International Journal of Engineering Sciences, 2(3) March 2013, Pages: 62-67

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Squeezing Properties in the Superposition of Coherent State and Squeezed State

Yongxin Zhan 1, Jie Pan 2 1 Department of Physics of SiChuan Agriculture University, SiChuan, Ya An, 625014, China. 2 College of Life and Science ,SiChuan Agricultural University, SiChuan, Ya An, 625014, China.

A R T I C L E I N F O A B S T R A C T

Keywords: coherent state squeezed state measurement operator squeezing

The squeezing properties in the superposition of coherent States and squeezed state are investigated by means of the measurement phase operator introduced by Barnett and S M and Pagg D T. A superposition of coherent state and squeezing state is defined, Results show that the states appears squeezing effect under the given conditions by means of numerical calculation.

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1. Introduction The study of measurement phase operator has given rise to a great many interest recently and attained an important position in quantum

optics [1~6] . As is well known, the amplitude of field in the quantum optics is directly proportional to square root of optic-quantum number operator and the phase of field is described as operator exp( )i by Susskind-Glogower. However the SG phase operators aren’t possessed of unitary, hence hermit operators can not be constructed by them. Though two hermit operators.

1cos [exp( ) exp( )]2

i i . (1)

1sin [exp( ) exp( )]2

i i . (2)

are composed by SG operators, they have not classical property because 2 2cos sin 1 . On purpose to overcome the difficulty, unitary exponential phase operator and measurement phase operator have been defined by Pegg and Branett in optical field, and

their essence is studied in progress [1 2] . In the quantum optics, coherent state and squeezed state are two very important states in the quantum optics [3 7]� . Squeezed state is squeezing transition vacuum state, whose one orthogonal amplitude variance is less than coherent state, however the other is more than coherent state. Squeezed state has very important application in interference measurement in high degree of accuracy, photo-communication as well as detection of gravitation wave and microwave signal [8~9] . According to superposition principle, a great many of new quantum states are constructed through superposition of arbitrary state and many favorable works have done. But these superposition states are usually composed of the same sort state, such as superposition of odd-even coherent state or squeezed odd-even coherent state. Some of their non-classical properties on measurement operator have discussed in detail. Recently the new superposition state composed of

coherent state and squeezed state is studied and its quantum effect is obtained [10] . In the paper, the squeezed properties of the superposition

state on measurement operator are researched by mean of method in the paper [6] . 2. The superposition of coherent state and squeezing state The superposition of coherent state and squeezing state is defined [10] ,

( )igN e . (3)

Squeezing Properties in the Superposition of Coherent State and Squeezed State Internat ional Journal of Engineeri ng Sciences, 2(3) March 2013

63

where is the coherent state and ( )g

S m is squeezed state. 2 21 1ˆ ˆ( ) exp[ ( ) ]2 2

S a a is squeezing

operator, ( 0,0 2 )ire r , r is squeezing factor, is squeezing angle. 1m and 1m denote phase difference

between vector states with phase 0 and respectively. N is normal coefficient. To not lose generality, assume that and 1m is real, 0 . According to the normal condition, The parameters of、 r 、 、 N as well as m satisfy the equation

2 21 2

cos cosh2 [1 exp( )] 1cosh cosh

m rNr r

. (4)

3. Squeezing effect of measurement phase operator First, Pegg and Barnnet have defined orthogonal measurement phase operators

1cos ,M X 2sinM X ， (5) where 1X and 2X are two orthogonal component of optics field

11 ˆ ˆ( ),2

X a a 21 ˆ ˆ( )2

X a ai

. (6)

According to the requirement of the classical condition: 2 2cos sin 1,M M is chosen 1 2( 1 2)n , where n is an

average photon number, i.e. +ˆ ˆ=< >n a a N

2 2 2

22

3 2

[ (1 cosh2 sinh2 ) sinh2 ( sinh )cos coshexp( )].

(cosh ) cosh

n N r r rm r m r

r r

(7)

By means of the relation of measurement phase operator, measurement phase operator and optical number operator, the important equations are obtained

2[cos ,sin ] ,2M Mi

(8)

[cos , ] sin ,M MN i (9)

[sin , ] - cos .M MN i (10)

In quantum-optical field, when two operators 1Y and 2Y don’t satisfy reciprocal relation, the accuracy of measurement is restricted by measurement indeterminacy principle.

