Optimized t-expansion method for the Rabi Hamiltonian

A polemic arose recently about the applicability of the t-expansion method to the calculation of the ground state energy E₀ of the Rabi model. For specific choices of the trial function and very large number of involved connected moments, the t-expansion results are rather poor and exhibit considerable oscillations. In this Letter, we formulate the t-expansion method for trial functions containing two free parameters which capture two exactly solvable limits of the Rabi Hamiltonian. At each order of the t-series, E₀ is assumed to be stationary with respect to the free parameters. A high accuracy of E₀ estimates is achieved for small numbers (5 or 6) of involved connected moments, the relative error being smaller than ¹⁰−4 (0.01%) within the whole parameter space of the Rabi Hamiltonian. A special symmetrization of the trial function enables us to calculate also the first excited energy E₁, with the relative error smaller than ¹⁰−2 (1%).     

Balance Equations-Based Properties of the Rabi Hamiltonian

A stationary physical system satisfies peculiar balance conditions involving mean values of appropriate observables. In this paper, we show how to deduce such quantitative links, named balance equations, demonstrating as well their usefulness in bringing to light physical properties of the system without solving the Schrödinger equation. The knowledge of such properties in the case of the Rabi Hamiltonian is exploited to provide arguments to make easier the variational engineering of the ground state of this model.     

A Class of Quasi-exact Solutions of Rabi Hamiltonian

A class of quasi-exact solutions of the Rabi Hamiltonian, which describes a two-level atom interacting with a single-mode radiation field via a dipole interaction without the rotating-wave approximation, are obtained by using a wavefunction ansatz. Exact solutions for part of the spectrum are obtained when the atom-field coupling strength and the field frequency satisfy certain relations. As an example, the lowest exact energy level and the corresponding atom-field entanglement at the quasi-exactly solvable point are calculated and compared to results from the Jaynes-Cummings and counter-rotating cases of the Rabi Hamiltonian.     

A Class of Quasi-exact Solutions of Rabi Hamiltonian

A class of quasi-exact solutions of the Rabi Hamiltonian, which describes a two-level atom interacting with a single-mode radiation field via a dipole interaction without the rotating-wave approximation, are obtained by using a wavefunction ansatz. Exact solutions for part of the spectrum are obtained when the atom-field coupling strength and the field frequency satisfy certain relations. As an example, the lowest exact energy level and the corresponding atom-field entanglement at the quasi-exactly solvable point are calculated and compared to results from the Jaynes-Cummings and counter-rotating cases of the Rabi Hamiltonian.     

Diagonalization of the Rabi Hamiltonian by means of a unitary transformation

A unitary transformation is derived which diagonalizes the Rabi Hamiltonian. While the solution of this diagonalization problem by standard methods has long been known, it is found that the unitary operator is of a form which has previously not been used in the connexion with spin-boson problems. The transformed Hamiltonian as well as some other characteristic transformed operators are calculated. A comparison is made between the exact unitary operator and a weak coupling transformation which is equivalent to second order perturbation theory.     

A generalized moments expansion

A new generalized moments expansion (GMX), which is based on the t-expansion, is derived. The well-known connected moments expansion (CMX) and alternate moments expansion (AMX) are shown to be special cases of the GMX.     

Fourier expansion of the Hamiltonian of crystals

The number of independent constants in the Fourier expansion of the matrix of the Hamiltonian of crystals can be diminished if the orthogonality of the states corresponding to different complex bands of the crystal is used in addition to taking account of the spatial symmetry, the symmetry relative to time inversion, and the fact that the matrix is Hermitian. Bands with the space group O⁷ (Si, Ge, cubic SiO₂) are examined as an illustration.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 101–105, July, 1981.In conclusion, the author is grateful to I. V. Abarenkov for useful discussions.     

A novel strong coupling expansion of the QCD Hamiltonian

Introducing an infinite spatial lattice with box length a, a systematic expansion of the physical QCD Hamiltonian in λ = g ^?2/3 can be obtained, with the free part being the sum of the Hamiltonians of the quantum mechanics of spatially constant fields for each box, and interaction terms proportional to λ ⁿ with n spatial derivatives connecting different boxes. As an example, the energy of the vacuum and the lowest scalar glueball is calculated up to order λ ² for the case of SU(2) Yang-Mills theory.     

Hamiltonian fluid reductions of electromagnetic drift-kinetic equations for an arbitrary number of moments

We present an infinite family of Hamiltonian electromagnetic fluid models for plasmas, derived from drift-kinetic equations. An infinite hierarchy of fluid equations is obtained from a Hamiltonian drift-kinetic system by taking moments of a generalized distribution function and using Hermite polynomials as weight functions of the velocity coordinate along the magnetic guide field. Each fluid model is then obtained by truncating the hierarchy to a finite number N+1$N+1$ of equations by means of a closure relation. We show that, for any positive N$N$, a linear closure relation between the moment of order N+1$N+1$ and the moment of order N$N$ guarantees that the resulting fluid model possesses a Hamiltonian structure, thus respecting the Hamiltonian character of the parent drift-kinetic model. An orthogonal transformation is identified which maps the fluid moments to a new set of dynamical variables in terms of which the Poisson brackets of the fluid models become a direct sum and which unveils remarkable dynamical properties of the models in the two-dimensional (2D) limit. Indeed, when imposing translational symmetry with respect to the direction of the magnetic guide field, all models belonging to the infinite family can be reformulated as systems of advection equations for Lagrangian invariants transported by incompressible generalized velocities. These are reminiscent of the advection properties of the parent drift-kinetic model in the 2D limit and are related to the Casimirs of the Poisson brackets of the fluid models. The Hamiltonian structure of the generic fluid model belonging to the infinite family is illustrated treating a specific example of a fluid model retaining five moments in the electron dynamics and two in the ion dynamics. We also clarify the connection existing between the fluid models of this infinite family and some fluid models already present in the literature.     

Magnetic moments of negative parity baryons from the effective Hamiltonian approach to QCD

The magnetic moments of the S ₁₁(1535) and S ₁₁(1650) baryons are studied in the framework of the relativistic three-quark Hamiltonian derived in the Field Correlation Method. The baryon magnetic moments are expressed via the average current quark energies which are defined by the fundamental QCD parameters: the string tension σ, the quark masses, and the strong coupling constant α _s . The resulting magnetic moments for the J ^P = 1/2^? nucleons are compared both to model calculations and to those from lattice QCD.     

Spin semi-invariant expansion for the Kondo Hamiltonian: A simple derivation of the scaling laws

The first- and second-order scaling laws for the Kondo Hamiltonian are rederived using a bandwidth cut-off scaling procedure that requires simply the invariance of electronic self-energy near the Fermi surface. To this end the latter quantity is calculated up to fourth-order in perturbation theory using the spin semi-invariant diagrammatic expansion technique that ensures the validity of linked-cluster theorem. The essential details of this technique, which can be applied as well in the study of the corresponding Kondo-lattice model, are presented.     

Integrability of the Rabi model

The Rabi model is a paradigm for interacting quantum systems. It couples a bosonic mode to the smallest possible quantum model, a two-level system. I present the analytical solution which allows us to consider the question of integrability for quantum systems that do not possess a classical limit. A criterion for quantum integrability is proposed which shows that the Rabi model is integrable due to the presence of a discrete symmetry. Moreover, I introduce a generalization with no symmetries; the generalized Rabi model is the first example of a nonintegrable but exactly solvable system.