By S S Schweber
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182) The states |α with α ∈ C are called ‘coherent states’. It is straightforward to verify that |α is an eigenstate of the annihilation operator aˆ : aˆ |α = α|α , 3 This is the most general pure state for a single mode. We will consider mixed states in Chapter 3. 183) 39 Fig. 7. 5 Quantum states of the electromagnetic field The complex state space of coherent states. The state |α is represented by a point 2α. It is surrounded by a circle, which indicates the region of uncertainty. The displacement operator moves the point 2α around the plane.
98) is a solution to the paraxial wave equation. In the quantum theory, the single-field mode in Eq. 98) is associated with the annihilation operator aˆ (k). The Gaussian function then describes the appropriate amplitude function in the transverse to the beam direction. A physical state of a photon would require a further regularization of the continuum of modes in the direction of propagation. For a mode function strongly peaked around wave vector k0 we can then assign the photon a frequency ω0 = ck0 .
In the exponential form of U (t), the Hamiltonian is often called the ‘generator’ of U . For convenience, we define μ = −it/ , such that we have A(μ) = exp(μH)A exp(−μH) . 118) A general Taylor expansion of A(μ) around μ = 0 can be written as A(μ) = A(0) + μ dA dμ μ=0 + μ2 d 2 A 2! 119) 26 The quantum theory of light Next, we evaluate the derivatives of A to μ in Eq. 119) using Eq. 118) and i dU (t) = HU (t) . 121) and so forth. This leads to the Baker–Campbell–Hausdorff relation A(μ) = A(0) + μ[H, A(0)] + μ2 [H, [H, A(0)]] + · · · 2!