NICOLE GARBERS AND ANDREAS RUFFING
Received 27 January 2006; Revised 11 April 2006; Accepted 12 April 2006
Starting from supersymmetric quantum mechanics and related supermodels within Schr¨odinger theory, we review the meaning of self-similar superpotentials which exhibit the spectrum of a geometric series. We construct special types of discretizations of the Schr¨odinger equation on time scales with particular symmetries. This discretization leads to the same type of point spectrum for the referred Schr¨odinger difference operator than in the self-similar superpotential case, hence exploiting an isospectrality situation. A dis-cussion is opened on the question of how the considered energy sequence and its gener-alizations serve the understanding of coherent states in quantum optics.
Copyright © 2006 N. Garbers and A. Ruffing. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis-tribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Items like “coherent states” or “squeezed states” can nowadays be found in many recent articles onquantum optics. The fact that the Nobel Prize in Physics 2005 has been awarded to pioneers on this area, like R. Glauber, gives insight how active this area is.
The kind of physical states behind coherent states or squeezed states are the so-called nonclassical states. They are minimal uncertainty states. These properties are essential for an efficient signal transmission in the quantum world. The theory of coherent states in physics has been developed all over the last decades, among others by Glauber, Klauder, and Sudarshan.
Coherent states play a major role inlaser physics. The mathematical modeling in laser physics allows three different approaches to coherent states: first by the method of trans-lation operators, second by the method of ladder operators, and third by the method of minimal uncertainty. Nonclassical states likesqueezed laser fieldsare very important for applications: the experimental methods when dealing with squeezed laser fields include for instance the so-calledhomodyne tomography. The mathematical modeling in self-homodyne tomography allows a tomographical reconstruction of the Wigner function, belonging to a set of probability densities of fluctuations in different field amplitudes.
Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 72768, Pages1–14
Squeezed laser states are—like coherent states—states of minimal uncertainty. From the viewpoint of statistics, semiconductor lasers have asuper-Poisson distributionup to thepumping level, that is, a distribution which goes beyond the non-normalized Pois-son distribution. But also the so-calledsub-Poisson distributionhas a particular mean-ing inmultiboson systems: the definition of coherent states is directly related to solu-tions of Stieltjes moment problems. In [4], Penson and Solomon could show that q-discretizations of orthogonality measures, solving the moment problems, allow to in-vestigate multiboson coherent states which could not so far be understood in the con-ventional quantum setting. The nature of these states is sub-Poissonian. Regarding the statistical properties of different physical quantum states in quantum optics, one also refers tosuper-Poisson statesandsub-Poisson states.
Apart from the development in quantum optics, there are also great achievements visi-ble in mathematical physics which support the theoretical understanding of the described phenomena. On the one hand, the mathematical frame for coherent states is steadily de-veloped throughout analysis, on the other hand one can see the development that super-symmetric quantum mechanicsanddiscrete Schr¨odinger theorybecome related to essential problems in quantum optics.
In this article, we are going to exploit the connections betweenself-similar supermodels within supersymmetric quantum mechanics, so-calledbasic versionsof coherent states, and their relations todiscretized Schr¨odinger equations on time scales. In Sections2and3, we review some fundamental facts on supersymmetric Schr¨odinger operators where we are going to exploit a generalizedsupermodel definitionat the end of Section 4. We are going to represent generalized supermodels through their creation and annihilation op-erators and characterize them as solutions to a new type of discretization for Schr¨odinger operators inSection 4. Finally, inSection 5, we are going to establish the connection of the obtained results and representations to basic items of coherent state theory in quan-tum optics.
2. Schr¨odinger equations and superpotentials
Supersymmetry is one of the most powerful tools being applied to problems of theoretical physics. In the last years, there were great achievements especially on the area of super-symmetry in quantum mechanics. For an excellent contribution to the topic see for in-stance the articles by Robnik [7] and Robnik and Liu [8]. The stationary one-dimensional version of Schr¨odinger’s equation (λbeing a fixed value inR) is
−ψ(x) +V(x)ψ(x)=λψ(x), x∈R. (2.1)
It can with some general success be factorized by using the concept of so-called superpo-tentials.
In Schr¨odinger theory, the following scenario is of particular interest. Given two Schr¨odinger equations with different potentialsV1andV2. Under some circumstances, it is possible to write them in the form
whereBandB+are formally adjoint to each other, being defined on some common do-main inᏸ2(R). Let us shortly review the method of how to address the stated factoriza-tion problem.
