Supercoherent states approach to the SUSY harmonic oscillator

The nonlinear supercoherent states, associated with a nonlinear generalization of the Kornbluth-Zypman (KZ) supersymmetric annihilation operator (SAO) of the supersymmetric harmonic oscillator, will be studied. We discuss as well the Heisenberg uncertainty relation σ2xσ2p for a special case which will allow us to compare our results with those obtained for the KZ linear supercoherent states.


Introduction
For the standard harmonic oscillator, the commutators between the HamiltonianĤ with the annihilation and creation operatorsâ,â † generate the well known Heisenberg-Weyl algebra.
On the other hand, for the SUSY harmonic oscillator [1,2] the analysis of the commutation relations between the HamiltonianĤ sĤ s = ω â †â 0 0ââ † (1) and the corresponding creationÂ † and annihilation operatorsÂ of the system requires the knowledge of the explicit form of these operators [3,4]. Recently [5], a quite general expression forÂ was introduced and its eigenvectors |Z with complex eigenvalues z were found (called supercoherent states), which became expressed in terms of the standard harmonic oscillator coherent states [6,7].
Despite the expression proposed in [5] forÂ is very general, it is not unique. In this work, we will consider some specific deformationsQ, which maintain 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 eigenvalues X. Due to the deformation assumed, the eigenstates ofQ turn out to be expressed in terms of nonlinear coherent states associated to deformed (nonlinear) Lie algebras [8][9][10][11], so they are called 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.
In the Fock basis, the normalized coherent states read Other definitions can be used to build this type of quantum states, which are equivalent to each other for the harmonic oscillator; nevertheless, in this work we will use the previous definition.
The structure of these states depends on which Fock states |n are annihilated byã; this is determined by the explicit form of the function f (N ) sincẽ We will consider the following linear form f (N ) =N − k1, where k ∈ N ∪ {−1, 0}. So, the nonlinear coherent states |α (k) associated to the operatorã (k) = (N − k1)â turn out to be where r = |α| and p F q is the generalized hypergeometric function Note that in Eq. (4) the contribution of the lowest energy Fock states to the nonlinear coherent state depends on the parameter k, i.e., somehow it is possible to isolate the lowest energy eigenstates according to the value of k.

Heisenberg uncertainty relation
Knowing the explicit form of the standard coherent states |α and the nonlinear ones |α (k) , the corresponding Heisenberg uncertainty relations can be straightforwardly calculated.
For the standard coherent states, the mean values of the operatorsx,p,x 2 ,p 2 become (with = m = ω = 1): Hence, the Heisenberg uncertainty relation turns out to be given by (σ x ) 2 α (σ p ) 2 α = 1/4. For the nonlinear coherent states |α (k) , the mean values of the operatorsx,p,x 2 ,p 2 are: where The Heisenberg uncertainty relation becomes: When k = −1, the expression for the Heisenberg uncertainty relation is (see Figure 1): where .
When k = 0, the expression for the Heisenberg uncertainty relation is now (see Figure 2 and [16]): where .
For k = 1 and k = 2 we have shown the Heisenberg uncertainty relation in Figure 3 and Figure 4, respectively.

Nonlinear supercoherent states
A very general form for the SAOÂ [5] and its simplest deformation are respectively: where k i ∈ C and we will take either f (N ) =N +1 or f (N ) =N . We will labelQ =Â when f (N ) =N +1, andQ =Â when f (N ) =N . Note thatQ =Â when f (N ) =1. Both linear and nonlinear supercoherent states are defined in the generic way aŝ From this equation an overall matrix relationship is found as where the quantitiesã n andc n are defined as 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 Eq. (2) while those ofÂ andÂ are determined by the ones of Eq. (6) or Eq. (7), 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 another set of supercoherent states 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
The supercoherent states are explicitly where (26b)

Singular
The corresponding supercoherent states become now .

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 straightforwardly the expressions for the uncertainties of the operatorŝ x andp and their squares.
where the quantities in Eq. (31) are evaluated taking X = z.

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 Eq. (14) takes the generic form: meanwhile the superposition (28) (with η = λ = π/4) becomes On the other hand, the eigenvalues ψ ± of the matrix K of Eq. (16) become now: so that for 0 < θ < π/2 both ψ ± are real while for π/2 < θ < π they turn out to be complex. By substituting equations (41) and (42) in expressions (32)-(39), the Heisenberg uncertainty relation for each form of the operatorQ is found.  [5] is shown in Figures 5 and 6. For z real (left plot, Figure 5) the uncertainty reaches a maximum value equal to 0.83 at |z| ∼ 0.5 for real eigenvalues ψ ± (0 < θ < π/2), while it shows a growing behavior as |z| increases for complex eigenvalues ψ ± (π/2 < θ < π). For complex z (right plot, Figure 6), the uncertainty relation behaves similarly as for real z.

5.2.
Heisenberg uncertainty relation forQ =Â . Figures 7 and 8 show the Heisenberg uncertainty relation σ for the nonlinear supercoherent states |Y associated to the operatorÂ . In case of Y real (left plot, Figure 8), for both real (0 < θ < π/2) as well as complex (π/2 < θ < π) eigenvalues ψ ± , the uncertainty starts from a minimum value and grows slowly. In the region of degenerate eigenvalues, for θ = π/2, the uncertainty reaches a maximum value. For complex Y (right plot, Figure 8), the uncertainty relation behaves similarly as for real Y .  for the nonlinear supercoherent states |Z associated to the operatorÂ . In case of Z real (left plot, Figure 9), for both real (0 < θ < π/2) as well as complex (π/2 < θ < π) eigenvalues ψ ± , the uncertainty starts from a value equal to 2.25, then it decreases smoothly and grows again. In the region of degenerate eigenvalues ψ ± , for θ = π/2, the uncertainty reaches a minimum value close to 0.25. For complex Z (right plot, Figure 10), the uncertainty relation behaves similarly as for real Z.
As shown in Figures 5-10, the behavior of the Heisenberg uncertainty relation for the linear supercoherent states |Z becomes more involved than for the nonlinear ones, |Y and |Z . Moreover, since the nonlinear supercoherent states |Z are expressed in terms of the nonlinear coherent states |α N L which isolate the eigenstate |0 , the corresponding uncertainty relation starts from a different value at Z = 0 than its counterparts associated toÂ andÂ .

Conclusions
The deformation of the supersymmetric annihilation operatorÂ given in [5] allowed us to find interesting results related with the nonlinear supercoherent states. As it was noted, the form of the operatorã = f (N )â in the corresponding nonlinear coherent states, enabled us to isolate the ground state contribution, which has interesting implications for the properties of the nonlinear supercoherent states and their associated Heisenberg uncertainty relation.
Let us remark that all the results obtained in this paper obey the Heisenberg uncertainty principle (σ 2 x σ 2 p ≥ 1/4), which ensures that the states constructed here are consistent with the quantum theory.
Finally, the freedom we have for choosing the form of the function f (N ) will allow us to consider a more detailed algebraic study, focused on the commutation relations of the operators {Ĥ s ,Â,Â † } for the supersymmetric harmonic oscillator and the possible relation with the polynomial deformations of the Heisenberg algebra [17,18], as it happens with the operators {Ĥ,â,â † } for the standard harmonic oscillator.