Coherent states for the supersymmetric partners of the truncated oscillator
David J FernÃ¡ndez VÃ©ronique Hussin VS Morales-Salgado

A Conservation Law Method in Optimization
    Bin Shi
    December 30, 2023
=========================================



 We build the coherent states for a family of solvable singular Schrödinger Hamiltonians 
 obtained through supersymmetric quantum mechanics from the truncated oscillator.
 The main feature of such systems is the fact that their eigenfunctions
 are not completely connected by their natural ladder operators.
 We find a definition that behaves appropriately in the complete 
 Hilbert space of the system, through linearised ladder operators. 
 In doing so, we study basic properties of such states like 
 continuity in the complex parameter, resolution of the identity, 
 probability density, time evolution and possibility of entanglement.




§ INTRODUCTION

When studying quantum systems few tools have proven to be as fruitful 
as those describing them through coherent states (CS).
These states were originally introduced to characterise the most classical 
behaviour for the harmonic oscillator, despite the system being quantum in nature <cit.>.
Since then they became a widespread tool in the understanding of quantum phenomena
<cit.>.

On the other hand, supersymmetric quantum mechanics is a spectral design method that allows 
obtaining solvable Hamiltonians departing from an initial one whose solution is already known.
Such a technique has been used to produce plenty of new solvable quantum systems
<cit.>.
Although most of the work has focused on non-singular potentials, 
recent efforts start to consider potentials with singularities in their domains <cit.>.
In particular, the truncated oscillator, 
a system described by a harmonic oscillator potential with an infinite barrier at the origin,
has shown to be an appropriate and simple model of this kind <cit.>.
Moreover, there is a natural definition of ladder operators 
for the supersymmetric partners of the truncated oscillator, 
even though the energy spectra of such systems are divided into subsets, that cannot be connected through the action of these operators.
In fact, the spectral manipulation achievable for such a system is 
highly affected by the boundary condition at the origin.

Previously, it has been possible to define CS for the supersymmetric partners of non-singular potentials <cit.>.
Thus, it is natural to proceed further and study such states for some singular cases.
However, earlier works have shown that a reduction process on the ladder operators is required in order to obtain appropriate CS for a kind of energy spectrum that some supersymmetric partners possess.
Linearised coherent states have been found to be a successful definition in these problematic cases <cit.>.
This was done for non-singular systems connected with the Painlevé IV equation.
In this work we shall extend the study as well to singular systems, that can be connected with the Painlevé V equation <cit.>.

This work is devoted to study several alternative definitions of coherent states 
for the truncated oscillator and its supersymmetric partners. 
In order to achieve this goal, the article is organised as follows:
In Section 2 we introduce the definitions of coherent states we are interested in.
In Section 3 we study the coherent states for the truncated oscillator, while in 
Section 4 we analyse these states for its supersymmetric partners. 
Finally, in Section 5 we present our final remarks and conclusions.



§ COHERENT STATES

In this section we will introduce in general terms several definitions of coherent states (CS) 
starting from a given set of operators {H,l^+,l^-} characterising the system under study, 
where H is a Schrödinger Hamiltonian and l^± are its ladder operators,
that are not necessarily hermitian conjugate to each other.
We suppose that these operators can be realised in a separable Hilbert space ℋ.
In the Fock representation, an element of this basis is given by |k⟩, 
where k∈ℕ_0={0,1,2,...,d-1}, where d is either finite or infinite,
and the action of these operators on a state |k⟩ is given by:

    H|k⟩   =   ξ(k) |k⟩ ,
     
     l^-|k⟩   =   √(f(k))  |k-1⟩ ,
    
     l^+|k-1⟩   =   √(g(k))  |k⟩ ,

where ξ(k) is a growing function of k that defines 
the energy eigenvalue associated to |k⟩ 
and f(k) and g(k) are functions of k such
that f(0)=0 and, if d is finite, g(d)=0.



 §.§ l^- coherent states

The first definition of coherent states |z⟩ to be considered, 
the l^--CS, is given by the condition

    l^- | z ⟩ =z|z⟩ ,

where z∈ℂ, i.e., |z⟩ is an eigenstate 
of l^- with complex eigenvalue z. 
Through a standard procedure <cit.>, |z⟩ 
can be formally expanded in the Fock basis as

    |z⟩=C_z∑_k=0^d-1z^k/√([f(k)]!)|k⟩ ,

where C_z=[∑_k=0^d-1|z|^2k/[f(k)]!]^-1/2
is a normalisation constant, and 

    [f(k)]!={1       for    k=0
    ∏_k'=1^k f(k')   for    k>0.

is a generalisation of the factorial function.
Normalisability of these states is attained as long as C_z remains finite.
For finite d this fulfillment is immediate, but for an infinite dimensional Hilbert space it depends on the behavior of the function f(k) when k→∞.

Notice that if the space ℋ is finite-dimensional (dim(ℋ)=d 
with d finite), the operator l^- has a d× d-matrix 
representation that is nilpotent of index d. 
Thus, the only solution to the exact eigenvalue equation (<ref>) is z=0 <cit.>.
Such a result makes the condition (<ref>) unsuccessful for defining 
families of CS in a finite dimensional space ℋ. 
Work has been done in order to avoid this problem for some physical models with such a finite energy spectrum (like the Morse potential <cit.>) by considering “almost eigenstates of l^-” .



 §.§ D_l(z) coherent states

An alternative definition of coherent states, the D_l(z)-CS, 
is given by the action of the so-called displacement operator D_l(z)
on an extremal state of the system, that we shall identify with |0⟩.
The extremal states of a quantum system are those non-trivial eigenstates of H that are in the kernel of the annihilation operator l^-.

Recall that the displacement operator for the harmonic oscillator is 

    D_a(z)= exp(za^+-z^*a^-)
           =exp(-1/2|z|^2)exp(za^+)exp(-z^*a^-) ,

where the second equation (the factorised expression) 
is derived from the commutation relation between a^- and a^+.
For systems described by more general algebraic structures,
we use the following definition for the displacement operator 
(possibly non-unitary) <cit.>

    D_l(z)=C̃_z exp(zl^+)exp(-z^*l^-).

It means that for the systems of interest in this work, 
the D_l(z)-CS are formally given by:

    |z⟩=D_l(z)|0⟩
              =C̃_z ∑_k=0^d-1√([g(k)]!)/k! z^k |k⟩.

