In Sec. 15.4 we introduced polychromatic single-photon states in Hilbert space [Eq. (15.80)], |Φ〉 = L− 32 ∑ i φi|Pi〉, i = (q, s), (16.1) by linear superposition of monochromatic plane (P) one-photon states |Pi〉 ≡ |1i〉 = aˆ†i |0〉. (16.2) In the one-photon subspace the |Pi〉 states satisfy the orthonormality, 〈Pj |Pi〉 = δij , (16.3) and completeness ∑ i |Pi〉〈Pi| = 1 (16.4) relations. Instead of taking the |Pi〉’s as a basis set for the expansion of |Φ〉, one may use different sets of one-photon wave-packet states, as we now shall see. The generalization to wave-packet expansion is of importance from a fundamental point of view, and in this chapter we shall realize the usefulness of such an expansion for single-photon correlation studies.
In Sec. 15.4 we introduced polychromatic single-photon states in Hilbert space [Eq. (15.80)], |Φ〉 = L− 32 ∑ i φi|Pi〉, i = (q, s), (16.1) by linear superposition of monochromatic plane (P) one-photon states |Pi〉 ≡ |1i〉 = aˆ†i |0〉. (16.2) In the one-photon subspace the |Pi〉 states satisfy the orthonormality, 〈Pj |Pi〉 = δij , (16.3) and completeness ∑ i |Pi〉〈Pi| = 1 (16.4) relations. Instead of taking the |Pi〉’s as a basis set for the expansion of |Φ〉, one may use different sets of one-photon wave-packet states, as we now shall see. The generalization to wave-packet expansion is of importance from a fundamental point of view, and in this chapter we shall realize the usefulness of such an expansion for single-photon correlation studies.
It appears from the description in Sec. 15.3 that a global state containing a single photon in each of the modes i = (q, s) and j = (q′, s′) has the form |ψ〉 = |ψij〉 ⊗ ∏ k 6=i,j |0k〉, (33.1) where |ψij〉 = |1i〉 ⊗ |1j〉 = aˆ†i (0)aˆ†j(0)|0i〉 ⊗ |0j〉. (33.2) A Fock state containing precisely two photons also can be constructed by putting the two quanta into the same mode (say i), that is |ψij〉 = |2i〉 ⊗ |0j〉 = 1√ 2 [ aˆ†i (0) ]2 |0i〉 ⊗ |0j〉. (33.3) By linear superposition of Fock modes of the type given in Eq. (33.1) a general polychromatic two-photon state, |Φ2〉, can be formed: |Φ2〉 = 1 2 L−3 ∑ i,j φij aˆ † i (0)aˆ † j(0)|0〉. (33.4) The factor 1/2 appearing in Eq. (33.4) originates in our wish to end up with a relation between the two-photon wave functions in direct and reciprocal space having the same structural form as in the one-photon case [Eq. (15.125)]. The quantities φij are amplitude weight factors characterizing the various |Φ2〉-states. The form of the superposition in Eq. (33.4) shows that only the sum φij + φji can play a physical role, and without loss of generality one therefore may set φji = φij . (33.5) The inner product of |Φ2〉 with itself is readily obtained utilizing that 〈0|aˆkaˆlaˆ†i aˆ†j|0〉 = δkiδlj + δkjδli. (33.6) of Thus, 〈Φ2|Φ2〉 =L −6 ∑ i,j,k,l = L−6 ∑ i,j,k,l φijφ ∗ kl (δkiδlj + δkjδli) = L−6 ∑ i,j φij ( φ∗ij + φ ) = L−6 ∑ i,j |φ∗ij |2. (33.7) The two-photon state is normalized, i.e., 〈Φ2|Φ2〉 = 1, (33.8) provided the weight factors satisfy the condition L−6 ∑ i,j |φij |2 → 1 |φs,s′(q,q′)|2 d 3q (2π)3 d3q′ (2π)3 = 1. (33.9) By acting on |Φ2〉 with the global number operator one obtains Nˆ |Φ2〉 = 1 2 L−3 ∑ i,k,l φklaˆ † i aˆiaˆ = 1 2 L−3 ∑ i,k,l ( aˆ†kaˆi + δik ) aˆ†l |0〉 = 1 2 L−3 ∑ i,k,l φkl [ aˆ†i aˆ † l δik + aˆ ( aˆ†l aˆi + δil )] |0〉 = 1 2 L−3 ∑ ∑ i,k = 1 2 L−3 ∑ i,j (φij + φji) aˆ † i aˆ † j |0〉. (33.10) In view of Eqs. (33.4) and (33.5), one finally has Nˆ |Φ2〉 = 2|Φ2〉. (33.11) The polychromatic state |Φ2〉 hence is an eigenstate for the global number operator with eigenvalue 2, a result which of course follows from the construction in Eq. (33.4).
A brief account of the connection between the photon little group and gauge in- variance is given, considering the four-potential as the wave function of the photon. In most textbooks single photons are associated to the quanta of monochromatic plane-wave components of the electromagnetic field. However, polychromatic superposition of such states may also represent single photons. Having established the framework for polychromatic single-photon states in Hilbert space, and upon a discussion of the mean value and variance of the photon Hamilton operator for polychromatic states, we turn our attention to- wards the single-photon description in direct space. A photon wave function in direct space is obtained via the inner product of the photon mean position state (introduced in Chapter 3 ) and the Hilbert space polychromatic single-photon state. Noting that the photon is a relativistic object a rigorous polychromatic photon wave function is, in the continuum superposition of monochromatic plane-wave components, obtained noting the form of the relativistically invari- ant volume element in wave-vector space. We establish a wave-packet basis for single-photon states in Hilbert space, and the related photon wave functions. Thereafter, multi-photon wave packets are discussed, and the equal-time com- mutator relations between the wave packets are discussed, and the equal-time commutator relations between the wave-packet annihilation and creation oper- ators set up. As an example, we focus on two-photon wave packet states. We finish the chapter by a discussion of the two-photon concept and two-photon wave mechanics, analysing in particular the physical difference between two photons and bi-photons (an entangled state of two photons).
