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Physical Review Letters

Test of the linearity of quantum mechanics by rf spectroscopy of the Be+9 ground state

J. J. Bollinger, Daniel J. Heinzen, Wayne M. Itano, S. L. Gilbert, David J. Wineland

Published Sep 4, 1989• 151 citations
Publisher

Abstract

A hyperfine transition in the ground state of $^{9}\mathrm{Be}^{+}$ was used to test a nonlinear generalization of quantum mechanics recently formulated by Weinberg. We searched for a dependence of the frequency of a coherent superposition of two hyperfine states on the populations of the states. We are able to set a limit of 4\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}27}$ on the fraction of binding energy per nucleon of the $^{9}\mathrm{Be}^{+}$ nucleus that could be due to nonlinear corrections to quantum mechanics.

DOI
10.1103/physrevlett.63.1031

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A hyperfine transition in the ground state of 9Be+^{9}\mathrm{Be}^{+}9Be+ was used to test a nonlinear generalization of quantum mechanics recently formulated by Weinberg. We searched for a dependence of the frequency of a coherent superposition of two hyperfine states on the populations of the states. We are able to set a limit of 4\ifmmode\times\else\texttimes\fi{}10\ensuremath−27{10}^{\mathrm{\ensuremath{-}}27}10\ensuremath−27 on the fraction of binding energy per nucleon of the 9Be+^{9}\mathrm{Be}^{+}9Be+ nucleus that could be due to nonlinear corrections to quantum mechanics.
Atomic physics experiments which test a nonlinear generalization of quantum mechanics recently formulated by Weinberg are described. The experiments search for a dependence of hyperfine transition frequencies or nuclear spin precession frequencies on the relative populations of the hyperfine or nuclear spin states. set limits less than 10 pHz on the size of the possible nonlinear contributions to these frequencies. In some cases this can be interpreted as a limit of less than- on the fraction of binding energy per nucleon that could be due to a nonlinear correction to a nuclear Hamiltonian. nonlinear addition to quantum mechanics violates causality is discussed. The experiments The possibility that a
It is proposed to set stringent limits on possible nonlinear corrections to ordinary quantum mechanics by searching for the detuning of resonant transitions. A suggested nonlinear generalization of quantum mechanics is used to show that such detuning would be expected in the rf transition in 9Be+^{9}\mathrm{Be}^{+}9Be+ ions that is used to set frequency standards. Measurements at the National Bureau of Standards already set limits of order ${10}^{\ensuremath{-}21}$ on the fraction of the energy of the 9Be^{9}\mathrm{Be}9Be nucleus that could be due to nonlinear corrections to quantum mechanics, with good prospects of improving this by 2-3 orders of magnitude.
Measurement of the free precession of nuclear spins in optically pumped ground-state 201Hg^{201}\mathrm{Hg}201Hg atoms has allowed a test of Weinberg's nonlinear generalization of quantum mechanics. Shifts among the three resolved precession frequencies in the spin-3/2 system were studied as a function of the projection of spin polarization along the precession axis. Our results set an upper limit on \ensuremath{\epsilon}, Weinberg's nonlinearity parameter, of \ensuremath{\Vert}\ensuremath{\epsilon}\ensuremath{\Vert}/2\ensuremath{\pi}\ensuremath{\Elzxh}3.8\ensuremath{\mu} Hz, which corresponds to 2.0\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}27}$ of the binding energy per nucleon. This is the most stringent such limit to date.
We have observed coherence among the four magnetic sublevels of freely precessing 21Ne^{21}\mathrm{Ne}21Ne for times of 4.5 h. This has allowed direct detection of the quadrupole component of coherence relaxation and a test of linearity of quantum mechanics. The quadrupole component of relxation is evident in the different relaxation rates for the three lines of the NMR spectrum. Nonlinear corrections to quantum mechanics are found to be less than 1.6\ifmmode\times\else\texttimes\fi{}${10}^{\mathrm{\ensuremath{-}}26}$ of the binding energy per nucleon of 21Ne^{21}\mathrm{Ne}21Ne.
We extend a nonlinear generalization of quantum mechanics, recently formulated by Weinberg, to systems of composite spin, such as atoms. We have used hydrogen (H) masers to set a limit of 3.7\ifmmode\times\else\texttimes\fi{}1020{10}^{20}1020 eV (8.9 \ensuremath{\mu} Hz) on the magnitude of a nonlinear correction to the quantum mechanics of atomic spins. This result is of comparable magnitude to the limit recently set in a single-valued (nuclear) spin system. In the absence of nonlinear effects, the experiment provides a new and stringent test of H-maser clock performance and the applicability of standard maser theory.
One hears about testing quantum electrodynamics or relativity theory, but the theoretical basis of quantum mechanics is rarely questioned. Nevertheless, Steven Weinberg (University of Texas) has recently called for high-precision tests of quantum mechanics that are independent of any particular quantum mechanical theory. To pave the way for an examination of quantum mechanics, he has suggested one possible way of generalizing quantum mechanics to make it nonlinear. As Eugene Wigner pointed out in a 1939 paper, the linearity of quantum mechanics is an important assumption and one that may not necessarily always prove true. Weinberg stressed to us that he does not really feel that quantum mechanics is in any immediate danger, but he does believe that we can learn by questioning it: If we find that quantum mechanics cannot be generalized any further, we may come to understand better why it works so well. If we find that the theory can be generalized in a plausible way, then we must ask why ordinary quantum mechanics is so nearly valid—and we may discover some hidden physics in the process.