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. 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.
B. Goss Levi, Phys. Today 46, 17 (1993).
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. Review of Geometric Phase
After a cyclic evolution, a particle returns to its initial state; its wave function acquires an extra phase
Dynamical 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
Gauge invariance

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, (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
Condition
Space
Berry’s Phase
M. Berry, Proc. R. Soc. Lond. A, p. 45 (1984).
Adiabatic and cyclic
Parameter space
Aharonov-Anadan Phase (A-A Phase)
Y. Aharonov and J. Anandan, PRL 58, 1593 (1987).
Cyclic
Ray Space
Pancharatnam Phase
S. Pancharatnam, Proc. Indian Acad. Sci., 247 (1956); J. Samuel and R. Bhandari, PRL 60, 2339 (1988).
General
Explain adiabaticity and cyclic

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
Example of spin half, only a sphere is enough for theta and phi

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

12. A-A phase
where
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
C(s)
F is gauge invariant
Non-cyclic evolution: open loop!
Need to close the loop to ensure local gauge invariant!

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

14. Pancharatnam Phase Relative phase can be measured by interference
To remove dynamical phase, define
Define a vector potential
use s as parameter instead of t
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!
The area is undefined/ interference gives nothing

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, (2001).

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

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

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

21. Projective Phase
First project two states onto i and then let them interfere
3 indices, gauge invariance of I and psi

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

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

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
Fibre bundle transition property, having the meaning of transform to j then back to I, Lorentz transformation
With this transformation, one projective phase can give all others

25. Bargmann invariant The difference between and is
where the Bargmann invariant
is defined by
The difference between and is
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. The monopole problem g e a b
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. Dirac monopole quantization:
At south pole:
Dirac monopole quantization:
Wu and Yang: 2 vector potentials ( ) to cover the sphere, and gauge transformation Sab to relate them
Infinitesimal loop giving finite phase-> need quantization
Int A at lower half sphere gives total charge non zero, but inf small area -> quantization

28. Monopole and projective phase
The 2-state system projective phase has the same fiber bundle structure as a monopole with g = 1/2
We showed that the geometric structure is the same, especially the transition function

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

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

31. Extracting a topological numberj
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: i

32. 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).
Resonance: →two-state system
Excellent approximate analytic solution using Rotating Wave Approximation (RWA)
resonances
Rabi Oscillation at resonance: RWA vs. numerical solution

33. Geometric phase at resonance
RWA solution for geometric phase:
T = Rabi oscillation period
Numerical solution
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. 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 t
Geometric phase
But: dynamical phase much larger!
Can be achieved by feshbach resonence to modify the strength of interaction of bec atoms

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 phases can 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, (2001).
Without B
With B to rotate the neutrons
Geometric phase of