arXiv:1001.4354v1 [cond-mat.mes-hall] 25 Jan 2010 Entanglement dynamics in the system of coupled quantum fields Mikhail Erementchouk and Michael N. Leuenberger NanoScience Technology Center and Department of Physics, University of Central Florida, Orlando, FL 32826 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 the field operators, the 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 with high number of photons. While the photon-atom interaction is assumed to conserve helicity the photons with different polarizations still get entangled with the characteristic entanglement time linearly increasing with the number of photons. PACS numbers: 03.67.Bg,03.65.Yz,32.90.+a 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. 4–6 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 H =  k  ǫ k a † k a k + e k (t)a † k + e ∗ k (t)a k  , (1) 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 7–9 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) K kq (t)=  a † k (t)a q (t)  . (2)