Generalized coherent states and their statistical characteristics in power- law potentials Shahid Iqbal and Farhan Saif Citation: J. Math. Phys. 52, 082105 (2011); doi: 10.1063/1.3626936 View online: http://dx.doi.org/10.1063/1.3626936 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v52/i8 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 Downloaded 06 Aug 2013 to 128.248.155.225. This article is copyrighted as indicated in the abstract. Reuse of AIP content is subject to the terms at: http://jmp.aip.org/about/rights_and_permissions
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Generalized coherent states and their statistical characteristics in power-law potentialsShahid Iqbal and Farhan Saif Citation: J. Math. Phys. 52, 082105 (2011); doi: 10.1063/1.3626936 View online: http://dx.doi.org/10.1063/1.3626936 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v52/i8 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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Coherent states, first introduced by Schrodinger in 1926, are the venerable objects in the moderntheories of physics, as these are quantum mechanical states which follow classical trajectories.1 In1963, Glauber introduced the coherent states (CSs) in the context of quantum theory of light, asthe states that are closest to classical description of radiation field.2, 3 These states contain importantconcepts in quantum optics and have many other applications in different fields of applied physicsand mathematical physics,4–7 ranging from solid state physics to cosmology. The standard CSsof a radiation field are usually constructed using one of the following three methods: (i) they areeigenstates of the standard annihilation operator, a, of the harmonic oscillator, i.e., a|α〉 = α|α〉,where |α〉 is coherent state; (ii) they can be reproduced by the action of a unitary displacementoperator on the vacuum of radiation field, i.e., exp(αa† − α∗a), α being a complex number; (iii)they minimize the Heisenberg uncertainty relation �z.�p ≥ 1
2 , with equal variances as the limit ofvacuum in both quadratures.
Because of their abundant applications in various branches of physics,4, 5 there have beencontinuous efforts to generalize the notion of coherent states for the systems other than harmonicoscillator. The early work on generalized CSs is based on a variety of algebraic groups, such asLie groups SU (1, 1)8 and non-compact groups SO(2, 1).9, 10 In 1996, Klauder suggested a new typeof CSs suitable for systems with degenerate discrete spectra (i.e., the hydrogen atom) without aexplicit dependence on any group structure.11 Later, Gazeau and Klauder (hereafter referred as GK)extended this idea to define CSs for non-degenerate discrete and continuous spectra.12
Generally, it is difficult to build coherent states for arbitrary quantum mechanical systems. ButGazeau and Klauder procedure provides us an elegant method of building coherent states for arbitraryquantum systems with known spectrums. In this paper we present generalized coherent states, basedon Gazeau and Klauder formalism, for power-law potentials13, 14 and investigate their statisticalproperties. The power-law potentials can be used to describe a large class of quantum mechanicalsystems13–16 that have many applications in theoretical and experimental physics. Therefore, thecoherent states of these potentials could be very helpful to bring more insights on these subjects.
In contrast to the coherent states of harmonic oscillator, generalized coherent states may ex-hibit various nonclassical properties such as squeezing and sub-Poissonian statistics, for the reason,nonclassical states of light are central to quantum optics. Their importance comes from poten-
a)Electronic mail: [email protected]. Also at Department of Physics, Govt. College University, Lahore 54000, Pakistan.b)Also at Center for Applied Physics and Mathematics, National University of Science and Technology, Islamabad, Pakistan.
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082105-2 S. Iqbal and F. Saif J. Math. Phys. 52, 082105 (2011)
tial applications in advanced areas of quantum optics, such as, quantum teleportation,20 quantumcomputation,21 quantum communication,22 quantum cryptography,23 and quantum lithography.24 Inparticular, a relation between coherent states and entanglement is of special interest in quantum infor-matics, where a careful knowledge of the properties of the initial states (i.e., their nonclassical nature)is needed.25, 26 In this article we investigate the probability distributions of Gazeau-Klauder gener-alized coherent states of power-law potentials and show that these states exhibit super-Poissonian,Poissonian, or sub-Poissonian distributions depending on the values of the power-law exponent.Moreover, we compare the statistical properties of GK coherent states of power-law potentials tothose of generalized Heisenberg algebra (GHA) coherent states of power-law potentials,27 publishedrecently, and report that some of the statistical properties presented in Ref. 27 are incorrect.
The paper is organized as follows: In Sec. II, we give a brief description of power-law potentialsand their underlying energy spectra. In Sec. III, we construct GK coherent states for power-lawpotentials and investigate their basic properties. In Sec. IV, we analyze the statistical characteristicsof these CSs by means of Mandel’s Q-parameter. We compare the statistical properties of GKcoherent states to those of generalized Heisenberg algebra coherent states in Sec. V and conclusionsare given in Sec. VI.
