ABSTRACT
Since each of the monochromatic (ωq = cq) plane-wave modes in the expansion of the
transverse vector potential operator (AˆT (r, t)) given in Eq. (15.30) satisfies the free-space
wave equation [ exp[i(q · r− cqt)] = 0], AˆT (r, t) also obeys this equation; that is,
AˆT (r, t) = 0. (18.1)
Starting from Eqs. (15.33), (15.34), and (18.1), one can derive a set of Maxwell operator equations for free space form-identical to the ones given for the classical fields in Eqs. (2.1) and (2.2). Thus, by taking the curl of Eq. (15.33) and using Eq. (15.34), one obtains
∇× EˆT (r, t) = − ∂ ∂t Bˆ(r, t). (18.2)
The operator equation associated to Eq. (2.2) is gotten by first acting with the curl operator on Eq. (15.34). Hence,
∇× Bˆ(r, t) =∇× ( ∇× AˆT (r, t)
) = −∇2AˆT (r, t) = −c−2 ∂
∂t2 AˆT (r, t). (18.3)
The third member of Eq. (18.3) follows because ∇ · AˆT (r, t) = 0, and the last member is a consequence of Eq. (18.1). Eliminating AˆT in favor of EˆT by utilizing Eq. (15.33), one finally obtains
∇× Bˆ(r, t) = c−2 ∂ ∂t EˆT (r, t), (18.4)
that is, the operator form of Eq. (2.2). Having established the Maxwell operator equations it is of interest to try to find the
quantum state |{αi}〉, i ≡ q, s, which in the best possible manner reproduces the classical Maxwell equations. Specifically, we require that the quantum mechanical mean values of the Maxwell operator equations in the state |{αi}〉 coincide with the classical Maxwell equations. We have seen above that the quantized Maxwell equations can be obtained starting from Eqs. (15.33), (15.34), and (18.1). It is obvious therefore that to reach our goal we just need to require that the mean value of the transverse vector potential operator coincides with the classical transverse vector potential, i.e.,
〈{αi}|AˆT (r, t)|{αi}〉 = AT ({αi}; r, t). (18.5)
In the notation on the right side of Eq. (18.5), {αi} refers to the set of amplitudes, {αi(0)} =
of
{αqs(0)}, which enter the monochromatic plane-wave expansion of a given AT (r, t); see Eq. (15.11). The canonical quantization procedure discussed in Sec. 15.1 results in the transcription in Eq. (15.30), and from this it is clear that the condition in Eq. (18.5) is equivalent to
〈{αi}|aˆi(0)|{αi}〉 = αi, ∀i. (18.6) In order that the mean values of the Hamilton operator for the transverse electromagnetic field [with the vacuum contribution omitted] and the transverse field momentum operator coincide with the corresponding classical quantities, it is required that
〈{αi}|aˆ†i (0)aˆi(0)|{αi}〉 = α∗iαi, ∀i, (18.7) as the reader readily may realize comparing Eqs. (15.94) and (15.44) to Eqs. (15.12) and (15.42). Introduction of the operator
bˆi(0) = aˆi(0)− αi1ˆ, (18.8)
where 1ˆ is the identity operator, allows one to write Eqs. (18.6) and (18.7) as follows:
〈{αi}|bˆi(0)|{αi}〉 =0, ∀i, (18.9) 〈{αi}|bˆ†i (0)bˆi(0)|{αi}〉 =0, ∀i. (18.10)
To obtain Eq. (18.10) use has been made of Eq. (18.9) and its complex conjugate form. Eq. (18.10) can also be written in the form
∥∥∥bˆi(0)|{αi}〉∥∥∥2 = 0, (18.11) where ‖ · · · ‖ denotes the norm. Since the norm of bˆi(0)|{αi}〉 is zero, one has
bˆi(0)|{αi}〉 = 0, ∀i. (18.12) A result which in turn implies that Eq. (18.9) is satisfied. Reintroducing aˆi(0) in the last equation, we get
aˆi(0)|{αi}〉 = αi(0)|{αi}〉, ∀i. (18.13) The result in Eq. (18.13) shows that |{αi}〉 is a tensor product state, namely,
|{αi}〉 = |α1〉 ⊗ |α2〉 ⊗ · · · ⊗ |αi〉 ⊗ · · · ≡ ∏ i
|αi〉, (18.14)
with
aˆi(0)|αi〉 = αi(0)|αi〉, ∀i. (18.15) Because of the requirement in Eq. (18.5), |{αi}〉 is called a quasi-classical state, or a globally coherent state. The state |αi〉 is called a single-mode (i) coherent state [155, 214, 77, 78]. It appears from Eq. (18.15) that |αi〉 is an eigenstate for the T-photon annihilation operator with eigenvalue αi. Since aˆi(0) is a nonhermitian operator the eigenvalues (αi) are not real in general. It follows from Eq. (18.15) that
〈αi|aˆ†i (0) = α∗i (0)〈αi|, ∀i, (18.16)
as well as
Aˆ (+) T (r, t)|{αi}〉 = A(+)T ({αi}; r, t)|{αi}〉, (18.17)
〈{αi}|Aˆ(−)T (r, t) = A(−)T ({αi}; r, t)〈{αi}|. (18.18) The quasi-classical ket (|{αi}〉) and bra (〈{αi}|) states hence are eigenstates for the positive (Aˆ
(+) T ) and negative (Aˆ
(−) T )-frequency parts of the transverse vector potential operator,
respectively. The associated eigenvalues are just the positive (A (+) T ) and negative (A
frequency parts of the corresponding classical potential. States containing a fixed number of photons cannot be quasi-classical states since the
mean value of AˆT (r, t) vanishes in any number state:
〈{ni}|AˆT (r, t)|{ni}〉 = 0, (18.19) for all |{ni}〉 states. If the coherent states are defined by Eq. (18.13), one may include the global vacuum state, |0〉, among the coherent states, but the photon vacuum is a concept outside the framework of classical electrodynamics. The single-photon states in focus in this book are not quasi-classical states, and as such these states have no corresponding classical fields.
