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 = 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 aiα*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{−(qiqi0)2 / 4σi2 + i pi0 (qiqi0)} .

$ 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)2p)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.

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