IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV - 71

Sorelli et al.
a^y
i jniii: ¼
p
ffiffiffiffiffiffiffiffiffiffiffiffi
n þ 1
jni þ 1ii:
It follows directly from (7) that ^ay
a consequence, we define ^ni ¼ ^ay
(7b)
i ^aijniii ¼ nijniii.As
i ^ai, and we call it the
photon number operator. Moreover, creation and annihilation
operators obey the following commutation relations:
hi
a^i; ^ay
j ¼ ^ai ^ay
j ^ay
j ^ai ¼ di;j:
(8)
Given the abovementioned definitions, we can write
the quantum mechanical operator describing the electromagnetic
field as
E^ðr;tÞ¼
XN
i¼1
1=2hvi
a^ifiðr;tÞþ ^ay
20 V
i f
i ðr;tÞ
:
(9)
Equation (9) is the conceptual core of quantum optics,
where the quantum part comes from the creation and
annihilation operators, their action on photon-number
states (7), and their commutation relation (8). On the
other hand, the optics part stems from the mode functions
fiðr;tÞ that are solutions of the wave (3), and
therefore, they evolve and propagate as classical electromagnetic
waves.
After having defined the quantum-mechanical electromagnetic
field operator, we now introduce the formalism
to describe quantum states of light. Let us start with
the vacuum state j0i, namely the state of the electromagnetic
field containing no photons. When no photons are
present, no photon can be removed (^aij0i¼ 0). On the
contrary, photons can be added to the vacuum, and we
can obtain the states jniii by applying ni times (7b):
jnii¼
the total field is then given by a linear superposition of
products of photon-number states of the individual
modes [20]
ni
ai^y
jCi¼
X
n1
X
nm
Cn1nmjn1i1 ...jnmim; ... ;
(10)
where the symbol denotes the tensor product.
In the linear superposition in (10), the phase relation
between the different multimode photon-number
states in jCi are perfectly defined by the complex
coefficients Cn1nm. However, knowing these phase
relations is not always possible in quantum optics, and
sometimes the only thing we can specify are a set of
probabilities for the field to be found in certain states.
When this is the case, the state of the field cannot be
written as a pure state, i.e., it is not given by (10), and
we refer to it as a statistical mixture or simply as a
mixed state.Amixedstate canbeexpressedasa density
operator [20]
MAY 2022
r ¼
X
i
Pijciihcij
(11)
where Pi represents the probability for the field to be
found in the pure state jcii. Accordingly, we have 0
Pi 1 and
P
mechanical operator ^O can then be expressed as
h ^Oi¼ tr
O^r ¼
X
i
Pihcij ^Ojcii
i Pi ¼ 1. The mean value of a quantum
(12)
where tr denotes the trace operation. Equation (12) can be
understood as the sum of the expectation values hcij ^Ojcii
of the operator
probabilities of these states to appear in the statistical mixture
r.
Let us conclude by pointing out that a pure state is a
special mixed state for which a particular probability Pi ¼
1, while all others vanish, i.e., r ¼jcihcj. Therefore,
when dealing with general expressions valid both for
mixed and pure states, especially when dealing with mean
values [see (12)], it is often convenient to use the density
matrix formalism.
PHASESPACEDISTRIBUTIONANDGAUSSIANSTATES
j0i=ðni!Þ. The most general pure state of
In this section, we will discuss the properties of the particular
quantum states of light which are relevant for QIs. All
these states fall in the class of the so called Gaussian
states, namely quantum states, which are fully characterized
by the first and second moments of the quadrature
operators [23]. In particular, mixed Gaussian states are the
quantum analogue of classical Gaussian noise. Those
states are better understood in terms of their Wignerfunction,
which we will introduce in the following.
Wigner function. Let us consider a physical system
consisting of N electromagnetic modes, we can arrange
the corresponding quadrature operators ^qi ¼ ^ai þ ^ay
p^i ¼ ið^ay
i and
i ^ai) in the operators vector
x^ ¼ ^q1; ^p1; ... ; ^qN; ^pNðÞ (13)
and define the operator
DðÞ¼ exp i^xTV
O^ for the states jcii weighted with the
(14)
where T denotes the transpose, 2 R2 N, and
V ¼
MN
i¼1
v ¼
@
v
. . .
v
1
A; with v ¼
(15)
01
10
is 2 N
2 N matrix known as the symplectic form. The
density operator r of an arbitrary quantum state is then
equivalent to the characteristic function [23]
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IEEE - Aerospace and Electronic Systems - May 2022 - Tutorial XV

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