About a new family of coherent states for some SU(1,1) central fieldpotentialsDusan Popov, Vjekoslav Sajfert, Nicolina Pop, and Viorel Chiritoiu Citation: J. Math. Phys. 54, 032103 (2013); doi: 10.1063/1.4795137 View online: http://dx.doi.org/10.1063/1.4795137 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v54/i3 Published by the AIP Publishing LLC. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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JOURNAL OF MATHEMATICAL PHYSICS 54, 032103 (2013)

About a new family of coherent states for some SU(1,1)central field potentials

Dusan Popov,1 Vjekoslav Sajfert,2 Nicolina Pop,1,a) and Viorel Chiritoiu1

1Department of Physical Foundations of Engineering, “Politehnica” University of Timisoara,Bv. Vasile Parvan No. 2, RO-300223 Timisoara, Romania2Technical Faculty “M. Pupin” Zrenjanin, University of Novi Sad, Djure Djakovica bb,23000 Serbia

(Received 19 September 2011; accepted 13 February 2013; published online 19 March 2013)

In this paper, we shall define a new family of coherent states which we shall callthe “mother coherent states,” bearing in mind the fact that these states are inde-pendent from any parameter (the Bargmann index, the rotational quantum numberJ, and so on). So, these coherent states are defined on the whole Hilbert space ofthe Fock basis vectors. The defined coherent states are of the Barut-Girardello kind,i.e., they are the eigenstates of the lowering operator. For these coherent states weshall calculate the expectation values of different quantum observables, the cor-responding Mandel parameter, the Husimi’s distribution function and also the P-function. Finally, we shall particularize the obtained results for the three-dimensionalharmonic and pseudoharmonic oscillators. C© 2013 American Institute of Physics.[http://dx.doi.org/10.1063/1.4795137]

I. INTRODUCTION

The aim of this paper is to examine the coherent states of some central field potentials, usingthe density matrix approach. These potentials have the SU(1,1) group symmetry and so it canbe examined in a common manner, bearing in mind that their energy spectrum is linear withrespect to the principal quantum number n and also that it depends on the rotational quantumnumber J (through an analytical function k(J)). An important observation is that, from the energeticpoint of view, these two quantum numbers are mutually disconnected, i.e., the vibrational androtational motions are decoupled. As it is well known, the coherent states approach is especiallyeffective for the physical systems for which the Hamiltonian has a dynamical symmetry group.1

We shall consider the case when the Hamiltonian H of the examined quantum system is linear inthe group generators Ki of an irreducible unitary representation of the corresponding Lie algebraH = �

(�0 K0 − �1 K1 − �2 K2

). There are three different cases depending on the form of the

vector � = (�0, �1, �2). Such a group is SU(1,1). Due to the fact that all these potentials havethe SU(1,1) group symmetry, it is possible to built the same kind of coherent states, with the sameinternal constant structure ρ(n, k), which are not dependent on the rotational quantum number J,i.e., to built on the whole infinite-dimensional Fock space of the energy eigenvectors { |n, k(J)〉; n,J = 0, 1, 2, . . . , ∞ }. For this reason, we shall call these coherent states “mother coherent states”(MCSs). In what follows we shall build the coherent states of the Barut-Girardello kind, i.e., thestates which are the eigenstates of the lowering operator K−. As far as we know such coherent stateshave not been built so far in the physical literature.

In Sec. II, we recall the general solving of the Schrodinger equation of the stationary states inthe three-dimensional case in order to introduce some constants that we need in the rest of the paper.Particularizing these constants, we have to examine some particular potentials: the three-dimensional

a)Author to whom correspondence should be addressed. Electronic mail: nicolina.pop@et.upt.ro. Tel.: + 400256403093.Fax: + 400256403091.

0022-2488/2013/54(3)/032103/21/$30.00 C©2013 American Institute of Physics54, 032103-1

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032103-2 Popov et al. J. Math. Phys. 54, 032103 (2013)

isotropic harmonic oscillator (HO-3D) potential and the pseudoharmonic oscillator (PHO) potential(in Sec. VI). In Sec. III, we build the MCSs and examine some of their properties, while Sec. IV isdevoted to the statistical properties of these states. In Sec. V, we deal with the thermal states and weobtain the expressions of the Husimi’s function and also of P-function. In the end, in Sec. VII, weformulate some concluding remarks.

II. DENSITY MATRIX APPROACH OF THE SU(1,1) CENTRAL FIELD POTENTIALS

Consider a time-independent quantum anharmonic oscillator with the reduced mass mr andangular frequency ω that moves in a central field potential with a singularity of the inverse squareinteraction kind, of the following form:

V (r ) = mrω2

2r2 + �

2

2mr

c

r2+ E0, (1)

where c is a constant which will be determined later and E0 is a term which will specify the originof energy scale.

This quantum system displays a dynamical symmetry associated with the SU(1,1) group andso we may introduce the standard basis |n, k〉 for the Lie algebra of this group. As we will showlater, n = 0, 1, 2, . . . is the energy quantum number and k = k(J) plays the role of Bargmann indexand it depends on the rotational quantum number J = 0, 1, 2, . . . . The three group generators are(K0, K1, K2) or, more convenient, (K0, K−, K+); their group structure is1

K± = K1 ± i K2,[K0, K±

] = ±K± , (2)[K+, K−

] = −2K0.

Consequently, the action of these operators on the basis vectors |n, k〉 is

K0|n, k〉 = (n + k) |n, k〉,K−|n, k〉 = √

n(n + 2k − 1)|n − 1, k〉,K+|n, k〉 = √

(n + 1)(n + 2k)|n + 1, k〉 ,

(3)

while the corresponding Casimir operator (i.e., an operator which commutes with all the groupgenerators) for this basis is

C = K 20 − 1

2

(K+ K− + K− K+

) = k(k − 1) I

C |n, k〉 = k(k − 1)|n, k〉. (4)

The operators K− and K+ behave as the lowering and the raising operators because actingon the basis vectors |n, k〉, the energy quantum number decreases and increases by one unit. Sincethe Casimir operator can be diagonalized simultaneously with its generators, the eigenvalues of theCasimir operator may be used to label the unitary irreducible representations (UIR) of this group.Between all UIR our interest is focused on the positive discrete series D+ (k), with the real numberk > 0 (called the Bargmann index), i.e., the states |n, k〉 that diagonalize the compact operator K0,as we have already seen earlier.2 The interest in positive discrete series is justified by the factthat coherent and squeezed states are concerned with these kinds of UIR. Since the SU(1,1) is anon-compact group, all its UIR (also the positive series) are infinite-dimensional.

