Quantum Geometric Phase
Ming-chung Chu
Department of Physics
The Chinese University of Hong Kong
Content
1.
2.
3.
4.
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)
1. Review of Geometric Phase
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.
Review of Geometric Phase
• After a cyclic evolution, a particle returns to its initial
state; its wave function acquires an extra phase  tot
itot
 (T )  e
• Dynamical phase:
 (t  0)
 dyn  
1

T
0
dt H
• Geometric phase is the extra phase in addition to the
dynamical phase geo   tot dyn
• It arises from the movement of the wave function and
contains information about the geometry of the space
in which the wave function evolves
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)!
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  .
Rectangular cavity: 3-state degeneracy
H.-M. Lauber, P. Weidenhammer, D. Dubbers, PRL 72, 1004 (1994).
Generalizations of Geometric phase
Berry’s Phase
M. Berry, Proc. R. Soc. Lond. A, p. 45 (1984).
Aharonov-Anadan Phase
(A-A Phase)
Condition
Space
Adiabatic
and cyclic
Cyclic
Parameter
space
Ray Space
General
Ray Space
Y. Aharonov and J. Anandan, PRL 58, 1593 (1987).
Pancharatnam Phase
S. Pancharatnam, Proc. Indian Acad. Sci., 247 (1956);
J. Samuel and R. Bhandari, PRL 60, 2339 (1988).
Ray space (projective Hilbert space)
• States with only an overall phase difference are
R  H / S1
identified to the same point
• Eg. Two-state systems: ray space = surface of a sphere
i  




i
 1  e  cos   e sin  
2
2





i / 3 
i
 2  e  cos   e sin  
2
2


i
j  
A-A Phase
Y. Aharonov and J. Anandan, PRL 58, 1593 (1987).
 e 
i
d i
i e   Hei 
dt
d
1
 i 
   H
dt
 (T )  e
i (T )
C(s)
 (0)
d
1 T
 (T )  i  dt 
   dt  H 
C
0
dt
geom
d
d
 i  dt  (t )
 (t )  i  ds 

C
C
dt
ds
Can use any parameterization of the loop: geometrical
A-A phase

C
Ads 

C
dS   A   dS F
C
d
 , F   A
where A  i 
ds
• 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
Non-cyclic evolution: open loop!
Need to close the loop to ensure
local gauge invariant!
C(s)
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
 (0)
Evolution path
geodesic
 (t )
Pancharatnam Phase
• Relative phase can be measured by interference
 (t )   (0)   (t )   (0)  2 Re  (0)  (t )
2
2
2
z  rei   (0)  (t )
• To remove dynamical phase, define
(t )   H (t )  E(t )  (t ) .
d
 ( s) ;
Define a vector potential As  Im  ( s)
ds
i
then z  re   (0)  (t ) where the geodesic is the curve
 
geodesic
As (s)ds
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).
2. Problems with orthogonal
states
Pancharatnam phase between
orthogonal states
When  (0) and  (t ) are orthogonal
z  ei   (0)  (t )  0   is undefined!
There are infinitely
many geodesics
(eg. 1, 2) possible
to close the path!
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  j ( s1 )   j ( s2 ) orthogonal to  j ( s1 ) ;
can still compare its phase to another eigenstate
• Off-diagonal geometric phases:
 jk   jk   kj , where  jk  arg  j ( s1 )  k ( s2 ) .
• Independent combinations of  ij’s are gauge invariant
and contain all phase information of the system
• Measurable by neutron interferometry
Y. Hasegawa et al., PRL 87, 070401 (2001).
Off-diagonal geometric phase
 j ( s1 )
 j ( s2 )
 j ( s1 )  j ( s2 )  0
 k ( s1 )  k ( s2 )  0
geodic
 jk
geodic
 jk  arg  j ( s1 )  k ( s2 )
 kj  arg  k ( s1 )  j ( s2 )
 jk   jk   kj
 k (s2 )
 k (s1 )
Off-diagonal geometric phases are measurable and complement
diagonal (Berry’s) phases. Y. Hasegawa et al., PRL 87, 070401 (2001).
3. Projective Phase: a new
formalism
Hon Man Wong, Kai Ming Cheng, and M.-C. Chu
Phys. Rev. Lett. 94, 070406 (2005).
Projective phase
• Two orthogonal polarized light cannot
interfere
y
polarizer
x
After inserting a polarizer, they can interfere!
Projective Phase
• First project two states onto i and then let them
interfere
i i  (0)
i i  (t )
i (0, t )  arg  (0) i i  (t )
When i   (0) , the projective phase reduces to
Pancharatnam phase i (0, t )  arg  (0)  (0)  (0)  (t) .
Geometrical meaning
Find a state |i > not
orthogonal to either one,
then join them with
geodesics.
1
1
0
0
Schrödinger evolution
i (0, t )   A1ds1   A2 ds2
d
Ak  Im k ( sk )
k ( sk )
dsk
i
Projective phase
i-dependent!
Gauge Transformation
i (0, t )  arg  (0) i i  (t )
 j (0, t )  arg  (0) j j  (t )
y
Polarizer i
Polarizer j
x
Gauge Transformation
• The gauge transformation at a point P is
j P Pi
1
Sij ( P)  S ji ( P) 
j P Pi
• This is the transition function in fiber bundle
• The two projective phases are related by
exp  ii (0, t )   Sij (0) exp  i j (0, t )  S ji (t )
• With this transformation, one projective phase
can give all others
Bargmann invariant
i
The difference between i and  j is
  arg 0 i i t t j j 0
 (0)
where the Bargmann invariant
is defined by
B  a, b, c, d   arg a b b c c d d a
which is equal to the –ve of the geometric
phase enclosed by the 4 geodesics
R. Simon and N. Mukunda, PRL 70, 880 (1993).
 (t )
j
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
 ie