The normalisation condition requires again that
C̃_z=[∑_k=0^d-1[g(k)]!/k!^2|z|^2k]^-1/2 
which, as mentioned before, relies on its finiteness.
If ℋ is infinite-dimensional, this cannot be guaranteed in general, since the main quantity to be controlled for obtaining an appropriate normalisation is the ratio
[g(k)]!/k!^2 and its behavior for k→∞.



 §.§ Linearised coherent states

In order to reconciliate both definitions of CS (l^--CS and D_l(z)-CS) it has been 
proposed to use a deformed version of the ladder operators l^± <cit.> that linearises their commutator by using the associated number operator. 
Recently such a reduction was used to obtain a definition for 
D_l(z)-CS working well in both finite- and infinite-dimensional cases <cit.>. 
One of the reasons to look for a definition of coherent states that 
works well for both types of Hilbert spaces is to be able to generate them 
for systems as those that will be described in the following sections.
Indeed, let us introduce the linearised ladder operators ℓ^± 
through their action on the Fock basis:

    ℓ^-|k⟩=√(α k)  | k-1⟩ ,    ℓ^+|k-1⟩=√(α k)  |k⟩ .


In the case of an infinite-dimensional Hilbert space the generalised factorial
becomes [f(k)]!=k! α^k and thus the ℓ^--CS are given by:

    |z⟩=C_z∑_k=0^∞1/√(k!)(z/√(α))^k|k⟩,

where C_z=exp(-|z|^2/2α).
As indicated in section <ref>, in the case of a finite dimensional Hilbert space
such a definition of ℓ^--CS gives only the state |0⟩.

On the other hand, having linearised the ladder operators 
makes the normalisability of the D_ℓ(z)-CS straightforward. 
In fact, the linearised D_ℓ(z)-CS are given by

    |z⟩=C̃_z ∑_k=0^d-11/√(k!) (√(α)z)^k |k⟩ ,

where 
C̃_z=[exp(α |z|^2)
     -|z|^2dα^d/Γ(d+1) _2F_2(1,d+1;d+1,d+1;α|z|^2)]^-1/2
and d= dim(ℋ). In the infinite-dimensional case we have that
C̃_z=exp(-α |z|^2/2). 
Note that, throughout this article
_pF_q(a_1,...,a_p;b_1,...,b_q;x) represents the generalised hypergeometric function.

From equations (<ref>) and (<ref>) we can see that when the Hilbert space
is infinite dimensional (d=∞) and α=1 both definitions coincide 
and the standard expression for the CS for the harmonic oscillator is reproduced.



§ TRUNCATED OSCILLATOR AND COHERENT STATES

Let us remember that a quantum harmonic oscillator truncated by an infinite barrier at the origin is described by the Schrödinger Hamiltonian 
H_0=-1/2 d^2/ dx^2+V_0(x), where

    V_0(x) = {[      x^2/2 if   x > 0;          ∞ if   x ≤ 0;            ].,

is the potential of the system. 

The general solution to the stationary Schrödinger equation H_0u(x)=ϵ u(x) 
for x>0 is a linear combination of two solutions of definite parity, an odd and even one,
given by:

    u(x,ϵ)=e^-x^2/2[ _1 F_1(1-2ϵ/4;1/2;x^2)+2νΓ(3-2ϵ/4)/Γ(1-2ϵ/4) x _1 F_1(3-2ϵ/4;3/2;x^2)] .

Moreover, the boundary conditions of the problem, ψ_k(0)=ψ_k(∞)=0, 
make the admissible eigenfunctions of H_0 to be 

    ψ_k(x)=A_k x  e^-x^2/2 _1_1(-k;3/2;x^2)
              =B_k  e^-x^2/2 H_2k+1(x) ,

where A_k=[√(π) 4^k-1 (k!)^2/(2k+1)!]^-1/2, 
B_k=[√(π) 4^k (2k+1)!]^-1/2 
and H_n is the n-th Hermite polynomial,
with corresponding eigenvalues E_k=2k+3/2, k=0,1,...
Note that the states ψ_k of the truncated oscillator are related 
to the states ψ_n^ HO of the harmonic oscillator restricted 
to the domain (0,∞) by ψ_k=√(2) ψ_2k+1^ HO.

The system described by H_0 has the natural ladder operators l^±=(a^±)^2, 
where a^± are the ladder operators for the standard harmonic oscillator.
We thus get the commutation relations: 

    [H_0,l^±]=± 2l^± ,    [l^-,l^+]=4H_0 .

Their action on a Fock state |k⟩ associated to the eigenfunction
ψ_k(x)=⟨ x|k⟩ of equation (<ref>) is given by 

    l^-|k⟩=√(2k(2k+1)) |k-1⟩ ,      
     l^+|k-1⟩=√(2k(2k+1)) |k⟩ .

It means that for this system we have f(k)=g(k)=2k(2k+1).



 §.§ Coherent states and their properties

Such a system has an infinite dimensional Hilbert space ℋ_0,
and we examine next the three definitions of CS presented in section <ref>.

The l^--CS are found as

    |z⟩=C_z∑_k=0^∞ z^k/√((2k+1)!) |k⟩,

where C_z=[sinh(|z|)/|z|]^-1/2.

The D_l(z)-CS become:

    |z⟩=C̃_z ∑_k=0^∞ √((2k+1)!)/k! z^k |k⟩ .

Such states are normalisable only for |z|<1/2, where the normalisation constant 
is C̃_z=(1-4|z|^2)^3/4, making this definition a poor choice.
The third choice leads to the linearised CS already given in 
equations (<ref>) and (<ref>) when α=2.

Let us study next some properties of the l^--CS.
First of all, through a standard procedure <cit.>, we can show that these coherent states are continuous in the complex parameter z.
Even more, they provide a resolution of the identity in ℋ, 
1=∫|z⟩⟨ z| μ(z) dz, where the integral measure is

    μ(z)=|z|^2/8 π C_z^2  exp(-|z|) .


The probability density distribution for a l^--CS |z⟩ 
is given by P_z(x)=|⟨ x|z⟩|^2, where

    ⟨ x|z⟩=∑_n=0^∞ ⟨ x|n⟩⟨ n|z⟩
                       =C_z ∑_n=0^∞z^n/√((2n+1)!) ψ_n(x) .
       
