Search results for "Hamiltonian"
showing 10 items of 662 documents
Explicit near-symplectic mappings of Hamiltonian systems with Lie-generating functions
2008
The construction of explicit near-symplectic mappings for generic Hamiltonian systems with the utilization of Lie transforms is presented. The method is mathematically rigorous and systematically extended to high order with respect to a perturbation parameter. The explicit mappings are compared to their implicit counterparts, which use mixed-variable generating functions, in terms of conservation of invariant quantities, calculation speed and accurate construction of Poincare surfaces of sections. The comparative study considers a wide range of parameters and initial conditions for which different time scales are involved due to large differences between internal and external frequencies of…
Gibbs states defined by biorthogonal sequences
2016
Motivated by the growing interest on PT-quantum mechanics, in this paper we discuss some facts on generalized Gibbs states and on their related KMS-like conditions. To achieve this, we first consider some useful connections between similar (Hamiltonian) operators and we propose some extended version of the Heisenberg algebraic dynamics, deducing some of their properties, useful for our purposes.
Hamiltonians defined by biorthogonal sets
2017
In some recent papers, the studies on biorthogonal Riesz bases has found a renewed motivation because of their connection with pseudo-hermitian Quantum Mechanics, which deals with physical systems described by Hamiltonians which are not self-adjoint but still may have real point spectra. Also, their eigenvectors may form Riesz, not necessarily orthonormal, bases for the Hilbert space in which the model is defined. Those Riesz bases allow a decomposition of the Hamiltonian, as already discussed is some previous papers. However, in many physical models, one has to deal not with o.n. bases or with Riesz bases, but just with biorthogonal sets. Here, we consider the more general concept of $\mat…
One-directional quantum mechanical dynamics and an application to decision making
2020
In recent works we have used quantum tools in the analysis of the time evolution of several macroscopic systems. The main ingredient in our approach is the self-adjoint Hamiltonian $H$ of the system $\Sc$. This Hamiltonian quite often, and in particular for systems with a finite number of degrees of freedom, gives rise to reversible and oscillatory dynamics. Sometimes this is not what physical reasons suggest. We discuss here how to use non self-adjoint Hamiltonians to overcome this difficulty: the time evolution we obtain out of them show a preferable arrow of time, and it is not reversible. Several applications are constructed, in particular in connection to information dynamics.
A Swanson-like Hamiltonian and the inverted harmonic oscillator
2022
We deduce the eigenvalues and the eigenvectors of a parameter-dependent Hamiltonian $H_\theta$ which is closely related to the Swanson Hamiltonian, and we construct bi-coherent states for it. After that, we show how and in which sense the eigensystem of the Hamiltonian $H$ of the inverted quantum harmonic oscillator can be deduced from that of $H_\theta$. We show that there is no need to introduce a different scalar product using some ad hoc metric operator, as suggested by other authors. Indeed we prove that a distributional approach is sufficient to deal with the Hamiltonian $H$ of the inverted oscillator.
Quantum simulation of quantum relativistic diffusion via quantum walks
2019
Two models are first presented, of one-dimensional discrete-time quantum walk (DTQW) with temporal noise on the internal degree of freedom (i.e., the coin): (i) a model with both a coin-flip and a phase-flip channel, and (ii) a model with random coin unitaries. It is then shown that both these models admit a common limit in the spacetime continuum, namely, a Lindblad equation with Dirac-fermion Hamiltonian part and, as Lindblad jumps, a chirality flip and a chirality-dependent phase flip, which are two of the three standard error channels for a two-level quantum system. This, as one may call it, Dirac Lindblad equation, provides a model of quantum relativistic spatial diffusion, which is ev…
Spin-1/2 sub-dynamics nested in the quantum dynamics of two coupled qutrits
2017
In this paper we investigate the quantum dynamics of two spin-1 systems, $\vec{\textbf{S}}_1$ and $\vec{\textbf{S}}_2$, adopting a generalized $(\vec{\textbf{S}}_1+\vec{\textbf{S}}_2)^2$-nonconserving Heisenberg model. We show that, due to its symmetry property, the nine-dimensional dynamics of the two qutrits exactly decouples into the direct sum of two sub-dynamics living in two orthogonal four- and five-dimensional subspaces. Such a reduction is further strengthened by our central result consisting in the fact that in the four-dimensional dynamically invariant subspace, the two qutrits quantum dynamics, with no approximations, is equivalent to that of two non interacting spin 1/2's. The …
A quantum particle in a box with moving walls
2013
We analyze the non-relativistic problem of a quantum particle that bounces back and forth between two moving walls. We recast this problem into the equivalent one of a quantum particle in a fixed box whose dynamics is governed by an appropriate time-dependent Schroedinger operator.
Newton algorithm for Hamiltonian characterization in quantum control
2014
We propose a Newton algorithm to characterize the Hamiltonian of a quantum system interacting with a given laser field. The algorithm is based on the assumption that the evolution operator of the system is perfectly known at a fixed time. The computational scheme uses the Crank-Nicholson approximation to explicitly determine the derivatives of the propagator with respect to the Hamiltonians of the system. In order to globalize this algorithm, we use a continuation method that improves its convergence properties. This technique is applied to a two-level quantum system and to a molecular one with a double-well potential. The numerical tests show that accurate estimates of the unknown paramete…
Integrable Hamiltonian systems with swallowtails
2010
International audience; We consider two-degree-of-freedom integrable Hamiltonian systems with bifurcation diagrams containing swallowtail structures. The global properties of the action coordinates in such systems together with the parallel transport of the period lattice and corresponding quantum cells in the joint spectrum are described in detail. The relation to the concept of bidromy which was introduced in Sadovski´ı and Zhilinski´ı (2007 Ann. Phys. 322 164–200) is discussed.