Generalized superposition of two squeezed states: generation and statistical properties

Physica A 280 (2000) 346–361www.elsevier.com/locate/physa

Generalized superposition of two squeezed states:generation and statistical propertiesYuri A. Barbosaa ;∗, G.C. Marquesa , B. Baseiab

aInstituto de F ��sica, Universidade de Sao Paulo Caixa Postal 66.318, CEP 05315-970,Sao Paulo (SP), Brazil

bIntituto de F ��sica, Universidade Federal de Goi�as Caixa Postal 131, CEP 74001-970,Goiania (GO), Brazil

Received 25 May 1999; received in revised form 30 September 1999

Abstract

Superpositions of squeezed states were introduced by Sanders [Phys. Rev. A 39 (1998) 4284],Schleich et al. [Phys. Rev. A 38 (1988) 1177], Xin et al. [Phys. Rev. A 50 (1994) 2865], toinvestigate the occurrence of nonclassical properties of the quantized radiation �eld. In thisreport we present a generalized superposition state which interpolates between two arbitrarysqueezed states. Nonclassical properties of this intermediate state as function of the interpolatingparameters are studied, the previous results in the literature becoming a particularization of ours.An experimental proposal to generate this state is also presented. c© 2000 Elsevier ScienceB.V. All rights reserved.

PACS: 42.50.Dv

Keywords: Squeezed states; Superposition state; Statistical properties;Quantum state engineering

1. Introduction

Growing interest in the study and preparation of new nonclassical states of quantizedlight �eld can be found in the recent literature, including those concerned with atomicsystems. In the realm of quantum optics one can cite the works related with whatis called as “quantum states engineering” [1–5], whereas in atomic physics the readeris referred to [6–9]. In quantum optics there are many nonclassical states

∗ Corresponding author.E-mail address: [email protected] (Y.A. Barbosa)

0378-4371/00/$ - see front matter c© 2000 Elsevier Science B.V. All rights reserved.PII: S 0378 -4371(99)00476 -8

Y.A. Barbosa et al. / Physica A 280 (2000) 346–361 347

which have been investigated in the last decade. Among them one can cite: (i)superposition of two particular number states, |0〉 and |1〉, or |0〉 and |2〉, studiedby Wodkiewicz et al. [10–12]; (ii) superposition of two coherent states, by Hillery[13–19]; (iii) superposition leading to even and odd squeezed coherent states, by Gerryet al. [20–24]; (iv) superposition yielding even and odd squeezed states by Xin et al.[25,26]; etc.The foregoing mentioned superpositions, involving two states of the light �eld, di�er

from other types of superpositions found in the literature, such as the binomial state,introduced by Stoler et al. [27–29], which interpolates between a number state |N 〉and a coherent state |�〉; the intermediate number-phase state, by Baseia et al. [30,31],which interpolates between a number state |N 〉 and the Pegg–Barnett phase state |�〉[32–34]; the intermediate number-squeezed state, by Baseia et al. [35,36], which in-terpolates between a number state |N 〉 and a squeezed state |z; �〉 = S(z)|�〉, whereS(z) = exp[(za†

2

+ z∗a2)=2] is the squeeze operator [37–39]; etc.In view of the large number of states recently disseminated in the literature, and

also due to the great di�culty in their experimental realization, a justi�able pessimismdominates part of the community. In fact, even the preparation of the number state,a “battle horse” of quantum optics, is di�cult to realize. Also, the realization of thebinomial states was not reached, the various suggestions to this end seeming to be in-e�cient [40,41]. In this scenario, various states of the quantized light �eld seem to bevery “arti�cial”, or “academic”. However, one should remember that the experimentalrealization of the squeezed state [42] had to wait for 15 years since its theoreticalprediction (see Ref. [37–39]). Recently, Barnett and Pegg [43,44] suggested an inter-esting proposal to measure the phase of an arbitrary (pure or mixed) state, in which anapparent “exotic” state, called reciprocal-binomial state (RBS), plays a crucial role inthe experiment. The authors then claimed that in view of recent advances in quantumstates engineering [1–5] it could be possible in the future to realize the RBS, allowingone to measure the �eld-phase [43,44]. Of course, following a line of pessimism, manyreaders would fail to see any relevance in an experiment which crucially depends onthe existence of an “exotic” state, such as the RBS, mainly remembering that eventhe realization of the binomial state (BS) is not available yet. Nevertheless, a feasibleexperiment realizing the RBS was proposed in the literature [45], opening a new frontof optimism in this area.In the present work we will introduce a Generalized Superposition of two Squeezed

States (GSSS), which interpolates between these two states. It is shown that this stateincorporates various states studied in the literature for convenient choices of the in-volved parameters. In Section 2 we de�ne our GSSS and in Section 3 we propose itsgeneration through the use of “quantum states engineering”, by a non-unitary evolutionapplied to an initial squeezed state previously prepared inside a cavity. In Section 4we study nonclassical properties of our GSSS. Section 5 contains the comments andconclusion.