Plots for some particular examples of such a probability density can be found in figure <ref>.


We could also compute the state probability, i.e., the probability 
p_n(z)=|⟨ n|z⟩|^2 of obtaining the eigenvalue ξ(n) 
as a result of an energy measurement when the system is in the coherent state 
|z⟩ (see equation (<ref>)), that is given by

    p_n(z)=C^2_z |z|^2n/(2n+1)! .


The time evolution of a l^--CS is obtained through the expression

    U(t)|z⟩=exp(-iH_0t)|z⟩ .

Since the energy eigenvalues E_k are linear in k,
these coherent states are temporally stable, i.e., 
an arbitrary l^--CS evolves into another state that is also a l^--CS. 

It is worth mentioning that these properties were analysed also for the ℓ^--CS 
and the results do not pose great difference from those of the l^--CS. 
The linearised coherent states are continuous in z and temporally stable.
They provide a resolution of the identity and their probability density
shows well localisation similar to that of the l^--CS depicted in figure <ref>.



 §.§ Measures of entanglement
    
Now, we shall turn to the question of whether these CS can be used to produce entanglement.
To deal with this possibility let us study two measures of entanglement:
the uncertainty relation and the linear entropy.
For this we need to compute explicitly expectation values in the coherent states, obtained here as follows. 

For an observable O it is possible to obtain its expectation value in a l^--CS
in the Schrödinger picture through the equation

    ⟨ O⟩=⟨ z|O|z⟩
                     =∑_m,n=0^∞ Λ_mn(z) ⟨ O_nm⟩ ,

where 
Λ_mn(z)=C_z^2 z^m(z^*)^n/√((2m+1)!(2n+1)!) 
and ⟨ O_nm⟩=∫ ψ^*_n(x) O ψ_m(x)  dx.
In order to simplify future calculations let us use the fact that O is hermitian
to rewrite equation (<ref>) as follows

    ⟨ O⟩=∑_n=0^∞ Λ_nn(z) ⟨ O_nn⟩ 
                      +∑_n=1^∞∑_m=0^n-1 2  Re(Λ_mn(z) ⟨ O_nm⟩) .

For example, if the observable O is chosen as the Hamiltonian H_0,
we obtain the expectation value of the energy ⟨ H_0⟩=1/2+|z|(|z|).

Before moving on, let us recall that on the domain (0,∞) the momentum operator p=-i d/ dx is not an observable,
as can be asserted by the fact that its deficiency indices are (1,0), and thus, by the deficiency theorem p has no self-adjoint extensions <cit.>.
Nonetheless, we are still interested in studying the behavior of the expectation value of the operator p.




  §.§.§ Uncertainty relation.

We can now compute the standard deviations of the position and momentum operators
σ_x=√(⟨ x^2⟩-⟨ x⟩^2),
σ_p=√(⟨ p^2⟩-⟨ p⟩^2), 
and ultimately their product σ_xσ_p.
The lower bound of this product characterises the uncertainty relation 
for the pair of operators x and p.
Let us recall that, since the ladder operators are now l^±=(a^±)^2,
then x and p cannot be written in a simple manner in terms of l^±.

In order to compute the uncertainty relation for the operators x and p, 
we use the matrix elements 


    ⟨ x_nm⟩   =   √((2n+1)!/(2m+1)!)(-2)^n-m-1Γ(n-m-1/2)/π (2n-2m)!
                  × _2F_1[-2m-1,n-m-1/2;2(n-m)+1;2] ,
    ⟨ x^2_nm⟩   =   1/2δ_nm+√( (2n+1)!(2m+1)!)/(n+m)!(1/2δ_n,m-1+δ_n,m+1/2δ_n,m+1) ,
    
    ⟨ p_nm⟩   =   
       i ⟨ x_nm⟩-i(2m+1)/π √((2n+1)!/(2m+1)!)(-2)^n-m(2n-2m-1)Γ(n-m-1/2)/(2n-2m+1)!
                  × _2F_1[n-m+1/2,-2m;2(n-m+1);2] ,
    ⟨ p^2_nm⟩   =   1/2δ_nm-2√(2m(2m+1))δ_m-1,n
                   +2√((2n+1)!(2m+1)!)/(n+m)!(3/2δ_n,m-1+δ_n,m-1/2δ_n,m+1) ,
 
where n≥ m.

Figure <ref> shows approximations of the corresponding standard deviations 
and their product as functions of the modulus of the complex parameter z. 
These were obtained by truncating the infinite sums in equation (<ref>) up to the 30th term.
We can see that the dispersion of the position x 
is always lower than that of the momentum p. 
However, as |z| increases both σ_x and σ_p approach the value
1/√(2) and thus their product tends to the value 1/2.
    

For the linearised coherent states, figure <ref> 
shows the results of the approximations carried out up to the 30th term of (<ref>).
Observe that σ_p is a decreasing function of |z| while σ_x is
a growing function of |z|.
For 0<|z|<1 the dispersion of the momentum is greater than that of the position
and they become equal for |z|=1. 
For 1<|z| the dispersion of the position surpasses that of the momentum.
Opposite to the results for the l^--CS, 
in this case we can see that the dispersions of 
x and p do not approach to each other as |z| increases.
However, once again the product σ_xσ_p tends to 1/2 as |z| increases.
  

In any case, these graphs yield an uncertainty relation such that

    σ_xσ_p≥1/2 ,

and, since σ_x and σ_p are different from each other 
in the case of linearised as well as non-linearised coherent states,
we can say that there is squeezing phenomena <cit.>, that suggests the possibility of producing entangled states by means of these coherent states.



  §.§.§ Entanglement and linear entropy.

As a measure of entanglement let us use the beam splitter tool 
and compute the linear entropy of the out-state <cit.>.
By definition the beam splitter operator is 
ℬ= e^θ/2(a^+b  e^iφ-ab^+  e^-iφ),
where θ is the angle of the beam splitter and φ 
is the phase difference between the reflected and transmitted states,
from which the reflection and transmission amplitudes are
r=- e^-iφsin(θ/2), t=cos(θ/2), respectively.

