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Saturation and rabi oscillations in resonant multiphoton ionization
Volume
22, number
August
OPTICS COMMUNICATIONS
2
SATURATION
AND RABI OSCILLATIONS
1977
IN RESONANT MULTIPHOTON IONIZATION
P.L. KNIGHT Department
of Physics, Royal Holloway College, University of London,
Received 2 May 1977 Revised manuscript received
We present shifts, induced
Egham Hill, Egham, Surrey, TW20 0E.Y England
6 June 1977
an analysis of saturation widths and bound-bound
in resonant nutational
two-photon ionization which includes bound Rabi frequencies at extreme saturation.
We describe an analysis of near-resonant two-photon ionization which reduces to the usual lowest order perturbation theory approach [ 1] away from boundbound resonances, exhibits saturation near resonance in the form of induced widths and shifts which themselves saturate (see also [2]), and for strong saturation demonstrates the importance of Rabi nutational oscillations [3]. In addition we predict an interesting (although difficult to observe) splitting of the ejected photoelectron energy spectrum. The saturation of participating bound-bound transitions is an important factor in multiphoton excitation experiments performed with tunable lasers. Multiphoton ionization probabilities may be greatly enhanced if the driving radiation field frequency is tuned close to a participating atomic resonance frequency [4]. This enhancement makes it possible to explore the intensity and frequency dependence of multiphoton ionization using the quite modest laser power available from tunable dye lasers. However the gain in probability is won at the expense of introducing nonlinear complications: it is easy to saturate a resonantly driven bound-bound transition. We consider a simplified model of an atom with two discrete bound states la) and lb) with energies E,, Eb such that (Eb - Ea) = uba, and a set of continuum states we label IE) (fig. 1) (we use units such that ti = c = 1). The atom interacts with a single highly occupied mode of the electromagnetic field, of wavenumber k and frequency w = Ik I which is detuned from the bound-bound ab transition by an amount A = w ba - w. We do not consider coupling of the atom to
and continuum
energy
b
0
t
__1..
ab
_
a
Fig. 1. Model for resonant two photon ionization involving two discrete atomic states a and b separated by Wab, and a set of continuum states of energy E.
the other empty radiation field modes: we will neglect for the moment all effects of spontaneous decay. The total Hamiltonian for the coupled atom-field system is given by H=HA
+ua+a+
p,
(1)
where HA is the atomic Hamiltonian, a+ and a are field creation and annihilation operators. The interactinn Hamiltonian p is
(2) where the vector potential
in dipole approximation
A(r) = (2n/wk F’)l12 [aS(kh) + h.c.] .
is (3) 173
Volume 22, number 2
OPTICS COMMUNICATIONS
Here Vis the quantization volume, c(kh) a polarization unit vector, k. 6(kh) = 0, h = 1,2. The radiation field is quantized in number states in); we use a quantized field description for convenience only. If we allow only energy-conserving stimulated absorption and emission processes from the initial state Ii) = In)la)
(4a)
the remaining participating
r/ii)
(10)
?,, will not shift the positions of bound-bound resonances, we will omit the tilda. It will affect the position of bound-free transitions however, and shifts the final photoelectron energy. Then from eqs. (4), (5) and (9), we find
(Jb, c>
where we choose the origin of energy to be such that Ei = 0, Ei = (tiba - w), E, = (WE - 2~). The choice of states in eq. (4) is equivalent to a rotating-wave approximation (RWA). The wavefunction at time t is expanded into the set of states given by eq. (4):
= viigi(E)
(E - Ei)gi(E)
= $igi (E> f 5: $~g#l~
(1 lb)
(E - Ef>@‘)
= ~pi(E)
Olc)
n
(1 la>
exp[-S&l I&)
.
These may be solved by elimination,
E-Ei-l~i~12/IE-Ei-Ri(E)}
(9
to give
1
&?jQ = IQ(t)) = C&(t)
+1)
(E - Ei)gi(E)
states are
IJ, = In-2)IE))
Ii)= In-l)lb),
i;j =
August 1917
’
(12a)
~jj
and using the time-dependent SchrGdinger equation and orthonormality we obtain the equations of motion
q(E) =
ii, (t) = c ?,,,, exp [i(E, - E,) m
and finally
[E-Ei-RiQ]
[E-Ei-l~i~12/{E-Ej-~j(~}]
’ (12b)
t] b, (t) + iF,$ (t) ,
(6) where k,,, z (nl ?lm). The last term‘in eq. (6) incorporates the initial condition bi(0) = 1, b,fi(O) = 0 directly in the equations of motion. To solve these linear coupled equations of motion we use standard Heitler-Ma techniques [5]. We express b,(t) in terms of Fourier amplitudes gn (E) through b, (t) = - &
s dEg, (E) exp [i(E, - E) t] -co
(7)
S tiexp -m
[i(Ei - E) t] .
