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Intensity dependence of the photoelectric effect induced by a circularly polarized laser beam
17June 1996
PHYSICS
ELSEVIER
Physics Letters A 216 (1996)
LETTERS
A
125-128
Intensity depe~de~~~ of the p~otoe~ect~c effect induced by a circufarly polarized laser beam * Qi-Ren Zhang
’
CCAST {World Laboratory),
Department
PO. Box 8730, Beijing 100080. China of Technical Physics, Peking University, Beijing 100871. China 2
Received 12 September 1995; revised manuscript received 27 February 1996; accepted for publication 21 March 1996 Communicated by P.R. Holland
Abstract
We show: ( 1) The photoelectric effect induced by a circularly polarized light beam is exactly equivalent to a quantum transition induced by a time independent perturbation between stationary states. The equivalent energy level of the target system is then light intensity dependent. (2) This energy shift, in the dipole approximation, is similar to that of a combined Zeeman-Stark effect in magnetic and electric fields perpendicular to each other. PACS:
32.90.ta;
33.60.-q
According to standard quantum mechanics course, the energy of photoelectrons is determined by the frequency but independent of the intensity of the incident light beam. This was a crucial point in the important advance of quantum theory made by Einstein [ I] at the beginning of this century. Now we know this is true only for the photoelectric effects induced by weak light beams. The energy level and the structure of an atom themselves may be disturbed by an intense light beam. Therefore the energy and the angular distribution of the photoelectrons ejected by a laser beam may be dependent on the beam intensity. This intensity dependence is an interesting nonperturbative effect, which has attracted considerable experimental and theoretical attention. For a few recent examples,
see Refs. [ 2-5 1, and references therein. To check this effect quantitatively one has to solve the Schriidinger equation for electrons in an atom interacting with an intense light wave, The problem of electron motion in a light wave was solved 60 years ago [ 61, but its motion in an atom and under light interaction is still not solved satisfactorily. This is because of the time dependent character of the problem. However, this problem may be solved for the photoelectric effect induced by a circularly polarized laser beam, since a time dependent rotation reduces the problem to a time independent one. Consider an electron in a central potential field V(r) and interacting with a circularly polarized electromagnetic plane wave described by the vector potential A=
*Work
partty supported by the Natural
China and by Science Foundation of Nuclear Indus~,
’ E-mail:
- wt) +ynsin(kz
- wt)]
(1)
China.
in the Coulomb
gauge. The HamiItonian
is
zhangqr~sun.ihep.ac.~n.
H=He+H’,
2 Mailing address.
1X75-9601
A[xocos(kz
Science Foundation of
/96/$12.00
@
PI1 SO375-9601(96)00259-9
1996 Elsevier Science B.V. AR rights reserved
(2)
(3)
H'=
e2A2 ~[p~cos(kz-ot)i-p+in(kz-~t)]+~. m (4)
Denote the nth bound eigenstate of Ho by in). Suppose that the electron was in the state 1~)at t = -co, and the interaction H’ switched on adiabatically afterwards. In the interaction picture, the interaction Hamiltonian is H/ = I
in which p is the density of final states at Ef = Ei. This is Fermi’s golden rule in our case. The energy of the photoelectrons is
elrNir/h)fH/e-i(HO/“)’
=
ei(H~/A)tH//e-i(H~/lZ)r,
(5) H,$= Hoi-wL,,
(6)
L, is the z component of the orbital angular momentum L for the eIectron. According to the theorem of GelI-Mann and Low [7], the evolution from t = -00 to t = 0 changes the state of the electron from In) to the corresponding eigenstate ii) = E, _ ~~ + i~H”li’
(8)
Efo=Ef-~fiiW=Ei-~Fiw.
(13)
Ei is the effective energy of the electron immersed in the laser beam. To be the eigenvalue of the effective Hamilton~an (9), it is apparently laser intensity dependent. The condition Ef = Ei could be satisfied by the state ]fO) with a definite quantum number ,u. If for the state ]fO) the quantum number ,u is uncertain, its projection IfOp) on the state space with definite p should be substituted. Suppose that the asymptotic form of the out-state IfO) is a plane wave of unit amplitude, the differential cross section of the photoelectric effect is
i ~cAlii)f,
+p,,sin(kz)
in which a = l/137.0 is the fine structure constant, u = hike/m is the velocity of the photoelectron, and k, = d-/A. In the coordinate representation, ( 14) takes the form
of the effective ~amiItonian
da He = H; + H”,
(9)
with the eigenvalue Ei. L:is a positive infinitesimal. On the other hand, in the time interval 0 < t < cm, the state If) satisfying the Lippmann-Schwinger equation
(14)
--
ii?i-
al
16~ffu li2w
=
(
-i)‘eiq’
2
O” Ulna c ~=I.4
(15)
1
syyl;(&Q e
x [pxcos(kz)
+p!,sin(kz)
+ ieA]Wi(r)dr.
