Effective Hamiltonian theory and its applications in quantum information

Abstract

This paper presents a useful compact formula for deriving an effective Hamiltonian describing the time-averaged dynamics of detuned quantum systems. The formalism also works for ensembleaveraged dynamics of stochastic systems. To illustrate the technique we give examples involving Raman processes, Bloch-Siegert shifts and Quantum Logic Gates.* To be published in Canadian Journal of Physics.

Quasi-Bell states in a strongly coupled qubit–oscillator system and their delocalization in the phase space

“…We qualitatively describe it below. Following [42] we may obtain physically observed quantities after a suitable coarse graining of the time scale that acts as a low-pass filter effectively eliminating the high frequency components in quantum oscillations. The long time behavior of physical quantities, say S p`q W , is set by the interaction modes typically having frequencies Op r ∆q, while the high oscillator frequency ω gives rise to the rapid fluctuations.…”

“…The squeezing is, however, present during part of the oscillations. The coarse-graining process [42] discussed before removes the squeezing property as the measured quantity is averaged over time T coarse graining . In the ultra-strong coupling regime the squeezing property is eliminated altogether since the phase coherence properties of the Q-distribution, which is necessary for the squeezing to appear, is destroyed in the domain λ " ω where the interaction modes tOpx n r ∆q|n " 0, 1, .…”

Atom-mediated effective interactions between modes of a bimodal cavity

“…In this work, we use the effective Hamiltonian approach [22][23][24] in order to obtain two effective interactions between the modes of a bimodal cavity, Hamiltonians (4) and (11).…”

“…Assuming large detunings, so that Ω ≪ |∆| and |∆| ≫ √ n i |λ| (i = a, b), where n i is the mean number of photons in the i-th cavity mode, H cav presents fast oscillating time dependence, which allows us to apply the effective Hamiltonian approach proposed in references [22][23][24]. From the high harmonic disturbance of H cav , we can determine the dynamical evolution by considering an averaged density matrix in a time resolution which eliminates the high-frequency feature explicitly.…”

Inelastic Multiple Scattering of Interacting Bosons in Weak Random Potentials

“…We will use here a formula derived by James and Jerke [220,221] which allows to extract an effective Hamiltonian from an existing interaction picture Hamiltonian if the interaction is sufficiently weak and if the perturbation has a harmonic time dependence.…”