A theory of transformation is presented for the diagonalization of a Hamiltonian that is quadratic in creation and annihilation operators or in coordinates and momenta. It is the systemization and theorization of Dirac and Bogoliubov-Valatin transformations, and thus provides us an operational procedure to answer, in a direct manner, the questions as to whether a quadratic Hamiltonian is diagonalizable, whether the diagonalization is unique, and how the transformation can be constructed if the diagonalization exists. The underlying idea is to consider the dynamic matrix. Each quadratic Hamiltonian has a dynamic matrix of its own. The eigenvalue problem of the dynamic matrix determines the diagonalizability of the quadratic Hamiltonian completely. In brief, the theory ascribes the diagonalization of a quadratic Hamiltonian to the eigenvalue problem of its dynamic matrix, which is familiar to all of us. That makes it much easy to use. Applications to various physical systems are discussed, with especial emphasis on the quantum fields, such as Klein-Gordon field, phonon field, etc..
We report the effects of off-diagonal coupling on eigenstates and eigenvalues of a realistic quantum Hamiltonian, a
121Sb (spin
I=5/2) nuclear spin Hamiltonian in an antimony single crystal. In our nuclear resonance experiment, off-diagonal matrix elements are generated and controlled by the angle $\ensuremath{\theta}$ between the directions of the electric field gradient and an external magnetic field. With a very small $\ensuremath{\theta}$, a two-state-like avoided level crossing is directly probed as a function of an external magnetic field. On the other hand, by increasing $\ensuremath{\theta}$ under a weak magnetic field, we find that the nuclear spin transitions undergo dramatic changes as $\ensuremath{\theta}$ approaches ${90}^{\ensuremath{\circ}}$. All our experimental observations are in excellent agreement with exact diagonalization calculations for the Hamiltonian, demonstrating how superposition states of a multistate quantum system evolve as a function of a well-defined control parameter.
We introduce a minimal set of physically motivated postulates that the Hamiltonian
H of a continuous-time quantum walk should satisfy in order to properly represent the quantum counterpart of the classical random walk on a given graph. We found that these conditions are satisfied by infinitely many quantum Hamiltonians, which provide novel degrees of freedom for quantum enhanced protocols, In particular, the on-site energies, i.e., the diagonal elements of
H, and the phases of the off-diagonal elements are unconstrained on the quantum side. The diagonal elements represent a potential-energy landscape for the quantum walk and may be controlled by the interaction with a classical scalar field, whereas, for regular lattices in generic dimension, the off-diagonal phases of
H may be tuned by the interaction with a classical gauge field residing on the edges, e.g., the electromagnetic vector potential for a charged walker.