In order to introduce the density matrix approach, we need to recall the resolution of theSchrodinger equation of stationary states

H (�r )�(�r ) ≡[− �

2

2mr∇2 + V (�r )

]�(�r ) = E�(�r ), (5)

which can be solved using the standard procedure, i.e., by separating the radial r and angular (θ , ϕ)variables

�(�r ) = R(r ) YJ M (θ, ϕ). (6)

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032103-3 Popov et al. J. Math. Phys. 54, 032103 (2013)

We do not repeat all the procedures after the variable separation, this can be found in all quantummechanics textbooks. Instead, we only point out to the radial equation[

− �2

2mr

(d2

dr2 + 2r

ddr

)+ mr ω

2

2 r2 + �2

2mr

c+J (J+1)r2

]R(r )

= (E − E0) R(r ). (7)

When we introduce the reduced radial function u(r) = rR(r) (which will be useful later in thepaper) then the corresponding equation becomes

Hred (r )u(r ) ≡[− �

2

2mr

d2

dr2+ +mrω

2

2r2 + �

2

2mr

c + J (J + 1)

r2

]u(r )

= (E − E0) u(r ). (8)

Finally, the solution of the radial equation is expressed in terms of associated Laguerrepolynomials

Rn J (r ) =[

2(mr ω

�

)3/2�(n + 1)

�(n + CJ + 1)

]1/2 [mrω

�r2] CJ

2 − 14 ·

· e− 12

mr ω�

r2LCJ

n

(mrω

�r2)

, (9)

where we shall use the following notation for an important constant in our approach:

.CJ =√

c +(

J + 1

2

)2

. (10)

Consequently, the energy eigenvalues can be expressed in the following manner:

En J = E0 + �ω CJ + �ω(2n + 1). (11)

The whole eigenfunction of the Hamiltonian is, then

�n J M (�r ) = �n J M (r, θ, ϕ) = (12)

= Rn J (r ) YJ M (θ, ϕ) = 〈�r | n, k〉.Identify now the connection between the Bargmann index k and the constant CJ. For this purpose

we return to the reduced radial equation which can be written in a following manner:3

2mr

�2Hred u(r ) ≡ (13)

≡[− d2

dr2+ ar2 + g

r2

]u(r ) = eu(r ),

where the following notations have been used: a = (mr ω�

)2, g = c + J (J + 1) > 0,

e = 2mr�2 (E − E0).

This manner of writing the equation for the reduced radial function allows us to express thegroup generators of SU(1,1) in terms of one-variable differential operators

K± = �2

4mrω

[d2

dr2+ ar2 − g

r2

]± i

�

2

(r

d

dr+ 1

2

), (14)

K0 = �

4mrω

[− d2

dr2+ ar2 + g

r2

]. (15)

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032103-4 Popov et al. J. Math. Phys. 54, 032103 (2013)

After some straightforward calculations, we obtain that the eigenvalue of the Casimir operator,which corresponds to the positive series of UIR is

k(k − 1) = g (16)

from which it yields that

k = 1

2

(1 +

√g + 1

4

)= 1

2

⎛⎝1 +

√c +

(J + 1

2

)2⎞⎠ . (17)

So a useful relation which often appears in what follows is

2k − 1 =√

c +(

J + 1

2

)2

≡ CJ . (18)

This relation shows that the Bargmann index k depends strongly on the rotational quantumnumber J, i.e., k = k(J). But, for reasons of simplicity, we will not emphasize this dependence, i.e.,we will not write k(J) or kJ, but we will just consider it and write only k. Because the rotationalquantum number has an infinite number of integer values J = 0, 1, 2, . . . , ∞, the Bargmann index khas also an infinite number of values. So, the Bargmann index k labels an infinite number of sheets.

With these group generators, the Hamiltonian Hred can now be rewritten simply as a linearcombination of a single group generator, according to the Perelomov’s case classification (case 1,i.e., with �0 ≡ 2�ω > 0), [1]

Hred = 2�ω K0 (19)

and thus the Schrodinger equation of stationary states becomes

Hred |n, k〉 = 2�ω (n + k) |n, k〉, (20)

which leads to the correct expression for the energy eigenvalues (11).We shall now consider that the quantum state of the system is the equilibrium state for a field

coupled to a reservoir at temperature T = (kBβ)− 1, where kB is the Boltzmann constant. In thissituation, the state of the system is a mixed state and the distribution function which describesa thermal state’s statistics is similar to the Maxwell–Boltzmann distribution. The correspondingnormalized density operator is

ρ = 1

Z (β)

∑n,J,M

e−βEn J |n, k〉〈n, k| , (21)

which can be written as, if we use the energy expression

ρ = 1

Z (β)e−β(E0+�ω)

∞∑J=0

e−β �ω CJ ·

·∞∑

n=0

(e−2β �ω

)n |n, k〉〈n, k|J∑

M=−J

1. (22)

The last sum, over the magnetic quantum number M leads to the rotational degeneration factor 2J+ 1. If we use the expression of the Bose–Einstein population number (i.e., the average number ofquanta),

n = 1

e2β�ω − 1, (23)

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032103-5 Popov et al. J. Math. Phys. 54, 032103 (2013)

then the expression for the density operator can be written as

ρ = 1

Z (β)e−βE0

√n

n + 1

∞∑J=0

(2J + 1)

(√n

n + 1

)CJ

·

·∞∑

n=0

(n

n + 1

)n

|n, k〉〈n, k|. (24)

The evaluation of the partition function Z(β) can be achieved if the density operator is normalizedto unity

Tr ρ =∑

n,J,M

〈n, k|ρ|n, k〉 = 1. (25)

After some simple calculation we obtain

Z (β) = e−βE0√

n(n + 1)∞∑

J=0

(2J + 1)

(√n

n + 1

)CJ

≡

≡ e−βE0√

n(n + 1)S (0; n) , (26)

where a useful notation has been introduced which often appears in what follows:

S (0; n) ≡∞∑

J=0

(2J + 1)

(√n

n + 1

)CJ

. (27)

As it is well admitted, if the partition function of a quantum system is known, then we cancalculate any thermo-dynamical function which characterizes this system.

Finally, the expression for the normalized density operator is

ρ = 1

(n + 1) S (0; n)

∞∑J=0

(2J + 1)

(√n

n + 1

)CJ

·

·∞∑

n=0

(n

n + 1

)n

|n, k〉〈n, k|. (28)

This expression will be used in order to calculate the expectation values for different physicalobservables which characterize the examined quantum system.

III. THE “MOTHER COHERENT STATES”

Starting with Perelomov’s well-known papers and book,1, 4 a series of authors (see, e.g.,Refs. 2 and 5 and references therein) have turned their attention to the SU (1,1) coherent states.

