IL NUOVO CIMENTO VOL. 108 B, N. 11 Novembre 1993

7:/2 Pulse Shaping v/a Analytic Inversion of Bloch Equations(*).

CHANG JAE LEE

Department of Chemistry, Sunghwa University - Chun An 330-150, Korea

(ricevuto il 29 Maggio 1993; approvato il 2 Luglio 1993)

Summary. - - We report here that for an amplitude-modulated 7:/2 pulse the detuning-independent initial pulse profile is found to be given by a delta-function. This was shown by applying the inverse-scattering transform to undamped Bloch equations and solving the Gel'fand-Levitan-Marchenko equation analytically.

PACS 42.30.Kq - Fourier transform optics. PACS 42.50.Md - Optical transient phenomena (including quantum beats, de- phasings and revivals, photon echoes, free induction decay, and optical mutation). PACS 33.20 - Molecular spectra, grouped by wavelength ranges.

In order to describe the dynamics of atom-field interactions in ,,thin- samples, Bloch equations have provided a convenient means, although their failure in certain optical processes has been noted [1]. The equations have been solved analytically for continuous wave and some pulse shapes [2]. For other pulse shapes the equations can be solved by numerical integration. Shaping pulses gives rich structure to resonance fluorescence spectra [3] and finds great practical importance in spectroscopy and in controlling a wide range of chemical and physical processes coherently [4]. A very powerful method for obtaining pulse shapes is the inverse-scattering transform [5] (IST). For example, a family of purely amplitude-modulated 2NT~(N= 1, 2, ...) soliton pulses, which brings the magnetization (or the Bloch) vector back to the initial equilibrium value regardless of detuning, has been found by IST[6]. In usual applications of IST, the calculation of pulse profiles starts with an assumed initial pulse shape [7]--an initial-value problem. However, for certain applications such as the one considered in this paper, one has to determine in the first place the initial pulse shape that gives rise to the desired response. The purpose of this paper is to determine, by applying IST, the right initial ~/2 pulse shape for broadband excitation, which is also of great significance in coherent optics and spectroscopy.

Let us consider the undamped Bloch equation describing the interaction of a

(*) The author of this paper has agreed to not receive the proofs for correction.

1299

1300 CHANG JAE LEE

two-level atom with a time-varying monochromatic field, which in a rotating frame is given by

d (1) - - M = - . Q x M,

dt

where M--- (M~, My, M~) is a magnetization vector and Y2 = (oJ~ (t), 0, A~o) with A(o and o~1 (t) being, respectively, the detuning and the amplitude of the driving field.

In terms of the dimensionless variable V defined by [6, 8]

(2) ~(t) - Mz + iM~

M~-~

eq. (1) can be recast into a generalized Riccati equation

i (3) F~ = io~1~ + -z Aco(V 2 - 1),

Z

which can be further reduced to a standard form

i A ~ - i 5oj[( ~ /2 + 1 2 ~ 1 ] (4) ~ - 2 2 [\ Ao~ / -- iho~ -----~ '

by modifying the variable as ~.-r~ + ~1/AoJ. Finally, the Riccati equation may be linearized as

1 (A(o 2 + ~o~ + 2 i&1)~ = 0 , (5) ~ + 4

in a standard manner by introducing a non-zero differentiable function ~(t) defined by

2i~ (6) ; -

A~o

Equation (5) is in a Sturm-Liouville form and with the change of variables f o x may be recognized immediately as a one-dimensional SchrSdinger equation with the potential V and energy E given by

(7)

- V i d

E = 1 hoj2 .

