The atomic behavior in a strong converging laser field of circular polarization

Optics Communications91 ( 1992) 474-480 North-Holland

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Full length article

The atomic behavior in a strong converging laser field of circular polarization Wenqing Z h a n g a n d Weihan T a n Shanghai Institute of Optics and Fine Mechanics, Academia Sinica, P.O. Box 800-211, Shanghai, 201800, China Received 23 September 199l; revised manuscript received 30 December 1991

We studied the atomic behavior in a strong converging laser field of circular polarization. We first proved that the modified potential in the range of r ~ Ro for circular polarization. The energylevelsof the atom under the influence of the laser field decrease and split, the wavefunctionundergoesa drastical stretching in space. Some related problems will be also considered here.

1. Introduction

It has received attention early that an atom placed in a strong laser field undergoes structure distortion [ 1-3 ]. Recently, as the laser has generated such a superintense field that it is comparable with the electrostatic field created by a nucleus in an atom, this effect has stimulated people's interests again [4-7 ]. Under the circumstance of an intense field, the standard perturbative theory is invalid, the alternative is to develop nonperturbative formulas to approach the problem. Up to now, people have used many complicated methods to deal with the atomic behavior in an intense laser field [ 6-10 ]. Pont et al. [6,7] used the dressed potential method developed by Gavrila and Kaminski [ 11 ] to study the atomic behavior in a strong laser field of circular and linear polarization. They considered the problem with one beam of a plane laser field of linear or circular polarization. When adopting "high-frequency approximation", the modified potential is determined by a line segment of charges (linear polarization) or ring charges (circular polarization) [6]. In their early works, for simplification, they adopted the "decoupled/-channels approximation" and obtained a spherical symmetric potential summation [ 7 ]. In this paper, we have studied the atomic behavior in a strong converging laser field [ 14] of circular polarization, which can be produced by an 474

optical focusing system. It is shown that the modified potential is decided by a spherical surface, on which the nuclear charges distribute uniformly, with radius of Ro. Based on such modified potential, we compute the hydrogen atomic energy shift and wavefunction distortion. The results are that the energy levels decrease and split, and that the wavefunction undergoes stretching in space drastically. The arrangement of this paper is as follows. In sec. 2 we first present the detailed description of the model of an atomic modified potential in a converging laser field. In sect. 3 a method for the computation of the energy eigenvalue and eigenfunction in a laser field is given. Then in sec. 4 we list numerical results, and make a comparison of our results with those of others. We finally discuss some related problems.

2. The modified potential of an atom in a strong converging laser field

When applying a phase-factor transformation [ 1,10 ] on a wavefunction, the Schr'Odinger equation describing the motion of an electron in an atom in a laser field can be written as ~mmV:+ V ( r - a ( t )

(I)

0030-4018/92/$ 05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.

with

e ~t a(t)= ~ A ( t ) dt,

(2)

where A ( t ) is the vector potential of the incident electromagnetic field. V ( r - a ( t ) ) is the expression of the atomic Coulomb potential transformed by r - - , r - a ( t ). In this paper, we assume that the atom is illuminated by a laser field which is a nearly spherical symmetric converging laser field produced by an optical focusing method from multiple-beam lasers. Such kind of laser field had been used in the study of inertial confined fusion [ 15 ]. A schematic diagram for obtaining such a converging laser field is shown in fig. 1. For the atom in the converging laser field, we must average V ( r - a ( t ) ) over the solid angle 12 decided by the vector a(t) and r (i.e. all the laser propagation directions) to get the effective potential. Thus, 4~

l

V(r) = -~ f V ( r - a ( t ) ) d~2.

