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27 December 1999

.Physics Letters A 264 1999 265269

www.elsevier.nlrlocaterphysleta

Statistical properties of photon-added and photon-subtractedtwo-mode squeezed vacuum state

Lu Hong )

Department of Physics, Uniersity of Science and Technology of Foshan, Guangdong, 528000, China

Received 4 June 1999; received in revised form 9 November 1999; accepted 12 November 1999

Communicated by P.R. Holland

Abstract

We introduce photon-added and photon-subtracted two-mode squeezed vacuum states and examine the quantum

statistical properties of these states by analytic calculating. Because of the entanglement between the modes of the field in a .two-mode squeezed state, photon adding or subtracting for one of the modes results in interesting outcome. q1999

Published by Elsevier Science B.V. All rights reserved.

1. Introduction

There is currently much interest in the production and properties of states of quantized electromagnetic fieldw xthat exhibit various nonclassical properties. The most familiar examples include squeezed states 1 , even and

w x w xodd coherent states 2 , correlated two-mode states 3,4 . Several years ago, Agarwal and Tara introducedqm < : w xphoton-added coherent states a a 5 and demonstrated that such states were nonclassical states, they can

exhibit phase squeezing and sub-Poissonian character of the field. These authors also suggested a scheme to

generate such states by passing excited atoms through a cavity containing pre-existing coherent states. Similarly,w xZhang, Dodonov, and Jones et al. have studied the properties of photon-added squeezed vacuum states 6 ,

w x w xphoton-added evenrodd coherent states 7 and photon-added thermal states 8 , respectively. Recently Dacna et

al. proposed preparing single-mode photon-added states by conditional output measurement on a beam splitterw x9 . Present author have already studied the statistical properties of photon-added pair coherent states and

qm qm < :suggested a possible scheme to generate two-mode photon-added states a b ,q by the nondegenerate

w xtwo-photon interaction between atom and cavity field 10 . It has been realized that adding photons to a state isan important method of obtain new quantum states.w x .Correlated two-mode states, for example, two-mode squeezed vacuum state 3 TMSVS and pair coherent

w xstate 4 are of interest because they display strong nonclassical properties. In this Letter, we introduce

photon-added and photon-subtracted TMSVS and examine the influence of photon adding or subtracting on the

)

E-mail: hlu@163.net

0375-9601r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved. .P I I : S 0 3 7 5 - 9 6 0 1 9 9 0 0 8 0 2 - 6

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properties of the field in the TMSVS. It is well known that the TMSVS is important nonclassical state and hasw xbeen realized in the laboratory 11 . Recently TMSVS was used as entangled EinsteinPodolskyRosen state in

w x .a teleportation experiment 12 . We show that because of its strong correlation, photon adding or subtracting

for one of modes of the field in the TMSVS results in interesting outcome. For TMSVS, the effect ofannihilation of m a-photon is just the same as creation of m b-photon. Annihilation of photons of mode a or

.mode b actually increases the mean number of photons of mode a and mode b. We also calculate Mandel Q

parameter, cross-correlation function and squeezing of the field.

2. Photon-added and photon-subtracted TMSVS

The TMSVS is obtained by acting squeezing operator on a two-mode vacuum. In its number-state

representation, it takes the form

=1 niu< : < : < :z s S0 ,0 s ye tanh r n, n , 1 . .a b

cosh rns0

where

Ss exp z)ab y zaqbq , 2 . .

there a, b are annihilation operators, s re iu is complex squeeze parameter. We define photon-added .two-mode squeezed vacuum state PATMSVS as

qm < : qm

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.Thus, Eq. 8 reduces to

=1 n!niu< : < :z,y m s ye tanh r n y m ,n .a (m' n y m ! .m! cosh rsinh rnsm

=1 m q n ! .niu < :s ye tanh r n ,m q n . 10 . .

(mq 1

' n!m! cosh rns0 . .m i muIn Eq. 10 , a phase factor y1 e has been neglected.

. . .Comparing Eqs. 3 , 7 and 10 , one can find

< : < :z,y m s z, m . 11 .a b

Namely, for a TMSVS, the effect of annihilation of m a-photons is just the same as creation of m b-photons! So

in the following we only discuss the properties of PATMSVS.