348 Y.A. Barbosa et al. / Physica A 280 (2000) 346–361

2. General superposition of two arbitrary squeezed states (GSSS)

We de�ne the (pure) GSSS as the superposition of two squeezed states (with � ∈[0; 1], ∈ [0; 2�])

|(�; )〉= �[√

�|z1; �1〉+√1− �ei |z2; �2〉] (1)

where � is a normalization factor

�= [1 + 2√

�(1− �)Re(ei 〈z1; �1|z2; �2〉)]−1=2

=

[1 + 2

√�(1− �)Re

(ei

∞∑n=0

C(1)∗

n C(2)n

)]−1=2: (2)

Next, we set

|zj; �j〉= |�j〉= S(zj)|�j〉=∞∑n=0

C( j)n |n〉; j = 1; 2 : (3)

where S(zj) is the squeeze operator [37–39]

S(zj) = exp[(zja†2 − z∗j a

2)=2] (4)

with zj = rj exp(i�j) and

C( j)n = 〈n|zj; �j〉

=(�j=2�j)n=2√

n!�jexp

[−|�j|22

+(

�∗j2�j

)�2j

]Hn

(�j√2�j�j

); (5)

�j = Rj exp(i�j); �j = �j�j + �j�∗j ; (6)

�j = cosh rj; �j = exp(i�j)sinh rj : (7)

The properties exhibited by the GSSS will be studied in Section 4. Note that thisstate includes, as particular cases: (i) a generalized superposition of two coherent states,for r1 = r2 = r = 0 [46]; (ii) the superposition of two squeezed-vacuum states, for�1 = �2 = 0; (iii) even and odd cat-states, composed by coherent states, for r1 = r2 = 0and �2 =−�1; (iv) the even and odd cat-states composed by squeezed-coherent states[47] including the single coherent state (�=1; z=0) and the single squeezed-coherentstate (�= 1; z 6= 0).

3. Generation of the GSSS

One of the main aims of quantum optics is the manipulation of quantum propertiesof light �eld. Since the achievement of a successful generation of squeezed states (seeRefs. [42,48–50]), considerable e�orts have been devoted towards the production ofother states having interesting properties. These methods consist in using a convenient�eld state entering a nonlinear media [51–58] or using what is called as quantum state


Fig. 1. Schematic representation of the experimental setup for generation of the GSSS, based on QSE.

engineering (QSE) [1–5]. In the latter case, a regular beam of atoms interact, one at atime, with a �eld inside a cavity. These atom-�eld interactions can be resonant [59],or dispersive (far o�-resonance) [60,61]. Here we will employ the (more interesting)second alternative since it employs a smaller number of atoms (only one in this case)in comparison with the resonant case. This is very important due to crucial problemsconcerning with decoherence of a state prepared in a cavity and e�ciency of detection,which decreases when the number of atoms is large [60–65].Fig. 1 displays the experimental setup for preparing our desired state. It consists of

a single two-level atom (levels |e〉 and |g〉 for excited and ground states, respectively),a Ramsey-zone R1, a high-Q cavity C, a second Ramsey-zone R2 and a state-selectivedetector De(Dg). The two-level atom is considered as a circular Rydberg atom, whichis suitable for preparing and detecting atom-�eld long-lived correlations [62,63]: theypresent a strong coupling to microwaves and a very long radiative decay time, of theorder of lifetimes for high-Q cavities (� ≈ 0:1 s). Furthermore, Rydberg atomic statesare counted with high e�ciency by state-selective �eld ionization detectors [64,65].The atom, in the excited state |e〉, traverses the �rst Ramsey-zone R1 su�ering a

classical pulse which prepares the atom in the superposition state

|A(0)〉= 1√2(Cg|g〉+ ei Ce|e〉) (8)

and enters the cavity C, in which a �eld is previously prepared in a state |F(0)〉,here assumed as a squeezed state

|F(0)〉= |�; z〉 : (9)

The dispersive interaction-Hamiltonian reads [60,61]

