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Contents
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1 The Quantization of the Electromagnetic Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Normal mode expansion of the electromagnetic field in a resonator . . . . . . . . . . . . . . . . . 3
1.2 The quantization of the radiation field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 Coherent States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.1 Definitions of coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Properties of coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Phase space diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Number statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 Physical systems represented by coherent states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Squeezed States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.1 Definitions of squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.2 Quadrature uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
3.3 Phase space diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3.4 Detection of squeezed states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Parametric Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4. Hamiltonian formulation of the parametric interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5. Generation of squeezed state of the electromagnetic field by parametric interaction . . . . 22
6 Higher Orders Squeezing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . .27
7 Photon Bunching and Photon Antibunching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.1 Second order correlation function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
7.2 Photon bunching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
7.2 Photon antibunching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1
Introduction:
From the early days of physics, we are interested to learn the interaction of electromagnetic
radiation with atoms and molecules. With the advent of laser the studies has received a
tremendous boost. From the classical electrodynamics we know that the material medium
gets polarized as the electric field is introduced. Depending upon the intensity of the
electromagnetic field the order of polarization will be involved. We know that the
polarization is proportional to the electric field and the proportionality constant is known as
linear susceptibility. The linear relation between the polarization and electric field is valid
only when the electric field is not enough strong. On the other hand the polarization gets
modified remarkably when the strength of electric field is quiet high. In such a situation the
polarization is thought of linear and non linear polarization. Therefore it is clear that the
nonlinear polarization is possible only when the electric field is quiet strong. It is to be
mention that the strong electric field is achievable with high power laser source. This is
exactly why the laser sources have revived the interest on nonlinear polarization and hence
the nonlinear susceptibility.
The electromagnetic field in a cavity is easily quantized. The energy of the quantized field
appears to be the sum of the harmonic oscillator Hamiltonian. It is easy to establish the
electric and magnetic field correspond the quadrature component and are canonically
conjugate.
We already mention that the electric field coupled to the medium produces polarization
having linear and nonlinear components. The interaction part of the Hamiltonian is
contributed by the non linear polarization. On the other hand nonlinear polarization
contributes free field energy. The temporal development of the field operators are calculated
by taking care of the interaction Hamiltonian. These field operators are useful to study
various nonlinear non classical phenomenon in quantum optics. These include squeezing,
higher order squeezing, photon antibunching.
In the present studies we investigate the degenerate parametric amplitude as a candidate for
producing squeezing, higher order squeezing and photon anti bunching. We observe that the
squeezing in one of the quadratures is obtained at the cost of the other quadrature. In other
word the simultaneous squeezing of both quadratures are ruled out. We establish the input
coherent light coupled to the degenerate parametric amplifier is a good candidate for
production of higher order squeezing in one of the quadrature components. We also
investigate the condition for which the antibunching of photon will appear in a degenerate
parametric amplifier coupled to the coherent radiation field.
2
1 The quantization of the electromagnetic field :
1.1
Normal mode expansion of the electromagnetic field in a resonator:
The Maxwell's equations and the current continuity equation are
⃗
⃗∇ × 𝐸⃗ = − 𝜕𝐵
𝜕𝑡
⃗
⃗∇ × 𝐻
⃗ = 𝑖 + 𝜕𝐷
𝜕𝑡
⃗ .𝐷
⃗ =𝜌
∇
(1.1)
⃗∇. 𝐵
⃗ =0
𝜕𝜌
⃗⃗⃗
∇. 𝑖 = − 𝜕𝑡
We will limit ourselves, for this case to charge-free, isotropic, and homogeneous media
so that
⃗ = µ𝐻
⃗
⃗⃗⃗ 𝐷
⃗ =0
⃗ = 𝜖𝐸⃗
𝑖=0
𝐵
∇.
𝐷
(1.2)
where 𝜖 is the dielectric constant. Let we consider that the electric field ⃗⃗⃗
𝐸 (𝑟, 𝑡) and
⃗ (𝑟,t) inside a volume V bounded by a surface S of perfect conductivity.
magnetic field 𝐻
⃗ , 𝑛⃗.𝐻
⃗ , must both
The tangential component of 𝐸⃗ , −𝑛
⃗⃗⃗⃗⃗ × 𝑛⃗ × 𝐸⃗ , and normal component of 𝐻
⃗⃗⃗ and 𝐻
⃗⃗⃗⃗ in terms of two
be zero on S (𝑛⃗ is the unit normal to the s). We will expand 𝐸
⃗ а, respectively. These sets, which were introduced
orthogonal sets of vector fields 𝐸⃗ a and 𝐻
originally by Slater obey the relations
⃗𝑎
𝑘а 𝐸⃗𝑎 = ⃗⃗⃗
∇ ×𝐻
(1.3)
⃗⃗ a = ∇
⃗ × 𝐸⃗𝑎
ka H
(1.4)
Where 𝑘а is to be considered, for the moment, a constant. The tangential component of 𝐸⃗𝑎 on
S is zero.
𝑛⃗ × 𝐸⃗𝑎 = 0
on
S
If we take the curl of both sides of (1.3) and (1.4) and using the identity
⃗⃗⃗∇ × ∇
⃗ ×𝐴= ∇
⃗ (∇
⃗ . 𝐴 ) - ∇2 𝐴
They become
∇2 𝐸⃗𝑎 + k 2a 𝐸⃗𝑎 = 0
3
(1.5)
⃗𝑎
∇2 𝐻
⃗a=0
k 2a ⃗H
(1.6)
⃗⃗ a.
These are the familiar wave equations involving vector fields 𝐸⃗𝑎 𝑎𝑛𝑑 H
⃗⃗ a , 𝑛⃗ . H
⃗⃗ a is zero on S. In
It follows from (1.3), (1.4) and (1.5) that the normal component of H
order to prove this we consider an arbitrary closed contour C on S surrounding a surface S.
𝑑𝑙 = ∮𝑐 (−𝑛⃗ × 𝑛⃗ × 𝐸⃗ ). ⃗⃗⃗
𝑑𝑙
∫𝑐 𝐸⃗𝑎 . ⃗⃗⃗
𝑑𝑙 =0
∮𝑐 (𝑛⃗ . 𝐸⃗𝑎 ) 𝑛⃗ . ⃗⃗⃗
(1.7)
where 𝐸⃗𝑎 is expressed as the vector sum of its tangential (-𝑛⃗ × 𝑛⃗ × 𝐸⃗𝑎 ) and normal (𝑛⃗. 𝐸⃗𝑎 ) 𝑛⃗
components. The first term on the right side of (1.7) is zero because of (1.5) whereas the
⃗⃗⃗ . Using stokes’ theorem on the left of (1.7)
second one is zero since 𝑛⃗ is perpendicular to 𝑑𝑙
gives
∮
𝑐
⃗⃗⃗⃗ × 𝐸⃗𝑎 ). 𝑛⃗ 𝑑𝑎= 𝑘а ∫ ( ⃗H
⃗ a . 𝑛⃗) 𝑑𝑎 = 0
𝐸⃗𝑎 . ⃗⃗⃗
𝑑𝑙 = ∫𝑐 (∇
𝑠′
and since C is arbitrary, it follows that
⃗⃗ a . 𝑛⃗ = 0
H
on S.
(1.8)
⃗ a are orthogonal in the sense
We will next prove that the functions 𝐸⃗𝑎 and ⃗H
∫𝑣 𝐸⃗𝑎 . 𝐸⃗𝑏 𝑑𝑣 = 0
a b
⃗⃗ b . H
⃗⃗ b 𝑑𝑣 = 0
∫𝑣 H
a≠ 𝑏
(1.9)
Now we have
⃗∇. (𝐸⃗𝑏 × ⃗∇ × 𝐸⃗𝑎 ) − ⃗∇. (𝐸⃗𝑎 × ⃗∇ × 𝐸⃗𝑏 )
= ⃗∇ × 𝐸⃗𝑎 . ⃗∇ × 𝐸⃗𝑏 − 𝐸⃗𝑏 . ⃗∇ × ⃗∇ × 𝐸⃗𝑎 − ⃗∇ × 𝐸⃗𝑏 . ⃗∇ × 𝐸⃗𝑎 + 𝐸⃗𝑎 . ⃗∇ × ⃗∇ × 𝐸⃗𝑏
⃗)= 𝐵
⃗ . ⃗∇ × 𝐴 − 𝐴 . ⃗∇ × 𝐵
⃗ , is used.
where the vector identity, ⃗∇. (𝐴 × 𝐵
⃗ a and ⃗∇ × ⃗∇ × 𝐸⃗𝑎 = k 2a 𝐸⃗𝑎 , which substituted in the last
From (1.4), we have ⃗∇ × 𝐸⃗𝑎 = k a ⃗H
equation gives
⃗ 𝑎 ) − k b ⃗∇. (𝐸⃗𝑎 × 𝐻
⃗ 𝑏 ) = (k 2b − k 2a )𝐸⃗𝑎 . 𝐸⃗𝑏
k a ⃗∇. (𝐸⃗𝑎 × 𝐻
That, after applying Gauss’s theorem, becomes
⃗ 𝑎 ) − k b 𝑛⃗ . (𝐸⃗𝑎 × 𝐻
⃗ 𝑏 ) ]𝑑𝑎 = (k 2b − k 2a ) ∫ 𝐸⃗𝑎 . 𝐸⃗𝑏 𝑑𝑣
∫𝑠 [ k a 𝑛⃗ . (𝐸⃗𝑏 × 𝐻
𝑣
⃗ ×𝐶 =
The left side of the last equality can be shown, with the aid of the identity 𝐴 . B
⃗ and (1.5) to be zero, so that for k b ≠ k a , the first of equations (1.9) is proved. If
𝐶 . ⃗A × 𝐵
k b = k a , that is when 𝐸⃗𝑏 and 𝐸⃗𝑎 are members of a degenerate set, it is possible to construct
linear superposition of the degenerate functions so that orthogonality is preserved. The proof
⃗ a functions follows along identical lines.
of orthogonality of the ⃗H
4
⃗ a andE
⃗ a so they are normalized according to
We are free to choose the magnitude of ⃗H
⃗⃗ a . H
⃗⃗ b 𝑑𝑣 = δa,b
∫𝑣 H
⃗ a.E
⃗ b 𝑑𝑣 = δa,b
∫𝑣 E
(1.10)
⃗⃗⃗ (𝑟, 𝑡) and 𝐻
⃗⃗⃗⃗ (𝑟, 𝑡) can be expanded as
The total resonator fields of 𝐸
⃗⃗⃗ (𝑟, 𝑡) = − ∑𝑎
𝐸
⃗⃗⃗⃗ (𝑟, 𝑡) = ∑𝑎
𝐻
1
√𝜖
1
√𝜖
⃗ a (r)
𝑝𝑎 (𝑡)E
⃗⃗ a (r)
𝑞𝑎 𝜔𝑎 (𝑡)H
(1.11)
where, 𝜔𝑎 = 𝑘𝑎 /√µ𝜖 . Substituting (1.11) in the first of Maxwell equations (1.1) and using
(1.3) result is
∑
1
⃗𝑎
𝑝 (𝑡)𝑘𝑎 𝐻
√𝜖 𝑎
= −𝜇 ∑
1
√𝜖
⃗𝑎
𝜔𝑎 𝑞̇ 𝑎 𝐻
Equating the coefficient of like term in both sides, we have
𝑝𝑎 = 𝑞̇ 𝑎
(1.12)
In a similar fashion we obtain
𝜔𝑎2 𝑞𝑎 = −𝑝̇𝑎
(1.13)
Eliminating 𝑞𝑎 from equation (1.13) with the equation (1.12) gives
𝑝𝑎̈ + 𝜔𝑎2 𝑝𝑎 = 0
This identifies
𝑘𝑎
√µ𝜖
(1.14)
= 𝜔𝑎 as the radian oscillation frequency of the 𝑎th mode.
1.2 The quantization of the radiation field:
In this section we will show that the electromagnetic field inside a resonator can be
considered, an ensemble of independent harmonic oscillators.
