Since the quantum number
0
∂t
m is conserved for both cases and the ground state is
where H is the non-relativistic field-free Hamiltonian chosen as initial state, the value of m is fixed to 1/2 for
0
H , the relativisticcaseandto0forthe non-relativisticcase.
0
Substituting Eq. (23) into Eq. (3) or TDSE (15) the
p2
latter onesarereducedto asystemofcoupledfirst-order
H = +U(r), (16)
0
2 ordinary differential equations,
and in contrast to the relativistic case, the interaction
withtheelectromagneticfieldV(t)isgiveninthevelocity CK′ (t)= ei(EK−EK′)tVKK′(t)CK′(t) , (24)
gauge by XK′
which is integrated numerically using a variable-order,
VV(t)=p·A(t)=p A(t). (17)
z variable-step Adams solver.
Theionizationyield(afterthepulse)isthendefinedas
Similar to the relativistic case, the eigenstates of H ,
0
Φ (r)=r−1R(r)Y (ˆr),areobtainedbyprojectingthe
lm l lm Y = |C (t=T/2)|2 (25)
radial function R(r) onto the B-spline basis ion K
l
K
X
n
R(r)= ρ B (r) (18) where the summation in Eq. (25) is performed over all
i i+1
discretized continuum states. The convergence can be
i=1
X
4
controlled by varying the value of l (or j ) which
max max
limitsthenumberofdifferentsymmetriesinvolvedinthe
summation (23). 1.5×10-3
InthecaseoftheTDDEthequestionofapropertreat- d
l
ment of the spurious states mentioned in Sec.IIA arises. e
i
y
The success of the spectral approach relies on the com- n 1.0×10-3
pletenessofthestatesincludedinthesummation,atleast o
i
for a given box size. In fact, already in the case of the at
non-relativistic Schr¨odinger equation the use of a finite iz L-gauge, no NE states
n
boxleadstotheoccurrenceofnon-physicalpseudostates Io5.0×10-4 L-gauge, with NE states
(see, e.g., [17]). In this case it turned out to be in fact V-gauge, no NE states
important to include those states in the spectral expan- V-gauge, with NE states
sion, since otherwise some relevant part of the Hilbert
0.0
space is omitted. In order to decide on the proper treat- 3/2 5/2 7/2 9/2 11/2 13/2 15/2 17/2
mentofthespuriousstatesofthe Diracequation(due to j
max
theB-splineexpansion),acarefulcheckoftherelativistic
sum rules [28] was performedthat also servedas a check
FIG.1: Convergencestudywithrespecttothemaximalvalue
of the proper implementation of the length and velocity ofthetotalangularmomentumjmax. Therelativisticcalcula-
forms of the dipole-operator matrix elements. On this tionsoftheionizationyieldforanionwiththenuclearcharge
basis it was concluded that the spurious states could be Z = 50 exposed to a 20-cycle cos2-shaped laser pulse with
omitted from the basis for solving the TDDE, since the peak intensity I = 5×1023W/cm2 and a photon energy of
sum rules were fulfilled in this case. The agreement of 500 a.u. are performed using the length (L) or velocity (V)
the TDDE solutions obtained within the length and the gaugeeitherincludingorexcludingthenegativeenergy(NE)
states in theexpansion (23).
velocitygaugediscussedbelowisanotherindicationthat
the omission of the spurious states is appropriate.
fect, cf. [13].
