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Journal of Optics B Quantum and Semiclassical Optics

Generation of a C-NOT gate using a trapped ion

Miguel Orszag, Francisco Larrain

Published Nov 23, 2005• 5 citations
Publisher

Abstract

In this paper we propose the use of a single ion to generate a C-NOT gate by coupling two vibrational degrees of freedom coupled with a suitable chosen laser excitation via the internal ionic states. We concentrate on a two-dimensional ion trap with different frequencies in the X and Y directions and demonstrate how the C-NOT gate can be achieved.

DOI
10.1088/1464-4266/7/12/045

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In this paper we propose the use of a single ion to generate a C-NOT gate by coupling two vibrational degrees of freedom coupled with a suitable chosen laser excitation via the internal ionic states. We concentrate on a two-dimensional ion trap with different frequencies in the X and Y directions and demonstrate how the C-NOT gate can be achieved.
We describe how two vibrational degrees of freedom of a single trapped ion can be coupled through the action of suitably chosen laser excitation. We concentrate on a two-dimensional ion trap with dissimilar vibrational frequencies in the xxx and yyy directions of motion and derive from first principles a variety of quantized two-mode couplings, concentrating on a linear coupling that takes excitations from one mode to another. We demonstrate how this can result in a state rotation, in which it is possible to transfer the motional state of the ion from, say, the xxx direction to the yyy direction without prior knowledge of that motional state.
We show that, under certain conditions, the two-qubit controlled-NOT logic operations can be realized exactly using one laser pulse. The target qubit and the control qubit are encoded by the internal and external degrees of freedom of a single trapped ion. The multiquantum process of the interaction between the internal and external degrees of freedom of the ion is considered. This scheme neither requires the cooled ion in its vibrational ground state nor necessitates performance of the Lamb-Dicke approximation.
We present an alternative scheme to exactly implement one-qubit and two-qubit quantum gates with a single trapped cold ion driven by a travelling laser field. The internal degree of freedom of the ion acts as the target qubit and the control qubit is encoded by two Fock states of the external vibration of the ion. The conditions to realize these operations, including the duration of each applied laser pulse and Lamb-Dicke parameter, are derived. In our scheme neither the auxiliary atomic level nor the Lamb-Dicke approximation is required. The multiquantum transition between the internal and external degrees of freedom of the ion is considered.
We study the realization of quantum logic gates in the interacting system of a two-dimensional trapped ion and two laser beams. We simplify the Hamiltonian of the system by Choosing the frequency detuning of laser fields and the transition frequency between the internal states of the ion, and furthermore，we describe the realization of a C-NOT gate, a swap gate and a phase gate. During these processes, the system needs to meet the Lamb-Dicke limit, and the Rabi oscillation frequency should be far smaller than the phonon frequency.
We propose a scheme to implement two-qubit controlled quantum phase gate(CQPG) via a single trapped two-level ion located in the standing wave field of a quantum cavity, in which the trap works beyond the Lamb–Dicke limit. When the light field is resonant with the atomic transition |g > ↔ |e> of the ion located at the antinode of the standing wave, we can perform CQPG between the internal and external states of the trapped ion; while the frequency of the light field is chosen to be resonant with the first red sideband of the collective vibrational mode of the ion located at the node of the standing wave, we can perform CQPG between the cavity mode and the collective vibrational mode of the trapped ion. Neither the Lamb–Dicke approximation nor the assistant classical laser is needed. Also we can generate a GHZ state if assisted with a classical laser.