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arXiv:0909.0046 · Physical Review A

Preparation of Dicke states in an ion chain

David B. Hume, C. W. Chou, Till Rosenband, David J. Wineland

Published Nov 2, 2009• 80 citations
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Abstract

We have investigated theoretically and experimentally a method for preparing Dicke states in trapped atomic ions. We consider a linear chain of $N$ ion qubits that is prepared in a particular Fock state of motion $|m⟩$. The $m$ phonons are removed by applying a laser pulse globally to the $N$ qubits and converting the motional excitation to $m$ flipped spins. The global nature of this pulse ensures that the $m$ flipped spins are shared by all the target ions in a state that is a close approximation to the Dicke state $|{D}_{N}^{(m)}⟩$. We calculate numerically the fidelity limits of the protocol and find small deviations from the ideal state for $m=1$ and $m=2$. We have demonstrated the basic features of this protocol by preparing the Bell state $|{D}_{2}^{(1)}⟩$ in two ${^{25}\text{M}\text{g}}^{+}$ target ions trapped simultaneously with an ${^{27}\text{A}\text{l}}^{+}$ ancillary ion.

DOI
10.1103/physreva.80.052302

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We have investigated theoretically and experimentally a method for preparing Dicke states in trapped atomic ions. We consider a linear chain of NNN ion qubits that is prepared in a particular Fock state of motion ∣m⟩|m⟩∣m⟩. The mmm phonons are removed by applying a laser pulse globally to the NNN qubits and converting the motional excitation to mmm flipped spins. The global nature of this pulse ensures that the mmm flipped spins are shared by all the target ions in a state that is a close approximation to the Dicke state ∣DN(m)⟩|{D}_{N}^{(m)}⟩∣DN(m)​⟩. We calculate numerically the fidelity limits of the protocol and find small deviations from the ideal state for m=1m=1m=1 and m=2m=2m=2. We have demonstrated the basic features of this protocol by preparing the Bell state ∣D2(1)⟩|{D}_{2}^{(1)}⟩∣D2(1)​⟩ in two 25Mg+{^{25}\text{M}\text{g}}^{+}25Mg+ target ions trapped simultaneously with an 27Al+{^{27}\text{A}\text{l}}^{+}27Al+ ancillary ion.
The quantum states of trapped atomic ions can be highly isolated from external perturbation, and precisely manipulated with applied laser fields. This makes them an excellent medium for quantum-limited experiments such as quantum information processing and precision spectroscopy. A relatively small number of ion species have been used for these types of experiments because most species are difficult to laser cool and detect directly. This thesis demonstrates a way to overcome this limitation by use of sympathetic cooling and state detection based on quantum logic. We apply these techniques to mixed-species arrays of Al+ and Be+ ions. A mathematical model for the motion of such a two-species array is presented in order to explore some features of the ion dynamics that are relevant for the experiments. Repetitive quantum nondemolition measurements of the electronic states of a Al+ ion show detection fidelities as high as 99.94 %. We also demonstrate the simultaneous detection of two Al+ ions and observe a detection fidelity of 99.8 %. The basic ideas behind the detection strategy are extended to potentially enable similar experiments on a more general class of atomic and molecular ions.\nWe have also investigated, theoretically and experimentally, a method for preparing entangled Dicke states in trapped atomic ions. We consider a linear chain of N ion qubits that is prepared in a particular Fock state of motion. The m phonons are removed by applying a laser pulse globally to the N qubits, and converting the motional excitation to m flipped spins. The global nature of this pulse ensures that the m flipped spins are shared by all the target ions in a state that is a close approximation to the Dicke state.\nWe calculate numerically the fidelity limits of the protocol and find small deviations from the ideal state for m = 1 and m = 2. We have demonstrated the basic features of this protocol by preparing the Bell state in two Mg+ target ions trapped simultaneously with an Al+ ancillary ion.
We propose a simple technique for the generation of arbitrary-sized Dicke states in a chain of trapped ions. The method uses global addressing of the entire chain by two pairs of delayed but partially overlapping laser pulses to engineer a collective adiabatic passage along a multi-ion dark state. Our technique, which is a many-particle generalization of the stimulated Raman adiabatic passage technique, is decoherence free with respect to spontaneous emission and robust against moderate fluctuations in the experimental parameters. Furthermore, because the process is very rapid, the effects of heating are almost negligible under realistic experimental conditions. We predict that the overall fidelity of synthesis of a Dicke state involving ten ions sharing two excitations should approach 98% with currently achievable experimental parameters.
We propose a fast and efficient technique to create classes of highly entangled states of trapped ions, such as arbitrary Dicke states and superpositions of them, e.g. NOON states. The ions are initialized in the phonon ground state and are addressed globally with a composite pulse that is resonant with the first motional sideband. The technique is fairly robust to parameter fluctuations and operates on comparatively short time scales, as resonant interactions allow one to use the minimum laser pulse area. The number of single pulses from the composite sequence is equal to the number of ions; thus the implementation complexity grows only linearly with the size of the system. The approach does not require individual addressing of the ions in the trap and can be applied both inside and outside the Lamb–Dicke regime.
We propose an implementation of the Grover's search algorithm in a nonclassical database using a linear chain of trapped ions. The database comprises all collective states with two ionic excitations; hence the number of database entries scales quadratically with the number of ions NNN. The system is initialized in an even superposition of all register states, i.e., in the symmetric Dicke state with NNN ions sharing two excitations. The reflection-about-the-mean operator is produced merely by global addressing of the ion string by an off-resonant pulse with a suitable area, whereas the oracle operator is a control-phase gate. This simplification should allow a demonstration of quantum search in a database of hundreds of elements without needing to synthesize multiqubit quantum gates. The technique does not rely on coherent addition or subtraction of phonons, and hence the ion chain can be sympathetically cooled throughout the process.
A simple and scalable scheme is proposed to generate a n-qubit W state in a trapped-ion system without the Lamb–Dicke limit. The n-qubit W state can be generated by the interaction between the ions and the laser field if the collective mode is initially prepared in the single-phonon state and each ion is in the ground state. The scheme only requires a single laser and avoids laser manipulation of the individual ion. The time required to complete the process decreases with the number of ions. The present scheme is not limited to small values of the LD parameter, which greatly enhances operation speeds.