D. Computational details and scaling of the TDSE
with Z
III. RESULTS AND DISCUSSION
Forthesakeofconsistency,thesameB-splinebasisset
is used for both the relativistic and the non-relativistic
treatment. Typically, the values k = 9 and n = 500 are A. Convergence behavior
usedtoconstructanalmostlinearknotsequenceinwhich
the first 40 intervals increase with a geometric progres- ThehandlingoftheNEsolutionsoftheDiracequation
sion using g = 1.05 and all following intervals have the is an important issue and steered considerable attention
length of the 40th one. Such a choice ensures an accu- in the literature, see [13, 33] and references therein. The
rate numerical description in the vicinity of the nucleus question is whether the NE states should be removed
and,atthe sametime,asufficientcompletenessforade- from the basis as long as the creation of real positron-
scription of the discretized continuum [32]. Depending electron pair is energetically out of reach. Such a sit-
on the nuclear charge Z, a box size of R=(250/Z)a0 is uation occurs, for example, in case of ionization of an
adopted. ThischoiceofRreflectsthewell-knownscaling ion with the nuclear charge Z = 50 exposed to a pulse
property of the time-independent Schr¨odinger equation with photon energy 500 a.u. (see Fig. 1). Whereas only
of hydrogenlikesystems. With the substitution r′ =r/Z 3photonsaresufficientforionization,morethan70pho-
and E′ = Z2E for the position r and the energy E, re- tons are required for positron-electron pair creation. As
spectively, the eigensolutions for a hydrogenlike system is demonstrated in Fig. 1, the answer to the question of
with charge Z reduces to the one of the hydrogen atom the proper treatment of the NE states depends already
with Z =1. within the dipole approximationadopted here clearlyon
Within the dipole approximation this property is pre- the gauge and thus the interaction operator used in the
served also for the TDSE, if the pulse parameters are time propagation. For the length gauge the exclusion of
also properly scaled, e.g., the time as t′ = t/Z2, the NE states has virtually no effect on the final result. In
laserfrequency asω′ =Z2ω,the amplitude ofthe vector contrast, for the velocity gauge the results are obviously
potential as A′ = ZA , and the laser peak intensity as different. IftheNEstatesareincludedinthetimepropa-
0 0
I′ =Z6I. Since this property does not persist in case of gation,theconvergedresultagreeswiththeoneobtained
the relativistic treatment, and the properly scaled solu- using the length gauge, whereas without the NE states
tions of the TDDE are thus not identical anymore, any theconvergedresultdiffersinthisexamplebyafactorof
deviation from the non-relativistic prediction based on about 1.5. Therefore, these states are absolutely neces-
the scaling relations can be classified as a relativistic ef- sary for obtaining the correct result.
5
From the practicalpoint of view the finding is also in-
teresting, since the omission of the NE states and thus 1
half of the total number of states reduces the numerical
efforts tremendously. Clearly, doubling the number of
0.8
states leads to an increase of the number of operations Z=40, L-gauge
per time step by a factor 4, since the number of transi- io Z=80, L-gauge
t
tion dipole matrix elements increases by this factor. In Ra0.6 Z=40, V-gauge
the present example (and for a given number of jmax) Z=80, V-gauge
the time propagation in length gauge without the NE
states is by about a factor of 6 faster compared to the 0.4
1 photon 2 photons 3 photons
one where the NE states are included. The additional
time difference (factor of about 1.5) arises from an in-
creasednumberoftime stepsrequiredforconvergencein 0.20 50 100 150 200 250 300
the used adaptive time propagation,if the NE states are λZ2, nm
included. In fact, the length-gauge time propagation by
itself is found to be about 6 times faster than the one
FIG. 2: Ratio of the ionization yields that are obtained by
performed in velocity gauge, even if the NE states are
either excluding or including the negative energy states in
included in both of them. This factor is due to the finer theexpansion (23) adoptingeitherlength (L)orvelocity (V)
timegridneededforconvergenceinvelocitygauge. Con- gauge. Thecalculationsareperformedforanionwithtwodif-
sideringbotheffectstogether,evenaspeed-upbyafactor ferent values of nuclear charge Z by varying the wavelength
of 50 is found when comparing the length-gauge calcu- λ of a 20-cycle cos2-shaped laser pulse with the peak inten-
lation without NE states and the velocity-gauge variant sity Z6 ×1013W/cm2. The character of the ionization pro-
with NE states that for sufficiently large value of j cess changes from single-photon ionization (λZ2 =40nm) to
max three-photon ionization (λZ2 =260nm).
both yield practically identical results.
