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Physical Review Letters

Precision Tests of Quantum Mechanics

Steven Weinberg

Published Jan 30, 1989• 382 citations
Publisher

Abstract

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 $^{9}\mathrm{Be}^{+}$ 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 $^{9}\mathrm{Be}$ nucleus that could be due to nonlinear corrections to quantum mechanics, with good prospects of improving this by 2-3 orders of magnitude.

DOI
10.1103/physrevlett.62.485

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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{10}^{\ensuremath{-}21}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.
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{}${10}^{\mathrm{\ensuremath{-}}27}$ 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
Quantum mechanics requires the time evolution of the wave function to be linear. While this feature has been associated with the preservation of causality, a consistent causal nonlinear theory was recently developed. Interestingly, this theory is unavoidably sensitive to the full physical spread of the wave function, rendering existing experimental tests for nonlinearities inapplicable. Here, using well-controlled motional superpositions of a trapped ion, we set a stringent limit of 5.4×10^{-12} on the magnitude of the unitless scaling factor ε[over ˜]_{γ} for the predicted causal nonlinear perturbation.
There are strong arguments that quantum mechanics may be nonlinear in its dynamics. A discovery of nonlinearity would hint at a novel understanding of the interplay between gravity and quantum field theory, for example. As such, experiments searching for potential nonlinear effects in the electromagnetic sector are important. Here we outline such an experiment, consisting of a stream of random bits (which were generated using Rigetti's Aspen-M-3 chip) as input to an RF signal generator coupled to a cryogenic detector. Projective measurements of the qubit state, which is originally prepared in an equal superposition, serve as the random binary output of a signal generator. Thereafter, spectral analysis of the RF detector would yield a detectable excess signal predicted to arise from such a nonlinear effect. A comparison between the projective measurements of the quantum bits vs the classical baseline showed no power excess. This sets a new limit on the electromagnetic nonlinearity parameter ∣ε∣⪅1.15×10−12|ε| \lessapprox 1.15 \times 10^{-12}∣ε∣⪅1.15×10−12, at a 90.0% confidence level. This is the most stringent limit on nonlinear quantum mechanics thus far and an improvement by nearly a factor of 50 over the previous experimental limit.
Inflation predicts that quantum fluctuations determine the large scale structure of the Universe. This raises the striking possibility that quantum mechanics, developed to describe nature at short distances, can be tested by studying nature at its most immense -- cosmology. In this note, we realize this possibility. A nonlinear generalization of quantum mechanics modifies predictions for the cosmological power spectrum. Observational cosmology is sufficiently precise to place a stringent limit, b<3×10−34b< 3\times 10^{-34}b<3×10−34 eV, on the size of the nonlinear term.
We analyze the free-space propagation of matter waves with a view to placing an upper limit on the strength of possible nonlinear terms in the Schr\"odinger equation. Such additional terms of the form $\ensuremath{\psi}F({|\ensuremath{\psi}|}^{2})$ were introduced by Bialynicki-Birula and Mycielski in order to counteract the spreading of wave packets, thereby allowing solutions which behave macroscopically like classical particles. For the particularly interesting case of a logarithmic nonlinearity of the form $F=\ensuremath{-}\mathit{b}\mathrm{ln}{|\ensuremath{\psi}|}^{2}$, we find that the free-space propagation of slow neutrons places a very stringent upper limit on the magnitude of b\mathit{b}b. Precise measurements of Fresnel diffraction with slow neutrons do not give any evidence for nonlinear effects and allow us to deduce an upper limit for $\mathit{b}<3.3\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}15}$ eV about 3 orders of magnitude smaller than the lower bound proposed by the above authors.