Differential topology of adiabatically controlled quantum processes
It is shown that in a controlled adiabatic homotopy between two Hamiltonians, H 0 and H 1, the gap or “anti-crossing” phenomenon can be viewed as the development of cusps and swallow tails in the region of the complex plane where two critical value curves of the quadratic map associated with the numerical range of H 0 + i H 1 come close. The “near crossing” in the energy level plots happens to be a generic situation, in the sense that a crossing is a manifestation of the quadratic numerical range map being unstable in the sense of differential topology. The stable singularities that can develop are identified and it is shown that they could occur near the gap, making those singularities of paramount importance. Various applications, including the quantum random walk, are provided to illustrate this theory.
KeywordsAdiabatic theorem Numerical range Cusps Swallow tails Gap Anti-crossing
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- 7.Jonckheere, E., Shabani, A., Rezakhani, A.: 4th International Symposium on Communications, Control, and Signal Processing (ISCCSP’2010) (Limassol, Cyprus, 2010), Special Session on Quantum Control I. ArXiv:1002.1515v2Google Scholar
- 9.Ahmad, F.: Differential topology of numerical range and robust control analysis. Ph.D. thesis, Department of Electrical Engineering, University of Southern California (1999)Google Scholar
- 10.Arnold V.I., Gusein-Zade S.M., Varchenko A.N.: Singularities of Differentiable Maps—The Classification of Critical Points, Caustics and Wave Fronts, vol. 1. Birkhäuser, Boston (1985)Google Scholar
- 12.Arnold, V.I.: The theory of singularities and its applications. Accademia Nationale Dei Lincei; Secuola Normale Superiore Lezioni Fermiane. Press Syndicate of the University of Cambridge, Pisa, Italy (1993)Google Scholar
- 13.Arnold, V.: Topological invariants of plane curves and caustics, University Lecture Series; Dean Jacqueline B. Lewis Memorial Lectures, Rutgers University, vol. 5. American Mathematical Society, Providence, RI (1994)Google Scholar
- 15.Cerf, J.: Publications Mathématiques, Institut des Hautes Etudes Scientifiques (I.H.E.S.) (39), p. 5 (1970)Google Scholar
- 16.Jonckheere E.A.: Algebraic and Differential Topology of Robust Stability. Oxford University Press, New York (1997)Google Scholar