The first step is the construction of a so-calledsuperpotentialW such that one can express thepartner potentialsV1,V2as follows:
V1=1
The superpotentialWis fixed by assuming that the potentialV1allows 0 as an eigenvalue, the corresponding eigenfunctionϕbeing positive. This leads to the condition
−ϕ(x) +1 2
W2(x)−√2W(x)ϕ(x)=0, x∈R. (2.4)
A solution to this equation is given by
W(x)= −√2(lnϕ)(x), x∈R. (2.5)
The aimed factorization is now achieved by the equalities
H1=B+B, H2=BB+, (2.6)
where the differential operatorsH1,H2are specified by thesupersymmetric ladder opera-tors
To illustrate this formalism, let us consider the two potentials
V1(x)=x2
As for the superpotentialW, we obtain justW(x)=√2x, leading to the well-understood conventional ladder operator formalism
The key message is now that one can determine the point spectrum ofH1,H2completely by using the operatorsB,B+. Further, more illustrative example is the so-called Rosen-Morse potential, in which a real parameteryoccurs:
V1(x,y)=y2− y(y+ 1)
cosh2(x), x∈R. (2.10)
Assuming that the equation
has a solutionϕ∈ᏸ2(R), we are first led to the superpotential
W(x)=√2ytanh(x), x∈R, (2.12)
as well as to the potential
V2(x,y)=y2− y(y−1)
cosh2(x), x∈R. (2.13)
The difference of the two partner potentials is given by
V2(x,y)−V1(x,y−1)=2y−1
2 , x∈R. (2.14)
Generalizing the observations made so far, the two potentials V1,V2 are called form-invariant if the following identity holds for different values ofx, y1,y2, the expression R(y1) being a continuous function ofy1:
V2x,y1=V1x,y2+Ry1. (2.15)
According to the above construction, the pairB,B+specifies two different Schr¨odinger equations, which together are referred to as asupersymmetric modelor just as a super-model.
3. Self-similar supermodels
In many important applications, it follows from the defining equation for form invari-ance,
V2x,y1=V1x,y2+Ry1, (3.1)
that y2=y1+h, wherehis a real constant. A completely new class of form-invariant potentials has been proposed in [3], where potentials were constructed whose parameters are related to each other by
y2=qy1, 0< q <1. (3.2)
In order to make apparent what kind of possible point spectrum is generated by the prop-erty (3.2), we follow the basic outline in [2], where the superpotential is expanded as follows:
W(x,y)= ∞
j=0
gj(x)yj (3.3)
for some suitable parameter,y∈R. The functionRfrom (3.3) is assumed to be given by an analytic ansatz
R(y)= ∞
j=0
Plugging this ansatz into formula (3.1) yields
R(y)=V2(x,y)−V1(x,qy)
=W2(x,y)−W2(x,qy) +√2∂
xW(x,y) +∂xW(x,qy).
(3.5)
Inserting now the expansion (3.4) for the functionR, one obtains, by comparing the coefficients,
Rn=1 2
∞
i=1
1 +qn−i1−qig
ign−i+√21 +qngn, n∈N, (3.6)
and the valueR0being given byR0=g0. With the abbreviations
rn:= Rn
1−qn, dn:= 1−qn
1 +qn, n∈N, (3.7)
one is led to nonlinear integral equations, given by
gn(x)=√dn 2
x
a
2rn− n−1
i=1
gi(t)gn−i(t) dt, x∈R,n∈N, (3.8)
where restrictions of the solutions of these equations are put by the conditions
R0=0, g0(x)=0, rn=zδn1, n∈N, (3.9)
δn1denoting the Kronecker symbol andzbeing a positive parameter. This nonlinear in-tegral equation allows now the solutions
R(y)=R1y=R, gn(x)=√1 2βnx
2n−1, x∈R,n∈N, (3.10)
where the coefficientsβnare fixed by the recurrence formula
β1= 2R1
1 +q, β0=0, βn= − dn 2n−1
n−1
i=1
βiβn−i, n∈N\ {1}. (3.11)
The superpotential now reads
W(x,y)= ∞
j=1 βjyj
x
√
2
2j−1
, x∈R. (3.12)
The formal ground-state, belonging toV1, is given by the formula
ψ0(x,y)=Ce−∞j=1(βj/2j)yj(x/√2)2j, x∈R. (3.13)
Direct calculation leads to the formula
showing some self-similar property. This type of superpotential is therefore also referred to by the nameself-similar superpotential. Applying a generalized version of the ladder operator formalism, one is led to the energies of the operatorH1, being given by
λn=R n−1
j=0
qj=R1−q n
1−q , n∈N0. (3.15)
Let us now arrive at an interesting isospectrality scenario.
