# An Elementary Construction on Nonlinear Coherent States Associated

Authors: Abdelkader Intissar
Journal: International Journal on Mathematics and Mathematical Sciences
 (2010)

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## Abstract

Consider the spaceL2(ℂ,dμ(z)), where dμ(z)=e−|z|2dz⋀dz¯ is the
Gaussian measure, and its generalized Bargmann subspaces  Em which are the null
kernels of the operator Δ=-∂2/∂z∂z¯+z¯(∂/∂z¯)-mI;
m=0,1,…. In this work, we present an other construction of  Em following the Hermite functions
which allows us to define a family of generalized Bargmann transform Bm which maps isometrically Em into L2(ℝ). The generalized coherent states ∣z〉m associated to Em are constructed and some properties of them are given.


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## Body

## 1. Introduction

The annihilation operatora and the creator operator a*  are well-known from the quantum theory of harmonic oscillator.They are defined by the commutation relation(1.1)[a,a*]=1.Of special interest is a representation of these operatorsa and a*  as linear operators in a separable Hilbert H spanned by the eigenvectors |n〉; n=0,1,… of the positive semidefinite number operator  N=a*a.One has the well-known relations(1.2)a|n〉=n|n-1〉,a*|n〉=n+1|n+1〉.The coherent states analysis is a very well-known tool in physics, in particular in quantum optics and in quantum mechanics. The name “coherent states” was first used by Glauber, Nobel prize in physics (2005), for his works in quantum optics and electrodynamics. In the book of Klauder and Skagerstam [1], the reader can get an idea of the coherent states application fields in physics and mathematical-physics.A general mathematical theory of coherent states is detailed in [2].Let us begin by reviewing the most important properties of the spaceI0 of coherent states spanned by the set {|z〉0}, where(1.3)|z〉0=e-1/2|z|2∑n=0∞znn!|n〉
for each complex z=x+iy.(i)
The vectors|z〉0 are the eigenvectors of the annihilation operator; one has a|z〉0=z|z〉0.(ii)
The spaceI0 of coherent states is a separable Hilbert space which is isomorphic to the Hilbert space H spanned by {|n〉}.(iii)
|n〉=a*n/n!|0〉.(iv)
|z〉0=e-|z|2/2eza*|0〉.(v)
0〈z|=〈0|ez̅ae-|z|2/2 is the adjoint vector of coherent state |z〉0, where z̅=x-iy.From the above properties, we obtain(vi)
a*|z〉0=(z¯/2+∂/∂z)|z〉0,(vii)
a|z〉0=(z/2-∂/∂z¯)|z〉0,where∂/∂z and ∂/∂z¯ are the linear partial differential operators on ℝ2 given by
(1.4)∂∂z=∂∂x-i∂∂y,z=x+iy,∂∂z¯=∂∂x+i∂∂y,z¯=x-iy.
Let Z=∂/∂z+z¯/2 and Z¯=∂/∂z¯-z/2Then we define the linear partial differential operatorL on ℝ2 by(1.5)L=-12(ZZ¯+Z¯Z).The vector fields Z andZ¯ and the identity operator form a basis for a Lie algebra in which the Lie bracket of two elements is their commutator. In fact, -Z¯ is the formal adjoint of Z and L is an elliptic partial differential operator on ℝ2 given by(1.6)L=-(∂2∂x2+∂2∂y2)+14(x2+y2)-i(x∂∂y-y∂∂x).Thus,L is the usual Hermite operator(1.7)-(∂2∂x2+∂2∂y2)+14(x2+y2)
perturbed by the rotation operator(1.8)-i(x∂∂y-y∂∂x).