3. Nonclassical properties of PATMSVS

< :For the state z,m , mean numbers of photons N and N are given, respectively, bya a b

mq 1mq1 q < < :z a a z .q 2 :N s aa y 1 s y 1 s m q 1 cosh ry 1 , 12 . .a 2 mm!cosh r

and

< y1 m y1 q y1 qm < :0 ,0 S a SS bb SS a S 0 ,0a b a bq 2 :N s bb y 1 s y 1 s m q 1 sinh r. 13 . .b 2 mm!cosh r

Thus

N yN s m . 14 .a b

.From Eq. 13 , we see that when photons are added to the mode a, the photon number of mode b increases

too. Furthermore, when we add one a-photon to TMSVS, N increases to twice of its original values, and addb . .two a-photons, N increases to three times; etc. Combining Eqs. 11 13 , we see that annihilation of photonsb

.of mode a or mode b only increase the mean number of photons of mode a and mode b! This is seemingly

beyond ones comprehension, but it is not surprising if we bear in mind that there is no conservation of the

mean energy. For instance, photon subtracting for a coherent state would not reduce its mean number of photons

owing to coherent state is the eigenstate of annihilation operator.

Then, we examine the statistical properties of photons in the individual modes by calculating the Mandel Qw xparameter. For mode a, this parameter is defined as 13

2 2q q : :a a y a a .Q s . 15 .a q :a a

q q .2It can easily be calculated by expressing a a and a a in the antinormally ordered form

2 q2 : q:2 q: 2 2a a y aa y aa m q 1 cosh r cosh ry 1 . .Q s s . 16 .a q 2 :aa y 1 m q 1 cosh ry 1 .

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. 2Sub-Poissonian photon statistics exist whenever Q - 1. From 16 we see that for m s 0, have Q s cosh r?1,a aso the statistics of mode a is always super-Poissonian. For m ) 0, the statistics may be sub-Poissonian, if

2 w 2 2 .xy1sinh r- 1 and m q 1 ) cosh r 1 y sinh r . For mode b,

2 q2 : q:2 q:b b y bb y bbQ sb q :bb y 1

24 2 4 2 2 2 2

2cosh rq msinh 2 rq m my 1 sinh ry cosh rq msinh r y cosh ry msinh r . . 2s s cosh r?1 .2m q 1 sinh r .

17 .

Therefore, in spite of N been associated with m, Q is not. The statistics of mode b is always super-Poisso-b bnian.

w xThe normalized cross-correlation function of a two-mode field is defined as 14

q q :a b a b2.g s . 18 .12 q q : :a a b b

If g 2. is more than unity, we say that the photons of mode a and mode b are correlated. For the PATMSVS,12

we find the result2q q y1 q y1 q y1 y1 2 2 4 : < < :a a b b s m ,0 S a SS b SS aSS bS m ,0 s m q 1 sinh rcosh rq m q 1 sinh r. . .

19 .

Hence

cosh2 r2.g s 1 q ) 1 . 20 .12 2m q 1 cosh ry 1 .

2. .As m increases, g decreases, so photon adding or subtracting weakens the correlation of mode a and mode12b.

w xFinally, we examine the squeezing associated with the two-mode quadrature operator 15

1q qXs a q a q b q b , 21 . .3r22

12where squeezing exists for DX - .4

< :For the state z,m , one can easily prove thata

: 2 : : 2 : q:a s a s b s b s ab s 0 , 22 .

: iuab s ye m q 1 sinh rcosh r, 23 . .

2 12 2 : :DX s X y X s m q 1 cosh2 ry sinh2 rcosu . 24 . . .4

Let us 0, we have

12 y2 rDX s m q 1 e . 25 . .4

.As a result, photon adding or subtracting decreases the squeezing of the field.

In conclusion, we have introduced PATMSVS and PSTMSVS, and have discussed their statistical properties

by analytic derivation. It is of interest to note that, for a TMSVS, the effect of adding m a-photon is just the

same as subtracting m b-photon. This result reflects strong entanglement of photons between the modes of the

field. We are studding whether the result hold for else correlation two-mode field. Finally, we indicate suchw xstates can be generated by the method of similar to Ref. 5 .

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Acknowledgements

The author would like to thank Professor G.C. Guo for valuable discussions.

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