4. Isospectral supermodels and strip discretizations
We now address the general question of how to construct discrete Schr¨odinger oper-ators, that is, Schr¨odinger difference operators whose wave functions are defined on a nonempty closed setΩ⊂Rwith Lebesgue measureμ(Ω)>0, leading to the same point spectrum (3.15).
We address this question in context of Schr¨odingerq-difference equations, where we study piecewise continuous solutions to these equations, having support on some kind of strip structures which are generated by the symmetries of the lattice{+qn,−qn|n∈Z}. This approach seems to fit naturally to the given point spectrum (3.15) and might be of importance for applications and numerical investigations of the underlying eigenvalue and spectral problems.
Let us now elucidate in some detail the philosophy of using the framework of q-difference operators for discretizing the Schr¨odinger equation. As indicated, we restrict our investigations to subsets of the real axis which we will call homogeneousq-strip dis-cretizations or just strip disdis-cretizations. To do so, we have to provide the tools that help us in formulating the special boundary conditions.
Let us refer throughout the sequel to a parameter 0< q <1, as it was motivated by the investigation of self-similar superpotentials in the previous section.
Definition 4.1 (strip discretization). Let Ω⊆R\ {0} be a nonempty closed set with Lebesgue measureμ(Ω)>0 as well as
∀x∈Ω, qx∈Ω, q−1x∈Ω, −x∈Ω. (4.1)
We call the time scaleΩa homogeneous strip discretization or juststrip discretizationof the configuration space. The Hilbert space of the strip discretization is introduced by the requirement
ᏸ2(Ω) :=f ∈ᏸ2(R)|f = f◦χ
Ω, (4.2)
and the scalar product of two functions f,g∈ᏸ2(Ω) is introduced by
(f,g)Ω:=
∞
−∞f(x)g(x)χΩ(x)dx=
∞
−∞f(x)g(x)dx, (4.3)
the Schr¨odinger equation with quadratic potential, given by
−ψ(x) +x2ψ(x)=λψ(x), x∈R. (4.4)
The following structure is one the most familiar facts within mathematical physics: let the sequence of functions (ψn)n∈N0be recursively given by the requirement
ψn+1(x) := −ψn(x) +xψn(x), x∈R,n∈N0, (4.5)
whereψ0:R→R,x→ψ0(x) :=e−(1/2)x2
. We then haveψn∈ᏸ2(R)∩C2(R) forn∈N0 and moreover
−ψn(x) +x2ψ
n(x)=(2n+ 1)ψn(x), x∈R,n∈N0. (4.6)
This result reflects the conventionalladder operator formalism. We now develop a result in discrete Schr¨odinger theory on strip structures which turns out to be aq-analog of the just described continuous situation. Let us therefore state in a next step some more useful tools for the strip discretization approach.
Definition 4.2. LetΩ⊆R\ {0}be a nonempty closed set with the propertyμ(Ω)>0 as well as
∀x∈Ω, qx∈Ω, q−1x∈Ω, −x∈Ω. (4.7)
Let for any f :Ω→Rthe right-shift, respectively, left-shift operations be defined by
(R f)(x) :=f(qx), (L f)(x) :=fq−1x, x∈Ω, (4.8)
respectively. The right-hand, respectively, left-handq-difference operations will for any f :Ω→Rbe given by
Dqf
(x) := f(qx)−f(x)
qx−x ,
Dq−1f(x) := f
q−1x−f(x)
q−1x−x , x∈Ω. (4.9)
Let moreoverα >0 and let
g:Ω−→R+, x−→g(x) :=
ϕ(qx)−ϕ(x)
ϕ(x)(q−1)x =
1 +α(1−q)x2−1
qx−x , (4.10)
where the positive even continuous functionϕ:Ω→R+is chosen as a solution to the q-difference equation
ϕ(qx)=1 +α(1−q)x2ϕ(x), x∈Ω. (4.11)
Thecreation operationA†q and, respectively,annihilation operationAqare introduced
We refer to thediscrete Schr¨odinger equationwith an oscillator potential onΩby
q−1−Dq+g(X)R
LDq+Lg(X)
ψ=λψ. (4.13)
The following result reveals that the discrete Schr¨odinger equation with an oscillator potential onΩshows similar properties than its classical analog does.