II. GENERAL POWER-LAW POTENTIALS
Power-law potentials describe a large class of physical systems, effectively divided into twotypes: loosely binding potentials and tightly binding potentials.14 The general form of a one-dimensional power-law potential14, 15 is
V (k)(x) = V0
∣∣∣ x
a
∣∣∣k, (1)
where V0 and a are constants with dimensions of energy and length, respectively. The exponentk determines the type of potential: tightly binding potentials correspond to k > 2, loosely bindingpotentials to k < 2, and the harmonic oscillator, to k = 2. The corresponding Hamiltonian,
H (k) = p2
2m+ V (k)(x), (2)
obeys the eigenvalue equation,
H (k)|n〉 = E (k)n |n〉, n ≥ 0. (3)
The eigenenergies E (k)n can be obtained within the WKB approximation,14, 15
E (k)n = ω(k)
(n + γ
4
)2k/(k+2), (4)
where
ω(k) =[
�π
2a√
2mV 1/k
0
�(1/k + 3/2)
�(1/k + 1)�(3/2)
]2k/(k+2)
(5)
is a constant for any particular value of k, in units of energy. Here, γ is the Maslov index, whichaccounts for boundary effects at the classical turning points. In particular, γ = 2, if both turningpoints lie at “soft boundaries” and γ = 4, if both turning points lie at “hard boundaries”, such as,harmonic oscillator and infinite square well, respectively.14, 17
The WKB method is often used to find large-n behavior of a quantum system. However, soobtained eigenenergies for power-law potentials, E (k)
n , given in Eq. (4), give exact quantized energiesfor k = 2, i.e,
E (2)n = �� (n + 1/2) , (6)
that correspond to harmonic oscillator with frequency �, where arbitrary constant V0 is m�2a2/2.For k → ∞, the eigen-energies are
E (∞)n = (n + 1)2π2
�2
8ma2, (7)
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082105-3 Generalized coherent states and their statistical J. Math. Phys. 52, 082105 (2011)
which correspond to infinite square well. Moreover, Eq. (4) with k = 1, gives WKB eigenenergiesfor symmetric linear potential,
E (1)n =
[(n + 1
2
)3�π
4a√
2mV0
]2/3
, (8)
which leads us to calculate the eigenenergies of a familiar system, namely, triangular well potentialor quantum bouncer,18, 19 i.e.,
E (qb)n =
[3�πg
√m
2√
2
(n + 3
4
)]2/3
, (9)
where, we have taken V0 = mga, with g being gravitational acceleration and γ = 3, to obtainEq. (9). It is worth noting that WKB energies in Eq. (9), give a very good estimate to exact quantizedenergies13, 18 of quantum bouncer. For the simplicity of our computations, we will use � = m = 1throughout the following discussion.
In order to gain insight into the structure of the energy spectrum given by Eq. (4), we take theenergy difference between adjacent levels,
�E (k)n = E (k)
n − E (k)n−1 = ω(k)
(2k
k + 2
)(n + γ
4
) k−2k+2
. (10)
Equation (10) shows that for k = 2, �E (k)n does not depend on n, so the energy spectrum is equally
spaced. For k > 2, the energy difference between adjacent levels increases with n (tightly bindingpotentials), while for k < 2, it decreases with n (loosely binding potentials).
The Hamiltonian H (k) can be factorized as
H (k) = ω(k) X (k)N , (11)
where X (k)N is a dimensionless Hamiltonian that obeys the eigenvalue equation,
X (k)N |n〉 = e(k)
n |n〉, (12)
where e(k)n are dimensionless eigenvalues given by
e(k)n = E (k)
n − E (k)0
ω(k)=
(n + γ
4
)2k/(k+2)−
(γ
4
)2k/(k+2). (13)
The energy spectrum e(k)n is an increasing function of n, e(k)
n+1 > e(k)n , with e(k)
0 = 0.
