^{1,a)}and Fredy Zypman

^{1,b)}

### Abstract

This study presents supersymmetric coherent states that are eigenstates of a general four-parameter family of annihilation operators. The elements of this family are defined as operators in Fock space that transform a subspace of a definite number of particles into a subspace with one particle removed. The emphasis is on classifying parameter space in various regions according to the uncertainty bounds of the corresponding coherent states. Specifically, the uncertainty in position-momentum is analyzed, with specific focus on characterizing regions of minimum uncertainty states, regions where the uncertainties are bounded from above, and where they grow unbound.

MK was partially supported by Kressel and Schottenstein fellowships. FZ laboratory received a grant from the Gamson Fund. We thank Yitzhak Kornbluth for fruitful comments.

I. INTRODUCTION

II. ENERGY EIGENSTATES

III. GENERALIZED SUPERSYMMETRIC ANNIHILATION OPERATOR

IV. SUPERCOHERENT STATES

A. Recursion relations

V. CLASSIFICATION: THREE FAMILIES

A. Degenerate

B. Singular

C. Generic

VI. SUPERPOSITIONS AND UNCERTAINTIES

A. General form

B. Boundedness

C. A numerical example

VII. ENERGIES

VIII. CONCLUSIONS

### Key Topics

- Coherent states
- 28.0
- Eigenvalues
- 21.0
- Uncertainty principle
- 12.0
- Subspaces
- 10.0
- Supersymmetry
- 10.0

## Figures

Uncertainty of the supercoherent state |*Z* _{θ}⟩, describing the superposition of two basis states |*Z* _{θ}⟩ = 2^{−1/2}(|*Z* _{+}⟩ + *e* ^{ iπ/4}|*Z* _{−}⟩). The uncertainty separates into two regions: (1) In 0 < θ < π/2, the uncertainty is bounded. The maximum is 0.83, at approximately *z* = 0.5*e* ^{ iπ/4}, θ = π/4. (2) In π/2 < θ < π, the uncertainty diverges with *z*. The rate of divergence depends upon the phase of *z*: |*z*|^{2} for , while |*z*|^{4} for .

Uncertainty of the supercoherent state |*Z* _{θ}⟩, describing the superposition of two basis states |*Z* _{θ}⟩ = 2^{−1/2}(|*Z* _{+}⟩ + *e* ^{ iπ/4}|*Z* _{−}⟩). The uncertainty separates into two regions: (1) In 0 < θ < π/2, the uncertainty is bounded. The maximum is 0.83, at approximately *z* = 0.5*e* ^{ iπ/4}, θ = π/4. (2) In π/2 < θ < π, the uncertainty diverges with *z*. The rate of divergence depends upon the phase of *z*: |*z*|^{2} for , while |*z*|^{4} for .

Graph of the parameter space of the supercoherent states. The blue-yellow surface describes the degenerate case, while the green surface corresponds to the singular matrix. The red-black circle describes the family , for which the red portions have bounded uncertainty for all eigenstates while the black portions have unbounded uncertainty for most eigenstates. All parameters *k* _{ i } are assumed to be real and divided by *k* _{1}, because the SUSY annihilation matrix effectively occupies a three-dimensional space.

Graph of the parameter space of the supercoherent states. The blue-yellow surface describes the degenerate case, while the green surface corresponds to the singular matrix. The red-black circle describes the family , for which the red portions have bounded uncertainty for all eigenstates while the black portions have unbounded uncertainty for most eigenstates. All parameters *k* _{ i } are assumed to be real and divided by *k* _{1}, because the SUSY annihilation matrix effectively occupies a three-dimensional space.

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