(10) evolves to the state IfO), which is an eigenstate of HO and Hk with eigenvabres E, and Ef = Efo -4 p&w, respectively. ,Uis the magnetic quantum number of the state /fO). According to the formal theory for collisions (see for example Ref. [8]), the transition amplitude induced by the laser beam is
(16) Here, Wi(r) is the coordinate representation of ii), r, B and (b are spherical coordinates of r. Fl( k,r) is a stationa~ radial wave function satisfying the boundary condition Fl(O) = 0 and the asymptotic form Fl(k,r)
-sinfk,r-
when r + 00, (,fli) = -2riS(
Ei - Ef) (fOl H”li).
7 LT
(17)
(11)
In the derivation, Eqs. (8) and (IO) together with eigenequations of He and Hk have been used. The transition probabiIity per unit time is P = ,!{folH”lij12p,
~In.+~~+~ln(Zk,r)l,
(12)
for an electron in the central field V(r). The term containing x = Zamc/i”Lk, is a contribution of the long Coulomb tail -Zfica/r in V(r). l&(&b) is the spherical ha~onic function. The quantum number p connected with the number -I* of photons absorbed in the photoelectric effect as shown in (13) now is
Q.-R. Zhnng/Physics
Letters A 216 (1996) 125-I28
connected with the angular distribution of photoelectrons. This twofold meaning of ,u is a result of angular momentum conservation. The generalization of this formulation to a multielectron system is trivial. To consider the electron spin, one needs only to replace Lz by J,, which is the z component of the total angular momentum J of the system. Our problem of solving the photoelectric effect induced by a circularly polarized light beam is thus reduced to a usual time independent quantum mechanical problem, only the potential energy is moditied. Consider the problem in more detail under the dipole approximation, which becomes exact in the long wave limit with k = 0 for light beams. This approximation is very good for the photoelectric effect induced by visible light on light atoms. In this approximation
He=
2 +V(r) +
e2A2
wJ,
Its eigcnvalue problem fective Hamiltonian H,,
=
+
spr+--_.
is equivalent
CI”Ax:‘hHee-'PA*ih =
(18)
to that of the ef-
11-7
This theorem shows a simple relation exactly in the long wave limit of the laser beam. It provides a way to check the nonperturbative dependence of the photoelectron energy on the laser intensity for a spin singlet target atom, by directly comparing two experiments instead of comparing the experiment with a theoretical calculation. The nonperturbative theoretical calculation is circumvented. This is a great advantage of the method. However, to compare the ~~rnan-Stark cffeet with the photoelec~ic effect induced by a visible light we need a magnetic field as high as 10” T. This is far from attainable at the moment. Therefore, in comparison, we have to observe the photoelectric effect of a far infrared laser on the highly excited atoms. This is still not easy, but it is already sensible. For an atom with nonsinglet spin, the use of this theorem is complicated by a correction of the Land6 factor for the Zeeman effect. Under the same dipole approximation
e&., H” = ------I-m
e2A2
( 20)
2m ’
the selection rules for its nonzero matrix elements
HO + w.17 + eAwy.
(19)
Here Ho is expressed in (3) for a single valence atom, but should be a sum of these kinds of single electron expressions and the Coulomb repulsion between electrons for a multi-valence atom. In the latter case, \’ in the last term of the last equation should also be understood to be a sum of y-coordinates of valence electrons. For a spin singlet atom ( 19) is formally identical to its Hamiltonian in a magnetic field of strength 2mo/e in the z-direction and an electric field of strength Aw in the y-direction. Since the modified energy Et of the atom interacting with the laser beam should be the eigenvalue of this effective Hamiltonian, we have the following theorem. Theorem of similarity. In the dipole approximation, the energy shift of an atom interacting with the circularly polarized laser beam (1) equals that in a combined normal Zeeman-Stark effect, with the magnetic field of strength 2mo/e along the direction of laser propagation, and the electric field of strength Aw perpendicular to it.
