Non-classical correlations in reducible Quantum Electrodynamics

The question is discussed whether the momentum of a photon has a quantum uncertainty or whether it is a classical quantity. The latter assumption is the main characteristic of reducible Quantum Electrodynamics (rQED). Recent experiments in Quantum Optics may resolve the question. The non-classical correlation of quantum noise in color-entangled beams cannot be explained by rQED without modification of the standing explanation. On the other hand, rQED explains uncertainty of the momentum of a single photon when it is entangled with a quantum spin residing in its environment. The explanation of the historical experiment with equally-polarized pairs of photons, showing violation of the Bell inequalities, invokes the argument of collapse of the wave function, also in rQED.


Introduction
In Fock space a superposition of a horizontally polarized photon with wave vector k and a vertically polarized photon with wave vector k ′ is described by Here, |0, 0 denotes the vacuum state and a † H , a † V are the creation operators for a horizontally, respectively vertically polarized photon. Such a superposition with equal vectors, k = k ′ , is for instance needed to describe circularly polarized states. If the wave vectors are not equal then the superposition describes a photon whose wave vector and polarization are undetermined.
The existence of color entanglement has been experimentally established in a convincing manner [1,2,3,4,5,6,7,8]. The existence of superpositions of the form involving a single photon in superposition of two distinct wave vectors, is required in the analysis of color entanglement as found in [9,10]. These superpositions describe photon states whose energy and momentum are undetermined. On the other hand, if (2) is not allowed then the wave vector k of an idealized plane wave photon is always well-determined and does not undergo quantum fluctuations. The present paper questions whether states of the form (1) or (2) do occur in Nature. If they don't then a modified theory of Quantum Electrodynamics (QED) is needed. The original picture behind QED is that Euclidean space is filled with two-dimensional quantum harmonic oscillators the excitations of which are photons. Marek Czachor suggested [11] that the frequency of these oscillators should be quantized as well. Together with his collaborators he developed a non-canonical theory of QED. See [11,12,13,14,15,16,17,18] and references quoted in these papers. One of the main results of the theory is that QED after renormalization is recovered as a limiting case, without the need to fall back on ad hoc procedures like renormalization. On the other hand, an attempt to force the formalism into a mathematically rigorous framework was not successful [19].
The formalism of Czachor is still compatible with the existence of superpositions of the form (2). In fact, it emphasizes the role of these superpositions by assuming that the frequency of the oscillators, and hence the momentum of the photons, is undergoing quantum fluctuations. Recently [20,21], the present author restored the historical assumption of the frequency being a parameter and added some simplifying assumptions to the Czachor formalism. The resulting theory is a reducible version of QED. It is shortly introduced in the next Section.
Reducible QED may also lead to a better understanding of the so-called collapse of the wave function, which is an essential ingredient in the interpretation of the historical experiments [25,26,27] on the violation of Bell's inequalities. This point is shortly discussed at the end of the paper.
The next Section contains a short introduction on rQED. Sections 3 and 4 treat the reconstruction of spectral modes following [28], but using the axioms of rQED. Section 5 analyzes the historical experiment demonstrating the entanglement of a photon pair. Section 6 introduces the covariance matrix following [29] and shows that with rQED the DGCZ inequalities are not violated by a pair of identically polarized subbeams. Section 7 considers entanglement of a one-photon state with a spin variable in the environment. At the end follows a section Discussion and Conclusions.

Reducible QED
In rQED a quantum electromagnetic field is described by a normalized wave function ζ k in the Hilbert space H of the two-dimensional quantum harmonic oscillator. The wave function ζ k depends on the wave vector k which is a non-vanishing vector in the three-dimensional Euclidean space R 3 . A relativistic description is obtained by adding |k| as the zeroth component of the 4-momentum.
Let us choose a bazis |m, n of eigenstates of the two-dimensional quantum harmonic oscillator and give it the interpretation of describing a state with m horizontally and n vertically polarized photons. These are photons in the sense of Einstein: interactions of the electromagnetic field with its environment occur by creation or annihilation of a photon. The ground state is |0, 0 . Excited states are obtained by the action of the creation operators a † H and a † V in the usual manner a † H |m, n = √ m + 1|m + 1, n and a † V |m, n = √ n + 1|m, n + 1 .
The annihilation operators a H and a V are the hermitian conjugates of the corresponding creation operators. A superposition of two fields ζ k and η k is of the form λζ k + µη k with |λ| 2 + |µ| 2 = 1.
Wave functions do only combine at equal wave vectors. For this reason the representation of the Poincaré group is reducible. Quantum averages such as the total energy of the field ζ k decompose into their irreducible components. Note that an arbitrary constant ℓ with the dimension of a length has been inserted to make both wave functions and creation and annihilation operators dimensionless. Convergence of the integral in (3) requires that the wave vectors ζ k converge to the vacuum vector |0, 0 for large values of |k|. A special role is reserved for the coherent electromagnetic fields. Let F H (k) and F V (k) be two complex functions of the wave vector k. Let |F H (k), F V (k) c denote the corresponding coherent wave function of the two-dimensional quantum harmonic oscillator. It describes a coherent electromagnetic field. A short calculation shows that its total energy equals This gives ℓ 3 |F H (k)| 2 and ℓ 3 |F V (k)| 2 the meaning of the density of horizontally respectively vertically polarized photons with wave vector k. Details of the formalism of rQED are found in the appendices of [21].

