IEEE Aerospace and Electronic Systems Magazine - November 2020 - 61

Bourassa and Wilson
probability for a given pFA can help reduce the required
number of measurements significantly.
To determine how r impacts the number of measurements
N while keeping the detection probability constant, we consider the situation where the correlation rp(
the probaﬃﬃﬃﬃﬃﬃ1 and
pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
bility of detection becomes PD % Q1 ðr 2N; À2 ln pFA Þ.
For a given pFA , improving the correlation r of the source by
a factor m then allows us to reduce the number of measurements by a factor 1=m2 to achieve the same detection performance. This fact is further demonstrated numerically in
Figure 3, where the open circles indicate that a system with
10-times more correlations with r ¼ 0:2 achieves the same
performance using, however, 100-times less measurements
with N = 5 000.
As we have shown, any improvement in the correlations emitted by the source can lead to large gains in
detection time to decide whether a target is there or not.
Because quantum sources of microwave signals can generate entangled signals with stronger correlations than traditionally allowed, the potential gains of microwave
quantum noise radar may be significant.

PROPAGATION AND MEASUREMENT OF QUANTUM
MICROWAVE SIGNALS
To better understand the properties of quantum signals and
their measurement, we here review the description of
quantum states of electromagnetic radiation using the language of quantum optics. A more in-depth description can
be found in numerous textbooks [14], [24], [27]. The
reader interested in the final results of this development
can skip directly to the " Quantum Description of Classical
Noise Radar " section.

QUANTUM DESCRIPTION OF ELECTROMAGNETIC
SIGNALS
In quantum optics, the electromagnetic field is modeled as
an infinite collection of independent traveling modes,
each oscillating at a unique frequency f, wavelength ;
and traveling at the speed of light c ¼ f % 3 Â 108 m/s.
The energy E of the field is quantized in photons of energy
hf, where h is Planck's constant. For a signal of frequency
f and power P integrated for a time T ¼ 1=B given by the
measurement bandwidth B, then the average number of
photons per second per unit bandwidth is given by
N ¼ P=ðhfBÞ.
The electromagnetic signal at a given frequency at the
output of a generator can be described by its initial state
vector jci determining the photon distribution statistics.
The evolution of the traveling field is given by a single
quantum matrix operator a^ðtÞ describing the field transformations as it interacts with the environment through
NOVEMBER 2020

scattering, absorption, amplification, etc. In this language,
the electric field (or equivalently, the voltage in circuits)
of amplitude E0 , angular frequency v ¼ 2pf; and phase f
is given by the complex operator function

E0
^
Eðv;
tÞ ¼ pﬃﬃﬃ a^yv ðtÞeiðvtþfÞ þ a^v ðtÞeÀiðvtþfÞ :
2

(3)

In quantum field theory, a^v and a^y v are complex linear
operators acting on a quantum state where a^v removes a
single photon of frequency v in a given state jci, whereas
its adjoint a^y v adds one. These operators are best
described in the photon number (or Fock) state basis
fjNiv g containing N ! 0 photons of frequency v. In this
representation, a^y v promotes a state jNiv to the state jN þ
1iv according to
a^y v jNiv ¼

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ
N þ 1 jN þ 1iv

(4)

whiles a^v demotes jNiv to the state jN À 1iv following
a^v jNiv ¼

pﬃﬃﬃﬃﬃ
N jN À 1iv :

(5)

It automatically follows that the composed operator N^v ¼
a^y v a^v counts the number of photons in the state as
N^v jNiv ¼ a^y v a^v jNiv ¼ N jNiv

(6)

following (4) and (5).
Given that photons are quantum particles known as
bosons, the operators f^
av ; a^y v g must obey the commutation relation
½^
av ðtÞ; a^y v ðt0 Þ a^v ðtÞ^
ay v ðt0 Þ À a^y v ðt0 Þ^
av ðtÞ ¼ dðt À t0 Þ
(7)
where dðxÞ is the Dirac delta function. This unique property entails two things. First, for a given field of frequency
v, only photons at equal times are statistically related.
Second, creating a photon first and destroying it afterward
does not lead to the same state as doing the opposite. As
we will see shortly, it is this property of boson statistics
that ultimately leads to the famous Heisenberg uncertainty
principle and to the fundamental quantum limits of measurement precision.
The field statistics are obtained from the expectation
values of different quantum operators acting on the quantum state vector jci of the initially generated field. For
instance, the average number of photons in a given pulse
is obtained by calculating the expectation value of the
number operator through hN^v ic hcjN^v jci. In the
microwave domain, one is usually interested in the expectation value of the electric field (or voltage) given by
hE^v ðtÞic whose variance is given by Var½E^v ðtÞc ¼
hE^v2 ðtÞic À hE^v ðtÞi2c , as one could expect.
Another common way to represent the field in (3)
is by its (unitless) in-phase I^v ðtÞ and quadrature
^v ðtÞ operators
Q

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