Báo cáo hóa học: "USING SUPERMODELS IN QUANTUM OPTICS NICOLE GARBERS AND ANDREAS RUFFING " pdf

NICOLE GARBERS AND ANDREAS RUFFINGReceived 27 January 2006; Revised 11 April 2006; Accepted 12 April 2006 Starting from supersymmetric quantum mechanics and related supermodels within Sc

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 on quantum 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 in laser 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 like squeezed laser fields are very important for

applications: the experimental methods when dealing with squeezed laser fields include

for instance the so-called homodyne 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 Di fference Equations

Volume 2006, Article ID 72768, Pages 1 14

DOI 10.1155/ADE/2006/72768

Squeezed laser states are—like coherent states—states of minimal uncertainty From

the viewpoint of statistics, semiconductor lasers have a super-Poisson distribution up to the pumping level, that is, a distribution which goes beyond the non-normalized Pois-son distribution But also the so-called sub-PoisPois-son distribution has a particular mean-ing in multiboson 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 to super-Poisson states and sub-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 mechanics and discrete Schr¨odinger theory become related to essential

problems in quantum optics

In this article, we are going to exploit the connections between self-similar supermodels within supersymmetric quantum mechanics, so-called basic versions of coherent states, and their relations to discretized 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 generalized supermodel definition at the end ofSection 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 (λ being 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 potentials V1andV2 Under some circumstances, it

is possible to write them in the form

whereB and B+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-called superpotential W such that one can express the partner potentials V1,V2as follows:

V1=1

2



W2− √2W 

, V2=1

2



W2+√

2W 

The superpotentialW is fixed by assuming that the potential V1allows 0 as an eigenvalue, the corresponding eigenfunctionϕ being 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

where the differential operators H1,H2are specified by the supersymmetric ladder opera-tors

B : = √1

2



W + √

2 d

dx



, B+:=√1

2



W − √2 d

dx



To illustrate this formalism, let us consider the two potentials

V1(x) = x2

4 −1

2, V2(x) = x2

4 +

1

As for the superpotentialW, we obtain just W(x) = √2x, leading to the well-understood

conventional ladder operator formalism

H1= B+B, H2= BB+,

B = √1

2



x + √

2 d

dx



, B+= √1

2



x − √2 d

dx



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 parametery occurs:

V1(x, y) = y2− y(y + 1)

cosh2(x), x ∈ R (2.10) Assuming that the equation

− ϕ (x) + V1(x, y)ϕ(x) =0 (2.11)

has a solutionϕ ∈ᏸ2(R), we are first led to the superpotential

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)=2 −1

Generalizing the observations made so far, the two potentials V1,V2 are called form-invariant if the following identity holds for di fferent values of x, y1, y2, the expression

R(y1) being a continuous function ofy1:

V2



x, y1



= V1



x, y2



+R

y1



According to the above construction, the pairB, B+specifies two different Schr¨odinger

equations, which together are referred to as a supersymmetric model or just as a super-model.

3 Self-similar supermodels

In many important applications, it follows from the defining equation for form invari-ance,

V2



x, y1



= V1



x, y2



+R

y1



that y2= y1+h, where h is 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

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

for some suitable parameter,y ∈ R The function R from (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

∂ x W

(x, y) +

∂ x W

(x, qy)

Inserting now the expansion (3.4) for the functionR, one obtains, by comparing the

coefficients,

R n =1

2

∞



i =1



1 +q n − i

1− q i

g i g n − i+√

2

1 +q n

g n , n ∈ N, (3.6)

and the valueR0being given byR0= g0 With the abbreviations

r n:= R n

1− q n, d n:=1− q n

one is led to nonlinear integral equations, given by

g n(x) = √ d n

2

x

a



2r n −

n−1

i =1

g i(t)g n − 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, r n = zδ n1, n ∈ N, (3.9)

δ n1denoting the Kronecker symbol andz being a positive parameter This nonlinear

in-tegral equation allows now the solutions

R(y) = R1y = R, g n(x) = √1

2β n x

2n −1, x ∈ R, n ∈ N, (3.10) where the coefficients βnare fixed by the recurrence formula

