Download - Quantum Geometric Phase
Quantum Geometric Phase
Ming-chung ChuDepartment of Physics
The Chinese University of Hong Kong
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)
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
• 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
• Dynamical phase:dyn 0
1 Tdt H
geo tot dyn
tot( ) ( 0)iT e t
• After a cyclic evolution, a particle returns to its initial state; its wave function acquires an extra phase
tot
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 degeneracyH.-M. Lauber, P. Weidenhammer, D. Dubbers, PRL 72, 1004 (1994).
Generalizations of Geometric phase
Condition Space
Berry’s PhaseM. 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 PhaseS. Pancharatnam, Proc. Indian Acad. Sci., 247 (1956); J. Samuel and R. Bhandari, PRL 60, 2339 (1988).
General Ray Space
Ray space (projective Hilbert space)
1/R H S
1 cos sin2 2
i ie e
/ 32 cos sin
2 2i ie e
• States with only an overall phase difference are identified to the same point
• Eg. Two-state systems: ray space = surface of a sphere
i
j
A-A Phase
( )( ) (0)i TT e
ie
1di H
dt
0
1( )
T
C
dT i dt dt H
dt
i idi e He
dt
C(s)
geom ( ) ( )C C
d di dt t t i ds
dt ds
Can use any parameterization of the loop: geometrical
Y. Aharonov and J. Anandan, PRL 58, 1593 (1987).
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
C C CAds dS A dS F
C(s)
,d
A i F Ads
where
Non-cyclic evolution: open loop!
Need to close the loop to ensure local gauge invariant!
Pancharatnam phase
• Pancharatnam phase ~ A-A phase
• For unclosed paths (non-cyclic evolutions), just join the states with a geodesic
geodesic
Evolution path
(0)
( )t
J. Samuel and R. Bhandari: just join the open points with a geodesic!
Pancharatnam Phase• Relative phase can be measured by interference
• To remove dynamical phase, define
( ) ( ) ( ) ( ) .t H t E t t
Define a vector potential Im ( ) ( ) ;s
dA s s
ds
S. Pancharatnam, Proc. Indian Acad. Sci., 247 (1956); J. Samuel and R. Bhandari, PRL 60, 2339 (1988).
(0) ( )iz re t
2 2 2( ) (0) ( ) (0) 2Re (0) ( )t t t
geodesic( )sA s ds
then (0) ( )iz re t where the geodesic is the curve connecting φ(0) and φ(t) in the ray space given by the geodesic equation.
2. Problems with orthogonal states
Pancharatnam phase between orthogonal states
When (0) and ( ) are orthogonal
(0) ( ) 0 is undefined!i
t
z e t
There are infinitely many geodesics (eg. 1, 2) possible to close the path!
Off-diagonal Geometric Phases
• 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).
1 2( ) ( )j js s 1( )j s
1 2, where arg ( ) ( ) .jk jk kj jk j ks s
ij
N. Manini and F. Pistolesi, PRL 85, 3067 (2000).
Off-diagonal geometric phase
jk jk kj
1 2arg ( ) ( )jk j ks s
1 2arg ( ) ( )kj k js s
1 2
1 2
( ) ( ) 0
( ) ( ) 0
j j
k k
s s
s s
1( )j s 2( )j s
1( )k s2( )k s
geodic geodicjk
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
x
y polarizer
After inserting a polarizer, they can interfere!
Projective Phase
• First project two states onto i and then let them interfere
(0, ) arg (0) ( )i t i i t
(0)i i ( )i i t
When (0) , the projective phase reduces to
Pancharatnam phase
i (0, ) arg (0) (0) (0) ( ) .i t t
Geometrical meaning
1 1
1 1 2 20 0(0, )i t A ds A ds
Im ( ) ( )k k k k kk
dA s s
ds
Pancharatnam phase
geodesic
Schrödinger evolutionFind a state |i > not orthogonal to either one, then join them with geodesics.
Projective phase
i
geodesic geod
esic
i-dependent!
Gauge Transformation
x
y Polarizer i
x
y
Polarizer j
(0, ) arg (0) ( )i t i i t
(0, ) arg (0) ( )j t j j t
Gauge Transformation
• The gauge transformation at a point P is
1( ) ( )ij ji
j P P iS P S P
j P P i
• This is the transition function in fiber bundle • The two projective phases are related by
exp (0, ) (0)exp (0, ) ( )i ij j jii t S i t S t
• With this transformation, one projective phase can give all others
Bargmann invarianti
j
(0)
( )t
which is equal to the –ve of the geometric phase enclosed by the 4 geodesics
, , , argB a b c d a b b c c d d a
arg 0 0i i t t j j
where the Bargmann invariantis defined by
jThe difference between and is i
R. Simon and N. Mukunda, PRL 70, 880 (1993).
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
expa
ieA dx
c
g
e
a
b
• At south pole:• Dirac monopole quantization:• Wu and Yang: 2 vector potentials ( ) to cover the
sphere, and gauge transformation Sab to relate them
4A dx g
2
n cg
e
1 cos
1 cos
0, ,
a
b
r i i
A g
A g
A A i a b
2expab
igeS
c
A
Monopole and projective phase
• The 2-state system projective phase has the same fiber bundle structure as a monopole with g = 1/2
( ) expij
j P P iS P i
j P P i
taking , and i j
1(1 cos )
21
(1 cos )2
i
j
A
A
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
1( )j s 2( )j s
1( )k s2( )k s
geod geod
i
1
2
1B 2B
1 2
1 2
1 2
1 2
1 ( ( ), ( ))
arg
2 ( ( ), ( ))
arg
i j j
j j
i k k
k k
s s
s i i s
s s
s i i s
Let
The off-diagonal geometricphase is:
1 2 1 2jk B B
1 1 2 2 1
2 1 2 2 1
arg
arg
B j k k j
B k j j k
s s s i i s
s s s i i s
where
1( )j s 2( )j s
1( )k s2( )k s
geod geodjk
Can be decomposed into projective phases and Bargmann Invariants
n projective phases = n(n-1) off-diagonal phases
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: j
1
1 2( , )i 1 2( , )j 2 1
2i j n
2n m,i m j m
i
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 systemE Excellent approximate analytic solution using Rotating Wave Approximation (RWA)
resonancesRabi Oscillation at resonance: RWA vs. numerical solution
Geometric phase at resonance
RWA solution for geometric phase:
Numerical solution
T = Rabi oscillation period
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
• With the projective phase formalism, we can show the existence of the πjump (and the condition for its occurence).
1 22 1( , )(0, ) (0, )
1 2( ) ( )ji ii t ti t i t
ij jie e S t e S t
Geometric phase of a BEC• Bose-Einstein Condensate (BEC): macroscopic wavefu
nction – 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 Sc
hrödinger equation: Gross-Pitaevskii equation (GPE)2 22
2202 2
i im xdi U
dt m
Numerical Results• Solving GPE with Crank-Nicholson algorithm• Initial state prepared by time-independent GPE solu
tion with • Time-evolve with • Resulting phases agree well with perturbative calcul
ation
t
Geo
met
ric
phas
e
But: dynamical phase much larger!
10g
8g
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 PhysicsThe Chinese University of Hong Kong
Eg. Neutron interferometry
Y. Hasegawa et al., PRL 87, 070401 (2001).
Without B
With B to rotate the neutrons
Geometric phase of