Journal Article•DOI•

Some inferential aspects of finite population sampling with additional resources

S. Sengupta¹, Bikas Kumar Sinha², Bikas Kumar Sinha³, Sastry G. Pantula³•Institutions (3)

University of Calcutta¹, Indian Statistical Institute², North Carolina State University³

01 Jan 1987-Journal of Statistical Planning and Inference (North-Holland)-Vol. 16, Iss: 2, pp 203-211

TL;DR: In this paper, the problem of extending a given sampling design, when additional resources are available, is considered and some existing methods of improving an initial sampling strategy, so that the use of the additional resources is justified, are critically reviewed.

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About: This article is published in Journal of Statistical Planning and Inference.The article was published on 1987-01-01 and is currently open access. It has received None citations till now. The article focuses on the topics: Sampling design & Sampling (statistics).

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Summary (2 min read)

Jump to: [1. Introduction] – [2. Hamiltonian model: implementation in trapped ions] – [3.1. Symmetries] – [3.2. Solving the eigenvalue problem] – [4. Time evolutions] and [5. Conclusive remarks]

1. Introduction

The Jaynes–Cummings (JC) model [1] provides a very fruitful description of the light–matter interaction, wherein the matter is usually modelled as a two-level (pseudo-spin) system coupled to a harmonic oscillator which describes one mode of the quantized radiation.
Exactly this kind of processes is taken into account in the JC model.
In the context of trapped ions, JC-like Hamiltonian models are used to describe the coupling between the internal and translational degrees of freedom of a confined particle.
The knowledge of the eigenstates and eigenvalues of such a Hamiltonian allows us to evaluate the time evolution of the system and to discuss the possibility of realizing interesting applications aimed at generating nonclassical states such as GHZ states and W-states [12].
Finally, some conclusive remarks are given in section 5.

2. Hamiltonian model: implementation in trapped ions

The electromagnetic Paul trap generates a time-dependent inhomogeneous (in particular quadrupole) electromagnetic (e.m.) field that induces the charged particle motion including two contributions, i.e., a ‘secular motion’ and a ‘micromotion’ [3].
The first one is the motion of a particle confined in a three-dimensional atomic well.
Moreover, suitably adjusting the e.m. fields in the trap, it is possible to render the three frequencies of oscillations of the ion centre of mass to be equal and obtain a degenerate trap [13].
The action of more than one laser is described via the sum of the Hamiltonian models related to each laser fields.
In other words, the interaction has the structure of a standard JC interaction term associated with the mode related to âx .

3.1. Symmetries

Each of these bosonic parts possesses a so(3) symmetry, being invariant under rotations around all axes.
As usual, in order to obtain a complete set of commuting operators (CSCO), the authors shall consider two commuting operators, i.e., the square of the angular momentum, L̂2 := L̂2x + L̂2y + L̂2z , and the third component L̂z.
Ĥ 0, so that they really are constants of motion, in the sense that no explicit time dependence has been introduced in the passage to the interaction picture.

3.2. Solving the eigenvalue problem

Because of the commutation between the interaction picture Hamiltonian and the ‘conserved operators’ in (7), each couple of states | (n, l,m, σ = ±)〉 (which are also eigenstates of Ĥ 0) constitutes an invariant subspace under the action of the Hamiltonian Ĥ int.
Alternatively, it is possible to find a unitary transformation that realizes such a diagonalization.
Once the canonical transformation in (11) has been realized, the diagonalization may be completed finding the common eigensolutions of the total excitation number N̂ , the angular momentum L̂2 and of the third Pauli matrix σ̂3 ).
To obtain a complete set of commuting operators (CSCO) the authors add the third component L̂z of the angular momentum operator.

4. Time evolutions

Indeed, once the system has been lead to the vibronic ground state |0〉x |0〉y |0〉z|−〉 through cooling techniques [4, 16], it is enough to induce an atomic population inversion.
As another example consider a non-spherically symmetric initial state which distinguishes the z-direction from the other two.

5. Conclusive remarks

The authors have described a physical scenario wherein a three-dimensional two-phonon Jaynes–Cummings-like Hamiltonian model is implementable and, thanks to its symmetries, solvable.
It is worth remarking that the success of their approach comes from the particular features of the Hamiltonian the authors have considered, and especially from the fact that it possesses a so(3) symmetry coming from the presence of only squares of the annihilation/creation operators in the interaction Hamiltonian.
In addition, the extension of the Hamiltonian-digonalization procedure from the two-dimensional to the three-dimensional anisotropic model is seemingly not straightforward.
This possibility relies on the fact that the dynamical behaviour of the system preserves the symmetries of the initial states.
In contrast, in (19), no instant of time exists at which it is possible to obtain a spherically symmetric state.

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In this paper, the problem of extending a given sampling design, when additional resources are available, is considered, and existing methods of improving an initial sampling strategy, so that the use of the additional resources is justified, are critically reviewed.