Multiplying the first of equation of (1.11) by 𝐸⃗𝑏 and integrate over the resonator volume we
get,
1
⃗ a (r). 𝐸⃗𝑏 (𝑟)𝑑𝑣
∫𝑣 𝐸⃗ (𝑟, 𝑡). 𝐸⃗𝑏 (𝑟)𝑑𝑣 = − ∫𝑣 ∑𝑎 √𝜖 𝑝𝑎 (𝑡)E
= − ∑𝑎
=−
1
√𝜖
𝑝𝑎 (𝑡)𝛿𝑎𝑏
𝑝𝑏
√𝜖
𝑝𝑏 (𝑡) = −√𝜖 ∫𝑣 ⃗⃗⃗
𝐸 (𝑟, 𝑡) . 𝐸⃗𝑏 (𝑟)𝑑𝑣
(1.15)
õ
⃗⃗⃗⃗ (𝑟, 𝑡) . ⃗⃗⃗⃗⃗
𝑞𝑏 (𝑡) = 𝜔 ∫𝑣 𝐻
𝐻𝑏 (𝑟)𝑑𝑣
(1.16)
Similarly
𝑏
5
The total energy (Hamiltonian) of the electromagnetic field is given by
1
⃗ .𝐻
⃗ + 𝜖𝐸⃗ . 𝐸⃗ )𝑑𝑣
H = 2 ∫𝑣 (µ 𝐻
(1.17)
⃗ 𝑎𝑛𝑑 𝐸⃗ by their expansion (1.11) gives
Substituting 𝐻
2
1
𝜔
⃗ a (r). ⃗H
⃗ b (r) 𝑑𝑣
𝐻 = 2𝜖 ∑ 𝑝𝑎 𝑝𝑏 ∫ ⃗Ea (r). ⃗Eb (r) 𝑑𝑣 + 2𝜇 ∑ 𝑞𝑎 𝑞𝑏 ∫ ⃗H
1
1
= 2 ∑ 𝑝𝑎2 + 2 ∑ 𝜔2 𝑞𝑎2
= ∑𝑎 12 (𝑝𝑎2 + 𝜔𝑎2 𝑞𝑎2 )
(1.18)
The dynamical variables 𝑝𝑎 𝑎𝑛𝑑 𝑞𝑎 constitute canonically conjugate variables. This can be
seen by considering Hamilton’s equations of motion relating to 𝑝̇𝑎 𝑡𝑜 𝑞𝑎 and 𝑞̇ 𝑎 𝑡𝑜 𝑝𝑎.
𝜕𝐻
𝑝̇𝑎 = −
𝜕𝑞𝑎
𝜕𝐻
𝑞̇ 𝑎 =
𝜕𝑝𝑎
= −𝜔𝑎2 𝑞𝑎
= 𝑝𝑎
(1.19)
These are identical with equations (1.12) and (1.13) obtained from Maxwell’s equations.
The quantization of electromagnetic radiation is achieved by considering 𝑝𝑎 𝑎𝑛𝑑𝑞𝑎 as
equivalent to momentum and coordinate operator of a quantum mechanical harmonic
oscillator. Thus taking the commutation relations connecting the dynamic variables as
[𝑝𝑎 , 𝑝𝑏 ] = [𝑞𝑎 , 𝑞𝑏 ] = 0
[𝑞𝑎 , 𝑝𝑏 ] = 𝑖ћ𝛿𝑎,𝑏
(1.20)
Now we define creation and annihilation operator by
1
𝑎𝑙† (𝑡) = (2ћ𝜔 )
1⁄2
𝑙
1⁄2
1
𝑎𝑙 (𝑡) = (2ћ𝜔 )
𝑙
[𝜔𝑙 𝑞𝑙 (𝑡) − 𝑖𝑝𝑙 (𝑡)]
[𝜔𝑙 𝑞𝑙 (𝑡) + 𝑖𝑝𝑙 (𝑡)]
(1.21)
(1.22)
The commutation relations of the operators (1.21) and (1.22) follows as
†
[𝑎𝑙 , 𝑎𝑚 ] = [𝑎𝑙† 𝑎𝑚
]=0
†
[𝑎𝑙 , 𝑎𝑚
] = 𝛿𝑙,𝑚
Solving equation (1.21) & (1.22) we get
𝑝𝑙 (𝑡) = 𝑖 (
ћ𝜔𝑙
2
1⁄
2
)
[𝑎𝑙† (𝑡) − 𝑎𝑙 (𝑡)]
1⁄
2
ћ
𝑞𝑙 (𝑡) = −𝑖 (2𝜔 )
𝑙
[𝑎𝑙† (𝑡) + 𝑎𝑙 (𝑡)]
(1.23)
(1.24)
Now we can express the Hamiltonian in terms of the creation and annihilation operator by
Substituting the equation (1.23) and (1.24) in (1.18). Hence, we have
1
𝐻 = ∑𝑙 ћ𝜔𝑙 (𝑎𝑙† 𝑎𝑙 + 2)
6
(1.25)
Quantum mechanically, a stationary state of the total radiation field can be characterized by
an eigen function 𝜓 , which is a product of the eigenfunctionsof the individual Hamiltonians.
The expectation value of the operator 𝑎𝑙† 𝑎𝑙 is
⟨𝜓|𝑎𝑙† 𝑎𝑙 |𝜓⟩ = ⟨𝑛𝑙 |𝑎𝑙† 𝑎𝑙 |𝑛𝑙 ⟩ = 𝑛𝑙
(1.26)
Where 𝑛𝑙 is number of quanta in the 𝑙𝑡ℎ mode of the resonator.
2 Coherent states:
In connection with classical states of quantum mechanical harmonic oscillator the idea of
coherent states was conceived way back in the year 1926 by Schrodinger. After a long period,
Glauber, Sudarshan and Klauder revived the interests in coherent states. The advancement in
the studies of coherent state is being accelerated with the invention of lasers. Interestingly,
the coherent states are of classical nature originating from quantum mechanical sources. In
the following sections we will discuss the definitions and few relevant properties of the
coherent states.
2.1 Definitions of coherent states:
Coherent states of harmonic oscillator can be defined in three ways. These are as follows.
Definition 1: Annihilation Operator Coherent State (AOCS). The coherent states
|𝛼〉 are eigenstates of the annihilation operator a of the harmonic oscillator,
𝑎|𝛼〉 = 𝛼|𝛼〉,
(2.1)
where 𝛼 = |𝛼|𝑒 𝑖𝜃 , is in general complex and |𝛼|2 is the average photon number.
Definition 2: Displacement Operator Coherent State (DOCS). The coherent states
|𝛼〉 can be obtained from the ground state of the harmonic oscillator by the application of an
operator 𝐷(𝛼). Therefore
|𝛼〉 = 𝐷(𝛼)|𝑜〉 ,
(2.2)
where 𝐷(𝛼) = exp(𝛼𝑎† − 𝛼 ∗ 𝑎) is the displacement operator and hence the coherent state
is termed as displaced ground state of the harmonic oscillator.
7
Definition 3: Minimum Uncertainty Coherent State (MUCS). The coherent states
|𝛼〉 are quantum states with a minimum uncertainty relationship,
1
∆𝑋∆𝑃 = ,
(2.3)
2
where (∆𝑋)2 and (∆𝑃)2 are the second order variances in the canonically conjugate self
adjoint quadrature operators X and P. It is to be noted that the third definition is not unique.
The so called squeezed states also qualify for this definition and so sometimes are called as
generalized coherent states. These states are discussed in the next section. However, only the
states, with uncertainties in both the quadratures equal to the standard quantum limit (SQL)
1
√2
, are referred as coherent states.
2.2 Properties of coherent states:
With the above definitions the coherent states of the harmonic oscillator, we would like to put
few properties of the said states. These are listed below.
.
1. Fock state expansion of coherent states:
Coherent states of harmonic oscillator can be expanded in the diagonal representation of Fock
states or number states,|𝑛〉, as
1
|𝛼〉 = 𝑒 −2
|𝛼|2
∑∞
0
𝛼𝑛
√𝑛!
|𝑛〉
(2.4)
2. Non-orthogonality:
By direct calculation and by using the Fock state expansion (2.4), we get
2
|⟨𝛼|𝛽⟩|2 = 𝑒 −|𝛼−𝛽| ,
(2.5)
The above relation shows that the coherent states are non-orthogonal.
However, for |𝛼 − 𝛽| = 𝑙𝑎𝑟𝑔𝑒, the coherent states will be orthogonal.
3. Over-completeness:
Glauber has shown that the resolution of the identity in terms of coherent states is not unique
and a common resolution is
∫|𝛼〉〈𝛼|𝑑2 𝛼 = 𝜋
Or,
1
𝜋
∫|𝛼〉〈𝛼|𝑑2 𝛼 = 𝐼
(2.6)
The above relation also shows that the coherent states are over complete.
8
2.3 Phase space diagram:
P
X
Figure 1: Phase space diagram of coherent and vacuum states of a harmonic oscillator
The coherent states have the same Gaussian dependence on both the quadratures as of ground
state of a harmonic oscillator, but the mean is displaced by a constant amount |𝛼| from the
zero value for the ground state. In the phase space diagram, in figure 1, the uncertainties of
these states are represented by a circle around the mean position. The plots represent the
uncertainties, quantitatively the full width half maxima (FWHM) of the quasiprobability
distribution function. The circle at the origin is the uncertainty of the vacuum state, which has
been displaced by an amount |𝛼| towards the angle 𝜃 the phase angle of the complex
coherence parameter 𝛼. The mean value of the quadratures rotate around the origin following
the bigger circular path showing sinusoidal temporal dependence like a classical oscillator of
same frequency.
2.4 Number statistics:
The probability distribution of particles in a coherent state follows Poissonian statistics
𝑃(𝑛) = |⟨𝑛|𝛼⟩|2 = |𝛼|2𝑛
where |𝛼|2 is the mean of the particle number.
9
𝑒 −|𝛼|
𝑛!
2
(2.7)
2.5 Physical systems represented by coherent states:
Two types of coherent physical systems are considered, the coherent radiation field (a system
of photons represented by above definitions) or laser, and coherently distributed ultra-cold
bosonic atoms. The first one is responsible for the revival and most of the developments of
the mathematical formulation of the coherent states. In the case of electromagnetic radiation,
the vector potential, electric or magnetic fields as well as the intensity are completely
described by the creation and annihilation operators. The measured experimental observables
are the quadratures of the electric field at
𝜋
2
phase interval, which later are converted to the
normalized quadratures X and P.
Though the development of coherent states followed the path of the optical system, recent
technological breakthrough succeeded in creating samples of atoms with thermal de Broglie
wavelength comparable to the sample size, fulfilled at very low temperature and high
densities. A careful look at the periodic table reveals that most of the atoms in their ground
state are bosonic. During the last decade rapid progress has been made in the area of cooling
and trapping of atoms. By combination of different cooling and trapping techniques the atoms
could be kept at very low temperature (1-100 𝜇K)and high density (1010 − 1013 𝑐𝑚−3 ).
More recently, further developed techniques has been reported to achieve Bose Einstein
condensates (BEC) at densities 1012 − 1014 𝑐𝑚−3 and temperatures 10−8 − 10−7 Kelvin in
samples of rubidium, lithium and sodium atoms. At such low temperatures and high densities
the atoms loose their individual identity and behave like a single macroscopic quantum state
and the effect of quantum statistics becomes crucially important. Such ultra cold atomic
ensembles are expected to have properties similar to the role of lasers in conventional
coherent optics. Exploiting this analogy it is possible to express ultra cold samples as
coherent state of harmonic oscillator. The quadratures represent dimensionless position and
momentum of the atomic system.
10
3 Squeezed states:
The concept of squeezed states has been first proposed by Yuen, in the name of two photon
Coherent states or generalized coherent states,the specific term squeezing came later. The
states have been first defined in the context of light, but other systems have also been studied
,though not comparably. All the definitions match with other for the single mode optical
signal.However there is a discrepencies in the definition in other systems. The experimental
reliazition of squeezed state have reduced our uncertainty of observation to a non-classical
systems. The availability of squeezed state enabled us to apply it for the technical uses of
optical communication, interferometry, noise reduction and graviton detection as well as to
understand some fundamental questions involving quantum mechanics.
3.1 Definitions Of squeezed states :
The squeezed states can be defined in the following three ways.
Definition 1 : Two photon coherent state or generalized coherent state. Yuen has
defined the generalized coherent state or two photon coherent states|𝛽⟩ as eigenstates of the
operator 𝑏 = 𝜇𝑎 + 𝜈𝑎 †
𝑏|𝛽⟩ = 𝛽|𝛽⟩
where μ ,ν and β are in general complex numbers.
