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An approach to the study of population transfer through the continuum

R. BUFFA(*)

Dipartimento di Fisica, Università di Firenze and Istituto Nazionale per la Fisica della Materia Largo Enrico Fermi 2, 50125 Firenze, Italy

(ricevuto il 19 Giugno 1997; approvato il 31 Luglio 1997)

Summary. — The question of how efficient the transfer of population through

continuum intermediate states in a two-photon resonant Raman transition can be is addressed. The problem is studied by using an optimal control technique, looking for the optimal laser pulse shapes which maximize the population transfer. A phe-nomenological analytical expression for the maximum transfer of population obtainable for a given value of the Fano parameter is provided.

PACS 32.80.Rm – Photon interactions with atoms: multiphoton ionization and excitation to highly excited states (e.g., Rydberg states).

PACS 42.50.Hz – Strong-field excitation of optical transitions in quantum systems; multi-photon processes; dynamic Stark shift.

1. – Introduction

When two laser pulses of proper frequency interact with a L or ladder three-level atomic or molecular system, the population can be transferred from the initially populated state Nab to the initially unpopulated final state Nbb through the excitation of the intermediate state Ncb. The efficiency of the transfer process is critically dependent on the frequency, intensity and temporal shape of the laser pulses. However, if the two laser pulses are applied in a counterintuitive temporal sequence (i.e. the laser pulse, coupling the initially unpopulated states Nbb and Ncb, is applied before the laser pulse coupling the states Nab and Nbb), then the high efficiency obtainable for the transfer process, under two-photon resonance condition, results quite independent of the frequency, intensity and temporal shape of the laser pulses. Moreover, since the transfer of population is accomplished without a significant population—at any time—of the intermediate state, the result is valid also with a decaying intermediate state. The effect, well established experimentally [1], has been explained as due to the

(*) E-mail: buffaHfi.infn.it

adiabatic temporal evolution of a non-decaying coherent superposition of states not involving the intermediate state [2].

In 1992, Carroll and Hioe [3] have argued that, even though the single intermediate state is replaced by a continuum of states, a population transfer approaching unity can be obtained with the counterintuitive temporal sequence for the laser pulses. This might have important implications for selective excitation of atoms and molecules. However, Nakajima et al. [4] have shown that the special assumptions used by Carroll and Hioe in their model are too restrictive to represent a real situation, and that a complete population transfer is not in general possible through continuum intermedi-ate stintermedi-ates. However, the work of Nakajima et al. [4] simply provides an example for atomic sodium, and leaves open the question whether the process is actually experimentally observable or rather washed away by incoherent photoionization induced by the high laser intensities used in their calculations.

More recently, Nakajima and Lambropoulos [5] have studied the effect of an autoionizion structure, Carroll and Hioe [6] suggested a proper choice of laser intensities and detuning to improve the population transfer, and Yatsenko et al. [7] proposed using auxiliary laser pulses to overcome the effect of dynamic Stark shifts.

In this paper I wish to present a different approach to the study of population transfer through continuum intermediate states, looking for the optimal laser pulse shapes which maximize the transfer of population.

2. – Model

Figure 1 shows a schematic energy-level diagram of the model considered. The atomic system is assumed to be described by two bound states Nab and Nbb, of same parity and energy Ej4 ˇvj( j 4a, b), and by a continuum of free states NEb, of opposite

parity and energy E 4ˇvE. A laser pulse, of frequency v1and intensity I1(t), couples the initially populated state Nab and the continuum, while a laser pulse, of frequency v2 and intensity I2(t), couples the initially unpopulated state Nbb and the continuum. This scheme is usually referred to as laser-induced continuum structure (LICS) and it has been studied both theoretically and experimentally [8, 9].

Writing the Schrödinger equation and following a procedure described in many papers (see for instance ref. [8]), it is possible to eliminate the probability amplitudes of the continuum states and to write the equations of motion for the probability

amplitudes a and b of the bound states as follows:

.

`

/

`

´

g

h

In (2.1), D 4 (v11 va) 2 (v21 vb) is the two-photon detuning of the Raman

tran-sition coupling the bound states, Gj(t) 4sjIj(t) are the photoionization rates of state

Nab due to laser 1 and state Nbb due to laser 2 (sjfphotoionization cross-sections

of the bound states), G 4 (G1G2)1 O2 is the two-photon Rabi frequency of the Raman transition, and q is a dimensionless atomic parameter, independent of the laser pulses parameters—usually referred to as Fano parameter—which describes the relative weight of the two interfering pathways leading to ionization of the atom. For NqNb1, photoionization of state Nab is the dominant process, while, for NqNc1, two-photon Raman excitation of state Nbb, followed by photoionization, is dominant.