Theorem4.3. Let the time scaleΩbe a strip discretization in the sense ofDefinition 4.1
and let the functionϕbe specified like inDefinition 4.2, satisfying theq-difference equation (4.11) onΩ,
ϕ(qx)=1 +α(1−q)x2ϕ(x), ϕ(x)=ϕ(−x)>0, x∈Ω. (4.14)
Forn∈N0, the functionsψn:Ω→R, given byψn(x) :=((A†q)n√ϕ)(x),x∈Ω, are well defined inᏸ2(Ω)and solve theq-Schr¨odinger equation (4.13) in the following sense:
q−1−D There exists the general observation
and the functions(ψn)n∈N0constitute an orthonormal system inᏸ
2(Ω).
Proof. Let us forϕ∈C(R) first consider the equation
ϕ(qx)xn=1 +α(1−q)x2ϕ(x)xn, x∈Ω,n∈N0, (4.19)
functionϕfulfiling (4.11) is inᏸ1(R).This implies
Using the substitution rule to the left-hand side, this directly implies
∞
We have shown earlier [1] that any probability densityψ which generates moments of type (4.23) yields an orthogonality measure to the polynomials (Hnq)n∈N0which are for
k∈Nfixed through the recurrence relation
Hkq+1(x)−αqkxH
the variablex being chosen in a suitable integration support. As a consequence of this general result, we obtain the following orthogonality relation:
∞
−∞H q
m(x)Hnq(x)ϕ(x)χΩ(x)dx=vn(Ω)δmn, m,n∈N0, (4.25)
with a sequence of positive numbers (vn(Ω))n∈N0. Direct calculations and induction show
ψn(x) :=
A†qnϕ◦χΩ(x)=Hnq(X)ϕ◦χΩ(x), x∈R,n∈N0. (4.26)
Let us from now on—without any restriction—refer to the special parameter choiceα=
1. The functions (ψn)n∈N0constitute an orthonormal system inᏸ
2(Ω). Let us show next that the ladder property (4.16) is fulfiled. The first equation in (4.16) is trivial due to the definition of the functions (ψn)n∈N0. We remember that the functiongis specified like in
Definition 4.2. We obtain in the sense of the multiplication operator notation
LDq+Lg(X)
Xnψ0=LD
which yields
This may be rewritten as
Using now however the formulas in (4.10) for the functiong, we obtain (Dqψ0+gψ0)=0 and therefore
generated by linear combinations of the firstm+ 1 monomials of the sequence (Xn) n∈N0.
with uniquely defined real numberscn
j, wherej=0,. . .,n−1 withn∈N. Applying again standard substitution techniques to the scalar product integral (4.3), we can derive for any functions f,g∈ᏸ2(Ω) which are both in the algebraic span of the functions (ψ
n)n∈N0the
following relation:
A†qf,gΩ=f,AqgΩ. (4.32)
In particular, this result implies
Using the first equation in (4.16) and because of the fact that the functions (ψn)n∈N0
constitute an orthogonal system in ᏸ2(Ω), the second relation in (4.16) follows from standard methods of calculating the norms of the functions (ψn)n∈N0.
Equation (4.15) now follows immediately from the first two relations in (4.16). Taking all the steps of the proof together, this finally confirms the statements ofTheorem 4.3.
Let us interpret the obtained results in context of quantum mechanical supermodels. First, we have obtained the desired isospectrality result, that is, we have found a rich class of discrete Schr¨odinger operators showing in (4.15) the same point spectrum that we obtain from the self-similar superpotentials in (3.15). This gives us an important tool at hand to extend the definition of asupermodel for purposes of quantum optics.
As the discrete ladder operator formalism that we have revealed and orthogonal eigen-systems for self-similar superpotentials lead to the same point spectrum, fixed by (4.15), both type of solutions may be considered as two different representations of one and the
This is the reason why we introduce the following general definition for aself-similar supermodel.
Definition 4.4. Let for 0< q <1 the bounded sequenceλ=(λn)n∈N0of eigenvalues
λn:= qn−1
q−1, n∈N0, (4.34)
be given, and let moreover the subspacel2(λ)⊆l2(N0) be canonically introduced as fol-lows:
ψ= ∞
n=0
cnen∈l2N0
| ∞
n=1
λnλn−1cn2<∞
. (4.35)
Let the pair of adjoint operators
A†:DmaxA†⊆l2(λ)−→l2N 0
, A:Dmax(A)⊆l2(λ)−→l2N 0
(4.36)
be given, being fixed by their actions on the standard orthogonal basis vectors ofl2(λ)⊆ l2(N0) as follows:
A†en:=
λn+1en+1, Aen:=
λnen, n∈N0. (4.37)
Then the triple (A,A†,l2(λ)) is calledself-similar supermodelin Fock space.