Definition 2: Squeezed vacuum. The squeezed states of a harmonic oscillator |𝛽⟩ =
|𝛼, 𝜉⟩ are obtained by operating the squeezing operator
1
2
𝑆(𝜉) = 𝑒𝑥𝑝 2 (𝜉 ∗ 𝑎2 − 𝜉𝑎† ), 𝜉 = 𝑟𝑒 𝑖𝜃 ,
on the coherent state |𝛼⟩. If it is operating on the vacuum state, which is a special case of the
coherent state with 𝛼 = 0, the result state is a squeezed vacuum. The Bogoliubov
transformation is generated by operating the unitary squeezing operator on the annihilation
and creation operator
𝑏 = 𝑆 −1 (𝜉)𝑎𝑆(𝜉),
𝑏 † = 𝑆 −1 (𝜉)𝑎† 𝑆(𝜉)
The quantities μ and ν used in the previous definition is related to the squeezing parameter ξ.
These are given by
𝜇 = cosh 𝑟 and 𝜈 = 𝑒 𝑖𝜙 sinh 𝑟 .
11
Definition 3: Uncertainty of any quadrature is less than the standard quantum
limit(SQL). The Squeezed states |𝛽⟩ are quantum states having uncertainty in one of the
quadratures less than that of the coherent or vacuum state i.e.,
∆𝑋 𝑜𝑟 ∆𝑃 <
1
2
,
But maintain the uncertainty relation. The first two definitions of squeezed states for the
harmonic oscillator qualify the third definition.
So far we know that coherent state is a minimum uncertainty state for which the two
quadrature uncertainties are equal. A generalized state, the squeezed state, can remain a
minimum uncertainty state if one variance is compressed at the expense of an expansion of
the complementary variance. The modified variances can be written as
1
〈(∆𝑋)2 〉 = exp(−2𝑟),
4
(3.1)
1
〈(∆𝑌)2 〉 = exp(2𝑟)
4
(3.2)
We will prove these relations later. When the squeezing parameter 𝑟 = 0, these give the
results for vacuum or coherent state, with phase insensitive equal variances. The squeezing
transformation can be written
𝑋 → 𝑋𝑠 = 𝑋𝑒𝑥𝑝(−𝑟),
𝑌 → 𝑌𝑠 = 𝑌𝑒𝑥𝑝(𝑟),
And the corresponding annihilation and creation operators obtained with the use of equations
1
𝑋 = 2 (𝑎† + 𝑎),
𝑖
𝑌 = 2 (𝑎† − 𝑎),
Are
𝑎𝑠 = 𝑎 cosh 𝑟 − 𝑎† sinh 𝑟,
𝑎𝑠† = 𝑎† cosh 𝑟 − 𝑎 sinh 𝑟
𝑖
These new operator preserve the commutation relations[𝑋𝑠 , 𝑌𝑠 ] = 2 and [𝑎𝑠 , 𝑎𝑠† ] = 1.
The transformed Hamiltonian is
1
𝐻 = (𝑎𝑠† 𝑎𝑠 + 2)ћ𝜔 ,
With the pseudo number states|𝑛𝑠 ⟩
Transformed coherent states may be defined analogously to equations
|𝛼⟩ = 𝐷(𝛼)|0⟩ ,
(3.3)
12
Where 𝐷(𝛼) = exp(𝛼𝑎† − 𝛼 ∗ 𝑎).
And through the action of a transformed Glauber displacement operator
|𝛼⟩ = 𝐷𝑠 (𝛼)|0𝑠 ⟩ ,
Where 𝐷𝑠 (𝛼) = exp(𝛼𝑎𝑠† − 𝛼 ∗ 𝑎𝑠 ).
The error contour in 𝑋𝑠 , 𝑌𝑠 space remains circular since the state defined in equation (3.3) is
a true coherent state in this space. However, in the original X, Y space, the exponentials in
equations (3.1) and (3.2) for s>0 compress the X variance and expand the Y variance to
produce an elliptical error contour. The squeezing condition for the X quadrature is that its
variance should be less than the vacuum value,
1
〈(∆𝑋)2 〉 < .
4
The state |0𝑠 ⟩ represents the squeezed vacuum state.
So far we have considered squeezing of a special kind in which the compression and
expansion of the canonical variables are oriented parallel to the X and Y axes respectively. A
more general quadratic generator of squeezed states is given by the unitary squeeze operator
1
2
𝑆(𝜉) = 𝑒𝑥𝑝 2 (𝜉 ∗ 𝑎2 − 𝜉𝑎† ),
(3.4)
Where ξ is the complex squeeze parameter
𝜉 = 𝑟𝑒 𝑖𝜃 ,0 ≤ 𝑟 ≤ ∞ , 0 ≤ 𝜃 ≤ 2𝜋 .
This more general transformation corresponds to compression and expansion of the canonical
variables in directions inclined at angles 𝜃⁄2 to the X and Y axes. The coherent-state error
circle
is now deformed into the squeezed-state error ellipse . The squeeze operator
transforms the annihilation and creation operators as
𝑆 † (𝜉)𝑎𝑆(𝜉) = 𝑎 cosh 𝑟 − 𝑎† exp(𝑖𝜃) sinh 𝑟
𝑆 † (𝜉)𝑎† 𝑆(𝜉) = 𝑎† cosh 𝑟 − 𝑎exp(−𝑖𝜃) sinh 𝑟.
(3.5)
Now we will prove these two relations and also the relations in equation (3.1) & (3.2)
Proof:
Taking Hermitian conjugate in equation (3.4) we have
1
1
2
2
𝑆 † (𝜉) = 𝑒𝑥𝑝 2 (𝜉𝑎† − 𝜉 ∗ 𝑎2 ) = exp(− 2 (𝜉 ∗ 𝑎2 − 𝜉𝑎† ) = 𝑆(−𝜉)),
So
(3.6)
𝑆 † (𝜉)𝑆(𝜉) = 1 , i.e. Squeeze operator is a unitary operator.
Now we have to calculate
1
1
2
2
𝑆 † (𝜉) 𝑎 𝑆(𝜉) = exp (− 2 (𝜉 ∗ 𝑎2 − 𝜉𝑎† )) 𝑎 exp 2 (𝜉 ∗ 𝑎2 − 𝜉𝑎† )
13
(3.7)
Using the identity 𝑒 𝛼𝐴 𝐵𝑒 −𝛼𝐴 = 𝐵 + 𝛼[𝐴, 𝐵] +
𝛼2
2!
[𝐴, [𝐴, 𝐵]] +
𝛼3
3!
[𝐴, [𝐴, [𝐴, 𝐵]]] + ⋯
(3.8)
we will able to calculate the above equation (3.7). Let us consider that
𝛼 = −1,
1
2
𝐴 = 2 (𝜉 ∗ 𝑎2 − 𝜉𝑎† ),
𝐵=𝑎,
1
1
2
1
1
2
[𝐴, 𝐵] = [ (𝜉 ∗ 𝑎2 − 𝜉𝑎† ), 𝑎] = [ (𝜉 ∗ 𝑎2 ), 𝑎] − [ (𝜉𝑎† ), 𝑎] = − 𝜉{𝑎† [𝑎† , 𝑎] +
2
2
2
2
Now
𝜉
[𝑎† , 𝑎]𝑎† } = 2𝑎 † = 𝜉𝑎 †
2
1
(3.9)
1
2
1
2
2
[𝐴, [𝐴, 𝐵]] = [ (𝜉 ∗ 𝑎2 − 𝜉𝑎† ), [ (𝜉 ∗ 𝑎2 − 𝜉𝑎† ), 𝑎]] = [ (𝜉 ∗ 𝑎2 − 𝜉𝑎† ), 𝜉𝑎 † ] =
2
2
1
1
2
1
2
[2 (𝜉 ∗ 𝑎2 ), 𝜉𝑎 † ] − [2 (𝜉𝑎† ), 𝜉𝑎 † ] = 2 𝜉 ∗ 𝜉{𝑎[𝑎, 𝑎† ] + [𝑎, 𝑎 † ]𝑎} =
Again
1
|𝜉|2
2
1
2
2𝑎 = |𝜉|2 𝑎 (3.10)
1
2
[𝐴, [𝐴, [𝐴, 𝐵]]] = [2 (𝜉 ∗ 𝑎2 − 𝜉𝑎† ), |𝜉|2 𝑎] = [2 (𝜉 ∗ 𝑎2 ), |𝜉|2 𝑎] − [2 (𝜉𝑎† ), |𝜉|2 𝑎] =
1
𝜉
− 2 𝜉|𝜉|2 {𝑎† [𝑎† , 𝑎] + [𝑎† , 𝑎]𝑎† } = 2 |𝜉|2 2𝑎† = |𝜉|2 𝜉𝑎†
(3.11)
Similarly [𝐴, [𝐴, [𝐴, [𝐴, 𝐵]]]] = |𝜉|4 𝑎
(3.12)
Using equation (3.8),(3.9),(3.10),(3.11),(3.12) we can write equation (3.7) as
1
1
2
2
𝑆 † (𝜉) 𝑎 𝑆(𝜉) = exp (− (𝜉 ∗ 𝑎2 − 𝜉𝑎† )) 𝑎 exp (𝜉 ∗ 𝑎2 − 𝜉𝑎† )
2
2
= 𝑎 + (−1)𝜉𝑎† +
(−1)2
2!
|𝜉|2 𝑎 +
𝜉 = 𝑟𝑒 𝑖𝜃
Putting
1
(−1)3 |𝜉|2 𝜉𝑎†
3!
+
in
(−1)4
4!
