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Journal of Physics B Atomic Molecular and Optical Physics

Invariants of motion for an N-level system

K. S. Mallesh, G. Ramachandran

Published Jul 28, 1989• 3 citations
Publisher

Abstract

For an N-level system interacting with N-1 laser fields, it is shown that there exist in general N 2 -N+1 independent invariants of motion when the Hamiltonian assumes a time-independent form under a suitable unitary transformation. Two different procedures to obtain these invariants are presented. The special case of N=3 is considered in detail and two new independent invariants are shown to exist in addition to the five already known. Explicit expressions for these two invariants are also given.

DOI
10.1088/0953-4075/22/14/013

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For an N-level system interacting with N-1 laser fields, it is shown that there exist in general N 2 -N+1 independent invariants of motion when the Hamiltonian assumes a time-independent form under a suitable unitary transformation. Two different procedures to obtain these invariants are presented. The special case of N=3 is considered in detail and two new independent invariants are shown to exist in addition to the five already known. Explicit expressions for these two invariants are also given.
We analyze the time-dependence of N-level systems under the Rotating Wave Approximation and dipole selection rules. Such systems can be solved straightforwardly if the Hamiltonian can be transformed into a time-independent form. The conditions under which a unitary transformation can be used to render time-dependent Hamiltonians into a time-independent form, thereby making the solution, are examined. After case-by-case analysis of different four and five-level systems, we conclude that systems having only one odd or even parity level achieve time-independence. In contrast, the others must satisfy a condition of laser detuning to achieve time-independence.
The time-dependent integrals of motion, linear in position and momentum operators, of a quantum system are extracted from Noether's theorem prescription by means of special time-dependent variations of coordinates. For the stationary case of the generalized two-dimensional harmonic oscillator, the time-independent integrals of motion are shown to correspond to special Bragg-type symmetry properties. A detailed study for the nonstationary case of this quantum system is presented. The linear integrals of motion are constructed explicitly for the case of varying mass and coupling strength. They are also obtained from Noether's theorem. The general treatment for a multi-dimensional quadratic system is indicated and it is shown that the time-dependent variations that give rise to the linear invariants, as conserved quantities, satisfy the corresponding classical homogeneous equations of motion for the coordinates.
The quantum Liouville equation for an nnn-level atomic system driven by external fields has a nontrivial kinematic structure; the quantities $\mathrm{tr}{\ensuremath{\rho}}^{j}$, j=1,2,…,nj\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}1,2,\dots{},nj=1,2,…,n remain constant in time, independent of the Hamiltonian. These invariants are physically significant; the qualitative character of the solution depends on their existence. A generic numerical method will not, in general, preserve these invariants. We present a numerical technique which evolves the density matrix via unitary transformations thus exactly preserving these invariants to all orders in the time step.
It is well known that the motion of a dynamical system can be pictured in two distinct ways, which Dirac names the Heisenberg picture and the Schrödinger picture. The equations of motion take different forms in the two pictures, but of course have identical physical consequences, since the motion of the system does not depend in any way on which picture we choose. It should therefore be possible to express the equations in an invariant form (independent, that is, of the picture used). It will be shown in this note that not only can this be done (equation (3)), but it can be done in such a way that it is not even necessary to introduce a picture at all (equation (3′)).
This paper develops a gauge-independent symmetry theory of non-relativistic quantum mechanical systems, in line with that previously considered in the context of classical mechanics. We first discuss at length the motivation for adopting the view that the invariance of a system K under a physical symmetry operation .<I' should be taken to mean invariance of the equation of motion of K under a certain gauge-independent unitary transformation U c( .<1'). The formal development of the theory is then carried through, and some detailed examples are presented. In particular, corresponding to every direction along which a system K happens to be translation invariant there exists a gauge-independent generator of translations which leaves K invariant but which is not, in general, a component of either the canonical or the kinetic momentum. The connexion between such invariant generators of translations and the so-called magnetic translation operators is referred to.
A strong laser field interacting with an atom or a molecule can induce a coherent process which includes a large number of energy levels. If we can find a transformation which eliminates the explicit time dependence of the interaction Hamiltonian (for example, the rotating-wave approximation), we can reduce the dynamic problem to the consideration of eigenvalues and eigenvectors of the Hamiltonian. However, the complexity of an arbitrary multilevel Hamiltonian usually forces us to use numerical methods, and, therefore, not much general understanding about multilevel systems can be achieved. We consider the multilevel problems in a more general setting. Our approach is based on the observation that an arbitrary Hamiltonian matrix can be transformed into a tridiagonal form by unitary transformation.1 A tridiagonal form represents a ladder-type multilevel system. From all W-level systems the ladder configuration is the simplest in the sense that the number of level connections is the smallest possible. This means that the most effective analysis of a given multilevel system can be achieved using the transformation to ladder configuration. The spectral analysis of ladders is facilitated by the connections to orthogonal polynomials and continued fractions. The investigations about the spectral properties of ladders are reported.