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arXiv:quant-ph/9812044 · Physical Review A

Decoherence and fidelity in ion traps with fluctuating trap parameters

Sara Schneider, Gerard J Milburn

Published May 1, 1999• 74 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.59.3766

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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.

Journal of Modern Optics•Apr 1, 2004•7 citations
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.
This thesis is concerned with various aspects of ion traps and their use as a quantum simulation and computation device. In its first part we investigate various sources of noise and decoherence in ion traps. As quantum information is very fragile, a detailed knowledge of noise and decoherence sources in a quantum computation device is essential. In the special case of an ion trap quantum computer we investigate the effects of intensity and phase noise in the laser, which is used to perform the gate operations. We then look at other sources of noise which are present without a laser being switched on. These are fluctuations in the trapping frequency caused by noise in the electric potentials applied to the trap and fluctuating electrical fields which will cause heating of the centre-of-mass vibrational state of the ions in the trap. For the case of fluctuating electrical fields we estimate the effect on a quantum gate operation. We then propose a scheme for performing quantum gates without having the ions cooled down to their motional ground state. The second part deals with various aspects of the use of ion traps as a device for quantum computation. We start with the use of ionic qubits as a measurement device for the centre-of-mass vibrational mode and investigate in detail the effect these measurements will have on the vibrational mode. If one wants to use quantum computation devices as systems to simulate quantum mechanics, it is of interest to know how to simulate say a k-level system with N qubits. We investigate the easiest case of this wider problem and look at how to simulate a three-level system (a so called trit) with two qubits in an ion trap quantum computer. We show how to get and measure a SU (3) geometric phase with this toy model. Finally we investigate how to simulate collective angular momentum models with a string of qubits in an ion trap. We assume that the ionic qubits are coupled to a thermal reservoir and derive a master equation for this case. We investigate the semiclassical limit of this master equation and, in the case for two qubits in the trap, determine the entanglement of the steady state. We also outline a way to find the steady state for the master equation using coherence vectors.
Summary form only given.To model the decoherence in ion traps resulting from laser intensity and phase fluctuations, we assume that the stochastic processes involved are white noise processes, which enables us to give a simple master equation description of this source of decoherence. We discuss the effects those fluctuations have on gate operations in ion-trap quantum computation and briefly address possible detection and error correction mechanisms. The master equation is then averaged over the noise and is sufficient to describe the results of experiments that probe the oscillations in the electronic populations as energy is exchanged between the internal and electronic motion. We get solutions for the probability of the ion to be in the ground state for both intensity and phase fluctuations. Our results predict that the decoherence rate resulting from intensity fluctuations will depend on the vibrational quantum number in different ways depending on which vibrational excitation sideband is used. So the dependence of the decoherence rate for the second red sideband depends quadratically on the vibrational energy state |n> the ion is prepared in, whereas for the first sideband, this effect is linear in n. Due to this result, the second red (or any higher-order) sideband is a better choice for testing the dependence of decoherence on the excitation in the vibrational state experimentally than just the first-order sideband. Because those fluctuations in the exciting laser pulse are not the only source of decoherence, we briefly discuss fluctuations in the trap potential itself. These fluctuations lead to a wobbling trap center point, and we discuss the effects due to them.
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 have investigated motional heating of laser-cooled 9Be+{}^{9}{\mathrm{Be}}^{+}9Be+ ions held in radio-frequency (Paul) traps. We have measured heating rates in a variety of traps with different geometries, electrode materials, and characteristic sizes. The results show that heating is due to electric-field noise from the trap electrodes that exerts a stochastic fluctuating force on the ion. The scaling of the heating rate with trap size is much stronger than that expected from a spatially uniform noise source on the electrodes (such as Johnson noise from external circuits), indicating that a microscopic uncorrelated noise source on the electrodes (such as fluctuating patch-potential fields) is a more likely candidate for the source of heating.
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.
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.