However, Fig. 1 reveals also that the calculations in
velocity gauge converge faster with respect to the quan-
tum numbers j included in the calculation. Therefore, is, of course, required also in length gauge. In fact, al-
the efficiency gain of the length gauge is smaller than readyfortheparametersdiscussedinthecontextofFig.1
the value givenabove. Inthe concreteexample shownin wefindarelativedeviationbetweenotherwiseconverged
Fig. 1 the (within better than 0.1%) converged length- length-gaugeresultswithandwithoutNEstatesofabout
gaugecalculation(jmax =15/2,withoutNEstates)isby 2.2 × 10−5. This appears reasonable, since it is of the
afactorofabout40fasterthanthe(withinabout10−3%) order of the (reciprocal) rest energy c−2 ≈ 5.3 × 10−5.
converged velocity-gauge result (jmax = 11/2, with NE Forveryhighlaserintensitiesthe gainfromomitting the
states). Even compared to the only about 0.5% con- NE states is thus lost and future calculations will have
verged velocity-gauge result with jmax = 9/2 (with NE toshowwhetherinthatregimelength-gaugecalculations
states)thereisstillafactorofabout30intime gain. On can still profit from the need of a sparser time grid, or
the other hand, the question of the most efficient choice whetherthefasterconvergencewithj persistsinvelocity
of the gauge depends on the laser parameters. If few- gauge and may make this gauge more efficient.
photon processes, in the most extreme case one-photon
In [13] it was concluded on the basis of the numerical
ionization at low intensities, are considered, the average
results that the inclusion of the NE states is not cru-
angular momentum j transferred to the ion is small. In
cial for solving the TDDE, if the dipole approximation
such a case the need for a larger value of j in the
max is adopted. This result was found despite the fact that
length-gauge calculation could even overcompensate the
the calculation was performed in velocity gauge. There-
gain from the exclusion of the NE states. However, if
fore, this finding appears to contradict our conclusions.
the average number of absorbed photons increases (due
However, it turns out that the importance of the inclu-
to a smaller ratio of the photon frequency with respect
sion of the NE states depends in fact on the character
to the ionizationpotential ordue to a higherlaserinten-
of the multiphoton ionization process, i.e. on the num-
sity), the relative increase in j due to the slower conver-
ber of photons. Figure 2 shows the ratio of the ioniza-
gence leads to a smaller increase of the total number of
tion yields obtained either with or without inclusion of
states than the inclusion of the NE states. In this mul-
the NE states for both gauges and two values of the nu-
tiphoton regime calculations in length gauge appear to
clearchargeZ. Inagreementwith the findings discussed
be muchmore efficient. This finding for the TDDE solu-
above, the inclusion of the NE states has practically no
tion seems to differs fromthe one for the non-relativistic
influence for the length-gauge results, independently of
TDSEwherethevelocitygaugeisoftensupposedtocon-
the number of photons involved or the nuclear charge.
verge faster than the length gauge.
