Entanglement dynamics of second quantized quantum fields

We study the entanglement dynamics in the system of coupled quantum fields. We prove that if the coupling is linear, that is if the total Hamiltonian is a quadratic form of field operators, entanglement can only be transferred between the fields. We show that entanglement is produced in the model of the two-mode self-interacting boson field with the characteristic Gaussian decay of coherence in the limit of high number of particles. The interesting feature of this system is that the particles in different modes become entangled even if there is no direct interaction between the modes. We apply these results for analysis of the entanglement dynamics in the two-mode Jaynes-Cummings model in the limit of large number of photons. While the photon-atom interaction is assumed to conserve helicity the photons with different polarizations still get entangled due to an effective interaction mediated by the atom with the characteristic entanglement time linearly increasing with the number of photons.


I. INTRODUCTION
Entanglement is a feature, which is quintessentially quantum. It signifies that several particles may form a new entity, a complex of particles, say, not two particles but a pair and so on. If, for example, one would attempt to separate the particles from such complex by performing a measurement on one individual particle, one would unavoidably modify the states of others even if the direct interaction between the particles is absent or negligible. Such departure from the classical properties not surprisingly attracts the significant attention (see e.g. the recent extensive review Ref. 1).
As the problem of special interest stands out the problem of sources of entangled states. For instance, nowadays the most developed and widely used method of generating the entangled photons is the parametric down conversion, 2,3 which is based on on the two-photon radiative decay of excited states of a matter. This method, however, suffers from intrinsic limitations -very low yield and rescaling the wavelength of the emitted photons. [4][5][6] Therefore, there is the constant search of alternative sources of entangled light.
The consideration of the problem of entanglement of photons in the course of interaction with the matter excitations naturally suggests the more general perspective of the systems with non-conserving number of particles, which are described in the field theoretical framework. In this approach the particles appear not as a predefined entities as, say, qubits in canonical quantum mechanical treatment, but rather as the excitations of the quantum field. In this context the difficulty of producing the entangled states is clearly demonstrated by the simple example of a bosonic field excited by an external source. Let the field dynamics be described by the Hamiltonian where k enumerates the modes of the field with the energies ǫ k , a † k and a k are the creation and the annihilation operators of the particles in the respective modes and e k are c-numbers determined by the projection of, generally speaking time dependent, external classical field on the modes. In order to avoid complications arising in the fermion case [7][8][9] in the present paper we deal with entanglement of only boson fields.
Before applying rigorous methods for analysis of this situation let us first consider it using often employed arguments based on the interference of different paths leading to the final state of the field. Let us assume for the sake of this discussion that either the system admits only two modes or for some reason we can restrict ourselves to considering only two of them out of the whole set. Thus the sum in Eq. (1) would contain only terms with k = 1, 2. The typical argument sounds as follows. One path is when one particle first goes under the action of the external source to mode 1 then to mode 2, while the sequence of the modes for the second particle is 0 → 2 → 1, where 0 denotes vacuum. The second path for the pair is obtained by "flipping" the particles: 0 → 2 → 1 for the first particle and 0 → 1 → 2 for the second one. The interference between these two paths is assumed to lead to entanglement.