As in ref.[6], the pulse envelope ~ol can be related to the potential V as o~1 = m. �9 [ R e ( - V ) ] 1/~, where the proportionality constant m is to be determined from the

r:/2 P U L S E SHAPING via ANALYTIC INVERSION ETC. 1301

pulse area

(8) ; o~1 ( t ) d t 7: 2

- a o

Of course, one could equally use the imaginary par t of the potential. Upon inverting eq. (6), one can express ~ in terms of ~ as

(9) [ I ] i [5~ov(t ') + (o~( t ' ) ]d t ' ~(t) = a(h~o)exp - ~

E > 0, so one may consider the eigenfunction only for the continuous spectrum. Since ~ (and hence V) vanishes as t ~ + ~ , the eigenfunction must be shown to be given

by a linear combination of exp[ +_iA~ot/2] asymptotically. Obviously, with the initial condition M -- (0, 0, - 1 ) and with the pulse turned off,

~z(t) behaves as

(10) ~(t) N a (k ) exp[ - ikt] as t ~ - ~ ,

where k = 5oJ/2 and a (k ) is the transmission coefficient. Now suppose that at the end of the 7:/2 pulse (t = tp ) the magnetization vector lies

along the y-axis of the rotat ing frame, so that My = 1 and Mx = M: = 0. Observe that then, at a later t ime t > tp, the magnetization vector will have precessed due to the detuning to give Mx = sin Ao~(t - tp), My = cos A~o(t - tp), and thus

(11) v(t) = i cos Aoj(t -- tp )

sin A~o(t - tp ) - 1

Consequently, the asymptotic behaviour of ~(t) is

(12) ~(t) = a (k)exp [s t ][ ; cos 2k( t ' - tp ) dt ' exp - i k v ( t ' ) d t k 1

tp - ~

[ ; ] E 1[ i ,) 7: �9 exp - ~ _ ~ (ol(t dt ' = a(k)s in k( t - tp) - -~ exp

] - i k I ~ ( t ' ) d t ' �9

(13) F ( x ) = - -

co

1 l 27:

b(k) exp [ ikx] dk = i~(x - 2 t p ) .

�9 exp - i ~ = e x p [ - i k t ] + i e x p [ - 2 i k t p ] e x p [ i k t ] , t o + oo

In the last line all the constants have been absorbed in the transmission coefficient a(k) . The reflection coefficient is seen to be given by b(k) = i exp [ - 2iktp ], which has no poles and hence we confirm that there is no discrete spectrum for eq. (5).

With the initial scattering data given by b(k) one can proceed to define

1302 CHANG J A E LEE

Plugg ing eq. (13) into the G e l ' f a n d - L e v i t a n - M a r c h e n k o (GLM) equat ion gives

(14) K ( x , z ) + i 3 ( x + z - 2 t p ) + i ~ K ( x , y ) 3 ( y + z - 2 t p ) d y = O , z > x > - or x

which is to be solved for K(x , z). Then the potent ia l is calculated f rom

(15) V(x) = - 2 d K(x , x) dx

I f x > 2tp - z, K ( x , z) = 0 and hence K ( x , x) = 0. Thus the potent ia l van ishes as it should, since the pulse is a s s u m e d to t e r m i n a t e a t t = tp.

But if x ~< 2tp - z, one has to solve

(16) K(x , z) + i3(x + z - 2tp) + iK(x , 2tp - z) = O.

To this end, let us decompose K(x , z) as K(x , z) = KR (x, z) + iKi (x, z), where KR and KI a re rea l functions. Then f rom eq. (16) one has

(17a) K a (x, z) - KI (x, 2tp - z) -- 0 ,

(17b) /{i (x, z) + KR (x, 2tp - z) + 3(x + z - 2tp) = 0 .

Solving eq. (17), one ge t s

1 (18a) KI (x, z) = - - d(x + z - 2 tp) ,

2

1 ( 1 8 b ) K R ( x , z ) = - - - 3 ( x - z ) .

2

and, consequent ly , upon se t t ing z = x,

(19a) KR (x, x) = - - - 1

3(0) , 2

1 (19b) KI (x, x) = - -- ~(x - tp) .

4

F r o m the 3-function p r o p e r t y [ 9 ] I d x 3 ( x - b )3 (x - a) = 3(b - a) and hence

f d x 3 ( x - a ) 3 ( x - a ) = 3(0), a being a r b i t r a r y . Especia l ly , for r ea sons t ha t will

b e c o m e evident immed ia t e ly below, one m a y choose a = tp and se t

(20) ~(0) = f d x ' t ' ( x ' - tp) 2 x ~< tp . x

Then

(21) 1 "

K ( x , x) = - -~ d x ' 3 (x ' - tp)2 + 3(x - tp) ,

7:/2 PULSE SHAPING via ANALYTIC INVERSION ETC. 1303

and the potential for x < tp is

(22) V(x) = - ~ ( x - tp)2 -2 dx ~(X - tp) .

