# Relation between coherent and Fock states of light

1. Sep 24, 2011

### sweet springs

Hi.

Coherent states of light, which correspond to classical em wave, are eigenstates of non-Hermite annihilation operator. Fock states are eigenstates of Hermite number operator. Are Fock states are expressed by combination of coherent states? If yes, how?

2. Sep 25, 2011

### Bill_K

Look for the Wikipedia page "Coherent States", and on that page search for "representation"

3. Sep 25, 2011

### Staff: Mentor

Hi Bill, on that page:
http://en.wikipedia.org/wiki/Coherent_states

It describes decomposing a coherent state as a superposition of Fock states, but as far as I can tell it doesn't describe the other way around, decomposing a Fock state as a superposition of coherent states. Do you know if that is possible?

4. Sep 25, 2011

### nightlight

The decomposition of any state (pure or mixed) into superposition of coherent states was the central result of the coherent states formalism (follow up on Glauber-Sudarshan P representation in that wiki article). Note also that the coherent states are not mutually orthogonal (since anihilation operator is not Hermitean) and that they form over-complete basis, hence the decomposition of arbitrary field state into coherent states is not unique (thus, there are other decompositions besides the canonical G-S P representation).

5. Sep 25, 2011

### sweet springs

Hi, nightlight. Thanks for your teaching.

From formula in Wiki |α> = e^(-|α|^2 /2) ( |0> + α|1> + ... ) disregarding higher orders of α　for |α|<<1
|α> = |0> + α|1> so coherent state of very weak light is superposition of almost vaccuum and small poriton linear to α of one photon Fock state, I think.

Though coherent states are eigenstates of non Hermitian annihilation operators, why we can regard them corresponding to classical em wave ? I believe only observables, i.e. Herimite operators, should have classical correspondents. Am I wrong?

May I write Fock state |0>=D^-1(α)|α> where |α> is coherent state and D(α)=exp(α a+ - α* a) or D^-1(α)=exp(|α|^2/2)exp( - α a+) ?
Though I do not know the way of practical calculation.

Regards.

Last edited: Sep 26, 2011
6. Sep 26, 2011

### tom.stoer

The idea is to prove something like

$1 = c\, \int_\mathbb{C}dz\,d\bar{z}\,|z\rangle\langle z|$

and to express the coherent states in terms of Fock states.

Using

$z^n = r^n\,e^{in\phi}$

one can rewrite the integral and use the phi-integration

$\int_0^{2\pi} d\phi\,e^{i(m-n)\phi} = \delta_{mn}$

Last edited: Sep 26, 2011
7. Sep 26, 2011

### nightlight

The "classicality" here means that the joint probabilities of photon counts at spacelike points factorizes (just as it would do if for the classical EM field). The largest class of "classical" states in that sense consists of all superpositions of coherent states with positive, non-singular P(alpha).

In the coherent state decomposition the vaccum state |0> uses displacement operator D(0). For calculations and usefulness of coherent state representation, check the original Glabuer's papers (from early 1960s) or a good QO textbook, e.g. chap 11 in L. Mandel, E. Wolf "Optical Coherence and Quantum Optics" Cambridge Univ. Press., 1995.

8. Sep 26, 2011

### DrChinese

Haven't seen your name in a while... welcome back.

-DrC

9. Sep 27, 2011

### nightlight

Oh, I drop by here every few days. But I am way too busy with my day job to join discussions.

10. Sep 29, 2011

### Staff: Mentor

Thanks, that is what I needed.