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Physics Letters A www.elsevier.com/locate/pla

Generalized Heisenberg algebra coherent states for power-law potentials K. Berrada a,b,∗ , M. El Baz b , Y. Hassouni b a b

Institut für Angewandte Physik, Technische Universität Darmstadt, D-64289, Germany Laboratoire de Physique Théorique, Faculté des Sciences, Université Mohammed V-Agdal, Av. Ibn Battouta, B.P. 1014, Agdal Rabat, Morocco

a r t i c l e

i n f o

Article history: Received 23 April 2010 Received in revised form 6 November 2010 Accepted 9 November 2010 Available online 13 November 2010 Communicated by A.P. Fordy

a b s t r a c t Coherent states for power-law potentials are constructed using generalized Heisenberg algebra. Klauder’s minimal set of conditions required to obtain coherent states are satisﬁed. The statistical properties of these states are investigated through the evaluation of the Mandel’s parameter. It is shown that these coherent states are useful for describing the states of real and ideal lasers. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Ever since the introduction of coherent states theory, several works have been focused on these states and they have turned out to be a very useful tool for the investigation of various problems in physics and have offered surprisingly rich structures [1–4]. Coherent states were ﬁrst introduced by Schrödinger [5] in the context of the harmonic oscillator, and were interested in ﬁnding quantum states which provide a close behavior to the classical one. Later, the notion of coherent states had become very important in quantum optics thanks to the works of Glauber [6]. He deﬁned these states as eigenstates of the annihilation operator aˆ of the harmonic oscillator, while demonstrating that they have the interesting property of minimizing the Heisenberg uncertainty relation. The notion of coherent states was extended afterwards to different systems other the harmonic oscillator for which they were originally built. Coherent states associated to SU (2) and SU (1, 1) algebras were introduced by Peremolov [7,8]. These states describe several systems and also have many applications in quantum optics, statistical mechanics, nuclear physics, and condensed matter physics [9–11]. Generally, there are several deﬁnitions of coherent states. The ﬁrst is to deﬁne the coherent states as eigenstates of an annihilation operator with complex eigenvalues. This deﬁnition is called Barut–Girardello coherent states [12]. The second deﬁnition, often called Peremolov coherent states [7,13], is to construct the coherent states by acting with a unitary z-displacement operator on the ground state of the system, with z ∈ C . The last deﬁnition, often called intelligent coherent states [14,15], is to consider that the coherent states give the minimum-uncertainty value

*

Corresponding author at: Laboratoire de Physique Théorique, Faculté des Sciences, Université Mohammed V-Agdal, Av. Ibn Battouta, B.P. 1014, Agdal Rabat, Morocco. E-mail addresses: [email protected] (K. Berrada), [email protected] (M. El Baz), [email protected] (Y. Hassouni). 0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.11.027

x p = h2¯ . These three deﬁnitions are equivalent only in the spe-

cial case of the Heisenberg–Weyl group (Harmonic oscillator). Due to this multitude of deﬁnitions of the notion of coherent states, Klauder [9] proposed a set of conditions to be obeyed by any set of coherent states. These conditions are in fact the common properties satisﬁed by coherent states in all the deﬁnitions presented before. In this way this set of condition constitutes the minimum set of requirements on a set of states in order to deserve the nomenclature coherent states. Generally, it is diﬃcult to build coherent states for arbitrary quantum mechanical systems. But for systems with discrete spectrums, there is an elegant method of building coherent states which has been recently proposed in [1] basing on generalized Heisenberg algebra. These coherent states are deﬁned as eigenstates of the annihilation operator of the generalized Heisenberg algebra having inﬁnite-dimensional representations and satisfying the minimum set of conditions required to construct Klauder’s coherent states. Coherent states for harmonic oscillator and other systems have attracted much attention for the past years [16–22] and play important roles in different contexts. The power-law potentials (a general class of potentials) have many applications in theoretical and experimental physics, which can be used to describe a large class of quantum mechanical systems [23–26]. The coherent states of these potentials could be very helpful and bring more insights on these subjects. In this Letter, we are going to construct the coherent states for these potentials following the formalism used in [1]. We, then, investigate the statistical properties of these coherent states. It will be shown that these states can exhibit a sub-Poissonian, Poissonian or a super-Poissonian distribution depending on the parameters of the system. The organization of the Letter is as follows. In Section 2 we describe the construction of generalized Heisenberg algebra coherent states; in Section 3 we build coherent states for power-law potentials and investigate their statistical properties using Man-

