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Atomic level shifts in laser fields and the space-translation Hamiltonian
Volume 60A, number 3
PHYSICS LETTERS
21 February 1977
ATOMIC LEVEL SHIFTS IN LASER FIELDS AND THE SPACE-TRANSLATION HAMILTONIAN P.L. KNIGHT Department of Physics, Royal Holloway College, University of London, Egha,n, Surrey, TW2O OEX, UK Received 29 November 1976
2, and compared with those Atomic energy level shifts due to the presence of a laser field are calculated to order e obtained recently using the space-translation Hamiltonian. Extensive cancellations amongst terms occur; for this reason all contributions to order e2 must be included in the description of such level shifts.
An external electromagnetic field incident upon an atom not only induces real transitions, but generates energy level shifts. This “dressing” effect has been studied using optical pumping methods for many years [1], but has assumed a new importance with the advent of the laser. Using high-intensity laser fields, multiphoton ionization has been extensively studied. There is both experimental and theoretical evidence that the atomic energy levels are shifted by the laser field and that this effect modifies the ionization behaviour. A straightforward theoretical description of these level shifts can be obtained using perturbation theory to the required order with the conventional interaction Hamiltonian describing the coupling of the laser field to the bound electron. An elegant alternative is to allow part of the Hamiltonian to dress the atomic states, while the remainder induces transitions between these dressed states [2]. One possible version of this approach is to transform the minimal substitution Hamiltonian (we use units such that h = = 1)
~
~
~R +
0(r) —-i-p A(r) +f_A2(r),
(1)
(where 0(r) is the atomic Coulomb potential) using the space-translation transformation [31 S
exp~—i(e/m)pZ}
where the Hertz vector Z (A
z= 182
~
k,x
/
2ir
~—~~-)
\1/2
(2)
±)is given by
[éxax(k)eI*r_
è~a~(k)e~”’] (3)
Then the transformed Hamiltonian is to order e2 knew
=
(r)
+ ~R
~
~1~new =
+ Øfr —
S~~JCS
eZ/m).
(4)
In the electric dipole approximation, we may use the Taylor expansion =
anew
p2 ~
+ ~R
~
(r) + ~(r) —-~-ZVØ(r) (5)
+
I
-~-~-
(Z’ V2~(r).
2 m2 The interaction of the bound atomic electron with the laser field is described using the space-translation Hamiltonian (4) in part by an intensity dependent effective potential. The transformation (2) forms the basis of mass-renormalization, the archetype for radiative “dressing” problems. Welton has given an intuitive interpretation of the Lamb shift based upon ~new The space-translation method has been extended to include tensity dependent externalionization fields by Henneberger energies have[2] been and cominputed using the concept by Choi,[4], Henneberger andeffective Sanderspotential [2]. Recently O’Connell exploiting the similarity between external-field induced level shifts and level shifts due to self-interactions, has presented for potentials intensity-dependent energy level shifts and results ionization using the approximate method of Welton [5]. He obtains the level shifts in essence from first order matrix elements of the last term of eq. (5), and in particular predicts a level shift ~b for a bound electron state proportional to the
Volume 60A, number 3
PHYSICS LETTERS
21 February 1977
fourth power of the radiation wavelength, and with resonance effects entireely absent. Ford and O’Connell [6] further argue that results obtained from the space-
such that h
translation method are applicable only if the applied frequency is much higher than characteristic atomic binding frequencies. This has been refuted [7] by an explicit and conventional perturbation calculation which shows that results obtained using the fUll ~new agree with those obtained using ~C.The present note is concerned moie with the validity of O’Connell’s resuits at fairly low intensities where both the Welton method and more importantly, low order perturbation theory is expected to be adequate. Our results confirm those of ref. [7]. It isoccur known extensive cancellations amongst terms in that the calculation of bound-state intensity-dependent level shifts using the conventional Hamiltonian of eq. (1) [8]. A totally misleading result may be obtained if only part ofthe interaction of the bound electron with the radiation field is considered. We should be prepared to expect similar cancellations amongst terms if we use knew and this has very recently been demonstrated by explicit calculation [7]. Indeed such effects have been noted earlier in a treatment of two-photon absorption [9]. To be certain