One way to define a coherent state is as an eigenstate of the lowering operator K− (thesecoherent states are called the Barut-Girardello coherent states6),

K−|z, k〉BG = z|z, k〉BG . (29)

For example, in the case of the PHO potential (which can be written as a sum of a HO-3Dpotential plus a centrifugal barrier and a free term), the normalized coherent states are deducedearlier as7

|z, k〉BG = (30)

=√

|z|2k−1

I2k−1(2 |z|)∞∑

n=0

zn

√n!�(n + 2k)

|n, k〉,

where �(. . . ) is the Euler gamma function and I2k − 1(. . . )- the Bessel function of the first kind.

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032103-6 Popov et al. J. Math. Phys. 54, 032103 (2013)

We must observe here that some authors refer to these coherent states as the “intelligent states.”2

It is clear that these states refer only to the vector Fock subspace Fn,k ≡ {|n, k〉} with fixedk = k(J), J = 0, 1, 2, . . . which is an infinite-dimensional subspace. This can be considered as a partof the whole or entire Fock space F = ∑∞

n,J=0 ⊕Fn,k , for which J and also n take an infinite numberof values. Thus, the whole Fock space is also infinite dimensional. The Fock basis vectors |n, k〉 arecomplete over the whole Fock space

∞∑n,J=0

|n, k〉〈n, k| = 1, (31)

and, consequently, the lowering operator K− can be written in the following manner (and similarly,the raising operator K+):

K− =∞∑

n,J=0

√n(n + 2k − 1)|n − 1, k〉〈n, k|, (32)

from which we can notice that this operator acts on the whole Fock space and does not depend onthe Bargmann index k = k(J).

Then, the next step in our examination is to check and deduce the expression of those coherentstates which are defined for the whole space, i.e., which will be independent of the Bargmann indexk. This is in fact the main purpose of our paper and, to our knowledge, these kinds of calculationshave not been done so far.

We begin with a similar definition of the coherent states for the whole space

K−|z〉 = z|z〉. (33)

We shall call these states MCSs because they refer to the whole Fock space of the basis vectors.The decomposition of the MCSs with respect to the Fock basis is

|z〉 = N (|z|)∑n,J

cnk(z)|n, k〉 (34)

and if we apply the definition of MCSs and the action of the lowering operator on the Fock vectors,after a recurrence relation, the development functions cnk(z) are obtained

cnk(z) = c0k(|z|) zn√n!�(n+2k)

�(2k)

, (35)

where, intuitively, having in mind the general structure of a usual coherent state,8 we notice that thefunction c0k(|z|) depends on the |z| and not on the z. Thus the following partial result yields:

|z〉 = N (|z|) · (36)

·∞∑

J=0

c0k(|z|)√

�(2k)∞∑

n=0

zn

√n!�(n + 2k)

|n, k〉.

The normalization condition leads to the relation:

1 = 〈z | z〉 = [N (|z|)]2 · (37)

·∑

J

|c0k(|z|)|2 �(2k)∞∑

n=0

(|z|2)n

n!�(n + 2k).

Using the definition of the Bessel functions of the first kind9

I2k−1(2 |z|) =∞∑

n=0

|z|2n+2k−1

n!�(n + 2k), (38)

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032103-7 Popov et al. J. Math. Phys. 54, 032103 (2013)

we get the relation

N (|z|) = (39)

=[∑

J

|c0k (|z|)|2 �(2k)1

|z|2k−1 I2k−1 (2 |z|)] 1

2

.

The normalization constant N (|z|) must not depend on the k(J), so the sum over J must be equalwith such a function. This decomposition of the sum with respect to the quantum number J is notunique; there exist some possibilities to perform this decomposition. But, as we will see later, thesedifferent decompositions do not influence the physical meaning of the coherent states we want tobuild. A possible decomposition can be performed by using the property of the Bessel functions9

∞∑J=0

(−2 |z|)J

J !IJ+ν(2 |z|) = Jν(2 |z|), (40)

where Jν(2|z|) is the Bessel function of the first kind. Evidently, we will choose ν = 0. Consequently,the searched constants are obtained

c0k(z) = (−2)J2√

� (2k)

|z| J2 +k− 1

2√J !

√IJ (2 |z|)

I2k−1(2 |z|) . (41)

Using this expression, the function N (|z|) can be determined from the normalization conditionof the MCSs, i.e., 〈z | z〉 = 1: N (|z|) = 1√

J0(2|z|) . So the final expression of the MCSs is

|z〉 = 1√J0(2 |z|)

∞∑J=0

(−2 |z|) J2|z|k− 1

2√J !

√IJ (2 |z|)

I2k−1(2 |z|) ·

·∞∑

n=0

zn

√n!�(n + 2k)

|n, k〉. (42)

The decomposition of the unity operator in terms of MCSs is∫dμ(z)|z〉〈z| = 1, (43)

where the integration measure dμ(z) must be determined. We will write it as a sum of projectorsonto the Fock states

dμ(z) =∑n,J

dμk(z) |n, k〉〈n, k| ≡ (44)

≡ d2z

π

∑n,J

hk(|z|) |n, k〉〈n, k|,

where the product between the Lebesgue measure d2zπ

and a (yet undetermined) function hk(|z|) hasbeen introduced

dμk(z) = d2z

πhk (|z|) = dϕ

π

1

2d(|z|2) hk (|z|) . (45)

The angular integration leads to the result∫ 2π

0

dϕ

π(z∗)n′

zn = |z|n′+n∫ 2π

0

dϕ

πe−i(n−n′)ϕ =

= 2 |z|2n δnn′ (46)

from which we obtain n′ = n. To get the condition which expresses the completeness of theeigenfunctions, it is compulsory to also have simultaneously k′ = k, i.e., J′ = J.

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032103-8 Popov et al. J. Math. Phys. 54, 032103 (2013)

Using the notation

gk(|z|) = hk(|z|) [N (|z|) ]2 [c0k(|z|) ]2 (47)

and inserting it in the equation for the decomposition of the unity operator in terms of MCSs, therelation yields ∫ ∞

0d(|z|2) gk (|z|) (|z|2)n =

= 1

�(2k)�(n + 1)�(n + 2k). (48)

This is a Stieltjes moment problem and the function gk(|z|) may be found using the substitutionn = s − 1,10

∫ ∞

0d(|z|2) gk (|z|) (|z|2)s−1 =

= 1

�(2k)�(s)�(s + 2k − 1). (49)

It is useful to express the unknown function through the Meijer’s G-functions,11, 12

gk(|z|) = 1

�(2k)G20

02

(|z|2 | 0, 2k − 1) =

= 21

�(2k)|z|2k−1 K2k−1(2|z|). (50)

Finally, the integration measure becomes

dμ(z) = 2d2z

πJ0(2 |z| ) · (51)

·∑n,J

J !

(−2 |z|)J

K2k−1(2 |z| ) I2k−1(2 |z| )

IJ (2 |z| )|n, k〉〈n, k| .