The beam splitter acts on an input that is a bipartite state 
| in⟩ yielding the out-state <cit.>:

    | out⟩=ℬ| in⟩
                      = e^θ/2(a^+b  e^iφ-ab^+  e^-iφ)| in⟩ ,

where the in-state is such that | in⟩∈ℋ_1⊗ℋ_2, a, a^+ and b, b^+ are the standard first-order ladder operators 
of the harmonic oscillator in each space of the tensor product.
Notice that the operators K_-=ab^+, K_+=a^+b, K_0=1/2(a^+a-b^+b) 
generate a su(2) algebra.

In our case, | in⟩=|0⟩⊗| z ⟩
where | z ⟩ is a l^--CS. We thus get

    | out⟩=𝒞_z∑_n 1/√((2n+1)!) z^n ℬ(|0,n⟩) .


Using this last state we can define the density operator 
ρ_AB=| out⟩⟨ out|, 
and by taking a partial trace we obtain ρ_A. 
The linear entropy is thus S=1- Tr(ρ_A^2), 
and we know that S=0 corresponds to no entanglement 
while S=1 corresponds to maximal entanglement.
           
Let us recall that |n⟩=√(2) |2n+1⟩_ HO; thus
         ℬ|0,n⟩=2 e^θ/2(a^+b e^iφ-ab^+ e^-iφ)|1,2n+1⟩_HO   
                =2 ∑_k=0^2n+21/k![e^iφtanθ/2]^k[(cosθ/2)^2n√((k+1)!(2n+1)!/(2n+1-k)!) |k+1,2n+1-k⟩_HO.   

                  .-tan(θ/2) e^-iφ√(2n+2)(cosθ/2)^(2n+2)√((k)!(2n+2)!/(2n+2-k)!) |k,2n+2-k⟩_HO ]   

where we have used the Baker-Campbell-Haussdorf formula for the su(2) algebra 
<cit.> generated by K_-=ab^+, K_+=a^+b, K_0=1/2(a^+a-b^+b): 

    e^τ K_+-τ^*K_-= e^τ/|τ|tan|τ|K_+ e^-2(lncos|τ|)K_0 e^-τ^*/|τ|tan|τ|K_- .


In terms of the reflection and transmission amplitudes we obtain

    | out⟩   =   𝒞_z∑_n=0^∞ z^n/√((2n+1)!) ∑_k=0^2n+2(-r/t)^k/k!  e^i2kφ 
                  ×[Q_1 |k+1,2n+1-k⟩_ HO+Q_2 |k,2n+2-k⟩_ HO] ,

where

    Q_1(r,t,n,k)   =   t^2n√((k+1)!(2n+1)!/(2n+1-k)!) ,
    
     Q_2(r,t,n,k)   =   r t^(2n+1)√(2n+2) √((k)!(2n+2)!/(2n+2-k)!) .

     
Remember that the bra-ket product is obtained by an integration over the domain (0,∞). 
In fact <cit.>

    (⟨α|β⟩)_ HO=∫_0^∞ e^-x^2H_α(x)H_β(x) dx
      =√(π)  _2F_1[-α,-β;1-α+β/2;1/2]/2^1-α-βΓ(1-α+β/2) .


Now the density operator ρ_AB=| out⟩⟨ out| is given by
         ρ_AB
 =𝒞_z^2∑_m,n=0^∞ z^n(z^*)^m/√((2n+1)!(2m+1)!) ∑_k=0^2n+2∑_l=0^2m+2(-r/t)^k/k!(-r^*/t)^l/l! e^i2(k-l)φ   
               ×[Q^*_2(r,t,m,l)Q_2(r,t,n,k) (|k,2n+2-k⟩⟨l,2m+2-l|)_HO.   
                   + Q^*_1(r,t,m,l)Q_2(r,t,n,k) (|k,2n+2-k⟩⟨l+1,2m+1-l|)_HO   
                   + Q^*_2(r,t,m,l)Q_1(r,t,n,k) (|k+1,2n+1-k⟩⟨l,2m+2-l|)_HO   
                   .+ Q^*_1(r,t,m,l)Q_1(r,t,n,k) (|k+1,2n+1-k⟩⟨l+1,2m+1-l|)_HO] .   

     
Computing now the partial trace ρ_A we get the linear entropy S=1- Tr(ρ_A^2).
Figure <ref> shows numerical approximations for S(|z|),
obtained by truncating the resulting infinite sum up to the 20th term.
 

It is worth noticing that the plot is overall flat and it shows that 
the out-sate is entangled, although not maximally entangled.
If we repeat the computations now for the linearised coherent states,
the plot of S(|z|) behaves in general in the same way as with the l^--CS.



§ THE SUPERSYMMETRIC TRUNCATED OSCILLATOR AND COHERENT STATES

In quantum mechanics, a supersymmetric transformation relates 
two Hamiltonians as follows <cit.>. 
Suppose that the Hamiltonian H=-1/2 d^2/ dx^2+V(x) 
is obtained through a q-th order supersymmetric transformation from an 
initial Hamiltonian H_0=-1/2 d^2/ dx^2+V_0(x), 
i.e., there exists a differential operator Q of order q, 
called intertwining operator, such that the following relation holds

    HQ=QH_0 .


Up to an overall shift in the energy, the set of eigenvalues of H differs 
in a finite number, κ=κ(q), of elements from that of H_0. 
This means that the supersymmetric transformation has added and/or erased 
levels in the energy spectrum of H_0 to obtain the one of H.

If the energy spectrum of H_0 is infinite countable, then the Hilbert space ℋ 
spanned by the eigenfunctions of H is the direct sum of two subspaces: 
ℋ=ℋ_ iso⊕ℋ_ new,
an infinite-dimensional one denoted by ℋ_ iso
and a κ-dimensional subspace denoted by ℋ_ new.
The eigenvectors of H are given by the union of those 
|E_n⟩∈ℋ_ iso and the ones
|ℰ_j⟩∈ℋ_ new.
Then, the energy spectrum of H is also given by a union of two sets.
The first one is an infinite set given by {E_0,E_1,E_2,⋯} while the second one is the finite set {ℰ_0,ℰ_1,⋯,ℰ_κ-1}.

If the initial system admits ladder operators l^±, 
then the system described by H has natural ladder operators in 
the subspace ℋ_ iso given by the product Q l^± Q^†, 
where Q^† is the hermitian adjoint of Q.
On the other hand, in ℋ_ new such a definition does not yield 
in general operators that connect the eigenenergies in this subspace. 
However, in the following section we will use a reduction theorem that will 
allow us to obtain well-behaved ladder operators in ℋ_ new.



