 Open Access
 Authors : Sudhir Dalal
 Paper ID : IJERTV11IS070178
 Volume & Issue : Volume 11, Issue 07 (July 2022)
 Published (First Online): 29072022
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
An Analytical Study of Pulse Squeezing in Fourth Harmonic Generation
Sudhir Dalal
Department of Physics
A.I.J.H.M. College, Rohtak 124001, India
AbstractSqueezing of electromagnetic field is investigated in fundamental mode in fourth harmonic generation under a shorttime approximation based on quantum mechanical approach. The occurrence of squeezing in field amplitude and amplitudesquared of fundamental mode has been investigated. The degree of squeezing is found to be dependent on interaction time, coupling parameter of interacting waves and phase values

SQUEEZED STATES AND HIGHERORDER SQUEEZING
Squeezed states of an electromagnetic field are the states with reduced noise below the vacuum limit in one of the canonical conjugate quadrature. Normal squeezing is defined in terms of the operators
of field amplitude of the fundamental mode during the process. The dependence of squeezing on photon number has also been
X 1 ( A A )
1 2
(1a)
noticed and found that it increases nonlinearly. The photon
statistics of the field in the fundamental mode is found to be sub Poissonian.
and
X2
1 ( A A ) , (1b)
2i
KeywordsNonlinear optis, squeezing, photon number,
where X1 and X2 are the real and imaginary parts of the
harmonic generation, subpoissonian.

INTRODUCTION
The squeezed states are minimum uncertainty states with reduced fluctuation in one quadrature at the expense of
field amplitude respectively. A and operators defined by
A aeit
and
A are slowly varying
(2a)
increased fluctuation in the other quadrature. It is a purely
quantum mechanical phenomenon without any classical counterpart.
Hong and Mandel [1] introduced the concept of Nth order squeezing of electromagnetic field as a generalization of secondorder squeezing. Zhan [2] extended the result presented by Hillery [3] for second harmonic generation to k th harmonic generation. Jawahar Lal and Jaiswal [4] extended the results obtained by Zhan [2] for amplitudecubed squeezing in the fundamental mode during second and third
A aeit . (2b)
The operators X1 and X2 obey the commutation relation
[ X , X ] i , (3)1 2 2
which leads to uncertainty relation
X X 1 . (4)
1 2 4
A quantum state is squeezed in Xi variable if
1
harmonic generations to kth order. Generations of squeezed
X1
2
for i = 1 or 2. (5)
states theoretically and experimentally have been investigated
in a numerous nonlinear processes [510] viz. fourwave mixing, second and third harmonic generation, parametric
Amplitudesquared squeezing is defined in terms of operators Y1 and Y2 as
amplification, Raman and hyperRaman. The importance of squeezing in gravitational wave detection [11], optical communication [12], interferometric technique [13], high
Y 1 ( A2 A 2 )
1 2
and
(6a)
precision measurement [14], dense coding [15], quantum cryptography [16] etc. is due to its low noise property [17].
In general, the two important nonclassical effects, squeezing and antibunching (or SubPoissionian photon statistics) are not interrelated i.e. some states exist that exhibit the first but not the second and vice versa. This paper shows
Y 1 ( A2 A 2 ) . (6b)
2 2i
The operators Y1 and Y2 obey the commutation relation [Y1 ,Y2 ] i(2N 1) , where N is the usual number operator which leads to the uncertainty relation
one of the distinguished example of nonlinear process when 1
light exhibits both squeezing and subPoissionian photon
Y1Y2
2
N
. (7)
statistics at the same time. This paper presents the squeezing
of electromagnetic field in the fundamental mode in fourth harmonic generation. The dependence of squeezing on photon
Amplitudesquared squeezing is said to exist in Yi variable
if
number and the Poissionian behaviour of the field has also
i
(Y )2
N
1
for i = 1 or 2. (8)
been investigated. 2