The strongly continuous one parameter semigroup e-tL,t>0 is ultracontractive in the sense that for any t>0, e-tL is a bounded linear operator from L2(ℝ2) to L∞(ℝ2), and it is hypercontractive in the sense that for all t>0, e-tL is a bounded linear operator from L2(ℝ2) to L4(ℝ2).The theory of ultracontractive semigroups can be found in Davie's book [3] and the connections of hypercontractivity with constructive quantum field theory are attributed to Nelson [4] and are explained in the Simon's book [5].Coming back to the spaceI0 of coherent states, it is closely related to Bargmann's space which was used in [6] for the canonical commutation rules as representation space of quantum mechanics. Since then, it had occurred in many different contents, that is, in representation theory of nilpotent Lie groups and in Reggeon field theory. This last theory is governed by a nonselfadjoint Gribov operator [7, 8] constructed as a polynomial in a and a*.For any|ϕ0〉∈I0, we can define an entire analytic function by(1.9)ϕ0(z)=e1/2|z|2〈ϕ0∣z〉0=∑n=0∞znn!〈ϕ0∣n〉.
As ∫ℂ|〈ϕ0∣z〉0|2dxdy〈∞· then(1.10)∫ℂ|ϕ0(z)|2e-|z|2dxdy<∞.
We denote the Bargmann space by:(1.11)E0={ϕ0:ℂ→ℂentire;∫ℂ|ϕ0(z)|2e-|z|2dxdy<∞}.E0 is closed in L2(ℂ,dμ(z)), where the measure dμ(z)=e-|z|2dxdy and is closed related to L2(ℝ) by an unitary transform of L2(ℝ) onto E0 given in [6] by the following integral transform (for some appropriate constant c):(1.12)ϕ0(z)=[B0f](z)=c∫ℝe-(1/2)z2+2uz-(1/2)u2f(u)du.Iff∈L2(ℝ), the integral converges absolutely.InL2(ℂ,dμ(z)), Askour et al. [9, 10] have introduced the so-called“generalized” Bargmann spaces Em, m=0,1,…, as null kernels of the operator(1.13)Δ=-∂2∂z∂z¯+z¯∂∂z¯-mI,m=0,1,…
and have proved that L2(ℂ,dμ(z)) is the direct sum of the Bargmann space E0 and all “generalized” Bargmann spaces Em.Otherwise, Vasilevski in [11] concerning the structure of polyanalytic Fock spaces has introduced the so-called “true-poly-Fock” spaces and proved that L2(ℂ,dμ(z)) is the direct sum of the Bargmann space (Fock) and all “true-poly-Fock” spaces.According to chapter 2 of Daubechies book [12], we give in this work a family of embedding operators Bm from L2(ℝ) into L2(ℂ,dμ(z)) such that the range B0(L2(ℝ)) is a Bargmann space and Bm(L2(ℝ)) are reproducing subspaces.The construction ofBm is based on the Fourier-Bargmann transform which is an isometry from L2(ℝ) into L2(ℝ2).To do this, we considerh∈L2(ℝ) as a windowed function and denote  hu,v(x)=eiuxh(x-v). Then we can consider the windowed transform (the Fourier-Bargmann transform or the Gabor Fourier transform):(1.14)Bh:L2(ℝ,dx)→L2(ℝ2),f→Bhf(u,v)=〈f,hu,v〉L2(ℝ).In the nice Folland's book [13], the Fourier-Bargmann transform is called wave packet transform and is very close to the Bargmann transform and FBI transform; see also [14].By takingh(x)=(1/π4)e-x2/2, Daubechies has pointed out that Bh(L2(ℝ)) is isomorphic to the Bargmann space.In this work, takingh0(x)=(1/π4)e-x2/2 and the Hermite functions hm(x)=(1/2mm!)(x-d/dx)mh0(x) as windowed functions, we point out that Bhm(L2(ℝ)) (Gabor space with the mth Hermite window hm) is isomorphic to a “generalized” Bargmann space Em.We adopt this last construction to give a “generalized” Bargmann transformBm associated to Em and we construct the phase spaces Im of “generalized coherent states” |z〉m associated to Em.To do this, letz=u+iv and m=0,1,…, we use the following integral transform:(1.15)Bhm:L2(ℝ,dx)→L2(ℝ2,dudv),f→Bhmf
such that(I)Bhmf:ℝ2→ℂ,(u,v)→Bhmf(u,v)=〈f,hmz〉L2(ℝ)=∫ℝe-iuxhm(x-v)f(x)dx,
where(1.16)hmz(x)=eiuxhm(x-v).
We try to be self-contained and elementary as far as possible in this paper which is organized as follows: in Section 2, we present a construction of “generalized” Bargmann spaces following the Hermite functions by using the integral transform (I) for all m, and we define the “generalized” Bargmann transform associated to Em. In Section 3, we construct the “generalized coherent states” |z〉m, m=0,1,…, associated to “generalized” Bargmann spaces Em and give some properties of them.