(8)
Cpnmg,(E)+6,i. m
(9)
In our problem p has diagonal matrix elements to the A2 term; this term will not drive transitions does shift energies. We incorporate these diagonal ments in E, by defining En = E,, + V,, . Since for
due but elelarge
n, 174
gf(E) =(E-Ef)
[E-E~-R~)] 1
’ [E-Ei-l~i~12/{E-E~-~j(E)~]
.
(12c)
The complex energy Ri (E) is Rj(E) = 7
1$~/(E-E~).
(E-E,
) gn (E) = Tn (E> Si (E)
(13)
together with the standard scattering theory results
The Fourier amplitudes gn (E)‘satisfy the linear algebraic equations (E-En)gn(E)=
^ qi
We obtain T-matrix elements from eq. (11) using the definition
together with i6 (t) = - &
^ ‘jj
[5,61
Tn(E)= pni•tmqi pn,3.(E-E,) T,(E)a ,
(15)
~i,C(E-E,)T~Q,
(16)
gi(E) = [E-Ei + ir(f?)/2]-l f Wl=im~i
(14)
where {(E-E,) = (E-E, + ilej)-l and 1~1is an infinitesimal positive number. For weak ? the perturba-
Volume
22, number
tive transition state If) is
August
OPTICS COMMUNICATIONS
2
rate for going from Ii) to any continuum
nators and energy conservation delta-functions. From eqs. (22) and (19) the induced width of the initial state (remember rj is intensity-dependent) is Yi/2 = (Yj)l pij12/A2 + $14
where p(Ef> is the density of final states and AE =
NYEf)/21. If we expand Tf(Ef> given by eq. (14) in a perturbation series, and retain only the lowest order contributing term, we find from eq. (17) ^ ^ V~ Vji 2 Y.If =2n P CEf) 6 (+Ei) dEf 3 (18) EfEi + i lel which on using the delta function to replace Ef by Ei in the denominator, is exactly the traditional perturbative two-photon ionization rate. If the effect of the field on the atom is not so small, we abandon the perturbation expansion of T,, (E), calculate T,(E) exactly within our model, but continue to use eq. (17) to find a transition rate. This is the method previously used to describe 4-photon ionization [7]. First we make a “pole” approximation of taking the on-shell value for Rj(E) [8] Rjo‘Rj(Ej)=~l~f12/(Ej-Ef)-~j+~yi
(19)
where Af$ and rj are the intensity-dependent shifts and widths of the intermediate state ]j) due to its coupling to the continuum. We incorporate aEj into Ej by defining Ei = Ej f AL$. We do not take the value of aEj in eq. (19) too seriously since the RWA has introduced a significant error here. The induced shift and width of the initial state is described by iI’( and itself saturates. The diagonal energy shifts do not shift the position of the bound-bound transition since ~ii = vij. From eqs. (12) and (13) we find
1911
(23)
and the induced shift of the initial state is aEi = 281 Vij12/A2 + r’/4 I .
(24)
For stronger fields the basic perturbative approximation behind eq. (17) becomes less trustworthy and the idea of a transition rate constant in time may be invalid: it is most unlikely that lb, (t)12 is exactly proportional to time for all times [3]. We know that when resonance fluorescence saturates, there are strong initial transient oscillations and a rate argument is inadequate [9] ; since this is the first step in resonant multiphoton ionization we similarly expect eq. (17) to break down. To see this we invert eqs. (12) using the pole approximation, eq. (19):
hi(t) = - &
bi(‘) = - &
J dE --m 7 -cc
(E-Ej-Rj(Ej))exp[i(Ei-E)
t]
(E-E,) (E-E- 1
&
vii exp [i(E.-E) t] J dE (E-E +)(4-E_)
(26)
where E* =$ [Ej+Rj(Ej)]
~~ [(Ej+Rj(Ej))2+41~ij12]1’2.