 §.§ Supersymmetric partners of the truncated oscillator

The supersymmetric partners of the truncated oscillator <cit.>,
obtained through a q-th order supersymmetric transformation, 
are described by Hamiltonians H=-1/2 d^2/ dx^2+V 
with potentials given by <cit.>

    V=V_0-{ln[W(u_1,...,u_q)]}” ,

where W(u_1,...,u_q) is the Wronskian of q seed solutions 
of the initial Schrödinger equation H_0 u_j=ϵ_j u_j, 
j=1,...,q, given in general by (<ref>).
These u_j's do not need to satisfy any boundary conditions, 
although they must not introduce new singularities to the superpartner potential V.

Without loss of generality, we can order the ϵ_j's as ϵ_1<...<ϵ_q.
Moreover, as is usual we shall suppose that ϵ_q<1/2.
Under these assumptions, the set of eigenvalues of H has 
κ=[q/2] (the integer part of q/2) 
new elements corresponding to the eigenvalues ℰ_i 
added by the non-singular supersymmetric transformation, 
which associated eigenfunctions indeed satisfy the boundary conditions <cit.>.
Hence, the eigenvalues in ℋ_ iso form an infinite ladder 
{E_0,E_1,E_2,...} such that E_k=E_0+2k, where E_0=3/2, 
while in ℋ_ new the eigenvalues 
{ℰ_0,ℰ_1,...,ℰ_κ-1}
are not in general equally spaced.

Now, in order to define ladder operators working appropriately on both subspaces,
we have to consider that the eigenvalues in ℋ_ new 
are related by ℰ_j=ℰ_0+2j. 
Thus, the partial spectrum {ℰ_0,ℰ_1,...,ℰ_κ-1} forms 
an equidistant set of eigenvalues, and by means of a reduction theorem 
applied to the operators Q (a^±)^2 Q^†,
where a^± are the standard first-order ladder operators of the harmonic oscillator,
it is possible to obtain now sixth-order ladder operators L^± in the whole space ℋ that satisfy <cit.>

    L^+L^-=(H-1/2)(H-3/2)
            (H-ϵ_1)(H-ϵ_2)
            (H-ϵ_q-1-2 )(H-ϵ_q-2 ).

Moreover, the commutation relations [H,L^±]=± 2L^± still hold,
and they yield the following action of L^± on the basis vectors of ℋ.

In ℋ_ iso:

    L^-|E_n⟩
      =[(E_n-1/2)(E_n-3/2)
             (E_n-ϵ_1)(E_n-ϵ_2 ).   
                   ×.(E_n-ϵ_q-1-2)(E_n-ϵ_q-2 )]^1/2  | E_n-1⟩ ,   
       L^+|E_n⟩
      =[(E_n+1-1/2)(E_n+1-3/2)
              (E_n+1-ϵ_1)(E_n+1-ϵ_2).   
                   ×.(E_n+1-ϵ_q-1-2)(E_n+1-ϵ_q-2 )]^1/2  | E_n+1⟩ .
 
In ℋ_ new:

    L^-|ℰ_j⟩
      =[(ℰ_j-1/2)(ℰ_j-3/2)
             (ℰ_j-ϵ_1)(ℰ_j-ϵ_2 ).
               .×(ℰ_j-ϵ_q-1-2)(ℰ_j-ϵ_q-2 )]^1/2  |ℰ_j-1⟩ ,
       L^+|ℰ_j⟩
      =[(ℰ_j+1-1/2)(ℰ_j+1-3/2)
             (ℰ_j+1-ϵ_1)(ℰ_j+1-ϵ_2 ).
               .×(ℰ_j+1-ϵ_q-1-2)(ℰ_j+1-ϵ_q-2 )]^1/2  |ℰ_j+1⟩ .
 
From these results we can see that L^- annihilates the eigenstates 
corresponding to 3/2 and ℰ_0, while L^+ 
annihilates the eigenstate corresponding to ℰ_κ-1.

It is well known that the differential operators L^-, L^+ and H defining second and third degree polynomial Heisenberg algebras are determined by solutions of the 
Painlevé IV and Painlevé V equations respectively
<cit.>.
In <cit.> it was explicitly shown how different supersymmetric partners of the truncated oscillator, 
ruled by appropriate L^-, L^+ and H, are thus connected to 
these non-linear second-order differential equations.
In <cit.> coherent states were built for systems connected to the Painlevé IV equation. 
Here we will build coherent states for supersymmetric partners of the truncated oscillator that are ruled by both, the second and fifth degree polynomial Heisenberg algebras.

When trying to define coherent states in ℋ using L^±, one realises that 
these ladder operators do not connect energy levels from 
ℋ_ iso with those in ℋ_ new, 
which justifies the direct sum decomposition of ℋ. 
Thus, we require a definition of coherent states appropriate 
for finite- and infinite-dimensional spaces simultaneously. 

From section <ref> we know that the L^--CS are appropriate for 
ℋ_ iso but fail  in ℋ_ new. 
The opposite happens with D_L(z) coherent states:
while this definition is appropriate for ℋ_ new 
it fails in general in ℋ_ iso.
Therefore, we will implement the definition of linearised coherent states from section <ref> for the supersymmetric partners of the truncated oscillator.
As previously remarked, each subspace, ℋ_ iso or ℋ_ new, 
possesses one extremal state given by its lowest energy state,
thus allowing one to apply the displacement operator in each case.
Before doing this, however, we will illustrate with an example how
the supersymmetry method applied to the truncated oscillator works.

Let us consider the case where a fourth order (q=4) 
supersymmetric transformation has been implemented,
with ϵ_1=-11/2, ϵ_2=-9/2, 
ϵ_3=-7/2, ϵ_4=-5/2.
Then we get that κ=2, and the eigenstates in ℋ_ new 
are those associated to the eigenvalues ℰ_0=ϵ_2=-9/2
and ℰ_1=ϵ_4=-5/2.
The supersymmetric partner potential thus obtained is given by


    V(x)=x^2/2-4(256 x^16-2560 x^14+10496 x^12-19584 x^10+27360 x^8-10080 x^6-10800 x^4+16200 x^2+2025)/(16 x^8-64 x^6+120 x^4+45)^2
  
where x∈(0,∞).

