The Born Rule in Quantum and Classical Mechanics

Considerable effort has been devoted to deriving the Born rule (e.g. that $|\psi(x)|^2 dx$ is the probability of finding a system, described by $\psi$, between $x$ and $x + dx$) in quantum mechanics. Here we show that the Born rule is not solely quantum mechanical; rather, it arises naturally in the Hilbert space formulation of {\it classical} mechanics as well. These results provide new insights into the nature of the Born rule, and impact on its understanding in the framework of quantum mechanics.Comment: 5 pages, no figures, to appear in Phys. Rev.

Floquet topological phases in a spin-1/2 double kicked rotor

The double kicked rotor model is a physically realizable extension of the paradigmatic kicked rotor model in the study of quantum chaos. Even before the concept of Floquet topological phases became widely known, the discovery of the Hofstadter butterfly spectrum in the double kicked rotor model [J. Wang and J. Gong, Phys. Rev. A 77, 031405 (2008)] already suggested the importance of periodic driving to the generation of unconventional topological matter. In this work, we explore Floquet topological phases of a double kicked rotor with an extra spin-1/2 degree of freedom. The latter has been experimentally engineered in a quantum kicked rotor recently by loading Rb87 condensates into a periodically pulsed optical lattice. Under the on-resonance condition, the spin-1/2 double kicked rotor admits fruitful topological phases due to the interplay between its external and internal degrees of freedom. Each of these topological phases is characterized by a pair of winding numbers, whose combination predicts the number of topologically protected 0 and \pi-quasienergy edge states in the system. Topological phases with arbitrarily large winding numbers can be easily found by tuning the kicking strength. We discuss an experimental proposal to realize this model in kicked Rb87 condensates, and suggest to detect its topological invariants by measuring the mean chiral displacement in momentum space.Comment: 10 pages, 4 figures, typos removed and references adde

Stabilizing Non-Hermitian Systems by Periodic Driving

The time evolution of a system with a time-dependent non-Hermitian Hamiltonian is in general unstable with exponential growth or decay. A periodic driving field may stabilize the dynamics because the eigenphases of the associated Floquet operator may become all real. This possibility can emerge for a continuous range of system parameters with subtle domain boundaries. It is further shown that the issue of stability of a driven non-Hermitian Rabi model can be mapped onto the band structure problem of a class of lattice Hamiltonians. As an application, we show how to use the stability of driven non-Hermitian two-level systems (0-dimension in space) to simulate a spectrum analogous to Hofstadter's butterfly that has played a paradigmatic role in quantum Hall physics. The simulation of the band structure of non-Hermitian superlattice potentials with parity-time reversal symmetry is also briefly discussed

Complete quantum control of the population transfer branching ratio between two degenerate target states

A five-level four-pulse phase-sensitive extended stimulated Raman adiabatic passage scheme is proposed to realize complete control of the population transfer branching ratio between two degenerate target states. The control is achieved via a three-node null eigenstate that can be correlated with an arbitrary superposition of the target states. Our results suggest that complete suppression of the yield of one of two degenerate product states, and therefore absolute selectivity in photochemistry, is achievable and predictable, even without studying the properties of the unwanted product state beforehand.Comment: 9 pages, 5 figures, to appear in J. Chem. Phy

Simulation of non-Abelian braiding in Majorana time crystals

Discrete time crystals have attracted considerable theoretical and experimental studies but their potential applications have remained unexplored. A particular type of discrete time crystals, termed "Majorana time crystals", is found to emerge in a periodically driven superconducting wire accommodating two different species of topological edge modes. It is further shown that different Majorana edge modes separated in the time lattice can be braided, giving rise to an unforeseen scenario for topologically protected gate operations. The proposed braiding scheme can also generate a magic state that is important for universal quantum computation. This study thus advances the quantum control in discrete time crystals and reveals their great potential arising from their time-domain properties.Comment: 10 pages, 6 figures. Finalized versio