Then b,(t) = (E+-E-J’
{[E+-Ej-Rj(Ej)J
_ [E_ -Ej-Rj(Ej)]
ei(-‘+E+)r
ei(Ei-E-)f}
(27)
and bj(f) = (E+-E-)-l
{ei(Ej-E+)’
_ ei(Ej-E-)t}
5i.
(28) Tf (E) = vfi pi/ {E-Ej-Rj(Ej) and
‘if
* ^ VfiJ$i = 2n
(20)
2
Eqs. (27) and (28) are immediately recognizable from the damped Rabi problem [lo]. We find for the state probabilities
‘(Ef-Ei-w”(Ef{%
,‘+~j-Rj(Ej)
where AE = Im [F(Ef)/2] - ~ r(Ef)
},
and
= I~ij12/{E~Ej-Rj(Ej)}
.
(22)
We note that W$ in eq. (21) is in effect the perturbative result corrected for saturation in energy denomi-
X 4101~ sin? [
sin y
+ 41fi12 cos y
_ 4k*a
sin?
t 4iaa*cos
cos s
‘q
cos y
sin F
1
(29)
175
Volume
22. number
August
OPTICS COMMUNICATIONS
2
1977
and
1
e-Yjf12
(30)
where C2= [&+41Q2]
l/2 = [(El tJ7#))2
Obviously our model of resonant two-photon ionization is a very special damped Rabi problem: the bound-bound transition is so strongly driven that Rabi nutational oscillations occur, damped only by an irreversible loss into the continuum at rate rj. This behaviour is familiar from the work of Beers and Armstrong [3]. What distinguishes this two-photon process from the traditional excited state transition described by the damped Rabi solution is that the damping is itself induced. The ionization probability is PI(t) = 1 - lbj(t)12 - Ibj(f)P .
(31)
We note that PI(t) -+ 1 as t -+ 00: all atoms eventually ionize in the absence of any decay channel back to the ground state. The final state amplitude as t + m is bf(“)
= Tf (Ef>/
{Ef Ei + iIYEJ2)
(32)
and using eq. (22) and the definitions of E,, the probability of two-photon ionization to a particular atomic continuum state If> is ^ ^
lwyy
If I Fiji is much larger than rj, AEj (extreme saturation), then lbf(m)12 describes a doublet: the photoelectron energies split into two groups centred at Ef= f A f f [A2 + 41 fiij12] li2. This “dynamic Stark splitting” proportional to the laser electric field strength is familar from work on saturated bound-bound absorption spectra [l l] and fluorescence spectra [12]. We note that the doublet profile of the photoelectron energy distribution is produced at fixed w;,there is no need to scan probe frequencies to discern each component as in the bound-bound absorption analogue [l 1 ] since both values of Ef are simultaneously produced. This can be seen best from a “dressed atom” viewpoint [13] where the effect of pan the bound-bound transition is to split the energy levels into doublets (see fig. 2). Then i’is further responsible for transitions from the dressed states to the continuum. We further 176
4
-I-41&12]1/2 .
I i
Ivij.\
Fig. 2. Dressed states of atom plus radiation field showing the dynamic Stark splitting of the ab transition at resonance: W = ‘Jab.
note that the final state energy is shifted by an amount tiffi due entirely to the A2 term. If we express the number of photons in terms of the mean square laser electric field L through the identity (4nwn/V) = & 2 then Vff = (a/m) &2/2w2 familiar from work on socalled “mass-shifts” of free electrons [ 141. It is not usual to energy analyze the emitted photoelectrons (see however ref. [1.5]), but if this were to be done, an electron spectrometer would measure lbf(-)12 summed over a “bandwidth” range of Ef’s. Of course, if the spectrometer is dispensed with and the photoelectrons just counted, our model predicts
yf(“p
=P&+=
1.