## 2. Construction of Generalized Bargmann Spaces Following the Hermite Functions

Let us consider the Hamiltonian oscillatorHosc=1/2(P2+Q2) on L2(ℝ), where Q, P are the usual position and momentum operators ([Q,P]=iI). It is well-known that the eigenvalues and the eigenvectors of Hosc are given by(2.1)Hoschm=(m+12)hm,
where hm are the following Hermite functions:(2.2)hm(x)=12mm!(x-ddx)m1π412mm!e-x2/2=1π412mm!e-x2/2Hm(x)
and Hm(x)=∑j=0[m/2](-1)j(m!/j!(m-2j)!)(2x)m-2j are the Hermite polynomials satisfying the recursion relations xHm(x)=mHm-1(x)+(1/2)Hm+1(x).Remark 2.1.
(i) It is well-known thatλm=m+1/2 may also be seen as the eigenvectors of the usual Schrödinger operator HS on ℝ2 in presence of a constant magnetic field. Namely HS acting in L2(ℝ2) is given by
(2.3)HS=-14[(∂∂x+iy)2+(∂∂y-ix)2].
(ii)HS has an infinite degeneracy.
(iii) It is well-known that the complex Hermite polynomials
(2.4)Hn,k(z,z¯)=(-1)m+ke|z|2∂m+k∂zm∂z¯ke-|z|2
are an orthogonal basis of the Hilbert space L2(ℂ,dμ(z)) and we can see in [15] how it was used to give some spectral properties of Cauchy transform on L2(ℂ,dμ(z)).If we formally takez to be real in (2.4), the complex Hermite polynomials Hn,k(z,z¯) become the well-known real Hermite polynomials Hn+k(x).Now, letz=u+iv and m=0,1,…, we define the following integral transforms:(2.5)Bhm:L2(ℝ,dx)→L2(ℝ2,dudv),f→Bhmf,
by(2.6)Bhmf(u,v)=〈f,hmz〉L2(ℝ)=∫ℝe-iuxhm(x-v)f(x)dx,
where(2.7)hmz(x)=eiuxhm(x-v).(1) If we take in (2.6) f(x)=hm+k(x) for k=0, 1,…, then we can use the formulas on the Hermite functions given in [16, 17], in particular the famous formula:(2.8)∫ℝhm+k(x-v2)e-iuxhm(x+v2)dx=(-i2)km!(m+k)!e-1/4[u2+v2](u-iv)kLmk(|z|22),
where Lmk(t)  is the Laguerre polynomial defined by the Rodriguez formula as(2.9)Lmk(t)=1m!t-ket(ddt)m(tk+me-t),t∈ℝ.The same formula appeared in [18, Chapter 23].After a simple variable change, the formula (2.8) can be also expressed as(2.10)∫ℝhm+k(x)e-iuxhm(x+v)dx=(-i2)km!(m+k)!e-1/4[u2+v2-2iuv](u-iv)kLmk(|z|22).From (2.10), we deduce for k=0,1,2,… that(2.11)Bhmhm+k(u,v)=(-i2)km!(m+k)!e-1/4[u2+v2+2iuv](u+iv)kLmk(|z|22)
or(2.12)Bhmhm+k(u,v)=ω(u,v)(-i2)km!(m+k)!(u+iv)kLmk(|z|22)