2

1 2 1 21( [ , ]4

Y Y Y Y 2 2） （ ） ， (11)

If the inequality is right ,

1 21 [ , ]2iY Y Y 2（ ） 1, 2)i （ ， (12)

then there is squeeze effect in component iY of optical field. With the view of description degree of squeeze, squeezed degree iS is defined,

Yongxin Zhan and Jie Pan Internat ional Journal of Engineeri ng Sciences, 2(3) March 2013

64

1 1 21 [ , ]2iS Y Y Y 2（ ） 1, 2)i （ . (13)

When 0iS , it indicated that there is squeezing effect in the iY . From (8) to (13), the squeezing degrees of measurement phase operator are

2

2 2 21 ˆ ˆ ˆ ˆ ˆ ˆ[2 ]

4csS a a a a a a . (14)

22 2 2

2 ˆ ˆ ˆ ˆ ˆ ˆ[2 ]4

csS a a a a a a . (15)

2

2 2

2

ˆ ˆ ˆ ˆ[24

ˆ ˆ ˆ ˆ1] .4

cnS a a a a

a a a a

(16)

2

2 2

2

ˆ ˆ ˆ ˆ[24

ˆ ˆ ˆ ˆ1] .4

snS a a a a

a a a a

(17)

Using the expression of coherent state and squeezed state in Fock space,

2

0exp( )

2 !

n

nn

n

. (18)

12 2

0

22

( ! ) ( )2

exp( ) ( ) .2 2 2

n

gn

n

vn uu

v H nu uv

(19)

and properties of hermite polynomial,

2

0

( )exp(2 )!

nn

n

H xxt t tn

, (20)

1 1( ) 2 ( ) 2 ( )n n nH x xH x tH x , (21) the expressions below are obtained,

2 2 2 2

2 22

1 2 5 2

21 2

ˆ ˆ( ) 2 { (1 cosh2 sinh2 )

cosh sinh [ ( sinh )(cosh ) (cosh )

tanh cosh] cos exp( )}.(cosh ) cosh

a a N r r

r r m rr r

r m rr r

(22)

Squeezing Properties in the Superposition of Coherent State and Squeezed State Internat ional Journal of Engineeri ng Sciences, 2(3) March 2013

65

2 2 4 2

22 2

3

3 2

2

ˆ ˆ( ) 4 [(1 cosh sinh )

( cosh sinh ) coshcos exp(2(cosh ) cosh

2(1 cosh sinh )( cosh sinh ) cos(cosh )

coshexp( ],cosh

a a N m r m r

m r r m rr r

m r m r m r rr

m rr

）

）

(23)

2 2 4 2

22

3

ˆ ˆ( ( ) ) 4 sin

(cosh sinh ) coshexp(2(cosh ) cosh

a a N

r r m m rr r

）,

(24)

2

23 2

ˆ ˆ( ) 2 [(1 cosh sinh )( cosh sinh ) coshcos exp( )],

(cosh ) cosh

a a N m r m rm r r m r

r r

(25)

2

23 2

ˆ ˆ( ) 2 sincosh sinh coshexp( ).

(cosh ) cosh

a a i Nr r m m r

r r

(26)

After (22)~(26) are put into (14)~(17)， some of figures which indicate squeezing degree csiS ( 1,2)i cnS and snS varying

with、 、 r and m are given by means of numerical calculation technique, Since curves are symmetric about longitudinal axis, the

below curves are given under 0 . If ( 1, 2)csiS i is negative, there is a kind of squeezing effect of cs . If cnS or snS is negative,

there is a kind of squeezing effect of cn or sn . In the fig 1, figures from (a) and (c) are drown under 0m . Figures from (c) are drown under 0m . They show squeeze degree curves

1csS varying with under given r、 and m . We find that 1

csS appears negative in these figures and that squeezed degree increase

accompanying with m ascending under 0m and m descending under 0m . Hence, there is a kind of squeezing effect of CS in

1Y under given parameters. In the fig 2, figures from (a) and (c) are drown under 0m . Figures from (c) are drown under 0m . They show squeeze degree curves

2csS varying with under give r 、 and m . We find that 2

csS appears negative when 0 or 0.5 and positive when

Hence, there is a kind of squeezing effect of CS in 2Y when 0 or 0.5 and squeezed degree increase accompanying with m

ascending under 0m and m descending under 0m . Under 0m , squeezing range increases when m increases and then squeezing range almost doesn’t change under 0m .