One-photon sources are important elements in quantum optics. The archetypal example is an atom raised to an excited state at time t = 0, then de-exciting with emission of a single photon. The development of this kind of source depends on progress with experimental techniques, e.g. the possibility of isolating a single atom, molecule or quantum well. In this complement, we present the formalism for describing the corresponding radiation, and use it to discuss some spectacular experiments which bring out properties quite incompatible with a classical description of the electromagnetic field. We begin in Section 5B.2 by describing the anti-correlation between detections on either side of a semi-reflecting mirror, establishing the quantitative difference with a classical field. Section 5B.3 discusses a quantum optical effect that was only demonstrated at the beginning of the twenty-first century, namely the quantum coalescence of two one-photon wave packets on a semi-reflecting mirror, which occurs even when the two photons were emitted by independent atoms. An analogous effect, the Hong–Hou–Mandel effect, is discussed in Chapter 7. These effects exemplify quantum interference involving two photons. Finally, Section 5B.4 is concerned with quantum calculations involving quasi-classical states. As we now know, this leads to results that are identical to the predictions of semi-classical theory.
The physics of the photon vacuum, one-photon wave packets, and bi-photons are the subjects of the books SECTION II. Here we present an overview of these subjects paying particular attention to fundamental aspects. Cavity elec- trodynamics is often used to study free-photon states. However, such states constitutes an idealization since photon-matter interaction always is present at least in the surface region of the cavity wall. This means that even a photon- empty state does not exist in a rigorous physical sense. The transverse gauge photon states coupled to the wall of a closed cavity is discussed in a novel man- ner on the basis of a microscopic version of the Ewald-Oseen extinction theorem. Single-photon physics is introduced with background in the mean position state introduced in Chapter 3 . It is shown that a wave-packet one-photon wave func- tion in direct space satisfies a dynamical equation of a a Hamilton-like form. Quantum states containing two photons may be entangled, and such states are here called bi-photons. Using wave-packet bases for the two photons, a brief discussion of entangled transverse gauge photon states is given both in Hilbert space and direct space.
However we have shown in Eqs.(18.29) and (18.30) that the square of the absolute value of a complex scalar wave function Ψ is a measure of the wave intensity, which is the quantity of primary interest for the theories of interference and diffraction for either mechanical or electromagnetic waves. The periodic spatial variation of intensity throughout a region of superposition of waves which originate in the same source is known as interference. Its theory is based essentially on the linearity of the wave equation, which allows a linear combination of partial waves to constitute the actual wave at any given point. Consider first the superposition of two monochromatic plane waves, having the complex representation (16.25), that is (18.1) Since we are usually interested in the relative variation of the intensity only, in a medium 0 ,Z and not in its absolute value, it is common practice to adopt simplified wave intensities, compared to those in Eqs.(18.29) or (18.30), which read ( ) ( )2 21 1 2 2, , ,I r t I r= Ψ = Ψ tG G (18.2) The superposition of monochromatic waves (18.1) results in an intensity ( ) ( ) ( )2 21 1 2, , ,tG ( )I r t r t r= Ψ = Ψ +ΨG G ( ) ( ) ( )2 2 *1 2 1 2, , 2Re ,r t r t r t r t,⎡ ⎤= Ψ + Ψ + Ψ Ψ⎣ ⎦G G G G ( )1 2 1 2 1 2 1 22 cosI I I I k r k r ϕ ϕ= + + ⋅ − ⋅ + −G GG G (18.3) where the last term is called the interference term. If the phase difference, given by ( )1 2 1k k r 2δ ϕ ϕ= − ⋅ + −G G G (18.4) is a constant, the two waves are said to be mutually coherent, and the interference term indicates a sinusoidal spatial variation between maxima and minima, of intensity max 1 2 1 22 ,I I I I I m2δ π= + + = (18.5) ( )min 1 2 1 22 , 2I I I I I m 1δ π= + − = + B A SS 21 Figure 18.1. Interference pattern produced by two monochromatic waves originating from sources 1 2andS S Hence, the stationary interference pattern consists of a succession of surfaces, defined by the condition Nδ π= where .12or2 += mmN It is usually produced by superposition of waves originating in two point sources , which can be assumed, for simplicity, to be of equal phase constants 21 and SS 1 2 0.ϕ ϕ− = If the two sources are far enough from a given point, at distances 1 2 1 2,r r r r>> − ,G G the incident waves at the point can be regarded as plane waves, as we shall show in Chapter 19. Under this assumption Eq.(18.4) reduces to ( ) ( )1 2 1 2 1 2or 2k k r k r r N r r N Nkπ λδ π= − ⋅ ≅ − = − = =G G G (18.6) Both the antinodal and nodal surfaces ( mN 2= ) ( )12 += mN are defined by Eq.(18.6) to be hyperboloids of revolution. Their intersection with a plane will produce alternate nodal and antinodal lines called fringes of the form of either hyperbolas or circular rings if the plane is parallel (A) or normal (B) to the line joining the sources (Figure 18.1). The interference fringes are not observed if the phase difference 1 2ϕ ϕ− has a random variation with time, since the interference term in Eq.(18.3) is averaged out. In this event the waves are said to be mutually uncoherent and do not interfere by superposition. Mutually coherent waves that originate in the same source can be obtained by splitting a single beam, so that on recombining partial waves interference effects occur.