where(2.13)ω(u,v)=e-1/4[u2+v2+2iuv]=ω(z).(2) If 0≤k≤m, we have the following formula:(2.14)∫ℝhm-k(x)e-iuxhm(x-v)dx=(-i2)k(m-k)!m!e-1/4[u2+v2+2iuv](u-iv)kLm-kk(|z|22).That permits us to write for0≤k≤m(2.15)Bhmhm-k(u,v)=ω(u,v)(-i2)k(m-k)!m!e-1/4[u2+v2+2iuv](u-iv)kLm-kk(|z|22).Denotingz′=u′+iv′, then the reproducing kernel Nhm of Bhm(L2(ℝ)) can be computed as(2.16)Nhm(u,v,u′,v′)=〈hmz,hmz′〉L2(ℝ2).Hence we have(2.17)Nhm(u,v,u′,v′)=∫ℝe-iuxhm(x-v)eiu′xhm(x-v′)dx.
After a simple variable change, for example x=y+v′, then Nhm(u,v,u′,v′) can be alsoexpressed as
(2.18)Nhm(u,v,u′,v′)=∫ℝhm(y)hm(y+v′-v)e-i(u-u′)(y+v′)dy.Hence(2.19)Nhm(u,v,u′,v′)=e-1/4[u2+v2+2iuv]e-1/4[u′2+v′2-2iu′v′]e(u+iv)(u′-iv′)/2Lm((u-u′)2+(v-v′)22)
or(2.20)Nhm(u,v,u′,v′)=ω(u,v)ω(u,v)¯e(u+iv)(u′-iv′)/2Lm((u-u′)2+(v-v′)22).The reproducing relation inBhm(L2(ℝ)) can be written as(2.21)Bhmf(u,v)=∫ℝ2Nhm(u,v,u′,v′)Bhmf(u′,v′)du′dv′,
then Bhm(L2(ℝ)) is a reproducing space with the reproducing kernel Nhm.Introducing the complex structuresz and z′, theabove formulas can be rewriten as(2.22)Bhmhm+k(z)=ω(z)(-iz2)km!(m+k)!Lmk(|z|22),Bhmhm-k(z)=ω(z)(-iz¯2)k(m-k)!m!Lm-kk(|z|22),Nhm(z,z′)=ω(z)ω(z)¯Nm(z,z′),with(2.23)Nm(z,z′)=ezz′¯/2Lm(|z-z′|22).We also can see that by changing the variablez into 2ξ and z′ into 2ξ′ that Nm(z,z′) can be transformed to kernels given in [9, Theorem 2](2.24)Km(ξ,ξ′)=eξξ′¯Lm(|ξ-ξ′|2).
Now, we can define the “generalized” Bargmann transforms by(2.25)Bmf(z)=1ω(z)Bhmf(z)
and the “generalized” Bargmann spaces by(2.26)Em=Bm[L2(ℝ)].
Now, the consequences of all the above computations are summarized in the following form.Theorem 2.2.
(i)Bm[L2(ℝ)] is a subspace of L2(ℂ,e-|z|2dudv).
(ii)(2.27)Bmhm+k(z)=(-iz2)km!(m+k)!Lmk(|z|22),k=0,1,….
(iii) Form=0,  1/k!(-iz/2)k is a basis of classical Bargmann space E0.
(iv)(2.28)Bmhm-k(z)=(-iz2)k(m-k)!m!Lm-kk(|z|22),k=0,…,m.
(v)(2.29)Bmf(z)=1ω(z)∫ℝeiuxf(x)(x-v-ddx)me-(x-v)2dx=2mm!ω(z)∫ℝe-1/2(x2+v2)+iz¯xHm(x-v)f(x)dx.
(vi)(2.30)Bmf(iz)=2mm!∫ℝe-(z/2-x)2Hm(x-z+z¯2)f(x)dx.Remark 2.3.
(i) The operatorA*=-∂/∂z+z¯ is the adjoint of the operator A=∂/∂z¯ in L2(ℂ,e-|z|2dudv) and Δ=A*A; see for example [19] or [20].
(ii) In [9, 10] the “generalized” Bargmann spaces Em are constructed as null spaces of the second-order differential operator Δ=-∂2/∂z∂z¯+z¯(∂/∂z¯)-mI in L2(ℂ,e-|z|2dudv) but in this section we have given another construction of these spaces which allows us to construct the “generalized” Bargmann transforms (2.25).