In the fig 3, figures from (a)and (c) are drown under 0m . Figures from (c) are drown under 0m . Under 0m , cnS changes from

positive to negative when m increase. Under 0m , cnS always appears negative. Hence there is cn squeezing effect at any 0m , and then there is cn squeezing effect when m is more than certain positive. Squeezing degree and squeezing range of cn and

sn increase accompanying m ascend. snS always appears negative at any m .

Yongxin Zhan and Jie Pan Internat ional Journal of Engineeri ng Sciences, 2(3) March 2013

66

4. Conclusion In the paper, the squeeze properties of measurement phase operator are investigated in Superposition of Coherent State and Squeezed state. The below results of different from [10] are obtained.

(1) There is a kind of squeezing effect of CS in 1Y under given parameters.

(2) There is a kind of squeezing effect of CS in 2Y when 0 or 0.5 and squeezed degree increase accompanying with

m ascending under 0m and m descending under 0m . Under 0m , squeezing range increases when m increases and then squeezing range almost doesn’t change under 0m .

(3) There is cn squeezing effect at any 0m , and then there is cn squeezing effect when m is more than certain positive.

Squeezing degree and squeezing range of cn and sn increase accompanying m ascend. snS always appears negative at

any m . References [1] Pegg D T, Barnett S M. “ Phase properties of single-mode electromagnetic field” [J]. Phys Rev A, 1989, a39: 1665. [2] Barnett S M, Pegg D T. “Phase in quantum optics” [J]. Phys (A) Math Gen, 1986, 19: 3849. [3] Lynch R. “Phase fluctuations in a squeezed state using measured phase operators” [J]. J Opt Soc Am (B), 1987, 4 : 1723. [4] Nath R, Kumerp. “Quasi-photon phase states [J]. J Mod Opt, 1991, 38: 263 [5] TU Hsiteh, Gong Chang-de. “Properties of the measured phase operators in squeezed number states” [J]. J Mod Opt, 1993, 40: 57. [6] Hillery M. “Sum and different squeezing of the electromagnetic field” [J]. Phys Rev A 1989, 40(6): 3147-3155. [7] Zeng Ai-hua, Zhao Yun-fei, Kung Zhen-hua. “Squeezing Properties of Entangled Coherent States” [J]. Acta Sinica Quantum Optica, 2004,10

(2) :67-72. [8] Loudon R, Knight P L. “ Squeezed light” [J]. J Mod Opt, 1987, 34(6-7): 709-759. [9] Wall D F. “Squeezed States of Light [J]. Nature, 1983, 306(10): 141-146 [10] Ren Chuan-hua,Song Tong-qiang “Squeezing Properties in the Superposition of Coherent State and Squeezed state” [J]. Acta Sinica Quantum

Optica, 2008, 14(4): 360-363.

Squeezing Properties in the Superposition of Coherent State and Squeezed State Internat ional Journal of Engineeri ng Sciences, 2(3) March 2013

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Appendix Figures

a. 0.6, 0r b. 0.6, 0.5r c. 1, 0.5r

Figure 1. The curve of degree of squeeze 1csS varying with 、 、 r and m

a. 0.6, 0r b. 1.2, 0.5r c. 1.2,r

Figure 2. The curve of degree of squeeze 2csS varying with 、 、 r and m

a. 1.2, 0r b. 1.2, 0.5r c. 1.2,r

Figure 3. The curve of degree of squeeze cnS and cnS varying with 、 、 r and m .Solid lines denote cnS and dot lines denote snS

1m

7m

1m

7m

1m

8m

1m

8m

1m

1m

1 0m

1 0m

1.8m 1.7m

1.81m

1m

4m

7m

10, 3m r

200, 4m r

2000, 5m r

0.3m

1m

1.8m

2.3m

40m

30m

20m

1m

10m

8m

6.8m