The effect of spontaneous emission upon the resonant two-photon ionization is not just to add damping rates to the atomic energy levels. We would expect for example that the spontaneous emission spectrum 1;) back to Ii) would exhibit the by now well known resonant Stark triplet structure [ 121. If only one decay photon into some mode h is allowed in the dynamics, and the correspondingly transition I{n-I }, l,)la> + I {rz-2}, 1 ,)lb> is neglected, then we incorrectly predict a doublet for the fluorescence spectrum. To allow this extra transition, and to account for cascades in the dressed two-level problem is straightforward [I 61; when transitions to the continuum are admitted the problem becomes more complicated. To summarise, we have presented a unified.treat-
Volume 22, number ‘2
OPTICS COMMUNICATIONS
ment of two-photon resonant ionization in a simple model using standard Heitler techniques. With this technique we are able to describe the ionization behaviour for weak, medium and strong interaction of the model atom with the radiation field. The weak interaction limit leads to precisely the usual lowest order perturbation theory result. For moderate intensities the bound-bound resonance saturates and this leads to the intensity-dependent widths and shifts in eqs. (23) and (24) which themselves saturate, and to a splitting of the final photoelectron energy distribution. The insaturated versions of the widths and shifts have, of course, been observed: AE in that case is just the AC Stark shift. The observation of saturation in AE seems to be quite feasible. At stronger field strengths the rate concept breaks down as shown by Beers and Armstrong [3], and the Rabi frequency exhibits itself not only in atomic dynamics but in a splitting of the photoelectron energy which may well be observable. I am grateful to Royal Holloway College for the award of a Jubilee Research Fellowship.
References [1] H.B. Bebb and A. Gold, Phys. Rev. 143 (1966) 1. [2] P. Lambropoulos, Phys. Rev. A9 (1974) 1992. (31 B.L. Beers and L. Armstrong Jr., Phys. Rev. Al2 (1975) 2447. [4] P. Lambropoulos and M. Lambropoulos, in Electron and photon interactions with atoms, eds. H. Kleinpoppen and M.R.C. McDowell (Plenum Press, New York, 1976) p. 528. [S] W. Heitler, The quantum theory of radiation (Clarendon Press, Oxford, 1954).
August 1977
[6] M.L. Goldberger and K.M. Watson, Collision theory (Wiley, New York, 1964). [7] F.H.M. Faisal, J. Phys. B: Atom. Molec. Phys. 9 (1976) 3009. [8] f.L. Knight and P.W. Milonni, Phys. Lett. 56A (1976) 275; also ref. [ 31. (91 For example C. Cohen-Tannoudji, Laser Spectroscopy, eds. S. Haroche, J.C. Pebay-Peyroula, T.W. Hansch and S.E. Harris (Springer-Verlag, 1975) p. 324. [lo] M. Sargent HI, M.O. Scully and W.E. Lamb Jr., Laser physics (Addison Wesley, Reading, Mass., 1974). [ll] S. Feneuille and M.-G. Schweighofer, J. de Physique (Paris) 36 (1975) 781; (121 A. Schabert, R. Keil and P.E. Toschek, Opt. Commun. 13 (1975) 265; A. Schabert, R. Keil and P.E. Toschek, Appl. Phys. 6 (1975) 181; J.L. Pique and J. Pinard, J. Phys. B: Atom. Molec. Phys. 9 (1976) L77; C. Delsart and J.-C. Keller, J. Phys. B: Atom. Molec. Phys. 9 (1976) 2769; Ph. Cahuzac and R. Vetter, Phys. Rev. Al4 (1976) 270. [12] B.R. Mollow, Phys. Rev. 188 (1969) 1969; F. Schuda, C.R. Stroud Jr. and M. Hercher, J. Phys. B: Atom. Molec. Phys. 7 (1974) L 198; F.Y. Wu, R.E. Grove and S. Ezekiel, Phys. Rev. Letters 35 (1975) 1426; W. Hartig, W. Rasmussen, R. Schieder and H. Walther, Z. Physik A 278 (1976) 205. 1131 S.H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703. E.T. Jaynes and F.W. Cummings, Proc. IRE 51 (1963) 89. [14] J.H. Eberly, Progress in Optics, Vol. VII, ed. E. Wolf (North-Holland, 1969), p. 359. [15] E.A. Martin and L. Mandel, Appl. Optics 15 (1976) 2378. [16] H.J. Carmichael and D.F. Walls, J. Phys. B: Atom. Molec. Phys. 8 (1975) L77. M.E. Smithers and H.S. Freedhoff, J. Phys. B: Atom. Molec. Phys. 8 (1975) 2911; S. Swain, J. Phys. B: Atom. Molec. Phys. 8 (1975) L437.
