Ultrashort Laser Pulse Phenomena, 2nd Edition

250 Coherent Phenomena

of Fig. 4.8(c), with amplitude:

˜

E(t − t

1

) +

˜

E(t − t

2

)e

iϕ

1

+

˜

E(t − t

3

)e

iϕ

2

+

˜

E(t − t

4

)e

iϕ

3

(4.69)

where t

i

denotes the pulse delay. The interaction with a single pulse can be

represented by a matrix that transforms any set of initial c

i

(t

0

) into a set of

coefficients c

i

(t

+

1

) (where t

+

1

designates the time just following the pulse). The

phase of each pulse corresponds to a rotation of that matrix. Between pulses, all

the

˜

E

i

in Eq. (4.68) are zero, and the interaction matrix is diagonal. The solution

of the system of equations (4.68) becomes in this case particularly simple. The

eigenvalues are the successive detunings given by Eq. (4.57). For the four-level

system, the coefficients c

i

after the first pulse will evolve according to:

c

0

(t