III. GK COHERENT STATES FOR POWER-LAW POTENTIALS
Gazeau and Klauder proposed a formalism12 to construct coherent states of Hamiltonians withdiscrete and non-degenerate energy spectra without any explicit dependence on group properties.These states are labelled by two real parameters, J and θ , with J ≥ 0 and −∞ < θ < ∞. Forgeneral power-law potentials they have the form,
∣∣(J, θ )(k)⟩ = 1√
N (k)(J )
∞∑n=0
Jn2 e−ie(k)
n θ√ρ
(k)n
|n〉. (14)
The quantities ρ(k)n are defined in terms of e(k)
n , given by Eq. (13) , as
ρ(k)n ≡
n∏j=1
e(k)j =
n∏j=1
[(j + γ
4
)2k/(k+2)−
(γ
4
)2k/(k+2)]
. (15)
Following Gazeau and Klauder’s definitions, CSs must obey the following properties:11, 12 (a)normalization; (b) continuity in J and θ ; (c) resolution of unity; (d) action identity, and (e) temporalstability. In the following we discuss these properties for the states given in Eq. (14).
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082105-4 S. Iqbal and F. Saif J. Math. Phys. 52, 082105 (2011)
The normalization⟨(J, θ )(k)
∣∣(J, θ )(k)⟩ = 1 requires that the normalization constant is
N (k)(J ) =∞∑
n=0
J n
ρ(k)n
. (16)
The overlap of the two CSs can be written as
⟨(J ′, θ ′)(k)
∣∣∣ (J, θ )(k)〉 = 1√N (k)(J ′)
√N (k)(J )
∞∑n=0
(J ′ J )n
ρ(k)n
e−i(θ−θ ′)e(k)n . (17)
The continuity in J and θ follows from the continuity of the overlap⟨(J ′, θ ′)(k)
∣∣(J, θ )(k)⟩
because
‖ ∣∣(J, θ )(k)⟩ − ∣∣(J ′, θ ′)(k)⟩ ‖2= 2[1 − Re
⟨(J ′, θ ′)(k)
∣∣(J, θ )(k)⟩] (18)
approaches zero as (J ′, θ ′) → (J, θ ).For the resolution of unity, we take an integration measure dμ (J, θ ), such that,12
∫dμ (J, θ )
∣∣(J, θ )(k)⟩ ⟨
(J, θ )(k)∣∣ = lim
�→∞1
2�
∫ R
0
d J h(J )∫ �
−�
dθ∣∣(J, θ )(k)
⟩ ⟨(J, θ )(k)
∣∣ , (19)
where h(J ) is defined as12
h(J ) = N (k)(J )w(k)(J ) 0 ≤ J < R, (20)
which is specified by a proper choice of the probability distribution function, wk(J ). The range ofallowed values of J , 0 ≤ J < R, is determined by the radius of convergence,
R = limn→∞
n
√ρ
(k)n . (21)
For k > 0, R is infinite. Integration over θ in Eq. (19) provides
lim�→∞
1
2�
∫ �
−�
(.)dθ = δe(k)n ,e(k)
m, (22)
which can be identified with the Kronecker delta δn,m , due to the non-degeneracy of the energyspectrum. Therefore we can write Eq. (19) as
∫dμ (J, θ )
∣∣(J, θ )(k)⟩ ⟨
(J, θ )(k)∣∣ = I =
∞∑n=0
1
ρ(k)n
{∫ ∞
0
h(J )
N (k)(J )J nd J
}|n〉 〈n| , (23)
provided there is a function w(k)(J ) which satisfies∫ ∞
0
J nw(k)(J )d J = ρ(k)n
. (24)
The positive constants ρ(k)n
are then power moments of the function w(k)(J ). For an appropriatechoice of w(k)(J ),28 the integral in Eq. (24) becomes a Stieltjes moment problem that can be solvedusing Mellin and inverse Mellin transformations.
The action identity can easily be obtained from Eqs. (13)–(16),
⟨(J, θ )(k)
∣∣ H∣∣(J, θ )(k)
⟩ = ω(k)
N (k)(J )
∞∑n=0
J n
ρ(k)n
e(k)n = ω(k) J, (25)
where we have used ρ(k)0 = 1 as in Ref. 12. However, in our earlier work,13 we have shown that
temporal stability, as exhibited by Glauber CS, ceases to occur in GK CS and they disperse inthe long-time evolution, which leads to the formation of space time structures, namely, quantumcarpets.29, 30
The particular case of the harmonic oscillator corresponds to k = 2 in Eq. (1). In this case,Eqs. (13) and (15) provide e(2)
n → n and ρ(2)n → n!, respectively, and the generalized CSs in Eq. (14)
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082105-5 Generalized coherent states and their statistical J. Math. Phys. 52, 082105 (2011)
become
∣∣(J, θ )(2)⟩ = 1√
N (2)(J )
∞∑n=0
Jn2 e−inθ
√n!
|n〉 , (26)
where the normalization constant N (2)(J ) is
N (2)(J ) =∞∑
n=0
J n
n!. (27)
It is important to note that the GK CSs of the harmonic oscillator, given by Eq. (26), are completelyequivalent to Glauber’s CSs.2, 3 This can easily be seen with a parameter transformation from (J, θ )to z, such that, z = √
Je−iθ . In this case, the normalization factor in Eq. (27) is, N (2)(J ) = e−|z|2/2,as for Glauber’s CSs.
