IEEE Aerospace and Electronic Systems Magazine - November 2020 - 63

Bourassa and Wilson

Figure 5.
Quantum optical analogue of an IQ-mixer measuring both quadratures of a quantum field a^. Figure adapted from [10].

Finally, one random waveform that can be used in noise
radar system is the light generated by a lamp, or the Johnson
noise emitted by a warm resistor. Such a thermal state jciT
represent the thermal radiation emitted by a black body emitter at temperature T . At a given frequency v, the thermal
state has a mean h^
aiT ¼ h^
ay iT ¼ 0 and an average number
y
of photons h^
a a^iT ¼ NT given by the Bose-Einstein occuÀ
ÁÀ1
pation number NT ¼ ehv=ðkB T Þ À 1
where kB is the
Boltzmann constant. The quadratures of a thermal field then
^
^
have the expected zero-mean hIðtÞi
T ¼ hQðtÞiT ¼ 0 and
^
^
variance Var½IðtÞT ¼ Var½QðtÞT ¼ NT þ 12 . The zerotemperature thermal state recovers the vacuum state and is,
therefore, not noiseless.

QUADRATURE MEASUREMENTS OF QUANTUM

output fields are down-converted to an intermediate frequency and then digitized and recorded.
To link the measured outcomes to the field quadratures
of the a^-field, we define the signal operator S^a ¼
^ 1 þ iP^2 . From the real and imaginary parts of complex
X
classical signal hS^a i along with (10) and (13), the quadra^a correspond to [10],
tures operators I^a and Q
^a þ X
^v
X
I^a ¼ pﬃﬃﬃ ;
2

^ À P^v
^ a ¼ Pap
ﬃﬃﬃ :
Q
2

(14)

With the assumption that the noise driving the second
input on the BS is a zero-mean Gaussian thermal
noise with a number of noise photons NT , the measured
^
quadratures
pﬃﬃﬃ have the desired
pﬃﬃﬃexpectation values hIa i ¼
^a i ¼ hP^a i= 2, with variances
^ a i= 2 and hQ
hX
Á
1À
^ a  þ NT þ 1=2 ;
Var½X
2
À
Á
^ a  ¼ 1 Var½P^a  þ NT þ 1=2
Var½Q
2
Var½I^a  ¼

(15)

where the prefactor of 1=2 comes from the definition of
the quadrature operators (14). As we can see, quantum
mechanics prohibits the production of identical copies of
the input signal as at least vacuum noise must unavoidably
be added in the process.

SIGNALS
Contrary to classical signals, quantum signals cannot be
measured arbitrarily as the back-action of the measurement apparatus unavoidably adds noise in the process, limiting the detection efficiency. While the amount of extra
(ideally vacuum) noise depends on the physical realization
of the detector, it can be mitigated by performing so-called
back-action evading measurements [9], complex measurement schemes that will not be addressed here.
We will here focus on the simpler and ubiquitous heterodyne detection using linear detectors and IQ-mixers.
Figure 5 depicts a simple quantum optical model for an
IQ-mixer where the field to be measured a^ is first sent
toward a 50/50 beam-splitter (BS), together with uncorrelated noise v^ on the second input [10]. At the outputs, the
BS produces two correlated fields a^1 and a^2 given by
a^ þ v^
a^1 ¼ pﬃﬃﬃ ;
2

a^ À v^
a^2 ¼ pﬃﬃﬃ :
2

(13)

The presence of uncorrelated noise input assures that the
output state is physical where both fields commute
½^
a1 ; a^y2  ¼ 0. In the ideal BS where only vacuum noise is
present, then the outputs are ideally correlated signals.
Afterward, both quadratures of the input a^ can be measured simultaneously, with some added noise, provided
only one quadrature is measured for both outputs. This is
done using standard heterodyne detection where the
NOVEMBER 2020

QUANTUM DESCRIPTION OF CLASSICAL NOISE RADAR
Now that the quantum description of quadrature measurements have been obtained, we can now proceed to the calculation of the covariance matrix of a two-signal state and
apply this to classical noise radar.
We first recall that the signal and idler are two classically correlated Gaussian thermal noise fields, which can be
obtained by mixing a broadband Gaussian noise with a continuous-wave signal [8]. In quantum optics, one of the simplest ways to produces classically correlated thermal states
corresponds to using a balanced BS as in Figure 5 using a
Gaussian thermal noise field (that here we will label c^) and
uncorrelated vacuum noise (^
v) as inputs. At the outputs, the
^
balanced BS produces the correlated signal (^
a) and idler (b)
fields, corresponding to a^1 and a^2 in (13), respectively.
When heterodyne measurement of both signals is performed, their respective quadrature operators are given
by (14). We then define the quadrature vector x^ ¼
^a ; I^b ; Q
^b ÞT and obtain the covariance matrix R of
ðI^a ; Q
the system from the expectation value R ¼ h^
xx^T i, or
more explicitly
0

1
^a i hI^a I^b i hI^a Q
^b i
hI^a2 i
hI^a Q
B ^ ^
^a I^b i hQ
^a Q
^b i C
hQ2a i
hQ
B hQ I i
C
R¼B a a
C:
2
^
^
^
^
^
^
^
@ hIb Ia i hIb Qa i
hIb i
hIb Qb i A
^b Q
^b I^b i
^2 i
^ a i hQ
^b I^a i hQ
hQ
hQ
b

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IEEE Aerospace and Electronic Systems Magazine - November 2020

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