Donoho and Stark have shown that a precise deterministic recovery of missing information contained in a time interval shorter than the time-frequency uncertainty limit is possible. We analyze this signal recovery mechanism from a physics point of vie
w and show that the well-known Shannon-Nyquist sampling theorem, which is fundamental in signal processing, also uses essentially the same mechanism. The uncertainty relation in the context of information theory, which is based on Fourier analysis, provides a criterion to distinguish Shannon-Nyquist sampling from compressed sensing. A new signal recovery formula, which is analogous to Donoho-Stark formula, is given using the idea of Shannon-Nyquist sampling; in this formulation, the smearing of information below the uncertainty limit as well as the recovery of information with specified bandwidth take place. We also discuss the recovery of states from the domain below the uncertainty limit of coordinate and momentum in quantum mechanics and show that in principle the state recovery works by assuming ideal measurement procedures. The recovery of the lost information in the sub-uncertainty domain means that the loss of information in such a small domain is not fatal, which is in accord with our common understanding of the uncertainty principle, although its precise recovery is something we are not used to in quantum mechanics. The uncertainty principle provides a universal sampling criterion covering both the classical Shannon-Nyquist sampling theorem and the quantum mechanical measurement.
Besides the conventional transverse couplings between superconducting qubits (SQs) and electromagnetic fields, there are additional longitudinal couplings when the inversion symmetry of the potential energies of the SQs is broken. We study nonclassic
al-state generation in a SQ which is driven by a classical field and coupled to a single-mode microwave field. We find that the classical field can induce transitions between two energy levels of the SQs, which either generate or annihilate, in a controllable way, different photon numbers of the cavity field. The effective Hamiltonians of these classical-field-assisted multiphoton processes of the single-mode cavity field are very similar to those for cold ions, confined to a coaxial RF-ion trap and driven by a classical field. We show that arbitrary superpositions of Fock states can be more efficiently generated using these controllable multiphoton transitions, in contrast to the single-photon resonant transition when there is only a SQ-field transverse coupling. The experimental feasibility for different SQs is also discussed.
It is well-known that some nonlinear phenomena such as strong photon blockade are hard to be observed in optomechanical system with current experimental technology. Here, we present a coherent feedback control strategy in which a linear cavity is coh
erently controlled by an optomechanical controller in a feedback manner. The coherent feedback loop transfers and enhances quantum nonlinearity from the controller to the controlled cavity, which makes it possible to observe strong nonlinear effects in either linear cavity or optomechanical cavity. More interestingly, we find that the strong photon blockade under single-photon optomechanical weak coupling condition could be observed in the quantum regime. Additionally, the coherent feedback loop leads to two-photon and multiphoton tunnelings for the controlled linear cavity, which are also typical quantum nonlinear phenomenon. We hope that our work can give new perspectives in engineering nonlinear quantum phenomena.
