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Quantum Geometric Phase

Quantum Geometric Phase. Ming-chung Chu Department of Physics The Chinese University of Hong Kong. Content. A brief review of quantum geometric phase Problems with orthogonal states Projective phase: a new formalism

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Quantum Geometric Phase

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  1. Quantum Geometric Phase Ming-chung Chu Department of Physics The Chinese University of Hong Kong

  2. Content • A brief review of quantum geometric phase • Problems with orthogonal states • Projective phase: a new formalism • Applications: off-diagonal geometric phases, extracting a topological number, geometric phase at a resonance, geometric phase of a BEC (preliminary)

  3. 1. Review of Geometric Phase

  4. B. Goss Levi, Phys. Today 46, 17(1993). Review of Geometric Phase Classic example of geometric phase acquired by parallel transporting a vector through a loop Parallel transport: at each small step, keep the vector as aligned to the previous one as possible. The blue vector is rotated by an angle which is equal to the solid angle subtended at the center enclosed by the loop: geometry of the space.

  5. After a cyclic evolution, a particle returns to its initial state; its wave function acquires an extra phase • Dynamical phase: Review of Geometric Phase • Geometric phase is the extra phase in addition to the dynamical phase • It arises from the movement of the wave function and contains information about the geometry of the space in which the wave function evolves

  6. Physical realization of geometric phase • Neutron interferometry – spin ½ systems evolving in changing external fields eg. A. Wagh et al., PRL 78, 755 (1997); B. Allman et al., PRA 56, 4420 (1997); Y. Hasegawa et al., PRL 87, 070401 (2001). • Microwave resonators – real-valued wave functions evolving in cavity with changing boundaries eg. H.-M. Lauber, P. Weidenhammer, D. Dubbers, PRL 72, 1004 (1994). • Quantum pumping – time-varying potential walls (gates) for a quantum dot: geometric phase number of electrons transported eg. J. Avron et al., PRB 62, R10618 (2000); M. Switkes et al., Science 283, 1905 (1999). • Level splitting and quantum number shifting in molecular physics • Intimately connected to physics of fractional statistics, quantized Hall effect, and anomalies in gauge theory • … Quantum geometric phase is physical, measurable, and can have non-trivial observable effects; it may even be useful for quantum computation (phase gates)!

  7. Eg. Quantum geometric phase observed in microwave cavity with changing boundaries (adiabatic) H.-M. Lauber, P. Weidenhammer, D. Dubbers, PRL 72, 1004 (1994). After a cyclic evolution, the wave function (states 13, 14) acquires a sign change = geometric phase of .

  8. Rectangular cavity: 3-state degeneracy H.-M. Lauber, P. Weidenhammer, D. Dubbers, PRL 72, 1004 (1994).

  9. Generalizations of Geometric phase

  10. Ray space (projective Hilbert space) • States with only an overall phase difference are identified to the same point • Eg. Two-state systems: ray space = surface of a sphere

  11. C(s) Can use any parameterization of the loop: geometrical A-A Phase Y. Aharonov and J. Anandan, PRL 58, 1593 (1987).

  12. A-A phase • The field strength F integrated over the area is the geometric phase • In a 2-state system, half of the solid angle included is the geometric phase where C(s) Non-cyclic evolution: open loop! Need to close the loop to ensure local gauge invariant!

  13. Pancharatnam phase ~ A-A phase For unclosed paths (non-cyclic evolutions), just join the states with a geodesic Evolution path geodesic Pancharatnam phase J. Samuel and R. Bhandari: just join the open points with a geodesic!

  14. Relative phase can be measured by interference Define a vector potential Pancharatnam Phase • To remove dynamical phase, define where the geodesic is the curve connecting φ(0) and φ(t) in the ray space given by the geodesic equation. S. Pancharatnam, Proc. Indian Acad. Sci., 247 (1956); J. Samuel and R. Bhandari, PRL 60, 2339 (1988).

  15. 2. Problems with orthogonal states

  16. Pancharatnam phase between orthogonal states There are infinitely many geodesics (eg. 1, 2) possible to close the path!

  17. Off-diagonal Geometric Phases N. Manini and F. Pistolesi, PRL 85, 3067 (2000). • A scheme to extract phase information for orthogonal states, by using more than 1 state, in adiabatic evolution • An eigenstate orthogonal to ; can still compare its phase to another eigenstate • Off-diagonal geometric phases: • Independent combinations of ’s are gauge invariant and contain all phase information of the system • Measurable by neutron interferometry Y. Hasegawa et al., PRL 87, 070401 (2001).

