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 (*q*^{i}_{0},
*p*_{i}^{0}) ∈ Γ in
phase space, with minimum uncertainty.

* __Notation__: Using a complex structure on Γ,
identify (*τ* = *mω* for the usual harmonic oscillator)

*z*^{i} =
(1/2\(\hbar\))^{1/2} (*τ*^{1/2} *q*^{i}
+ i *τ*^{–1/2} *p*_{i})
, *α*_{i}
= (1/2\(\hbar\))^{1/2} (*τ*^{1/2} *q*^{i}_{0} + i *τ*^{–1/2} *p*_{i}^{0}) .

$ __Fock space representation__:
An eigenstate of the annihilation operators, defined up to normalization by

*a*_{i} |*α*\(\rangle\)
= *α*_{i} |*α*\(\rangle\)
, or

|*α*\(\rangle\) =
exp{*α*_{i} *a*^{†}_{i} – *α**_{i }*a*_{i}}
|0\(\rangle\) =
exp{–(|*α*_{1}|^{2} +
... + |*α*_{N}|^{2})/2} ∑_{n=0}^{∞} {*α*^{n1} (*n*_{1}!)^{–1/2}} ··· {*α*^{nN} (*n*_{N}!)^{–1/2}}
|*n*_{1}, ...,* n*_{N}\(\rangle\).

$ __Configuration space representation__:
A Gaussian wave function *ψ*_{0} centered
at a phase space point (*q*_{0}, *p*^{0}),
of the form

*ψ*_{0}(*q*)
= *N* Π_{i=1}^{N} exp{–(*q*^{i}–*q*^{i}_{0})^{2}/4*σ*_{i}^{2} +
i *p*_{i}^{0}(*q*^{i}–*q*^{i}_{0})}
.

$ __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*\(\rangle\),

|*q*_{0}, *p*^{0}\(\rangle\):= *U*(*q*_{0}, *p*^{0})
|*η*\(\rangle\) , where *U*(*q*_{0}, *p*^{0}):=
exp{–i *q*_{0} *p*/\(\hbar\)}
exp{i* p*^{0} *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 = ∫ (d*p*d*q*/2π\(\hbar\))
|*p*, *q*\(\rangle\langle\)*p*, *q*| ;

(iii) Among the squeezed states, for which
(Δ*q*)^{2}(Δ*p*)^{2}
= (\(\hbar\)/2)^{2}, they are the ones with
Δ*q* = Δ*p* = (\(\hbar\)/2)^{1/2}.

**Applications, Special Topics** > s.a. Darboux
Transformation; entanglement; 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{–i*Ht*/\(\hbar\)} |*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].

@ __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 a1212; Kordas et al PRA(14)-a1408 [with a continuum time].

@ __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-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 a1506 [stable coherent states]; > s.a. types of coherent states.

@ __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.

@ __Measurement__: Diósi qp/96; Das & Arvind 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 2
jun
2016