This result can be also proved by the direct calculations of the unity operator decomposition.Klauder introduces a set of physical criteria that the generalized coherent states should have.13

So, the coherent states: (a) are normalized and parameterized continuously in the label z; (b) admit aresolution of unity with a positive measure dμ(z), (c) enjoy the property that the temporal evolutionof any coherent state by the Hamiltonian of quantum system remains a coherent state for all time,and (d) minimize the uncertainty relations.

The first two criteria have already been demonstrated in the present paper. The latter assertioncan be easily proved, e.g., for the linear energy spectra En = a n + b, due to the fact that

e− i�

H t |n, k〉 = e− i�

En t |n, k〉 == e− i

�b t(

e− i�

a t)n |n, k〉, (52)

we obtain successively

e− i�

H t |z〉 = e− i�

b t |z(t)〉, (53)

where the variable zis yet time-dependent, according to the relation z(t) = z e− i�

a t .The fourth Klauder’s requirement or condition will be demonstrated in Sec. IV.

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032103-9 Popov et al. J. Math. Phys. 54, 032103 (2013)

IV. STATISTICAL PROPERTIES

In order to examine the statistical properties of the MCSs for SU(1,1) central field potentials, wemust derive the expression of the expectation values of the different physical observables A whichcharacterize the quantum system.

Generally, the expectation value of an observable A in a MCS |z〉 is defined as

〈z| A|z〉 ≡ ⟨A⟩z = 1

J0(2|z|) · (54)

·∞∑

J,J ′=0

(−2|z|) J+J ′2√

J !J ′!|z|k+k ′−1

√IJ (2|z|)IJ ′(2|z|)

I2k−1(2|z|)I2k ′−1(2|z|)

·∞∑

n,n′=0

(z∗)n′zn

√n!�(n + 2k)

√n′!�(n′ + 2k ′)

⟨n′, k ′∣∣ A|n, k〉.

An important category of observables are those which are diagonal in the energy basis |n, k〉. Forsuch an observable we have the following eigenvalue equation:

A|n, k〉 = ank |n, k〉 (55)

so the corresponding expectation value is

〈z| A|z〉 ≡ ⟨A⟩z = 1

J0(2|z|) · (56)

·∞∑

J=0

(−2|z|)J

J !|z|2k−1 IJ (2|z|)

I2k−1(2|z|)∞∑

n=0

(|z|2)n

n!�(n + 2k)ank .

As a first application example, let we consider the particle number operator N , which is diagonalin the energy basis and independent on the rotational quantum number J,

N s |n, k〉 = ns |n, k〉, s = 0, 1, 2, 3, .... (57)

For this case, we have

⟨N s⟩z = 1

J0(2|z|) · (58)

·∞∑

J=0

(−2|z|)J

J !|z|2k−1 IJ (2|z|)

I2k−1(2|z|)∞∑

n=0

(|z|2)n

n!�(n + 2k)ns .

Denote the following sum as the unnormalized expectation value of the operator N s in thecoherent state |z, k〉 which refers to the Fock subspace vectors Fn,k ≡ { |n, k〉, n = 0, 1, 2, . . . , ∞ ; k− fixed }:

S(s)k (|z|) ≡ ⟨

N s⟩k ≡

∞∑n=0

(|z|2)n

n!�(n + 2k)ns . (59)

The fundamental sum S(0)k (|z|) can be evaluated and expressed through the Bessel function

I2k − 1(2|z|):9, 11

S(0)k (|z|) ≡

∞∑n=0

(|z|2)n

n!�(n + 2k)= I2k−1(2|z|)

|z|2k−1. (60)

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032103-10 Popov et al. J. Math. Phys. 54, 032103 (2013)

Consequently, the other sums are

S(s)k (|z|) =

(|z|2 d

d|z|2)s

S(0)k (|z|) = (61)

=(

|z|2 d

d|z|2)s [ I2k−1(2|z|)

|z|2k−1

].

In order to simplify the above calculations, we will use our ansatz from an earlier paper.14 Thedemonstration is performed in Appendix A, and the result is

⟨N s⟩k =

(|z|2 d

d|z|2)s

S(0)k (|z|) = (62)

s∑l=1

c(s)l (|z|2)l

(d

d|z|2)l

S(0)k (|z|) =

= 1

|z|2k−1

s∑l=1

c(s)l |z|l I2k−1+l (2|z|).

Because after the first differentiation of the fundamental sum S(0)k (|z|) the factor n is obtained, after

the second one the factor n(n − 1) is obtained and so on, then the coefficients c(s)l can be obtained

easily from the equality:14

ns =s∑

l=1

c(s)l

n!

(n − l)!. (63)

Using some differentiation properties referring to the Bessel functions,9, 11 we finally obtain

⟨N s⟩|z| = 1

J0(2|z|)s∑

l=1

c(s)l |z|l · (64)

·∞∑

J=0

(−2|z|)J

J !IJ (2|z|) I2k−1+l(2|z|)

I2k−1(2|z|) .

This expression cannot be simplified due to the presence of three Bessel functions and thedependence k = k(J).

By particularizing this equation for s = 1 and s = 2, i.e.,

⟨N⟩|z| = 1

J0(2|z|) |z| · (65)

·∞∑

J=0

(−2|z|)J

J !IJ (2|z|) I2k(2|z|)

I2k−1(2|z|) ,

⟨N 2⟩|z| = ⟨

N⟩|z| +

1

J0(2|z|) |z| · (66)

·∞∑

J=0

(−2|z|)J

J !IJ (2|z|) I2k(2|z|)

I2k−1(2|z|) ,

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032103-11 Popov et al. J. Math. Phys. 54, 032103 (2013)

we can express the Mandel parameter15 (which is a measure of statistical behavior of the coherentstates) for the MCSs,

Q|z| =⟨N 2⟩|z| −

(⟨N⟩|z|)2

⟨N⟩|z|

− 1 = (67)

= |z|∑∞

J=0(−2|z|)J

J ! IJ (2|z|) I2k+1(2|z|)I2k−1(2|z|)∑∞

J=0(−2|z|)J

J ! IJ (2|z|) I2k (2|z|)I2k−1(2|z|)

−

− |z|J0(2|z|)

∞∑J=0

(−2|z|)J

J !IJ (2|z|) I2k(2|z|)

I2k−1(2|z|) .

In order to verify the behavior of the MCSs, i.e., to see if there are sub-Poissonian (for which,Q|z| < 0), Poissonian (with Q|z| = 0), or super Poissonian (with Q|z| > 0), the behavior of the Mandelparameter Q|z| with respect to variable |z| can be examined. Thus, the statistical properties of theMCSs are dependent on the analytical properties of the mathematical operations (products, sum,ratios) of the above Bessel functions. The analytical expression of the Mandel parameter Q|z| withrespect to variable |z| is difficult to express, so this parameter must be calculated numerically.

