Quantum-Mechanical Description

So far the effect of the cross-Kerr nonlinearity has been considered in a purely classical context. That is, the quantities considered are all measurable for classical fields. Let us now consider the effect of the cross-Kerr nonlinearity on quantum states of the light field. The Hamiltonian of the cross-Kerr nonlinearity is given by [35]

ˆ

H =~_{Knˆanˆb. (1.60)}

We can derive this Hamiltonian using a method very similar to the quantisation of the free electromagnetic field. In this case we consider the energy shift of the atom due to the electric-dipole interaction with two orthogonal electromagnetic fields subject to the cross-Kerr interaction. The total energy is given by the volume integral over the electric field energy density

H= 1 2

Z

V

16 CHAPTER 1. INTRODUCTION TO QUANTUM OPTICS

Here the electric field is assumed to consist of two components E(r, t) =Ea(r, t) +

Eb(r, t), which are given by

Eα(r, t) = iωαeα

Aαei(k·r−ωαt)−A∗αe−i(k·r−ωαt)

, (1.62)

where ea·eb = 0. The polarisation is given by the cross-Kerr interaction only

P(r, t) = 3 20χ (3) |E(ωb)|2Ea(r, t) + 3 20χ (3) |E(ωa)|2Eb(r, t). (1.63)

The nonlinear susceptibility is assumed to be real, and therefore lossless. Now, we choose to integrate the electric field energy density of a volume _V. For each of the fields we find the interaction energy is given by

H = 60Vχ(3)ωa2ωb2AaA∗aAbA∗b. (1.64)

It is now possible to construct the quantum-mechanical Hamiltonian by using the relationships (1.29-1.30) and (1.33-1.34). After dropping terms associated with the zero-point energy we find that each of the electromagnetic fields will experience an interaction with the atom of the form

ˆ HI = − 3~2_ωaωcχ(3) 20V ˆ nanˆc. (1.65)

The interaction strength is therefore clearly given by

K =₋3~ωaωcχ (3)

20V

. (1.66)

We now ask what effect will this Hamiltonian have on quantum states of the field. Consider the evolution of two electromagnetic fields, both of which are in Fock states. If _|ψ(0)_i=_|nai ⊗ |nbi then at a later time the combined state is given by

|ψ(t)_i=eiKnanbt_|n

ai ⊗ |nbi. (1.67)

Thus, the Fock state experiences a phase shift that is proportional to the product of the photon numbers. This simple interaction forms the basis of many applications of the cross-Kerr effect in quantum information/optics.

Chapter 2

Introduction to Quantum

Electronics

In the first chapter we developed a quantum-mechanical description of light in the presence of a dielectric material. This is the domain of quantum optics. Very closely related, and nowadays seldom differentiated, is the topic of this chapter:

quantum electronics. Whereas quantum optics focuses on the optical fields, quantum electronics considers the effect of photons on the quantum state of electrons from which matter is composed. An understanding of these atom-field interactions has led to important technological developments such as the laser, optical amplifiers and laser cooling.

2.1

The Schr¨odinger Equation

One of the most common problems in life is working out what will happen next. Given that we can estimate the initial state of a system and know approximate rules for its evolution, then we can determine its configuration at a later time.

However, in many of the sciences the discovery of the evolutionary rules remains an outstanding problem. Even when these laws are known estimating the initial conditions or evaluating the result is often impractical. Nonetheless no discipline has developed a greater quantitative understanding than that achieved in physics.

18 CHAPTER 2. INTRODUCTION TO QUANTUM ELECTRONICS

Fortunately in the case of quantum optics, the systems studied can often be modelled with remarkable accuracy using quite straightforward methods.

As physicists we appeal to the framework of mathematics and physical intuition to form equations from which predictions can be made. In the case of quantum mechanics the starting point of our investigations is usually the Sch¨odinger equation

i~∂

∂t|ψ(t)i= ˆH|ψ(t)i, (2.1)

where ˆH is the Hamiltonian, or “total energy operator”. The Hamiltonian defines the energy eigenstates

ˆ

H_|φn(0)i=En|φn(0)i. (2.2)

The Hamiltonian has particular significance in both classical and quantum mechan- ics. In addition to giving the energy the Hamiltonian also generates the evolution of the system via either the classical Hamilton-Jacobi equation [23] or the Sch¨odinger equation. This dual role means that the eigenstates of the Hamiltonian are also steady states of the probability distribution. The number of eigenstates of the Hamiltonian is equal to the dimension of the quantum system. The evolution of each is simply given by

|φn(t)i= exp(−iEnt/~)|φn(0)i. (2.3)

Since these eigenstates form a basis for solutions of the Schr¨odinger equation, then the evolution of any pure quantum state can be decomposed in terms of these func- tions. This provides a powerful and straightforward method for determining solu- tions of the Schr¨odinger equation.