AI =0,&l,
A~=O,&l.
arc (21)
Therefore, to make the photoelectron energy ( 13) with arbitrary integer -p possible, we need an in~nitive series
for the initial wave function. lution for the Zeeman-Stark eigenfunction
But, as m the usual soeffect, the approximate
of Hef is a sum containing only a few harmonic terms each with definite quantum numbers 1 and F. However. corresponding to the unitary transformation ( 19) 01‘ the effective Hamiltonian from H, to H,f, there IS 3 unitary transformation of the wave function qj(r)
= e+A”‘n&(r).
It is the factor
(24)
128
Q.-R. Zhang/Physics
e-ieA.r/h
=
47r E
that makes Y;(r) with
Letters A 216 11996) 125-128
number /_Lcompatible with this condition can be really observed. Since in the dipole approximation j,( kr) = j/(O) = 810, and in the plane wave approximation for the outgoing wave of photoelectrons Fl( k,r) /k,r = j~(k,r), the integralsin thecoefficients (16) can all be calculated analytically by an eiementary method. The
i’jl (eAr/ti)
an infinitive
harmonic
series (22),
photoelectron angular distribution can be worked out whenever a comparison with experiment is planned. References
x 5zp2 ( ~rn)h2(eAr/Wur,,,(r),
(26)
where jl is the spherical Bessel function of order 1. Of course, the factor j12(eAr/ A) in (26) shows that only terms with 12 6 eAa/A are important, a is the linear dimension of the electron orbit in the atom. This means that only a photoelectron with a quantum
[I] A. Einstein, Ann. Phys. (Leipzig) 17 (1905) 132. f2f P. Krstic and Y. Hahn, Phys. Rev. A 50 ( 1994) 4629. [3] R.R. Jones, D. You and PH. Bucksbaum, Phys. Rev. Lett. 70 (1993) 1236. 141 D.-S. Guo and G.W.F. Drake, Phys. Rev. A 45 ( 1992) 6622. [5] P.H. Bucksbaum,D.W. Schumacherand M. Bashkansky, Phys. Rev. Lett. 61 (1988) 1162. 161 D.M. Volkov, 2. Phys. 94 (1935) 25. [7] M. Gell-Mann and E Low, Phys. Rev. 84 ( 1951) 350. 18] D. Luri&, Particles and fields (Interscience, New York, 1968) Ch. 7.
PHYSICS
ELSEVIER
Physics Letters A 216 (1996)
LETTERS
A
125-128
Intensity depe~de~~~ of the p~otoe~ect~c effect induced by a circufarly polarized laser beam * Qi-Ren Zhang
’
CCAST {World Laboratory),
Department
PO. Box 8730, Beijing 100080. China of Technical Physics, Peking University, Beijing 100871. China 2
Received 12 September 1995; revised manuscript received 27 February 1996; accepted for publication 21 March 1996 Communicated by P.R. Holland
Abstract
We show: ( 1) The photoelectric effect induced by a circularly polarized light beam is exactly equivalent to a quantum transition induced by a time independent perturbation between stationary states. The equivalent energy level of the target system is then light intensity dependent. (2) This energy shift, in the dipole approximation, is similar to that of a combined Zeeman-Stark effect in magnetic and electric fields perpendicular to each other. PACS:
32.90.ta;
33.60.-q
According to standard quantum mechanics course, the energy of photoelectrons is determined by the frequency but independent of the intensity of the incident light beam. This was a crucial point in the important advance of quantum theory made by Einstein [ I] at the beginning of this century. Now we know this is true only for the photoelectric effects induced by weak light beams. The energy level and the structure of an atom themselves may be disturbed by an intense light beam. Therefore the energy and the angular distribution of the photoelectrons ejected by a laser beam may be dependent on the beam intensity. This intensity dependence is an interesting nonperturbative effect, which has attracted considerable experimental and theoretical attention. For a few recent examples,
see Refs. [ 2-5 1, and references therein. To check this effect quantitatively one has to solve the Schriidinger equation for electrons in an atom interacting with an intense light wave, The problem of electron motion in a light wave was solved 60 years ago [ 61, but its motion in an atom and under light interaction is still not solved satisfactorily. This is because of the time dependent character of the problem. However, this problem may be solved for the photoelectric effect induced by a circularly polarized laser beam, since a time dependent rotation reduces the problem to a time independent one. Consider an electron in a central potential field V(r) and interacting with a circularly polarized electromagnetic plane wave described by the vector potential A=
*Work
partty supported by the Natural
China and by Science Foundation of Nuclear Indus~,
’ E-mail:
- wt) +ynsin(kz
- wt)]
(1)
China.
in the Coulomb
gauge. The HamiItonian
is
zhangqr~sun.ihep.ac.~n.