As a direct consequence, we see thatAandA†fulfil on a common maximum domain ofl2(λ)⊆l2(N0) the commutation relation
AA†−qA†A=E, (4.38)
the operatorEdenoting the identity map on the maximal common domain ofA,A†. ByDefinition 4.4, we give an abstract formulation of the ladder operator formalism for which the self-similar supermodels fromSection 3and the discrete Schr¨odinger models fromSection 4so far are two different realizations. Therefore, an eigenvalue distribution of type (4.34) has indeed a physical interpretation in Schr¨odinger theory. We now may ask what is the impact of this type of spectrum for coherent state theory within quantum optics. A discussion on this topic will be started by the next section.
5. Discussion of applications to coherent state theory
In the introduction, we have mentioned the role of super-Poisson states respectively, sub-Poisson states in quantum optics. A mathematical model for distinguishing between super-Poisson states and sub-Poisson states is given by the Mandel functional, see for instance [5,6]. The characterization of Mandel’s functional is associated with Jacobi op-erators inl2(N0). The physical states of the quantum system are modelled by the elements ofl2(N0).
a standard orthonormal basis ofl2(N0). Moreover, let
l2⊆l1⊆l2N 0
(5.1)
denote the particular C-linear subspace which is defined by all the elements ψ =
∞
the nonlinearMandel functional f :l2→Ris defined by
f(ψ) :=∞ 1
Mandel’s functional now allows to distinguish between super-Poisson states and sub-Poisson states of the quantum system. On super-sub-Poisson statesψ∈l2, we have f(ψ)>0, and on sub-Poisson statesψ∈l2, we have f(ψ)<0. States which are associated with the Poisson distribution itself, are characterized by a vanishing Mandel functional.
Let nowU⊆Cbe a connected set and let
ΨU:=ψz∈l2N0|z∈U. (5.4)
ΨUis modeling quantum states on which a continuous labelzfromUhas been fixed. The elements ofΨUare referred to by the name coherent statesif
zn−→z=⇒ψzn−ψzl2N0−→0 (5.5)
for any convergent sequence (zn)n∈NinUand if they allow a decomposition of the iden-tity map in the following sense:
∀ϕ∈l2N
0→R+is a suitable weight function andψz∗denotes a canonically constructed dual form onl2(N0), its construction being inherited from the structure of
ΨU. Physical states which arise in the simplest quantum systems, for instance, are mod-eled by
In the sense of the above-given definition, these functions indeed are coherent states. They are states which are eigenvectors of the linear map being fixed by
on the maximal possible definition range for theC-linear mapA. The mapAis modeling the transition from a particular energy level of the quantum system to its lower neigh-bor. In this way, the coherent states which are fixed by (4.26) are stable towards energy perturbations. This is one of their most important properties, being fundamental for the respective applications in quantum optics. This stability is mathematically modeled by the following eigenvalue equation:
Aϕz=zϕz, z∈C. (5.9)
The definition of a so-calledsqueezed stateis given by the variancevQ(ψ) of a linear op-eratorQinψ∈l2(N0). LetQandQ2be defined on a common dense domainΔ⊆l2(N0). The nonlinear variance functionalvQ:Δ→Cis expressed in terms of the canonical scalar product as follows:
vQ(ψ) :=ψ,Q2ψ−(ψ,Qψ)2, ψ∈Δ. (5.10)
Let nowA∗be the adjoint of the operatorAin (5.8). Let moreover the following operators be given, sharing the joint definition ranges ofA, respectively,A∗inl2(N0):
P:=√−i
One has then for all elementsψof a common definition rangeΔ:
Heisenberg uncertainty relation vP(ψ)vX(ψ)≥1
4, ψ∈Δ. (5.12) In this situation, one calls
ψ∈Δ squeezed state:⇐⇒vX(ψ)<1
2. (5.13)
Thus it is a state which has only a small freedom of variation in the spatial sense. Starting from a suitable sequenceλ=(λn)n∈N0of positive numbers modeling the
en-ergies of a given quantum system, on can investigate generalizations of the Mandel func-tional fλ:l2(λ)→Rand study their properties:
The subspacel2(λ)⊆l2(N0) is canonically given as a generalization of the situation (5.1),
The following question is of particular interest, given a special energy sequence (λn)n∈N0.
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Nicole Garbers: Fakult¨at f¨ur Mathematik, Technische Universit¨at M¨unchen, Boltzmannstraße 3, Garching 85747, Germany
E-mail address:ngarbers@web.de
Andreas Ruffing: Fakult¨at f¨ur Mathematik, Technische Universit¨at M¨unchen, Boltzmannstraße 3, Garching 85747, Germany