Quantum Zeno effect as a topological phase transition in full counting statistics and spin noise spectroscopy

When the interaction of a quantum system with a detector is changing from weak to strong coupling limits, the system experiences a transition from the regime with quantum mechanical coherent oscillations to the regime with a frozen dynamics. In addition to this quantum Zeno transition, we show that the full counting statistics of detector signal events experiences a topological phase transition at the boundary between two phases at intermediate coupling of a quantum system to the detector. We demonstrate that this transition belongs to the class of topological phase transitions that can be classified by elements of the braid group. We predict that this transition can be explored experimentally by means of the optical spin noise spectroscopy.Comment: 5 pages, 2 figure

Topologically protected Grover's oracle for the partition problem

The Number Partitioning Problem (NPP) is one of the NP-complete computational problems. Its definite exact solution generally requires a check of all $N$ solution candidates, which is exponentially large. Here we describe a path to the fast solution of this problem in $\sqrt{N}$ quasi-adiabatic quantum annealing steps. We argue that the errors due to the finite duration of the quantum annealing can be suppressed if the annealing time scales with $N$ only logarithmically. Moreover, our adiabatic oracle is topologically protected, in the sense that it is robust against small uncertainty and slow time-dependence of the physical parameters or the choice of the annealing protocol.Comment: v2: final version; to appear in Physical Review

Stochastic pump effect and geometric phases in dissipative and stochastic systems

The success of Berry phases in quantum mechanics stimulated the study of similar phenomena in other areas of physics, including the theory of living cell locomotion and motion of patterns in nonlinear media. More recently, geometric phases have been applied to systems operating in a strongly stochastic environment, such as molecular motors. We discuss such geometric effects in purely classical dissipative stochastic systems and their role in the theory of the stochastic pump effect (SPE)