H= ˝��za†a : (10)

where � is a coupling constant, �z= |e〉〈e|− |g〉〈g| is the atomic inversion operator anda†(a) is the creation (annihilation) operator for photons. As a result of this interaction,the e�ective action of Hamiltonian (10) upon the initial state of the whole system,|AF(0)〉= |A(0)〉 ⊗ |F(0)〉, is given by

|AF(�)〉= (e−i�(�)n|e〉〈e| − |g〉〈g|)(Cg|g〉+ ei Ce|e〉)|�; z〉=Cg|g〉|�; z〉+ ei Ce|e〉|e−i�(�)�; e−2i�(�)z〉 (11)


with �(�) = 2�=�, where is the Rabi frequency, � is the interaction time betweenatom and �eld in the cavity and �=!−!0 is the detuning parameter [!(!0) being the�eld (atomic) frequency]. In Eq. (11) the equality e−i�(�)n|�; z〉 = |e−i�(�)�; e−2i�(�)z〉was employed [66]. Next, the atom traverses the second Ramsey zone R2, su�ers againa �=2 pulse, leading the components |e〉 → |e〉+ |g〉 and |g〉 → |g〉− |e〉. As result, thestate in Eq. (11) is transformed into

|AF(�)〉= |g〉(Cg|�; z〉+ ei Ce|e−i�(�)�; e−2i�(�)z〉)−|e〉(Cg|�; z〉 − ei Ce|e−i�(�)�; e−2i�(�)z〉) : (12)

Thus, note that the detection of the atom in the state |g〉 or |e〉 results in the �eld-stateinside the cavity, |+

F (�)〉 or |−F (�)〉, given by

|±F (�)〉=N±(Cg|�; z〉 ± ei Ce|e−i�(�)�; e−2i�(�)z〉) (13)

where N± is a normalization constant.Now, the state in Eq. (13) coincides with our GSSS for the identi�cation

Cg =√

�; Ce =√1− � ;

�1 = �; �2 = e−i�(�)� ;

z1 = z; z2 = e−2i�(�)z : (14)

An interesting case occurs when �= 12 (hence Cg = Ce = 1=

√2), = 0 and �= �. In

this situation, we �nd from Eqs. (13) and (14) that

|±F (�)〉=N±(|�; z〉 ± | − �; z〉) (15)

which is an even (+) or odd (−) squeezed Schr�odinger’s cat state, whereN± = {2[1± exp(−2|�′|2)]}−1=2 (16)

with �′ = ��+ ��∗; � = cosh r; �= ei�sinh r

4. Statistical properties of the GSSS

4.1. Oscillations in the photon number distribuition

By expanding our GSSS in Fock’s basis

|(�; )〉=∞∑n=0

Cn(�; )|n〉 ; (17)

where

Cn(�; ) = 〈n|(�; )〉= �[

√�〈n|�1〉+

√1− � ei 〈n|�2〉]

= �[√

� C(1)n +√1− � ei C(2)n ] : (18)


Fig. 2. Plots of the photon-number distribution Pn; versus n, for di�erent values of the parameter �.

we obtain the photon number distribution Pn(�; ) for our state. We have

Pn(�; ) = |〈n|(�; )〉|2= �2[�P(1)n + (1− �)P(2)n + 2

√�(1− �)Reei C(1)

∗n C(2)n ] ; (19)

where P(i)n = |C(i)n |2 = |〈n|�i〉|2 stands for the probability of �nding n photons in thecomponent |�i〉; given in Eq. (3).Fig. 2 shows the behaviour of Pn versus n for R1=R2=5; r1=r2=0:5; �1=0; �2=

�; �1 = �2 = 0 and = 0: Note that when � = 0 (or � = 1) only a single componentremains in the scenario. In this case Pn shows no oscillation (Fig. 2a). When � 6= 0,say � = 0:05 (or � = 0:95) the distribution Pn shows some oscillations (Fig. 2b). For� → 0:5 the oscillations in Pn become complete, namely, for certain values of n, Pn

vanishes. As is known in the literature [67], a zero value of Pn corresponds to theexistence of some kind of nonclassical e�ect in the light �eld represented by this state.Solid (dotted) line in Fig. 2c is for =0 ( = �): note that Pn=0 for n odd (n even).In this case, the superposition corresponds to an even (odd) squeezed state. A betterunderstanding of such interference e�ects can be obtained from the Eq. (19) written


in the form

Pn(�; ) = �2�[1 +W 2n + 2Wn cos(�n)] ; (20)

where we have employed

C( j)n = �( j)n ei�( j)n ; (21)