(3)

0

For the laser field of circular polarization,

.4 ( t ) = ( a / x/~ ) [ i cos ( t o t - Z ) + j sin ( t o t - X ) ] ,

(4)

a(t) =Ro[isin(a~t-X) - j c o s ( o J t - Z ) ] ,

(5)

with Ro-

1 e a l _ m c V /e~

I 1/2 to - 2

v T ~ rnc to

°

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Volume 91, number 5,6

(6)

the parameter R0 represents the quiver motion of the classical electron in the field. Z is a random and irrelevant phase factor with respect to each laser propagation direction. Inserting eq. (5) into eq. (3), we get the effective potential,

V(r) = - Ze2 4n X

dO 0

d~ 0

sin 0 ' [ r2 + R 2 - 2rRo sin Ocos ( t o t - X - ~ ) ]'/2'

(7)

where 0, ~0is the angle decided by the solid angle 12. It is proved that the ir~tegral result of eq. (7) (see appendix) is,

~'(r)=-Ze2/Ro,

(r~Ro) ,

= -Ze2/r,

(r>Ro) .

(8)

Z is the nuclear charge. It is obvious that the modified potential becomes a uniform field characterized by Ro in the range ofr~Ro, and also a Coulomb field in r> Ro. Thus the potential is the same as that of a spherical surface on which the nuclear charges distribute uniformly, with radius Ro. In fig. 2, the potential configuration for the laser field of circular polarization is depicted. By comparing the exact potential (8) without taking any approximation in the strong converging laser field with the spherical potential, which was obtained by taking the high-frequency and decoupled /-channels approximations in a plane laser field [ 7 ], we find that Pont's approximatical spherical potential containing many terms has a much complicated form. Only in the case of quantum number l=0, it is the same as ours.

\/ 9(0 Ro

I

Ze 2 RO

Fig. I. Schematic diagram to obtain a nearly spherical symmetric converging laser field from multiple-beam lasers, and the interaction picture of an atom with the converging laser field. A, B, C, D: laser beams, a, b, c, d: optical lens. l: atom.

Fig. 2. Modified atomic potenlial configuration in strong converging laser field of circular polarization.

475

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3. The computation of energ~ eigenvalue and eigenfunctions

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RO

f,F---i--~R,,t(ot,,r)R,.,l(o6,,r)

A,,,,, =

r2dr,

0

The time-independent Sch iSdinger equation of an electron in a spherically sym~ aetric potential IV(r) is

Ro [,t

B,,,,, = j R,,t(ot,,r) R,,t(ot,,,r) rdr.

(15)

0

- - - V 2 + IV(r) q/=E~u, 2m

(9)

where E is a new eigenvalue i nd q/is the corresponding eigenfunction. For an ek ztron in the spherically symmetric potential 17(r), th ~angular momentum is conservative, so the wavef ruction V may be expanded with the hydrogen at( m wavefunction V.t~ (r, 0, ~o) with the same l [16]. Are let

Letting

flt=2rn/h 2 , p2=c~2=2Z/ao, eq. (14) can be expressed as

~ C,.{ f12 (A,,,,,-B,,,,,)+ 1 1

)

= [EIC,,.

~'= Z c.~,.,,,,(r, o, ~o) nhn

= YT'(0, ~o) ( ~ C,,R,,t(ot,,')).

(10)

Using the equation sati: fled by YT'(O, (o) and R~j(a,r), we get the coeffic ent equation by substituting eq. (10) into eq. (9),

c°r(L\F(r) where

.---R 0 ,

(r>R0) ,

(r<~Ro) ,

1 1 4/Yl

a ,2=

Z2e 2

1

2ao n 2'

(17)

which are just the eigenlevels of the hydrogen atom [16].

4. Numerical results

4.1. Energy eigenvalues

2-

2mZ~ 2 ah 2 ,

2Z a,-

nao

.

ao is the Bohr radius, and w,., have taken into account that E = - I E I for the bind ing energy level. By making the transforr lation f~rZR,,t(a,,r)dr in eq. ( 11 ), we get

T. C.[a~(A..,-B...)+.~(,~-o~2),~...]=O, n

(14)

476

....