177
22, number
August
OPTICS COMMUNICATIONS
2
SATURATION
AND RABI OSCILLATIONS
1977
IN RESONANT MULTIPHOTON IONIZATION
P.L. KNIGHT Department
of Physics, Royal Holloway College, University of London,
Received 2 May 1977 Revised manuscript received
We present shifts, induced
Egham Hill, Egham, Surrey, TW20 0E.Y England
6 June 1977
an analysis of saturation widths and bound-bound
in resonant nutational
two-photon ionization which includes bound Rabi frequencies at extreme saturation.
We describe an analysis of near-resonant two-photon ionization which reduces to the usual lowest order perturbation theory approach [ 1] away from boundbound resonances, exhibits saturation near resonance in the form of induced widths and shifts which themselves saturate (see also [2]), and for strong saturation demonstrates the importance of Rabi nutational oscillations [3]. In addition we predict an interesting (although difficult to observe) splitting of the ejected photoelectron energy spectrum. The saturation of participating bound-bound transitions is an important factor in multiphoton excitation experiments performed with tunable lasers. Multiphoton ionization probabilities may be greatly enhanced if the driving radiation field frequency is tuned close to a participating atomic resonance frequency [4]. This enhancement makes it possible to explore the intensity and frequency dependence of multiphoton ionization using the quite modest laser power available from tunable dye lasers. However the gain in probability is won at the expense of introducing nonlinear complications: it is easy to saturate a resonantly driven bound-bound transition. We consider a simplified model of an atom with two discrete bound states la) and lb) with energies E,, Eb such that (Eb - Ea) = uba, and a set of continuum states we label IE) (fig. 1) (we use units such that ti = c = 1). The atom interacts with a single highly occupied mode of the electromagnetic field, of wavenumber k and frequency w = Ik I which is detuned from the bound-bound ab transition by an amount A = w ba - w. We do not consider coupling of the atom to
and continuum
energy
b
0
t
__1..
ab
_
a
Fig. 1. Model for resonant two photon ionization involving two discrete atomic states a and b separated by Wab, and a set of continuum states of energy E.
the other empty radiation field modes: we will neglect for the moment all effects of spontaneous decay. The total Hamiltonian for the coupled atom-field system is given by H=HA
+ua+a+
p,
(1)
where HA is the atomic Hamiltonian, a+ and a are field creation and annihilation operators. The interactinn Hamiltonian p is
(2) where the vector potential
in dipole approximation
A(r) = (2n/wk F’)l12 [aS(kh) + h.c.] .
is (3) 173
Volume 22, number 2
OPTICS COMMUNICATIONS
Here Vis the quantization volume, c(kh) a polarization unit vector, k. 6(kh) = 0, h = 1,2. The radiation field is quantized in number states in); we use a quantized field description for convenience only. If we allow only energy-conserving stimulated absorption and emission processes from the initial state Ii) = In)la)
(4a)
the remaining participating
r/ii)
(10)
?,, will not shift the positions of bound-bound resonances, we will omit the tilda. It will affect the position of bound-free transitions however, and shifts the final photoelectron energy. Then from eqs. (4), (5) and (9), we find
(Jb, c>
where we choose the origin of energy to be such that Ei = 0, Ei = (tiba - w), E, = (WE - 2~). The choice of states in eq. (4) is equivalent to a rotating-wave approximation (RWA). The wavefunction at time t is expanded into the set of states given by eq. (4):
= viigi(E)
(E - Ei)gi(E)
= $igi (E> f 5: $~g#l~
(1 lb)
(E - Ef>@‘)
= ~pi(E)
Olc)
n
(1 la>
exp[-S&l I&)
.
These may be solved by elimination,
E-Ei-l~i~12/IE-Ei-Ri(E)}
(9
to give
1
&?jQ = IQ(t)) = C&(t)
+1)
(E - Ei)gi(E)
states are
IJ, = In-2)IE))
Ii)= In-l)lb),
i;j =
August 1917
’
(12a)
~jj
and using the time-dependent SchrGdinger equation and orthonormality we obtain the equations of motion
q(E) =
ii, (t) = c ?,,,, exp [i(E, - E,) m
and finally
[E-Ei-RiQ]
[E-Ei-l~i~12/{E-Ej-~j(~}]
’ (12b)
t] b, (t) + iF,$ (t) ,
(6) where k,,, z (nl ?lm). The last term‘in eq. (6) incorporates the initial condition bi(0) = 1, b,fi(O) = 0 directly in the equations of motion. To solve these linear coupled equations of motion we use standard Heitler-Ma techniques [5]. We express b,(t) in terms of Fourier amplitudes gn (E) through b, (t) = - &
s dEg, (E) exp [i(E, - E) t] -co
(7)
S tiexp -m
[i(Ei - E) t] .