Wn =

√1− ��

�(2)n =�(1)n ; (22)

�n = + �(1)n − �(2)n = +��n : (23)

As a consequence we have from Eq. (20),

Pn(�; ) = �2�(1 +Wn)2 (24)

when �n = +��n = 2M�; andPn(�; ) = �2�(1−Wn)2 (25)

when +��n=(2M+1)�; M=0; 1; 2; : : : . Hence, we obtain constructive (destructive)interference when +��n=2M�[ +��n=(2M +1)�]. In both cases, the interferenceis partial: to be complete we must have Wn=1: For example, according to Eq. (22) if�=0:5 (�=0:2) this is accomplished for �(2)n =�(1)n (�(2)n =2�(1)n ). Oscillations in Pn

are interpreted as interference in the phase space [68–71]. The in uence of the squeezeparameter r upon these oscillations can be seen in Fig. 3, showing the plot of Pn versusn for various values of r. Note that when r goes from r = 0:5 (Fig. 3a) → r = 1:0(Fig. 3b) → r = 1:5 (Fig. 3c) oscillations in Pn increase. Hence, interference in thephase space increases when r increases. The parameter R is responsible for the positionof the “envelope” of oscillations, as shown in Fig. 4, for R=5(a); R=6(b); R=7(c);with R= R1 = R2:

4.2. Sub-Poissonian statistics

What about the average value 〈n(�; )〉 in the superposition state | (�; )〉? We have〈n(�; )〉= 〈(�; )|n|(�; )〉

= �2[√

�〈�1|+√1− �e−i 〈�2|]n[

√�|�1〉+

√1− �ei |�2〉]

= �2[�〈�1|n|�1〉+ (1− �)〈�2|n|�2〉+ 2√

�(1− �)R (ei 〈�1|n|�2〉)] :(26)

The application of Eqs. (3) and (5) in Eq. (26) gives

〈n(�; )〉= �2[�

∞∑n=0

n|C(1)n |2 + (1− �)∞∑n=0

n|C(2)n |2

+ 2√

�(1− �)Re

(ei

∞∑n=0

nC(1)∗

n C(2)n

)]: (27)


Fig. 3. Same as in Fig. 2, for di�erent values of the parameter r.

Similarly, we have

〈n2(�; )〉= �2[�

∞∑n=0

n2|C(1)n |2 + (1− �)∞∑n=0

n2|C(2)n |2

+ 2√

�(1− �)Re

(ei

∞∑n=0

n2C(1)∗

n C(2)n

)]: (28)

From Eqs. (27) and (28) we obtain the Mandel Q-factor

Q(�; ) =�n2(�; )− 〈n(�; )〉

〈n(�; )〉 ; (29)

where �n2(�; ) = 〈n2(�; )〉 − 〈n(�; )〉2 is the variance of the photon-number in thestate |(�; )〉, obtained from Eqs. (27) and (28).Fig. 5 shows the plots of Q versus R for: (i) the GSSS, for r = 0:5; �1 = 0; �2 =

�; �1 = �2 = 0; � = 0:5 and = � (solid line); (ii) only a single squeezed state, forr=0:5; �1 =0; �1 =0 (dotted line). Note that in this case the sub-Poissonian statisticsis greater in the GSSS than that exhibited by a single component of the GSSS, with


Fig. 4. Same as in Fig. 2, for di�erent values of the parameter R.

Fig. 5. Plots of the Mandel Q-factor as function of R, for a single component (dotted line) and for the GSSS(solid line).

a maximum value Q ' −0:8 being obtained for R = 0:65. Fig. 6 stands for Q versusr, for R= 3; �1 = 0; �2 = �; �1 = 0; �= 0:5 and = 0: Note that the sub-Poissonianstatistics increases when �2 diminishes, the e�ect reaching a maximum near Q ' −0:6when �2 = 0: Fig. 7 shows Q versus � and , for R = 0:55; r = 0:12; �1 = 0; �2 = �


Fig. 6. Plots of the Mandel Q-factor as function of r; for various values of the parameter �2.

and �1 = �2 = 0 (Figs. 7a–c). For these parameters, sub-Poissonian statistics reachesa maximum near Q ' −1 for � = 0:5 and = �. Fig. 8a, shows the ellipse repre-senting each squeezed state composing the GSSS, whose Wigner function is shown inFig. 8b. Maximum sub-Poissonian e�ect (Q ' −1) is also obtained for other values ofthe parameters involved. For example, Q ' −1 for R = 0 (corresponding to the caseof two squeezed-vacuum states), r = 0:10; �1 = 0; �2 = �; �1 = 0; �2 = �=2; � = 0:5and = 0 (not shown in �gures).