The subspace isospectral to the truncated oscillator ℋ_ iso is spanned by

    ϕ_n =Qψ_n/√(∏_i=1^4(E_n-ϵ_i)),

where ψ_n is given by equation (<ref>) and the intertwining operator 

    Q=1/4( d^4/ dx^4+η_3(x) d^3/ dx^3
       +η_2(x) d^2/ dx^2+η_1(x) d/ dx+η_0(x))

is a fourth-order differential operator whose coefficients are given by

    η_3   =   -4 x (16 x^8-32 x^6+24 x^4+120 x^2+45)/16 x^8-64 x^6+120 x^4+45,
    η_2   =   6 (16 x^10-16 x^8+88 x^6-120 x^4+165 x^2-105)/16 x^8-64 x^6+120 x^4+45,
    η_1   =   4 x (-16 x^10+16 x^8-216 x^6+48 x^4+915 x^2+315)/16 x^8-64 x^6+120 x^4+45,
    η_0   =   16 x^12-32 x^10+360 x^8+240 x^6-795 x^4-7110 x^2+1935/16 x^8-64 x^6+120 x^4+45.

The lowering operator L^- acts on the basis of this subspace as follows

    L^-|E_n⟩=√((E_n-1/2)(E_n-3/2)(E_n+11/2)(E_n+9/2)(E_n+3/2 )(E_n+1/2 ))  | E_n-1⟩.


On the other hand, the only two eigenfunctions in ℋ_ new are given by

    ϕ_ℰ_0   =   4 √(3) e^-x^2/2 x (8 x^6-4 x^4+10 x^2+15)/√(π)(8 (2 x^4-8 x^2+15) x^4+45),
    ϕ_ℰ_1   =   2 e^-x^2/2 x (16 x^8+72 x^4-135)/√(3)√(π)(8 (2 x^4-8 x^2+15) x^4+45),

associated to ℰ_0=-9/2, ℰ_1=-5/2, respectively.
The lowering operator L^- acts on the corresponding basis vectors as follows

    L^-|ℰ_j⟩
      =√((ℰ_j-1/2)(ℰ_j-3/2)(ℰ_j+11/2)(ℰ_j+9/2)(ℰ_j+3/2)(ℰ_j+1/2))  |ℰ_j-1⟩.




 §.§ Linearised D_ℒ(z) coherent states

A linearisation of L^± can be obtained by defining 
the following linearised ladder operators

    ℒ^+:= γ(H) L^+ ,      ℒ^-:= γ(H+2) L^- ,

where γ(H)=[(H-1/2)(H-ϵ_1)(H-ϵ_2)(H-ϵ_q-1-2)(H-ϵ_q-2)]^-1/2.
 
The action of ℒ^± on the basis vectors of ℋ_ iso is

    ℒ^-|E_n⟩=√(E_n-3/2)  | E_n-1⟩=√(2n)  | E_n-1⟩,
    ℒ^+|E_n⟩=√(E_n+1-3/2)  | E_n+1⟩=√(2n+2)  | E_n+1⟩.

Thus, the operator ℒ^- annihilates the state |E_0⟩, 
which is the extremal state in ℋ_ iso.
Also, note that the commutators between ℒ^± and H 
are those of a Heisenberg-Weyl algebra in this subspace

    [H,ℒ^±]=±2ℒ^±,     [ℒ^-,ℒ^+]=2.


Acting the displacement operator
D_ℒ(z)=exp(-|z|^2)exp(zℒ^+)exp(-z^*ℒ^-)
on the extremal state |E_0⟩ we obtain the desired linearised CS in ℋ_iso:

    |z⟩_iso=D_ℒ(z)|E_0⟩
      =exp(-|z|^2)∑_n=0^∞(√(2)z)^n/√(n!)|E_n⟩.


On the other hand, the action of ℒ^± on 
the basis vectors of ℋ_new is given by

    ℒ^-|ℰ_j⟩   =   (1-δ_j,0)√(ℰ_j-3/2)  |ℰ_j-1⟩
       =(1-δ_j,0)√(2j-Δ_1)  |ℰ_j-1⟩,
    ℒ^+|ℰ_j⟩   =   (1-δ_j,κ-1)√(ℰ_j+1-3/2)  |ℰ_j+1⟩
       =(1-δ_j,κ-1)√(2j+2-Δ_1)  |ℰ_j+1⟩ ,

where Δ_1=3/2-ℰ_0.
Note that ℒ^- annihilates the state |ℰ_0⟩,
while ℒ^+ annihilates the state |ℰ_κ-1⟩ so that |ℰ_0⟩ is the extremal state in ℋ_ new.
Although one needs to be cautious when these operators act on
|ℰ_0⟩ and |ℰ_κ-1⟩, 
in general, the commutators in (<ref>) still hold when acting on the other basis vectors of ℋ_ new.

Displacing now the state |ℰ_0⟩ gives us 
the following CS in ℋ_new:

    |z⟩_new=D_ℒ(z)|ℰ_0⟩
      =Ĉ_z ∑_j=0^κ-1(√(2)z)^j/j!√(Γ(-Δ_1/2 +j)/Γ(-Δ_1/2)) |ℰ_j⟩ ,

where

    Ĉ_z=  _1F_1[-Δ_1/2;1;2|z|^2]
         -|z|^2κ 2^κΓ(2κ-Δ_1/2)/[Γ(κ+1)]^2 Γ(-Δ_1/2) _2F_2[1,κ-Δ_1/2;κ+1,κ+1;2|z|^2] .


The D_ℒ(z)-CS are continuous in the complex parameter z 
and they lead to a resolution of the identity, where the integral measures in each subspace ℋ_ iso and ℋ_ new are given respectively by

    μ_ iso(z)=2/π ,     μ_ new(z)=2 Γ(-Δ_1/2)/πĈ_z^2 G^2,0_1,2(-(Δ_1+2/2)
    0 , 0 | 2|z|^2 ) ,

where G^m,n_p,q(a_1,...,a_p
b_1,...,b_q | x) is the Meijer G-function. 
As in the non-supersymmetric case, these states are temporally stable.