More rigorous description of entanglement uses the notion of the single-particle observables. Each such observable is represented by a single-particle operator O = kq O kq a † k a q (see e.g. Ref. 10), which motivates introducing the single-particle density matrix (SPDM) Throughout the paper we incorporate the time dependence into the Heisenber representation of the field operators a k (t) = exp(iHt)a k exp(−iHt), therefore, the average in formulas similar to Eq. (2) is taken with respect to the initial state . . . = ψ(t = 0)| . . . |ψ(t = 0) . For the purpose of this consideration it suffices to assume that entanglement manifests itself in incoherent single-particle states, that is the SPDM has rank more than one. 7,9 The solution of the operator equation of motions can be easily seen to have the form where are c-number functions. Substituting this representation into Eq. (2) and assuming that the initial field state is the eigenstate of the operator of the total number of particles, N = k a † k a k , we express the time dependence of the SPDM in terms of its initial value K kq (0) and the contribution due to the Generally the dynamics of entanglement in the presence of the external source is not trivial. The addition E kq (t) may lead to the change of entanglement depending on the structure of the initial state. If, however, initially the system is in vacuum state then matrix E(t) (throughout the paper the hats denote matrices in the space of the modes of the quantum field) with the matrix elements E kq (t) is at most of rank one. Hence, if K(0) = 0 then the rank of the SPDM is no greater than one implying the absence of entanglement. This simple example shows that the problem of the dynamics of entanglement should be treated with certain care. First, the naive arguments based on the picture of interference of different paths connecting initial and final manyparticle states may be misleading, when it is applied to the field dynamics. One has to study the actual structure of the field state. Second, no matter how complex is the spectrum, ǫ k , of the (boson) quantum field the states, which are reached out of vacuum under the action of the classical external excitation, are disentangled. Thus, in the discussion above regarding the effect of the excitation on an initial entangled state that state should be created by other means than the external excitation.
The rest of the paper is organized as follows. In Section II we consider the entanglement dynamics for the system of linearly coupled boson fields. In Section ??e study the entanglement for the two-mode self-interacting boson field. In Section IV we apply the results obtained for the analysis of the entanglement dynamics in the two-mode Jaynes-Cummings model in the limit with high number of photons.

II. ENTANGLEMENT TRANSFER BETWEEN COUPLED FIELDS
We start the analysis of the entanglement dynamics from the consideration of two coupled fields described by the Hamiltonian where k and κ enumerate the modes of the fields, ǫ κ and d k,κ are the spectra of the fields and the coupling constants between them, respectively, and the operators a k and b κ are assumed to obey the boson commutation relations.
The important feature of the time evolution of entanglement in this system is that the total entanglement remains constant and is solely determined by the initial state. More precisely, instead of the operators a k and b κ let us introduce the combined operators u i . That is instead of two fields with two sets of modes M a and M b we consider the single field, whose modes are the direct sum When both i and j belong to, say, M a the respective matrix elements K ij (t) give the SPDM for field a and so on. Furthermore, let the matrix K have the spectral representation where λ l and v l are the eigenvalues and the unit eigenvectors of K, respectively, and ⊗ denotes the tensor product, which is defined as Then the total entanglement can be found as the von Neumann entropy where λ l (t) = λ l (t)/ m λ m (t). The total entanglement can be shown to remain constant, so that it is determined by the initial state E N (t) = E N (0), and the entanglement evolution restricts purely to its redistribution. We prove this in a bit more general context. Namely, we consider the system described by the Hamiltonian where the field operators obey the commutation relation with v ij being a symmetric c-number matrix. The total entanglement is defined as the von Neumann entropy of the SPDM. Its conservation follows from the fact that the evolution SPDM is governed by the equation where L ij = k h ik v kj . Indeed, the spectrum of the matrices, whose time dependence is determined by an equation of motion with the commutator in the right-hand side, remains constant. Hence, in the spectral representation similar to presented in Eq. (7) only the eigenvectors will be functions of time.
For example, if initially there was only single non-zero eigenvalue (i.e. entanglement was zero) it remains the only one later on implying no production of entanglement. The same result holds for the SPDM corresponding to either fields a or b. Indeed, the SPDM of the field a is obtained from the SPDM of the combined field u applying the respective projection operators, Π a , so that Using for K its spectral representation one can see that such projection cannot increase the rank of the SPDM. At the same time the circumstance that the time evolution of entanglement of specific particles is determined by the projections of the total SPDM results in non-trivial time evolution of initially entangled state. As an example let us consider the situation of small total entanglement, more specifically, when there are only terms with l = 1 and 2 in Eq. (7) with λ 1 ≫ λ 2 . According to Eq. (12) the SPDM for the field a is given by where v Thus, the value of entanglement depends on the magnitude of the projections Π a v l , which is determined by the internal dynamics of the coupled fields. In particular, if Π a v l ≪ 1 2 (t) resulting in the significant entanglement of particles a, either with each other or with particles b. For more concrete information one has to take into account that, generally speaking, Eq. (13) may not be the spectral representation of the matrix K (a) because the vectors v Thus, if at some particular instant one has λ  N ≈ 1. Strong entanglement is produced when the "weak" component of the SPDM ∼ λ 2 is more effectively transferred into the field a than the major component ∼ λ 1 , whose only small part is moved into the field a during the evolution. At the same time, as can be seen from Eq. (14) after slight change of notations, such spike of entanglement between particles a is accompanied with disentanglement of particles b.