This changes, however, if the velocity gauge is used in
For extremely high intensities around and above the the time propagation. The importance of the NE states
critical field strength F = c3 ≈ 2.57 × 106 where real increases with the order of the multiphoton process (the
cr
pairproductionispossible,theinclusionoftheNEstates number of photons required for ionization) and with the
6
nuclear charge. Within the one-photon regime there is
1
practically no influence of the NE states. Thus the find- Non-relativistic, any Z
ing in [13] is confirmed that even in velocity gauge an
0.9
inclusionoftheNEstatesisnotrequiredforcalculations Z=40
in the dipole approximation. However,it is to be under- Z=50
0.8
sotitnnoeuoupdmhto.htaoQtnutiihtseossrueeffimficanierdkniantbgtlsoearispeapacllyhsoiontnhtloeytsthtoreotinohgneidzceaaptsieeonnwdechnoecnree- Ratio0.7 n yield10-4 ZZZ===867000
o
of the importance of the NE states on Z. For exam- 0.6 zati10-5
ple, for the wavelength λ = 260/Z2nm the exclusion of ni
o
I
the NE states leads to a three times smaller value of the 0.5 10-6
ionization yield for Z = 80, whereas for Z = 40 the de-
15 20 25 30 35 40 45 50
crease is only about 10%. The Z dependence can also 0.4
15 20 25 30 35 40 45 50
explainthenegligibleeffectofNEstateswithinthedipole
ω/Z2, eV
approximation found in Fig.1 of [13] where Z = 1 was
used. Furthermore, only the probability of remaining in
FIG.3: Ratiosoftherelativisticionization yieldstothenon-
the initialgroundstate wasconsideredandthis quantity
relativistic ones(shownin theinsert) obtainedfor 5different
is found to converge much faster than, e.g., the ioniza-
valuesofthenuclearchargeZbyvaryingthecarrierfrequency
tion yield. This couldpossibly make it also less sensitive ωofa20-cyclecos2-shapedlaserpulsewiththepeakintensity
to the NE states. of Z6×1011W/cm2.
B. Single-photon ionization
of a single photon with the energy 15Z2eV is not suf-
As has been discussed in Sec. IID, the solution of the ficient for ionization anymore and the character of the
TDSE for different nuclear chargesZ givesidentical ion- ionization process changes from single-photon to two-
ization yields, if the laser pulse parameters are scaled photonionization. For the consideredintensities the lat-
properly. For example, the insert in Fig. 3 shows the ter process possesses of course a much smaller proba-
non-relativisticionizationyieldsobtainedforanionwith bility. In fact, the ratio would be even smaller, if the
the nuclear charge Z exposed to a 20-cycle cos2-shaped ratio at 15Z2eV would correspond to pure two-photon
laser pulse with a peak intensity of Z6×1011W/cm2 for ionization. The finite spectral width of the adopted 20-
photon energies varying in the range between 15Z2eV cycle laser pulse allows, however, one-photon ionization
and 45Z2eV. Since the ionization potential of the ion to occur even at this energy and increases thus the ion-
is equal to 13.6Z2eV, the ionization should occur via ization yield. Therefore, a finer photon-frequency grid
absorption of a single photon. In order to study the rel- in between 15Z2eV and 20Z2eV would show a much
ativistic effects in this one-photonionization regime, the smoother behavior than the one visible from Fig. 3.
TDDE is solved for the same system and for 5 different
In [13] (Fig.5) relativistic effects were considered by
values of Z (in between 40 and 80 with a stepsize of 10).
a comparison of the ionization rates obtained with the
The ratio of the relativistic to the non-relativistic ion-
TDSE and the TDDE where the latter was solved for 8
ization yield is shown in Fig. 3. Two features can be
different values of Z. However, there the behavior was
observed. First, above 18Z2eV the ratio decreases with
studied as a function of the laser field amplitude (also
increasing photon energy. Thus relativistic effects are
scaled by the corresponding non-relativistic scaling re-
more pronounced for higher photon frequencies ω. As
lations) and for a fixed photon frequency. The authors
couldbeexpected,thiseffectbecomesmoreandmoreim-
discussstabilization,sincetheionizationratedoesnotin-
portant as Z increases. Second, a discontinuity develops
creasemonotonously with the field strength, but instead
with increasing Z at a photon energy of about 18Z2eV. decreases for intensities beyond about F = 1Z3. The
0
Especially for large values of Z the ratio for the lowest
for large values of Z increasing value of F for which the
0
considered photon energy drops down substantially for
ionizationratehasitsmaximumisthenexplainedbythe
increasingZ. This isa manifestationofthe factthatthe
stabilization criterion of Gavrila (see [34] and references
ionizationpotentialofthe ionincreasesasaconsequence
therein) and the conjecture that due to the lowerenergy
of the relativistic velocity of an electron in the vicinity
of the ground state in the relativistic case the condition
ofstrongCoulombpotentials. Theshiftoftheionization
for the occurrence of stabilization shifts to higher field
potential
strengths.
