IEEE Aerospace and Electronic Systems Magazine - November 2020 - 64

Progress Toward an All-Microwave Quantum Illumination Radar
From an initial single-mode thermal-noise signal with
hcy ciT ¼ 2NS , assuming f ¼ 0 for simplicity and vacuum
noise inputs throughout, the measured covariance matrix
of the ideal classically correlated noise state RC is
calculated to be
0

0
NS þ 1
1B
0
NS þ 1
B
RC ¼ @
NS
0
2
0
NS

NS
0
NS þ 1
0

1
0
NS C
C:
0 A
NS þ 1

(17)

By comparing with the expected classical result in (1), the
measured powers at the receiver are identical with P a ¼
P b ¼ NS þ 1. For each channel, it corresponds to half of
the initial emitted thermal noise power plus 1 unit of vacuum noise, where one half comes from the generation
stage, and another half from the detection stage. In the
nonideal case where thermal noise is added instead of vacuum during the preparation and measurement steps, then
the measured power for each mode at the receiver will be
P ¼ NS þ NT þ 1. In the ideal setting, the covariance for
classically correlated signals is CC ¼ NS or, equivalently,
corresponds to the measured power minus one unit of vacuum noise, or
CC ¼ P À 1

(18)

and is the optimal upper bound for classically correlated
signals.
Assessing the presence or absence of a target is done
by estimating the signal-idler covariance. While under
both hypothesis, the idler remains the same, both the
received signal power and the covariance will change. If
the target is absent, then only the bright thermal atmospheric noise is received with a noise power NB . Given
that it is not correlated with the idler, the covariance
matrix reads
0

NB þ 1=2
0
1B
0
NB þ 1=2
B
Rabsent
¼
C
0
0
2@
0
0

0
0
NS þ 1
0

1
0
0 C
C:
0 A
NS þ 1

(19)
On the other hand, if the target is present, then a small
fraction h of the transmitted signal makes its way back to
the receiver, in addition to the bright thermal background.
The covariance matrix in this case is given in (20) (assuming f ¼ 0 for simplicity).
In the situation where the received signal to the background noise ratio is small hNS =NB ( 1, then the

0

hðNS þ 1=2Þ þ NB þ 1=2
0
present 1 B
B
pﬃﬃﬃ
R
¼ @
C
h NS
2
0

64

measured power is almost identical in both cases, unlike
the covariance which is nonzero only if the target is
present.

QUANTUM-ENHANCED NOISE RADAR
At the heart of QI, there are special quantum states of light
known as TMSSs. These entangled Gaussian states can be
generated by SPDC, a nonlinear optical three-wave mixing process occurring in an optically active element continuously driven by a pump field. There, a three-wave
mixing process converts one pump photon at frequency
vp into a pair of signal and idler photons at different frequencies vS and vI respecting energy conservation with
vP ¼ vS þ vI . Since the photons in the pair are created
from the same vacuum, both share entanglement in the
form of quantum correlations that are stronger than classically allowed. In the microwave domain, SPDC can be
engineered in superconducting microwave quantum circuits using phase-preserving JPAs [8], [13]. These quantum-limited amplifiers can be designed to operate across
the 3-10 GHz range with signal and idler frequencies that
can be tuned in situ by around 0.1-1 GHz.
In the language of quantum optics, a JPA can be seen
as an active four-port device mixing two vacuum signals at
the input (^
m and ^n) and producing two amplified sideband
^ The
signals at the output, the signal and idler (^
a and b).
generation of entanglement from uncorrelated vacuum
noise inputs can be modeled by the following equations [9]:
a^ ¼
b^ ¼

^y
coshðrÞ ^n þ sinhðrÞ m
y
^:
sinhðrÞ ^n þ coshðrÞ m

(21)

In essence, the two independent vacuum noise inputs are
mixed and amplified with power gain g ¼ cosh2 r, where
the parameter r is referred to as the squeezing parameter.
By performing heterodyne measurements on the signal
and idler exactly in the same way as before (see
" Quadrature Measurements of Quantum Signals " ), the
covariance matrix for TMSS reads
0
1B
RQ ¼ B
4@

1
coshð2rÞ þ 1
0
sinhð2rÞ
0
0
coshð2rÞ þ 1
0
À sinhð2rÞ C
C:
A
sinhð2rÞ
0
coshð2rÞ þ 1
0
0
À sinhð2rÞ
0
coshð2rÞ þ 1

(22)
Contrary to the classical result (17), the quantum signal and
idler fields are anticorrelated, as indicated by the negative
covariance. At first sight, the signal and idler quadratures
of a TMSS are similar to zero-mean Gaussian thermal

pﬃﬃﬃ
0
hNS
hðNS þ 1=2Þ þ NB þ 1=2
0
0
NS þ 1
pﬃﬃﬃ
h NS
0

IEEE A&E SYSTEMS MAGAZINE

1
0
pﬃﬃﬃ
h NS C
C:
0 A
NS þ 1

(20)

NOVEMBER 2020
IEEE Aerospace and Electronic Systems Magazine - November 2020

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