Re f(wk) =
that we have included all effectsscattering of the radiation field 2, we use Feynman’s amplitude to order e method to describe level shifts [10], and find agreement with the results of ref. [7]. For an external field of fixed intensity and polarization, the intensity-dependent level shift of an atomic energy level is Re f(wk) = Li n (6) — —
k
k
~k
where V is the quantization volume, ~k the number of photons in mode (k) and Ref(~ok) the realamplitude part of the spin-averaged forward Compton scattering from the atom. The scattering amplitude f(~ok)may be computed using either 1C or ~new~ Not surprisingly, in view of the unitary nature of S, the scatteringamplitude computed using ~Cagrees with that using ~new- In both cases, well-known sum rules must be employed. Unfortunately, it turns out that more labour is needed to compute ~ using ~new than if ~JC is used. For a single-electron atom with an for infinitely massive nucleus, the non-relativistic result Re f(~k)~ in dipole approximation and with units
=
c = 1, is the Rayleigh amplitude 2cz
2 ~
11(w~)= ~
E~0lr 2 iO 2’ E~—
(7)
where the Cauchy principal 2, r value is employed if wk = E~0. In eq. (7), a = e 10 ~(ilrI0>, E~o E1— E0, 10> is the state whose shift we are calculating, and 11(c~)is the real part of the electric polarizability of the atom. If the laser field is approximated by a single mode (k) with occupation number ~ then eqs. (6) and (7), together with the identity 4lrwk ‘~ki V = ~2 (8) where e2 is the mean square electric field, predict a resonance-dominated level shift ‘~bfor the bound electron state 10>: = — ~T11 (w~)e2. (9) This is immediately recognizable as a Stark shift induced by the oscillating field, as we may have expected. If the field frequency is much less than the average E 10, as it could be if the atom under consideration is to tonized absorption from an infraredbe glass laser by or amultiphoton CO 2 laser field, we may neglect the frequency dependence in the resonance denominator, and
~b
~b
(Wk
~
E.0)
—
~ fl(0)e~
(10)
where 11(0) is the static polarizabiity of the atom. An order of magnitude estimate for 11(0) is Iv(0)l where a0272. is the Bohr So eq. (10) ~b This is toradius. be compared withgives O’Connell’s result for the ls state of hydrogen, ~Eb cr 214. Our —a~ result agrees with the result of O’Connell to the extent that resonance effects are absent (but only if wk ~E 10), but has a totally different wavelength dependence. In the opposite limit wk ~ E10, an electron in the bound state 10> scatters as if it were free:
4
2E, 2k’ H(wk ~“E10) = (2a/3) Ir10l 01w which reduces to (—a/m)(l /w~)using the Thomas— —
E
183
Volume 60A, number 3
PHYSICS LETTERS
Reiche—Kuhn rule. Then from (9), Ej~) (a/m)e2i2~
(11)
-
~‘
This is exactly the same as the change in energy of a free electron: for a free electron, f(wk) is given by the Thomson amplitude f(wk)=—a/m, and the energy change of a free electron, is =
(a/m)e2/2w~
21 February 1977
conclusion is that any proper calculation of level shifts using either ~Cor anew is laborious (and tedious), and that the use Of~Wnewcertainly offers no short-cuts in the intensity and frequency regime we have confined ourselves. The further results of [6] similarly contain only part of the correct order e2 shift, as is demonstrated in [7].
(12) ~,
from (6)
(13)
This work was carried out during the tenure of an SRC Research Fellowship at the University of Sussex, for which I am grateful. I would like to thank G. Barton for discussions, and W. Henneberger, R.F. O’Connell and E.A. Power for their comments, and E.A. Power for a preprint of ref. [7].
