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Ex t

Z. Ficek, Mohamed Ridza Wahiddin

Published Apr 19, 2016• 0 citations
Publisher

Abstract

It is easy to show that the expectation value of the amplitude of the EM field in the photon number state |n〉 is equal to zero. Since 〈n| Eˆ x |n〉 = E 〈n|aˆ + aˆ†|n〉 sin(kz), (4.5) and aˆ |n〉 = √n |n− 1〉 , aˆ† |n〉 = √n+ 1 |n+ 1〉 , (4.6) with 〈n|m〉 = δnm, we obtain 〈n| Eˆ x |n〉 = 0. (4.7) Note that 〈n|Eˆ x |n〉 = 0 independent of n, that is, 〈n|Eˆ x |n〉 = 0 even if n → ∞. This is unusual property of the quantum field as in the limit of n → ∞ the properties of the photon-number state should convert into that corresponding to a classical state. However, we know that the classical field performs harmonic oscillations in time.

DOI
10.1201/b16875-9

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It is easy to show that the expectation value of the amplitude of the EM field in the photon number state |n〉 is equal to zero. Since 〈n| Eˆ x |n〉 = E 〈n|aˆ + aˆ†|n〉 sin(kz), (4.5) and aˆ |n〉 = √n |n− 1〉 , aˆ† |n〉 = √n+ 1 |n+ 1〉 , (4.6) with 〈n|m〉 = δnm, we obtain 〈n| Eˆ x |n〉 = 0. (4.7) Note that 〈n|Eˆ x |n〉 = 0 independent of n, that is, 〈n|Eˆ x |n〉 = 0 even if n → ∞. This is unusual property of the quantum field as in the limit of n → ∞ the properties of the photon-number state should convert into that corresponding to a classical state. However, we know that the classical field performs harmonic oscillations in time.
At the end of the preceding chapter, we showed that the photon number states | n 〉 have a uniform phase distribution over the range 0 to 2π. Essentially, then, there is no well-defined phase for these states and, as we have already shown, the expectation value of the field operator for a number state vanishes. It is frequently suggested (see, for example, Sakurai) that the classical limit of the quantized field is the limit in which the number of photons becomes very large such that the number operator becomes a continuous variable. However, this cannot be the whole story since the mean field 〈 n | Ê x | n 〉 = 0 no matter how large the value of n . We know that at a fixed point in space a classical field oscillates sinusoidally in time. Clearly this does not happen for the expectation value of the field operator for a number state. In this chapter we present a set of states, the coherent states, which do give rise to a sensible classical limit; and, in fact, these states are the “most classical” quantum states of a harmonic oscillator, as we shall see. Eigenstates of the annihilation operator and minimum uncertainty states In order to have a non-zero expectation value of the electric field operator or, equivalently, of the annihilation and creation operators, we are required to have a superposition of number states differing only by ±1.
Suppose, the field Eˆ2 is a coherent laser field (local oscillator) of a large amplitude α, whereas the signal (detected) field is weak. As a result, the terms proportional to 〈bˆ†bˆ〉 and 〈bˆ†〉, 〈bˆ〉 dominate over those without bˆ† and bˆ. Hence, the terms independent of the amplitude of the local oscillator can be discarded. Thus, we can ignore the term β〈aˆ†aˆ〉, and denoting 〈bˆ〉 = |α| exp(iφ), 〈bˆ†〉 = |α| exp(−iφ), the resultant light intensity at the detector is then I ≈ (1− β) |α|2 + 2 |α| √ β (1− β) 〈Eˆφ〉 , (6.102) where 〈 Eˆφ 〉 = 1 2i (〈aˆ〉e−iφ − 〈aˆ†〉eiφ) , (6.103) and φ is the phase of the laser. The first term on the right-hand side of Eq. (6.102) is equal to the intensity of the reflected coherent beam. The second term is an interference term between the coherent and the signals beams. This term contains the phase-dependent quadrature amplitude of the signal beam.
Starting from the well-known plane-wave decomposition of the operator associ- ated with the transverse part of the vector potential (Aˆ T) , and by making use of the operator relations defining the relations between and magnetic (Bˆ [with ∇ · Bˆ = 0]) field operators, the plane-wave mode expansions for Eˆ T and Bˆ are established. Subsequently, the general operator relation ∇ × Eˆ T = −∂ Bˆ/ ∂t follows. From the Heisenberg equation of motion for the magnetic field operator the inhomogeneous Maxwell operator equation ∇ × Bˆ = µ 0 Jˆsym + c − 2 ∂ Eˆ T/∂t is established. Here the operator Jˆsym T is the symmetrized transverse current den- sity operator in the space-time domain. The transverse set of Maxwell operator equations are of utmost importance for the subsequent photon energy operator wave function formalism. For globally coherent quantum fields, where the par- ticle source current density distribution is quasi-classical, dynamical equations for the two helicity species of Riemann-Silberstien field operators are obtained. Neglect of the quantum field fluctuations in the coherent state makes the bridge to photon wave mechanics in the Oppenheimer-Riemann-Silberstein-Bialynicki approach. The chapter is closed with a discussion of response theory and Ein- stein causality.