Equations (2.1) neglect the multiphoton ionization of state Nab due to laser pulse 2, the single-photon ionization of state Nbb due to laser pulse 1 and the dynamic Stark shift of continuum states. Depending on the intensities of laser pulses, these processes can play a decisive role in actual experiments where they should be carefully evaluated in order to tie theoretical prediction to experimental results. The presence of other off-resonance bound states modifies the atomic expression of q but leaves the structure of eqs. (2.1) unchanged [10].

For overlapping laser pulses such that G1(t) 4G2(t), and under two-photon resonance condition, the analytical solution of (2.1) provides the following expression for the population Pbof state Nbb [8]:

Pb(t) 4 ]11e22 U(t)2 2 cos [qU(t) ] e2U(t)( O4 ,

(2.2) where U_{(t) 4}



The plot of (2.2) as a function of U shows a damped oscillation whose peak value depends on the q parameter. In figs. 2 and 3 such peak value Pbm for the population

transfer and the corresponding value Um of U, where (2.2) reaches its maximum, are reported vs. the q Fano parameter. For two-photon detunings D c 0 , the solution of (2.1) provides reduced peak values for the population transfer. Then, fig. 2 gives the value of the maximum transfer of population obtainable with overlapping pulses, provided that G1(t) 4G2(t) and U(tf) 4Um. For q KQ the transfer of population approaches unity while UmB pOq .

2.1. Adiabatic states. – In terms of the new variables . / ´ a 4cos ua2sin ub , b 4sin ua1cos ub , (2.4)

Fig. 2. – Maximum transfer of population vs. q Fano parameter obtainable with overlapping laser pulses such that G1(t) 4G2(t).

with tan u 4 (G1OG2)1 O2, which are the probability amplitudes of the adiabatic states .

/ ´

Nab 4 cos uNab 2 sin uNbb , Nbb 4 sin uNab 1 cos uNbb , (2.5)

Fig. 3. – Area of the two-photon Rabi frequency producing the transfer of population shown in fig. 2 vs. Fano parameter.

eqs. (2.1) can be cast, under two-photon resonance condition, in the form

.

/

´

g

h

.

`

/

`

´

which shows how, for laser pulses such that G1(t) 4G2(t), the non-decaying Nab state and the decaying Nbb state are uncoupled for any value of the q parameter. However, as shown by fig. 2, the transfer of population approaches unity only for q KQ. On the contrary, for non-overlapping laser pulses, the adiabatic states (2.5) are uncoupled only if q 40 [4], and Ndu(t)OdtNbG1(t) 1G2(t) (adiabatic interaction). In this case, a transfer of population approaching unity can be obtained with a counterintuitive laser pulse sequence [4].

In this paper the question of what is the maximum transfer of population obtainable for a finite q is addressed. The problem is studied using an optimal control technique [11], looking for the optimal laser pulse shapes which maximize the transfer of population from Nab to Nbb. A similar technique was developed by Wang and Rabitz [12] in a study of optical pulse propagation in a medium of three-level systems.

3. – Optimal control

Having introduced a real state vector X, whose components Xj are obtained from

a 4X11 iX2 and b 4X31 iX4 and four real Lagrange multipliers lj, we define a

Hamiltonian functional as

H [X , l , G] 4lTAX ,

(3.1)

where l is the column vector whose j-component is given by lj, A is the real 4 3 4

matrix obtained by writing (2.1) as dXOdt4AX, and G is the control vector, whose components are given by G1and G2.The minimum of Pontryagin theorem [11] requires,

for G to be the optimal control Gopt, that the following equations must be satisfied: X n j4 ¯H ¯lj , (3.2a) l n j4 2 ¯H ¯Xj , (3.2b) 0 4 ¯H ¯Gj

N

The initial conditions for (3.2a)—which are equivalent to (2.1)—are given by

X(ti) 4

C

`

`

`

D

E

`

`

`

F

describing the atom initially in state Nab, while the final conditions for (3.2b) are given by lj(tf) 4 ¯f ¯Xj

N

where F(X) is the objective functional which has to be maximized at the final time

t 4tf. In our case, F(X) 4 (X3)21 (X4)2is the population of state Nbb.

The two-point boundary-value problem, represented by eqs. (3.2a), (3.2b) with boundary conditions (3.3) and (3.4), can be numerically solved by using the following iterative gradient algorithm:

i) choose an initial trial control vector G;

ii) compute from (3.2a) the temporal evolution of the state vector X with initial conditions (3.3) and compute the objective functional F(X);

iii) compute from (3.2b) the temporal evolution of the Lagrange multipliers lj

with final conditions lj(tf) obtained from (3.4);

iv) update the new control vector as G JG1K˜G(H), where ˜G(H) is a column

vector whose j-component is given by ¯HO¯Gjand K is a properly chosen 2 32 diagonal

matrix which maximizes the objective functional F(X) at time t 4tf;

v) repeat ii) to iv) until F[X(tf) ] converges and some stopping condition is satisfied.