H=He+H’,
2 Mailing address.
1X75-9601
A[xocos(kz
Science Foundation of
/96/$12.00
@
PI1 SO375-9601(96)00259-9
1996 Elsevier Science B.V. AR rights reserved
(2)
(3)
H'=
e2A2 ~[p~cos(kz-ot)i-p+in(kz-~t)]+~. m (4)
Denote the nth bound eigenstate of Ho by in). Suppose that the electron was in the state 1~)at t = -co, and the interaction H’ switched on adiabatically afterwards. In the interaction picture, the interaction Hamiltonian is H/ = I
in which p is the density of final states at Ef = Ei. This is Fermi’s golden rule in our case. The energy of the photoelectrons is
elrNir/h)fH/e-i(HO/“)’
=
ei(H~/A)tH//e-i(H~/lZ)r,
(5) H,$= Hoi-wL,,
(6)
L, is the z component of the orbital angular momentum L for the eIectron. According to the theorem of GelI-Mann and Low [7], the evolution from t = -00 to t = 0 changes the state of the electron from In) to the corresponding eigenstate ii) = E, _ ~~ + i~H”li’
(8)
Efo=Ef-~fiiW=Ei-~Fiw.
(13)
Ei is the effective energy of the electron immersed in the laser beam. To be the eigenvalue of the effective Hamilton~an (9), it is apparently laser intensity dependent. The condition Ef = Ei could be satisfied by the state ]fO) with a definite quantum number ,u. If for the state ]fO) the quantum number ,u is uncertain, its projection IfOp) on the state space with definite p should be substituted. Suppose that the asymptotic form of the out-state IfO) is a plane wave of unit amplitude, the differential cross section of the photoelectric effect is
i ~cAlii)f,
+p,,sin(kz)
in which a = l/137.0 is the fine structure constant, u = hike/m is the velocity of the photoelectron, and k, = d-/A. In the coordinate representation, ( 14) takes the form
of the effective ~amiItonian
da He = H; + H”,
(9)
with the eigenvalue Ei. L:is a positive infinitesimal. On the other hand, in the time interval 0 < t < cm, the state If) satisfying the Lippmann-Schwinger equation
(14)
--
ii?i-
al
16~ffu li2w
=
(
-i)‘eiq’
2
O” Ulna c ~=I.4
(15)
1
syyl;(&Q e
x [pxcos(kz)
+p!,sin(kz)
+ ieA]Wi(r)dr.
(10) evolves to the state IfO), which is an eigenstate of HO and Hk with eigenvabres E, and Ef = Efo -4 p&w, respectively. ,Uis the magnetic quantum number of the state /fO). According to the formal theory for collisions (see for example Ref. [8]), the transition amplitude induced by the laser beam is
(16) Here, Wi(r) is the coordinate representation of ii), r, B and (b are spherical coordinates of r. Fl( k,r) is a stationa~ radial wave function satisfying the boundary condition Fl(O) = 0 and the asymptotic form Fl(k,r)
-sinfk,r-
when r + 00, (,fli) = -2riS(
Ei - Ef) (fOl H”li).
7 LT
(17)
(11)
In the derivation, Eqs. (8) and (IO) together with eigenequations of He and Hk have been used. The transition probabiIity per unit time is P = ,!{folH”lij12p,
~In.+~~+~ln(Zk,r)l,
(12)
for an electron in the central field V(r). The term containing x = Zamc/i”Lk, is a contribution of the long Coulomb tail -Zfica/r in V(r). l&(&b) is the spherical ha~onic function. The quantum number p connected with the number -I* of photons absorbed in the photoelectric effect as shown in (13) now is
Q.-R. Zhnng/Physics
Letters A 216 (1996) 125-I28
connected with the angular distribution of photoelectrons. This twofold meaning of ,u is a result of angular momentum conservation. The generalization of this formulation to a multielectron system is trivial. To consider the electron spin, one needs only to replace Lz by J,, which is the z component of the total angular momentum J of the system. Our problem of solving the photoelectric effect induced by a circularly polarized light beam is thus reduced to a usual time independent quantum mechanical problem, only the potential energy is moditied. Consider the problem in more detail under the dipole approximation, which becomes exact in the long wave limit with k = 0 for light beams. This approximation is very good for the photoelectric effect induced by visible light on light atoms. In this approximation
He=
2 +V(r) +
e2A2
wJ,
Its eigcnvalue problem fective Hamiltonian H,,
=
+
spr+--_.