  18. geodic geodic Off-diagonal geometric phase Off-diagonal geometric phases are measurable and complement diagonal (Berry’s) phases. Y. Hasegawa et al., PRL 87, 070401 (2001).

  19. 3. Projective Phase: a new formalism Hon Man Wong, Kai Ming Cheng, and M.-C. Chu Phys. Rev. Lett. 94, 070406 (2005).

  20. y x polarizer After inserting a polarizer, they can interfere! Projective phase • Two orthogonal polarized light cannot interfere

  21. Projective Phase • First project two states onto i and then let them interfere

  22. Schrödinger evolution geodesic Pancharatnam phase geodesic geodesic Projective phase i-dependent! Geometrical meaning Find a state |i > not orthogonal to either one, then join them with geodesics.

  23. y y x x Polarizer i Polarizer j Gauge Transformation

  24. Gauge Transformation • The gauge transformation at a point P is • This is the transition function in fiber bundle • The two projective phases are related by • With this transformation, one projective phase can give all others

  25. The difference between and is where the Bargmann invariant is defined by Bargmann invariant which is equal to the –ve of the geometric phase enclosed by the 4 geodesics R. Simon and N. Mukunda, PRL 70, 880 (1993).

  26. a g e b The monopole problem • A monopole with magnetic charge g is placed at the origin • When a charged particle moves in a closed loop, it gains a phase factor

  27. At south pole: • Dirac monopole quantization: • Wu and Yang: 2 vector potentials () to cover the sphere, and gauge transformation Sab to relate them

  28. Monopole and projective phase • The 2-state system projective phase has the same fiber bundle structure as a monopole with g=1/2

  29. 4. Applications Off-diagonal geometric phases Extracting a topological number Geometric phase at a resonance Geometric phase of a BEC (preliminary)

  30. geod geod geod geod Off-diagonal geometric phase Can be decomposed into projective phases and Bargmann Invariants Let The off-diagonal geometric phase is: n projective phases = n(n-1) off-diagonal phases where

  31. i i j Extracting a topological number • The difference between and as (closed loop) can be used to extract the first Chern number n: • The loop can be smoothly deformed and n is not changed • n is a topological number of the ray space • Eg. spin-m systems:

  32. Excellent approximate analytic solution using Rotating Wave Approximation (RWA) Resonance: →two-state system Rabi Oscillation at resonance: RWA vs. numerical solution resonances Geometric phase at resonance: Schrödinger particle in a vibrating cavity K.W. Yuen, H.T. Fung, K.M. Cheng, M.-C. Chu, and K. Colanero, Journal of Physics A 36,11321 (2003).

  33. RWA solution for geometric phase: Numerical solution Geometric phase at resonance T = Rabi oscillation period Similar solution for an electron in a rotating magnetic field.

  34. π phase change • In monopole problem, when the particle enters a region Aa is undefined it should be switched to Ab • In projective phase, at a state orthogonal to the initial state, the covering should be switched • the phase factor is • With the projective phase formalism, we can show the existence of the πjump (and the condition for its occurence).

  35. Geometric phase of a BEC • Bose-Einstein Condensate (BEC): macroscopic wavefunction – can we see its geometric phase? • The phase of a BEC can be measured recently • The evolution of a BEC is governed by a non-linear Schrödinger equation: Gross-Pitaevskii equation (GPE)

  36. Geometric phase t Numerical Results • Solving GPE with Crank-Nicholson algorithm • Initial state prepared by time-independent GPE solution with • Time-evolve with • Resulting phases agree well with perturbative calculation But: dynamical phase much larger!

  37. Summary • We have constructed the formalism of projective phase, with geometrical meaning and fiber-bundle structure • It can be used to compute the phase between any two states (even orthogonal, non-adiabatic, non-cyclic) • Off-diagonal geometric phasescan be decomposed into projective phases and Bargmann invariants • We show that a topological number can be extracted from the projective phases • We have analyzed the π phase change with projective phase, showing only 0 or π phase change can occur at orthogonal states

  38. Quantum Geometric Phase Hon Man Wong, Kai Ming Cheng, Ming-chung Chu Department of Physics The Chinese University of Hong Kong

  39. Eg. Neutron interferometry Y. Hasegawa et al., PRL 87, 070401 (2001). With B to rotate the neutrons Without B Geometric phase of

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