Quantum Geometric Phase

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Quantum Geometric Phase

• Ming-chung Chu
• Department of Physics
• The Chinese University of Hong Kong

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Content

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

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1. Review of Geometric Phase
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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.
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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

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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)!
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Generalizations of Geometric phase
Condition Space
Berrys 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 Ray Space
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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

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A-A Phase
Y. Aharonov and J. Anandan, PRL 58, 1593 (1987).
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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)
Non-cyclic evolution open loop!
Need to close the loop to ensure local gauge
invariant!
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Pancharatnam 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

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Pancharatnam Phase

• Relative phase can be measured by interference

• To remove dynamical phase, define

where the geodesic is the curve connecting f(0)
and f(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).
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2. Problems with orthogonal states
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Pancharatnam phase between orthogonal states
There are infinitely many geodesics (eg. 1, 2)
possible to close the path!
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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).

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Off-diagonal geometric phase
Off-diagonal geometric phases are measurable and
complement diagonal (Berrys) phases. Y. Hasegawa
et al., PRL 87, 070401 (2001).
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3. Projective Phase a new formalism

• Hon Man Wong, Kai Ming Cheng, and M.-C. Chu
• Phys. Rev. Lett. 94, 070406 (2005).

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Projective phase

• Two orthogonal polarized light cannot interfere

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Projective Phase

• First project two states onto i and then let them
  interfere

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Geometrical meaning
Find a state i gt not orthogonal to either one,
then join them with geodesics.
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Gauge Transformation
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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

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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).
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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

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

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Monopole and projective phase

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

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4. Applications

• Off-diagonal geometric phases
• Extracting a topological number
• Geometric phase at a resonance
• Geometric phase of a BEC (preliminary)

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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
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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

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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).
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Geometric phase at resonance
T Rabi oscillation period
Similar solution for an electron in a rotating
magnetic field.
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p 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 pjump (and the condition for
  its occurence).

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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)

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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!
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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 p phase change with
  projective phase, showing only 0 or p phase
  change can occur at orthogonal states

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Quantum Geometric Phase

• Hon Man Wong, Kai Ming Cheng, Ming-chung Chu
• Department of Physics
• The Chinese University of Hong Kong

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