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arXiv:quant-ph/0203111 · Physical Review A

Measuring the mean value of vibrational observables in trapped-ion systems

Benedetto Militello, ANTONINO MESSINA, Anna Napoli

Published Aug 5, 2002• 4 citations
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Abstract

The theoretical foundations of a general approach to the measurement problem of vibrational observables in trapped ion systems is reported. The method rests upon the introduction of a simple vibronic coupling structure appropriately conceived to link the internal ionic state measurement outcomes to the mean value of a motional variable of interest. Some applications are provided and discussed in detail, bringing to light the feasibility and the wide potentiality of the proposal.

DOI
10.1103/physreva.66.023402

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The theoretical foundations of a general approach to the measurement problem of vibrational observables in trapped ion systems is reported. The method rests upon the introduction of a simple vibronic coupling structure appropriately conceived to link the internal ionic state measurement outcomes to the mean value of a motional variable of interest. Some applications are provided and discussed in detail, bringing to light the feasibility and the wide potentiality of the proposal.
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We present a simple method that can be used to directly measure the squeezing of the quantum motional states of a trapped ion. Through the use of the interaction between the trapped ion and classical lasers, one can design a required coupling between the internal electronic and external vibrational degrees of freedom of the ion and can transfer information of the expectation value of a vibrational operator to the atomic internal populations. Thus measurement of squeezing on the quantum motional state can directly be realized. By adjusting the phases of the interacting lasers, one can measure the squeezing of both position and momentum quadratures.
The preparation of a wide class of entangled vibrational states involving two or three phononic modes of a three-dimensional trapped ion has been reported in the literature from both theoretical and experimental points of view. Here, the time evolution of such a system from an initial condition wherein two oscillatory modes (M x, M y) are reciprocally entangled and both are disentangled to the third mode (M z) is studied. By coupling one of the entangled oscillators (M x) with the third oscillator (M z), a correlation between the two uncoupled modes (M y, M z) is induced, well visible when the mean value of a suitable operator is considered. A method of measuring the expectation value of a vibrational observable is briefly sketched and then exploited in order to reveal such nonclassical behavior.
Vibrational degrees of freedom in trapped-ion systems have recently been gaining attention as a quantum resource, beyond the role as a mediator for entangling quantum operations on internal degrees of freedom, because of the large available Hilbert space. The vibrational modes can be represented as quantum harmonic oscillators and thus offer a Hilbert space with infinite dimensions. Here we review recent theoretical and experimental progress in the coherent manipulation of the vibrational modes, including bosonic encoding schemes in quantum information, reliable and efficient measurement techniques, and quantum operations that allow various quantum simulations and quantum computation algorithms. We describe experiments using the vibrational modes, including the preparation of non-classical states, molecular vibronic sampling, and applications in quantum thermodynamics. We finally discuss the potential prospects and challenges of trapped-ion vibrational-mode quantum information processing.
We propose a simplified scheme for measuring the entangled state of the motional and electronic degrees of freedom of an ion trapped in a one-dimensional harmonic potential. The scheme is based on the polychromatic excitation of the ion by three lasers tuned to the carrier, the first lower, and the upper vibrational sidebands, respectively. The method directly yields the elements of the Wigner characteristic function matrix in the electronic basis. The scheme can be easily generalized to the two-dimensional case.