K. Berrada et al. / Physics Letters A 375 (2011) 298–302

del’s Q parameter; in the last section, we present our conclusion. 2. Coherent states of generalized Heisenberg algebra

†

H A = A f ( H ),

(1)

A H = f (H ) A,

(2)

A, A† = f (H ) − H ,

(3)

where its generators ( H , A , A † ) obey the Hermiticity properties A = ( A † )† , H = H † . The Casimir element of the GHA is given by

C = A † A − H = A A † − f ( H ).

(5)

†

A |m = N m |m + 1,

(6)

A |m = N m−1 |m − 1

(7)

with

(8)

where E m = f m ( E 0 ), the mth iterate of E 0 under f , are the eigenvalues of the Hamiltonian H . A † and A are the generalized creation and annihilation operators of the GHA. This GHA describes a class of quantum systems having energy spectra written as

E n +1 = f ( E n )

(9)

where E n+1 and E n are successive energy levels. The coherent states for the GHA were previously investigated in Ref. [1]. They are deﬁned as the eigenvalues of the annihilation operator of the GHA

z ∈ C,

| z = N ( z)

n =0

zn N n −1 !

|n.

N ( z) =

∞ | z|2n n =0

N n2−1 !

| z − z → 0;

(14)

(iii) Resolution of unity: there must exist a positive Weight function W (| z|2 ) such that

d2 z | z z| W | z|2 = 1.

(15)

C

In general, the ﬁrst two conditions are easily satisﬁed. This is not the case for the third. In fact, this condition restricts considerably the choice of the states to be considered. Now, we are going to analyze the above minimal set of conditions to obtain Klauder’s coherent states for power-law potentials. 3. Coherent states for power-law potentials The general expression of a one-dimensional power-law potential is [23]

k x ˆ V (x, k) = V 0 , a

(16)

where V 0 and a are constants with the dimensions of energy and length, respectively. Here k is the power-law exponent. These power-law potentials can be used to describe a large class of quantum systems by a proper choice of the exponent k. For k > 2 we ﬁnd tightly binding potentials, for k = 2 we have harmonic oscillator potentials and k < 2 expresses loosely binding potentials. The Hamiltonian corresponding to different potentials can be written in the form

pˆ 2 2m

+ Vˆ (x, k),

(17)

where the eigenvalue equations are given by

ˆ (k)|n = E n,k |n, H

n 0.

(18)

The energies E n,k may be obtained with the help of W K B approximation [24–26]

2k γ k +2 E n,k = ω(k) n + ,

(19)

4

where

π

1 k

Γ ( 1k + 32 )

k2k +2

ω(k) =

(11)

is the effective frequency. Here γ deﬁnes the Maslov index which accounts for boundary effects at the classical turning points. The above equation shows that the energy difference between

√

2a 2m

V0

Γ ( 1k + 1)Γ ( 32 )

γ

The normalization factor is given by

⇒

(10)

and can be expressed as ∞

(13)

z − z → 0

ˆ = H

2 Nm = E m +1 − E 0 ,

A | z = z| z,

z | z = 1;

(4)

Here H is the Hamiltonian of the physical system under consideration and f ( H ) is an analytic function of H , called the characteristic function of the algebra. It was shown in Ref. [27] that a large class of Heisenberg algebras can be obtained by choosing the appropriate function f ( H ). For example, by choosing the characteristic function as f ( H ) = H + 1, the algebra described by Eqs. (1)–(3) becomes the harmonic oscillator algebra and for f ( H ) = qH + 1 we obtain the deformed Heisenberg algebra. The irreducible representation of the GHA is given through a general vector |m which is required to be an eigenstate of the Hamiltonian. It is described as follows