β1= 2R1

1 +q, β0=0, β n = − d n

2n −1

n−1

i =1

β i β n − i, n ∈ N \ {1} (3.11)

The superpotential now reads

W(x, y) =

∞



j =1

β j y j

 x

√

2

 2j −1

The formal ground-state, belonging toV1, is given by the formula

ψ0(x, y) = Ce − ∞ j =1 (β j /2 j)y j(x/ √

2) 2j

Direct calculation leads to the formula

W

x, y 

=qW

qx, y 

showing some self-similar property This type of superpotential is therefore also referred

to by the name self-similar superpotential Applying a generalized version of the ladder

operator formalism, one is led to the energies of the operatorH1, being given by

λ n = R

n−1

j =0

q j = R1− q n

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{+ q n,−q n | 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 scaleΩ a homogeneous strip discretization or just strip discretization of

the configuration space The Hilbert space of the strip discretization is introduced by the requirement

ᏸ2(Ω) :=f ∈ᏸ2(R)|f = f ◦ χΩ , (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 =

∞

using the characteristic function χΩ of the time scaleΩ By construction, it is clear that

ᏸ2(Ω) is a Hilbert space overC, being a proper subspace of the square-integrable func-tions themselves, that is, ofᏸ2(R) In order to proceed, let us first review some facts on

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 (ψ n)n ∈N0be recursively given by the requirement

ψ n+1(x) : = − ψ n (x) + xψ n(x), x ∈ R, n ∈ N0, (4.5) whereψ0:R → R,x → ψ0(x) : = e −(1/2)x2

We then haveψ n ∈ᏸ2(R)∩ C2(R) forn ∈ N0

and moreover

− ψ n (x) + x2ψ n(x) =(2n + 1)ψ n(x), x ∈ R, n ∈ N0 (4.6)

This result reflects the conventional ladder 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) : = f

q −1x

respectively The right-hand, respectively, left-handq-difference operations will for any

f :Ω→ Rbe given by



D q f

(x) : = f (qx) − f (x)

qx − x ,



D q −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

where the positive even continuous functionϕ :Ω→ R+is chosen as a solution to the

q-difference equation

ϕ(qx) =1 +α(1 − q)x2

We are now able to define discrete ladder operators on strip structures.

The creation operation A † q and, respectively, annihilation operation A qare introduced

by their actions on anyψ :Ω→ Ras follows:

A † q ψ =− D q+g(X)R

ψ, A q ψ = q −1

LD q+Lg(X)

We refer to the discrete Schr¨odinger equation with an oscillator potential onΩ by

q −1

− D q+g(X)R

LD q+Lg(X)

The following result reveals that the discrete Schr¨odinger equation with an oscillator potential onΩ shows similar properties than its classical analog does

Theorem 4.3 Let the time scale Ω be a strip discretization in the sense of Definition 4.1 and let the function ϕ be specified like in Definition 4.2 , satisfying the q-difference equation ( 4.11 ) on Ω,

ϕ(qx) =1 +α(1 − q)x2 

ϕ(x), ϕ(x) = ϕ( − x) > 0, x ∈ Ω. (4.14)

For n ∈ N0 , the functions ψ n:Ω→ R , given by ψ n(x) : =((A † q)n √ ϕ)(x), x ∈ Ω, are well defined inᏸ2(Ω) and solve the q-Schr¨odinger equation ( 4.13 ) in the following sense:

q −1 

− D q+g(X)R

LD q+Lg(X)

ψ n = α q

q n −1

q −1ψ n . (4.15)

Moreover, the linear maps A q , A † q act as ladder operators on the functions (ψ n)n ∈N0in the following sense (n ∈ N0, ψ −1:=0):

A † q ψ n = ψ n+1, A q ψ n = α

q

q n −1

q −1ψ n −1, ψ n(x) = H n q(x)ψ0(x), x ∈Ω, (4.16)

where for n ∈ N0 , the functions H n q:Ω→ R are given by

H n+1 q (x) − αq n xH n q(x) + α q n −1

q −1 H

q

n −1(x) =0, H0q(x) =1, H1q(x) = αx.