|𝜉|4 𝑎 + ⋯
equation
1
(3.13),
1
(3.13)
we
will
get
1
𝑆 † (𝜉) 𝑎 𝑆(𝜉) = 𝑎 − 𝑟𝑒 𝑖𝜃 𝑎† + 2! 𝑟 2 𝑎 − 3! 𝑟 3 𝑒 𝑖𝜃 𝑎† + 4! 𝑟 4 𝑎 − 5! 𝑟 5 𝑒 𝑖𝜃 𝑎† +
1
1
1
1
= 𝑎 (1 + 2! 𝑟 2 + 4! 𝑟 4 + ⋯ ) − 𝑒 𝑖𝜃 𝑎† (𝑟 + 3! 𝑟 3 + 5! 𝑟 5 + ⋯ )
=𝑎 cosh 𝑟 − 𝑎† 𝑒 𝑖𝜃 sinh 𝑟
So 𝑆 † (𝜉) 𝑎 𝑆(𝜉) = 𝑎 cosh 𝑟 − 𝑎 † 𝑒 𝑖𝜃 sinh 𝑟
(Proved)
(3.14)
Now taking Hermitian conjugate on both side of equation (3.14) the second relation will be
proved easily
(𝑆 † (𝜉) 𝑎 𝑆(𝜉))† = (𝑎 cosh 𝑟 − 𝑎† 𝑒 𝑖𝜃 sinh 𝑟)†
Or 𝑆 † (𝜉)𝑎† 𝑆(𝜉) = 𝑎† cosh 𝑟 − 𝑎exp(−𝑖𝜃) sinh 𝑟 (proved)
14
(3.15)
3.2 Quadrature uncertainties:
Here we calculate the quadrature uncertainties in terms of the squeezed state. The quadrature
operators are given by
1
𝑋 = 2 (𝑎† + 𝑎),
𝑖
𝑌 = 2 (𝑎† − 𝑎),
(3.16)
The variance of the of the quadrature operator X is given by
(∆𝑋)2 = ⟨𝜉|𝑋 2 |𝜉⟩ − ⟨𝜉|𝑋|𝜉⟩2
(3.17)
1
1
= ⟨𝜉|(2 (𝑎† + 𝑎))2|𝜉⟩ − ⟨𝜉| 2 (𝑎† + 𝑎)|𝜉⟩
1
= ⟨𝜉| 4 (𝑎
†2
2
1
+ 𝑎 + 𝑎𝑎 + 𝑎 𝑎)|𝜉⟩ − ⟨𝜉| 2 (𝑎† + 𝑎)|𝜉⟩
2
†
2
†
2
(3.18)
2
⟨𝜉|𝑎† |𝜉⟩ = ⟨0|𝑆 † 𝑎† 𝑆|0⟩ = ⟨0|𝑆 † 𝑎† 𝑆𝑆 † 𝑎† 𝑆|0⟩ = ⟨0|(𝑆 † 𝑎† 𝑆)2 |0⟩ =
Now
⟨0|(𝑎† cosh 𝑟 − 𝑎exp(−𝑖𝜃) sinh 𝑟)2|0⟩
(3.19)
For the calculation of previous equation we have used equation (3.15) and the following
relations
𝑆𝑆 † = 1 , |𝜉⟩ = 𝑆|0⟩ 𝑎𝑛𝑑
⟨𝜉| = ⟨0|𝑆 †
Now only the term ⟨0|−𝑎𝑎 † exp(−𝑖𝜃) sinh 𝑟 cosh 𝑟 |0⟩ in equation (3.19) will contribute
and the remaining terms will be vanished. Therefore the equation reduces to
2
⟨𝜉|𝑎† |𝜉⟩ = − exp(−𝑖𝜃) sinh 𝑟 cosh 𝑟 ⟨0|𝑎𝑎† |0⟩ =
− exp(−𝑖𝜃) sinh 𝑟 cosh 𝑟 (3.20)
⟨𝜉|𝑎2 |𝜉⟩ = ⟨0|𝑆 † 𝑎2 𝑆|0⟩ = ⟨0|𝑆 † 𝑎𝑆𝑆 † 𝑎𝑆|0⟩ = ⟨0|(𝑆 † 𝑎𝑆)2 |0⟩ =
⟨0|(𝑎 cosh 𝑟 − 𝑎 † 𝑒 𝑖𝜃 𝑠𝑖𝑛ℎ 𝑟) 2 |0⟩
(3.21)
Now only the term ⟨0|−𝑎𝑎 † exp(𝑖𝜃) sinh 𝑟 cosh 𝑟 |0⟩ in equation (3.21) will contribute and
the other terms will be vanished. So
⟨𝜉|𝑎2 |𝜉⟩ = − exp(𝑖𝜃) sinh 𝑟 cosh 𝑟 ⟨0|𝑎𝑎† |0⟩ = − exp(𝑖𝜃) sinh 𝑟 cosh 𝑟
(3.22)
⟨𝜉|𝑎† 𝑎|𝜉⟩ = ⟨0|𝑆 † 𝑎† 𝑎𝑆|0⟩ = ⟨0|𝑆 † 𝑎† 𝑆𝑆 † 𝑎𝑆|0⟩ =
⟨0|(𝑎† cosh 𝑟 − 𝑎exp(−𝑖𝜃) sinh 𝑟)(𝑎 cosh 𝑟 − 𝑎† 𝑒 𝑖𝜃 𝑠𝑖𝑛ℎ 𝑟) |0⟩
(3.23)
Now only the term ⟨0|𝑎𝑎† (sinh 𝑟) 2 |0⟩ in equation (3.21) will contribute and the other
terms will be vanished. So
sinh2 r⟨𝜉|𝑎† 𝑎|𝜉⟩ = ⟨0|𝑎𝑎† |0⟩ = sinh2 𝑟
(3.24)
Again,
15
⟨𝜉|𝑎𝑎 † |𝜉⟩ = ⟨0|𝑆 † 𝑎𝑎† 𝑆|0⟩ = ⟨0|𝑆 † 𝑎𝑆𝑆 † 𝑎† 𝑆|0⟩
⟨0|(𝑎 cosh 𝑟 − 𝑎† 𝑒 𝑖𝜃 𝑠𝑖𝑛ℎ 𝑟)(𝑎† cosh 𝑟 − 𝑎exp(−𝑖𝜃) sinh 𝑟) |0⟩
=
= 𝑐𝑜𝑠ℎ2 𝑟⟨0|𝑎𝑎† |0⟩ = cosh 2 𝑟
2
1
(3.25)
2
1
Now ⟨𝜉|𝑋|𝜉⟩2 = ⟨𝜉| 2 (𝑎† + 𝑎)|𝜉⟩ = ⟨0|𝑆 † 2 (𝑎† + 𝑎)𝑆|0⟩ = 0
(3.26)
Using equation (3.20), (3.22), (3.24), (3.25), (3.26), we get from equation (3.18),
〈(∆𝑋)2 〉 = ⟨𝜉|𝑋 2 |𝜉⟩ − ⟨𝜉|𝑋|𝜉⟩2
1
= 4 {𝑐𝑜𝑠ℎ2 𝑟 + 𝑠𝑖𝑛ℎ2 𝑟 − exp(𝑖𝜃) sinh 𝑟 cosh 𝑟 − exp(−𝑖𝜃) sinh 𝑟 cosh 𝑟}
1
= 4 {𝑐𝑜𝑠ℎ2 𝑟 + 𝑠𝑖𝑛ℎ2 𝑟 − sinh 𝑟 cosh 𝑟(exp(𝑖𝜃) + exp(−𝑖𝜃)}
1
= {𝑐𝑜𝑠ℎ2 𝑟 + 𝑠𝑖𝑛ℎ2 𝑟 − 2sinh 𝑟 cosh 𝑟 cos 𝜃}
(3.27)
4
Similarly the variance of the quadrature operator Y is given by
〈(∆𝑌)2 〉 = ⟨𝜉|𝑌 2 |𝜉⟩ − ⟨𝜉|𝑌|𝜉⟩2
𝑖
𝑖
= ⟨𝜉|(2 (𝑎† − 𝑎))2|𝜉⟩ − ⟨𝜉| 2 (𝑎† − 𝑎)|𝜉⟩
1
2
𝑖
2
= ⟨𝜉|− 4 (𝑎† + 𝑎2 − 𝑎𝑎† − 𝑎† 𝑎)|𝜉⟩ − ⟨𝜉| 2 (𝑎† − 𝑎)|𝜉⟩
2
Using equation (3.20), (3.22), (3.24), (3.25), (3.26) we get
1
{〈(∆𝑌)2 〉 = − {− exp(𝑖𝜃) sinh 𝑟 cosh 𝑟 − exp(−𝑖𝜃) sinh 𝑟 cosh 𝑟 − 𝑐𝑜𝑠ℎ2 𝑟 − 𝑠𝑖𝑛ℎ2 𝑟}
4
1
= 4 {𝑐𝑜𝑠ℎ2 𝑟 + 𝑠𝑖𝑛ℎ2 𝑟 + sinh 𝑟 cosh 𝑟(exp(𝑖𝜃) + exp(−𝑖𝜃)}
1
= 4 {𝑐𝑜𝑠ℎ2 𝑟 + 𝑠𝑖𝑛ℎ2 𝑟 + 2sinh 𝑟 cosh 𝑟 cos 𝜃}
So the variances of quadrature operators are given by
1
〈(∆𝑋)2 〉 = {𝑐𝑜𝑠ℎ2 𝑟 + 𝑠𝑖𝑛ℎ2 𝑟 − 2sinh 𝑟 cosh 𝑟 cos 𝜃}
4
1
〈(∆𝑌)2 〉 = {𝑐𝑜𝑠ℎ2 𝑟 + 𝑠𝑖𝑛ℎ2 𝑟 + 2sinh 𝑟 cosh 𝑟 cos 𝜃}
4
3.3 Phase space diagram of squeezed states:
16
(3.28)
The variances are independent of the coherent amplitude α and they can be represented
pictorially by the uncertainty ellipses . These ellipses can be defined in terms of contours of
the squeezed-state Wigner function , but we use here a simple definition that is illustrated in
figure1. With an ellipse having the sizes and orientations of major and minor axes shown, a
simple application of elliptical theory shows that the projections of the ellipse on to the X and
Y axes are equal to the square roots of the corresponding variances.
Y
1
〈(∆𝑌)2 〉2
X
1
〈(∆𝑥)2 〉2
Figure 1: Uncertainty ellipse of the ideal squeezed vacuum state showing how 𝑟 𝑎𝑛𝑑 𝜃
determine the axis lengths and orientation, and with the quadrature operator variances
obtained from projections on to the X and Y axes.
3.3 Detection of squeezed states:
17
3.3.1 Ordinary homodyne detection:
Detector2
Balanced H.D
Ordinary H.D.
Detector 1
signal, 𝑎
B.S
.
b
Local oscillator
Figure2: Schematic arrangement of a homodyne detector
Here are many theoretical predictions where the squeezed state is obtained. Now we would
like to discuss the possibility of detecting the squeezed state in a laboratory. In this discussion
we will consider the ordinary homodyne detection technique.
Figure 2 shows the schematic arrangement of a homodyne detector. The input signal mode
with operator 𝑎 is superposed on a local oscillator mode with operator 𝑏 by a lossless
symmetric beam-splitter, with reflectivity R and transmittivity T such that 𝑅 + 𝑇 = 1. Then
denoting the out modes reaching photo detectors D1 and D2 by 𝑐 𝑎𝑛𝑑 𝑏 respectively.
Therefore we have
𝑐 = √𝑇𝑎 + 𝑖√1 − 𝑇𝑏
𝑑 = 𝑖√1 − 𝑇𝑎 + √𝑇𝑏
(3.29)
The quantum theory of homodyne detection was originated principally by Yuen and Shapiro.
Two basic varieties of detector are capable in principle of measuring quantum effects in the
input signal noise. In ordinary homodyne detection, the beam-splitter coefficients satisfy
𝑇≫𝑅
(ordinary)
(3.30)
.
and only the output current from photo detector 1 is utilized . In balanced homodyne
detection, the beam-splitter coefficients satisfy
1
𝑅=𝑇=2
(balanced)
And the output current for processing is formed from the difference of the two photo detector
currents.
18
We will consider ordinary homodyne detection first, where the measured signal is determined
by the operator
𝑐 † 𝑐 = (√𝑇𝑎† − 𝑖√1 − 𝑇𝑏 † )(√𝑇𝑎 + 𝑖√1 − 𝑇𝑏)
= 𝑇𝑎† 𝑎 + (1 − 𝑇)𝑏 † 𝑏 + 𝑖√𝑇(1 − 𝑇)(𝑎† 𝑏 − 𝑏 † 𝑎)
(3.31)
Similarly the measured signal in photo detector 2 determined by the operator
𝑑 † 𝑑 = (−𝑖√1 − 𝑇𝑎† + √𝑇𝑏 † )(𝑖√1 − 𝑇𝑎 + √𝑇𝑏)
= (1 − 𝑇)𝑎† 𝑎 + 𝑇𝑏 † 𝑏 − 𝑖√𝑇(1 − 𝑇)(𝑎† 𝑏 − 𝑏 † 𝑎)
(3.32)
The input and local oscillator modes have the same frequency in homodyne detection, and the
above operators are therefore time independent. The local oscillator light is excited into a
large amplitude coherent state |𝛽𝑙 〉 with phase 𝜙𝑙 .
So the operator expectation value is
〈𝑐 † 𝑐〉 = 𝑇〈𝑎 † 𝑎〉 + (1 − 𝑇)|𝛽𝑙 |2 − 2√𝑇(1 − 𝑇)𝛽𝑙 〈𝜒(𝜙𝑙 + 𝜋⁄2)〉
Where
1
𝜒(𝜙) = 2 (𝑎𝑒 −𝑖𝜙 + 𝑎† 𝑒 𝑖𝜙 ), and
(3.33)
(3.34)
〈𝑏〉 = |𝛽𝑙 |𝑒 𝑖𝜙𝑙
𝑏|𝛽𝑙 〉 = 𝛽𝑙 |𝛽𝑙 〉 , scince local oscillator signal is in coherent state.
Now
𝜒(𝜙𝑙 + 𝜋⁄2) =
1
2
=
=
(𝑎𝑒 −𝑖(𝜙𝑙+
1
2
𝑖
2
𝜋⁄ )
2
+ 𝑎 † 𝑒 𝑖(𝜙𝑙 +
𝜋⁄ )
2 ),
(−𝑖𝑎𝑒 −𝑖𝜙𝑙 + 𝑖𝑎 † 𝑒 𝑖𝜙𝑙 ),
(𝑎† 𝑒 𝑖𝜙𝑙 − 𝑎𝑒 −𝑖𝜙𝑙 )
So 〈𝑐 † 𝑐〉 = 𝑇〈𝑎† 𝑎〉 + (1 − 𝑇)|𝛽𝑙 |2 − 2√𝑇(1 − 𝑇)𝛽𝑙 〈𝜒(𝜙𝑙 + 𝜋⁄2)〉
We see that the output signal 𝑐 † 𝑐 contains the transmitted part of the input photon and
reflected part of the local oscillator field, most importantly an interference between the input
field and local oscillator signal. It is precisely the interference term that contains a quadrature
of the input field depending upon the phase of the local oscillator.
In the case of a ordinary homodyne detection normally a strong local oscillator is used that
satisfies the condition
√𝑅 |𝛽𝑙 | ≫ √𝑡|𝛼| ,
(3.35)
Where 𝛼 is the input signal complex amplitude. The inequalities (3.30) and (3.35) taken
together imply that almost all of the input light reaches the photo detector, but even the small
fraction of local-oscillator light reaching the detector still dominates the input contribution .
19
Therefore we have
𝑇〈𝑎† 𝑎〉 ≪ (1 − 𝑇)|𝛽𝑙 |2
Therefore we have only the two term.