(12)

(13)

with

E=-IEI

In this section, we find the modified atomic energy levels and wavefunction by diagonalizing the coefficient matrix of eq. (16).

and

012__ 8m[EI hE ,

From here on, we can compute the eigenvalue and eigenfunction of the atom in a given laser field. As Ro=0,

n~n') + ] ( c 2 -ot2) ] Rnt( oGr) =O , (11)

F(r) =r,

(16)

In fig. 3, we give the hydrogen atomic levels of the principal quantum number n = 1-4 for different values of Ro. It is found that the atomic binding energy decreases as the laser intensity increases. The smaller the quantum number is, the larger the energy levels shift. It is because the state with smaller quantum number is nearer to the nucleus, that a change of the potential in r
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Volume 9 l, number 5,6

1.0

1.C~

n=l 1=0

n=2 (a)l=l

I" .5

o.s

i.

5

10

15

20

5

10

Ro(BOhr R a d i u s )

15

20

Ro(BOhr R a d i u s )

1 .O

sl.O

5

5 =

(a)l=3 (b)l=2

rl=3 (a)l=2 (b)l=l (c)l=O

(c)l=l

(d)l=O .

5

10

15

20

Ro(BOhr Radius)

5

F

10

i

15

20

Ro(BOhr Radius)

Fig. 3. Energy shift versus Ro (in Bohr radius). E. is the atomic eigenvalue with Ro=0. with respect to the magnetic quantum number m is still degenerate.

4.2. Energy eigenfunction Apart from the energy shift, the wavefunction for all q u a n t u m states will undergo distortion. In fig. 4, we give out the configuration o f the wavefunction o f n = l, n = 3 ( l = 0 - 2 ) with different Ro. It is obvious that the wavefunction spreads in space.

In addition, we calculate the average atomic radius gfor n = 1-4 with R 0 = 0 - 2 0 (in Bohr radius) as given in fig. 5, and using ? as a criterion o f state expanding. For the state with same principal quantum number n, the state with smaller angular quantum number l expands more.i But, for the state with different n and same l, the 18rger n is, the more the state expands. This is because the electron distribution with smaller r expands t~ outer space much quicker as the node density o f the wavefunction increases 477

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1 August 1992

l" n=l,l=O

,2 '

(a)Ro=O. (b)Ro-5. (C)Ro=lO. (.d)Ro=20.

i a)

..o ~ (c) I . (5

no3:l=o

J

A

1

/\

'a>R°'°" --/--~

/i ",,.

%~.2

~o

(C)Ro=lO" '°'°" (d) RO=20.

a

L.

/

/\

I

.4

'-',

!

;

•

I

',

"-"---_____.-."---'-

'' , @

20

J'O r(B(

20

Ir

F(BOhF

~ b

n=3.1 =2 (a)Ro= 0 .

c

d

(d)R°='O"

Radii)

(d}

~=3,1=1 a

(a)Ro=O'

•

(b)Ro=5. (C)Ro=IO '

I

!

:

..oi

(b} ,

1.6

40

20

Radii)

(b)Ro=5, I-~

1.6 • ~~

,

(C)Ro=IO"

,d,Ro-.O

,,, -

f,;., ' ,

=

!

/ ,' \',

ho 1.2.

~.

b

•~

•

.8,

d

;',.

/

',

/,/ • ,g

.~-1

•

•

•

i

•

•

1

•

r(B(

ir

/

|

40

20

0

Radii)

20

40

r ( Bobr

Radii)

Fig. 4. Wavefunction distortion w th different values of Ro (in Bohr radius). (a) Ro= 0, n = 1, 1= 0. ( b ) Ro= 5, n = 3, 1= 2, (c) Ro= 10, n=3, I=0, (d) Ro= 20, n=3, l= 1 with increasing n. We can u lderstand such behavior from figs. 4c, d.

paring with those of N = 20, the relative deviation is lower than 2% for a binding energy in the range o f our numerical simulation.