H |m = E m |m,

(i) Normalizability

(ii) Continuity in the label

In this Letter, we are going to use the generalized Heisenberg algebra (GHA) which has been introduced in Ref. [27] and construct the coherent states associated. The version of the GHA we are going to describe is deﬁned by the following commutation relations †

299

− 12 ,

(12)

where by deﬁnition N n ! = N 0 N 1 · · · N n and by consistency N −1 ! = 1. According to Klauder [9,28], the minimal set of requirements to be imposed on a state | z to be a coherent state is:

k −2

adjacent levels, E n,k = E n,k − E n−1,k ∝ (n + 4 ) k+2 , increases for k > 2 as n increases (tightly binding potentials). In contrast, for k < 2 it decreases with the increase of n (loosely binding potentials), while for k = 2 it does not depend on n, so the energy spectrum is equally spaced. The question now is to ﬁnd the characteristic function of the generalized Heisenberg algebra associated with these spectrums. From the above expression, we have 2k

n = ( E n,k ) k+2 −

γ 4

(20)

300

K. Berrada et al. / Physics Letters A 375 (2011) 298–302

leading us to the expression

E n+1,k = n + 1 +

k2k +2

γ

(21)

.

4

The substitution of Eq. (20) in Eq. (21) allows us to obtain the characteristic equation

E n+1,k = ( E n,k )

k +2 2k

+1

2k k +2

(22)

.

The characteristic function, then, is

f (x, k) = (x)

k +2 2k

+1

2k k +2

(23)

.

Here, we have used unities such that h¯ = ω = 1 as is usually the case in quantum mechanics. This actually amounts to normalizing the energies by dividing by ω(k). According to the scheme prescribed in the previous section, the coherent states associated to power-law potentials are given by

| z , k = N | z |2 , k

∞

n =0

zn g (n, k)

|n

(24)

g (n, k) =

n

i+

γ

k2k +2

4

i =1

k2k γ +2 , −

M−1 [ g (s − 1, k); x] , π N 2 (x, k)

N | z| , k =

4

g (0, k) = 1,

(25)

and Eq. (27) holds. Now, we are going to build the coherent states of physical systems corresponding to particular choices of the exponent k. For k = 2, the GHA associated to the power-law potential spectrum reduces to the Heisenberg algebra with characteristic function f (x, 2) = x + 1. In this case, we have N n2−1 = n and Eq. (24) becomes the standard coherent states for the harmonic oscillan ∞ tor, | z, 2 = N (| z|2 , 2) n=0 √z |n, with normalization function n!

N (| z|2 , 2) = exp(− |z2| ) and the Weight function π1 . In the limit case k → ∞, the power-law potential becomes the 2

inﬁnite square-well potential, and the energies deﬁned in Eq. (19) are written as

E n,∞ = (n + 1)2 ,

n = 1, 2, 3, . . . .

∞ | z|2n g (n, k)

− 12 (26)

.

E n+1,∞ = ( E n,∞ + 1)2 .

H , A† = 2 A†

√

√

Now, we can investigate the overcompleteness (or resolution of unity operator) of the GHA coherent states for power-law potentials given by Eq. (24). To this end, we assume the existence of a positive Weight function W k (| z|2 ) so that an integral over the complex plane exists and gives the result

d2 z| z, k z, k| W k | z|2 = 1.

(27)

C

Indeed, substituting Eq. (24) into Eq. (27), adopting the polar coordinate representation z = r exp(i θ) of complex numbers and using the completeness of the states |n, the resolution of unity can be expressed by

∞ xn W k (x) dx = g (n, k),

n = 0, 1, 2, . . .

