On the continuous spectral component of the Floquet operator for a periodically kicked quantum system

Abstract

By a straightforward generalisation, we extend the work of Combescure [J. Stat. Phys. 59, 679 (1990)] from rank-1 to rank-N perturbations. The requirement for the Floquet operator to be pure point is established and compared to that in Combescure. The result matches that in McCaw and McKellar [J. Math. Phys. 46, 032108 (2005)]. The method here is an alternative to that work. We show that if the condition for the Floquet operator to be pure point is relaxed, then in the case of the δ-kicked Harmonic oscillator, a singularly continuous component of the Floquet operator spectrum exists. We also provide an in-depth discussion of the conjecture presented in the work of Combescure of the case where the unperturbed Hamiltonian is more general. We link the physics conjecture directly to a number-theoretic conjecture of Vinogradov [The Method of Trigonometrical Sums in the Theory of Numbers (Interscience, London, 1954)] and show that a solution of Vinogradov's conjecture solves the physics conjecture. The result is extended to the rank-N case. The relationship between our work and the work of Bourget [J. Math. Anal. Appl. 276, 28 (2002); 301, 65 (2005)] on the physics conjecture is discussed.

Markovian theory of dynamical decoupling by periodic control

“…Equivalently, one can choose another time frame for propagator with time 0 chosen "just after" the first kick, i.e., at t = . In this new time frame, all kicks are slightly displaced, by , in the direction of negative t. It must be noted, however, that many sources use seemingly different approach [22,23,25,26] where U (T ) is calculated as the evolution from state ψ(nT − ) to ψ((n + 1)T − ) with t − denoting time just before time t, e.g., t − = t − with again positive and small. This, being a rather opposite convention results in different Floquet operator, which turns out to be completely equivalent in most applications to ours.…”

“…A formal expression for the Floquet operator in the case of periodically kicked systems has been known for quite a long time already (see, e.g., Refs. [22][23][24][25] and references therein). Because of the singular nature of H(t), heuristically speaking, one is interested in a unitary evolution taking the state ψ(t) not from time 0 to T , but rather from a time 0 + just after the time of the first kick to a time T + just after the time of the second kick.…”

“…It must be noted, however, that many sources use seemingly different approach [22,23,25,26] where U (T ) is calculated as the evolution from state ψ(nT − ) to ψ((n + 1)T − ) with t − denoting time just before time t, e.g., t − = t − with again positive and small. This, being a rather opposite convention results in different Floquet operator, which turns out to be completely equivalent in most applications to ours.…”

On the Energy Growth of Some Periodically Driven Quantum Systems with Shrinking Gaps in the Spectrum

“…Unfortunately the quantum dynamics in the time-dependent case proved itself to be rather difficult to analyze in its full generality and complexity. The systems which allow for at least partially analytical treatment and whose dynamics has been perhaps best studied from various points of view are either driven harmonic oscillators [4,17,10,15] or periodically kicked quantum Hamiltonians [11,12,5,7,8,25]. On a more general level, it is widely believed that there exist close links between long time behavior of a quantum system and its spectral properties.…”