4.3. Quantum noise and squeezing

De�ning the quadrature operators,

X 1 = (a+ a†)=2 ; (30)

X 2 = (a− a†)=2i ; (31)

we can calculate their variances in the GSSS,

�X2j = 〈X 2

j 〉 − 〈Xj〉2; j = 1; 2 : (32)

To this end, we �rstly obtain

⟨X 1(�; )

⟩=

⟨(�; )

∣∣∣∣∣ (a+ a†)2

∣∣∣∣∣(�; )⟩

=�2

2

{�

( ∞∑n=0

√n+ 1C(1)n+1(C

(1)n )∗ + c:c:

)


Fig. 7. Three-dimensional plots of the Mandel Q-factor as function of � and (a); (b) and (c) show cutsin (a), where sub-Poissonian e�ect is maximum (Q ' −1).

+ (1− �)

[ ∞∑n=0

√n+ 1C(2)n+1(C

(2)n )∗ + c:c:

]

+√

�(1− �)

[ei

∞∑n=0

√n+ 1(C(1)

∗n C(2)n+1 + C(1)

∗

n+1C(2)n ) + c:c:

]}

(33)

and

⟨X21(�; )

⟩=

⟨(�; )

∣∣∣∣∣∣[(a+ a†)2

]2∣∣∣∣∣∣(�; )⟩

=�2

4

{�

[ ∞∑n=0

C(1)∗n C(1)n+2

√(n+ 1)(n+ 2) + c:c:


Fig. 8. Ellipses representing each squeezed state composing our GSSS (a) and the Wigner function corre-sponding to their superposition yielding the GSSS (b).

+∞∑n=0

|C(1)n |2(2n+ 1)]+ (1− �)

×[ ∞∑

n=0

C(2)∗n C(2)n+2

√(n+ 1)(n+ 2) + c:c:+

∞∑n=0

|C(2)n |2(2n+ 1)]

+√

�(1− �)Re

[ei ( ∞∑

n=0

√(n+ 1)(n+ 2)(C(1)

∗n C(2)n+2

+C(1)∗

n+2C(2)n +

∞∑n=0

(2n+ 1)C(1)∗

n C(2)n ))]}

: (34)

The substitution of Eq. (34) and (square of) Eq. (33) into Eq. (32) gives the variancein the quadrature X 1. In a similar way, we obtain

⟨X 2(�; )

⟩=

⟨(�; )

∣∣∣∣∣ (a− a†)2i

∣∣∣∣∣(�; )⟩

=�2

2

{�

[ ∞∑n=0

√n+ 1i

C(1)n+1C(1)∗n − c:c:

]

+(1− �)

[ ∞∑n=0

√n+ 1i

C(2)∗n C(2)n+1 − c:c:]

+√

�(1− �)

[ei

∞∑n=0

√(n+ 1)(C(1)∗n C(2)n+1 − C(1)∗n+1C

(2)n ) + c:c:

]}

(35)


and

⟨X22(�; )

⟩=

⟨(�; )

∣∣∣∣∣∣[(a− a†)2i

]2∣∣∣∣∣∣(�; )⟩

=�2

4

{�

[ ∞∑n=0

∣∣C(1)n

∣∣2 (2n+ 1)−( ∞∑

n=0

C(1)∗n C(1)n+2

√(n+ 1)(n+ 2)

+c:c:

) ]+ (1− �)

[ ∞∑n=0

∣∣C(2)n

∣∣2 (2n+ 1)

−( ∞∑

n=0

C(2)∗n C(2)n+2

√(n+ 1)(n+ 2) + c:c:

)]

+√

�(1− �)Re

[ei ( ∞∑

n=0

(2n+ 1)C(1)∗n C(2)n

−∞∑n=0

√(n+ 1)(n+ 2)(C(1)

∗n C(2)n+2 + C(1)

∗

n+2C(2)n )

)]}: (36)

The substitution of Eqs. (35), (36) into Eq. (32) allows one to obtain the variance inthe quadrature X 2. Squeezing e�ect occurs if �X

