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CONTINUOUS VS DISCRETE PROCESSES: THE PROBABILISTIC EVOLUTION OF SINGLE TRAPPED IONS

Storrs McCall, Andrew Charles Whitaker, Glyn George

Published Sep 1, 2000• 2 citations
OpenAlex

Abstract

The evolution of a single trapped ion exhibiting intermittent fluorescence and dark periods may be described either as a continuous process, using differential rate equations, or discretely, as a Markov process. The latter models the atom as making instantaneous transitions from one energy eigenstate to another, and is open to the objection that superpositions of energy states will form which are not covered by the Markov process. The superposition objection is replied to, and two new mathematical elements, Markov vectors and Markov matrices, are proposed as additions to quantum theory. The paper concludes by attributing the cause of dark periods in the ion's history to instantaneous transitions in the ion itself, rather than to photon detection or other components of measurement.

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The evolution of a single trapped ion exhibiting intermittent fluorescence and dark periods may be described either as a continuous process, using differential rate equations, or discretely, as a Markov process. The latter models the atom as making instantaneous transitions from one energy eigenstate to another, and is open to the objection that superpositions of energy states will form which are not covered by the Markov process. The superposition objection is replied to, and two new mathematical elements, Markov vectors and Markov matrices, are proposed as additions to quantum theory. The paper concludes by attributing the cause of dark periods in the ion's history to instantaneous transitions in the ion itself, rather than to photon detection or other components of measurement.
We use a recent stochastical diffusion model of quantum evolution to represent the evolution of a three-level quantum system undergoing quantum jumps. This is possible because the continuous change in the quantum state in this diffusion model is so rapid that it appears to be instantaneous in comparison with the time between transitions. Experimental data from a study of the intermittent fluorescence of a single trapped 24Mg+ ion and equivalent theoretical data are shown to be strikingly similar. Statistical comparisons of the data are also made.
Abstract When a quantum state is subjected to a measurement and the state is not an eigenstate of the dynamical variable being measured, the outcome is unpredictable. Only the probabilities of the various possible outcomes are predicted by theory. This phenomenon is sometimes discussed in terms of “wavefunction collapse.” Trapped ions can be used for real, as opposed to gedanken demonstrations of this basic process. In the experiments described here, a single ion, or a few identical ions, were prepared in well defined superpositions of two internal energy eigenstates. The populations of the energy levels were then measured. When the state amplitudes were equal, the population fluctuations were greater than when one of the amplitudes was nearly zero, in agreement with the predictions of quantum mechanics. In other experiments, such as those with atomic beams, the number of atoms under observation fluctuates, and this obscures the fluctuations from other sources. However, if the number of atoms is small and constant, the fundamental quantum mechanical fluctuations can be observed.
The apparent dichotomy between quantum jumps on the one hand, and continuous time evolution according to wave equations on the other hand, provided a challenge to Bohr's proposal of quantum jumps in atoms. Furthermore, Schrödinger's time-dependent equation also seemed to require a modification of the explanation for the origin of line spectra due to the apparent possibility of superpositions of energy eigenstates for different energy levels. Indeed, Schrödinger himself proposed a quantum beat mechanism for the generation of discrete line spectra from superpositions of eigenstates with different energies. However, these issues between old quantum theory and Schrödinger's wave mechanics were correctly resolved only after the development and full implementation of photon quantization. The second quantized scattering matrix formalism reconciles quantum jumps with continuous time evolution through the identification of quantum jumps with transitions between different sectors of Fock space. The continuous evolution of quantum states is then recognized as a sum over continually evolving jump amplitudes between different sectors in Fock space. In today's terminology, this suggests that linear combinations of scattering matrix elements are epistemic sums over ontic states. Insights from the resolution of the dichotomy between quantum jumps and continuous time evolution therefore hold important lessons for modern research both on interpretations of quantum mechanics and on the foundations of quantum computing. They demonstrate that discussions of interpretations of quantum theory necessarily need to take into account field quantization. They also demonstrate the limitations of the role of wave equations in quantum theory, and caution us that superpositions of quantum states for the formation of qubits may be more limited than usually expected.
The evolution of the discrete Wigner function is formally similar to a probabilistic process, but the transition probabilities, like the discrete Wigner function itself, can be negative. We investigate these transition probabilities, as well as the transition rates for a continuous process, aiming particularly to give simple criteria for deciding when a set of such quantities corresponds to a legitimate quantum process. We also show how the transition rates for any Hamiltonian evolution can be worked out by expanding the Hamiltonian as a linear combination of displacement operators in the discrete phase space.