However, in examining the asymptotic properties, the asymptotic expressions of Besselfunctions16, 17 may be useful: for small arguments |z| → 0,

Iν(2|z|) ≈ 1

�(ν + 1)|z|ν , (68)

for large arguments |z| → ∞,

Iν(2|z|) ≈ e2|z|√

4π |z|[

1 + O

(1

2|z|)]

. (69)

If the observable A is just the Hamiltonian H of the quantum system, then the expectation valuein a MCS is

⟨H⟩|z| = 1

J0(2|z|)∞∑

J=0

(−2|z|)J

J !|z|2k−1 IJ (2|z|)

I2k−1(2|z|) ·

·∞∑

n=0

(|z|2)n

n!�(n + 2k)(E0 + �ω + �ω CJ + 2�ω n) , (70)

which can be split into three terms⟨H⟩|z| ≡ ⟨

H1⟩|z| +

⟨H2⟩|z| +

⟨H3⟩|z| , (71)

where ⟨H1⟩|z| = E0 + �ω, (72)

⟨H2⟩|z| = �ω

1

J0(2|z|)∞∑

J=0

CJ(−2|z|)J

J !IJ (2|z|). (73)

The third term contains in the last sum exactly the sum Slk(|z|), so that we obtain⟨

H3⟩|z| = 2�ω

⟨N⟩|z| = (74)

= 2�ω1

J0(2|z|) |z|∞∑

J=0

(−2|z|)J

J !IJ (2|z|) I2k(2|z|)

I2k−1(2|z|) .

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032103-12 Popov et al. J. Math. Phys. 54, 032103 (2013)

Consequently, the final expression of the expectation value of Hamiltonian is

⟨H⟩|z| = E0 + 2�ω

(⟨N⟩|z| +

1

2

)+ (75)

+ �ω1

J0(2|z|)∞∑

J=0

CJ(−2|z|)J

J !IJ (2|z|).

Yet, consider some non-diagonal observables in the energy basis. It will also be useful to calculate theexpectation values for the group generators, their powers and products. By considering the actionsof these operators on the Fock basis vectors, after the straightforward calculations we obtain (see,Appendix B), ⟨

K−⟩z = z, (76)

⟨K+⟩z = z∗, (77)

⟨K 2

−⟩z= z2, (78)

⟨K 2

+⟩z = (z∗)2, (79)

⟨K+ K−

⟩|z| = |z|2, (80)

⟨K0⟩|z| = 1

2

1

�ω

⟨Hred

⟩|z| , (81)

⟨K− K+

⟩|z| = |z|2 + ⟨

2K0⟩|z| , (82)

where Hred = H − E0.With these relations we can examine the behavior of the MCSs versus the uncertainty relations.

Introduce two generalized quadrature operators X1 and X2,2

X1 = 1

2

(K− + K+

), X2 = 1

2i

(K− − K+

)(83)

with the commutation relation [X1 , X2

] = i K0 (84)

and the variances ⟨( Xi )

2⟩ = ⟨

X2i

⟩2 − (⟨Xi⟩)2

. (85)

The generalized Robertson’s uncertainty relation18 for these generalized quadrature operatorstakes the following form:

⟨( X1)2

⟩z

⟨( X2)2

⟩z ≥ 1

4|〈K0〉z|2 . (86)

It is easy to verify that the variances of these quadrature operators can be expressed through theexpectation values of the generators of SU(1,1) algebra,2

⟨( X1)2

⟩z = 1

2

[⟨K+ K−

⟩|z| +

⟨K0⟩|z|]

+

+ 1

4

[⟨K 2

+ + K 2−⟩z − ⟨

K− + K+⟩2z

], (87)

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032103-13 Popov et al. J. Math. Phys. 54, 032103 (2013)

⟨( X1)2

⟩z = 1

2

[⟨K+ K−

⟩|z| +

⟨K0⟩|z|]

−

− 1

4

[⟨K 2

+ + K 2−⟩z + ⟨

K− + K+⟩2z

]. (88)

Using the above calculated expectation values of the combinations of generators in a MCS |z〉 wefind that the variances are ⟨

( X1)2⟩z = 1

2

⟨K0⟩|z| ,

⟨( X2)2

⟩z = 1

2

⟨K0⟩|z| (89)

so that the generalized Robertson’s uncertainty relation takes the minimum value

⟨( X1)2

⟩z

⟨( X2)2

⟩z = 1

4

∣∣∣⟨K0⟩|z|∣∣∣2 . (90)

By analogy with the coherent states of a simple one-dimensional harmonic oscillator (whichminimizes the uncertainty relation in position and momentum), we can say that our states, i.e., theMCSs |z〉 are really the coherent states. So the last condition imposed to coherent states by theKlauder prescription is accomplished. Given these considerations, sometimes these classes of statesare called intelligent states.2, 19, 20 However, we prefer to call these states as MCSs.

So all Klauder’s criteria are accomplished and our construction |z〉 is really a coherent state.

V. THERMAL STATES

One of the most important kinds of distribution function, from the practical point of view, isthe canonical distribution, i.e., the distribution which is applicable to systems in thermodynamicequilibrium with a much larger system (a “reservoir” or a “bath”) at the temperature T. As we sawin Sec. II, the normalized density operator for a mixed state of such a system (27) can be written as

ρ =∞∑

n,J=0

ρn,k(n)|n, k〉〈n, k| ≡∞∑

n,J=0

1

(n + 1) S (0; n)

· (2J + 1)

(√n

n + 1

)CJ (n

n + 1

)n

|n, k〉〈n, k|. (91)

Generally, the diagonal elements of the normalized density operator in the representation of thecoherent states are called the Q-function (in the context of quantum optics) or the Husimi’s Qdistribution function since, in 1940, it was introduced for the first time by Husimi as the simplestdistribution of quasi-probability in the phase space.21, 22 We need to examine the whole or entireFock space F = ∑∞

n,J=0 ⊕Fn,k , which includes an infinite number of subspaces or sheets Fn,k.Consequently, the whole or entire Husimi’s Q distribution function will be written as a sum ofprojectors onto the individual Fock states |n, k〉,

Q(|z|) =∞∑

n, J=0

Qnk(|z|)|n, k〉〈n, k|. (92)

In other words, the Husimi’s distribution function is defined as the expectation value of thenormalized density operator ρnk which corresponds to the state |n, k〉 in the representation of theMCS |z〉,

Qnk(|z|) ≡ 〈z|ρnk |z〉 = 1

(n + 1) S (0; n)(2J + 1) ·

·(√

n

n + 1

)CJ (n

n + 1

)n

|〈n, k | z〉|2 , (93)

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032103-14 Popov et al. J. Math. Phys. 54, 032103 (2013)

where

〈n, k | z〉 = 1√J0(2 |z|) (−2 |z|) J

2|z|k− 1

2√J !

