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1 September 1997

Physics Letters A 233 (1997) 291-296

PHYSICS LETTERS A

Even and odd nonlinear coherent states

Stefano Mancini a,b a e Fisica. Universita di &merino. I-62032 Camerino, Italy

b Istituto Nazionale di Fisica Nucleare. Sezione di Peru&a, Peru&a, Italy

Received 28 May 1997; accepted for publication 24 June 1997 Communicated by V.M. Agranovich

Abstract

The concept of even and odd nonlinear coherent states is introduced. They result as Schtidinger cat states for deformed fields. The statistical properties of such states are investigated with particular attention to their nonclassical effects. @ 1997 Elsevier Science B.V.

PACS: 42.50.D~; 03.65.Ca

1. Introduction

The concept of coherent state was introduced by Glauber [ 11, and since then attained an important position in

the study of quantum optics. This is because the coherent states not only have physical substance but also yield a very useful representation. The usual coherent states advanced by Glauber are eigenstates of the annihilation operator a of the harmonic oscillator. Based on this work, the even and odd coherent states were introduced in Ref. [ 21. The even (odd) coherent states are the symmetric (antisymmetric) combination of coherent states. They are two orthonormalized eigenstates of the square of the annihilation operator a, and essentially have two kinds of nonclassical effects: the even coherent state has a squeezing but no antibunching effect, while the odd

coherent state has an antibunching but no squeezing effect [ 31. On the other hand, quantum groups [4], introduced as a mathematical description of deformed Lie algebras,

have given the possibility of generalizing the notion of creation and annihilation operators of the usual oscillator

and to introduce q-oscillators [ 561. The latter were interpreted [7,8] as nonlinear oscillators with a very specific type of nonlinearity, in which

the frequency of vibration depends on the energy of these vibrations through the hyperbolic cosine function

containing a parameter of nonlinearity. This interpretation of q-oscillators becomes obvious if one uses the classical counterpart of the original

quantum q-oscillators. This observation suggests that there might exist other types of nonlinearity for which the frequency of oscillation varies with the amplitude via a generic function f; this leads to the concept of f-oscillators devised in Ref. [ 91. Then, the notion of f-coherent states was straightforwardly introduced [ 91, and the generation of such nonlinear coherent states enter in the real possibilities of trapped systems [ lo].

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292 S. Mancini/Physics L.etters A 233 (1997) 291-296

The aim of the present paper is to introduce the generalized even and odd coherent states by means of superposition of f-coherent states, and to study their statistical properties and their nonclassical effects.

2. Even and odd f-coherent states

Let the operators A and A+ represent the dynamical variables to be associated with the quantum f-

oscillators [ 91. They result as distortions of the usual annihilation and creation operators a and at,

A=af(A) = f(ii+ ~)a, t A=aa, (1)

A+ = f(k>a+ =a+f@+ I), (2)

where f is intended as an operator-valued function of the number operator (here it is assumed Hermitian). In

general, it can be made dependent on continuous parameters, in such a way that, for given particular values,

the usual algebra is reconstructed. This is the case of q-deformations [ Ill. As a consequence of the noncanonical transformation ( 1) , the commutation relation is not preserved,

[A,A+] = (A + 1) f'(ii+ 1) - Af2(A).

Of course the usual algebra is restored whenever f + i. A coherent state for the above f-deformed algebra should satisfy the equation [9]

(3)

Ala, f) = +, f),

and looking for the decomposition of ((Y, f) in the Fock space, one can find

Nf =

with the convention

f(n)! =f(O)f(l>...f(n>.

Now the even (+) and odd ( -) f-coherent states may be defined in a straightforward manner as

b,_f)k = N+(lwf) f I - wf)),

where the constants Nk are determined from the normalization condition

h(a,fl%f)+ = 1,

and the result is

(4)

(5)

(6)

(7)

(8)

(9)

Nh = (10)

S. Mancini/Physics Letters A 233 (1997) 291-296 293

It is easy to check that the states (8) are eigenstates of the square of the f-annihilation operator

A2)o, f)k = a2Ja, &.

They can be considered as Schrodinger cat states for a deformed field.

(11)

3. Statistical properties of even and odd f-coherent states

3.1. Photon distribution

By using Eqs. (8) and (5)) the even and odd f-coherent states can be expanded in the number basis as

(12)

From this expression it becomes evident that the even (odd) f-coherent state has a vanishing probability of containing an odd (even) number of photons. As a consequence the number probability is found to be strongly oscillating, which is a peculiarity of highly nonclassical states. These oscillations are also present in the nondeformed even and odd coherent states, but here the profile of the distribution will be determined by f.

The mean number of quanta is given by

and the dispersion is

Ita)” * (-Ly)“/2n2 _ N2 N2 n![f(n)!]2 f f .

Then, the second order correlation function [ 121 follows immediately from the relation

and explicitly it is given by

gg’(()) = e I(aY * (-a)“12 n=2 (n -2)![f(n)!12 (

N*Nf 2 iKy-o,;~~12n -2. n=o . 1

(13)

(14)

(16)

The r.h.s. of Eq. ( 16) can be less or greater than 1, depending on the particular form of f, producing either bunching (super-Poissonian statistics) or antibunching (sub-Poissonian statistics) independently of the symmetry of the state, in contrast with the usual algebra where antibunching effects are shown only by odd coherent states [ 31.