〈𝑐 † 𝑐〉 = (1 − 𝑇)|𝛽𝑙 |2 − 2√𝑇(1 − 𝑇)𝛽𝑙 〈𝜒(𝜙𝑙 + 𝜋⁄2)〉
(3.36)
The 1st term of the right hand side constitutes a known constant value which can be
subtracted from the signal and the remaining signal constructs the quadrature of the input.
Parametric interaction:
5. Hamiltonian formulation of parametric interaction:
20
𝐸1 (𝑟, 𝑡), 𝐸2 (𝑟, 𝑡) 𝑎𝑛𝑑 𝐸3 (𝑟, 𝑡)𝑎𝑡𝜔1 , 𝜔2 𝑎𝑛𝑑 𝜔3 ,
Let us consider three interacting fields
respectively, where
𝜔3 = 𝜔1 + 𝜔2
The Field interaction energy density is given by
2
𝑈 = − 3 𝑑𝐸1 𝐸2 𝐸3
(4.1)
where 𝑑 is an effective nonlinear coefficient involving the third order nonlinear
susceptibility.
So the interaction Hamiltonian is given by
2
𝐻1 = ∫ 𝑈 𝑑𝑣 = ∫ − 3 𝑑𝐸1 𝐸2 𝐸3 𝑑𝑣
(4.2)
Since all of the parametric amplification and oscillation utilize an intense pump, we consider
pump field 𝐸3 (𝑟, 𝑡) as classical one. We quantize the signal and idler fields by replacing
𝐸1 𝑎𝑛𝑑 𝐸2 by their corresponding quantum mechanical operators in the following manner.
ћ𝜔
1⁄
2
𝐸1 (𝑟𝑡) = −𝑖 ( 2𝜖1 )
𝐸1 (𝑟)[𝑎1† (𝑡) − 𝑎1 (𝑡)]
1⁄
2
ћ𝜔
𝐸2 (𝑟𝑡) = −𝑖 ( 2𝜖2 )
(4.3)
𝐸2 (𝑟)[𝑎2† (𝑡) − 𝑎2 (𝑡)]
(4.4)
where 𝑎𝑖 (𝑡) 𝑎𝑛𝑑 𝑎𝑖† (𝑡) are annihilation and creation operator respectively of the i-th mode.
Now the classical pump field is given by
𝐸3 (𝑟, 𝑡) = 𝐵3 𝐸3 (𝑟) cos 𝜔3 𝑡
(4.5)
where 𝐵3, is the amplitude of the classical pump.
Hence
𝑑𝐵3 ћ
3
interaction
𝜔1 𝜔2
√𝜖
1 𝜖2
Hamiltonian
assumes
the
form
2
𝐻1 = ∫ − 3 𝑑𝐸1 𝐸2 𝐸3 𝑑𝑣 =
†
†
∫𝑣 𝐸1 (𝑟)𝐸2 (𝑟) 𝐸3 (𝑟)[𝑎1 (𝑡) − 𝑎1 (𝑡)][𝑎2 (𝑡) −
𝑎2 (𝑡)] cos 𝜔3 𝑡 𝑑𝑣
= 2𝑠ћ cos 𝜔3 𝑡 [𝑎1† (𝑡) − 𝑎1 (𝑡)][𝑎2† (𝑡) − 𝑎2 (𝑡)]
where 𝑠 =
𝑑𝐵3
6
𝜔1 𝜔2
√𝜖
1 𝜖2
(4.6)
∫ 𝐸1 (𝑟)𝐸2 (𝑟)𝐸3 (𝑟)𝑑𝑣 is termed as interaction parameter.
Therefore the total Hamiltonian is the sum of the unperturbed(s=0) Hamiltonian and the
interaction
Hamiltonian
and
is
given
by
1
1
𝐻 = ћ𝜔1 (𝑎1† 𝑎1 + 2) + ћ𝜔2 (𝑎2† 𝑎2 + 2) + 2𝑠ћ cos 𝜔3 𝑡 [𝑎1† (𝑡) − 𝑎1 (𝑡)][𝑎2† (𝑡) − 𝑎2 (𝑡)] (4.7)
21
6. Generation of squeezed state of the electromagnetic field by
degenerate parametric interaction:
The quantum mechanical operators that correspond to the amplitudes of the two quadratures
of the field, called quadrature phase amplitudes. They are noncommuting Hermitian
operators. The two quadrature phase amplitudes thus obey an uncertainty relation due to
Heisenberg. When the field is in a vacuum state or in a coherent state, the uncertainty product
for the two quadrature amplitudes is a minimum. The uncertainties in the two quadratures
are equal for coherent state and for the ground state of the harmonic oscillator. On the other
hand, if the field is in a squeezed state, the variance of one quadrature is less than that of the
vacuum or coherent state.
In order to use the squeezed states in, for example, a communication system, we can
modulate the amplitude of the squeezed quadrature component with the contents of some
information bearing waveform. On the receving side of the channel, a phasesensitive detector
responding only to the squeezed qudrature component can recover the signal with a signal-tonoise ratio in excess of that corresponding to full shot noise. In what follows we show that the
output of a parametric amplifier is a squeezed state.
The
electric
field
1
ћ𝜔 ⁄2
𝐸𝑠 (𝑟𝑡) = −𝑖 ( 2𝜖 )
for
a
single-mode
electromagnetic
field
is
𝐸𝑠 (𝑟)[𝑎† (𝑡) − 𝑎(𝑡)]
given
by,
(5.1)
Here we consider the degenerate parametric amplifier i.e. the frequency of the input signal
and idler are equal(𝜔1= 𝜔2 ).The pump field is given by,
𝐸𝑝 (𝑟𝑡) = 𝐵3 𝐸3 (𝑟) cos(𝜔3 𝑡 + 2𝜃) = 𝐵3 𝐸3 (𝑟) sin(2𝜔𝑡 + 2𝜃 + 𝜋⁄2) = 𝐵3 𝐸3 (𝑟)𝑠𝑖𝑛𝜙𝑝 (𝑡)
(5.2)
The field Hamiltonian is
1
1
𝐻 = ћ𝜔 (𝑎† 𝑎 + 2) + ћ𝜔 (𝑎 † 𝑎 + 2) + 2𝑠ћcos(2𝜔𝑡 + 2𝜃)(𝑎† − 𝑎)2
(5.3)
Where, s is called interaction parameter.
We choose that 𝜃 = 0. So the Hamiltonian in equation (5.3) modifies as,
1
𝐻 = ћ𝜔 (𝑎† 𝑎 + 2) + 𝑠ћcos(2𝜔𝑡)(𝑎† − 𝑎)2
Now the equation of motion for the field operator 𝑎 is
𝑑𝑎
𝑑𝑡
𝑖
𝑖
1
= − ћ [𝑎, 𝐻] = − ћ [𝑎, ћ𝜔 (𝑎† 𝑎 + 2) + 𝑠ћcos(2𝜔𝑡)(𝑎† − 𝑎)2 ]
𝑖
1
= − ћ [[𝑎, ћ𝜔 (𝑎† 𝑎 + 2)] + [𝑎, 𝑠ћ cos(2𝜔𝑡) (𝑎† − 𝑎)2 ]]
22
(5.4)
𝑖
ћ𝜔
ћ
2
= − {[𝑎, ћ𝜔𝑎† 𝑎] + [𝑎,
2
] + 𝑠ћ cos(2𝜔𝑡) {[𝑎, 𝑎† ] + [𝑎, 𝑎2 ] − [𝑎, 𝑎𝑎† ] −
[𝑎, 𝑎† 𝑎]}}
𝑖
= − ћ {ћ𝜔𝑎 + 𝑠ћ cos(2𝜔𝑡) (2𝑎† − 𝑎 − 𝑎)}
= −𝑖𝜔𝑎 − 2𝑖𝑠 cos 2𝜔𝑡 (𝑎† − 𝑎)
Hence,
𝑑𝑎
= −𝑖𝜔𝑎 − 2𝑖𝑠 cos 2𝜔𝑡 (𝑎† − 𝑎)
𝑑𝑡
(5.5)
Taking Hermitian conjugate of equation (5.5) we have,
𝑑𝑎†
𝑑𝑡
= 𝑖𝜔𝑎† − 2𝑖𝑠 cos 2𝜔𝑡 (𝑎† − 𝑎)
(5.6)
In the absence of interaction (i.e., 𝑠 = 0) between the signal and the pump modes, the
Heisenberg equations of motion (5.5) and (5.6) takes the simple form
𝑑𝑎
𝑑𝑡
= −𝑖𝜔𝑎
𝑑𝑎†
𝑑𝑡
(5.7)
= 𝑖𝜔𝑎†
(5.8)
The solutions of equation (5.7) and (5.8) give the free evolution of 𝑎 𝑎𝑛𝑑 𝑎† and are given by
𝑎(𝑡) = 𝑎(0)𝑒 −𝑖𝜔𝑡
(5.9)
𝑎† (𝑡) = 𝑎† (0)𝑒 𝑖𝜔𝑡
(5.10)
By dropping the non synchronous terns from equations (5.5) & (5.6), we obtain
𝑑𝑎
𝑑𝑡
= −𝑖𝜔𝑎 − 𝑖𝑠𝑎† 𝑒 −𝑖2𝜔𝑡
𝑑𝑎†
𝑑𝑡
(5.11)
= 𝑖𝜔𝑎† − 𝑖𝑠𝑎𝑒 𝑖2𝜔𝑡
(5.12)
In order to get the solution of equation (5.11) & (5.12), we have to decouple these two
equations. Before doing this, we define a quadrature phase amplitude operators in the
following way,
𝜋
1
𝜋
𝑋1 (𝑡) = 2 [𝑎(𝑡)𝑒 𝑖(𝜔𝑡+ 4 ) + 𝑎 † (𝑡)𝑒 −𝑖(𝜔𝑡+ 4 ) ]
𝑖
𝜋
(5.13)
𝜋
𝑋2 (𝑡) = − 2 [𝑎(𝑡)𝑒 𝑖(𝜔𝑡+ 4 ) − 𝑎† (𝑡)𝑒 −𝑖(𝜔𝑡+ 4 ) ]
(5.14)
Solving above two equations we obtain,
𝜋
𝑎(𝑡) = [𝑋1 (𝑡) + 𝑖𝑋2 (𝑡)]𝑒 −𝑖(𝜔𝑡+ 4 )
(5.15)
𝜋
𝑎† (𝑡) = [𝑋1 (𝑡) − 𝑖𝑋2 (𝑡)]𝑒 𝑖(𝜔𝑡+ 4 )
23
(5.16)
Substituting equation (5.15) and (5.16) into equation (5.1), we can write the signal electric
field operator as
ћ𝜔
1⁄
2
ћ𝜔
1⁄
2
𝐸𝑠 (𝑟, 𝑡) = −𝑖 ( 2𝜖 )
= −𝑖 ( 2𝜖 )
𝐸𝑠 (𝑟, 𝑡) = (
2ћ𝜔
𝜖
1⁄
2
)
𝜋
𝜋
𝐸𝑠 (𝑟){[𝑋1 (𝑡) − 𝑖𝑋2 (𝑡)]𝑒 𝑖(𝜔𝑡+ 4 ) − [𝑋1 (𝑡) + 𝑖𝑋2 (𝑡)]𝑒 −𝑖(𝜔𝑡+ 4 ) ]}
𝜋
𝜋
𝐸𝑠 (𝑟)[2𝑖𝑋1 (𝑡) sin (𝜔𝑡 + 4 ) − 2𝑖𝑋2 (𝑡)cos(𝜔𝑡 + 4 )]
𝜋
𝜋
𝐸𝑠 (𝑟)[𝑋1 (𝑡) sin (𝜔𝑡 + 4 ) − 𝑋2 (𝑡)cos(𝜔𝑡 + 4 )]
(5.17)
𝑋1 (𝑡) is the operator corresponding to the amplitude in phase component, that is the one in
phase with a field 𝐸0 𝑠𝑖𝑛 [𝜙𝑝 (𝑡)⁄2], Where 𝜙𝑝 (𝑡) = 2𝜔𝑡 + 2𝜃 + 𝜋⁄2. 𝑋2 (𝑡), is thus the
amplitude of the out of phase amplitude.
