Coherent States| Coherent States |
In General
> s.a. quantum states; semiclassical
quantum mechanics / representations of quantum mechanics [Bargmann].
* History: 1926, Coherent states
of the harmonic oscillator introduced by Schrödinger, in reply to a remark
by Lorentz on the classical interpretation of the wave function; 1972, Simultaneous
discovery by Gilmore and Perelomov that coherent states are rooted in group theory.
* Idea:
A "semiclassical" state for bosonic particles, peaked at a point a point
(qi0,
pi0)
∈ Γ in phase space, with minimum uncertainty.
* Notation: Using a complex structure on
Γ, identify (τ = mω for the harmonic oscillator)
zi = (1/2\(\hbar\))1/2 (τ1/2 qi + i τ−1/2 pi) , αi = (1/2\(\hbar\))1/2 (τ1/2 qi0 + i τ−1/2 pi0) .
$ Fock space representation: An eigenstate of the annihilation operators, defined up to normalization by
ai |α\(\rangle\) = αi |α\(\rangle\) , or
|α\(\rangle\) = exp{αi a†i − α*i ai} |0\(\rangle\) = exp{−(|α1|2 + ... + |αN|2)/2} ∑n=0∞ {αn1 (n1!)−1/2} ··· {αnN (nN!)−1/2} |n1, ..., nN\(\rangle\).
$ Configuration space representation: A Gaussian wave function ψ0 centered at a phase space point (q0, p0), of the form
ψ0(q) = N Πi=1N exp{−(qi − qi0)2 / 4σi2 + i pi0 (qi − qi0)} .
$ As group orbits: Given an initial vector |η\(\rangle\) ∈ \(\cal H\) in the Hilbert space, usually taken to satisfy \(\langle\)η| p |η\(\rangle\) = 0 and \(\langle\)η| q |η\(\rangle\) = 0 and thought of as the vacuum |0\(\rangle\), a set of coherent states is defined by the action of a unitary operator on |η\(\rangle\),
|q0, p0\(\rangle\):= U(q0, p0) |η\(\rangle\) , where U(q0, p0):= exp{−i q0 p/\(\hbar\)} exp{i p0 q/\(\hbar\)} ;
More generally, they are labelled by |ξ, k\(\rangle\), where
ξ belongs to a coset space and k is the label for an irreducible
representation of G.
* Properties: Coherent states (i) Are
continuously parametrized by points (p, q) ∈ Γ; (ii) Form an
(overcomplete) basis for the Bargmann representation, and define a partition of unity,
1 = ∫ (dpdq/2π\(\hbar\)) |p, q\(\rangle\langle\)p, q| ;
(iii) Among the squeezed states, for which (Δq)2 (Δp)2 = (\(\hbar\)/2)2, they are the ones with Δq = Δp = (\(\hbar\)/2)1/2.
Applications, Special Topics
> s.a. Darboux Transformation; entangled
states; hilbert space [triplets]; quantum states [geometry];
uncertainty.
* Idea: Coherent states
allow us to "quantize'' any space X of parameters that
has a measure; If X is a phase space, i.e., it has a symplectic
structure and Hamiltonian, this leads to the usual quantum mechanics, but
the procedure is much more general; It can simply be considered as a choice
of resolution for the system, in analogy with data handling, where
coherent states (e.g., under the form of wavelets) are very efficient.
* And approaches to quantum theory:
They are used as basis elements in the coherent state Bargmann, Husimi, or
stellar representations, and in coherent-state phase space path integrals,
\(\langle\)p'', q''| exp{−iHt/\(\hbar\)}
|p', q'\(\rangle\).
@ General references: Isidro PLA(02)qp [and complex structures on phase space, and duality];
Campoamor-Stursberg et al PLA(12) [from classical action-angle variables];
Gouba a1909,
a1912-ln [generalized coherent-state quantization].
@ Coherent-state path integrals: Marchioro JMP(90) [as sums over classical paths];
Klauder qp/98-proc,
FP(01)qp/00;
Shibata & Niizeki JMP(01) [periodic potential];
Torre PRD(05)qp [linear systems];
dos Santos & de Aguiar JPA(06)qp [in the Weyl representation];
Gazeau et al JPA(07)qp/06 [and fuzzy sphere];
Yamashita IJTP(10) [stochastic approach];
Wilson & Galitski PRL(11)-a1012 [breakdown of method];
Rivas PRA(13)-a1207 [semiclassical approximation];
Yanay & Mueller EPJst(15)-a1212;
Kordas et al PRA(14)-a1408 [with a continuum time];
Bruckmann & Urbina a1807 [rigorous construction using dual representation];
Rançon a1906 [and stochastic calculus];
Berra-Montiel a2007 [star product representation].