4. 3. Numerical deviation Taking R,t(ot,r) as basis, we find the energy levels and wavefunction o f the at )m in a converging laser field by finding the eigenvl lue and eigenfunction o f the coefficient matrix. O f c )urse, the numerical precision is limited by the nm aber N of the basis used. We computed s o m e group,, o f results with different N. The results given here belong to N = 2 5 . Corn478

5. Discussion and conclusions In the first place, we gave a comparison of our modified atomic potential in a converging laser field with that in a plane laser field [ 1-2,6 ]. The m o s t important feature o f our model is that we get an exact spherical symmetric modified atomic potential with

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Volume 91, number 5,6

2./*

2.0

1.6

o '~

1.2

..5,

I0

9-0

Ro~BOtLr Radii )

Fig. 5. Averagedorbit radius ~versusR0. no approximation, which is not the case in a plane laser field. For a converging laser field of circular polarization, because the laser field acts on the atom isotropically in space, the time factor is annihilated by the integral over all directions of laser propagation. This makes the time average of the potential unnecessary in our model, and we have given a clear explanation in the appendix. In other works, the potential is averaged over the optical period and the high-frequency components are neglected [1-2,6]. Also, by taking the "decoupled/-channels approximation" to neglect the off-diagonal matrix elements in the same time [ 7 ], Pont's spherical symmetric potential is just an approximatical result. Just due to the spherical symmetry of the modified atomic potential, we observe some new atomic behavior in the external field. First, at given Ro, i.e. given laser wavelength and intensity, the atomic energy shift in the converging laser field is larger than that of the corresponding atomic level in the plane laser field [2,6 ]. Second, in the converging laser field, the atom size increases isotropieally in space, which is different from that the atom undergoes a dramatic

1 August 1992

anisotropic distortion of shape, and the atomic wavefunction also manifests anisotropy in the plane laser field [6 ]. Third, the azimuthal and magnetic quantum number can also be used to indicate the quantum state of an atom in a converging laser field, only the principal quantum number n is not a good quantum number because the new wavefunction includes all the atomic states with n >i l + 1. However, for the atom in a plane laser field, all of the three parameters n,/, m are notlgood quantum numbers [6]. Starting from eq. (6), we can discuss directly the relation between the atomic structure distortion and laser intensity and frequency. For a laser of wavelength 1.06 Jim, and intensity I = 1 × 1013 W/cm -2, the corresponding Ro is equal to 9.1. From fig. 3 and fig. 5, we see that the change of the atomic binding energy and the averaged atomic radius g are so large that the influence of the strong laser field on the atom must be considered when dealing with the problem related to the interaction process of an atom with a strong laser field [ 12,13 ]. So far we adopted an ideal model and neglected the influence of possible experimental conditions. In the first place, the high-intensity laser can be achieved only in the form of short pulse, so the infinite planewave description of the radiation field is just an idealized situation. But this is not a difficulty as discussed by another researcher [6]. Secondly, for the converging laser field, there is a phase difference or phase fluctuation of all the laser beams with different propagation directions. It is fortunate that such an effect has no influence on the results as proved in the appendix. Thirdly, !he atom is in fact always in random motion, and d0es not rest in the laser field as in our model. This will make our results have some deviation. The best situation is to diminute the atomic random motion to one's best in the experiment [ 17 ]. It is interesting to note that our model has some similarities wilh the "optical molasses" in laser cooling of an atom [ 18 ]. Perhaps, such factor could decrease the random motion of the atom itself.

Appendix The integral result of eq. (7) Rewrite the formula of eq. (7), 479

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dO o

1 August 1992

I= I p,(o)

d~o

i2; dX

o

0

dOsinOP,(cosO)

0 2n

sin 0 × [r2+R 2-2rRo sin 0co (mt-X-~o)

1 " (n-m)!p~(o) + ~--~m~=l ( n + m )-------~.

