Geometric Phase| 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: Rohrlich a0708-in;
Garg AJP(10)jul
[Berry curvature and Chern number, near degeneracies];
in Chang & Ge 17;
Cohen et al nRev(19)-a1912;
in Dittrich & Reuter 20.
@ 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];
Bae et al Chaos(18)
+ Fitzgerald pt(18)aug [bead on a hoop experiment].
@ 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; Thomas
Precession; 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 \(\int_S \langle\psi | {\rm d}\psi\rangle\).
* Relationships: It
generalizes the Aharonov-Bohm effect to loops in abstract parameter space;
> s.a. formulations of quantum theory.
@ General 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].
@ Generalizations:
Kapustin & Spodyneiko a2001 [higher-dimensional];
Hsin et al a2004 [quantum field theory].
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];
Mukhopadhyay & Ganguly a1802 [spinors and neutrinos, gravitational Zeeman effect].
@ 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];
Do et al a1903 [Bose-Einstein condensate];
Yang et al a1904 [simple system, interpretation];
> s.a. neutrino oscillations; 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,
JPA(15)-a1411;
Sjöqvist IJQC(15)-a1503 [and quantum information, computation, and entangled systems];
Andersson et al PTRS(16)-a1507 [Kitaev chain, Uhlmann's geometric phase];
Sjöqvist a1909 [geometry].
@ 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];
Banks et al PRX(17) [condensed matter].
@ 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];
Garcia-Chung a2004 [n-partite Gaussian states];
> s.a. topology in physics [topological quantum phase].
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