The potential is quite singular and not integrable, originating from the fact that the kernel K(x, x) is infinite. Thus the existence of the solution to the Sturm-Liouvflle equation and hence the applicability of the GLM equation may become questionable (see chapt. 3 of ref.[5]). However, since we gave the explicit continuous solution (eq. (9)) to the Sturm-Liouville equation, the problem of non-integrability of the potential does not enter here. Furthermore, remember that it is not the potential but the pulse amplitude that is of direct physical significance. Upon comparing eq. (22) with eq. (7) and the discussion following it, one can see that the conjecture given by eq. (20) is consistent and the pulse envelope is meaningfully given by

= (23) o~(t) = ~- ~(t - tp), - ~ < t < ~ .

It is interesting to note that the scattering data for the =/2 pulse satisfies I b(k) I = 1, meaning that the potential is totally reflecting. This is to be compared with 2N= pulses, which correspond to reflectionless potentials. The cases b(k) = + 1 have been treated by Lamb[10] and Ghosh Roy[ l l ] and the initial potential is given by ~'(x), which is equal to the imaginary part of eq. (22).

In conclusion, the desired amplitude-modulated =/2 pulse has initially a delta-function pulse envelope. By virtue of the uniqueness of the solution K(x, z) of the GLM equation, this is the only initial pulse profile that works as an ideal (i.e., detuning-independent) =/2 pulse. Of course, the motivation for shaping pulses is to avoid brute-force application of such a pulse, which is familiar to spectroscopists as a ,,hard, pulse. Other pulse shapes may be obtained if we allow the potential to evolve according to, for example, the Korteweg-de Vries (KdV) equation. This is certainly possible since both the Bloch and the KdV equations can be transformed into a Sturm-Liouville form. The initial-value problem with the initial scattering data b(k) given by eq. (12) along with the KdV and other evolution equations is under way.

The author wishes to thank Profi C. Turfus for helpful discussions. This work was supported by a grant from Korean Research Foundation.

R E F E R E N C E S

[1] R. G. DEVoE and R. G. BREWER: Phys. Rev. Lett., 50, 1269 (1983). [2] L. ALLEN and J. H. EBERLY: Optical Resonance and Two-Level Atoms (Dover, New York,

N.Y., 1987); B. W. SRORE: Theory of Coherent Atomic Excitation, Vol. I (Wiley, New York, N.Y., 1990).

[3] J. H. ERERLY and K. WODKIEWICZ: J. Opt. Soc. Am., 67, 1252 (1977). [4] For a review see, W. S. WARREN and M. S. SILVER: Adv. Mag. Res., 12, 247 (1988); W. S.

WARREN, H. RABITZ and A. C. DAHLEH: Science, 300, 123 (1993).

1304 CHANG JAE LEE

[5] For an introduction to the method of inverse scattering see, e. g., P. G. DRAZIN and R. S. JOHNSON: Solitons: an Introduction (Cambridge University Press, Cambridge, 1989).

[6] A. GRUNBAUM and A. HASENFELD: Inverse Probl., 2, 75 (1986); 4, 485 (1988). [7] R. MICHALSKA-TRAUTMAN: Phys. Rev. A, 22, 2738 (1981); 23, 352 (1981) and references

therein. [8] G. L. LAM~ jr.: Rev. Mod. Phys., 43, 99 (1971). [9] P. A. M. DIRAC: The Principles of Quantum Mechanics, 4th edition (Oxford University

Press, Oxford, 1958). [10] G. L. LAM~ jr.: Elements of Soliton Theory (Wiley, New York, N.Y., 1980). [11] D. N. GnOSH RoY: Inverse Problems in Physics (CRC Press, Boca Raton, Fla.,

1991).