in qualitative agreement with O’Connell’s result. L~fis
entirely due to the quadratic A2 term if~Cis used. The change in ionization energy due to the laser field i~ = ~ is then from eqs. (9)—(12) zero if wk E 10 and a complicated function of wk otherwise. Our results depend only upon the validity of the Rayleigh and Thomson amplitudes (in other words that the intensity is not too high and that we are not too close to resonance). To derive the scattering amplitude from either 1C2)or(forexample ~‘new we need to2)retain the A termall cancontributions to 0(e cels part of the p A term). It is not sufficient to choose one part of the Hamiltonian even at the modest intensities for which our results are valid. Of course at high intensities our perturbative expressions are madequate and very different behaviour may result. Certainly if the laser electric field strength is greater than the inherent Coulomb electric field responsible for the binding, resonanee effects may be unlikely, but such strong fields are well outside the scope of our treatment. At lower intensities, perturbation techniques do allow a fair comparison between the results obtained using ~JC and anew The level shifts of O’Connell [4] are obtained using only the (z. v)20
References
—
~‘
term in ~new’ whereas all terms should be included to
order e2 and combined using sum rule techniques to obtain reliable results at low intensities. The inevitable
184
[II G.W. Series, in: Quantum optics, Proc. Scottish Universities Summer School in Physics, 1969 (Academic Press, London, 1970). [2] L.V. Keldysh, Z. Eksp. Teor. Fiz. (USSR) 47 (1964) 1945 (Engi. Translation: Soviet Phys. JETP 20 (1965) 1307); W.C. Henneberger, Phys. Rev.and Lett. 21(1968) C.K. Choi, W.C. Henneberger F.C. Sanders,838; Phys. Rev. A9 (1974) 1895.
[ 3] H.A. Kramers, in: Les particules élémentaires, Report to the Eigth Solvay Conference (Editions Stoops, Brussels, 1950) p. 241; 1. Schwinger, Phys. Rev. 73 (1948) 416; E.A. Power, Introductory quantum electrodynamics (Longmans, London, 1964) p. 124. [4] R.F. O’Connell, Phys. Rev. Al2 (1975) 1132. [5] T. Welton, Phys. Rev. 74(1948)1157. [6] G.W. Ford and R.F. O’Connell, Phys. Rev. A13 (1976) 1281. [7] P. Lambropoulos, E.A. Power and T. Thirunamachandran, Phys. Rev. A. [8] P.L. Knight, J. Phys. A: Gen. Phys. 5 (1972) 417. [9] E.A. T. Thirunamachandran, AtomPower Molec.and Phys. 8 (1975) L167; L170.J. Phys. B: [10] R.P. Feynman, in: La théorie quantique des champs, (Interscience, New York and London, 1961), p.61; G. Barton, Phys. Rev. AS (1972) 468.
PHYSICS LETTERS
21 February 1977
ATOMIC LEVEL SHIFTS IN LASER FIELDS AND THE SPACE-TRANSLATION HAMILTONIAN P.L. KNIGHT Department of Physics, Royal Holloway College, University of London, Egha,n, Surrey, TW2O OEX, UK Received 29 November 1976
2, and compared with those Atomic energy level shifts due to the presence of a laser field are calculated to order e obtained recently using the space-translation Hamiltonian. Extensive cancellations amongst terms occur; for this reason all contributions to order e2 must be included in the description of such level shifts.
An external electromagnetic field incident upon an atom not only induces real transitions, but generates energy level shifts. This “dressing” effect has been studied using optical pumping methods for many years [1], but has assumed a new importance with the advent of the laser. Using high-intensity laser fields, multiphoton ionization has been extensively studied. There is both experimental and theoretical evidence that the atomic energy levels are shifted by the laser field and that this effect modifies the ionization behaviour. A straightforward theoretical description of these level shifts can be obtained using perturbation theory to the required order with the conventional interaction Hamiltonian describing the coupling of the laser field to the bound electron. An elegant alternative is to allow part of the Hamiltonian to dress the atomic states, while the remainder induces transitions between these dressed states [2]. One possible version of this approach is to transform the minimal substitution Hamiltonian (we use units such that h = = 1)
~
~
~R +
0(r) —-i-p A(r) +f_A2(r),
(1)
(where 0(r) is the atomic Coulomb potential) using the space-translation transformation [31 S
exp~—i(e/m)pZ}
where the Hertz vector Z (A
z= 182
~
k,x
/
2ir
~—~~-)
\1/2
(2)
±)is given by
[éxax(k)eI*r_
è~a~(k)e~”’] (3)
Then the transformed Hamiltonian is to order e2 knew
=
(r)
+ ~R
~
~1~new =
+ Øfr —
S~~JCS
eZ/m).
(4)
In the electric dipole approximation, we may use the Taylor expansion =
anew
p2 ~
+ ~R
~
(r) + ~(r) —-~-ZVØ(r) (5)
+
I
-~-~-
(Z’ V2~(r).