## 3. Construction of Generalized Coherent States|z〉m Associated to Generalized Bargmann Spaces Em

In the separable Hilbert spaceH spanned by the eigenvectors |n〉n=0,1,2,… of the positive semidefinite number operator N=a*a, one can define for each complex number z=x+iy(3.1)|z〉m=e-1/2|z|2∑n=0∞A*mznn!|n〉,
where(3.2)A*=-∂∂z+z¯.The spaceIm spanned by the set {|z〉m} is a separable Hilbert space which is in fact isomorphic to H. This space is called the “generalized” space of coherent states. It is closely related to the “generalized” Bargmann space Em.For any|ϕm〉∈Im, we can define a function by(3.3)ϕm(z)=e1/2|z|2〈ϕm∣z〉m=∑n=0∞A*mznn!〈ϕm∣n〉.As∫ℂ|〈ϕm∣z〉m|2dxdy<∞,  then(3.4)∫ℂ|ϕm(z)|2e-|z|2dxdy<∞.Theorem 3.1.
LetA=∂/∂z¯ and A*=-∂/∂z+z¯, then the following statements hold.(i)
AA*mf(z)=mA*m-1f(z); m=1,2,… for any holomorphic function f(z).(ii)
Letϕm(z)=e1/2|z|2〈ϕm∣z〉m, then ϕm(z) is in the “generalized” Bargmann space Em; m=0,1,…(iii)
(3.5)|z〉m=∑j=0m(-1)jCmjz¯m-ja*j|z〉0,
where a*|n〉=n+1|n+1〉 and Cmj=m!/j!(m-j)!Proof.
(i) Letf(z) be holomorphic function; then by applying the commutation relation [A,A*]=I and the Cauchy-Riemann equation, we deduce that (i) is satisfied for m=1 and by induction that
(3.6)AA*m+1f(z)=A*AA*mf(z)+A*mf(z)=(m+1)A*mf(z).
(ii) Form=0, we have |z〉0=e-1/2|z|2∑n=0∞(zn/n!)|n〉, and ϕ0(z) is in the classical Bargmann space E0.
Form≠0 and ϕm(z)=∑n=0∞A*m(zn/n!)〈ϕm∣n〉 we have
(3.7)A*Aϕm(z)=∑n=0∞A*AA*mznn!〈ϕm∣n〉.
As the function f(z)=zn is holomorphic, we can apply property (i). Premultiplying by A* the equation AA*mf(z)=mA*m-1f(z), we obtain
(3.8)A*Aϕm(z)=∑n=0∞A*AA*mznn!〈ϕm∣n〉=∑n=0∞mA*mznn!〈ϕm∣n〉=mϕm(z).
(iii) Let(3.9)a*j|z〉0=e-1/2|z|2∑n=0∞znn!a*j|n〉=e-1/2|z|2∑n=0∞(∂∂z)jznn!|n〉,|z〉m=e-1/2|z|2∑n=0∞A*mznn!|n〉.
As∂/∂z(z¯f(z))=z¯(∂/∂z)f(z), we have
(3.10)A*m=(z¯-∂∂z)m=∑j=0m(-1)jCmjz¯m-j(∂∂z)j,
where Cmj=m!/j!(m-j)!Then
(3.11)|z〉m=∑j=0m(-1)jCmjz¯m-ja*j|z〉0.
The action of the annihilation operator a on |z〉m for m≠0 is presented below, in the form of a theorem.Theorem 3.2.
(i)a|z〉m=z|z〉m-m|z〉m-1.
(ii) There exists an operatorTm such that Tm|z〉m=z|z〉m.Proof.
(i) Let|z〉m=e-1/2|z|2∑n=0∞A*mzn/n!|n〉, where  A*=-∂/∂z+z¯.
By applying the operatora=z/2-∂/∂z¯ on |z〉m, we get
(3.12)a|z〉m=z2|z〉m-∂∂z¯(e-1/2|z|2∑n=0∞A*mznn!|n〉)=z2|z〉m+z2|z〉m-e-1/2|z|2∂∂z¯(∑n=0∞A*mznn!|n〉)=z|z〉m-e-1/2|z|2∑n=0∞∂∂z¯A*mznn!|n〉
Now asf(z)=zn is a holomorphic function and A=∂/∂z̅, we have ∂/∂z¯A*mf(z)=mA*m-1f(z) and we get
(3.13)a|z〉m=z|z〉m-me-1/2|z|2∑n=0∞A*m-1znn!|n〉;
that is,
(3.14)a|z〉m=z|z〉m-m|z〉m-1.