(8)
Cpnmg,(E)+6,i. m
(9)
In our problem p has diagonal matrix elements to the A2 term; this term will not drive transitions does shift energies. We incorporate these diagonal ments in E, by defining En = E,, + V,, . Since for
due but elelarge
n, 174
gf(E) =(E-Ef)
[E-E~-R~)] 1
’ [E-Ei-l~i~12/{E-E~-~j(E)~]
.
(12c)
The complex energy Ri (E) is Rj(E) = 7
1$~/(E-E~).
(E-E,
) gn (E) = Tn (E> Si (E)
(13)
together with the standard scattering theory results
The Fourier amplitudes gn (E)‘satisfy the linear algebraic equations (E-En)gn(E)=
^ qi
We obtain T-matrix elements from eq. (11) using the definition
together with i6 (t) = - &
^ ‘jj
[5,61
Tn(E)= pni•tmqi pn,3.(E-E,) T,(E)a ,
(15)
~i,C(E-E,)T~Q,
(16)
gi(E) = [E-Ei + ir(f?)/2]-l f Wl=im~i
(14)
where {(E-E,) = (E-E, + ilej)-l and 1~1is an infinitesimal positive number. For weak ? the perturba-
Volume
22, number
tive transition state If) is
August
OPTICS COMMUNICATIONS
2
rate for going from Ii) to any continuum
nators and energy conservation delta-functions. From eqs. (22) and (19) the induced width of the initial state (remember rj is intensity-dependent) is Yi/2 = (Yj)l pij12/A2 + $14
where p(Ef> is the density of final states and AE =
NYEf)/21. If we expand Tf(Ef> given by eq. (14) in a perturbation series, and retain only the lowest order contributing term, we find from eq. (17) ^ ^ V~ Vji 2 Y.If =2n P CEf) 6 (+Ei) dEf 3 (18) EfEi + i lel which on using the delta function to replace Ef by Ei in the denominator, is exactly the traditional perturbative two-photon ionization rate. If the effect of the field on the atom is not so small, we abandon the perturbation expansion of T,, (E), calculate T,(E) exactly within our model, but continue to use eq. (17) to find a transition rate. This is the method previously used to describe 4-photon ionization [7]. First we make a “pole” approximation of taking the on-shell value for Rj(E) [8] Rjo‘Rj(Ej)=~l~f12/(Ej-Ef)-~j+~yi
(19)
where Af$ and rj are the intensity-dependent shifts and widths of the intermediate state ]j) due to its coupling to the continuum. We incorporate aEj into Ej by defining Ei = Ej f AL$. We do not take the value of aEj in eq. (19) too seriously since the RWA has introduced a significant error here. The induced shift and width of the initial state is described by iI’( and itself saturates. The diagonal energy shifts do not shift the position of the bound-bound transition since ~ii = vij. From eqs. (12) and (13) we find
1911
(23)
and the induced shift of the initial state is aEi = 281 Vij12/A2 + r’/4 I .
(24)
For stronger fields the basic perturbative approximation behind eq. (17) becomes less trustworthy and the idea of a transition rate constant in time may be invalid: it is most unlikely that lb, (t)12 is exactly proportional to time for all times [3]. We know that when resonance fluorescence saturates, there are strong initial transient oscillations and a rate argument is inadequate [9] ; since this is the first step in resonant multiphoton ionization we similarly expect eq. (17) to break down. To see this we invert eqs. (12) using the pole approximation, eq. (19):
hi(t) = - &
bi(‘) = - &
J dE --m 7 -cc
(E-Ej-Rj(Ej))exp[i(Ei-E)
t]
(E-E,) (E-E- 1
&
vii exp [i(E.-E) t] J dE (E-E +)(4-E_)
(26)
where E* =$ [Ej+Rj(Ej)]
~~ [(Ej+Rj(Ej))2+41~ij12]1’2.
