Geometric Phase  

In General > s.a. holonomy.
* 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)dec; Berry PT(90)dec; Von Baeyer ThSc(90)may; 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]; Aharonov et al JPCS(09)-a0907; Katanaev RPJ-a1212 [geometric interpretation].
@ Textbooks / reviews: in Dittrich & Reuter 94; Rohrlich a0708-in; Garg AJP(10)jul [Berry curvature and Chern number, near degeneracies]; in Chang & Ge 17.
@ Aharonov-Anandan phase: Segre mp/05 [and Hannay's angle]; Bracken IJQC(08)-mp/06 [geometrical]; Giscard a0901/PRL [operator].
@ Related topics: Montgomery CMP(88) [mathematical]; Robbins & Berry PRS(92) [chaotic systems]; Simon & Mukunda PRL(93) [applications]; Anandan et al AJP(97)mar-qp [resource letter]; Katanaev RPJ(11)-a0909 [geometrical]; Zygelman PRA-a1205 [and forces, qubit model].
@ Experiments: Bhandari & Samuel PRL(88), Chiao et al PRL(88), Suter et al PRL(88); Hariharan AJP(93)jul [simple optical demo]; Price & Cooper PRA(12) [mapping the Berry curvature].

Classical (Hannay's angle) > s.a. duality [electromagnetic field]; Pendulum [Foucault].
* Examples: Foucault's pendulum; Spins turning in a magnetic field.
@ In mechanics: Spallicci et al Nonlin(05)ap/03 [3-body problem]; Spallicci NCB(04)ap [satellite measurement]; Gil AJP(10)apr [mechanical device].
@ In optics: Bhandari PRP(97) [polarization]; Samuel & Sinha Pra-qp/97 [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. entanglement; quantum systems [non-trivial topology]; realism; wigner function.
* Idea: The holonomy around a closed loop c in the projective Hilbert space P with respect to the natural connection given by the inner product, or the area enclosed by c with respect to the natural symplectic structure on P; It can be expressed as the integral of the symplectic form of the Fubini-Study geometry over a surface S spanning c, i S \(\langle\)ψ | dψ\(\rangle\).
* Relationships: Generalizes the Aharonov-Bohm effect to loops in abstract parameter space; > s.a. formulations of quantum theory.
@ References: 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]; Filipp et al PRL(09) [experimental test of robustness]; Vutha & DeMille a0907/AJP [without geometry]; Ben-Aryeh a0909; Dennis et al ed-JPA(10) [25 years]; Capolupo & Vitiello NCC(15)-a1512 [applications, CPT violation, Unruh effect and temperature]; Moore a1706 [non-uniqueness of the Berry connection].

Specific Types of Systems > s.a. magnetism [momentum-space magnetic field]; 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 SQUIDs, neutron interferometry.
@ Open systems: Carollo et al PRL(03)qp, MPLA(05); Marzlin et al PRL(04) [distributions]; Burić & Radonjić PRA(09); Hu & Yu PRA(12) [accelerated two-level atom, and Unruh effect].
@ 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]; Freedman a0812-conf [quantum gravity model]; Papini PLA(12)-a1202 [gravitational Berry phase and Zitterbewegung].
@ Other field theories: Martinez PRD(90) [gauge theory + fermion]; Carollo et al PRA(03)qp/02 [cavity QED]; Baggio et al a1701.
@ 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]; Filipp et al PRL(09)-a0812 [spin-1/2 particle, robustness, experimental]; Niu et al PRA(10)-a1003 [interacting bipartite system]; Muminov & Yousefi a1103 [for coherent states]; Jafari PLA(13) [spin chains, quantum renormalization-group approach]; Aguilar et al MPLA(16)-a1609 [coupled to a quantum vector operator].
@ Other types of systems: 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]; Sinitsyn & Saxena JPA(08) [non-Hermitian Hamiltonian]; Kaufherr et al a0907-wd [1D scattering of heavy + light particle]; Sjöqvist PLA(10) [composite systems, from correlations]; Xiao et al RMP(10) [electrons in solids]; Du et al PRA(11), a1301 [non-Abelian geometric phase with 3-level atomic system]; Mousolou & Sjöqvist PRA(14) [coupled quantum bits]; Zhang et al a1410 [single solid-state spin qubit]; > s.a. neutron; spin models; quantum computation.

Other References > s.a. coherent states; Commutation Relations; generalized particle statistics; spin-statistics theorem.
@ Non-adiabatic: Aharonov & Anandan PRL(87); Anandan AIHP(88), PLA(88), & Aharonov PRD(88).
@ Non-cyclic evolutions: García de Polavieja & Sjöqvist AJP(98)may-qp; Pati AP(98)qp.
@ Arbitrary quantum evolutions: Anandan & Aharonov PRL(90); Gosselin & Mohrbach JPA(10)-a1008 [beyond the adiabatic regime].
@ Mixed states: & Uhlmann; Dittmann LMP(98) [connection]; Sjöqvist et al PRL(00); Ercolessi et al IJMPA(01)qp [multilevel quantum systems]; 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 EPJC(04)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]; Andersson & Heydari NJP(13)-a1302, a1411; Sjöqvist IJQC(15)-a1503 [and quantum information, computation, and entangled systems]; Andersson et al PTRS(16)-a1507 [Kitaev chain, Uhlmann's geometric phase].
@ 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 theory].
@ 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]; Horsley & Babiker PRL(07) [effect of time average and statistical variance of electromagnetic quantity]; Low JPCS(12)-a0903 [relativistic implications]; Chou & Wyatt AP(10) [in Bohmian mechanics]; > s.a. topology in physics [topological quantum phase].

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