Z2 Z2 In fact, the dependence of the ionization rates on Z
∆I =Irel(Z)−Inr(Z)=c2 1− 1− − (26)
p p p c2 2 discussed in [13] may be quantitatively understood from
r !
the scaling relations together with the lowering of the
rapidlyincreaseswithincreasingZ. (Forexample,∆I ≈ ground-state energy due to relativistic effects. This is
p
Z4/(8c2) for small Z.) Thus, for Z = 80 the absorption illustrated in the following way. From the condition
7
0.15
Non-rel. Non-rel., Z
n rate 0.1 ZZZ===358646 ield, a.u.1100--45 NReolna-trievl..,, ZZ’
o y
Ionizati0.05 ization 10-6
n
o10-7
I
2 photons 3 photons 4 photons
0 10-8
0 0.5 1 1.5 2 2.5 3 0.06 0.08 0.1 0.12 0.14
Electric field Wavelength, nm
FIG.4: Ionizationrate(inunitsofZ2 a.u.) ofahydrogenlike FIG. 5: Relativistic (blue circles) and non-relativistic (black
ionexposedtoamonochromaticlaserfieldwiththefrequency circles) ionization yields obtained for an ion with the nu-
ω = Z2 as a function of the amplitude of the electric field clear charge Z = 50 exposed to a 20-cycle cos2-shaped laser
(givenin unitsof Z3 a.u.) Theblacksolid curvepresentsthe pulse with a peak intensity of 5×1022 W/cm2 and various
(non-relativistic) Floquet ionization rate, whereas the other laser wavelengths. Additionally, the non-relativistic ioniza-
curvespresenttheionization ratesobtainedusingthescaling tion yields (red squares) for an ion with the nuclear charge
relation28inordertoestimatetherelativisticionizationrates Z′ = 50.88 are shown whose non-relativistic ionization po-
for three different values of Z. tential is equal to the relativistic ionization potential of the
ion with Z = 50. The non-relativistic N-photon thresholds
(N = 2−4) for Z and Z′ are indicated by black and red
vertical dashed lines, respectively.
Irel(Z)=Inr(Z′) one finds a scaled nuclear charge
p p
Z′ = 2c2 1− 1−Z2/c2 , (27)
r (cid:16) p (cid:17) by anorderofmagnitude ormore. This is demonstrated
for a given true charge Z. This allows to estimate the in Fig. 5 where relativistic and non-relativistic ioniza-
relativistic ionization rate from a non-relativistic calcu- tion yields are compared for an ion with the nuclear
lation using charge Z =50. The laser wavelengthvaries in the range
from0.05nm(2-photonionization)to0.15nm(5-photon
Z′ 2 Z 3 ionization). Whereas the general structure of the wave-
Γ′(F )= Γ F , (28)
0 Z 0 Z′ lengthdependenceissimilarforbothrelativisticandnon-
(cid:18) (cid:19) (cid:20) (cid:21) !
relativistic treatments, the difference in the positions of
ifthedependenceofthenon-relativisticionizationrateon allpronouncedfeatureslikepeaksandminimamayresult
the photon frequency is ignored. In order to obtain non- in a substantial discrepancy, if the ion yields are com-
relativistic ionization rates Γ(F ), Floquet calculations paredatasinglephotonfrequency. Obviously,the sharp
0
were performed as a function of the peak amplitude F steps are due to the N-photon channel closings and the
0
of the electric field using program STRFLO [35]. They peaks are signatures of resonantly enhanced multipho-
areshowninFig.4andareinreasonableagreementwith ton ionization (REMPI) processes. Their positions are
theTDSEratesin[13]. Basedonthenon-relativisticFlo- directly related to excitation energies and the ionization
quetratesthescalingrelation(28)allowstoestimatethe potential, respectively, and both are higher in the rela-
relativisticratesthatarealsoshowninFig.4forZ =36, tivistic than in the non-relativistic case.