Then b,(t) = (E+-E-J’
{[E+-Ej-Rj(Ej)J
_ [E_ -Ej-Rj(Ej)]
ei(-‘+E+)r
ei(Ei-E-)f}
(27)
and bj(f) = (E+-E-)-l
{ei(Ej-E+)’
_ ei(Ej-E-)t}
5i.
(28) Tf (E) = vfi pi/ {E-Ej-Rj(Ej) and
‘if
* ^ VfiJ$i = 2n
(20)
2
Eqs. (27) and (28) are immediately recognizable from the damped Rabi problem [lo]. We find for the state probabilities
‘(Ef-Ei-w”(Ef{%
,‘+~j-Rj(Ej)
where AE = Im [F(Ef)/2] - ~ r(Ef)
},
and
= I~ij12/{E~Ej-Rj(Ej)}
.
(22)
We note that W$ in eq. (21) is in effect the perturbative result corrected for saturation in energy denomi-
X 4101~ sin? [
sin y
+ 41fi12 cos y
_ 4k*a
sin?
t 4iaa*cos
cos s
‘q
cos y
sin F
1
(29)
175
Volume
22. number
August
OPTICS COMMUNICATIONS
2
1977
and
1
e-Yjf12
(30)
where C2= [&+41Q2]
l/2 = [(El tJ7#))2
Obviously our model of resonant two-photon ionization is a very special damped Rabi problem: the bound-bound transition is so strongly driven that Rabi nutational oscillations occur, damped only by an irreversible loss into the continuum at rate rj. This behaviour is familiar from the work of Beers and Armstrong [3]. What distinguishes this two-photon process from the traditional excited state transition described by the damped Rabi solution is that the damping is itself induced. The ionization probability is PI(t) = 1 - lbj(t)12 - Ibj(f)P .
(31)
We note that PI(t) -+ 1 as t -+ 00: all atoms eventually ionize in the absence of any decay channel back to the ground state. The final state amplitude as t + m is bf(“)
= Tf (Ef>/
{Ef Ei + iIYEJ2)
(32)
and using eq. (22) and the definitions of E,, the probability of two-photon ionization to a particular atomic continuum state If> is ^ ^
lwyy
If I Fiji is much larger than rj, AEj (extreme saturation), then lbf(m)12 describes a doublet: the photoelectron energies split into two groups centred at Ef= f A f f [A2 + 41 fiij12] li2. This “dynamic Stark splitting” proportional to the laser electric field strength is familar from work on saturated bound-bound absorption spectra [l l] and fluorescence spectra [12]. We note that the doublet profile of the photoelectron energy distribution is produced at fixed w;,there is no need to scan probe frequencies to discern each component as in the bound-bound absorption analogue [l 1 ] since both values of Ef are simultaneously produced. This can be seen best from a “dressed atom” viewpoint [13] where the effect of pan the bound-bound transition is to split the energy levels into doublets (see fig. 2). Then i’is further responsible for transitions from the dressed states to the continuum. We further 176
4
-I-41&12]1/2 .
I i
Ivij.\
Fig. 2. Dressed states of atom plus radiation field showing the dynamic Stark splitting of the ab transition at resonance: W = ‘Jab.
note that the final state energy is shifted by an amount tiffi due entirely to the A2 term. If we express the number of photons in terms of the mean square laser electric field L through the identity (4nwn/V) = & 2 then Vff = (a/m) &2/2w2 familiar from work on socalled “mass-shifts” of free electrons [ 141. It is not usual to energy analyze the emitted photoelectrons (see however ref. [1.5]), but if this were to be done, an electron spectrometer would measure lbf(-)12 summed over a “bandwidth” range of Ef’s. Of course, if the spectrometer is dispensed with and the photoelectrons just counted, our model predicts
yf(“p
=P&+=
1.