This different behavior is shown in Fig. I, where the function g?‘(O) for an even q-deformed coherent state is compared with that obtained in absence of deformation. The q-deformation is easily constructed by taking [9]

f(n) = J 19” - 4-” n q-q-1 ’ f(O) = 1, q E Ii%. (17)

294 S. Mancini/Physics Letters A 233 (1997) 291-296

0.6

Fig. 1. The second order correlation function gy’ ( ) 0 versus the field amplitude IGJ( (when argcY = 0) for an even q-deformed coherent state (solid line) when q = 2.5 and for an undeformed even coherent state, i.e. q = 0 (dashed line).

3.2. Squeezing and correlation

The dispersion and correlation of the quadratures

a+ a+ a- a+

x=Jz’ P= iv5 ’

can be obtained by noting that

(x)* = (P)i = 07

and

O” (a2)* = NiN+ c

[ (,*)n--2 f (-,*)n-*] [ (Cy)n Zk (-(Y)“]

n=2 (n-2)!f(n)f(n- I)[f(n-2)!]2 .

Hence, by collecting Eqs. ( 13) and (20), the x quadrature dispersion can be written as

O3 + ;N;N;c

[(a*)“-* -f (-a*y-*] [ (Cy)n Ik (-a)” ] + C.C.

n=2 (n-2>!f(n)f(n-l)[f(n-2)!]2 .

For the other quadrature one can get analogously

O” ((a)“* (-a)“12 cP.h =; +N:N:C n![f(n)‘]2 n

n=o

O” [((Y*)n-21t (-a*)“-*][(a)“k (-a)“] +c.c. - ~N:I+ C

n=2 (n-2)!f(n)f(n- l)[f(n-2)!]2 .

(18)

t 19)

(20)

(21)

(22)

Depending on the function f(n) the dispersion u,, l (a,,*) may become less than l/2. This means squeezing independently of the symmetry of the state, again in contrast with the nondeformed algebra where only the even coherent states exhibit squeezing. The deformation leading to the harmonious states [ 131, i.e.

S. ManciniIPhysics Letters A 233 (1997) 291-296 295

1.5

1.25

1.

arg c1 @-ad) 0.75

0.5

0.25

0~

I

0 0.5 1 1.5 2

-

-

I al

0 0.1 0.2 0.3 0.4 0.5

Ial

Fig. 2. The p quadrature dispersion function a,,- versus the field amplitude Ial (when arga = 0) for an odd harmonious coherent state

(solid line) and for an undeformed odd coherent state (dashed line),

Fig. 3. The phase space lines along which the correlation functions qxP, + (solid line) and rxP, - (dashed line) take zero value are shown

in the case of a deformation given by F.q. (25) with 7 = 0.88.

f(n) = 5, f(O) = 1,

represents a particular example of such a properties. Then, in Fig. 2 the function up,- for an odd harmonious coherent state is contrasted with the one obtained in the absence of deformation.

Finally one can calculate the correlation of quadrature components in even or odd f-coherent states as

O” [(cY*)“-Q(-a*)“-*][(a)“&(-a)“] -c.c. gxy,f = &‘J;x

n=2 (n-2>!f(n)f(n- 1)If(n>!J2

(24)

In general this is not equal to zero, so the introduced states have the property of being correlated states [ 141; it could also be negative leading to contractive states [ 151. However, the invariant cX, fgp, * - a$ f is larger than l/4, thus the even and odd f-coherent states do not minimize the Schriidinger uncertainty relation [ 161.

The quantity in Eq. (24) is investigated for the even and odd nonlinear coherent states determined by

f(n) =Lf,(g*)m+ 1>~11(17*>1-‘~ rl E R (25)

where Lz, is an associated Laguerre polynomial. Such a nonlinearity can be generated in a trapped and bichromatically laser-driven ion far from the Lamb-Dicke regime [ lo]. Then, for both cases, deformed and nondeformed, (T,~, * remains zero ‘v’lal when argcy = mr/2, m E N. But, as shown in Fig. 3, the quadrature correlations, in the deformed case, can have a zero vanishing value at a given field amplitude also for arg LY # mn-12, while in the undeformed case they cannot.

4. Conclusions

Summarizing, the introduced even and odd nonlinear coherent states have rather different statistical properties from those of the usual even and odd coherent states. Several examples are presented showing that nonclassical effects no longer depend on the symmetry of the states but rather on the introduced nonlinearity. In view of

296 S. MancitdPhysics Letters A 233 (1997) 291-296

their singular properties, states of the type considered might be of general interest, for example in the optical and microwave fields, in molecular vibrations or nuclei vibrations for polyatomic molecules etc. On the other

hand, they turned out to be interesting from the point of view of quantum groups too.

Acknowledgement

The author gratefully acknowledges V.I. Man’ko and P. Tombesi for fruitful discussions.

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