The main condition for this geometric effect is the smallness of the projection of the respective eigenvector of the total SPDM. As a result the characteristic feature of the SPDM of the strongly entangled states in this case is That is the states with E (a) N ≈ 1 developed from the states with low total entanglement are characterized by low excitation, while the strongly excited field, say the field b in the considered example, for which one has Tr[ K (b) ] ≈ Tr[ K], remain only weakly entangled.
The fact that the entanglement of the specific particles is determined by the projections of the total SPDM may lead not only to increased entanglement of the particles but also to disentanglement. Indeed, if for the SPDM given by Eq. (7) with l = 1, 2 at some instant the vectors v 1 and v 2 belong to different subspaces (say, Π a v 1 = v 1 and Π b v 2 = v 2 ) then the entanglements of both particles a and b are zero. One can understand this effect introducing the isospin quantum number, so that one value of isospin corresponds to the particle a and another one stands for the particle b. Such situation, when non-zero total entanglement coexists with zero entanglement as seen from specific particles' SPDM, corresponds to the entanglement stored in isospin.
It should be noted that while we have limited our attention only to boson fields the same formal results can be easily checked to hold also for fermion, i.e. obeying the anticommutation relations, fields.

III. ENTANGLEMENT PRODUCED BY INTERACTION
The consideration in the previous section showed that simple linear coupling of the quantum fields is not enough for the fields to become entanglement. The situation, as we will show in the present section, is different when there is an interaction in the system, that is when the energy depends non-linearly on the number particles. More specifically, we consider the self-interacting two-mode boson field described by the Hamiltonian Here the first two terms describe the internal dynamics of the modes with ǫ k and U k being the energies of the modes and the interaction parameters, respectively, and the last term represents the interaction between the modes, which is chosen in the form admitting conservation of particles within each mode. In order to evaluate the entanglement we need to inspect closer its relation to the matrix elements of the SPDM and the state of the field. We start from the case of a two-particle case, which provides the clear connection with the standard quantum-mechanical consideration. In the basis of the population numbers any two-particle state can be presented as where |n + , n − denotes the state with n + particles in the "+"-mode and n − particles in the "−"-mode, and α n+,n− are the respective amplitudes. For analysis of the SPDM it is convenient to introduce an alternative representation of the state in terms of the creation operators 7 where |0 is vacuum and w kq is a symmetric matrix. Comparing Eqs. (16) and (17) The SPDM is a 2 × 2 matrix and, therefore, its eigenvalues are completely determined by the determinant of the SPDM det[ K] = λ 1 λ 2 and its trace Tr[ K] = λ 1 + λ 2 = N with N being the number of particles. The entanglement, in turn, is determined by the normalized eigenvalues λ 1,2 = λ 1,2 /N , which are found as where |C| 2 = det K is the concurrence 11 . In order to see the relation with the standard definition of the concurrence in the two-particle case we introduce the "spin flip" transformation σ y |+ = −i |− , σ y |− = i |+ and the spin flip state ψ = (σ y ⊗ σ y |ψ ) * , then Thus the two-particle case completely fits into the canonical quantum-mechanical description. As follows from Eq. (20) the concurrence is expressed in terms of the amplitudes α kq as For a two-particle state to be disentangled the amplitudes have to meet the condition C = 0. Let initially the state be disentangled. It is seen that this condition not necessarily holds for all t > 0 if the dynamics of the system inhomogeneously depends on the population numbers. Since the Hamiltonian (15) conserves the number of particles in each mode, the states |n + , n − are the eigenstates and the time dependence of the amplitudes can be easily found where ∆ǫ n+,n− = U + n + (n + − 1) + U − n − (n − − 1) + U +− n + n − . Substituting Eq. (22) into Eq. (21) we find for the initially disentangled state where Ω = 2ǫ + + 2ǫ − + U + + U − + U +− and Several interesting conclusions can be drawn from this result. First, the entanglement oscillates between 0 and the maximum value determined by the contribution of |1, 1 into the initial state. The origin of the oscillations can be traced to the structure of the concurrence, Eq. (21), and the frequencies of the many-body amplitudes in Eq. (22). The interaction leads to the energy shifts ∆ǫ nm , which depends on the population of the particular modes. The phase mismatch between the amplitudes results in the nontrivial time dependence of |C(t)| 2 . Second, the interplay between the effects of the intra-mode, ∝ U ii , and inter-mode, U +− , interactions is not straightforward. Let the interaction between the modes be absent, U +− ≡ 0. As follows from Eq. (23), even in this case initially disentangled states become entangled.
The related effect is the mutual cancelation of the phase desynchronization if U +− = U + + U − , when despite the interaction, which changes the energies of the many-body states comparing to multiples of the single-particle states, initially disentangled states remain disentangled.
The two-particle case is useful for establishing the relation with the standard description of entanglement. However, in order to grasp the general structure of the dynamics of entanglement of the quantum field with interaction it is constructive to consider more general case with an arbitrary number of particles N . In this case it is convenient to introduce Schwinger's angular momentum model. 12 Defining the components of the operator of the angular momentum in terms of the creation and annihilation operators as one can enumerate the states by the total angular momentum j = (n + + n − )/2 and its projection m = (n + − n − )/2 instead of the population numbers n + and n − , that is |n + , n − = |j, m S . Expressing the SPDM in terms of the mean values of the operator of the angular momentum one finds the concurrence C(t) = N 2 /4 − | J(t) | 2 , so that the normalized eigenvalues of the SPDM can be expressed as where J = 2| J |/N . Thus, the entanglement can be written as In particular, disentangled states are the states with the maximum magnitude of the average angular momentum | J | = j and completely entangled ones are those with | J | = 0.
In terms of the operator of angular momentum Hamiltonian (15) can be presented as where f 1 = N k ǫ k U k /2 + N 2 (U + + U − + U +− )/4 and f 2 = ǫ + − ǫ − + (U + − U − )(1 + N ) depend on the operator of the total number of particles, N = a † + a + + a † − a − , and are irrelevant for the entanglement dynamics and ω is given by Eq. (24).