@ And decoherence, classical limit:
Zurek et al PRL(93);
Meinrenken JPA(94);
> s.a. decoherence.
@ And geometric phases: Nesterov & Sabinin IJTP(97)ht/00 [loops and geometric phases];
Field & Anandan JGP(04).
@ Semiclassical evolution: Hagedorn CMP(80);
Stone IJMPB(01)qp/00,
et al JMP(00)qp [spin];
Novaes JMP(05),
Ribeiro & de Aguiar AP(08)-a0704 [propagator];
Novaes & de Aguiar PRA(05)qp;
Mar-Sarao & Moya-Cessa a0806 [non-Markovian dissipation];
Viscondi & de Aguiar JMP(11)-a1103 [propagator SU(n) coherent states];
Hertz et al Symm(16)-a1606 [generalized coherent states, non-classical behavior];
> s.a. types of coherent states [spin].
@ Evolution, other: Dias et al JMP(06)ht/05 [anharmonic oscillator];
Zipfel & Thiemann PRD(16)-a1506 [stable coherent states];
> s.a. types of coherent states.
@ Entangled coherent states:
Fan & Chen PLA(02);
Sanders JPA(12)-a1112 [rev];
Zhong et al OL(19)-a1905 [non-locality];
> s.a. types of coherent states [entangled spins].
@ And information theory:
Fujii qp/01-talk,
qp/02-talk;
Kim et al PhyA(12)-a1104 [information geometry and entangled Gaussian states].
@ Coherent state superpositions:
Glancy & Vasconcelos JOSAB(08)-a0705 [optical cat states, production];
Lee & Jeong PRA(09) [effects of squeezing];
Messina & Draganescu a1306 [unitary representations].
@ Experiments: Marquardt et al PRA(07) [macroscopic quantum coherence];
> s.a. experiments in quantum mechanics.
@ Other topics: Klauder LNP(87) [approximation of solutions of wave equation];
Dass & Ganesh qp/01-wd [cloning];
Bashkirov & Sukhanov qp/01-conf [thermodynamics and entropy];
Penson & Solomon qp/01-conf [from combinatorial sequences];
Das IJTP(02) [interacting Fock space];
Ali et al JPA(04) [and change of basis];
Andersen et al PRA(05)qp [purification];
Wolf et al PRL(06)qp/05 [extremality];
Ashhab PRA(07)-a0706 [and entanglement detection];
Chakraborty et al a0805 [and quantizable observables];
Schliemann PRA(15)-a1503 [properties of fluctuations, and examples].
> Related topics: see first-class
constrained systems and dirac quantization;
non-commutative geometry [spectral distance];
pilot-wave theory [trajectories].
References
> s.a. geometric quantization; modified coherent
states and specific systems; fock space.
@ General: Rohrlich in(70);
Klauder & Skagerstam 85;
Zhang et al RMP(90);
Klauder IJTP(94);
Ali et al RVMP(95) [rev];
in Hannabuss 97;
Klauder qp/01 [rev];
Isidro ht/02 [conditions for existence];
Panigrahi et al qp/03 [general procedure];
Johansen PLA(04) [non-classical properties];
Nemoto & Braunstain PLA(04) [significance];
Milonni & Nieto in(09)-a0903 [rev];
Klauder O&S(11)-a1008-proc [and quantum foundations];
issue JPA(12)#24;
Rosas-Ortiz a1812-in [intro];
Martin-Dussaud a2003
[from Schrödinger to quantum optics and quantum gravity];
Bagchi et al IJMPA-a2004 [pedestrian introduction].
@ Measurement: Diósi qp/96;
Das & Arvind JPA(17)-a1601 [based on weak measurements].
@ Geometry of coherent-state space:
Field & Hughston JMP(99);
Brody & Graefe JPA(10)-a1001;
Martinetti & Tomassini CMP(13)-a1110 [spectral distance].
@ Related topics: Ali et al JPA(08) [and incorporated statistical distributions];
Bannai & Tagami JPA(11) [anticoherent states];
Chen & Lin JMP(13) [categorification];
Bojowald & Tsobanjan CQG(14)-a1401 [group coherent states and effective Casimir conditions];
> Segal-Bargmann Transform.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 11 jul 2020