By combining equation (5.11), (5.12) with equation (5.13) we obtain
𝑑𝑋1 (𝑡)
𝑑𝑡
𝜋
1 𝑑𝑎
𝜋
= 2 [ 𝑑𝑡 𝑒 𝑖(𝜔𝑡+ 4 ) + 𝑎(𝑡)(𝑖𝜔)𝑒 𝑖(𝜔𝑡+ 4 ) +
𝑑𝑎†
𝑑𝑡
𝜋
𝜋
1
𝜋
𝑒 −𝑖(𝜔𝑡+ 4 ) + 𝑎† (−𝑖𝜔)𝑒 −𝑖(𝜔𝑡+ 4 ) ]
𝜋
= 2 [(−𝑖𝜔𝑎 − 𝑖𝑠𝑎† 𝑒 −𝑖2𝜔𝑡 )𝑒 𝑖(𝜔𝑡+ 4 ) + 𝑎(𝑡)(𝑖𝜔)𝑒 𝑖(𝜔𝑡+ 4 ) + (𝑖𝜔𝑎† −
𝜋
𝜋
𝑖𝑠𝑎𝑒 𝑖2𝜔𝑡 )𝑒 −𝑖(𝜔𝑡+ 4 ) + 𝑎† (−𝑖𝜔)𝑒 −𝑖(𝜔𝑡+ 4 ) ]
=
𝜋
𝑖𝑠
=
=
𝜋
[(𝑎𝑒 𝑖(𝜔𝑡− 4 ) − 𝑎† 𝑒 −𝑖(𝜔𝑡− 4 ) ]
2
𝜋
𝑖𝑠
𝜋
𝜋
𝜋
[𝑎𝑒 𝑖(𝜔𝑡+ 4 ) 𝑒 −𝑖 2 − 𝑎† 𝑒 −𝑖(𝜔𝑡+ 4 ) 𝑒 𝑖 2 ]
2
𝜋
𝑖𝑠
𝜋
[−𝑖𝑎𝑒 𝑖(𝜔𝑡+ 4 ) − 𝑖𝑎† 𝑒 −𝑖(𝜔𝑡+ 4 ) ]
2
𝜋
𝑠
𝜋
= 2 [𝑎𝑒 𝑖(𝜔𝑡+ 4 ) + 𝑎† 𝑒 −𝑖(𝜔𝑡+ 4 ) ]
= 𝑠𝑋1 (𝑡)
(5.18)
Similarly combining equation (5.11), (5.12) with equation (5.14) we can get
𝑑𝑋2 (𝑡)
𝑑𝑡
= −𝑠𝑋2 (𝑡)
(5.19)
The solutions of Equation (5.18) & (5.19), are respectively
𝑋1 (𝑡) = 𝑋1 (0)𝑒 𝑠𝑡
(5.20)
𝑋2 (𝑡) = 𝑋2 (0)𝑒 −𝑠𝑡
(5.21)
From the above two equation we can say that a degenerate parametric amplifier will amplify
the in phase input field (for s>0) and attenuate the out- phase component.
24
Now inserting equations (5.20) & (5.21) in to the equation (5.15) we get
𝜋
𝑎(𝑡) = [𝑋1 (0)𝑒 𝑠𝑡 + 𝑖𝑋2 (0)𝑒 −𝑠𝑡 ]𝑒 −𝑖(𝜔𝑡+ 4 )
𝜋
1
𝜋
𝜋
𝑖
𝜋
𝜋
= [2 [𝑎(0)𝑒 𝑖( 4 ) + 𝑎† (0)𝑒 −𝑖( 4 ) ] 𝑒 𝑠𝑡 + 𝑖(− 2)[𝑎(0)𝑒 𝑖( 4 ) − 𝑎† (0)𝑒 −𝑖( 4 ) ]𝑒 −𝑠𝑡 ]𝑒 −𝑖(𝜔𝑡+ 4 )
𝜋
1
𝜋
𝜋
= 2 [𝑎(0)𝑒 𝑖( 4 ) (𝑒 𝑠𝑡 + 𝑒 −𝑠𝑡 ) + 𝑎† (0)𝑒 −𝑖( 4 ) (𝑒 𝑠𝑡 −𝑒 −𝑠𝑡 )]𝑒 −𝑖(𝜔𝑡+ 4 )
= [𝑎(0)𝑐𝑜𝑠ℎ𝑠𝑡 − 𝑖 𝑎† (0)𝑠𝑖𝑛ℎ𝑠𝑡]𝑒 −𝑖𝜔𝑡
(5.22)
So the annihilation operator can be written as
𝑎(𝑡) = 𝜇𝑎(0) + 𝜈𝑎† (0)
Again taking Hermitian conjugate in equation (5.22) we can get
𝑎† (𝑡) = [𝑎† (0)𝑐𝑜𝑠ℎ𝑠𝑡 + 𝑖 𝑎(0)𝑠𝑖𝑛ℎ𝑠𝑡]𝑒 𝑖𝜔𝑡
or,
(5.23)
𝑎† (𝑡) = 𝜇 ∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0)
where, 𝜇 = 𝑒 −𝑖𝜔𝑡 cosh 𝑠𝑡 & 𝜈 = −𝑖𝑒 −𝑖𝜔𝑡 sinh 𝑠𝑡 with |𝜇|2 − |𝜈|2 = 1.
Now we will calculate the uncertainties of the quadratures 𝑋1 (𝑡) 𝑎𝑛𝑑 𝑋2 (𝑡) in terms of the
initial coherent state. The corresponding variances are given by
〈(Δ𝑋1 )2 〉 = ⟨𝛼|𝑋1 2 (𝑡)|𝛼⟩ − ⟨𝛼|𝑋1 (𝑡)|𝛼⟩2
Or, 〈(Δ𝑋1 )2 〉 = ⟨𝛼|𝑋1 2 (0)|𝛼⟩𝑒 2𝑠𝑡 − ⟨𝛼|𝑋1 (0)|𝛼⟩2 𝑒 2𝑠𝑡
And
Or
(5.24)
〈(Δ𝑋2 (𝑡))2 〉 = ⟨𝛼|𝑋2 2 (𝑡)|𝛼⟩ − ⟨𝛼|𝑋2 (𝑡)|𝛼⟩2
〈(Δ𝑋2 (𝑡))2 〉 = ⟨𝛼|𝑋2 2 (0)|𝛼⟩𝑒 −2𝑠𝑡 − ⟨𝛼|𝑋2 (0)|𝛼⟩2 𝑒 −2𝑠𝑡
(5.25)
From equation (5.13) & (5.14) we have
𝜋
1
𝜋
𝑋1 (0) = 2 [𝑎(0)𝑒 𝑖( 4 ) + 𝑎† (0)𝑒 −𝑖( 4 ) ]
𝜋
𝑖
(5.26)
𝜋
𝑋2 (0) = − [𝑎(0)𝑒 𝑖( 4 ) − 𝑎† (0)𝑒 −𝑖( 4 ) ]
2
(5.27)
Now,
𝜋
1
𝜋
2
𝑋1 2 (0) = 4 [𝑎2 (0)𝑒 𝑖( 2 ) + 𝑎† (0)𝑒 −𝑖( 2 ) + 2𝑎† (0)𝑎(0) + 1}
1
𝜋
2
𝜋
𝑋2 2 (0) = 4 [𝑎2 (0)𝑒 𝑖( 2 ) + 𝑎† (0)𝑒 −𝑖( 2 ) − 2𝑎 † (0)𝑎(0) − 1]
So combining equation (5.26) & (5.28), with equation (5.24), we have
25
(5.28)
(5.29)
〈(Δ𝑋1 )2〉 = ⟨𝛼|𝑋1 2 (0)|𝛼⟩𝑒 2𝑠𝑡 − ⟨𝛼|𝑋1 (0)|𝛼⟩2 𝑒 2𝑠𝑡
1
⟨𝛼| 4 [𝑎2 (0)𝑒
𝜋
𝑖( )
2
†2
+ 𝑎 (0)𝑒
𝜋
−𝑖( )
2
𝜋
1
=
+ 2𝑎† (0)𝑎(0) + 1|𝛼⟩ 𝑒 2𝑠𝑡 −
2
𝜋
⟨𝛼| 2 [𝑎(0)𝑒 𝑖( 4 ) + 𝑎† (0)𝑒 −𝑖( 4 ) ]|𝛼⟩ 𝑒 2𝑠𝑡
1
𝜋
𝜋
1
𝜋
𝜋
= 4 [𝛼 2 𝑒 𝑖( 2 ) + 𝛼 ∗ 2 𝑒 −𝑖( 2 ) + 2|𝛼|2 + 1] 𝑒 2𝑠𝑡 − [2 (𝛼𝑒 𝑖( 4 ) + 𝛼 ∗ 𝑒 −𝑖( 2 ) ]2 𝑒 2𝑠𝑡
1
= 4 𝑒 2𝑠𝑡
Similarly, combining equation (5.27) & (5.29), with equation (5.25), we have
1
〈(Δ𝑋2 )2 〉 = 𝑒 −2𝑠𝑡
4
The output of a degenerate parametric amplifier with a coherent state input is squeezed, as is
seen by examining equations (5.20) & (5.21). For such an input the quadrature phase
uncertainties at the output are
1
1
〈(Δ𝑋1 )2〉2 = 𝑒 𝑠𝑡
2
1
(5.30)
1
〈(Δ𝑋2 )2 〉2 = 𝑒 −𝑠𝑡
2
(5.31)
From the above two equation we see that, the quadrature X1 is squeezed if 𝑠 < 0 and the
quadrature X2 is squeezed if 𝑠 > 0 .
6. Higher order squeezing:
Let we define the amplitude squared squeezing in terms of operator 𝑌1 𝑎𝑛𝑑 𝑌2 as
2
1
𝑌1 = 2 (𝐴2 + 𝐴† )
2
𝑖
𝑌2 = − 2 (𝐴2 − 𝐴† )
26
(6.1)
(6.2)
where, 𝐴 = 𝑎(𝑡)𝑒 𝑖𝜔𝑡 .
The uncertainty relationship of the above two quadrature operator is given by
1
∆𝑌1 . ∆𝑌2 ≥ 〈(𝑁 + 2)〉
(6.3)
The quadrature 𝑌1 will exhibit amplitude squared squeezing when,
1
(∆𝑌1 )2 < 〈(𝑁 + )〉
2
(6.4)
On the other hand 𝑌2 squeezing is possible for
1
(∆𝑌2 )2 < 〈(𝑁 + )〉
2
Now we will calculate the commutation relation between 𝑌1 𝑎𝑛𝑑 𝑌2 .
2
𝑖
2
[𝑌1 , 𝑌2 ] = − [(𝐴2 + 𝐴† ) , (𝐴2 − 𝐴† )]
4
2
𝑖
2
2
2
= − 4 [[𝐴2 , 𝐴2 ] − [𝐴2 , 𝐴† ] + [𝐴† , 𝐴2 ] − [𝐴† , 𝐴† ]]
2
𝑖
2
= − 4 [− [𝐴2 , 𝐴† ] + [𝐴† , 𝐴2 ]]
2
𝑖
= 2 [𝐴2 , 𝐴† ]
2
𝑖
2
= 2 [𝐴 [𝐴𝐴† ] + [𝐴, 𝐴† ] 𝐴]
𝑖
= 2 [2𝐴𝐴† + 2𝐴† 𝐴]
= 𝑖[𝐴† 𝐴 + 1 + 𝐴† 𝐴]
= 𝑖[2𝐴† 𝐴 + 1]
̂ + 1]
= 𝑖[2𝑁
(6.5)
In the context of degenerate parametric interaction we established that the output quadrature
1
𝑖
2
2
amplitude X1 ,(𝑋1 = (𝑎 + 𝑎 † )), is squeezed if 𝑠 < 0 and the quadrature X2 (𝑋2 = − (𝑎 −
𝑎† )) is squeezed if 𝑠 > 0 . Now we investigate the second order squeezing condition for
the quadrature components of the degenerate parametric interaction. In order to find out the
̂ + 1) in terms
condition, we calculate the variances of 𝑌1 , 𝑌2 and the expectation value of (𝑁
2
of the initial coherent state.
The final solution of the degenerate parametric interaction is already available in equation
(5.22) & (5.23). we have,
27
𝑎(𝑡) = [𝑎(0)𝑐𝑜𝑠ℎ𝑠𝑡 − 𝑖 𝑎† (0)𝑠𝑖𝑛ℎ𝑠𝑡]𝑒 −𝑖𝜔𝑡
where, 𝑎(𝑡) is the annihilation operator. The operators 𝑎(𝑡) 𝑎𝑛𝑑 𝑎† (𝑡)
following forms
or,
assumes the
𝑎(𝑡) = (𝜇𝑎(0) + 𝜈𝑎† (0))𝑒 −𝑖𝜔𝑡
(6.6)
𝑎† (𝑡) = (𝜇 ∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0))𝑒 𝑖𝜔𝑡
(6.7)
where, 𝜇 = 𝑐𝑜𝑠ℎ𝑠𝑡 & 𝜈 = −𝑖𝑠𝑖𝑛ℎ𝑠𝑡.