The effect of spontaneous emission upon the resonant two-photon ionization is not just to add damping rates to the atomic energy levels. We would expect for example that the spontaneous emission spectrum 1;) back to Ii) would exhibit the by now well known resonant Stark triplet structure [ 121. If only one decay photon into some mode h is allowed in the dynamics, and the correspondingly transition I{n-I }, l,)la> + I {rz-2}, 1 ,)lb> is neglected, then we incorrectly predict a doublet for the fluorescence spectrum. To allow this extra transition, and to account for cascades in the dressed two-level problem is straightforward [I 61; when transitions to the continuum are admitted the problem becomes more complicated. To summarise, we have presented a unified.treat-
Volume 22, number ‘2
OPTICS COMMUNICATIONS
ment of two-photon resonant ionization in a simple model using standard Heitler techniques. With this technique we are able to describe the ionization behaviour for weak, medium and strong interaction of the model atom with the radiation field. The weak interaction limit leads to precisely the usual lowest order perturbation theory result. For moderate intensities the bound-bound resonance saturates and this leads to the intensity-dependent widths and shifts in eqs. (23) and (24) which themselves saturate, and to a splitting of the final photoelectron energy distribution. The insaturated versions of the widths and shifts have, of course, been observed: AE in that case is just the AC Stark shift. The observation of saturation in AE seems to be quite feasible. At stronger field strengths the rate concept breaks down as shown by Beers and Armstrong [3], and the Rabi frequency exhibits itself not only in atomic dynamics but in a splitting of the photoelectron energy which may well be observable. I am grateful to Royal Holloway College for the award of a Jubilee Research Fellowship.
References [1] H.B. Bebb and A. Gold, Phys. Rev. 143 (1966) 1. [2] P. Lambropoulos, Phys. Rev. A9 (1974) 1992. (31 B.L. Beers and L. Armstrong Jr., Phys. Rev. Al2 (1975) 2447. [4] P. Lambropoulos and M. Lambropoulos, in Electron and photon interactions with atoms, eds. H. Kleinpoppen and M.R.C. McDowell (Plenum Press, New York, 1976) p. 528. [S] W. Heitler, The quantum theory of radiation (Clarendon Press, Oxford, 1954).
August 1977
[6] M.L. Goldberger and K.M. Watson, Collision theory (Wiley, New York, 1964). [7] F.H.M. Faisal, J. Phys. B: Atom. Molec. Phys. 9 (1976) 3009. [8] f.L. Knight and P.W. Milonni, Phys. Lett. 56A (1976) 275; also ref. [ 31. (91 For example C. Cohen-Tannoudji, Laser Spectroscopy, eds. S. Haroche, J.C. Pebay-Peyroula, T.W. Hansch and S.E. Harris (Springer-Verlag, 1975) p. 324. [lo] M. Sargent HI, M.O. Scully and W.E. Lamb Jr., Laser physics (Addison Wesley, Reading, Mass., 1974). [ll] S. Feneuille and M.-G. Schweighofer, J. de Physique (Paris) 36 (1975) 781; (121 A. Schabert, R. Keil and P.E. Toschek, Opt. Commun. 13 (1975) 265; A. Schabert, R. Keil and P.E. Toschek, Appl. Phys. 6 (1975) 181; J.L. Pique and J. Pinard, J. Phys. B: Atom. Molec. Phys. 9 (1976) L77; C. Delsart and J.-C. Keller, J. Phys. B: Atom. Molec. Phys. 9 (1976) 2769; Ph. Cahuzac and R. Vetter, Phys. Rev. Al4 (1976) 270. [12] B.R. Mollow, Phys. Rev. 188 (1969) 1969; F. Schuda, C.R. Stroud Jr. and M. Hercher, J. Phys. B: Atom. Molec. Phys. 7 (1974) L 198; F.Y. Wu, R.E. Grove and S. Ezekiel, Phys. Rev. Letters 35 (1975) 1426; W. Hartig, W. Rasmussen, R. Schieder and H. Walther, Z. Physik A 278 (1976) 205. 1131 S.H. Autler and C.H. Townes, Phys. Rev. 100 (1955) 703. E.T. Jaynes and F.W. Cummings, Proc. IRE 51 (1963) 89. [14] J.H. Eberly, Progress in Optics, Vol. VII, ed. E. Wolf (North-Holland, 1969), p. 359. [15] E.A. Martin and L. Mandel, Appl. Optics 15 (1976) 2378. [16] H.J. Carmichael and D.F. Walls, J. Phys. B: Atom. Molec. Phys. 8 (1975) L77. M.E. Smithers and H.S. Freedhoff, J. Phys. B: Atom. Molec. Phys. 8 (1975) 2911; S. Swain, J. Phys. B: Atom. Molec. Phys. 8 (1975) L437.
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