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arXiv:quant-ph/9804048 · Physical Review Letters

Theory of Heating of the Quantum Ground State of Trapped Ions

Daniel F. V. James

Published Jul 13, 1998• 93 citations
arXivHTMLPDF

Abstract

Using a displacement operator formalism, I analyze the depopulation of the vibrational ground state of trapped ions. Two heating times, one characterizing short time behavior and the other long time behavior, are found. The long time behavior is analyzed for both single and multiple ions, and a formula for the relative heating rates of different modes is derived. The possibility of correction of heating via the quantum Zeno effect, and the exploitation of the suppression of heating of higher modes to reduce errors in quantum computation, is considered.

DOI
10.1103/physrevlett.81.317

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Using a displacement operator formalism, I analyze the depopulation of the vibrational ground state of trapped ions. Two heating times, one characterizing short time behavior and the other long time behavior, are found. The long time behavior is analyzed for both single and multiple ions, and a formula for the relative heating rates of different modes is derived. The possibility of correction of heating via the quantum Zeno effect, and the exploitation of the suppression of heating of higher modes to reduce errors in quantum computation, is considered.
The rate at which external random thermal fluctuations drive transitions between, and cause decoherence of, the near ground vibrational (harmonic oscillator) quantum states of trapped cold ions is of crucial importance in relation to quantum computing. Presented here is an estimate of these rates, for a single trapped ion, based on a quantum perturbation approach where an external thermal energy reservoir is electrically coupled to the ion. The results are for a general system, and it is shown that the relevant parameter in the interpretation of experimentally observed heating rates is the correlation time of the fluctuations. The rates due to the fluctuating electric field associated with blackbody radiation are also considered and shown to be negligible.
We study the application of decoupling techniques to the case of a damped vibrational mode of a chain of trapped ions, which can be used as a quantum bus in linear ion trap quantum computers. We show that vibrational heating could be efficiently suppressed using appropriate ``parity kicks.'' We also show that vibrational decoherence can be suppressed by this decoupling procedure, even though this is generally more difficult because the rate at which the parity kicks have to applied increases with the effective bath temperature.
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.
In this article I review some results obtained during my PhD work in the group of Professor Messina, at the University of Palermo. I discuss some proposals aimed at exploring fundamental issues of quantum theory, e.g. entanglement and quantum superpositions, in the context of single trapped ions. This physical context turns out to be extremely well suited both for studying fundamental features of quantum mechanics, such as the quantum–classical border, and for technological applications such as quantum logic gates and quantum registers. I focus on some procedures for engineering nonclassical states of the vibrational motion of the centre of mass of the ion. I consider both the case in which the ion interacts with classical laser beams and the case of interaction with a quantized mode of light. In particular, I discuss the generation of Schrodinger cat-like states, Bell states and Greenberger–Horn–Zeilinger states. The schemes for generating nonclassical states stem from two different quantum processes: the parity effect and the quantum state manipulation via quantum non-demolition measurement. Finally, I consider a microscopic theory of the interaction of a quantum harmonic oscillator (the centre of mass of the ion in the trapped ion context) with a bosonic thermal environment. Using an exact approach to the dynamics, I discuss a quantum theory of heating of trapped ions able to describe both the short time non-Markovian regime and the thermalization process. I conclude showing briefly how the trapped ion systems can be used as simulators of key models of open quantum systems such as the Caldeira–Leggett model.
We study the application of decoupling techniques to the case of a damped vibrational mode of a chain of trapped ions, which can be used as a quantum bus in linear ion trap quantum computers. We show that vibrational heating could be efficiently suppressed using appropriate 'parity kicks'. We also show that vibrational decoherence can be suppressed by this decoupling procedure, even though this is generally more difficult because the rate at which the parity kicks have to applied increases with the effective bath temperature.