An Analytical Study of Pulse Squeezing in Fourth Harmonic Generation

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 short-time approximation based on quantum mechanical approach. The occurrence of squeezing in field amplitude and amplitude-squared 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

1. SQUEEZED STATES AND HIGHER-ORDER 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, sub-poissonian.

   1. 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 N-th order squeezing of electromagnetic field as a generalization of second-order 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 amplitude-cubed 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 k-th order. Generations of squeezed

      X1

      2

      for i = 1 or 2. (5)

      states theoretically and experimentally have been investigated

      in a numerous non-linear processes [5-10] viz. four-wave mixing, second and third harmonic generation, parametric

      Amplitude-squared squeezing is defined in terms of operators Y1 and Y2 as

      amplification, Raman and hyper-Raman. 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 non-classical effects, squeezing and antibunching (or Sub-Poissionian 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 non-linear process when 1

      light exhibits both squeezing and sub-Poissionian 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

      Amplitude-squared 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

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 amplitude-squared 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 sub-Poissonian, given as

   Thus, the higher-order 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

3. 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 amplitude-squared states than in field amplitude states. Squeezing occurs in the fundamental mode of fourth harmonic generation obeying sub-Poissonian 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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2

-8000

Fig. 3: Dependence of amplitude squared squeezing Sy on a .