Fluctuations of the electric field operator
In rQED the electric field operator at space-time position x is given by Here, λ is a constant which determines the units in which the electric field is measured. The polarization vectors are denoted ε The electric field operator can be decomposed into two so-called quadratures Fix a field ζ(k). Then the quantum uncertainties are given by In particular, if the wave function ζ is coherent then it follows from The intensity of a light beam is measured with a photo detector. It produces an electric signal proportional to the intensity. The quantum uncertainties ∆ ζ E c k (x) and ∆ ζ E s k (x) of the electric field result in a contamination of the electric signal with noise. In the case of a coherent field this is shot noise and its intensity is referred to as the quantum noise level. The noise is a direct evidence of the quantum nature of light. In particular, squeezing of light results in a reduction of noise in one of the quadratures. Reid [9] proposed to measure the noise levels of the two quadratures in order to demonstrate the Einstein-Podolsky-Rosen (EPR) paradox. The analysis of noise levels is considered later on in Sections 6 and 7.

Reconstruction of spectral modes
The photon current I(x) measured by a detector at space-time position x is proportional to the expectation E (−) E (+) ζ , where E (±) are the positive and negative frequency parts of the electric field operator. For a given field ζ one obtains Consider now a signal field with wave functions ζ s k . It is the field which one wants to investigate by letting it interfere with a coherent field. The superposition of both fields is described by wave functions The photon current becomes Relevant information is obtained by spectral analysis of the time dependence of I(x). Let A denote the set of wave vectors k for which |F H (k ′ ), F V (k ′ ) c = |0, 0 and B the set of wave vectors k for which ζ s k = 0. Assume that A and B do not overlap and select a frequency ω such that |k| − |k ′ | + ω/c = 0 for any pair of wave vectors k, k ′ in A. Then the contribution of leading order in ǫ vanishes and one obtains Experimentally, it is feasible to measure at once this spectrum and the one with F (k) replaced by iF (k) [29]. In this way also the imaginary part ofĨ(ω) is obtained. By varying experimental parameters one can obtain values for the func- The coherent states form an overcomplete basis of the Hilbert space of wave functions. One concludes that, in principle, a reconstruction of the wave vectors a H |ζ s k ′ and a V |ζ s k ′ is feasible. This shows that the results of [29] translate to the context of rQED.

Entanglement of polarized photon states
Let us convene in the present Section that horizontally polarized photons are independent from vertically polarized photons. Then the wave function |ζ of the two-dimensional harmonic oscillator is said to be separable when it can be written in the form |ζ H ⊗ |ζ V .
The measurement at position x destroys a photon of either horizontal or vertical polarization, using one of the following two annihilation operators The angle φ takes into account that the basis of polarization can be rotated. The wave function |ζ k can be written in terms of these rotated operators as If a horizontally polarized photon is taken out then the remaining field is Without restriction assume that the second measurement occurs in the original basis of the polarization vectors. Then (4) implies that the probability P (H, H) that the second measurement returns a horizontal polarization is proportional to while the probability P (H, V ) of a vertical polarization equals In the case that d 11 = 0, this result coincides with that discussed in [27]. This case is realized for instance when c 11 = 0 and c 20 = c 02 , which means that the two photons contributing to ζ k have the same polarization and both polarizations H and V have the same weight.
Note that the analysis here, as well as in [27], relies on the assumption that the detection of a photon causes a collapse of the wave function which prevents a subsequent measurement to detect the same photon once again. A more thorough analysis requires the use of measurement theory and is not elaborated here.

The covariance matrix
Here and in the next Section we focus on a polarized light beam. To simplify the notations all references to the polarization are omitted.
In the presence of interactions with the environment, the electromagnetic field is described in rQED by a wave vector-dependent density matrix ρ k instead of a wave function ζ k . Fix two wave vectors k and k ′ and consider measurements at wavelength k to be independent from measurements at wavelength k ′ . Any observable A at wavelength k, respectively k ′ , maps onto an operator A ⊗ I, respectively I ⊗ A. Because the representation is reducible the density matrix ρ k,k ′ of the product space is separable and can be written as with density matrices σ n , τ n and classic probabilities p n ≥ 0, n p n = 1. They satisfy ρ k = n p n σ n and ρ k ′ = n p n τ n . Following [29] we consider a column vector X with 4 elements given by

Averages are given by
The operators p and q are defined by and satisfy the commutation relation [p, q] = −i. A covariance matrix is then defined by One has The criterion used in the Literature to decide whether the two subbeams are entangled is based on violation of the DGCZ inequality [32]. Note that the density matrix ρ(k, k ′ ), given by (5), is separable. Therefore the inequality cannot be violated by this density matrix.

