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