The mean value of the energy in each subspace reads

    ⟨ H⟩_ iso=3/2+4|z|^2 ,    ⟨ H⟩_ new=
      ℰ_0+2|Ĉ_z|^2∑_j=0^σ-1j(2 |z|^2)^j/(j!)^2Γ(Δ_1/2-j) ,

while the state probability is given by

    P_n(z)=(2|z|^2)^n/n!exp(-2|z|^2) ,    
     P_j(z)=(2 |z|^2)^j-1/[(j-1)!]^2|Ĉ_z|^2/Γ(Δ_1/2+1-j) ,

in ℋ_ iso and ℋ_ new, respectively.

Consider again the previous example, 
where the levels ℰ_0=-9/2 and ℰ_0=-5/2 
have been added to the energy spectrum of the supersymmetric partner.
The probability density P(x)=|⟨ x|z⟩|^2 in each subspace of 
ℋ is depicted in figure <ref>, showing fairly good localisation.
We notice that for |z⟩_ iso this localisation 
is somewhat lost for small values of |z|. 
In the case of |z⟩_ new, 
for big values of |z| two distinct peaks occur,
that however remain close to each other. 
Greater localisation is achieved for small values of |z| since the peaks start to merge.




 §.§ Measures of entanglement
  
Following the results and notation in equation (<ref>) we obtain the matrix elements

    Λ_m,n(z)   =    exp(-2|z|^2) (√(2)z)^m(√(2)z^*)^n/√(m!n!) ,
    Λ_i,j(z)   =   Ĉ_z^2 (√(2)z)^i(√(2)z^*)^j/i!j! √(Γ(-Δ_1/2+i)Γ(-Δ_1/2+j))/Γ(-Δ_1/2) ,

in ℋ_ iso and ℋ_ new, respectively.



  §.§.§ Uncertainty relation.

Let us rely on the example introduced at the end of section <ref>, where a fourth 
order supersymmetric transformation was carried out for the truncated oscillator.
Recall that in this example ℋ_ new was two-dimensional,
since the added eigenfunctions spanning it were those associated to 
the eigenvalues ℰ_0=-9/2 and ℰ_1=-5/2, 
while the space ℋ_ iso remained infinite-dimensional,
as it is isospectral to the truncated oscillator.
Figure <ref> shows the behavior of σ_x, 
σ_p and their product σ_xσ_p, 
as functions of the modulus of the complex parameter z.


In both subspaces we can observe squeezing phenomena:
while in ℋ_ new the uncertainty in the momentum is always 
greater than the uncertainty in the position, in ℋ_ iso we find
the squeezing of σ_x for |z|<1 and the squeezing of σ_p for |z|>1.
This indicates once again that entanglement is attainable by means of these coherent states.



  §.§.§ Entanglement and linear entropy.

Let us obtain the linear entropy in each of the two subspaces 
ℋ_ iso and ℋ_ new.

The in-states in these subspaces are given by 

    | in⟩_ iso= e^-|z|^2 ∑_i=0^∞(√(2)z)^i/√(i!) |ℰ_0,E_i⟩ ,
        
     | in⟩_ new=Ĉ_z ∑_j=0^κ-1(√(2)z)^j/j!√(Γ(-Δ_1/2 +j)/Γ(-Δ_1/2)) |ℰ_0,ℰ_j⟩ ,

respectively.
By means of equation (<ref>) we can obtain the out-states
| out⟩_ iso, | out⟩_ new.
Through a similar treatment as in section <ref>, for which φ=0, 
we find that, in coordinates representation the out-states are given by
       ⟨x,y|out⟩_iso
  =𝒞̃_z ∑_j=0^∞ (√(2)z)^j/√(j!)
      ∑_l,m,n=0^∞1/l!m!n![tanθ/2]^n[ln(cosθ/2)]^m[- tanθ/2]^l   
                   ×∑_i=0^mmi (-1)^-i∂_x^n x^i ∂_x^i[x^l ⟨x|ℰ_0⟩] y^m+n-i⟨y|E_j⟩^(l+m-i) ,             

       ⟨x,y|out⟩_new
  =Ĉ_z ∑_j=0^κ-1 (√(2)z)^j/j!√(Γ(-Δ_1/2 +j)/Γ(-Δ_1/2))
      ∑_l,m,n=0^∞1/l!m!n![tanθ/2]^n[ln(cosθ/2)]^m[- tanθ/2]^l   
                   ×∑_i=0^mmi (-1)^-i∂_x^n x^i ∂_x^i[x^l ⟨x|ℰ_0⟩] y^m+n-i⟨y|ℰ_j⟩^(l+m-i) ,   

in ℋ_ iso and ℋ_ new, respectively, where 𝒞̃_z and Ĉ_z are normalisation constants.      
      
As before, we use the out-state to obtain the operator 
ρ=| out⟩⟨ out|
and then we compute the linear entropy S=1- Tr(ρ_A^2).

For the system described by the potential in equation (<ref>),
a numerical approximation to the linear entropy in ℋ_ new 
as function of |z| can be found in figure <ref>. 
The infinite series were truncated at the 30th term.
      
Although being flat, with values close to 0.5, 
this plot show evidence of entanglement, but not of a maximally entangled out-state.
In the case of the subspace ℋ_ iso the behavior of the linear entropy 
is similar to that depicted in figure <ref>.



§ CONCLUSIONS

In this work we studied several definitions of coherent states 
for the truncated oscillator and its supersymmetric partners.
The main difficulty in defining coherent states for systems obtained through
a supersymmetric transformation lays in the fact that the resulting energy spectrum 
is not completely connected by the action of the natural ladder operators,
leading to a decomposition of ℋ as a direct sum of two subspaces: 
a finite dimensional one and other infinite dimensional.

It has been found that the conventional CS definitions that work well for finite 
dimensional spaces fail when applied to infinite dimensional ones and vice versa.
We surpassed such a difficulty by using a reduced linearised version of the natural ladder operators for the system.
Thus we arrived to a common definition of coherent states, that can be applied to both subspaces.
Consequently, we studied some properties of these coherent states.
We found the production of squeezing in the position and momentum of these coherent states,
as well as the arising of entanglement in the out-state of a beam splitter whose in-state 
was one of these coherent states.



§ ACKNOWLEDGMENTS

This work has been supported in part by research grants from 
Natural sciences and engineering research council of Canada (NSERC).
The authors acknowledge the financial support of the Spanish MINECO 
(project MTM2014-57129-C2-1-P) and Junta de Castilla y León (VA057U16).
VS Morales-Salgado also acknowledges the Conacyt fellowship 243374
and the Department of Foreign Affairs, Trade and Development of Canada
for the Emerging Leaders in the Americas Program (ELAP) scholarship provided.