For the optimal control Gopt, as a consequence of (3.2) and boundary conditions, the Hamiltonian functional (3.1) must be equal to zero over time. Then, some scalar function of H can be used to set the stopping condition in v).

4. – Results and discussion

Equations (3.2) have been numerically solved by using the algorithm discussed in sect. 3 for different values of the q Fano parameter. As initial trial control, a family of temporally overlapping laser pulses has been used, such that G1(t) 4G2(t) and U(tf) 4 Um. These pulses, with sin2 rise and fall edge, cover from squared to smooth shapes with same peak value, duration (FWHM) and total energy. Two pairs of such pulses are shown in fig. 4 for the case q 44. This value of the Fano parameter corresponds to the most studied case of atomic sodium [9], where Nab 4N3sb and Nbb 4N5sb. G0and t0are normalization factors such that G0t04 1 . For t04 1 ns, the laser pulses shown in fig. 4 have peak intensities of the order of 1 GWOcm2for atomic sodium [4].

As a stopping condition for the solving algorithm, the following dimensionless control parameter:

CP 4



ti

tf

NH(t) Ndt

has been introduced, and the iterative algorithm interrupted when CP has been considered sufficiently near to zero.

In fig. 5 the population transfer vs. the number of iterations of the solving algorithm is reported for the initial trial sin2_{laser pulses shown in fig. 4 with solid lines. Starting} from the value of 0.55 (see fig. 2), the population transfer increases quite rapidly towards the limit value of 0.69, reaching almost 99% of the final value after 10 iterations. At the same time the control parameter decreases to negligible values (fig. 6). Initial trial pulses of different shapes show the same behaviour and lead to the same value of 0.69. In fig. 7, the optimal pulses, obtained after 70 iterations starting from the initial trial pulses shown in fig. 4, are reported. The two pairs of optimal pulses overlap in a counterintuitive temporal sequence, with comparable shapes in the overlapping region and almost identical two-photon Rabi frequency (fig. 8). Within the framework of the present model, which neglects ionization of state Nab (Nbb) by laser 2

Fig. 5. – Population transfer vs. number of iterations for q 44 and for the initial smooth trial laser pulses shown in fig. 4 with solid lines.

(1), the non-overlapping parts of the optimal pulses shown in fig. 7 are unessential. In fact, for a counterintuitive temporal sequence, only the overlapping part of the laser pulses produces population transfer. It has to be expected that the inclusion in the model of incoherent ionization channels, as well as dynamic Stark shifts, might lead to a unique pair of optimal pulses. Moreover, the relatively weak peak intensity found for the optimal laser pulses—almost two orders of magnitude lower than the one used by Nakajima et al. [4] in their calculations (I10B 3 GWOcm2 for the smooth pulses of

Fig. 6. – Control parameter vs. number of iterations for q 44 and for the initial smooth trial laser pulses shown in fig. 4 with solid lines.

Fig. 7. – Optimal laser pulses obtained after 70 iterations for q 44 and the initial trial laser pulses shown in fig. 4.

fig. 7)—indicates that incoherent photoionization might have in most of the cases only a minor effect on population transfer. This aspect, as well as the effect of dynamic Stark shift, chirping and non-resonance condition, is at present time under investigation, and will be discussed elsewhere.

The analysis presented in detail for q44 has been repeated for different values of the Fano parameter, and the main result of this systematic investigation is reported in fig. 9, where the maximum population transfer obtained for a given q is reported vs. the Fano

Fig. 8. – Optimal two-photon Rabi frequency Gopt(t) obtained after 70 iterations for q 44 and the

Fig. 9. – Maximum population transfer vs. Fano parameter. The solid line reproduces the phenomenological expression F(q) 4exp [22qO(1.351q1q2_{) ].}

parameter. The solid line represents the phenomenological analytical expression

F(q) 4exp [2aqO(b1q1q2_{) ]}

which has been found to fit very well the numerical data for a 42 and b41.35. Compared to the data of fig. 2, the data of fig. 9 show a remarkable increase of population transfer achievable with laser pulses, of proper intensity and temporal shape, applied in a counterintuitive temporal sequence, especially for low values of the Fano parameter. The efficiency is in any case larger than 50%. It may also be worth underlining that the minimum of the curve shown in fig. 9 occurs for q 41, corresponding to the physical condition where the two interfering pathways leading to ionization of the atoms have equal weight. The data reported in fig. 9 confirm some of the results obtained by Carroll and Hioe [6].

5. – Conclusions

In conclusion, the question of how efficient the transfer of population through continuum intermediate states in a two-photon resonant Raman transition can be has been addressed. A phenomenological expression for the maximum transfer of population obtainable for a given value of the q Fano parameter has been provided. The problem has been studied applying an optimal control technique, which appears as a promising tool for the optimization of coherent interaction processes between laser fields and atomic systems.

* * *

It is a pleasure to thank Dr. MANLIO MATERA for a critical reading of the manuscript.

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