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Assuming the scattering center of the atoms at r = 0 and ap- plying the far field approximation, g reads as 

where d is the atomic dipole moment and ν a scaling factor. ( n ) 

The atomic flip operators ˆ A ab obey the equal time rela- tion ˆ A ( n ) ab ˆ A ( n ) cd = δ$_{bc}$ ˆ A ( n ) ad , with δ being the Kronecker- δ . Their quantum averages are obtained from the optical Bloch equa- tions for the density matrix of a single atom. One has to include phase factors exp( − iϕ ( n ) j ) , according to [6], to ac- count for the interference between all atoms and the positions of the detectors. Here n ( n = 1 , . . . , N ) numbers the atoms and j labels the positions r$_{j}$ of the used points of observa- tion. Since the atoms are separated by linear distances large compared to the pump-laser wavelength, the correlation func- tions of products of atomic operators of different atoms fac- torize, 〈 ˆ A ( n ) ab ˆ A ( m ) a $^{′}$b′ 〉 = 〈 ˆ A ( n ) ab 〉〈 ˆ A ( m ) a $^{′}$b′ 〉 for m ̸ = n . That is, under such conditions cooperative effects are negligible. 

In the balanced homodyne correlation measurements under consideration, the interference of the signal field with the lo- cal oscillator is the observed signal, for details see [12]. The recorded correlation functions contain the slowly-varying sig- nal field operators, ˆ ˜ E ( ± $^{)}$( r , t ) = ˆ E ( ± $^{)}$( r , t ) e ± iω$_{lo}$$^{t}$, where ω$_{lo}$ is the frequency of the local oscillator. The fields at the points of observation r$_{j}$ (in the following we choose j = 1 , 2 ) can be related to the Bosonic operators ˆ a$_{j,λ}$ as 

The mode functions f$_{λ}$ ( r$_{j}$ ) describe those fields which are recorded by homodyne correlation measurements within the spatio-temporal resolution of the detectors. 

An important class of entangled states can be characterized by the negativity of the partial transposition of the quantum state under study [20]. The Peres condition for entanglement has been reformulated as [21] 

in terms of the partial transposition of an operator ˆ f † ˆ f , with ˆ f being a function of the Bosonic operators describing the sys- tem [21], for the multi-partite extension see [22]. Expanding the operator ˆ f into a Taylor series of ˆ a$_{j}$ , the entanglement con- ditions lead to a hierarchy of negativity conditions in terms of minors whose entries are moments of ˆ a$_{j}$ and ˆ a † $_{j}$. For a bipar- tite system consisting of two modes, a simple example of such an entanglement condition is 

This condition can be generalized for the multi-mode fields defined in Eq. (4). Now the operator ˆ f in the condition (5) is 

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chosen as a Taylor series in terms of the fields in two space- time points ( i = 1 , 2) , ˆ E ( ± $^{)}$( r$_{j}$ , t ) , for which we use the short hand ˆ E ( ± ) j . We can again formulate the entanglement condi- tions in terms of a hierarchy of minors. The quantum-field theoretical counterpart of the two-mode entanglement condi- tion (6) reads, in terms of the minor µ , as 

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The occurring correlations of the fields in two space-time points relative to the radiating source describe the measured behavior for the often used assumption of the detector areas being small compared to the spatial coherence area of the fields and the time resolution being smaller than the coher- ence time. For practical applications the corresponding fields can also be collimated by apt optics and propagated to two different regions, e.g. to Alice and Bob, which are far apart from each other and far from the source. Both these possibili- ties, the detection and the propagation of the entangled fields, are specified by choosing the mode functions in Eq. (4) to de- scribe localized and propagating fields, respectively. 

Expanding the minor µ in Eq. (7), one readily gets as con- dition for entanglement 

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Here we have introduced the symbol Δ ˆ O = ˆ O − 〈 ˆ O 〉 for an arbitrary operator ˆ O . If one would further reduce the minor in the condition (7) to the dimension 2x2, it is easy to see that the resulting minors are always non-negative. Hence the entanglement condition (8) under study is a rather simple ex- ample, but it will turn out in the following to be sufficient for demonstrating entanglement in resonance fluorescence. 

For the resonance fluorescence from identical two-level atoms, the dependence on the number N of atoms in µ is in the phase factors ϕ ( n ) j . After some straightforward algebra, the left hand side of the condition (8) reads as 

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where σ$_{ba}$ = 〈 ˆ A$_{ab}$ 〉 are the slowly varying density matrix ele- ments of the identical two-level atoms and ϑ is the angle be- tween the two functions g$_{1}$ and g$_{2}$ , with g$_{j}$ = g ( r$_{j}$ ) . For random positions of the atoms, the phase combinations in the second and third line in Eq. (9) are averaging to zero, so that only the first line contributes. Under these conditions the mi- nor µ is always positive semidefinite, and hence entanglement does not exist or it cannot be identified by the condition (8). 

