Quantum Phenomena in Simple Systems in Nonlinear Optics

Abstract In this chapter we will analyse some simple processes in nonlinear optics where analytic solutions are possible. This will serve to illustrate how the formalism developed in the preceding chapters may be applied. In addition, the simple exam- ples chosen illustrate many of the quantum phenomena studied in more complex systems in later chapters.
This chapter will serve as an introduction to how quantum phenomena such as
photon antibunching, squeezing and violation of certain classical inequalities may occur in nonlinear optical systems. In addition, we include an introduction to quan- tum limits to amplification.



5.1 Single-Mode Quantum Statistics


A single-mode field is the simplest example of a quantum field. However, a num- ber of quantum features such as photon antibunching and squeezing may occur in a single-mode field. To illustrate these phenomena we consider the degenerate para- metric amplifier which displays interesting quantum behaviour.



5.1.1 Degenerate Parametric Amplifier


One of the simplest interactions in nonlinear optics is where a photon of frequency 2? splits into two photons each with frequency ?. This process known as para-
metric down conversion may occur in a medium with a second-order nonlinear sus- ceptibility ? (2). A detailed discussion on nonlinear optical interactions is left until
Chap. 9.
We shall make use of the process of parametric down conversion to describe a parametric amplifier. In a parametric amplifier a signal at frequency ? is amplified by pumping a crystal with a ? (2) nonlinearity at frequency 2?. We consider a simple


73


model where the pump mode at frequency 2? is classical and the signal mode at frequency ? is described by the annihilation operator a. The Hamiltonian describing
the interaction is
H = k?a†a ? ik ? .a2e2i?t ? a†2e?2i?t . , (5.1) where ? is a constant proportional to the second-order nonlinear susceptibility and
the amplitude of the pump. If we work in the interaction picture we have the time-
independent Hamiltonian


HI = ?ik 2 .a

The Heisenberg equations of motion are


? a†2.


. (5.2)




da 1
=
dt ik


[a, HI]= ? a†,

da† dt

= .a†, HI . = ? a . (5.3)
ik


The interaction picture can be viewed equivalently as transforming to a frame rotat- ing at frequency ?.
These equations have the solution
a (t)= a (0) cosh ?t + a† (0) sinh ?t , (5.4) which has the form of a generator of the squeezing transformation, see (2.60). As
such we expect the light produced by parametric amplification to be squeezed. This can immediately be seen by introducing the two quadrature phase amplitudes
a ? a†

X1 = a + a†, X2 =
i

which diagonalize (5.2 and 5.3)

(5.5, 5.6)


dX1 = +? X , dX2 =


? X . (5.7, 5.8)

dt 1

dt ? 2


These equations demonstrate that the parametric amplifier is a phase-sensitive am- plifier which amplifies one quadrature and attenuates the other:
X1 (t)= e?t X1 (0) , (5.9)
X2 (t)= e??t X2 (0) . (5.10)

The parametric amplifier also reduces the noise in the X2 quadrature and increases the noise in the X1 quadrature. The variances V (Xi,t) satisfy the relations
V (X1,t)= e2?tV (X1, 0) , (5.11)
V (X2,t)= e?2?tV (X2, 0) . (5.12)


5.1 Single-Mode Quantum Statistics 75
For initial vacuum or coherent states V (Xi, 0)= 1, hence
V (X1,t)= e2?t ,
V (X2,t)= e?2?t ,

(5.13)
and the product of the variances satisfies the minimum uncertainty relation V (X1) V (X2)= 1. Thus the deamplified quadrature has less quantum noise than the vacuum level. The amount of squeezing or noise reduction is proportional to the strength of the nonlinearity, the amplitude of the pump and the interaction time.