IV. QUANTUM STATISTICAL PROPERTIES
The statistical features of GK CSs of power-law potentials can be probed through the weightingdistribution
∣∣〈n| (J, θ )(k)〉∣∣2, such that
P (k)n ≡ ∣∣〈n| (J, θ )(k)〉∣∣2 = 1
N (k)(J )
J n
ρ(k)n
. (28)
For the harmonic oscillator (k = 2), using the transformation z = √Je−iθ , one obtains
P (2)n = |z|2ne−|z|2/n!, (29)
which is a Poisson distribution. However, Eq. (28) reveals that for k = 2, the distribution is notPoissonian. In order to characterize the weighting distributions one needs to calculate the expectationvalues,
⟨(J, θ )(k)
∣∣ N m∣∣(J, θ )(k)
⟩ =∞∑
n=0
nm J n
N (k)(J )ρ(k)n
≡ ⟨nm
⟩, (30)
where, N is bosonic number operator. For m = 1, 2 Eq. (30) leads us to calculate the variance of thedistribution, as
σ 2 = ⟨n2
⟩ − 〈n〉2 . (31)
The Poisson distribution for Glauber’s coherent states2, 3 (k = 2), as given in Eq. (30), has themean and the variance that are related31 as
〈n〉 = σ 2 = |z|2 = J. (32)
However, the distribution of generalized (k = 2) GK coherent states for power-law potentials iseither narrower (σ 2 < 〈n〉) or broader (σ 2 > 〈n〉) than the Poisson distribution depending on thevalue of power-law exponent k.
For generalized coherent states Mandel’s Q-parameter, in general, is a good measure to de-cide whether the weighting distribution is Poissonian, super-Poissonian, or sub-Poissonian. It isdefined34, 35 as
Q = σ 2
〈n〉 − 1. (33)
The weighting distribution of CSs is Poissonian if Q = 0, super-Poissonian if Q > 0 and sub-Poissonian if Q < 0. In Fig. 1, we plot the Mandel’s Q-parameter as a function of J for:(a) loosely binding potentials (k < 2) and (b) tightly binding potentials (k > 2). It is obviousfrom the plot that GK coherent states for power-law potentials always exhibit super-Poissoniandistribution for loosely binding potentials (k < 2), Poissonian for harmonic oscillator (k = 2), and
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082105-6 S. Iqbal and F. Saif J. Math. Phys. 52, 082105 (2011)
FIG. 1. (Color online) We plot the Mandel’s parameter for GK (GHA) coherent states of power-law potentials as a functionof J (or |z|2) for: (a) loosely binding potentials k < 2; (b) tightly binding potentials k > 2.
sub-Poissonian for tightly binding potentials (k > 2), for all values of J . In the following we showthat generalized Heisenberg algebra coherent states for power-law potentials also exhibit the samestatistical features as displayed in Fig. 1, for all values of |z|2.
V. GENERALIZED HEISENBERG ALGEBRA COHERENT STATES
Generalized coherent states for power-law potentials have been constructed based on GHA.27
In order to compare the statistical properties of GK coherent states,13 presented in the last section,with that of generalized Heisenberg algebra coherent states, we rewrite the generalized Heisenbergalgebra coherent states for power-law potentials, i.e.,
|z, k〉 = 1√N (|z|, k)
∞∑n=0
zn√ρ
(k)n
|n〉, (34)
where ρ(k)n is the same as given by Eq. (15) and the normalization function,
N (|z|, k) =∞∑
n=0
|z|2n
ρ(k)n
. (35)
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082105-7 Generalized coherent states and their statistical J. Math. Phys. 52, 082105 (2011)
FIG. 2. (Color online) We plot the mean (dashed line) and the variance (solid line) of GHA coherent states of power-lawpotentials as a function of |z|2 for: (a) loosely binding potentials; (b) tightly binding potentials.
