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Journal of Mathematical Physics

Prolate spheroidal quantum cloak

Cheng-De Syue, De‐Hone Lin

Published Apr 1, 2015• 1 citation
Publisher

Abstract

To understand the propagation behavior of an oblique incident matter wave in a three-dimensional non-spherical quantum cloak, we perform the transformation design for the prolate spheroidal coordinate system and obtain a quantum cloak with an ellipsoidal shape. The mass parameters and effective potential for the creation of a perfect prolate spheroidal invisibility region are given. The analytic representations of the cloaked matter wave and probability current in the cloaking shell are presented. Special attention is paid to the discussions of the probability current in the cloaking shell for only that current can manifestly exhibit how the wave vector of the matter wave is curved, rotated, and guided in the cloaking shell to flow around the non-spherically invisible region. With the current analysis, one shows that the presented cloak can perfectly guide the matter wave in the situation of any oblique incidence. The proposed prolate spheroidal cloak for matter waves provides the first non-spherically three-dimensional setup for quantum cloaking.

DOI
10.1063/1.4918367

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To understand the propagation behavior of an oblique incident matter wave in a three-dimensional non-spherical quantum cloak, we perform the transformation design for the prolate spheroidal coordinate system and obtain a quantum cloak with an ellipsoidal shape. The mass parameters and effective potential for the creation of a perfect prolate spheroidal invisibility region are given. The analytic representations of the cloaked matter wave and probability current in the cloaking shell are presented. Special attention is paid to the discussions of the probability current in the cloaking shell for only that current can manifestly exhibit how the wave vector of the matter wave is curved, rotated, and guided in the cloaking shell to flow around the non-spherically invisible region. With the current analysis, one shows that the presented cloak can perfectly guide the matter wave in the situation of any oblique incidence. The proposed prolate spheroidal cloak for matter waves provides the first non-spherically three-dimensional setup for quantum cloaking.
Parabolic quantum cloaks with unbounded and bounded invisible regions are presented with the method of transformation design. The mass parameters of particles for perfect cloaking are shown to be constant along the parabolic coordinate axes of the cloaking shells. The invisibility performance of the cloaks is inspected from the viewpoints of waves and probability currents. The latter shows the controllable characteristic of a probability current by a quantum cloak. It also provides us with a simpler and more efficient way of exhibiting the performance of a quantum cloak without the solutions of the transformed wave equation. Through quantitative analysis of streamline structures in the cloaking shell, one defines the efficiency of the presented quantum cloak in the situation of oblique incidence. The cloaking models presented here give us more choices for testing and applying quantum cloaking.
An obstacle to constructing a quantum cloak with the transformation design method for steering a matter wave is the device involving the anisotropic modulation of the particle mass. It is the purpose of this paper to show that a narrow-acceptance-angle quantum cloak generated by a kind of conformal mapping could be mass-independent. This greatly simplifies the construction of the device since it can now be finished only by applying a potential field to a region of the shell. As demonstrations of the conclusion, an elliptical and a parabolic conformal quantum cloak are designed from the elliptical and parabolic coordinate systems. Their performance is inspected by probability currents. It is shown that the conformal devices can serve as novel splitters for a 2D matter wave. Finally, we discuss the difficulty of constructing a steering device from the hyperbolic coordinate system.
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A physical construct for the cloaking of relativistic spin-12\frac{1}{2}21​ matter waves is proposed. It is shown that when the effective energy and mass of relativistic spin-12\frac{1}{2}21​ particles moving in an effective vector field in a spherical shell are controlled, their matter waves can be perfectly guided through the shell without any distortion or loss; that is, the construct provides a three-dimensional cloaking shell for relativistic spin-12\frac{1}{2}21​ matter waves. The proposal serves as the basis for some interesting applications such as providing a method to guide the matter waves of spin particles and an ideal setup to exhibit spin-spin interactions as well as perfect quantum interferences of some global effects in spin-12\frac{1}{2}21​ matter waves.
As a probe to explore the ability of invisibility cloaks to conceal objects in the quantum mechanics domain, we study the spontaneous emission rate of an excited two-level atom in the vicinity of an ideal invisibility cloaking. On this base, first, a canonical quantization scheme is presented for the electromagnetic field interacting with atomic systems in an anisotropic, inhomogeneous, and absorbing magnetodielectric medium which can suitably be used for studying the influence of arbitrary invisibility cloak on the atomic radiative properties. The time dependence of the atomic subsystem is obtained in the Schrodinger picture. By introducing a modified set of the spherical wave-vector functions, the Green tensor of the system is calculated via exact and discrete methods. In this formalism, the decay rate and as well the emission pattern of the aforementioned atom are computed analytically for both weak and strong coupling interaction, and then numerically calculations are done to demonstrate the performances of cloaking in the quantum mechanics domain. Special attention is paid to different possible orientations and locations of the atomic system near the spherical invisibility cloaking. Results in the presence and the absence of the invisibility cloak are compared. We find that the cloak works very well far from its resonance frequency to conceal a macroscopic object, whereas at near the resonance frequency the object is more visible than the situation where the object is not covered by the cloak.