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International Journal of Modern Physics A

ANALYSIS OF IMPOSSIBILITY OF TESTING SPATIAL NONCOMMUTATIVITY USING RYDBERG ATOMS IN THE VICINITY OF SADDLE-POINT IN CROSSED ELECTRIC AND MAGNETIC FIELDS

Jian-zu Zhang A B, Li-Ming He, Zhu Yun-Xia

Published Jul 30, 2010• 2 citations
Publisher

Abstract

In relation to the current controversy over the need for an experimental test of spatial noncommutativity at the normal quantum scale, the spectra of a Rydberg electron in the vicinity of the saddle-point in a suitable combination of crossed electric and magnetic fields are investigated in both cases of commutative space and noncommutative space. Its local behavior around the saddle-point is clarified. An unbounded mode is revealed. It describes the unbounded direction along which the electron will escape the vicinity of the saddle-point, which means that it is impossible to locate electronic states. This finding indicates that the possibility of testing spatial noncommutativity in the vicinity of the saddle-point is ruled out. Because this unbounded mode applies only to states near the saddle-point, not to those from the outer well, it is suggested that if any test of spatial noncommutativity can be performed by using atoms in the crossed-field, then the test must be performed using states from the outer well, not those close to the saddle-point.

DOI
10.1142/s0217751x1004975x

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In relation to the current controversy over the need for an experimental test of spatial noncommutativity at the normal quantum scale, the spectra of a Rydberg electron in the vicinity of the saddle-point in a suitable combination of crossed electric and magnetic fields are investigated in both cases of commutative space and noncommutative space. Its local behavior around the saddle-point is clarified. An unbounded mode is revealed. It describes the unbounded direction along which the electron will escape the vicinity of the saddle-point, which means that it is impossible to locate electronic states. This finding indicates that the possibility of testing spatial noncommutativity in the vicinity of the saddle-point is ruled out. Because this unbounded mode applies only to states near the saddle-point, not to those from the outer well, it is suggested that if any test of spatial noncommutativity can be performed by using atoms in the crossed-field, then the test must be performed using states from the outer well, not those close to the saddle-point.
The possibility of testing spatial noncommutativity via Rydberg atoms is explored. An atomic di-pole of a cold Rydberg atom is arranged in appropriate electric and magnetic fields, so that the motion of the dipole is constrained to be planar and rotationally symmetric. Spatial noncommutativity leads the canonical angular momentum to possess fractional values. In the limit of vanishing kinetic energy, the dominate value of the lowest canonical angular momentum takes variant Planck's over h/2. Furthermore, in the limit of eliminating the magnetic field, the dominate value of the lowest canonical angular momentum changes from variant Planck's over h/2 to variant Planck's over h/4. This result is a clear signal of spatial noncommutativity. An experimental verification of this prediction is suggested.
We describe a novel experiment on Rydberg atoms in crossed electric and magnetic fields, recently performed at Imperial College. The novelty resides in a combination of a crossed-field geometry with a time-gated detection technique, by which separation of the circularly polarized components is now achieved. The importance of achieving such a separation and the new opportunities it opens up are explained. By this approach, a map of the diamagnetic Stark patterns is obtained. The data exhibit interesting behaviour in that the hydrogenic and non-hydrogenic groups within the n-shell now become distinguishable. Their evolution as a function of field strength brings out the difference between linear and quadratic effects. The levels with a quadratic dependence on electric field strength eventually invade and dominate the spectrum. We surmise that the trend from order to chaos is different in crossed fields from the trend in the diamagnetic limit. We comment on further theory required to clarify this point. Our observations extend up to and slightly above the saddle-point energy.
The excitation spectrum of atomic sodium in crossed electric and magnetic fields has been recorded. Data for states near n = 11 are found to be in good agreement with theoretical estimates of line positions. The agreement with estimates of line strengths, however, is only fair. The lack of complete agreement between simple hydrogenic theory and experiment can be traced t o a breakdown of the assumption that the influence of the nearby 12p state is negligibly small. The significance of this experiment is that, for the first time, clear evidence of m-mixing between bound states has been obtained. The dominant effect of the m-mixing has been to make accessible optically many Rydberg states. These states in the presence of electric or magnetic fields alone would not have been accessible because of selection rules on m.
The possibility of testing spatial noncommutativity via a Penning trap is explored. The case of both space-space and momentum-momentum noncommuting is considered. Spatial noncommutativity leads to the spectrum of the orbital angular momentum of a Penning trap possessing fractional values, and in the limits of vanishing kinetic energy and subsequent vanishing magnetic field, this system has non-trivial dynamics. The dominant value of the lowest orbital angular momentum is ℏ/4\hbar/4ℏ/4, which is a clear signal of spatial noncommutativity. An experimental verification of this prediction by a Stern-Gerlach-type experiment is suggested.
An approach to solve the critical problem of testing quantum effects of spatial noncommutativity is proposed. Magnetic hyperfine structures in a Rydberg system induced by fractional angular momentum originated from spatial noncommutativity are discussed. The orders of the corresponding magnetic hyperfine splitting of the spectrum ~10−7–10− 8 eV lie within the limits of accuracy of current experimental measurements. Experimental tests of physics beyond the standard model are the focus of broad interest. We note that the present approach is reasonably achievable with current technology. The proof is based on very general arguments involving only the deformed Heisenberg-Weyl algebra and the fundamental property of angular momentum. Its experimental verification would constitute an advance in understanding of fundamental significance, and would be a key step towards a decisive test of spatial noncommutativity.
The tunneling between double wells of atom in crossed electromagnetic fields is investigated by a one-dimensional Hamiltonian model. The crossed fields induced outer well is apart from the nuclear origin and it is very difficult to access by means of spectroscopy but it will be possible if there exists the tunneling of the electron between the outer well and the Coulomb potential predominated well at the nuclear origin. A one-dimensional quantum calculation with B-spline basis has been performed for hydrogen atom in crossed fields accessible in our laboratory, at B = 0.8 T and F = −220 V·cm −1 . The calculation shows that the wavefunctions of some excited states close to the Stark saddle point in the outer well extend over to the Coulomb potential well, making it possible to penetrate the quantum information of the outer well. However, the tunneling rate is very small and the spectral measurement of the transitions from the ground state should be of a high resolution and high sensitivity.