Casimir companion: An invariant of motion for Hamiltonian systems

Abstract

In this paper an invariant of motion for Hamiltonian systems is introduced: the Casimir companion. For systems with simple dynamical algebras (e.g., coupled spins, harmonic oscillators) our invariant is useful in problems that consider adiabatically varying the parameters in the Hamiltonian. In particular, it has proved useful in optimal control of changes in these parameters. The Casimir companion also allows simple calculation of the entropy of nonequilibrium ensembles.

Optimal efficiency of a noisy quantum heat engine

“…Our goal is to find the frequency profile ω(t) which minimizes the final energy E(T ) = E c , maximizing thus the performance of the engine [3]. The von Neumann entropy of the system is a monotonically increasing function of the following quantity, called the Casimir companion [50] …”

Time evolution of two-dimensional quadratic Hamiltonians: A Lie algebraic approach

“…which precisely correspond to Eqs (56)- (59). In these equations it is clear that there is a correspondence between the transformation parameters and the classical position and momentum.…”

“…The key element behind the Lie algebraic approach is that the general form of the evolution operator of such type of Hamiltonian can be expressed as [55][56][57][58][59] …”

Exponential bound in the quest for absolute zero

“…Here we highlight the most important points. For u 2 = 1, the starting point (1, 0) is an equilibrium point for system (24), (25), thus the candidate optimal trajectories (extremals) should start with u = u 1 . The solution of the transcendental equation in Theorem 2 from [7] is restricted in the interval 0 < s ≤ Min{(1 − u 1 ) 2 /4, (u 2 γ 2 − 1/γ 2 ) 2 /4}, which is simplified to 0 < s ≤ (1 − u 1 ) 2 /4 for the control bounds given in (27).…”

“…The physical intuition behind these "spiral" optimal solutions can be understood if Eqs. (24), (25) are interpreted as describing the onedimensional Newtonian motion of a unit mass particle, …”