Analytical solutions for the Rabi model

The Rabi model that describes the fundamental interaction between a two-level system with a quantized harmonic oscillator is one of the simplest and most ubiquitous models in modern physics. However, this model has not been solved exactly because it is hard to find a second conserved quantity besides the energy. Here we present a unitary transformation to map this unsolvable Rabi model into a solvable Jaynes-Cummings-like model by choosing a proper variation parameter. As a result, the analytical energy spectrums and wavefunctions including both the ground and the excited states can be obtained easily. Moreover, these explicit results agree well with the direct numerical simulations in a wide range of the experimental parameters. In addition, based on our obtained energy spectrums, the recent experimental observation of Bloch-Siegert in the circuit quantum electrodynamics with the ultrastrong coupling can be explained perfectly. Our results have the potential application in the solid-state quantum information processing.Comment: 5 pages, 4 figure

Solution for (1+1) dimensional surface solitons in thermal nonlinear media

Analytical solutions for (1+1)D surface fundamental solitons in thermal nonlinear media are obtained. The stationary position and the critical power of surface solitons are obtained using this analytical solutions. The analytical solutions are verified by numerical simulations. The solutions for surface breathers and their breathing period, and solutions for surface dipole and tripole solitons are also given.Comment: 7 pages, 8 figures, 28 reference, Accepted by Phys. Rev.

Relativistic Expansion of Magnetic Loops at the Self-similar Stage II: Magnetized outflows interacting with the ambient plasma

We obtained self-similar solutions of relativistically expanding magnetic loops by assuming axisymmetry and a purely radial flow. The stellar rotation and the magnetic fields in the ambient plasma are neglected. We include the Newtonian gravity of the central star. These solutions are extended from those in our previous work (Takahashi, Asano, & Matsumoto 2009) by taking into account discontinuities such as the contact discontinuity and the shock. The global plasma flow consists of three regions, the outflowing region, the post shocked region, and the ambient plasma. They are divided by two discontinuities. The solutions are characterized by the radial velocity, which plays a role of the self-similar parameter in our solutions. The shock Lorentz factor gradually increases with radius. It can be approximately represented by the power of radius with the power law index of 0.25. We also carried out magnetohydrodynamic simulations of the evolution of magnetic loops to study the stability and the generality of our analytical solutions. We used the analytical solutions as the initial condition and the inner boundary conditions. We confirmed that our solutions are stable over the simulation time and that numerical results nicely recover the analytical solutions. We then carried out numerical simulations to study the generality of our solutions by changing the power law index \delta of the ambient plasma density \rho_0 \propto r^{-\delta}. We alter the power law index \delta from 3.5 in the analytical solutions. The analytical solutions are used as the initial conditions inside the shock in all simulations. We observed that the shock Lorentz factor increases with time when \delta is larger than 3, while it decreases with time when \delta is smaller than 3. The shock Lorentz factor is proportional to t^{(\delta-3)/2}. These results are consistent with the analytical studies by Shapiro (1979).Comment: 19 pages, 13 figures, Accepted for publication in MNRA

Analytical Solutions of Open String Field Theory

In this work we review Schnabl's construction of the tachyon vacuum solution to bosonic covariant open string field theory and the results that followed. We survey the state of the art of string field theory research preceding this construction focusing on Sen's conjectures and the results obtained using level truncation methods. The tachyon vacuum solution can be described in various ways. We describe its geometric representation using wedge states, its formal algebraic representation as a pure-gauge solution and its oscillator representation. We also describe the analytical proofs of some of Sen's conjectures for this solution. The tools used in the context of the vacuum solution can be adapted to the construction of other solutions, namely various marginal deformations. We present some of the approaches used in the construction of these solutions. The generalization of these ideas to open superstring field theory is explained in detail. We start from the exposition of the problems one faces in the construction of superstring field theory. We then present the cubic and the non-polynomial versions of superstring field theory and discuss a proposal suggesting their classical equivalence. Finally, the bosonic solutions are generalized to this case. In particular, we focus on the (somewhat surprising) generalization of the tachyon solution to the case of a theory with no tachyons.Comment: Invited review for Physics Reports. v1: 106 p., 8 fig. v2: 108 p., minor changes. v3: 117 p., 9 fig., presentation modified and expanded in several places, typos corrected, ref. added and updated. v4: Published version. 125 p., 10 fig., further modifications of the presentation, ref. added and update

New chemical evolution analytical solutions including environment effects

In the last years, more and more interest has been devoted to analytical solutions, including inflow and outflow, to study the metallicity enrichment in galaxies. In this framework, we assume a star formation rate which follows a linear Schmidt law, and we present new analytical solutions for the evolution of the metallicity (Z) in galaxies. In particular, we take into account environmental effects including primordial and enriched gas infall, outflow, different star formation efficiencies, and galactic fountains. The enriched infall is included to take into account galaxy-galaxy interactions. Our main results can be summarized as: i) when a linear Schmidt law of star formation is assumed, the resulting time evolution of the metallicity Z is the same either for a closed-box model or for an outflow model. ii) The mass-metallicity relation for galaxies which suffer a chemically enriched infall, originating from another evolved galaxy with no pre-enriched gas, is shifted down in parallel at lower Z values, if compared the closed box model. iii) When a galaxy suffers at the same time a primordial infall and a chemically enriched one, the primordial infall always dominates the chemical evolution. iv) We present new solutions for the metallicity evolution in a galaxy which suffers galactic fountains and an enriched infall from another galaxy at the same time. The analytical solutions presented here can be very important to study the metallicity (oxygen), which is measured in high-redshift objects. These solutions can be very useful: a) in the context of cosmological semi-analytical models for galaxy formation and evolution, and b) for the study of compact groups of galaxies.Comment: Accepted for publication in MNRA