54, and 86. The qualitative agreement with the TDDE In view of the scaling relation (27) it is interesting to
results in Fig.5 of [13] is satisfactory and especially the compare the TDDE result (for Z = 50) with the non-
shift of the maximum to higher fields is wellreproduced. relativistic TDSE result for the scaled nuclear charge
However, in agreement with the discussion in [13] the Z′ = 50.88. As can be seen from Fig. 5, the ob-
model predicts that the maximum of the ionization rate tained TDSE results deviate from the relativistic ones
increases monotonically with Z. This is in contrast to for Z = 50 only by a few percent. This indicates that
the numerical findings in [13]. the change of the ionization potential is the by far dom-
inating relativistic effect, at least if multiphoton ioniza-
tionisdescribedwithinthedipoleapproximation. Other
C. Multiphoton ionization possible effects like the splitting of (resonant) intermedi-
ate states due to spin-orbitcoupling arethus verysmall,
If many photons are required for ionization, the rela- but can be quantified more easily, after the scaling re-
tivisticresultsmaydeviatefromthenon-relativisticones lation has accounted for the main effect. The dominant
8
influence of the shift of the ionization potential is likely cies are too low for allowing non-negligible real pair cre-
to depend on the considered laser intensity and photon ation. Ifionizationoccursviaabsorptionofasinglepho-
frequency. It should be reminded that the Keldysh pa- ton or the time propagation is performed in the length
rameter γ = 2I ω/F [36] varies in Fig. 5 in between form, the role of negative energy states is much less sig-
p 0
38.17and12.72. Thisisdeepinthe multiphotonregime, nificant.
p
in fact even in the perturbative one. Comparing solutions of the time-dependent Dirac and
Schr¨odinger equations for the same ion and laser pulses,
the relativistic change of the ionization potential is
IV. CONCLUSION demonstratedtodominateotherrelativisticeffectsinthe
caseofmultiphotonionization. Itisshownthatthiseffect
Single- and multiphoton ionization of highly charged is successfully accountedfor by a simple scalingrelation.
atomic ions has been numerically studied by a direct so-
lution of the time-dependent Dirac equation within the
dipole approximation. The stationary Dirac equation is
Acknowledgments
solvedbyprojectingtheradialpartontoaB-splinebasis
and the obtainedfield-free eigensolutionsare used in the
subsequent time propagation. Results for both length The authors acknowledge financial support by the
and velocity gauges for describing the ion-field interac- German Ministry for Research and Education (BMBF)
tion are obtained and compared. The inclusion of the within the SPARC.de network under grant 06BY9015,
negative-energyDiracstatesforthedescriptionoftherel- by the COST programme CM0702, and by the Fonds
ativistic dynamics is shown to be important in the case der Chemischen Industrie. This work was supported in
of the multiphoton ionization, if the velocity gauge is parts by the National Science Foundation under Grant
adopted, even if the considered intensities and frequen- No. NSF PHY05-51164.
[1] N. J. Kylstra, A. M. Ermolaev, and C. J. Joachain, 3973 (2005).
J.Phys.B 30, L449 (1997). [20] A. Palacios, H. Bachau, and F. Mart´ın, Phys.Rev.Lett.
[2] U. W. Rathe, C. H. Keitel, M. Protopapas, and P. L. 96, 143001 (2006).
Knight,J.Phys.B 30, L531 (1997). [21] Y. V. Vanne and A. Saenz, Phys.Rev.A 80, 053422
[3] G.R.MockenandC.H.Keitel,J.Comp.Phys.199,558 (2009).