Now with the combination of equation (6.6) & (6.7) 𝐴 𝑎𝑛𝑑 𝐴† assume the form
𝐴 = 𝑎(𝑡)𝑒 𝑖𝜔𝑡 = [𝜇𝑎(0) + 𝜈𝑎† (0)]𝑒 −𝑖𝜔𝑡 𝑒 𝑖𝜔𝑡 = [𝜇𝑎(0) + 𝜈𝑎† (0)]
(6.8)
𝐴† == 𝑎† (𝑡)𝑒 −𝑖𝜔𝑡 = [𝜇 ∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0)]𝑒 𝑖𝜔𝑡 𝑒 −𝑖𝜔𝑡 = [𝜇 ∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0)]
(6.9)
Hence, we have
2
𝐴2 = [𝜇𝑎(0) + 𝜈𝑎† (0)]2 = [𝜇 2 𝑎2 + 𝜈 2 𝑎† + 𝜇𝜈(𝑎𝑎† + 𝑎† 𝑎)]
2
2
𝐴† = [𝜇 ∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0)]2 = [𝜈 ∗ 2 𝑎2 + 𝜇 ∗ 2 𝑎† + 𝜇 ∗ 𝜈 ∗ (𝑎𝑎† + 𝑎† 𝑎)]
(6.10)
(6.11)
The equation (6.1) assumed the following form
2
1
𝑌1 = 2 (𝐴2 + 𝐴† )
2
1
2
= 2 ([𝜇 2 𝑎2 + 𝜈 2 𝑎† + 𝜇𝜈(𝑎𝑎† + 𝑎† 𝑎)] + [𝜈 ∗ 2 𝑎2 + 𝜇 ∗ 2 𝑎† + 𝜇 ∗ 𝜈 ∗ (𝑎𝑎† + 𝑎 † 𝑎)])
2
1
= 2 (𝑎2 (𝜇 2 + 𝜈 ∗ 2 ) + 𝑎† (𝜇 ∗ 2 + 𝜈 2 ) + (𝑎𝑎 † + 𝑎† 𝑎)(𝜇𝜈 + 𝜇 ∗ 𝜈 ∗ ))
2
1
= 2 (𝑎2 + 𝑎† )
(6.12)
Where the equation (6.10) & (6.11), are used.
In order to calculate 𝑌1 we use the following relation.
𝜇 2 + 𝜈 ∗ 2 = 𝜇 ∗ 2 + 𝜈 2 = 𝑐𝑜𝑠ℎ2 𝑠𝑡 − 𝑠𝑖𝑛ℎ2 𝑠𝑡 = 1 And
𝜇𝜈 + 𝜇 ∗ 𝜈 ∗ = −𝑖 sinh 𝑠𝑡 cosh 𝑠𝑡 + 𝑖 sinh 𝑠𝑡 cosh 𝑠𝑡 = 0
Now,
1
2
𝑌12 = [2 (𝑎2 + 𝑎 † )]
2
28
4
1
2
2
= (𝑎4 + 𝑎† + 𝑎2 𝑎† + 𝑎† 𝑎2 )
(6.13)
4
2
2
Now, we have 𝑎2 𝑎† = 𝑎† 𝑎2 + 4𝑎† 𝑎 + 2. Therefore, the equation (6.13) reduces to
4
1
2
2
𝑌12 = 4 (𝑎4 + 𝑎† + 𝑎† 𝑎2 + 4𝑎 † 𝑎 + 2 + 𝑎† 𝑎2 )
4
1
2
= 4 (𝑎4 + 𝑎† + 2𝑎† 𝑎2 + 4𝑎† 𝑎 + 2)
(6.14)
We calculate the second order variance of the quadrature 𝑌1 in terms of the initial coherent
state.the corresponding variance,
〈(Δ𝑌1 )2 〉 = ⟨𝛼|𝑌1 2 |𝛼⟩ − ⟨𝛼|𝑌1 |𝛼⟩2
4
1
2
2
1
= ⟨𝛼| 4 (𝑎4 + 𝑎† + 2𝑎† 𝑎2 + 4𝑎† 𝑎 + 2) |𝛼⟩ − ⟨𝛼| 2 (𝑎2 + 𝑎† ) |𝛼⟩
1
1
= [4 (𝛼 4 + 𝛼 ∗ 4 + 2|𝛼|4 + 4|𝛼|2 + 2)] − [2 (𝛼 2 + 𝛼 ∗ 2 )]
2
2
1
= 4 [(𝛼 4 + 𝛼 ∗ 4 + 2|𝛼|4 + 4|𝛼|2 + 2) − (𝛼 4 + 𝛼 ∗ 4 + 2|𝛼|4 )]
1
= 4 [(4|𝛼|2 + 2)]
= |𝛼|2 +
1
2
1
=𝑁+2
(6.15)
In order to calculate above variance we have made the following relations
𝑎|𝛼〉 = 𝛼|𝛼〉 ,
𝑎† |𝛼〉 = 𝛼 ∗ |𝛼〉,
(6.16)
and
(6.17)
|𝛼|2 = 𝑁, is the average photon number.
(6.18)
1
1
2
2
Now we have to calculate expectation value of (𝑁 + ). In order to calculate 〈(𝑁 + )〉, at
first we will calculate the value of 𝑁 = 𝐴† 𝐴 in terms of the output of the degenerate
parametric interaction. So with the combination of equation (6.6) & (6.7) we get,
𝑁 = 𝐴† 𝐴 = 𝑎† (𝑡)𝑒 −𝑖𝜔𝑡 𝑎(𝑡)𝑒 𝑖𝜔𝑡
= (𝜇 ∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0))𝑒 𝑖𝜔𝑡 (𝜇𝑎(0) + 𝜈𝑎† (0))𝑒 −𝑖𝜔𝑡
= (𝜇 ∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0)) (𝜇𝑎(0) + 𝜈𝑎† (0))
29
2
= (𝜇𝜈 ∗ 𝑎2 + 𝜇 ∗ 𝜈𝑎† + |𝜈|2 𝑎𝑎† + |𝜇|2 𝑎† 𝑎)
2
= (𝜇𝜈 ∗ 𝑎2 + 𝜇 ∗ 𝜈𝑎† + |𝜈|2 (𝑎† 𝑎 + 1) + |𝜇|2 𝑎† 𝑎)
(6.19)
Hence the required expectation value in terms of the initial coherent state is given by
2
1
1
〈(𝑁 + )〉 = ⟨𝛼| (𝜇𝜈 ∗ 𝑎2 + 𝜇 ∗ 𝜈𝑎 † + |𝜈|2 (𝑎† 𝑎 + 1) + |𝜇|2 𝑎† 𝑎) + |𝛼⟩
2
2
1
= ((𝜇𝜈 ∗ 𝛼 2 + 𝜇 ∗ 𝜈𝛼 ∗ 2 + |𝜈|2 (|𝛼|2 + 1) + |𝜇|2 |𝛼|2 ) + 2)
1
= (|𝜇|2 |𝛼|2 + |𝜈|2 (|𝛼|2 + 1) + 𝜇𝜈 ∗ 𝛼 2 + 𝜇 ∗ 𝜈𝛼 ∗ 2 + )
2
1
= (|𝜇|2 𝑁 + |𝜈|2 (𝑁 + 1) + (𝜇𝜈 ∗ 𝛼 2 + 𝐶. 𝐶) + 2)
𝜋
1
= (𝑁𝑐𝑜𝑠ℎ2 𝑠𝑡 + 𝑠𝑖𝑛ℎ2 𝑠𝑡(𝑁 + 1) + (|𝜇||𝜈|𝑒 𝑖 2 |𝛼|2 𝑒 2𝑖𝜃 + 𝐶. 𝐶) + 2)
𝜋
1
= (𝑁𝑐𝑜𝑠ℎ2 𝑠𝑡 + 𝑠𝑖𝑛ℎ2 𝑠𝑡(𝑁 + 1) + |𝜇||𝜈||𝛼|2 (𝑒 𝑖 2 𝑒 2𝑖𝜃 + 𝐶. 𝐶) + 2)
𝜋
𝜋
1
= (𝑁𝑐𝑜𝑠ℎ2 𝑠𝑡 + 𝑠𝑖𝑛ℎ2 𝑠𝑡(𝑁 + 1) + |𝜇||𝜈||𝛼|2 (𝑒 𝑖 2 +2𝑖𝜃 + 𝑒 −𝑖 2 −2𝑖𝜃 ) + 2)
𝜋
1
= (𝑁𝑐𝑜𝑠ℎ2 𝑠𝑡 + 𝑠𝑖𝑛ℎ2 𝑠𝑡(𝑁 + 1) + 2𝑁𝑐𝑜𝑠ℎ𝑠𝑡sinh𝑠𝑡 cos (2 + 2𝜃) + 2)
1
= (𝑁(𝑠𝑖𝑛ℎ2 𝑠𝑡 + 1) + 𝑠𝑖𝑛ℎ2 𝑠𝑡(𝑁 + 1) − 2𝑁𝑐𝑜𝑠ℎ𝑠𝑡 sinh𝑠𝑡 sin(2𝜃) + 2)
1
= ((𝑁 + 2) + 𝑠𝑖𝑛ℎ2 𝑠𝑡(2𝑁 + 1) − 2𝑁𝑐𝑜𝑠ℎ𝑠𝑡 sinh𝑠𝑡 sin(2𝜃))
1
= ((𝑁 + 2) + sinh𝑠𝑡(sinh𝑠𝑡(2𝑁 + 1) − 2𝑁𝑐𝑜𝑠ℎ𝑠𝑡 sin(2𝜃)))
(6.20)
In order to calculate above relation we have used equations (6.16), (6.17), (6.18), & (6.19)
and the following relations
𝛼 = |𝛼|𝑒 𝑖𝜃 ,
𝜇 = |𝜇| = cosh 𝑠𝑡,
𝜋
𝜈 = |𝜈|𝑒 −𝑖 2 , 𝑤ℎ𝑒𝑟𝑒 |𝜈| = sinh 𝑠𝑡,
From equation (6.4), (6.15) & (6.20), we can say that the quadrature Y1 will be squeezed
for 𝑠 > 0 if
30
1
1
2
2
𝑁 + < ((𝑁 + ) + sinh𝑠𝑡(sinh𝑠𝑡(2𝑁 + 1) − 2𝑁𝑐𝑜𝑠ℎ𝑠𝑡 sin(2𝜃)))
Or, (sinh𝑠𝑡)(sinh𝑠𝑡(2𝑁 + 1) − 2𝑁𝑐𝑜𝑠ℎ𝑠𝑡 sin(2𝜃)) > 0
(6.21)
The above equation will be satisfied if the both bracketed term of the right side of the
equation will either positive or negative. So the two conditions for the squeezing of qadrature
𝑌1 are
1.
2.
(sinh𝑠𝑡) > 0 , and (sinh𝑠𝑡(2𝑁 + 1) − 2𝑁𝑐𝑜𝑠ℎ𝑠𝑡 sin(2𝜃)) > 0
(sinh𝑠𝑡) < 0 , and (sinh𝑠𝑡(2𝑁 + 1) − 2𝑁𝑐𝑜𝑠ℎ𝑠𝑡 sin(2𝜃)) < 0
The quadrature Y2 will be squeezed for the above two conditions if 𝑠 < 0.
7 Photon Bunching and Photon Anti-Bunching:
If a beam of light from an ordinary light source is allowed to fall on a photo detector tube, the
phenomenon of photon bunching can be observed In this experiment photons are detected by
the photo tube in the form of bunches or cluster. The photon bunching effect was first
experimentally observed by Hanbury-Brown and Twiss in 1950.
31
But the invention of the laser create a debate concerning the existence of bunches of photon
in laser light. The coherent light from a laser operating above threshold does not show the
photon bunching effect. In this detection by the photo tube photons are randomly distributed.
The photon bunching is a property of incoherent or chaotic light from ordinary light sources.
At later time one another effect for the light which can be generated by nonlinear
interaction of laser light with matter was observed. The effect was termed as photon
antibunching. In this case photons have a tendency not to arrive together. The arrival of one
photon is detected by the phototube then another photon.