(4.17)

These recurrence relations apply for x ∈ Ω and n ∈ N0 , where ψ −1:=0, H − q1:=0 is set There exists the general observation



A † q ψ m,ψ n



Ω=ψ m,A q ψ n



and the functions (ψ n)n ∈N0constitute an orthonormal system inᏸ2(Ω)

Proof Let us for ϕ ∈ C(R) first consider the equation

ϕ(qx)x n =1 +α(1 − q)x2

ϕ(x)x n, x ∈ Ω, n ∈ N0, (4.19) which directly follows from (4.11) Using standard arguments, one can show that the

functionϕ fulfiling (4.11) is inᏸ1(R) This implies

∞

−∞ ϕ(qx)x n χΩ(x)dx =

∞

−∞



1 +α(1 − q)x2 

ϕ(x)x n χΩ(x)dx, n ∈ N0 (4.20) Using the substitution rule to the left-hand side, this directly implies

∞

−∞ ϕ(t)t n q − n χΩ(q −1t)q −1dt =

∞

−∞



1 +α(1 − q)x2 

ϕ(x)x n χΩ(x)dx, n ∈ N0

(4.21) Because of (4.7) we haveχΩ(q −1t) = χΩ(t) for any t ∈ Rand, therefore, (4.21) is equiva-lent to

∞

−∞ ϕ(t)t n q − n χΩ(t)q −1dt =

∞

−∞



1 +α(1 − q)x2 

ϕ(x)x n χΩ(x)dx, n ∈ N0 (4.22) Using the abbreviationμ n(Ω) :=Ωx n ϕ(x)dx for n ∈ N0we obtain the following result:

μ2n+2(Ω)= q −2n −1−1

α(1 − q) μ2n(Ω), μ2 n+1(Ω)=0, n ∈ N0 (4.23)

We have shown earlier [1] that any probability densityψ which generates moments of

type (4.23) yields an orthogonality measure to the polynomials (H n q)n ∈N0which are for

k ∈ Nfixed through the recurrence relation

H k+1 q (x) − αq k xH k q(x) + α q

k −1

q −1H

q

k −1(x) =0, H0q(x) =1, H1q(x) = αx,

(4.24) the variablex being chosen in a suitable integration support As a consequence of this

general result, we obtain the following orthogonality relation:

∞

−∞ H m q(x)H n q(x)ϕ(x)χΩ(x)dx = v n(Ω)δmn, m, n ∈ N0, (4.25) with a sequence of positive numbers (v n(Ω))n ∈N0 Direct calculations and induction show

ψ n(x) : =A † qn

ϕ ◦ χΩ

(x) =H n q(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 (ψ n)n ∈N0constitute an orthonormal system inᏸ2(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 (ψ n)n ∈N0 We remember that the functiong is specified like in

Definition 4.2 We obtain in the sense of the multiplication operator notation



LD q+Lg(X)

X n ψ0



= LD q X n ψ0+Lg(X)X n ψ0, n ∈ N0, (4.27)

which yields



LD q+Lg(X)

X n ψ0



= L

q n −1

q −1X

n −1Rψ0+X n D q ψ0



+Lg(X)X n ψ0, n ∈ N0

(4.28) This may be rewritten as



LD q+Lg(X)

X n ψ0



= q n −1

q −1q

− n+1 X n −1ψ0+LX n

D q ψ0+gψ0



, n ∈ N0 (4.29)

Using now however the formulas in (4.10) for the functiong, we obtain (D q ψ0+gψ0)=0 and therefore



LD q+Lg(X)

X n ψ0



= q n −1

q −1q

− n+1 X n −1ψ0, n ∈ N (4.30)

Form ∈ N0, the first m + 1 polynomials of the sequence (H n q(X)) n ∈N0 can uniquely be generated by linear combinations of the firstm + 1 monomials of the sequence (X n)n ∈N0

We therefore conclude as

A q ψ n =

n−1

j =0

c n

with uniquely defined real numbersc n

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 (ψn)n ∈N0the following relation:



A † q f , g

Ω=f , A q g

In particular, this result implies



A † q ψ m,ψ n

Ω=ψ m,A q ψ n

Using the first equation in (4.16) and because of the fact that the functions (ψ n)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 (ψ n)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 a supermodel 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

functionϕ fulfiling (4.11) is in? ??1(R)...

We are now able to define discrete ladder operators on strip structures.