As follows from Eqs. (26) and (28) the dynamics of entanglement is subject to the general constraint J z (t) = J z (0) , which follows from [H, J z ] = 0. Thus, the variation of entanglement is determined by the change with time of the "transversal" component J 2 In particular, this imposes the upper limit on the value of entanglement produced by the interaction, Let us consider the dynamics of initially disentangled states. As follows from Eq. (26), any such state can be obtained by rotating the state |j, j S where β i are the Euler angles. Indeed, the average J transforms under rotations as a 3D vector. Hence, by rotating around z-and y-axes we can set in the new frame only J z = 0. Since disentangled states have the maximum magnitude of the averaged angular momentum, this implies that in the new frame the state is proportional to |j, j S Rotations around the z-axis do not affect J 2 ⊥ and, therefore, the dynamics of entanglement of the states related through such rotations are identical. Hence, it suffices to consider only the states |ψ(β) = |ψ(0, β, 0) . From [H, J z ] = 0 it immediately follows that where the index β emphasizes the structure of the initial state, i.e. . . . β = ψ(β)| . . . |ψ(β) . The nontrivial part of the time dependence of J ± (t) β is given by the last term in Eq. (28) because the first two terms yield only the phase factor, which does not contribute to J 2 (t). Next, we notice that the solutions of the operator equations of motioṅ Using these solutions we obtain (see Appendix A for the details) J 2 = cos 2 (β)+ + 4 sin 2 (β/2) cos 2N −2 (θ/2) × e iγ sin(β/2) sin(θ/2) + cos(β/2) cos(θ/2) 2 , where sin(θ/2) = sin(β) sin(ωt) and cot(γ) = cos(β) tan(ωt). The overall dependence of J on time and on the structure of the initial state parametrized by the angle β is shown in Fig. 1. It follows from Eq. (32) that for β = 0 or π the transverse component of the effective angular momentum oscillates with the period T = 2π/ω. The maxima of J, which, according to Eq. (26), correspond to the minima of entanglement, are reached at t min = T n/2 with integer n. As follows from Eq. (32), J(t min ) = 1 yielding E N = 0 in agreement with the definition of the states |ψ(β) . There are also two sets of maxima of the entanglement at t respectively. One can see that J 2 (t With increasing the number of particles the ratio monotonously tends to 1. In order to better understand the effect of the number of particles on the time evolution of entanglement we consider the case β = π/2, that is the initially disentangled states with symmetrically populated modes ( ψ| Jz |ψ = 0). As follows from Eq. (32) and (26) these are the only states (among all initially disentangled states) that yield maximum entanglement E N = 1 in the course of the time evolution. It can be shown that all such states are spanned by two orthogonal states corresponding to β 1 = 0 and π/2 in Eq. (29). From Eq. (32) one finds that for these states The time dependence of entanglement following from Eq. (34) has two specific features. First, this is the periodic function of time. The entanglement evolves from 1 to 0 and back within the vicinities t min . When the number of particles increases these regions narrow. It is constructive to consider the limiting form of the time dependence when the number of particles becomes very large. In the limit N ≫ 1 one can approximate Thus, away from the points where J(t) = 0, it can be regarded as the sequence of the Gaussian bumps localized near t min . As a result, when √ N ≫ 1 one can consider the system as being in the completely entangled state most of the time.
The Gaussian decay of J(t) is similar to the Gaussian decay of the coherence of the central system, 13,14 two spins 1/2 coupled to the bath. In particular, the same dependence of the decay rate on the number of particles ∝ √ N in the environment [notice N − 1 in Eq. (35)] should be emphasized. There are, however, two important differences between this situation and our case. First, the Gaussian decay of the coherence of the central system appears in the limit of the slow dynamics of the environment. In the opposite limit the decay follows the Lorentsian law and in the intermediate case both types of decays present at different time scales. 15 For the two-mode boson field it is meaningless to separate particular particles and environment due to indistinguishability, however, it is worth noting that J(t) does not depend on the single-particle energies. The second important difference is that the concurrence of the two-mode boson field exhibits oscillations while the loss of coherence of the central system is irreversible.

IV. ENTANGLEMENT DYNAMICS IN THE JAYNES-CUMMINGS MODEL
In the previous section we considered two basic features of the entanglement dynamics in the system of coupled quantum fields -redistribution of the initial total entanglement and the production of entanglement by the interaction. In the present section we apply these ideas for the analysis of entanglement in a more complex situation.
We consider a single two-level atom interacting with the quantized electromagnetic field. The electron transitions in the atom are assumed to be characterized by definite helicity. The excitation of the electron state with the spin down at the ground level, |g ↓ , into the spin up state at the excited level |e ↑ occurs through the absorption of "+"-polarized photon and so on. The dynamics of the system is described by the Hamiltonian of the two-mode Jaynes-Cummings (JC) model, 16 which we write down in terms of the creation and annihilation operators Here the first two terms describe the dynamics of the free atom and the free field, respectively, with κ and k running over the atomic states, {g ↑, g ↓, e ↑, e ↓}, and the photon polarizations, + and −, respectively. The interaction between the atom and photons is described by the Hamiltonians H i = ω i a i σ † i + h.c., where σ † + = c † e↑ c g↓ , σ † − = c † e↓ c g↑ and ω ± are the respective Rabi frequencies.
