Supersymmetric harmonic oscillator and nonlinear supercoherent states

We find and analyze the (nonlinear) supercoherent states associated with the generalized nonlinear supersymmetric annihilation operator (SAO) of the supersymmetric harmonic oscillator. We discuss as well the uncertainty relation for a special case in order to compare our results with those obtained for the linear supercoherent states.


Introduction
For the standard harmonic oscillator with angular frequency ω, the commutation relations between the Hamiltonian and the annihilation and creation operatorsĤ,â,â † generate the well known Heisenberg-Weyl algebra: However, if a generalized form for the annihilation and creation operators is employed, which respects the commutation relations with the Hamiltonian, it is possible to establish a connection with the so-called polynomial deformations of the Heisenberg algebra.
Following the above ideas, the analysis of the algebraic commutation relations between the HamiltonianĤ s of the SUSY harmonic oscillator [1,2], and the corresponding creationÂ † and annihilation operatorsÂ of the system requires the knowledge of the explicit form of these operators. However, as it was shown in [3,4], this form is not unique.
Recently [5], it was introduced a general expression forÂ and its eigenvectors |Z⟩ with complex eigenvalues z were found (called supercoherent states) which become expressed in terms of the standard harmonic oscillator coherent states [6,7].
Although the expression found in [5] forÂ is very general, it is still not unique. In this work, we will consider some of its deformationsQ, which maintain however the structure given in [5]. We will analyze also a particular deformation of the SAO and we will find the explicit form of its eigenvectors |X ⟩ with complex eigenvalue X. Due to the deformation assumed, the eigenstates ofQ turn out to be expressed in terms of nonlinear coherent states, associated to deformed Lie algebras (called nonlinear algebras) [8,9,10,11]. Therefore, it is natural to call the states |X ⟩ nonlinear supercoherent states.

Standard coherent states (CS)
A standard coherent state |α⟩ can be defined as an eigenstate of the annihilation operatorâ with complex eigenvalue α, i.e., a|α⟩ = α|α⟩, α ∈ C, ⟨α|α⟩ = 1. ( In the Fock basis, the normalized coherent states read There are other definitions that can be used to build this type of quantum states, which are equivalent to each other for the harmonic oscillator; nevertheless, in this paper we will use the previous definition.

Nonlinear coherent states (NLCS)
The nonlinear coherent states |z⟩ f can be defined as eigenstates of the deformed annihilation operatorã = f (N )â [12], i.e.,ã f (N ) being a well behaved real function of the number operatorN =â †â . These states depend strongly on how many Fock states |n⟩ are annihilated byã, which in turn depend on the explicit form of the function f (N ) sincẽ For instance, for f (N ) =N +1 it turns out that f (n − 1) = n ̸ = 0, ∀ n = 1, 2, . . . which implies thatã|0⟩ = 0. Thus: where r = |α| and p F q is the generalized hypergeometric function On the other hand, for f (N ) =N we have that f (0) = 0. This assumption implies that a|0⟩ =ã|1⟩ = 0. Hence, the corresponding NLCS become now This result indicates that the contribution of the ground state is removed from |α⟩ N L since this eigenstate |0⟩ is isolated from the remaining ones due toã|0⟩ =ã † |0⟩ = 0.
Note that, if the form of the function f (N ) is chosen appropriately, it is possible to isolate more energy eigenstates.  Heisenberg uncertainty relation (σ x ) 2 α (σ p ) 2 α as function of α for the nonlinear coherent states of Eq. (9).

Heisenberg uncertainty relation
Knowing the explicit form of the standard coherent states |α⟩ and the nonlinear ones |α⟩ nl and |α⟩ N L , the corresponding Heisenberg uncertainty relations can be straightforwardly calculated.

Nonlinear supercoherent states
The generalized form for the SAOÂ [5] and its deformation are respectively: where Both linear and nonlinear supercoherent states are defined in the generic way aŝ From this equation, after several calculations and simplifications, an overall matrix relationship is found as where the quantitiesã n andc n are defined as a n = β n √ n!X 1−n a n ,c n = β n−1 forQ =Â ′ and n ≥ 1 (n − 1)!, forQ =Â ′′ and n ≥ 2 . ( As can be seen, both the linear and nonlinear supercoherent states depend on the eigenvalues ψ ± of the matrix K. This induces a classification of the supercoherent states |X ⟩ into three different families as follows: degenerate (ψ + = ψ − ≡ ψ ̸ = 0); singular (ψ + ψ − = 0); generic (everything else).
Furthermore, by rewriting a n and c n in terms of ψ ± , the eigenstates ofÂ turn out to be expressed in terms of the CS in (4) while those ofÂ ′ andÂ ′′ are determined by the ones of  (9), when choosing f (N ) =N +1 or f (N ) =N respectively. Then we will identify the respective supercoherent states as follows: where with and Besides the set {|X A ⟩, |X C ⟩}, we can choose a new basis for the state space formed by the elements whereby it is possible to pass from a parameter space {k 1 , k 2 , k 3 , k 4 } to a new one, formed by {ψ + , ψ − , k 1 , k 2 }.

Degenerate. Explicitly the supercoherent states are
where The corresponding supercoherent states are now . (31)

Superposition and uncertainties
Consider a superposition of states |X ± ⟩ with parameters η and λ as follows: where γ 1± and γ 2± are given by The mean value of an arbitrary observableĉ is then with This allows us to find the expressions for the uncertainties of the operatorsx andp and their squares.

Uncertainties forQ =Â.
The expressions for the uncertainties are [5]: where the quantities in Eq. (35) are evaluated taking X = z.
where X = Y in Eq. (35) and

A particular case
To analyze the behavior of the supercoherent states (linear and nonlinear) the following particular values for the parameters k i are taken: k 1 = k 4 = 1, k 2 = cos θ and k 3 = sin θ. Thus, the deformed SAO in (18) takes the generic form: ,