(ii)In the above property (i) of the present theorem, it is shown that the “generalized” coherent states are not eigenstates of the annihilation operator. Nevertheless, we will show that these “generalized” coherent states can be interpreted as nonlinear coherent states, see for example the theory of Photon-added coherent states used in [21] or the approach of nonlinear coherent states given in [22].
To do this, we have to show that|z〉m obeys to the equation
(3.15)Tm|z〉m=z|z〉m
with a suitable choice for the operator Tm. Let us now construct the explicit form of Tm.
From (3.11), we have z〉m=∑j=0m(-1)jCmjz¯m-ja*j|z〉0, then z〉m is in the space Fm spanned by {z¯m-ja*j|z〉0}j=0,1,…,m.
By definition, the coherent states|z〉0 satisfy
(3.16)a|z〉0=z|z〉0.
Premultiplying the both sides of this equation bya*j leads
(3.17)a*ja|z〉0=za*j|z〉0.
Using the commutation relation[a,a*]=I, the above equation is written as
(3.18)aa*j-ja*j-1|z〉0=za*j|z〉0.
which, making use of the identity (a*a+I)-1aa*=I, leads to
(3.19)[a-j(a*a+I)-1a]a*j|z〉0=za*j|z〉0
In the following, we introduce
(3.20)Rj=(I-j(a*a+I)-1)a
and Pj the projection operator on the space spanned by the system {z¯m-ja*j|z〉0}, j=0,1,…,m(3.21)Pkz¯m-ja*j|z〉0=δjkz¯m-ja*j|z〉0,
where δjk=1ifk=j and δjk=0ifk≠j.
Let
(3.22)Tm=∑j=0mRjPj.
Acting Tm on |z〉m leads to
(3.23)Tm|z〉m=∑j=0mRjPj∑j=0m(-1)jCmjz¯m-ja*j|z〉0=∑j=0m(-1)jCmjRjz¯m-ja*j|z〉0=∑j=0m(-1)jCmjz¯m-jRja*j|z〉0=∑j=0m(-1)jCmjz¯m-jza*j|z〉0=z∑j=0m(-1)jCmjz¯m-ja*j|z〉0,
that is,
(3.24)Tm|z〉m=z|z〉m.
To sum up, we have used the Gabor spaces Bhm(L2(ℝ)) with the mth Hermite window hm to give another construction of generalized Bargmann spaces Em which is different from that given in [9, 10] or [11] and have constructed a family of “generalized” Bargmann transforms attached to these spaces. The states |z〉m constructed in this work with m≠0 are nonclassical and can be interpreted as nonlinear coherent states in nonanalytic representation.        The spaces ⋀m:=Em-1,m=1,2,…, are constituted by the polyanalytic functions ϕ(z) of order m, that is,
(3.25)(∂∂z̅)mϕ(z)=0,m=1,2,…
which are not polyanalytic of any lower order and satisfy
(3.26)∫ℂ|ϕ(z)|2e-|z|2dxdy〈∞.
The functions  ϕ(z) in ⋀m,m=1,2,…, can be uniquely expressed in the form
(3.27)ϕ(z)=∑0≤k≤m-1z̅k∑j=0∞aj,kzj.
Finally, the spaces Em defined in this work are the “true polyanalytic Fock spaces” of [11] or the “generalized Bargmann spaces” of [9, 10].
Whenm is not zero, we give (in a further work) a complete spectral analysis of the nonselfadjoint operator HI=B*(B+B*)B on ⋀m or on ⊕j=1m⋀m (the natural generalization of the Bargmann space, since (3.25) generalizes the Cauchy-Riemann equation (∂/∂z̅)ϕ(z)=0), where B=∂/∂z and B*=-∂/∂z¯+z.
The restriction ofHI on the Bargmann space E0 is the Heun's operator and it has been studied in detail in [23].



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