As the consideration in the previous sections suggests, the main effect on the entanglement dynamics is due to the coupling between the systems rather than due to their internal dynamics. However, the interplay between different characteristic frequencies may lead to appearance of beatings. Therefore, in order to eliminate the technical complications and to concentrate on the main features, we adopt the resonant approximation and set ǫ (e) Additionally we assume that the symmetry between transitions with different helicities is not broken so that ω + = ω − = ω R .
We consider the photon entanglement, which is defined as the von Neumann entropy of the single-photon density matrix with the matrix elements This is 2 by 2 matrix and the problem of entanglement can be approached using the same description as in the previous section. As well as before our main objective is to study the time evolution of entanglement of initially disentangled photonic states. This closely corresponds to the situation when, for example, the cavity in the ground state is pumped by an external source. Taking into account that initially the atom is not excited the state of the system can be presented as where |ψ(β) is the disentangled photon state obtained by rotations from the state |j, j S [see Eq. (29)] and |0 e denotes the state of the atom with both electrons at the ground level. The time dependence of the photon entanglement of the states with only a few photons is highly nontrivial comparing to the few particle case of the two-mode boson field considered in the previous section. Here we would like to note only one characteristic feature leaving more detailed discussion for a separate publication. Drawing the analogy with the consideration of the two-particle case in the previous section one can expect that in the present case as well the time dependence of the entanglement will be determined by the mismatch between the amplitudes of states with different population numbers. In fact, as will be evident shortly, the typical frequencies are determined by the square roots of the population numbers. Taking this circumstance into consideration one can expect that the time dependence of the concurrence is the result of superposition of several harmonics with incommensurate frequencies. Thus it is a quasi-periodic function with a complex profile.
With increasing the number of photons, however, the contribution of specific frequencies becomes less important and the overall shape of J(t) changes toward some general regular pattern shown in Fig. 2. In order to describe it we use the explicit form of the Heisenberg representation for the photon operators a k (t) = exp(itH k )a k exp(itH k ). Taking into account the separability of the dynamics we have 17 where It should be noted that the second term in Eq. (39) does not contribute to K kq (t) because it vanishes while acting on the initial state of the atom.
First, we consider J z (t) . Taking into account the identity f C k a k = a k f (C k ), which holds for any well-behaving function f , we find where the average is taken over the initial photons state, . . . = ψ(β)| . . . |ψ(β) . The second term in the r.h.s. of Eq. (40) plays significant role only in the case when all photons have the same polarization, that is when the effect of the interaction is obviously small. In this case the interaction produces small entanglement ∼ 1/N oscillating with the frequency N ω R . In the unpolarized case, when J z = 0 the last term in Eq. (40) vanishes identically because of the symmetry of such disentangled states with respect to flipping all photon spins. While in the weakly polarized case the ratio of the last term to J z can be shown to remain limited from above and to decrease with the number of particles ∝ 1/ √ N . Thus, while due to absorption of the photons J z (t) is not a constant, as it was for the two-mode boson field, its total variation is small in the limit N ≫ 1. Therefore, the initial value J z (0) can be considered as imposing a limitation on the highest entanglement, which can be reached. Because of these reasons, we consider in more details the case when J z (0) = 0, that is when the initial photon state is given by Eq. (29) with β 2 = π/2. In this case the eigenvalues of the photon SPDM, and, hence, the entanglement, are determined by the magnitude of J + (t) . In order to calculate it we assume that the main contribution is due to the states with small m, that is n + , n − ≫ 1 and n + ≈ n − . This approximation improves with increasing the number of particles.