§ REFERENCES

100
s26 Schrödinger E 1926 Naturwiss 14 664
k63a Klauder J R 1963 J. Math. Phys. 4 1955
k63b Klauder J R 1963 J. Math. Phys. 4 1958
g63a Glauber R J 1963 Phys. Rev. 130 2529
g63b Glauber R J 1963 Phys. Rev. 131 2766
p86 Perelomov A 1986 Generalized Coherent States and their Applications (Springer) 
gk99 Gazeau J P and Klauder J R 1999 J. Phys. A: Math. Gen. 32 123
aag00 Ali S T, Antoine J P and Gazeau J P 2014 Coherent States, Wavelets and their Generalizations, 2nd Ed (Springer)
q01 Quesne C 2001 Ann. Phys. 293 147
ah08 Angelova M and Hussin V 2008 J. Phys. A 41 30416

wi81 Witten E 1981 Nucl. Phys. B 185 513
wi82 Witten E 1982 Nucl. Phys. B 202 253
mi84 Mielnik B 1984 J. Math. Phys. 25 3387
ais93 Andrianov A A, Ioffe M V and Spiridonov V P 1993 Phys. Lett. A 174 273
aicd95 Andrianov A A, Ioffe M V, Cannata F and Dedonder J P 1995 Int. J. Mod. Phys. A 10 2683
cks95 Cooper F, Khare A and Sukhatme U 1995 Phys. Rep. 251 267
bs97 Bagrov V G and Samsonov B F 1997 Phys. Part. Nucl. 28 374
fgn98 Fernández D J, Glasser M L and Nieto L M 1998 Phys. Lett. A 240 15
fhm98 Fernández D J, Hussin V and Mielnik B 1998 Phys. Lett. A 244 309
jr98 Junker G and Roy P Ann. Phys. 270 155
qv99 Quesne C and Vansteenkiste N 1999 Helv. Phys. Acta 72 71
sa99 Samsonov B F 1999 Phys.Lett. A 263 274
mnr00 Mielnik B, Nieto L M and Rosas-Ortiz O 2000 Phys. Lett. A 269 70
crf01 Cariñena J F, Ramos A and Fernández D J 2001 Ann. Phys. 292 42
ast01 Aoyama H, Sato M and Tanaka T 2001 Nucl. Phys. B 619 105
mr04 Mielnik B and Rosas-Ortiz O, J. Phys. A: Math. Gen. 37 10007
cfnn04 Carballo J M, Fernández D J, Negro J and Nieto L M 2004 J. Phys. A: Math. Gen. 37 10349
ff05 Fernández D J and Fernández-García N 2005 AIP Conference Proceedings 744 236
cf08 Contreras-Astorga A and Fernández D J 2008 J. Phys. A: Math. Theor. 41 475303
ma09 Marquette I 2009 J. Math. Phys. 50 095202
f10 Fernández D J 2010 AIP Conf. Proc. 1287 3
qe11 Quesne C 2011 Mod. Phys. Lett. A 26 1843
bf11a Bermúdez D and Fernández D J 2011 SIGMA 7 025
ma12 Marquette I 2012 J. Math. Phys. 53 012901
ai12 Andrianov A A and Ioffe M V 2012 J. Phys. A: Math. Theor. 45 503001
ggm13 Gómez-Ullate D, Grandati Y and Milson R 2014 J. Phys. A: Math. Theor. 47 015203

mnn98 Márquez I F, Negro J and Nieto L M 1998 J. Phys. A: Math. Gen. 31 4115
fgn11 Fernández D J, Gadella M and Nieto L M 2011 SIGMA 7 029
fm14 Fernández D J and Morales-Salgado V S 2014 J. Phys. A: Math. Theor. 47 035304
fm16 Fernández D J and Morales-Salgado V S 2016 J. Phys. A: Math. Theor. 49 195202
fm18 Fernández D J and Morales-Salgado V S 2018 Ann. Phys. 388 122

fhr07 Fernández D J, Hussin V and Rosas-Ortiz O 2007 J. Phys. A: Math. Theor. 40 6491 
bcf14 Bermudez D, Contreras-Astorga A and Fernández D J 2014 Ann. Phys. 350 615
bfn16 Bermudez D, Fernández D J and Negro J 2016 J. Phys. A: Math. Theor. 49 335203

s92 Shabat A 1992 Inverse Problems 8 303
vs93 Veselov A P and Shabat A B 1993 Funct. Anal. Appl. 27 81
a94 Adler V E 1994 Physica D 73 335
dek94 Dubov S Y, Eleonskii V M and Kulagin N E 1994 Chaos 4 47
ekk94 Eleonskii V M, Korolev V G and Kulagin N E 1994 Chaos 4 583
srk97 Sukhatme U P, Rasinariu C and Khare A 1997 Phys. Lett. A 234 401
acin00 Andrianov A A, Cannata F, Ioffe M and Nishnianidze D 2000 Phys. Lett. A 266 341
mn08 Mateo J and Negro J 2008 J. Phys. A: Math. Theor. 41 045204

W10 Weyl H 1910 Math. Ann. 68 220
vN29 von Neumann J 1929 Math. Ann. 102 49
t85 Truax D R 1985 Phys. Rev. D 31 1988
pbm90 Prudnikov A P, Brychkov Y A and Marichev O I 1990 Integrals and Series, Vol. 2: Special Functions (Gordon and Breach)

w83 Walls D F 1983 Nature 306 141 
lk87 Loudon R and Knight P L 1987 J. Mod. Opt. 34 709

fl87 Fearn H and Loudon R 1987 Opt. Commun. 64 485
twc91 Tan S M, Walls D F and Collett M J 1991 Phys. Rev. Lett. 66 252
sw01 Scheel S and Welsch D G 2001 Phys. Rev. A 64 063811 
ksbk02 Kim M S, Son A, Bužek V and Knight P L 2002 Phys. Rev. A 65 032323 
x02 Xiang-Bin W 2002 Phys. Rev. A 66 024303 

gk05 Gerry C and Knight P 2005 Introductory Quantum Optics (Cambridge University Press)