FOURTHHARMONIC GENERATION
X 2 (t) 1 [a2 a*2 2 a 2 1
The Fourth harmonic generation model has been adopted
from the works of Chen et al. [19] and is shown in Fig. 1. This process involves absorption of four photons, each having frequency 1 going from state 1 to 2 and emission of one photon of frequency 2 , where 2 41 .
1A
and
4
2g2t2 (2a2 a 6 3a2 a 4
2a*2 a 6 3a*2 a 4 4 a 8 )]
(16)
2 1 2
X 2 (t) [a2 a*2 2 a
1A
Therefore
4
2g2t2 (2a2 a 6 2a*2 a 6 4 a 8 ]
(17)
[X1A(t)]2 1 3g 2t 2 a 6 cos 2 , (18)
4
where is the phase angle, with
a* a exp(i) .
a a exp(i)
and
Fig. 1: Fourth harmonic generation model We can write Hamiltonian for this process as
4 4
The right hand side of expression (18) is negative, indicating that squeezing occurs in the field amplitude of fundamental mode in fourth harmonic generation for which cos 2 0 .
Using equation (13), the second order amplitude in fundamental mode is expressed as
A2 (t) A2 4igt(2 A3 AB 3A 2 B)
H 1a a 2b b g(a b
a b) , (9)
16g 2t 2 A 6 B2 2g 2t 2 (2 A3 A5 3A 2 A4 )
(19)
in which g is coupling constant for fourth harmonic
generation.
A a exp(i1t)
For amplitudesquared squeezing, the real quadrature component for the fundamental mode is given as
and
B b exp(i2t)
Y1A
(t) 1 [ A2 (t) A 2 (t)] . (20)
2
are the slowly varying operators at frequency 1
and 2
Using equations (15), (19) and (20), we get the expectation
value as
respectively, where
a(a )
and
b(b )
are the usual number
2 1 4 *4 4 2
operator with the relation 2 41 .
Y1A (t) [a a
4

2 a
4 a 2
2 2 6 4 2
4 *4
The Heisenberg equation of motion for fundamental mode A
4g t {(2 a
9 a
12 a
3)(a a )
is as
A dA i[H , A]. (10)
dt
and
4 a
10 10 a 8}]
(21)
Using equation (9) in equation (10), we obtain
A 4igA3 B
and
(11)
2 1 4
X 2 (t) [a4 a*4 2 a
1A 4
4g2t2 (a4 a*4 2 a 4 )(2 a 6 3 a 4 )] .(22)
B igA4 . (12)
With the short interaction time approximation, we can write
Therefore
A(t) in Taylors series and retaining the terms up to g 2t 2 as
A(t) A 4igtA3 B
[Y1A(t)]2 1 [4 a 2 2 4g 2t 2
4
{(6 a 4 12 a 2 3)(a4 a*4 ) 4 a 8}] . (23)
2g 2t 2 [(12 A 2 A3 36 A A2 24 A)B B A3 A4 ] (13)
For squeezing of field amplitude in fundamental mode, we can write the quadrature component as
The number of photons in mode A may be expressed as
1A
N (t) A (t) A(t) A A 4igt( A 4 B A4 B )
4g 2t 2 A 4 A4 16g 2t 2 A3 A3 B B
(24)
X1A
1 [ A(t) A (t)] . (14)
2
Using equation (15), the average value of
N (t) 1 is
A 2
Initially, we consider the quantum state as a product of
coherent state for the fundamental mode A and the vacuum state for the harmonic mode B i.e.
givn by
NA
(t) 1
1 [4 a 2 2 16g2t2 a 8 ] . (25)
A(0) a
a a
; B(0)
0 . (15) 2 4
Using equations (13)(15), the expectation values are derived
as
Subtracting equation (25) from equation (23), we get
[Y1A(t)]2
N1A
(t) 1
2
V. CONCLUSIONS
The squeezing increases nonlinearly with 2, which directly depends on the number of photons. The squeezing in
6g2t2 (2 a 8 4 a 6 a 4 ) cos 4 . (26)
The right hand side of above equation is negative for all
any order during stimulated interaction is higher than the squeezing in corresponding order in spontaneous interaction.
values of for which cos 4 0
and thus shows the existence
The squeezing is higher in higher orders in both processes.
of squeezing in the second order of field amplitude of the fundamental mode under short time approximation. The photon statistics of field amplitude in fundamental mode in fourth harmonic generation is subPoissonian, given as
Thus, the higherorder squeezing associated with higher order
nonlinear optical processes makes it possible to achieve significant noise reduction
[N(t)]2 N
(t)
12g2t2 a 8 . (27)
ACKNOWLEDGMENT
A A


RESULTS AND DISCUSSION
The results show the squeezing of field amplitude and square of the field amplitude of fundamental mode in fourth harmonic generation. To study squeezing, we denote the right hand side of equations (18) and (26) by Sx and Sy respectively. Taking
2
gt 104 and 0 , the variations are shown in Figure 2 and
2
3. Figures show that squeezing increases nonlinearly with a ,
i.e. with the number of photons.
It is obvious that the degree of squeezing is greater in amplitudesquared states than in field amplitude states. Squeezing occurs in the fundamental mode of fourth harmonic generation obeying subPoissonian photon statistics.
2
0 10 20 30 40 50
Sx
10
20
30
40
2
Fig. 2: Dependence of field amplitude squeezing Sx on a .
2
The author is thankful to Dr. Manjeet Singh, Department
of Physics, Govt. College, Matanhail, Jhajjar (Haryana) for his kind cooperation.
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Fig. 3: Dependence of amplitude squared squeezing Sy on a .