Geometric
Phase| Geometric
Phase |
In General
* Idea: The anholonomy
observed when a system undergoes a cyclic transformation in some parameter
space; It depends only on the geometry of the circuit in
parameter space.
* Geometrical analog: Parallel transport of a vector on a spherical
surface.
References > s.a. connection;
Parallel Transport.
@ II: Berry SA(88)dec; Holstein AJP(89);
Berry PT(90)dec; Von Baeyer ThSc(90);
Holstein CP(95).
@ General: Wilczek & Zee PRL(84);
Berry JPA(85);
Hannay JPA(85);
Anandan PRD(86);
Anandan & Stodolsky PRD(87);
Berry PRS(87);
Gozzi & Thacker PRD(87),
PRD(88);
Li PRL(87);
Stone & Goff pr(87); Anandan
PLA(88), PRL(88);
Anandan & Aharonov PRD(88);
Jackiw CPAM(88), IJMPA(88);
Samuel & Bhandari
PRL(88);
Giavarini et al PLA(89),
JPA(89); Shapere & Wilczek ed-89;
Aitchison & Wanelik
PRS(92);
Sudarshan et al PLA(92);
Batterman SHPMP(03)
[conceptual, and gauge].
@ Textbooks and reviews: in Dittrich & Reuter 94; Rohrlich a0708-in.
@ Related topics: Montgomery CMP(88)
[mathematical]; Robbins & Berry PRS(92)
[chaotic systems]; Simon & Mukunda
PRL(93)
[applications]; Anandan et al AJP(97)qp [resource
letter]; Segre mp/05 [Hannay's
angle and Aharonov-Anandan phase]; Bracken mp/06 [Aharonov-Anandan,
geometrical].
@ Experiments: Bhandari & Samuel PRL(88),
Chiao et al PRL(88),
Suter et
al PRL(88);
Hariharan AJP(93)
[simple optical demo].
Classical (Hannay's angle) > s.a. duality [electromagnetic
field]; Pendulum [Foucault].
* Examples: Foucault's pendulum; turning spins in a magnetic field.
@ In mechanics: Spallicci et al Nonlin(05)ap/03 [3-body
problem]; Spallicci NCB(04)ap [satellite
measurement].
@ In optics: Bhandari PRP(97) [polarization]; Samuel & Sinha qp/97/Pra
[Thomas precession]; Ghose & Samal qp/01 [gravity-induced].
@ Scalar field in curved spacetime: Mostafazadeh ht/96,
JPA(98)qp [charged
Klein-Gordon field].
Quantum (Berry phase) > s.a. quantum
systems [non-trivial
top]; wigner
function.
* Idea: The holonomy
around a closed loop c in
the projective Hilbert space P wrt the natural
connection given by the inner product, or
the area
enclosed by c wrt
the natural symplectic structure on P;
Can
be expressed as the integral of the symplectic form of the Fubini-Study
geometry
over a surface S spanning c,
i 
S 
|
d
.
* Relationships: Generalizes
the Aharonov-Bohm effect to loops
in abstract parameter space.
Specific Types of Systems > s.a. phase
transitions; quantum
computing; semiclassical evolution.
* Open systems: The geometric
phase should be described by a distribution; This
distribution is in
general ambiguous, but the imposition of reasonable physical constraints on
the environment and its coupling with the system yields a unique geometric
phase distribution.
* Examples: Aharonov-Bohm and Aharonov-Casher effects, rotating SQUID's,
neutron interferometry.
@ Open systems: Carollo et al PRL(03)qp,
MPLA(05); Marzlin et al PRL(04)
[distributions].
@ And gravity: Anandan PLA(94), gq/95;
Corichi & Pierri PRD(95)gq/94 [Klein-Gordon
particle around cosmic string]; Casadio & Venturi CQG(95); Ho & Morgan
PLA(97) [particle in Newtonian potential]; de Assis et al gq/03 [around
rotating massive body].
@ Other field theories: Martinez PRD(90)
[gauge theory + fermion]; Carollo et al PRA(03)qp/02 [cavity
QED].
@ Relativistic: Wang & Li PRA(99).
@ Spin: Hannay JPA(98)
[spin-j]; Fuentes-Guridi et al PRL(02)qp [spin-1/2, B];
Carollo et al PRL(04)qp/03 [spin-1/2,
decohering quantum fields]; Pachos & Carollo PTRS(06)qp [and
criticality].
@ Other: Solem & Biederharn FP(93);
Giller et al PLA(94)
[and degeneracies of Hamiltonian]; Strahov JMP(01)
[compact Lie groups]; Dreisigmeyer et al FPL(03)qp/01 [spinors];
Bertlmann et al PRA(04)
[entangled neutrons]; > s.a. neutron; spin
models; quantum computation.
Other References > s.a. coherent
states; Commutation
Relations; particle
statistics.
@ General: Simon PRL(83);
Berry PRS(84);
Page PRA(87);
Herdegen PLA(89);
Anandan PLA(90)
[cyclic motions, and state space metric]; Pati
PLA(91);
Stanley PLA(91);
Mukunda & Simon AP(93),
AP(93);
Bohm et al 03; Cabrera a0705 [geometric
features].
@ Non-adiabatic:
Aharonov & Anandan PRL(87); Anandan AIHP(88),
PLA(88), & Aharonov PRD(88).
@ Non-cyclic evolutions: García de Polavieja & Sjöqvist
AJP(98)qp;
Pati AP(98)qp.
@ Arbitrary quantum evolutions: Anandan & Aharonov PRL(90).
@ Mixed states: & Uhlmann; Dittmann LMP(98)
[connection]; Sjöqvist
et al PRL(00);
Slater LMP(02)mp/01;
Ericsson et al PRL(03)qp/02;
Filipp & Sjöqvist
PRL(03)qp/02 [off-diagonal];
Du et al PRL(03)
[observation]; Chaturvedi et al qp/03 [geometric
approach]; Ericsson et al PRL(05)qp/04 [measurement];
Rezakhani & Zanardi PRA(06)qp/05 [general
setting], PRA(06)
[T effects]; Fujikawa AP(07) [hidden local gauge symmetry].
@ And quantum Zeno effect: Facchi et al PLA(99)qp.
@ Geometric vs dynamical: Anastopoulos & Savvidou IJTP(02)qp/00 [and
consistent histories].
@ Classical vs quantum: Giavarini et al PRD(89); Giller et al PLA(93);
Biswas et al IJMPA(94).
@ Relationships: Rabei et al PRA(99)qp [and
Bargmann invariants]; Zeng & Lei
PLA(96) [Lewis phase]; Viennot et al JPA(06) [and time-dependent wave operators].
@ In interferometry: Bhandari & Samuel PRL(88)
[Pancharatnam phase, using laser polarization]; Sjöqvist et al PRL(06).
@ And measurement: Pati & Lawande PLA(96), qp/98/PRL;
Sjöqvist & Carlsen
PRA(97)
[pilot wave].
@ Related topics: Newton PRL(94)
[and S-matrix]; Pati PLA(95)
[projective Hilbert
space]; Sjöqvist et al PLA(97)
[Galilean non-invariance]; Martinez JPA(06)
[role
of space
symmetries]; Buniy & Kephart ht/06 [second-order
topological quantum phase]; Horsley & Babiker PRL(07)
[effect of time average
and statistical variance of electromagnetic quantity].
Main page – Abbreviations – Journals – Comments – Other
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
11 jul 2008