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Journal of Modern Optics

Effect of an external field on decoherence: Part II

Jian‐Min Zuo, Robert F. O’Connell

Published Apr 1, 2004• 7 citations
Publisher

Abstract

A previous paper analysed the experimental results of Myatt et al. [Nature, 403, 269 (2000)] on the decoherence of superposition states of Be ions confined in a harmonic potential by a linear Paul trap and subject to a random external electric field. Based on a rigorous derivation in the absence of an external force which had previously been obtained, a heuristic approach to the problem was presented. Whereas this approach was simple and physically appealing, a more rigorous analysis is clearly desirable, which is the content of this paper. It also makes clear that, for the case of a random force, decoherence is intrinsic to quantum mechanics and does not require a dissipative environment.

DOI
10.1080/09500340408233599

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A previous paper analysed the experimental results of Myatt et al. [Nature, 403, 269 (2000)] on the decoherence of superposition states of Be ions confined in a harmonic potential by a linear Paul trap and subject to a random external electric field. Based on a rigorous derivation in the absence of an external force which had previously been obtained, a heuristic approach to the problem was presented. Whereas this approach was simple and physically appealing, a more rigorous analysis is clearly desirable, which is the content of this paper. It also makes clear that, for the case of a random force, decoherence is intrinsic to quantum mechanics and does not require a dissipative environment.
``Decoherence of quantum superpositions through coupling to engineered reservoirs'' is the topic of a recent paper by Myatt et al. [Nature 403, 269 (2000)] which has attracted much interest because of its relevance to current research in fundamental quantum theory, quantum computation, teleportation, entanglement, and the quantum-classical interface. However, the preponderance of theoretical work on decoherence does not consider the effect of an external field. Here, we present an analysis of such an effect in the case of the random $\ensuremath{\delta}$-correlated force discussed by Myatt et al.
Decoherence is the suppression of the interference of quantum states. It plays important roles in the study of the quantum-classical transition. It is also the major obstacle in the implementation of various schemes of quantum computers. Decoherence can be affected by many factors. The interactions between quantum systems of interest and dissipative environments cause the extensively studied environment-induced-decoherence. Decoherence can also occur as the result of a temperature effect. A recent experiment investigated the engineering of decoherence, which involves applying an external field to the quantum system. In this thesis, we study the effect of an external field on decoherence in the case of a harmonic oscillator coupled to a heat bath by calculating its contribution to the attenuation coefficient, which is a measure of decoherence in coordinate space and involves directly observable probability distributions. It is found that, while non-random external force does not result in decoherence, a random external force can lead to intrinsic decoherence that does not require a dissipative environment. The attenuation coefficient for a free harmonic oscillator in a dissipative environment is also calculated using the solution of the initial value quantum Langevin equation. One of the key results obtained is that ”decoherence without dissipation” does not occur in the case of a free harmonic oscillator, in contrast to the case of a free particle.
We consider two different kinds of fluctuations in an ion trap potential: external fluctuating electrical fields, which cause statistical movement (``wobbling'') of the ion relative to the center of the trap, and fluctuations of the spring constant, which are due to fluctuations of the ac component of the potential applied in the Paul trap for ions. We write down master equations for both cases and, averaging out the noise, obtain expressions for the heating of the ion. We compare our results to previous results for far-off resonance optical traps and heating in ion traps. The effect of fluctuating external electrical fields for a quantum gate operation (controlled-NOT) is determined and the fidelity for that operation derived.
For an ion in a Paul trap, the effect of the micromotion on the heating by stray electric fields is studied analytically. A sequence of unitary transformations, set up from the solutions to the classical dynamics, leads to the exact quantum time propagator for each realization of the random classical field; subsequently, a statistical average is performed to obtain the fidelity of the motional ground state. In this nonperturbative approach, the role of the micromotion in the depopulation is understood as an effective change in the time dependence of the external field and an intrinsic modulation of the heating rate; it is shown that the consequent enhanced complexity of the dynamics can result in a reduction of the heating time.
Open system dynamics of an electron is studied in the presence of radiation field, confined between two parallel conducting pates. It has been suggested in previous works that the quantized zero-point modes of this field lead to finite decoherence effects, possibly due to the Casimir force. However, in this work it is shown that the decoherence found in previous works is due to the sudden switching on of the system-environment interaction and due to the acceleration of the electron enforced by the background paths whose superposition was analyzed. The work discusses important theoretical aspects of the setup and shows that, while coherence might be lost due to bremsstrahlung induced by an external or the image potential, it cannot be lost due to the mere presence of the quantum vacuum fluctuations between the plates.