Entanglement spatial

The different length scales involve different time scales with different types of motion. For short times corresponding to spatial distances shorter than the entanglement distance, we expect entropy-determined dynamics described by the so-called Rouse model [6, 35.]. As the spatial extent of motion increases and... 

Recently, des Cloizeaux has conceived a rubber -like model for the transition regime to local reptation [56]. He considered infinite chains with spatially fixed entanglement points at intermediate times. In between these... 

The spatial size of entanglement is too large to be included within the crystalline lattice. Therefore, we can assume that the entanglements, which are expressed by cross marks in Fig. 28, can exist only on the surface of a crystal or in the amorphous layers between lamellae. Therefore, thick extended chain single crystals (ECSCs) include little entanglement (ve 0), while FCCs composed of stacked thin lamellae include a lot of entanglements (Fig. 28) [49,50]. 

Rubber-like models take entanglements as local stress points acting as temporary cross finks. De Cloizeaux [66] has proposed such a model, where he considers infinite chains with spatially fixed entanglement points at intermediate times. Under the condition of fixed entanglements, which are distributed according to a Poisson distribution, the chains perform Rouse motion. This rubber-like model is closest to the idea of a temporary network. The resulting dynamic structure factor has the form ... 

From the analyses of NMR and electron-spray ionization mass (ESI-MS) spectroscopy, the polymers obtained from the polymerization of cyclic disulfides were found to be a cyclic structure [202], The cyclic structure consisting of poly(DT) is assumed to be formed by a backbiting reaction of propagating species [203]. Thermal and mechanical properties of the polymers, and decomposition behaviors of the polymers demonstrate that the polymers obtained from thermal polymerization of cyclic disulfides include a polycatenane structure. From polymerization of cyclic disulfides in the presence of cyclic polyethylene oxide), a polycatenane consisting of two different cyclic polymers was obtained [199]. Thus, poly(DT) contains spatial entanglements of cyclic polymers with each other (a polycatenane structure was presumed) (Fig. 61). 

Sufficiently dilute polymer solutions may be viewed as systems in which islands of polymer coils scattered in the sea of a liquid solvent occasionally impinge and interpenetrate. By this way, the spatial distribution of chain segments in them is quite heterogeneous and undergoes appreciable fluctuations from time to time. As the polymer concentration increases, the collision of the islands becomes more frequent and causes the chains to overlap and entangle in a complex fashion. 

An important role in the present model is played by the strongly non-linear elastic response of the rubber matrix that transmits the stress between the filler clusters. We refer here to an extended tube model of rubber elasticity, which is based on the following fundamental assumptions. The network chains in a highly entangled polymer network are heavily restricted in their fluctuations due to packing effects. This restriction is described by virtual tubes around the network chains that hinder the fluctuation. When the network elongates, these tubes deform non-affinely with a deformation exponent v=l/2. The tube radius in spatial direction p of the main axis system depends on the deformation ratio as follows ... 

To summarize, it must be assumed that amorphous PC does not exhibit strong intermolecular spatial and orientational correlations between chain segments. This result is consistent with the fact that PC chains are highly entangled (see Sect. 4.2). It is believed that, up to the present time, insufficient experimental evidence has been provided to prove the existence of some sort of molecular organization in glassy poly-... 

If entanglements acted like ordinary crosslinks (vN/2) per unit volume) the stored energy function would be given by the usual expression for tetrafunctional phantom networks with the spatial fluctuations of junctions suppressed ... 

This scheme performs the CNOT operation on the input photons in spatial modes a and a 2 the output qubits are contained in spatial modes b and 62-The ancilla photons in the spatial modes 03 and o are in the maximally entangled Bell state... 

Figure 2 A scheme to obtain a photonic realization of CNOT gate with two independent qubits. The qubits are encoded in the polarization of the photons. The scheme makes use of linear optical components, polarization entanglement and postselection. When one and only one photon is detected at the polarization sensitive detectors in the spatial modes 63 and 64 and in the polarization H, the scheme works as a CNOT gate.

Figure 3 The experimental setup. A type II Spontaneous parametric down-conversion is used both to produce the ancilla pair (in the spatial modes <23 and a4) and to produce the two input qubits (in the spatial modes ai and 0,2). In this case initial entanglement polarization is not desired, and it is destroyed by making the photons go through polarization filters which prepare the required input state. Half-wave plates have been placed in the photon paths in order to rotate the polarization compensators are able to nullify the birefringence effects of the non-linear crystal and of the polarizing beam splitters. Overlap of the wavepackets at the PBSs is assured through spatial and spectral filtering.

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