1
Physics 498-QOI: Topics in Quantum Optics and Quantum Information
Problem set #2 [29]
{Distributed 1/23; due 1/30}
1. [2] Why we need normal ordering. In class we introduced the 2nd order coherence
function, G(2)(x1, x2) = ·E(-)(x1) E(-)(x2) E(+)(x2) E(+)(x1) Ò ~ ·a† a† a a Ò {I apologize
that I cannot easily put operator hats on all these.}. I told you that this is what we use
to calculate the probability of a coincidence count. You might wonder why it is
·a† a† a a Ò, and not ·a† a a† a Ò. Or not. In any event, let’s say we have only one
photon, for which we know the probability of detecting 2 photons should be 0.
Calculate G(2) , and ·a† a a† a Ò, assuming the state |1 Ò.
2. [12] The result of the Hanbury-Brown Twiss experiment, that there is photon
“bunching” for thermal light fields, is quite important. Here we will calculate it
(well, you will!). Specifically, for each of the following states, calculate the
normalized coherence function: g(2)(0) = G(2)(0)/|G(1)(0)|^2, where the “0” means that
we are looking for two simultaneous photons (and effectively at the same place), i.e.,
so that G(1)(0) = · a† a Ò, G(2)(0) = ·a† a† a a Ò:
a). |
y
Ò = |
1
Ò (yes, it’s as easy as it looks) [1]
b). |
y
Ò = |nÒ, a general number state [2]
c). |
y
Ò = |
a
Ò, a coherent state [2]
d). “|
y
Ò” = thermal state. Now, this last case is a bit more tricky. One reason is that a
thermal light source emits a mixed state, namely
rn,m = e- n (hn/kT) [1 - e- (hn/kT)] dn,m, i.e.,
r = (1- A) |0
Ò·0| + (1- A)A |1
Ò·1| + (1- A)A2 |2
Ò·2|+ …(1- A)An |n
Ò·n|,
where A = e- (hn/kT).
i). Show that G(1) (= · a† a Ò = ·n_hatÒ) = 1/( e(hn/kT) -1). [2]
ii). Calculate G(2) [4]
Hint #1: Use commutation relations to write a† a† a a only in terms of n_hat.
Hint #2:
k qk=q
1-q
( )
2
k=0
•
Â
Last hint:
k2qk
k=0
•
Â
can be readily derived from Hint#2 by differentiating.
iii). Show that g(2) = 2. [1]
