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arXiv:2112.12487 · Physical Review A

Quantum parameter estimation of nonlinear coupling in a trilinear Hamiltonian with trapped ions

Peter A. Ivanov

Published Mar 28, 2022• 6 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physreva.105.032617

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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.
Abstract We propose a laser-free method for creation of a phonon two-mode squeezed state and a beam-splitter transformation, using time-varying electric fields and non-linear couplings between the normal modes in a linear ion crystal. Such non-linear Coulomb-mediated interactions between the collective vibrational modes arise under specific trap-frequency conditions in an ion trap. We study the quantum metrological capability for parameter estimation of the two quantum states and show that a Heisenberg limit of precision can be achieved when the initial state with n phonons evolves under the action of the beam-splitter transformation. Furthermore, we show that the phonon non-linearity and the spin-dependent force can be used for creation of a three-qubit Fredkin gate.
We propose an adiabatic method for optimal phonon temperature estimation using trapped ions which can be operated beyond the Lamb-Dicke regime.The quantum sensing technique relies on a time-dependent red-sideband transition of phonon modes, described by the nonlinear Jaynes-Cummings model in general.A unique feature of our sensing technique is that the relevant information of the phonon thermal distributions can be transferred to the collective spin-degree of freedom.We show that each of the thermal state probabilities is adiabatically mapped onto the respective collective spin-excitation configuration and thus the temperature estimation is carried out simply by performing a spin-dependent laser fluorescence measurement at the end of the adiabatic transition.We characterize the temperature uncertainty in terms of the Fisher information and show that the state projection measurement saturates the fundamental quantum Cramér-Rao bound for a quantum oscillator at thermal equilibrium.
We propose an adiabatic Ramsey interferometry technique for detecting weak nonlinearities with trapped ions. The method relies on using the quantum Rabi model as a probe, which is sensitive to nonlinear symmetry-breaking perturbations. We show that the couplings which arise either from anharmonic terms of the trapping potential or due to higher order terms in the Coulomb interaction expansion can be efficiently estimated by measuring the spin state probabilities alone. We show that the spin signal is amplified by the mean-phonon excitations, which results in the estimation precision reaching the super-Heisenberg limit. Notably, achieving such high-precision estimation does not require specific entangled state preparation and can be reached even for initial thermal motion state. Furthermore, we show that the super-Heisenberg scaling can be observed even in the presence of weak spin-dephasing.
We introduce a quantum sensing protocol for detecting the parameters characterizing the phase-space displacement by using a single trapped ion as a quantum probe. We show that, thanks to the laser-induced coupling between the ion's internal states and the motion mode, the estimation of the two conjugated parameters describing the displacement can be efficiently performed by a set of measurements of the atomic state populations. Furthermore, we introduce a three-parameter protocol capable of detecting the magnitude, the transverse direction, and the phase of the displacement. We characterize the uncertainty of the two- and three-parameter problems in terms of the Fisher information and show that state projective measurement saturates the fundamental quantum Cram\'er-Rao bound.
In traditional mechanics, harmonic oscillators can be used to measure force, acceleration, or rotation. Herein, we describe a quantum harmonic oscillator based on a Penning-trapped calcium ion crystal. Similar to traditional oscillators, the Coriolis force-induced axial oscillation amplitude is precisely measured to determine the input velocity. We show that the magnetron motion can be controlled through the rotating wall driving and treated as the driving oscillator. The Coriolis force couples with the magnetron motion and induces vibration in the axial direction or the z direction. The center of mass motion of the ion crystal in the axial direction could be precisely detected by the entanglement between the spins of the ions and the harmonic motion through lasers. The frequency of the magnetron motion needs to meet that of the axial motion under certain conditions, and, thus, the axial motion could be tuned to the resonance peak for maximum detection signal. We provided the parameter spaces for the matching of the magnetron frequencies with those of the axial frequencies. The measurement sensitivity was calculated in detail, and results show that a rotation angular velocity of 3.0×10−9rad/s/Hz could be achieved with 10 000 ions. Amplitude sensing could reach a sensitivity of 0.4 pm/Hz. With spin squeezing, the sensitivity could be further improved.
We introduce quantum sensing schemes for measuring very weak forces with a single trapped ion. They use the spin-motional coupling induced by the laser-ion interaction to transfer the relevant force information to the spin-degree of freedom. Therefore, the force estimation is carried out simply by observing the Ramsey-type oscillations of the ion spin states. Three quantum probes are considered, which are represented by systems obeying the Jaynes-Cummings, quantum Rabi (in 1D) and Jahn-Teller (in 2D) models. By using dynamical decoupling schemes in the Jaynes-Cummings and Jahn-Teller models, our force sensing protocols can be made robust to the spin dephasing caused by the thermal and magnetic field fluctuations. In the quantum-Rabi probe, the residual spin-phonon coupling vanishes, which makes this sensing protocol naturally robust to thermally-induced spin dephasing. We show that the proposed techniques can be used to sense the axial and transverse components of the force with a sensitivity beyond the range, i.e. in the (xennonewton, 10(-27)). The Jahn-Teller protocol, in particular, can be used to implement a two-channel vector spectrum analyzer for measuring ultra-low voltages.