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arXiv:1611.08700 · Physical Review Letters

NOON States of Nine Quantized Vibrations in Two Radial Modes of a Trapped Ion

Junhua Zhang, Mark Um, Dingshun Lv, Jing-Ning Zhang, L.-M. Duan, Kihwan Kim

Published Oct 18, 2018• 90 citations

Abstract

We develop a deterministic method to generate and verify arbitrarily high NOON states of quantized vibrations (phonons), through the coupling to the internal state. We experimentally create the entangled states up to N=9 phonons in two vibrational modes of a single trapped ^{171}Yb^{+} ion. We observe an increasing phase sensitivity of the generated NOON state as the number of phonons N increases and obtain the fidelity from the contrast of the phase interference and the population of the phonon states through the two-mode projective measurement, which are significantly above the classical bound. We also measure the quantum Fisher information of the generated state and observe Heisenberg scaling in the lower bounds of phase sensitivity as N increases. Our scheme is generic and applicable to other photonic or phononic systems such as circuit QED systems or nanomechanical oscillators, which have Jaynes-Cummings-type of interactions.

DOI: 10.1103/physrevlett.121.160502

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In the first part of this thesis, by using the two-photon Jaynes - Cummings Model (JCM),\nwe show how NOON states may be generated entangling two cavities, by passing atoms\nthrough them. The atoms interact with each cavity via two-photon resonant transitions.\nWe take advantage of the fact that depending on the state the atom enters (excited or\nground), it leaves or takes two photons per interaction and leaves the cavities in a pure\nstate.\nIn the second part of this thesis, we consider the Hamiltonian of a single ion vibrating in\ntwo dimensions interacting with a laser, we show how NOON states may be generated in\nion traps. We use the individual interaction of light with each of two vibrational modes of\nthe ion to entangle them. This allows us to generate NOON states with N = 8.

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.

Trapped atomic ions are a versatile platform for studying interactions between spins and bosons by coupling the internal states of the ions to their motion. Measurement of complex motional states with multiple modes is challenging, because all motional state populations can only be measured indirectly through the spin state of ions. Here we present a general method to determine the Fock state distributions and to reconstruct the density matrix of an arbitrary multimode motional state. We experimentally verify the method using different entangled states of multiple radial modes in a five-ion chain. This method can be extended to any system with Jaynes-Cummings-type interactions.

We propose a generation method of Bell-type states involving light and the vibrational motion of a single trapped ion. The trap itself is supposed to be placed inside a high-

Q

cavity sustaining a single mode, quantized electromagnetic field. Entangled light-motional states may be readily generated if a conditional measurement of the ion's internal electronic state is made after an appropriate interaction time and a suitable preparation of the initial state. We show that all four Bell states may be generated using different motional sidebands (either blue or red), as well as adequate ionic relative phases.

I propose an efficient method for measuring nonlinear coupling between the collective axial breathing mode and the radial rocking mode induced by the mutual Coulomb repulsion in linear ion crystal. The quantum sensing technique is based on the laser-induced coupling between one of the vibrational modes and the internal ion's spin states which allows to estimate the nonlinear coupling either by measuring the phonon probability distribution or by directly observing the Ramsey-type oscillations of the ion spin states. I show that due to the presence of nonlinear phonon coupling the off-resonance interaction between the ion spin states and the axial breathing mode leads to spin-dependent phonon squeezing of the radial rocking mode. Thus the nonlinear coupling can be estimated by measuring population distribution of the motional squeezed state. Furthermore, I show that the off-resonance interaction between the spin and the radial rocking mode creates a spin-dependent beam splitter operation between the two vibrational modes. Thus the parameter estimation can be carried out by detecting the ion spin populations. Finally, I show that the measurement uncertainty precision can reach the Heisenberg limit by using an entangled state between the two collective modes.