In order to investigate the statistical features of GHA coherent states of power-law potentials,we calculate the weighting distribution |〈n| z, k〉|2, that is,
Pn(|z|, k) ≡ |〈n| z, k〉|2 = 1
N (|z|, k)
|z|2n
ρ(k)n
, (36)
which can be characterized through the expectation values,
〈z, k| N m |z, k〉 = 1
N (|z|, k)
∞∑n=0
nm |z|2n
ρ(k)n
≡ ⟨nm
⟩. (37)
It is important to note that the statistics defining factors, such as, the weighting distribution functionand the expectation values, of GHA coherent states, given by Eqs. (36) and (37), are completelyequivalent to those of GK coherent states, given by Eqs. (28) and (30), respectively, under theparameter transformation |z|2 → J . Hence, the GHA coherent states of power-law potentials exhibitthe same statistical properties as those of GK coherent states.
In order to get more insight on the weighting distributions of generalized coherent states ofpower-law potentials, we plot, in Fig. 2, the mean, 〈n〉, and the variance, σ 2, of distribution asa function of coherent state parameter |z|2. It is obvious from Fig. 2(a) that for loosely bindingpotentials the variance of the distribution is always greater than the mean of the distribution, i.e.,super-Poissonian distribution. On the other hand, the variance is always less than the mean of thedistribution for tightly binding potentials, as displayed by Fig. 2(b), i.e., sub-Poissonian distribution.It is worth mentioning that the mean and the variance are directly proportional to |z|2. However,their rate of variation as a function of |z|2 increases as the value of power-law exponent k decreases
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082105-8 S. Iqbal and F. Saif J. Math. Phys. 52, 082105 (2011)
FIG. 3. (Color online) The weighting distributions as a function of n for different values of |z|2 for: (a) Harmonic oscillator(k = 2), (b) infinite square well (k → ∞), and (c) linear potential (k = 1).
and vice versa. As a result, the distributions of the coherent states of power-law potentials for lowervalues of k (k < 2, i.e., loosely binding potentials) take several times higher values of mean, 〈n〉,and variance, σ 2, for a particular value of |z|2, than those of larger values of k (k > 2, i.e., tightlybinding potentials), as shown in Figs. 2 and 3.
As a matter of caution, we mention here a possible source of error while computing weightingdistributions of these coherent states numerically. As indicated by Fig. 3, the terms that effec-tively contribute to a distribution can be specified by its mean and variance. Therefore, in the
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082105-9 Generalized coherent states and their statistical J. Math. Phys. 52, 082105 (2011)
FIG. 4. (Color online) Mandel’s Q-parameter for k = 1.5 (loosely binding potential) versus |z|2 for different values of nmax .
computation of the weighting distribution for a particular value of |z|2, one can safely truncatethe infinite sums involved in the expressions of normalization function and expectation values(Eqs. (16) and (30), respectively) at value of n = nmax , such that, nmax ≥ 〈n〉 + σ 2/2 to include allthe effective terms of the summations. The situation where nmax is much smaller than 〈n〉 + σ 2/2causes truncation error on neglecting important terms of the summation, as in Fig. 4, that leads tomisguided numerics and wrong conclusions, as in Ref. 27.
VI. CONCLUSIONS
We have developed Gazeau-Klauder coherent states for general Hamiltonian systems defined bypower-law, which are further classified as loosely binding potentials and tightly binding potentials.We study the statistical characteristics of these states, together with generalized Heisenberg algebracoherent states. As an important result we show that the GK coherent states as well as GHA coherentstates for power-law potentials always exhibit super-Poissonian distribution for loosely bindingpotentials (k < 2), Poissonian for harmonic oscillator (k = 2), and sub-Poissonian for tightly bindingpotentials (k > 2), for all values of J or |z|2.
Glauber’s CSs have been used to describe the photon statistics of ideal lasers, which is a Poissondistribution.31 On the other hand, real lasers do not adhere this statistics,32, 33 and therefore couldnot be modelled by Glauber states. However, GK coherent states of power-law potentials are usefulto describe the states of ideal and real lasers by a proper choice of power-law exponent k. Thenonclassical nature, described by sub-Poissonian or super-Poissonian behavior, is useful in variousbranches of quantum physics, such as quantum information. In this context, it has been shown thatin order to have entangled states as output states of a beam splitter require nonclassical states at itsinput.25, 26 Indeed in such studies, we are interested in the nonclassical nature of power-law potentialstates. Hence, by a proper choice of parameters, one can find an adequate set of states in orderto generate the entanglement of bipartite composite systems using a beam splitter.38 This leads usto conclude that these states may be useful to extend the horizons of quantum informatics such asentanglement generation,36 and exploit the entangled states in the context of quantum teleportation,20
quantum cryptography,23 and dense coding.37
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