(28)

xs−1 W k (x) dx = g (s − 1, k) = M W k (x); s ,

†

A |n = A |n =

(n

(n

+ 2)2

+ 1)2

0

n = 1, 2, . . . ,

− 1|n + 1,

1 2π i

g (s − 1, k)x−s ds = M−1 g (s − 1, k); x

C −i ∞

(39)

zn N n −1 !

|n

(40)

where

1 √ N n −1 ! = √ n! (n + 2)!. 2

(41)

The normalization condition requires that

2 N 2 | z |2 , ∞

∞

| z|2n = 1. n!(n + 2)!

(42)

Noting that

n =0

(30)

(37) (38)

− 1|n − 1.

∞

| z, ∞ = N | z|2 , ∞

∞

on the other hand

(36)

Therefore, the coherent states are

n =0

(29)

(35)

Fock space representations of the algebra generated by H , A and A † , as in Eqs. (34)–(36), are obtained by considering the eigenstates |n of H . We can verify that N n2−1 = (n + 1)2 − 1, and the action of this algebra generators on |n is given by

n =0

where x = r 2 and W k (x) = π W k (x)N 2 (x, k). The positive quantities g (n, k) are the power moments of the function W k (x) and the problem stated in Eq. (28) is the Stieltjes moment problem which can be solved by means of Mellin and inverse Mellin transforms. To determine the form of the function W k (x), we extend the natural integers n to complex values s (n → s − 1) and we rewrite Eq. (28) as [29,30]

(34)

[ H , A ] = −2 H A − A , √ A , A † = 2 H + 1.

0

C +i ∞

(33)

H + A†,

H |n = (n + 1)2 |n,

W k (x) =

(32)

From Eq. (33) we ﬁnd that the √ characteristic function for this physical system is f (x, ∞) = ( x + 1)2 and the corresponding GHA generated by the Hamiltonian H and the creation and annihilation operators A † , A is given by

n =0

∞

(31)

and the normalization function 2

W k (x) =

We can check that

where

where M[ W k (x); s] is the Mellin transform of W k (x) and M−1 [ g (s − 1, k); x] is the inverse Mellin transform of g (s − 1, k). Finally, the Weight function may be written as

| z|2n I 2 (2| z|) , = n!(n + 2)! | z |2

(43)

for 0 | z| < 1, where I n ( z) is the modiﬁed Bessel function of the ﬁrst kind of order n. Thus, the normalization function may be writ-

K. Berrada et al. / Physics Letters A 375 (2011) 298–302

301

ten as

N | z| , ∞ =

2

| z |2 2I 2 (2| z|)

12 (44)

where 0 | z| < ∞. To resolve the identity operator in terms of the coherent states corresponding to the square-well potential we need to ﬁnd the Weight function W ∞ (x) satisfying

∞ xn W ∞ (x) dx = n!(n + 2)!,

n = 0, 1, 2, . . .

(45)

0

where x = r 2 and W ∞ (x) = π W ∞ (x)N 2 (x, ∞). From Eq. (30), the function W ∞ (x) becomes C +i ∞

1

W ∞ (x) =

2π i

Γ (s)Γ (s + 2)x−s ds.

(46)

C −i ∞

For determining this function, we shall mainly use the following relations satisﬁed by the Mellin transform

C +i ∞

1 2π i

∗

1 − s+b ∗ s + b , a h f h h

M xb f axh ; s = ∗

−s

f (s) g (s)x

{a, h} > 0,

∞ ds =

C −i ∞

Fig. 1. Mandel’s Q parameter for loosely binding potentials versus | z| for k = 0.5 (dotted line), k = 1 (dashed line) and k = 1.5 (solid line).

x

f

t

g (t )

dt t

(47)

(48)

0

where M [ f (x); s] = f ∗ (s) and M [ g (x); s] = g ∗ (s). Employing the above relations and using the modiﬁed Bessel function of second kind K 2 (x) [31], we ﬁnd that

W ∞ (x) =

1 2π i

C +i ∞

Γ (s)Γ (s + 2)x−s ds

C −i ∞

∞

t exp(−t ) exp −

=

x t

dt

0

√ = 2xK 2 (2 x ).

