The Jaynes-Cummings model without the rotating-wave approximation can be solved exactly by an extended Swain ansatz with conserved parity. Analytical approximations are then performed at different levels. The well-known rotating-wave approximation (RWA) is naturally covered in the present zeroth-and first-order approximations. A first-order correction to the RWA can be obtained in a second-order approximation, by which the effect of the counter-rotating-wave term emerges clearly. Concise analytical expressions are given explicitly and can be applicable up to the ultrastrong-coupling regime. A preliminary application to vacuum Rabi splitting is shown to be very successful.
An effective scheme within two displaced bosonic operators with equal positive and negative displacements is extended to study qubit-oscillator systems analytically in an unified way. Many previous analytical treatments, such as generalized rotating-wave approximation (GRWA) [Phys. Rev. Lett. 99, 173601 (2007)] and an expansion in the qubit tunneling matrix element in the deep strong coupling regime [Phys. Rev. Lett. 105, 263603 (2010)] can be recovered straightforwardly within the present scheme. Moreover, further improving GRWA and extension to the finite-bias case are implemented easily. The analytical expressions are then derived explicitly and uniquely, which work well in a wide range of the coupling strengthes, detunings, and static bias including the recent experimentally accessible parameters.
Degree distributions of many real networks are known to follow the Mandelbrot law, which can be considered as an extension of the power law and is determined by not only the power-law exponent, but also the shifting coefficient. Although the shifting coefficient highly affects the shape of distribution, it receives less attention in the literature and in fact, mainstream analytical method based on backward or forward difference will lead to considerable deviations to its value. In this Letter, we show that the degree distribution of a growing network with linear preferential attachment approximately follows the Mandelbrot law. We propose an analytical method based on a recursive formula that can obtain a more accurate expression of the shifting coefficient. Simulations demonstrate the advantages of our method. This work provides a possible mechanism leading to the Mandelbrot law of evolving networks, and refines the mainstream analytical methods for the shifting coefficient.PACS numbers: 89.75.Hc, 89.75.Fb, 02.50.-r Many systems can be described as networks [1][2][3][4], in which, the nodes correspond to the elements and the links to the relations between elements. Uncovering the mechanisms underlying the structural features of real networks is one of the most significant challenges in network science. Two pioneering models, respectively for small-world [5] and scalefree networks [6], give explanations for many real phenomena, such as, the logarithmic growth of average distance, the power-law degree distribution, and the high clustering coefficient. With the idea of 'rich get richer', the Barabási-Albert (BA) network [6] embodies two mechanisms: growth and preferential attachment. That is, at each time step, a new node is added and connected to a few old nodes with probability proportional to their degree as:
This paper set out from Holstein model of the quantization, by means of the method of incoherent state expansion, get the non-Linear Schrdinger equation of the polaron in ground state of one-dimensional molecular crystal, and get the soliton solution of fixed state,the ground state energy, the lattice displacement.
We study the entanglement dynamics of T-C (Tavis-Cummings) model without rotating wave approximation. By using displaced coherent state method, the influence of initial state and coupling strength to concurrence is numerically studied. Our result demonstrates that the entanglement between two atoms always keep maximum when the initial state is antisymmetric while the non-entangled initial state produce entanglement periodically due to the effect of non-rotating terms. We also show that the coupling strength between the cavity field and atoms play a critical role in the entanglement dynamics.
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