A mapping is established in connecting density matrices, associated with an evolution of a quantum open system, with vector ray in a complex projective Hilbert space. By using the corresponding vector ray to represent the open two-level system, we may observe the geometric phase for the open two-level system. The geometric phase of the open two-level system depends only on the smooth (open or closed) curve in the complex projective Hilbert space of ray, which is formulated entirely in terms of geometric structures on this space.

A different way to realize nonadiabatic geometric quantum computation is proposed by varying parameters in the Hamiltonian for nuclear-magnetic resonance, where the dynamical and geometric phases are implemented separately without the usual operational process. Therefore the phase accumulated in the geometric gate is a pure geometric phase for any input state. In comparison with the conventional geometric gates by rotating operations, our approach simplifies experimental implementations making them robust to certain experimental errors. In contrast to the unconventional geometric gates, our approach distinguishes the total and geometric phases and offers a wide choice of the relations between the dynamical and geometric phases.

Artificial molecular walkers beyond burn-bridge designs are important for nanotechnology, but their systematic development remains difficult. Herein, we have reported a new rationally designed DNA walker-track system and experimentally verified a previously proposed general expulsion regime for implementing non-burn-bridge nanowalkers. The DNA walker has an optically powered engine motif that reversibly extends and contracts the walker via a quadruplex-duplex conformational change. The walker's extension is an energy-absorbing and force-generating process, which drives the walker's leg dissociation off-track in a piston-like expulsion stroke. The unzipping-shearing asymmetry provides the expulsion stroke a bias, which decides the direction of the walker. Moreover, three candidate walkers of different sizes were fabricated. Fluorescence motility experiments indicated two of them as successful walkers and revealed a distinctive size dependence that was expected for these expulsive walkers, but was not observed in previously reported walkers. This study identifies unique technical requirements for expulsive nanowalkers. The present DNA design is readily adapted for making similar walkers from other molecules since the unzipping-shearing asymmetry is common.

Nanowalkers take either inchworm (IW) or hand-over-hand (HOH) gait. The IW nanowalkers are advantageous over HOH ones in force generation, processivity and high-density integration, though both gaits occur in intracellular nanowalkers from biology. Artificial IW nanowalkers have been realized or proposed, but all rely on different 'head' and 'tail' to gain an adventitious direction. Here we report an inherently unidirectional IW nanowalker that is a biped with two identical legs (i.e., indistinguishable 'head' and 'tail'). This walker is made of DNA, and driven by a light-powered G-quadruplex engine. The directional inchworm motion is confirmed by operating the walker on a DNA duplex track that is designed to show a distinctive fluorescence pattern for IW walkers as compared to HOH ones. Interestingly, this walker exhibits stride-controlled IW-to-HOH gait switch and direction reversal when the track's periodic binding sites have wider and wider separation. The results altogether present an integrated mechanism for implementing nanowalkers of different gaits and directions on molecular tracks, optical potentials or even solid-state surfaces.

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