(49)

The Weight function W ∞ (x) given in Eq. (45), allowing the resolution of the identity operator, is written as

Fig. 2. Mandel’s Q parameter for tightly binding potentials versus | z| for k = 5 (dotted line), k = 10 (dashed line), k = 15 (dashed–dotted line), and k → ∞ (solid line).

√

W ∞ (x) =

2xK 2 (2 x )

π N 2 (x, ∞)

(50)

.

Finally,

W ∞ (x) =

4

π

√

√

K 2 (2 x ) I 2 (2 x ).

(51)

In order to complete the discussion, we should investigate the statistical properties of the GHA coherent states for power-law potentials. To this end, we study Mandel’s Q -parameter deﬁned by [32]

Q =

( N )2 − N , N

(52)

where ∞

| z|2n N = N 2 | z |2 , k n g (n, k)

(53)

n =0

is the average photon number of the state, and

( N )2 = N 2 − N 2 .

(54)

This parameter is a good indication to determine whether a state has a sub-Poissonian (if −1 Q < 0), Poissonian (if Q = 0) or super-Poissonian photon number distribution (if Q > 0). In Fig. 1, we plot the Mandel’s Q -parameter in terms of the GHA coherent states amplitude | z| for loosely binding potentials (k < 2). From the ﬁgure we ﬁnd that the states exhibit a superPoissonian behavior for lower values of | z| and sub-Poissonian for high | z|. In the case of tightly binding potentials (k > 2), it is noticed that the Mandel’s parameter is always negative for all values of | z| (see Fig. 2), so the distribution is sub-Poissonian. However, we observe that the state becomes in some sense less classical as | z| becomes large. By less classical we mean here that the states in question get farther from the states exhibiting Poissonian statistics in particular Glauber’s coherent states. It is well known that the study of the statistical properties is an important topic in quantum optics. In this way, our results show that those coherent states may be useful to describe the states of ideal and real lasers by a proper choice of the power-law exponent. This parameter plays then the role of a tuning parameter deﬁning how far the realized real laser is from the ideal one.

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K. Berrada et al. / Physics Letters A 375 (2011) 298–302

4. Conclusion In this Letter, using an algebraic approach, we constructed coherent states for power-law potentials. The minimum set of conditions required to obtain Klauder’s coherent states is investigated. We have shown that these states describe a large class of quantum systems (harmonic oscillator, loosely binding potentials and tightly binding), which can be used in several branches of quantum physics. Finally, we investigated the statistical properties of these coherent states using Mandel’s Q parameter. We have shown that they exhibit Poissonian, sub-Poissonian or super-Poissonian distributions. However, these states are useful to describe the states of ideal and real lasers by a proper choice of the different parameters (i.e., z and k). These properties are useful in branches of quantum physics, such as quantum information. In this context, it has been shown that in order to have entangled states as output states of a beam splitter require nonclassical states (sub-Poissonian or super-Poissonian) at its input [33]. Indeed in such studies, we are interested in the quantum behavior of power-law potential states. So by a proper choice of the different parameters, one can choose an adequate set of states in order to generate the entanglement of bipartite composite systems using a beam splitter. This leads us to conclude that these states may be helpful to extend the horizons of quantum information such as entanglement generation [34], and exploit the entangled states in the context of quantum teleportation [35], dense coding [36] and entanglement swapping [37]. This idea is a topic for future research. Acknowledgements K.B. is supported by the National Centre for Scientiﬁc and Technical Researcher of Morocco. K.B. pleased to express his sincere gratitude for the hospitality at the Institut für Angewandte Physik in Darmstadt where part of this work is done. He acknowledges ﬁnancial support by the DAAD. The authors wish to thank the referees for their valuable comments that resulted in improvements of the paper in many aspects.

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