2 m2 The interaction of the bound atomic electron with the laser field is described using the space-translation Hamiltonian (4) in part by an intensity dependent effective potential. The transformation (2) forms the basis of mass-renormalization, the archetype for radiative “dressing” problems. Welton has given an intuitive interpretation of the Lamb shift based upon ~new The space-translation method has been extended to include tensity dependent externalionization fields by Henneberger energies have[2] been and cominputed using the concept by Choi,[4], Henneberger andeffective Sanderspotential [2]. Recently O’Connell exploiting the similarity between external-field induced level shifts and level shifts due to self-interactions, has presented for potentials intensity-dependent energy level shifts and results ionization using the approximate method of Welton [5]. He obtains the level shifts in essence from first order matrix elements of the last term of eq. (5), and in particular predicts a level shift ~b for a bound electron state proportional to the
Volume 60A, number 3
PHYSICS LETTERS
21 February 1977
fourth power of the radiation wavelength, and with resonance effects entireely absent. Ford and O’Connell [6] further argue that results obtained from the space-
such that h
translation method are applicable only if the applied frequency is much higher than characteristic atomic binding frequencies. This has been refuted [7] by an explicit and conventional perturbation calculation which shows that results obtained using the fUll ~new agree with those obtained using ~C.The present note is concerned moie with the validity of O’Connell’s resuits at fairly low intensities where both the Welton method and more importantly, low order perturbation theory is expected to be adequate. Our results confirm those of ref. [7]. It isoccur known extensive cancellations amongst terms in that the calculation of bound-state intensity-dependent level shifts using the conventional Hamiltonian of eq. (1) [8]. A totally misleading result may be obtained if only part ofthe interaction of the bound electron with the radiation field is considered. We should be prepared to expect similar cancellations amongst terms if we use knew and this has very recently been demonstrated by explicit calculation [7]. Indeed such effects have been noted earlier in a treatment of two-photon absorption [9]. To be certain
Re f(wk) =
that we have included all effectsscattering of the radiation field 2, we use Feynman’s amplitude to order e method to describe level shifts [10], and find agreement with the results of ref. [7]. For an external field of fixed intensity and polarization, the intensity-dependent level shift of an atomic energy level is Re f(wk) = Li n (6) — —
k
k
~k
where V is the quantization volume, ~k the number of photons in mode (k) and Ref(~ok) the realamplitude part of the spin-averaged forward Compton scattering from the atom. The scattering amplitude f(~ok)may be computed using either 1C or ~new~ Not surprisingly, in view of the unitary nature of S, the scatteringamplitude computed using ~Cagrees with that using ~new- In both cases, well-known sum rules must be employed. Unfortunately, it turns out that more labour is needed to compute ~ using ~new than if ~JC is used. For a single-electron atom with an for infinitely massive nucleus, the non-relativistic result Re f(~k)~ in dipole approximation and with units
=
c = 1, is the Rayleigh amplitude 2cz
2 ~
11(w~)= ~
E~0lr 2 iO 2’ E~—
(7)
where the Cauchy principal 2, r value is employed if wk = E~0. In eq. (7), a = e 10 ~(ilrI0>, E~o E1— E0, 10> is the state whose shift we are calculating, and 11(c~)is the real part of the electric polarizability of the atom. If the laser field is approximated by a single mode (k) with occupation number ~ then eqs. (6) and (7), together with the identity 4lrwk ‘~ki V = ~2 (8) where e2 is the mean square electric field, predict a resonance-dominated level shift ‘~bfor the bound electron state 10>: = — ~T11 (w~)e2. (9) This is immediately recognizable as a Stark shift induced by the oscillating field, as we may have expected. If the field frequency is much less than the average E 10, as it could be if the atom under consideration is to tonized absorption from an infraredbe glass laser by or amultiphoton CO 2 laser field, we may neglect the frequency dependence in the resonance denominator, and
~b
~b
(Wk
~
E.0)
—
~ fl(0)e~
(10)
where 11(0) is the static polarizabiity of the atom. An order of magnitude estimate for 11(0) is Iv(0)l where a0272. is the Bohr So eq. (10) ~b This is toradius. be compared withgives O’Connell’s result for the ls state of hydrogen, ~Eb cr 214. Our —a~ result agrees with the result of O’Connell to the extent that resonance effects are absent (but only if wk ~E 10), but has a totally different wavelength dependence. In the opposite limit wk ~ E10, an electron in the bound state 10> scatters as if it were free:
4
2E, 2k’ H(wk ~“E10) = (2a/3) Ir10l 01w which reduces to (—a/m)(l /w~)using the Thomas— —
E
183
Volume 60A, number 3
PHYSICS LETTERS
Reiche—Kuhn rule. Then from (9), Ej~) (a/m)e2i2~
(11)
-
~‘
This is exactly the same as the change in energy of a free electron: for a free electron, f(wk) is given by the Thomson amplitude f(wk)=—a/m, and the energy change of a free electron, is =
(a/m)e2/2w~
21 February 1977
conclusion is that any proper calculation of level shifts using either ~Cor anew is laborious (and tedious), and that the use Of~Wnewcertainly offers no short-cuts in the intensity and frequency regime we have confined ourselves. The further results of [6] similarly contain only part of the correct order e2 shift, as is demonstrated in [7].