·

·√

IJ (2 |z|)I2k−1(2 |z|)

zn

√n!�(n + 2k)

. (94)

So, considering that

2k − 1 = CJ (95)

after some straightforward calculations, we obtain the Husimi’s function

Q(|z|) = 1

(n + 1) S(0; n)

1

J0 (2|z|) · (96)

·∞∑

J=0

(2J + 1)(−2|z|)J

J !IJ (2|z|) ·

·I2k−1

(2|z|

√n

n+1

)I2k−1 (2|z|)

∞∑n=0

|n, k〉〈n, k|.

Using the integrals involving the Bessel functions,9 it is easy to verify that the Husimi’s functionis really a distribution function, because it fulfills the equation∫

dμ(z) Q(|z|) = 1. (97)

The next step in our examination of the properties of MCSs is to build the P-distribution functionfrom the diagonal representation of the MCSs,22

ρ =∫

dμ(z) |z〉P(|z|)〈z|. (98)

In the Fock basis, the density operator is written in the diagonal representation, as it is showby Eq. (27), so in order to find the P-distribution function, we must equalize the right side ofEqs. (98) and (27). Like the integration measure dμ(z) and the Husimi’s distribution function Q(|z|),the P-distribution function P(|z|) should not depend on the quantum numbers n and J, so we chosethe later in the similar form

P(|z|) =∞∑

J=0

Pk(|z|)∞∑

n=0

|n, k〉〈n, k|, (99)

where the function Pk(|z|) must be determined. We write the two sides of Eq. (98) in the followingmanner:

ρL H S=

∑n,J

⎡⎣ 2J + 1

(n + 1) S (0; n)

(√n

n + 1

)CJ⎤⎦ ·

·[(

n

n + 1

)n]|n, k〉〈n, k| RH S= (100)

∑n,J

[2∫

d2z

πK2k−1(2|z|)Pk(|z|)|z|2n+2k−1

]·

·[

1

n!�(n + 2k)

]|n, k〉〈n, k|.

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032103-15 Popov et al. J. Math. Phys. 54, 032103 (2013)

Evidently, the two large brackets must be equal. This means that the integral from the left mustbe equal to the quantity from right hand brackets

2∫∞

0 d(|z|2) (|z|2)n+k− 12 Pk(|z|) K2k−1(2|z|) =

= 2J+1(n+1) S(0;n)

(√n

n+1

)CJ (n

n+1

)n�(n + 1)�(n + 2k).

(101)

After the function changes

Gk(|z|) = Pk(|z|) K2k−1(2|z|)2J+1

(n+1) S(0;n)

(√n

n+1

)CJ(102)

and the variable changes x = |z|2, we get the integral∫ ∞

0dx xn+k− 1

2 Gk(|z|) = 1

2

(n

n + 1

)n

�(n + 1)�(n + 2k). (103)

The infinite upper limit of integration is justified due to the convergence criterion,22 accordingto which the convergence radius is

R = limn→∞

n+k√

�(n + 1)�(n + 2k) → ∞, (104)

i.e., our problem is just the Stieltjes moment problem, if we change the exponent n + k − 12 = s − 1,

∫ ∞

0dx xs−1Gk(|z|) = 1

2

(n

n + 1

)−k− 12

· (105)

· 1(n+1

n

)s �

(s − k + 1

2

)�

(s + k − 1

2

).

The unknown function Gk(|z|) can be found by using the Meijer’s G-function method11 andthere are

Gk(|z|) = 1

2

(n

n + 1

)−k− 12

· (106)

· G2002

(|z|2 n + 1

n

∣∣∣∣−k + 1

2, k − 1

2

)

=(

n

n + 1

)−k− 12

K2k−1

(2|z|

√n + 1

n

). (107)

Using Eq. (102), we obtain

Pk(|z|) = 1

n S (0; n)(2J + 1)

K2k−1

(2|z|

√n+1

n

)K2k−1(2|z|) . (108)

So, the final expression of the P-function becomes

P(|z|) = 1

n S (0; n)

∞∑J=0

(2J + 1)K2k−1

(2|z|

√n+1

n

)K2k−1(2|z|) ·

·∞∑

n=0

|n, k〉〈n, k|. (109)

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032103-16 Popov et al. J. Math. Phys. 54, 032103 (2013)

It is not difficult to verify that the P-function fulfills the property of a distribution function,which results from the normalization condition of the whole density operator ρ. In order to simplifythe calculations, we use the expressions of the MCSs and the density operator, as they are written inEqs. (34) and (91),

Trρ =∞∑

n,J=0

∫dμk(|z|) [N (|z|)]2 |cnk(z)|2 Pk(z) = 1. (110)

The knowledge of the P-function is also important to calculate the thermal expectation values.For this purpose write the operator A as a sum of projectors onto the subspaces Fn,k,

A =∞∑

n,J=0

ank |n, k〉〈n, k|, (111)

⟨A⟩ = Trρ A =

∫dμ(z′)

⟨z′ |ρ A| z′⟩ = (112)

=∞∑

n,J=0

ank

∫dμk(z) Pk(|z|) [N (|z|)]2 |cnk(z)|2 .

By calculating the integral, we obtain the final result

⟨A⟩ = 1

(n + 1)S (0; n)(113)

·∞∑

J=0

(2J + 1)

(√n

n + 1

)CJ ∞∑n=0

(n

n + 1

)n

ank,

where we have used the eigenequation of the operator A with respect to the Fock basis |n, k〉. Thethermal expectation value for the Hamiltonian H is then

⟨H⟩ = 1

(n + 1)S (0; n)

∞∑J=0

(2J + 1)

(√n

n + 1

)CJ

·

·∞∑

n=0

(n

n + 1

)n

(E0 + �ω + �ωCJ + 2�ω n) , (114)

which leads to the result ⟨H⟩ = E0 + �ω coth β�ω − ∂

∂βln S(0; n). (115)

VI. SOME PARTICULAR POTENTIALS

These general results may be particularized for some known potentials which have the SU(1,1).Also, these results will serve as a check on the correctness of the calculations made in Secs. II–V.These potentials are: the HO-3D potential and the PHO.

A. The three dimensional isotropic harmonic oscillator potential

The potential is of the following manner:

VH O−3D(r ) = mrω2

2r2, (116)

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032103-17 Popov et al. J. Math. Phys. 54, 032103 (2013)

so the constants are c = 0 and E0 = 0 and, consequently, 2k − 1 = CJ = J + 12 . The corresponding

MCSs are

|z〉H O−3D = 1√J0(2 |z|)

∞∑J=0

(−2 |z|) J2|z| J

2 + 14√

J !· (117)

·√

IJ (2 |z|)IJ+ 1

2(2 |z|)

∞∑n=0

zn√n!�

(n + J + 3

2

) |n, J 〉H O−3D,

while the integration measure is

dμH O−3D(z) = 2d2z

πJ0(2 |z| )

∑n,J

J !