The coherent case
Consider the case that the density matrices σ n and τ n are orthogonal projections onto coherent states |F n (k) + iG n (k) c , respectively |F n (k ′ ) + iG n (k ′ ) c , where F n (k) and G n (k) are real functions of the wave vector k. Then one calculates and One concludes that in this case the covariance matrix Σ(k, k ′ ) equals Σ (0) . In particular, the two subbeams are not correlated. From it follows that in the case of a superposition of coherent fields one has ∆ 2 p − = ∆ 2 q + = 1/2. Hence, in this case the DGCZ inequality is actually an equality, as expected.

Entanglement with the environment
The field of density matrices ρ k , introduced in the previous Section, cannot explain all experimental data. This point is discussed in the final Section. Let us therefore consider an explicit example of a photon field ζ k entangled with a Pauli spin in its environment. The state of the system is described by with λ ↑ and λ ↓ complex numbers satisfying |λ ↑ | 2 + |λ ↓ | 2 = 1. The quantum expectation of a field operator A is given by Repeat the construction of the previous Section. Consider the wave vectors k and k ′ as belonging to independent subbeams. Then the quantum expectation of an observable of the form A ⊗ B ⊗ I, where A refers to the wave vector k and B to k ′ , is given by

Coherent case
Calculate the covariance matrix Σ(k, k ′ ), assuming coherent wave functions with F ↑ (k), F ↓ (k), G ↑ (k), G ↓ (k) real functions of the wave vector k. One finds and a similar definition for Y ↓ . The second order expression is One obtains

Now calculate
The inequality follows because the function f (x) = x 2 is convex and |λ ↑ | 2 +|λ ↓ | 2 = 1. Similarly is One concludes that the DGCZ inequality is not violated. This is not unexpected. See the comments in [33].

One-photon states
In the other extreme case the wave function is a one-photon state entangled with a spin state. It is described by where c ↑ (k) and c ↓ (k) are complex functions satisfying |c ↑↓ (k)| ≤ 1, and with a similar definition of γ ↓ (k). For the sake of simplicity the two spin states have the same weight. Let X be as before. A tedious calculation yields the DGCZ inequality

Take for instance
The DGCZ inequality becomes which is equivalent with u 2 + v 2 ≥ 1/3. One concludes that for small intensities the inequality can be violated.

Discussion and Conclusions
Let A and B denote two non-intersecting regions in the space of wave vectors k. They define two subbeams with different 'color'. Experimental papers on color entanglement observe a non-classical correlation between the noise signals which show up when the two subbeams are measured each with their own photo detector. The quantum nature of the correlations is established by demonstrating the violation of a DGCZ inequality [32]. See Section 6. This theoretical explanation cannot be reproduced here without modification because in rQED the density matrices ρ k and ρ k ′ , as given by (5), are separable right from the start.
At first sight one may conclude that rQED is not compatible with experiment. An easy way out to conciliate both is to take into account that entanglement with the environment is unavoidable. As proved in Section 7, entanglement with a quantum spin suffices to allow violation of the DGCZ inequalities under certain circumstances.
There is a more radical solution as well. Section 5 discusses the historical experiment [34,25] demonstrating the entanglement of a pair of equally polarized photons. The standard explanation of this experiment relies on the probabilistic interpretation of quantum mechanics and on the assumption of a collapse of the wave function. A similar reasoning is used in Section 5. The assumption is made that the measurement of a photon by one detector destroys the photon and modifies the wave function by the action of the annihilation operator a. A subsequent measurement by the second detector yields a correlated result. The elaboration of this argument requires the study of measurement theory, which is not considered here.
Conclusions In mainstream QED two photons with distinct wave vectors k = k ′ are treated as distinct particles. This is a consequence of the canonical commutation relations, which postulate that the commutator of creation and annihilation operators is proportional to a Dirac delta function δ (3) (k − k ′ ). As a consequence it is meaningful to study the bi-partite entanglement of photons and to interpret the observed [2] quantum correlations in terms of entanglement. In reducible QED the entanglement of photons with different wave vectors is not allowed. Never the less, a bipartite experiment can reveal non-classical correlations. This is the main result of the present paper.
Some difficulties remain. The analysis of the historical two-photon experiment [34,25] invokes the argument of the collapse of the wave function. The same explanation works in the context of rQED. See Section 5. However, a more in depth analysis requires the use of measurement theory, which is not considered here. The main question of the present paper, whether superpositions of photon states with unequal wave vectors are physical, and hence whether reducible QED, as presented here, is an incomplete description of quantum Electrodynamics, is left open.