(12) ~,
from (6)
(13)
This work was carried out during the tenure of an SRC Research Fellowship at the University of Sussex, for which I am grateful. I would like to thank G. Barton for discussions, and W. Henneberger, R.F. O’Connell and E.A. Power for their comments, and E.A. Power for a preprint of ref. [7].
in qualitative agreement with O’Connell’s result. L~fis
entirely due to the quadratic A2 term if~Cis used. The change in ionization energy due to the laser field i~ = ~ is then from eqs. (9)—(12) zero if wk E 10 and a complicated function of wk otherwise. Our results depend only upon the validity of the Rayleigh and Thomson amplitudes (in other words that the intensity is not too high and that we are not too close to resonance). To derive the scattering amplitude from either 1C2)or(forexample ~‘new we need to2)retain the A termall cancontributions to 0(e cels part of the p A term). It is not sufficient to choose one part of the Hamiltonian even at the modest intensities for which our results are valid. Of course at high intensities our perturbative expressions are madequate and very different behaviour may result. Certainly if the laser electric field strength is greater than the inherent Coulomb electric field responsible for the binding, resonanee effects may be unlikely, but such strong fields are well outside the scope of our treatment. At lower intensities, perturbation techniques do allow a fair comparison between the results obtained using ~JC and anew The level shifts of O’Connell [4] are obtained using only the (z. v)20
References
—
~‘
term in ~new’ whereas all terms should be included to
order e2 and combined using sum rule techniques to obtain reliable results at low intensities. The inevitable
184
[II G.W. Series, in: Quantum optics, Proc. Scottish Universities Summer School in Physics, 1969 (Academic Press, London, 1970). [2] L.V. Keldysh, Z. Eksp. Teor. Fiz. (USSR) 47 (1964) 1945 (Engi. Translation: Soviet Phys. JETP 20 (1965) 1307); W.C. Henneberger, Phys. Rev.and Lett. 21(1968) C.K. Choi, W.C. Henneberger F.C. Sanders,838; Phys. Rev. A9 (1974) 1895.
[ 3] H.A. Kramers, in: Les particules élémentaires, Report to the Eigth Solvay Conference (Editions Stoops, Brussels, 1950) p. 241; 1. Schwinger, Phys. Rev. 73 (1948) 416; E.A. Power, Introductory quantum electrodynamics (Longmans, London, 1964) p. 124. [4] R.F. O’Connell, Phys. Rev. Al2 (1975) 1132. [5] T. Welton, Phys. Rev. 74(1948)1157. [6] G.W. Ford and R.F. O’Connell, Phys. Rev. A13 (1976) 1281. [7] P. Lambropoulos, E.A. Power and T. Thirunamachandran, Phys. Rev. A. [8] P.L. Knight, J. Phys. A: Gen. Phys. 5 (1972) 417. [9] E.A. T. Thirunamachandran, AtomPower Molec.and Phys. 8 (1975) L167; L170.J. Phys. B: [10] R.P. Feynman, in: La théorie quantique des champs, (Interscience, New York and London, 1961), p.61; G. Barton, Phys. Rev. AS (1972) 468.