7.1 Second order correlation function:
The intensity fluctuations of the optical field are described by the Glauber correlation
function. The quantum degree of second order coherence is a measure of the correlation of
light intensities.let we define the positive frequency part of the field 𝐸⃗ + as
1
𝐸⃗ + (𝑡) = 2 𝜀𝑐 𝑎̂𝑒 −𝑖𝜔𝑡
(7.1)
with the negative-frequency part of the field
†
1
𝐸⃗ − (𝑡) = (𝐸⃗ + (𝑡)) = 2 𝜀𝑐 𝑎̂† 𝑒 −𝑖𝜔𝑡
(7.2)
Now we have to compute the field correlation at a time t and later time 𝑡 + 𝜏 measured at the
same space point. The relevant correlation function is
𝐺 (2) (𝑡, 𝑡 + 𝜏) = 〈𝐸⃗ − (𝑡)𝐸⃗ − (𝑡 + 𝜏)𝐸⃗ + (𝑡 + 𝜏)𝐸⃗ + (𝑡)〉
(7.3)
or in normalized form
𝑔(2) (𝑡, 𝑡 + 𝜏) =
〈𝐸⃗ − (𝑡)𝐸⃗ − (𝑡+𝜏)𝐸⃗ + (𝑡+𝜏)𝐸⃗ + (𝑡)〉
〈𝐸⃗ − (𝑡)𝐸⃗ + (𝑡)〉2
(7.4)
In the terms of normally ordered intensity correlations,
𝑔(2) (𝑡, 𝑡 + 𝜏) =
〈𝐼(𝑡+𝜏)𝐼(𝑡)〉
〈𝐼(𝑡)〉2
(7.5)
For a single mode radiation field
𝑔(2) (𝑡, 𝑡) = 𝑔(2) (0) =
𝑔
(2) (
𝜏) =
〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉
〈𝑎̂† 𝑎̂〉2
〈𝑎̂† (0)𝑎̂† (𝜏)𝑎̂(𝜏)𝑎̂(0)〉
〈𝑎̂† (0)𝑎̂(0)〉2
32
(7.6)
(7.7)
7.2 Photon bunching:
The photon will be bunched for ordinary light sources. The photons have a tendency to arrive
together. In other words, if the arrival of one photon is detected then there is an increased
likelihood of another photon arriving soon afterwards. This phenomenon is called photon
bunching. Light from thermal sources exhibits photon bunching.
The bunched photon always obey the following correlation
𝑔(2) (0) > 1
7.3 Photon antibunching:.
On the other hand, photons have a tendency not to arrive together. If the arrival of one photon
is detected then there is a reduced likelihood of another photon arriving soon afterwards. This
situation is called photon anti-bunching. Here the photons are emitted at regular time interval.
For example, squeezed state exhibits photon anti-bunching on a very short time scale.
The antibunched photon always obey the following correlation
𝑔(2) (0) < 1
Now the second order correlation function for the coherent light is given by,
𝑔
(2) (
0) =
〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉
〈𝑎̂† 𝑎̂〉2
=
̂
⟨𝛼 |𝑎
† †
𝑎̂ 𝑎̂𝑎̂|𝛼 ⟩
̂
⟨𝛼 |𝑎
†
𝑎̂|𝛼 ⟩
2
|𝛼|4
= |𝛼|4 = 1
Now we calculate 𝑔(2) (0) in terms of initial coherent state. From the definition of 𝑔(2) (0) in
(7.6) we get
𝑔(2) (0) =
〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉
〈𝑎̂† 𝑎̂〉2
=1+
=
〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉
〈𝑎̂† 𝑎̂〉2
−1+1
〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉−〈𝑎̂† 𝑎̂〉2
(7.8)
〈𝑎̂† 𝑎̂〉2
At first we calculate the expectation value of 〈𝑎̂† 𝑎̂ † 𝑎̂𝑎̂〉. With the combination of equation
(5.22) and (5.23) we have
〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉 = 〈(𝜇 ∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0))(𝜇∗ 𝑎† (0) + 𝜈 ∗ 𝑎(0)) (𝜇𝑎(0) + 𝜈𝑎† (0)) (𝜇𝑎(0) +
𝜈𝑎 † (0))〉
33
2
2
= 𝑐𝑜𝑠ℎ4 𝑠𝑡 (𝑎† 𝑎2 ) + 𝑠𝑖𝑛ℎ4 𝑠𝑡 (𝑎† 𝑎2 +4𝑎† 𝑎 + 2)
| +𝑠𝑖𝑛ℎ2 𝑠𝑡 𝑐𝑜𝑠ℎ2 𝑠𝑡 (4𝑎† 2 𝑎2 + 8𝑎† 𝑎 − 𝑎4 − 𝑎† 4 ) + |
= ⟨𝛼
𝛼⟩
3
† 3
2
†3
†2
𝑖
sinh
𝑠𝑡
𝑐𝑜𝑠ℎ
𝑠𝑡
(2𝑎
𝑎
+
𝑎
−
2𝑎
𝑎
−
𝑎
)
|
|
3
2
+𝑖𝑠𝑖𝑛ℎ3 𝑠𝑡 cosh 𝑠𝑡 (2𝑎† 𝑎3 − 2𝑎† 𝑎 + 5𝑎2 − 5𝑎† )
= |𝛼|4 𝑐𝑜𝑠ℎ4 𝑠𝑡 + 𝑠𝑖𝑛ℎ4 𝑠𝑡(|𝛼|4 + 4|𝛼|2 + 2) + 𝑠𝑖𝑛ℎ2 𝑠𝑡 𝑐𝑜𝑠ℎ2 𝑠𝑡(4|𝛼|4 +
8|𝛼|2 − 𝛼 4 − 𝛼 ∗ 4 ) + 𝑖 sinh 𝑠𝑡 𝑐𝑜𝑠ℎ3 𝑠𝑡(2𝛼 ∗ 𝛼 3 + 𝛼 2 − 2𝛼 ∗ 3 𝛼 − 𝛼 ∗ 2 ) +
𝑖𝑠𝑖𝑛ℎ3 𝑠𝑡 cosh 𝑠𝑡 (2𝛼 ∗ 𝛼 3 − 2𝛼 ∗ 3 𝛼 + 5𝛼 2 − 5𝛼 ∗ 2 )
(7.9)
Again, using equation (5.22) and (5.23) we can calculate the expectation value of 〈𝑎† 𝑎〉. The
result is
〈𝑎† 𝑎〉 = 𝜇𝜈 ∗ 𝛼 2 + 𝜇 ∗ 𝜈𝛼 ∗ 2 + |𝜈|2 (|𝛼|2 + 1) + |𝜇|2 |𝛼|2
So,
〈𝑎† 𝑎〉2 = (𝜇𝜈 ∗ 𝛼 2 + 𝜇 ∗ 𝜈𝛼 ∗ 2 + |𝜈|2 (|𝛼|2 + 1) + |𝜇|2 |𝛼|2 )(𝜇𝜈 ∗ 𝛼 2 + 𝜇 ∗ 𝜈𝛼 ∗ 2 +
|𝜈|2 (|𝛼|2 + 1) + |𝜇|2 |𝛼|2 )
= 𝑐𝑜𝑠ℎ4 𝑠𝑡(|𝛼|4 ) + 𝑠𝑖𝑛ℎ4 𝑠𝑡(|𝛼|4 + 2|𝛼|2 + 1) + 𝑠𝑖𝑛ℎ2 𝑠𝑡 𝑐𝑜𝑠ℎ2 𝑠𝑡(4|𝛼|4 +
2|𝛼|2 − 𝛼 4 − 𝛼 ∗ 4 ) + 𝑖 sinh 𝑠𝑡 𝑐𝑜𝑠ℎ3 𝑠𝑡(2𝛼 ∗ 𝛼 3 − 2𝛼 ∗ 3 𝛼) +
𝑖𝑠𝑖𝑛ℎ3 𝑠𝑡 cosh 𝑠𝑡 (2𝛼 ∗ 𝛼 3 − 2𝛼 ∗ 3 𝛼 + 2𝛼 2 − 2𝛼 ∗ 2 )
(7.10)
Now subtracting equation (7.10) from (7.9) we have
〈𝑎̂ † 𝑎̂† 𝑎̂𝑎̂〉 − 〈𝑎† 𝑎〉2
= 𝑠𝑖𝑛ℎ4 𝑠𝑡(2|𝛼|2 + 1) + 𝑠𝑖𝑛ℎ2 𝑠𝑡 𝑐𝑜𝑠ℎ2 𝑠𝑡( 6|𝛼|2 )
+ 𝑖 sinh 𝑠𝑡 𝑐𝑜𝑠ℎ3 𝑠𝑡(𝛼 2 − 𝛼 ∗ 2 ) + 𝑖𝑠𝑖𝑛ℎ3 𝑠𝑡 cosh 𝑠𝑡 (3𝛼 2 − 3𝛼 ∗ 2 )
= (2𝑁 + 1)𝑠𝑖𝑛ℎ4 𝑠𝑡 + 6𝑁𝑠𝑖𝑛ℎ2 𝑠𝑡 𝑐𝑜𝑠ℎ2 𝑠𝑡 − 2 Nsinh 𝑠𝑡 𝑐𝑜𝑠ℎ3 𝑠𝑡 sin 2𝜃 −
6𝑁 𝑠𝑖𝑛ℎ3 𝑠𝑡 cosh 𝑠𝑡 sin 2𝜃
(7.11)
Since the denominator of the right side of equation (7.8) is always positive, depending upon
the value of (〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉 − 〈𝑎† 𝑎〉2 ), 𝑔(2) (0) will be positive or negative. Here will discuss
three case
Case 1: when 𝜃 = 0, 〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉 − 〈𝑎† 𝑎〉2 = (2𝑁 + 1)𝑠𝑖𝑛ℎ4 𝑠𝑡 + 6𝑁𝑠𝑖𝑛ℎ2 𝑠𝑡 𝑐𝑜𝑠ℎ2 𝑠𝑡 it is
always positive. Hence from equation (7.8) we have, 𝑔(2) (0) > 1. So the bunching photon
will appear from the degenerate parametric amplifier.
Case 2: For the vacuum field with 𝑁 = 0, 〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉 − 〈𝑎† 𝑎〉2 = 𝑠𝑖𝑛ℎ4 𝑠𝑡, which is always
positive. Here we will get bunching photon.
34
𝜋
Case 3: when, 𝜃 = ,
4
〈𝑎̂† 𝑎̂† 𝑎̂𝑎̂〉 − 〈𝑎† 𝑎〉2 = (2𝑁 + 1)𝑠𝑖𝑛ℎ4 𝑠𝑡 + 6𝑁𝑠𝑖𝑛ℎ2 𝑠𝑡 𝑐𝑜𝑠ℎ2 𝑠𝑡 − (2 Nsinh 𝑠𝑡 𝑐𝑜𝑠ℎ3 𝑠𝑡 +
6𝑁 𝑠𝑖𝑛ℎ3 𝑠𝑡 cosh 𝑠𝑡)
When (2 Nsinh 𝑠𝑡 𝑐𝑜𝑠ℎ3 𝑠𝑡 + 6𝑁 𝑠𝑖𝑛ℎ3 𝑠𝑡 cosh 𝑠𝑡) dominates over the remaining terms,
𝑔(2) (0) < 1. Hence antibunching photon will appear in a degenerate parametric amplifier
coupled to a coherent radiation field.
8 Conclusion
In this study, we have studied three nonlinear nonclassical phenomenons. We have observed
that the production of squeezing is possible by nonlinear atom/molecule field interaction. We
investigate that the squeezing in one of the quadratures is possible at the cost of the remaining
qudratures when the interaction parameter(s) is positive. The squeezing of the remaining
qudrature is obtained when the interaction parameter(s) is negative. In other word we can say
that the simultaneous squeezing of both the quadratures is not possible. We have also seen
that second order squeezing of both the quadratures is possible at the cost of remaining
quadrature under a specific condition. But the production of simultaneous second order
squeezing is not possible. We have also studied that the antibunching photon can be obtained
in a degenerate parametric amplifier coupled to a coherent radiation field.
9 References:
[1] Amnon Yariv, Quantum Electronics. John Wiley & Sons.
[2] R.Loudon & P.L. Knight(1987): Squeezed Light, Journal of Modern Physics. vol 34, nos.
6/7, 709-759.
[3] Coherent States, ed. J. R. Klauder and B. S. Skagerstam, World Scientific ,Singapore,
35
1985.
[4] P.L.Knight & L.Allen, Concept of Quantum Optics, Pergamon Press.
[5] R.Loudon, Photon Bunching and Antibunching, Journal of Modern Physics.
36