(−2 |z|)J · (118)

·K J+ 1

2(2 |z| ) IJ+ 1

2(2 |z| )

IJ (2 |z| )|n, k〉H O−3D H O−3D〈n, k|.

The expectation value of the integer power of the number operator is then

⟨N s⟩H O−3D

|z| = 1

J0(2|z|)s∑

l=1

c(s)l |z|l (119)

·∞∑

J=0

(−2|z|)J

J !IJ (2|z|)

IJ+ 12 +l(2|z|)

IJ+ 12(2|z|)

through which we can easily deduce the corresponding Mandel parameter Q H O−3D|z| . Unfortunately,

the sums cannot be calculated analytically, so it is necessary to perform numerical calculations.The fundamental rotational sum for the HO-3D is then

SH O−3D(0; n) =∞∑

J=0

(2J + 1)

(√n

n + 1

)J+ 12

= 1

2 sinh β �ω2

coth β�ω

2(120)

and, consequently, the partition function becomes:14

Z H O−3D(β) =√

n(n + 1) SH O−3D(0; n) =

=(

1

2 sinh β �ω2

)3

, (121)

where the parenthesis contains the well-known expression for the partition function of the one-dimensional harmonic oscillator (HO-1D). The expectation value of the particle number operator ina MCS is

⟨N⟩H O−3D

|z|= 1

J0(2|z|) |z| · (122)

·∞∑

J=0

(−2|z|)J

J !IJ (2|z|)

IJ+ 32(2|z|)

IJ+ 12(2|z|) ,

while the Hamiltonian expectation value in a MCS is

⟨H⟩H O−3D

|z| = 2�ω

(〈N 〉H O−3D

|z + 3

4

)− 2�ω|z| J1(2|z|)

J0(2|z|) .

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032103-18 Popov et al. J. Math. Phys. 54, 032103 (2013)

The thermal expectation value is, according to Eq. (115):

⟨H⟩H O−3D = �ω coth β�ω − ∂

∂βln SH O−3D(0; n).

After some trigonometrical calculations, the final result is14

⟨H⟩H O−3D = 3

2�ω coth β

�ω

2= 3

(�ω

2+ �ω

eβ�ω − 1

),

where we have evinced in the parenthesis the internal energy which corresponds to a HO-1D. TheHusimi’s distribution function Q(|z|)HO − 3D and the P-distribution function are, respectively,

Q H O−3D(|z|) = 1

(n + 1) SH O−3D(0; n)

1

J0 (2|z|) ·

·∞∑

J=0

(2J + 1)(−2|z|)J

J !IJ (2|z|)

IJ+ 12

(2|z|

√n

n+1

)IJ+ 1

2(2|z|) ·

·∞∑

n=0

|n, J 〉H O−3D H O−3D〈n, J |, (123)

P H O−3D(|z|) = 1

n SH O−3D (0; n)·

·∞∑

J=0

(2J + 1)K J+ 1

2

(2|z|

√n+1

n

)K J+ 1

2(2|z|) ·

·∞∑

n=0

|n, J 〉H O−3D H O−3D〈n, J | . (124)

B. The pseudoharmonic oscillator potential

In this case the potential has the form

VP H O (r ) = mrω2

2r2

0

(r

r0− r0

r

)2

= (125)

= mrω2

2r2 + mrω

2

2r4

01

r2− mrω

2r20 ,

where r0 specifies the equilibrium position of the quantum system (e.g., the diatomic molecule).So, the constants are: c = (mr ω

�r2

0

)2and E0 = −mrω

2r20 and, consequently, 2k − 1 = CJ

=√(mr ω

�r2

0

)2 + (J + 1

2

)2.14

Due to the structure of the index 2k − 1, the expressions deduced in Secs. II–V cannot besimplified but it can be observed that the HO-3D can be considered as the harmonic limit of thePHO,14 so we can write

limr0 → 0

CJ → J + 12

AP H O ≡ lim AP H O

H O−3D= AH O−3D . (126)

This relation can also be used to verify the correctness of the obtained equations.

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032103-19 Popov et al. J. Math. Phys. 54, 032103 (2013)

VII. CONCLUDING REMARKS

In the present paper, we have defined a new kind of coherent state which we have called theMCSs. Even if these states are of the Barut-Girardello kind, i.e., they are defined as the eigenvectorsof the lowering operator K−, they are defined on the whole Fock space F = ∑∞

n,J=0 ⊕Fn,k of theenergy basis vectors, which contain all vectors |n, k〉, with all n = 0, 1, . . . , ∞ and all Bargmannindices k = k(J), where J = 0, 1, . . . , ∞ is the rotational quantum number. The MCSs defined in thismanner depend only on the complex variable z, without depending on any other parameters (e.g., theBargmann index and so on) as in the case of the coherent states defined in a common or “orthodox”way.

This manner of defining the MCSs is claimed by the need to calculate the thermal expectationvalues in the coherent states representation, having in mind that the thermal expectation values areindependent of these parameters.

In our knowledge, this approach has not appeared until now in the specific literature referringto the coherent states and from this point of view the calculations are original.

As a new approach in the scientific literature, we wrote the whole integration measure dμ(z), thewhole Husimi’s distribution function Q(|z|), as well as the whole P-function as sum of the projectorsonto the individual Fock states |n, k〉. This ansatz allows the calculation of correct expectation values(for a MCS |z〉 and also for thermal states). Implicitly, this is a reliable way to verify the accuracy ofthe obtained results. We believe that our approach will enrich the specialized literature dealing withcoherent states.

APPENDIX A: DEDUCTION OF THE⟨Ns⟩k

We will use the ansatz from an earlier paper (see, Ref. [14], Appendix B of this paper),

⟨N s⟩k =

∞∑n=0

(|z|2)n

n!�(n + 2k)ns = (A1)

=(

|z|2 d

d|z|2)s( ∞∑

n=0

(|z|2)n

n!�(n + 2k)

)=

=(

|z|2 d

d|z|2)s ( I2k−1(2|z|)

|z|2k−1

)=

=s∑

l=1

c(s)l (|z|2)l

(d

d|z|2)l ( I2k−1(2|z|)

|z|2k−1

).

