Coherent
States| Coherent
States |
In General > s.a. [quantum states,
semiclassical quantum mechanics]; representations
of quantum mechanics [Bargmann].
* Idea: A "semiclassical" state
for bosonic particles, peaked at a point a point (qi0,
pi0) 
in
phase space, with minimum uncertainty.
* Notation: Using a complex
structure on ,
identify (
= m
for the
usual harmonic oscillator)
zi =
(1/2
)1/2 (1/2 qi
+ i –1/2 pi)
, 
i
= (1/2)1/2 (1/2 qi0 + i –1/2 pi0) .
$ Fock space representation: An eigenstate of the annihilation operators, defined up to normalization by
ai |
= i |
, or
| =
exp{i a
i – *i ai}
|0 =
exp{–(|1|2 +
... + |N|2)/2} 
n=0infty {n_1/(n1!)1/2} ··· {n_N/(nN!)1/2}
|n1, ..., nN .
$ Configuration space representation:
A Gaussian wave function 
0 centered
at a phase space point (q0, p0),
of the form
0(q)
= N 
i=1N exp{–(qi–qi0)2/4
i2 +
i pi0(qi–qi0)}
.
$ As group orbits: Given an
initial vector |
,
usually taken to satisfy 
|
p | =
0 and |
q | =
0 and thought of as the vacuum |0,
a set of coherent states is defined
by the action of a unitary operator on |,
|q0, p0:= U(q0, p0)
| , where U(q0, p0):=
exp{–i q0 p/}
exp{i p0 q/}
;
More generally, they are labelled by |
, k,
where belongs
to a coset space and k is the label for an IRR of G.
* Properties: Coherent
states (i) Are continuously parametrized by points (p, q) ;
(ii) Form an (overcomplete) basis for the Bargmann representation, and
define a partition of unity,
1 = 
(dpdq/2
)
|p, q p, q|
;
(iii) Among the squeezed states, for which (
q)2(p)2 =
(/2)2,
they are the ones with q = p =
(/2)1/2.
Applications, Special Topics > s.a. Darboux
Transformation; entanglement; hilbert
space [triplets]; quantum states [geometry].
* 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,
p'', q''|
exp{–iHt/}
|p', q'.
@ Coherent state path integrals: Marchioro JMP(90) [as sums over classical
paths]; Klauder qp/98-in,
FP(01)qp/00;
Shibata & Niizeki
JMP(01)
[periodic potential]; Torre PRD(05)qp [linear
systems]; dos Santos & de Aguiar JPA(06)qp [in
Weyl representation]; Gazeau et al JPA(07)qp/06 [and
fuzzy sphere].
@ And decoherence, classical limit: Zurek et al PRL(93); Meinrenken JPA(94); > s.a.
decoherence.
@ And geometric phases: Nesterov & Sabinin IJTP(97)ht/00;
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]; > s.a. types
of coherent states [spin].
@ Evolution, other: Dias et al JMP(06)ht/05 [anharmonic
oscillator]; > s.a. types of coherent states.
@ Coherent state superpositions: Glancy & Vasconcelos JOSAB(08)-a0705 [optical
cat states, production].
@ 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-in
[thermodynamics and entropy]; Fujii qp/01-in, qp/02-in
[and information theory]; Penson & Solomon
qp/01-in
[from combinatorial sequences]; Das IJTP(02)
[interacting Fock space]; Isidro
PLA(02)qp [and
complex structures on phase space, and duality];
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].
>
Related topics: see first-class constrained
systems and dirac
quantization; uncertainty.
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);
Diósi qp/96 [and
measurement]; 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 a0903 [rev].
@ Related topics: Ashtekar & Magnon Pra(80)
[in quantum field theory]; Hall JFA(94), JFAA(01)mp,
Hall & Mitchell a0710 [Segal-Bargmann
transform]; Twareque Ali et al JPA(08) [and incorporated statistical distributions].
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2009