  exp    A  dx  
a
 c

a
g
e
b
• At south pole:  A dx   4 g
n c
g
• Dirac monopole quantization:
2e
• Wu and Yang: 2 vector potentials ( A) to cover the
sphere, and gauge transformation Sab to relate them
A 
A 
 a
 g 1  cos 
 b
  g 1  cos 
 Ar i   A i  0,
i  a, b
 2ige 
Sab  exp 

 c 
Monopole and projective phase
• The 2-state system projective phase has the
same fiber bundle structure as a monopole
with g = 1/2
taking i   , and j  
Sij ( P) 
j P Pi
j P Pi
 exp  i 
1
 A i  2 (1  cos  )
1
 A  j   2 (1  cos  )
4. Applications
- Off-diagonal geometric phases
- Extracting a topological number
- Geometric phase at a resonance
- Geometric phase of a BEC (preliminary)
Off-diagonal geometric phase
Can be decomposed into projective phases and Bargmann Invariants
Let
 (s j)( s1 )
 j((ss2))
1  i ( j ( s1 ), j ( s2 ))
j
1
j
1
 arg  j  s1  i i  j  s2 
 2  i ( k ( s1 ), k ( s2 ))
 arg  k  s1  i i  k  s2 
The off-diagonal geometric
phase is:
 jk  B1  B 2  1   2
2
geod
geod
 B1
 jk
B2
geod
geod
i
2
 k (s2 )
n projective phases = n(n-1)
 k (s2 )
off-diagonal phases
where   arg   s    s    s  i i   s 
B1
j
1
k
2
k
2
j
1
B 2  arg  k  s1   j  s2   j  s2  i i  k  s1 
 k (s1 )
 k (s1 )
Extracting a topological number
• The difference between i (1 , 2 ) and  j (1,2 ) as
2  1 (closed loop) can be used to extract the
first Chern number n: i   j  2n
i
• The loop can be smoothly
deformed and n is not changed
• n is a topological number of
the ray space
• Eg. spin-m systems: n  2m
i  m , j  m
1
j
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:   E →two-state system
Excellent approximate analytic solution using
Rotating Wave Approximation (RWA)
resonances
Rabi Oscillation at resonance:
RWA vs. numerical solution
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.
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 eii (0,t2 )  eii (0,t1 ) Sij (t1 )ei j (t1 ,t2 ) S ji (t2 )
• With the projective phase formalism, we can show the existence
of the πjump (and the condition for its occurence).
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)
2 2
2

m

d

2
2
i xi
i
 
 
 U 0  
dt
2
 2m

Numerical Results
But: dynamical
phase much larger!
Geometric phase
• Solving GPE with Crank-Nicholson algorithm
• Initial state prepared by time-independent GPE
solution with g  10
• Time-evolve with g  8
• Resulting phases agree well with perturbative
calculation
t
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
Quantum Geometric Phase
Hon Man Wong, Kai Ming Cheng,
Ming-chung Chu
Department of Physics
The Chinese University of Hong Kong
Eg. Neutron interferometry
Y. Hasegawa et al., PRL 87, 070401 (2001).
With B to
rotate the
neutrons
Without B
Geometric phase of
