IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 73

Sorelli et al.
Therefore, in our discussion ofQI, we will refer to coherent
states every time we want to compare with classical light.
Thermal states a.k.a. the noise:
Let us now discuss another important class of singlemode
Gaussian states, namely the quantum state of the
black body radiation, which is often associated with thermal
noise. Such a state is a statistical mixture of photon
number states [21]
rT ¼
X1
n¼1
ðNT þ 1Þnþ1 jnihnj
ðNTÞn
(30)
with NT ¼ trð^nrÞ the mean photon number. The mean
photon number for a mode with frequency v, at a given
temperature T is determined by the Bose-Einstein distribution
[21]
Figure 1.
Wigner distributions of the vacuum state j0i and a coherent state
jai with a ¼ 4 þ 4i.
simultaneous measurements of the q and p quadratures.
The Wigner distribution of the vacuum and of a coherent
state with nonzero photon number are compared in
Figure 1.
Coherent states are the quantum states that most
closely resemble a classical electromagnetic field. In fact,
the mean value of the electromagnetic field operator (9)
for a coherent state is given by
hiEðrr; tÞ ¼haj ^Eðrr; tÞjai
¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
hv=20 VðÞafðrr; tÞþ afðrr; tÞ
(28)
where we have used (24), and the fact that the state is single
mode.
If we consider the particular case of a plane wave
fðrr; tÞ¼ exp½iðk r vtÞ, we obtain
hiEðrr; tÞ ¼
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
hv=20 Vjaj cos ðk r vt 'Þ:
(29)
Let us recall that the mean energy of a quantum state
of light is given byhvðh^niþ 1=2Þ,wherehv=2 is the
zero-point energy of the vacuum, which manifest itself
as noise when measuring the field quadratures. By
using (27), we see that this term becomes negligible
when jaj 1. In this limit, a field measurement on a
coherent state results in a noiseless signal equal to
hiEðrr; tÞ , which according to (29) is exactly what we
would expect for a classical electromagnetic signal: a
cosine wave with amplitude proportional to square root
of the energy carried by the wave.
To convince oneself that this is a peculiarity of coherent
states, the reader should notice that for photon number
states jni, hnj ^Eðrr; tÞjni¼ 0, for every value of n.
MAY 2022
NT ¼
1
exp½hv=ðkBTÞ 1
(31)
where kB is the Boltzmann constant. Thermal states
(30) describes the noise background in QI. In this context,
T is assumedtobethe skytemperature at
the
radar receiver.
Thermal states (30) are Gaussian states with a Wigner
function of form (23) [23] with mean vectorx ¼ð0; 0Þ
and covariance matrix V ¼ð2NT þ 1Þ12. Therefore, as
illustrated in Figure 2 (note that the color scale is different
with respect to thats in Figure 1), their phase space representation
is always centred at the origin and have a width
which is determined by the mean photon number NT.
Accordingly, the Gaussian quasi-probability distribution
associated with a thermal state with NT > 0 is broader
than the one of a coherent state.
Simply looking at (31) one can notice a striking difference
between the intensity of noise backgrounds at optical
and microwave frequencies that will be relevant in our discussion
about quantum radar. For optical frequencies NT
is negligible, on the contrary for microwave frequencies
NT is significantly larger than one.4
Two-mode squeezed vacuum a.k.a. twin beams:
Let us now consider a quantum state involving two
modes ofthe electromagnetic field defined by the annihilation
operators
a^1 and ^a2. A pure quantum state of a two
mode field is called separable if it can be written as a
product of a state of mode a1 and a state of mode a2:
jCi¼jci1jfi2 [21]. A quantum state that cannot be written
in this form is called entangled [21]. A two-mode
entangled state that is essential for QI is the two-mode
squeezed vacuum
4An estimate of the noise background based only on (31) is actually
very rough, especially at optical frequency. However, a more accurate
calculation taking in account parameters other than temperature,
as for example the Sun irradiance, would lead to NT 106
at THz frequencies and NT 103 at 100 MHz, which does not
change the substance of the above statement.
IEEE A&E SYSTEMS MAGAZINE
73
IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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