Supplementary Material
I. TWO PHOTON FORMALISM
Assuming an initial phase of
0
,
an EM field can be written as [1]:
ˆ
E
(
t
) = (
A
(
t
) + ˆ
a
1
(
t
)) cos (
ω
0
t
) + ˆ
a
2
(
t
) sin (
ω
0
t
)
where
ˆ
a
1
,
2
(
t
)
are the Hermitian amplitude and phase quadrature operators respectively. They describe the amplitude
and phase modulation of the field (
ω
0
is the carrier frequency). They satisfy
⟨
ˆ
a
1
,
2
(
t
)
⟩
= 0
and their commutation
relations are given by:
[ˆ
a
1
(
t
)
,
ˆ
a
2
(
t
′
)] =
−
iδ
(
t
−
t
′
)
We can further define
ˆ
a
1
,
2
(Ω)
as the Fourier transform of
ˆ
a
1
,
2
(
t
)
:
ˆ
a
1
,
2
(Ω) =
1
√
2
π
Z
ˆ
a
1
,
2
(
t
)
e
−
i
Ω
t
dt
.
Here, we observe that
ˆ
a
1
,
2
(Ω)
are not Hermitian but they commute with the Hermitian conjugate of themselves.
This means that they have an orthonormal eigenbasis but their eigenvalues are complex. Therefore, the commutation
relations between them is given by:
ˆ
a
1
ˆ
a
2
,
ˆ
a
†
1
,
ˆ
a
†
2
:=
[ˆ
a
1
,
ˆ
a
†
1
] [ˆ
a
1
,
ˆ
a
†
2
]
[ˆ
a
2
,
ˆ
a
†
1
] [ˆ
a
2
,
ˆ
a
†
2
]
=
i
0 1
−
1 0
.
where the
Ω
dependence is suppressed, and we used the following notation for the commutation relations matrix:
h
ˆ
Q
,
ˆ
Q
†
i
:=
h
ˆ
Q
i
,
ˆ
Q
j
i
i,j
. Hereafter we will use this notation.
Therefore, we can interpret the above as two harmonic oscillators with:
ˆ
X
R
(Ω) =
√
2
Re
(ˆ
a
1
)
,
ˆ
P
R
(Ω)
=
√
2
Re
(ˆ
a
2
)
ˆ
X
I
(Ω) =
√
2
Im
(ˆ
a
1
)
,
ˆ
P
I
(Ω)
=
√
2
Im
(ˆ
a
2
)
It follows that the commutation relations of two harmonic oscillators are given by:
ˆ
X
R
ˆ
P
R
ˆ
X
I
ˆ
P
I
,