Abstract In this chapter we study the limit ħ → 0 of quantum mechanics. The passage from quantum mechanics to classical mechanics involves precisely this limit, and the approximation schemes developed for small ħ are known as the semi-classical approximation. The semi-classical approximation has a predecessor: the so-called old quantum theory. Many of the questions concerning the connection between the quantum and classical mechanics were already raised during the years of foundation of quantum mechanics, 1900–1927, but satisfactory answers have not always been obtained.
The classical cross section for scattering by a 1/r2 potential is linear in the strength of the potential, while from the Born approximation to quantum mechanics one might expect it to be quadratic. To resolve this puzzle, we explicitly carry out the passage to the classical limit of the quantum-mechanical partial-wave series for this potential. In the process, we discover the conditions under which quantum scattering becomes classical; under these conditions the Born approximation is not applicable.
Achieving high-fidelity direct two-qubit gates over meter-scale quantum interconnects is challenging, in part due to the multimode nature of such systems. One alternative scheme is to combine local operations with remote quantum state transfer or remote entanglement. Here, we theoretically study quantum state transfer and entanglement generation for two distant qubits, equipped with tunable interactions, over a common multimode interconnect. We model the performance of the superadiabatic transitionless driving (SATD) protocol for adiabatic passage and demonstrate various favorable improvements over the standard method. In particular, by suppressing leakage to a select (resonant) interconnect mode, SATD breaks the speed-limit relation imposed by the qubit-interconnect interaction
g, where instead the operation time is limited by leakage to the adjacent modes, i.e., the free spectral range ${\mathrm{\ensuremath{\Delta}}}_{c}$ of the interconnect, allowing for fast operations even with weak
g. Furthermore, we identify a multimode error mechanism for Bell-state generation using such adiabatic protocols, in which the even/odd modal dependence of qubit-interconnect interaction breaks down the dark-state symmetry, leading to detrimental adiabatic overlap with the odd modes growing as $(g/{\mathrm{\ensuremath{\Delta}}}_{c}{)}^{2}$. Therefore, adopting a weak coupling, imposed by a multimode interconnect, SATD provides a significant improvement in terms of operation speed and consequently sensitivity to incoherent error.