is equivalent
CI”Ax:‘hHee-'PA*ih =
(18)
to that of the ef-
11-7
This theorem shows a simple relation exactly in the long wave limit of the laser beam. It provides a way to check the nonperturbative dependence of the photoelectron energy on the laser intensity for a spin singlet target atom, by directly comparing two experiments instead of comparing the experiment with a theoretical calculation. The nonperturbative theoretical calculation is circumvented. This is a great advantage of the method. However, to compare the ~~rnan-Stark cffeet with the photoelec~ic effect induced by a visible light we need a magnetic field as high as 10” T. This is far from attainable at the moment. Therefore, in comparison, we have to observe the photoelectric effect of a far infrared laser on the highly excited atoms. This is still not easy, but it is already sensible. For an atom with nonsinglet spin, the use of this theorem is complicated by a correction of the Land6 factor for the Zeeman effect. Under the same dipole approximation
e&., H” = ------I-m
e2A2
( 20)
2m ’
the selection rules for its nonzero matrix elements
HO + w.17 + eAwy.
(19)
Here Ho is expressed in (3) for a single valence atom, but should be a sum of these kinds of single electron expressions and the Coulomb repulsion between electrons for a multi-valence atom. In the latter case, \’ in the last term of the last equation should also be understood to be a sum of y-coordinates of valence electrons. For a spin singlet atom ( 19) is formally identical to its Hamiltonian in a magnetic field of strength 2mo/e in the z-direction and an electric field of strength Aw in the y-direction. Since the modified energy Et of the atom interacting with the laser beam should be the eigenvalue of this effective Hamiltonian, we have the following theorem. Theorem of similarity. In the dipole approximation, the energy shift of an atom interacting with the circularly polarized laser beam (1) equals that in a combined normal Zeeman-Stark effect, with the magnetic field of strength 2mo/e along the direction of laser propagation, and the electric field of strength Aw perpendicular to it.
AI =0,&l,
A~=O,&l.
arc (21)
Therefore, to make the photoelectron energy ( 13) with arbitrary integer -p possible, we need an in~nitive series
for the initial wave function. lution for the Zeeman-Stark eigenfunction
But, as m the usual soeffect, the approximate
of Hef is a sum containing only a few harmonic terms each with definite quantum numbers 1 and F. However. corresponding to the unitary transformation ( 19) 01‘ the effective Hamiltonian from H, to H,f, there IS 3 unitary transformation of the wave function qj(r)
= e+A”‘n&(r).
It is the factor
(24)
128
Q.-R. Zhang/Physics
e-ieA.r/h
=
47r E
that makes Y;(r) with
Letters A 216 11996) 125-128
number /_Lcompatible with this condition can be really observed. Since in the dipole approximation j,( kr) = j/(O) = 810, and in the plane wave approximation for the outgoing wave of photoelectrons Fl( k,r) /k,r = j~(k,r), the integralsin thecoefficients (16) can all be calculated analytically by an eiementary method. The
i’jl (eAr/ti)
an infinitive
harmonic
series (22),
photoelectron angular distribution can be worked out whenever a comparison with experiment is planned. References
x 5zp2 ( ~rn)h2(eAr/Wur,,,(r),
(26)
where jl is the spherical Bessel function of order 1. Of course, the factor j12(eAr/ A) in (26) shows that only terms with 12 6 eAa/A are important, a is the linear dimension of the electron orbit in the atom. This means that only a photoelectron with a quantum
[I] A. Einstein, Ann. Phys. (Leipzig) 17 (1905) 132. f2f P. Krstic and Y. Hahn, Phys. Rev. A 50 ( 1994) 4629. [3] R.R. Jones, D. You and PH. Bucksbaum, Phys. Rev. Lett. 70 (1993) 1236. 141 D.-S. Guo and G.W.F. Drake, Phys. Rev. A 45 ( 1992) 6622. [5] P.H. Bucksbaum,D.W. Schumacherand M. Bashkansky, Phys. Rev. Lett. 61 (1988) 1162. 161 D.M. Volkov, 2. Phys. 94 (1935) 25. [7] M. Gell-Mann and E Low, Phys. Rev. 84 ( 1951) 350. 18] D. Luri&, Particles and fields (Interscience, New York, 1968) Ch. 7.