]1/2"

~ dXcosmX 0

(A.1) Noting that the influence o the result o f eq. (A. 1 ) is an over angle (0, qT), such a cor by a numerical m e t h o d by t; as a r a n d o m p a r a m e t e r wh~ over (0, ~0). We have

('(r) = - Ze2 4~

dO o

the factor ( t o t - Z ) on ihilated by the integral lusion could be p r o v e d ring the factor ( t o t - X ) i integrating eq. (A. 1 )

(A.2)

1

[r2+R~-2rRo sin 0cos X] ~

~

P.(sinOc sX),

~

Pn(sinOco X ) ,

(r~Ro), (r>Ro). (A.3)

F r o m eqs. ( A . 2 ) a n d ( A . ) , we see that the result o f eq. ( A . l ) is decided by n integral as follows,

I= ~

dO 0

dXsinOPn(s: L 0 c o s X ) .

(A.4)

0

Making use o f the p r o p e r , o f the Legendre polynomials, we have

P~(sinOcosX)=P~(O) P,( os0) "

(n-m)!

+ 2 m~__1 ( n + m)~ P ~ ( 0 )

're(cos 0) cos (reX) .

(A.5) Substituting eq. ( A . 5 ) into eq. ( A . 4 ) , we have

480

It can be seen that the result o f e q . (A.6) is

I=1,

(n=0),

=0,

(n~0).

(A.7)

o

We can expand the integral f action ofeq. (A.2) with Legendre polynomials,

-

(A.6)

0

Inserting eq. (A.7) into eqs. (A.2) a n d ( A . l ) , we get the result as given by eq. ( 8 ) in the article.

dX

sin 0 × [r2+R2-2rRo s i n 0 c c X] 1/2"

-

X i d 0 P T ( c o s 0) sin 0 .

References [ 1] W.C. Henneberger, Phys. Rev. Ixn. 21 (1968) 838. [ 2 ] J.I. Gerstem and M.H. Mittleman, J. Phys. B 9 (1976) 2561. [3] Chan K. Choi, Walter C. Henneberger and Franck C. Sanders, Phys. Rev. A 9 (1974) 1895. [4] D. Normand, L.-A. Lompre, A'Huillier, J. Morellec, M. Ferray, J. Lavancier, G. Mainfray and C. Manus, J. Opt. Soc. Am. B 6 (1989) 1513. [ 5 ] S. Liberman, J. Pinard and A. Taleb, Phys. Rev. Lett. 50 (1983) 888. [6l M. Pont, N.R. Walet and M. Gavrila, Phys. Rev. A 41 (1990) 477; Phys. Rev. Lett. 22 (1988) 939; M. Pont, Phys. Rev. A 40 (1989) 5659. [ 7 ] M. Pont and M. Gavrila, Phys. Lett. A 123 ( 1987 ) 469. M. Pont, M.J. Offerhaus and M. Gavrila, Z. Phys. D 9 (1988) 297. [8] Shih-I Chu and J. Cooper, Phys. Rev. A 32 (1985) 2769. 19] K.L. Kulander, Phys. Rev. A 35 (1987) 445. [ 10] F.H.M. Faisal, J. Phys. B 6 (1973) L89. [ 11 ] M. Gavrila and J.Z. Kaminski, Phys. Rev. Len. 52 (1984) 613. [ 12] M. Pont and M. Gavrila, Phys. Rev. Lett. 65 (1990) 2362. [ 13] V.C. Reed and K. Burnen, Phys. Rev. A 42 (1990) 3152. [ 14] W.Q. Zhang and W.H. Tan, Acta Optica Sinica 12 (1992) 342. [ 15] T.P. Hughes, Plasmas and laser light (Adam Hilger Ltd., Bristol, 1975) p. 411. [ 16 ] L.I. Schiff, Quantum Mechanics (3rd Ed. ) (McGraw-Hill, New York, 1968) p. 88. [ 17 ] W.D. Phillips, P.L. Gould and P.D. Lett, Science239 (1988) 877. I181 S.C. Chu, L. Hollberg, J.E. Bjorkholm, A. Cable and A. Ashkin, Phys. Rev. Lett. 55 (1985) 48.