Having in mind that

d

d|z|2 = 2

(1

2|z|d

d(2|z|))

(A2)

and the differential property of the Bessel functions

(d

xdx

)m ( Iν(x)

xν

)= Iν+m(x)

xν+m,(A3)

we obtain the final result

⟨N s⟩k = 1

|z|2k−1

s∑l=1

c(s)l |z|l I2k−1+l (2|z|). (A4)

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032103-20 Popov et al. J. Math. Phys. 54, 032103 (2013)

APPENDIX B: DEDUCTION OF SOME EXPECTATION VALUES IN MCSs

Calculate (and, implicitly to verify) the first relation

⟨K−⟩z= 1

J0(2|z|)∞∑

J,J ′=0

(−2|z|) J+J ′2√

J !J ′!|z|k+k ′−1 ·

·√

IJ (2|z|)IJ ′(2|z|)I2k−1(2|z|)I2k ′−1(2|z|) · (B1)

·∞∑

n,n′=0

(z∗)n′zn

√n!�(n + 2k)

√n′!�(n′ + 2k ′)

⟨n′, k ′∣∣K−|n, k〉.

The raising operator K− satisfies the equation

⟨n′, k ′∣∣K−|n, k〉 =

√n(n + 2k − 1)

⟨n′, k ′ ∣∣ n − 1, k

⟩=√

n(n + 2k − 1) δn′,n−1 δk ′,k, (B2)

⟨K−⟩z = 1

J0(2|z|)∞∑

J,J ′=0

(−2|z|) J+J ′2√

J !J ′!|z|k+k ′−1 ·

·√

IJ (2|z|)IJ ′(2|z|)I2k−1(2|z|)I2k ′−1(2|z|) ·

·∞∑

n′=0

(z∗)n′zn′+1

√(n′ + 1)!�(n′ + 2k + 1)

√n′!�(n′ + 2k)

·

·√

(n′ + 1)(n′ + 2k + 1), (B3)

⟨K−⟩z = |z| 1

J0(2|z|) · (B4)

·∞∑

J=0

(−2|z|)J

J !IJ (2|z|) = |z|.

By analogy, the expectation value of the product operators K+ K− is

⟨n′, k ′∣∣ K+ K−|n, k〉 = n(n + 2k − 1)

⟨n′, k ′ ∣∣ n, k

⟩= n(n + 2k − 1)δn′,nδk ′,k, (B5)

⟨K+ K−

⟩z = 1

J0(2|z|)∞∑

J,J ′=0

(−2|z|) J+J ′2√

J !J ′!|z|k+k ′−1 ·

·√

IJ (2|z|)IJ ′(2|z|)I2k−1(2|z|)I2k ′−1(2|z|) · (B6)

·∞∑

n′=0

(z∗)n′zn′

√n′!�(n′ + 2k)

√n′!�(n′ + 2k)

n′(n′ + 2k − 1).

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032103-21 Popov et al. J. Math. Phys. 54, 032103 (2013)

After the substitution n′ − 1 = m and the elimination of the first term which corresponds tom = − 1, the last sum becomes

∞∑n′=0

(z∗)n′zn′

(n′ − 1)!�(n′ + 2k − 1)= (B7)

= |z|2∞∑

m=0

(|z|2)m

m!�(m + 2k)= |z|2 I2k−1(2|z|)

|z|2k−1.

Finally, the result is ⟨K+ K−

⟩z = |z|2. (B8)

1 A. M. Perelomov, “Coherent states for arbitrary Lie group,” Commun. Math. Phys. 26, 222 (1972).2 V. Buzek, “SU(1,1) squeezing of SU(1,1) generalized coherent states,” J. Mod. Opt. 37(3), 303 (1990).3 M. Novaes, “Some basics of su(1,1),” Rev. Bras. Ensino Fis. 26(4), 351–357 (2004).4 A. M. Perelomov, Generalized Coherent States and Their Applications (Springer-Verlag, Berlin, 1986).5 J. K. Kim and S. P. Kim, “Group Theoretical Approach to the Coherent and the Squeeze States of a Time-Dependent

Harmonic Oscillator with a Singular Term,” J. Korean Phys. Soc. 28(1), 7–15 (1995).6 A. O. Barut and L. Girardello, “New ‘coherent’ states associated with non-compact groups,” Commun. Math. Phys. 21(1),

41–55 (1971).7 D. Popov, “Barut-Girardello coherent states of the pseudoharmonic oscillator,” J. Phys. A: Math. Gen. 34, 5283 (2001).8 J. R. Klauder, K. A. Penson, and J.-M. Sixdeniers, “Constructing coherent states through solutions of Stieltjes and Hausdorff

moment problems,” Phys. Rev. A 64(1), 013817 (2001).9 I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 7th ed. (Academic Press, Amsterdam, 2007).

10 J.-P. Antoine, J.-P. Gazeau, P. Monceau, J. R. Klauder, and K. A. Penson, “Temporally stable coherent states for infinitewell and Poschl–Teller potentials,” J. Math. Phys. 42, 2349 (2001).

11 A. M. Mathai and R. K. Saxena, Generalized Hypergeometric Functions with Applications in Statistics and PhysicalSciences, Lecture Notes in Mathematics Vol. 348 (Springer, Berlin, 1973).

12 D. Popov, “Some Properties of Generalized Hypergeometric Thermal Coherent States,” EJTP, Electronic J. Theoret. Phys.3(11), 123–132 (2006). http://www.ejtp.com/articles/ejtpv3i11p123.pdf.

13 J. R. Klauder, “Coherent states for the hydrogen atom,” J. Phys. A 29(12), L293–L298 (1996).14 D. Popov, “Photon-added Barut–Girardello coherent states of the pseudoharmonic oscillator,” J. Phys. A 35(33), 7205–7223

(2002).15 L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, Cambridge, 1994).16 A. Nikiforov and V. Ouvarov, lments de la thorie des Fonctions Spciales (ditions Mir, Moscou, 1976).17 Abramowitz, M. and I. A. Stegun, Handbook of Mathematical Functions, National Bureau of Standards Applied Mathe-

matics Series Vol. 55 (Tenth Printing, Washington, DC, 1972).18 H. P. Robertson, “An indeterminacy relation for several observables and its classical interpretation,” Phys. Rev. 46(9),

794–801 (1934).19 V. V. Dodonov, “‘Nonclassical’ states in quantum optics: A ‘squeezed’ review of the first 75 years,” J. Opt. B: Quantum

Semiclassical Opt. 4(1), R1–R33 (2002).20 V. V. Dodonov, M. A. Man’ko, V. I. Man’ko, and A. Vourdas, “Squeezed states and uncertainty relations since 1991,”

J. Russ. Laser Res. 28(5), 404–428 (2007).21 K. Husimi, “Some formal properties of the density matrix,” Proc. Phys.-Math. Soc. Japan, III. Ser. 22, 264–314 (1940).22 J.-M. Sixdeniers, K. A. Penson, and A. I. Solomon, “Mittag-Leffler coherent states,” J. Phys. A 32(43), 7543–7564 (1999).

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