(2004). [22] G. Sansone, F. Kelkensberg, J. F. Perez-Torres,
[4] G.R.MockenandC.H.Keitel,Comp.Phys.Comm.178, F. Morales, M. F. Kling, W. Siu, O. Ghafur, P. Johns-
868 (2008). son, M. Swoboda, E. Benedetti, et al., Nature 465, 763
[5] H.R. Reiss, JOSA B 7, 574 (1990). (2010).
[6] F. H. M. Faisal and S. Bhattacharyya, Phys.Rev.Lett. [23] Y. V. Vanne and A. Saenz, Phys.Rev.A 82, 011403
93, 053002 (2004). (2010).
[7] M. Klaiber, K. Z. Hatsagortsyan, and C. H. Keitel, [24] M. Awasthi, Y. V. Vanne, A. Saenz, A. Castro, and
Phys.Rev.A 73, 053411 (2006). P. Decleva, Phys.Rev.A 77, 063403 (2008).
[8] F. H.M. Faisal, J.Phys.B 42, 171003 (2009). [25] S.Petretti,Y.V.Vanne,A.Saenz,A.Castro,andP.De-
[9] N. Milosevic, V. P. Krainov, and T. Brabec, J.Phys.B cleva, Phys.Rev.Lett. 104, 223001 (2010).
35, 3515 (2002). [26] I. Grant, Relativistic Quantum Theory of Atoms and
[10] C. H. Keitel and P. L. Knight, Phys.Rev.A 51, 1420 Molecules, vol. 40 of Atomic, Optical, and Plasma
(1995). Physics (Springer, New York,2007).
[11] L. N. Gaier and C. H. Keitel, Phys.Rev.A 65, 023406 [27] W. R. Johnson, S. A. Blundell, and J. Sapirstein,
(2002). Phys.Rev.A 37, 307 (1988).
[12] H. G. Hetzheim and C. H. Keitel, Phys.Rev.Lett. 102, [28] G. W. F. Drake and S. P. Goldman, Phys.Rev.A 23,
083003 (2009). 2093 (1981).
[13] S. Selstø, E. Lindroth, and J. Bengtsson, Phys.Rev.A [29] V. M. Shabaev, I. I. Tupitsyn, V. A. Yerokhin, G. Plu-
79, 043418 (2009). nien, and G. Soff,Phys.Rev.Lett.93, 130405 (2004).
[14] M. S. Pindzola, J. A. Ludlow, and J. Colgan, [30] K. Beloy and A. Derevianko, Comp.Phys.Comm. 179,
Phys.Rev.A 81, 063431 (2010). 310 (2008).
[15] X. Tang, H. Rudolph, and P. Lambropoulos, [31] I. P. Grant, J.Phys.B 42, 055002 (2009).
Phys.Rev.Lett. 65, 3269 (1990). [32] Y. V.Vanneand A. Saenz, J.Phys.B 37, 4101 (2004).
[16] X. Tang, T. N. Chang, P. Lambropoulos, S. Fournier, [33] D. V. Fursa and I. Bray, Phys.Rev.Lett. 100, 113201
and L. F. DiMauro, Phys.Rev.A 41, 5265 (1990). (2008).
[17] P. Lambropoulos, P. Maragakis, and J. Zhang, [34] M. Gavrila, J.Phys.B 35, R147 (2002).
Phys.Rep. 305, 203 (1998). [35] R. M. Potvliege, Comp.Phys.Comm. 114, 42 (1998).
[18] T.Nubbemeyer,K.Gorling,A.Saenz,U.Eichmann,and [36] L. V. Keldysh,Sov.Phys.JETP 20, 1307 (1965).
W. Sandner,Phys.Rev.Lett. 101, 233001 (2008).
[19] M. Awasthi, Y. V. Vanne, and A. Saenz, J.Phys.B 38,
