By François Norguet

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**Example text**

It is commonly thought to be difficult to represent pulses of light using quantum field theory; indeed it is impossible to do so with internal consistency if a number state basis is employed. This is because such states are associated with infinite phase uncertainty (Loudon 2000), precluding their coherent superposition as a wavepacket. 3 Pump Photonics 49 uct of coherent states with a definite phase relationship, it is a relatively straightforward matter to model pulsed laser radiation (Andrews 1978).

These, characterised for any given radiation mode by minimisation of the uncertainty in phase and occupation number (Louisell 1973, Loudon 2000), are eigenstates of the corresponding annihilation operators, satisfying the equation k ak ja kY ki a kY kja kY ki Y 3X2X1 where a kY k is a complex number whose modulus relates to the mean photon number hqi through y k k hqi ha kY kjak ak ja kY ki ja kY kj2 X 3X2X2 3) For a given radiation mode, the coherent states are expressible in terms of the I n Á À a corre sponding number states through jai exp À 12 jaj2 n3 jni.

Annihilation of one photon from the pump mode is associated by the transition of a molecule, initially in its ground state j0i, to a virtual state jri. With the annihilation of a second pump photon the molecule proceeds to a state jsi. The sequence concludes with the emission of a second harmonic photon, the mol- 35 36 2 Perturbation Theory Fig. 2 The three time-ordered diagrams for SHG. Pump photons of wave-vector k and polarisation k impinge on molecular world line from the left and the harmonic kH Y kH leaves from the right.