The best way to employ this approximation is to derive the effective photon Hamiltonian directly from Eq. (39). Taking into account that initially the atom is in the ground state we can effectively substitute Eq. (39) by a k (t) = exp(iC k t)a k exp(−iC k t), which is the Heisenberg representation induced by the Hamiltonian Using the assumption regarding the main contribution due to small m we can expand this Hamiltonian with respect to J z /j finding in the first nonvanishing order Comparing this expression with Eq. (28) we come to the conclusion that in this limit the photons behave as the two-mode self-interacting boson field with the effective interaction strength ω = ω R /4j 3/2 . The characteristic feature of the effective interaction is that its intensity decreases with the number of particles ∝ 1/N 3/2 . This "spreading" is caused by sharing the interaction with the single atom among all photons and leads to different dynamics of J(t) comparing to the one we have seen in the previous section. Using the results obtained there we find the period of the long-scale oscillations T = √ 2N 3 /ω R and the decay time τ = √ 2N/ω R (see Fig. 2a). It should be noted that, as can be seen from Fig. 2, the shape of the bumps changes with their number -they become asymmetrical and acquire the side oscillations, the revival of coherence occurs non-monotonously. The approximation we have used misses these long-time changes, which are the reminiscent features of the quasi-periodicity mentioned above. If, however, one restricts to the initial growth of entanglement (the drop of coherence) these features are not important, and one has in the limit N ≫ 1 (43)

V. CONCLUSION
We have considered the basic dynamics of entanglement in the system of coupled quantum fields. We have quantified the entanglement as the von Neumann entropy of the single particle density matrix (SPDM) and studied its time evolution for different situations.
The first question, which has to be answered is how is it possible to produce an entangled state of a quantum field. The circumstance, which makes this question non-trivial, is that by classical source entangled states cannot be excited. Following the straightforward analogy with the canonical quantum mechanical picture one can consider the system of coupled quantum fields. We show, however, that if the coupling is linear, i.e. if the total Hamiltonian is a quadratic function of the field operators, the total entanglement conserves and one, at most, can only have the transfer of the entanglement between the coupled fields.
The simplest system demonstrating entanglement of initially disentangled states is the two-mode boson field with self-interaction. Reformulating the problem using the formalism of Schwinger's model of angular momentum we show that there is the direct relation between J, the magnitude of the average angular momentum J and the concurrence. More specifically, the states with maximum possible magnitude of the average angular momentum are disentangled and those with J = 0 are completely entangled, meaning that the SPDM is proportional to the identity matrix.
We show that in the limit of high number of particles J(t) has the form of the periodic sequence of Gaussian bumps, whose width (inverse entanglement time) decreases with the number of particles ∝ 1/ √ N . The interesting feature is that the entanglement is produced even if the interaction is between the particles within the same mode but not between different modes. This phenomenon is in the striking contrast with the standard quantum mechanical picture, where separable dynamics cannot entangle initially disentangled particles. The origin of this effect lies in the structure of the many-body states of the field. Because of indistinguishability of the particles one cannot say which particle belongs to which mode. As a result only states with amplitudes meeting the special condition are disentangled. The dynamics nonlinearly depending on the number of particles may break this condition resulting in entanglement.
We apply these results for analysis of the two-mode Jaynes-Cummings model. The photon-atom interaction has been assumed to be helicity preserving thus the dynamics of "+"-and "−"-polarized photons are separable. However, absorption and re-emission of photons by the atom introduces an effective interaction between the photons of the same polarizations. This interaction, similarly to the case of the two-mode boson field, entangles photons with the typical Gaussian drop of coherence in the limit of large number of photons N ≫ 1. Since the interaction with the single atom is shared among the photons the strength of the effective interaction drops ∝ 1/N 3/2 . This leads to prolonged both the oscillations of entanglement T = √ 2N 3 /ω R and the entanglement time τ = N √ 2/ω R .
For small φ and large j the magnitude of this function decays with φ following the Gaussian law. Let us denote J(t) = cos 2j−1 (φ/2), then For φ ≪ 1 we can keep only the leading term in the Taylor expansion of tan(φ/2) and obtain Such representation is meaningful for sufficiently large j when jφ 2 > 1.