2j ¡

14 ; j = 1 or 2 (not both simul-

taneously), which means reduction of quantum noise below that found in a vacuumstate |0〉, concerning one of the two quadratures X 1 and X 2.Fig. 9 shows the plots of variances as function of the interpolating parameter �:

for the variance �X21 the squeezing e�ect occurs around � ≈ 0 and � ≈ 1 whereas

for variance �X22 the e�ect occurs around � ≈ 0:5. So, the squeezing e�ect in the

quadrature X 1, already shown by a single component (obtained for � = 0 or � = 1),diminishes when the superposition is set on (� ≈ 0:5). However, the reverse occursfor the quarature X 2: two squeezed states not showing the e�ect in X 2; will exhibitit when superposed. The squeezing e�ect in �X

21 is small for = 0 and = 2� but

increases when → � (about 30%). The reverse occurs for �X22.

5. Comments and conclusion

We have studied a generalized superposition of two squeezed states, its genera-tion (Section 3) and nonclassical properties (Section 4) such as oscillations in thephoton-number distribution (Section 4.1), sub-Poissonian statistics (Section 4.2) andsqueezing (Section 4.3). The strength of each e�ect depends on the choice of the setof involved parameters (�; ; R; r; � and �), as shown in the various plots [Figs. (2)–(10)]. The present GSSS allows one to interpolate continuously between two arbitrarysqueezed states, by making the parameter � vary from �=0 to �=1. In this way, wemay pass, e.g., from a squeezed state having super-Poissonian statistics to a squeezed


Fig. 9. Three-dimensional plots of variances �X21 and �X

22 versus � and (a), showing cuts (b) and (c)

where the squeezing e�ect is maximum and minimum.

state having sub-Poissonian statistics. In view of the various particular cases includedby the GSSS, many other routes of interpolation are allowed. This variety of alternativesand characteristics makes the GSSS potentially interesting. For example, one specialcase of the GSSS is the even and odd squeezed “cat-state” (see Eq. (15)). If, for somechoice of the parameters involved, this state becomes resistent to decoherence [72] thenit would be important for teleporting processes [73–77], where the loss of coherence(due to the interaction of the �eld with its “environment”) is crucial in the experiment.Another potential application comes from Fig. 7, where Q ' −1 for the GSSS. So,we are faced with an interesting result: a state which is not a Fock state (see Fig. 8b)showing almost the same Sub-Poissonian e�ect. On the other hand, the generation ofthe GSSS can be achieved through the “standard” method of QSE (Section 3), using adispersive atom-�eld interaction in a high-Q cavity, where a �eld initially prepared in asingle squeezed state |z; �〉 evolves nonunitarily to our wanted state. For the generationof a single squeezed state inside a cavity, see Refs. [78,79]. As shown in Section 3,the arbitrary coe�cients of the initial atomic state (see Eq. (8)) are transferred to the�nal �eld state prepared in the cavity (see Eq. (1)). So, there is a one-to-one corre-spondence between these two states, the knowledge of one of them allowing one toknow the other, thus constituting a monitoring scheme for atomic state, or for the �eldstate, depending on which of them is experimentally easier to be controlled. Concerning


the role played by the parameters of the GSSS (�; ; Rj; rj; �j; �j; j = 1; 2) upon thecharacteristics of this state, we should emphasize that, e.g., with respect to oscillationsin the photon-number distribution Pn the parameter � controls the depth of the holesin Pn (Fig. 2) whereas the parameter controls their positions (cf. Eqs. (20) and(23)); the parameter r controls the number of holes of Pn (Fig. 3) whereas the param-eter R controls the position of the whole envelope (Fig. 4). As mentioned before inSection 4.1, when |C(1)n =C(2)n | = 1 the destructive interference in Pn is complete for� = 0:5; this is not the case when |C(1)n =C(2)n | 6= 1, a result coming from Eqs. (20)–(25). Whatever be the situation, a zero in Pn will correspond to some nonclassicale�ect of the light �eld [67]. Now, concerning the sub-Poissonian e�ect the in uenceof the parameters seems to be more complicated, the role played by one of them de-pending on the values of the others. For example, the sub-Poissonian e�ect reachesits maximum (Q ' −1) for � = 0:5 if ' �, but this e�ect disappears if ¡�=2 or ¿ 3�=2 (Fig. 7). The same is valid for the squeezing e�ect (Fig. 9).

Acknowledgements

This paper was partially supported by the Brasilien agency, CNPq.

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Generalized superposition of two squeezed states: generation and statistical properties

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Transcript of Generalized superposition of two squeezed states: generation and statistical properties