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arXiv:1807.07535 · Physical Review A

Interferometer with a driven trapped ion

S. Martínez-Garaot, Alvaro Rodriguez-Prieto, Juan Gonzalo Muga

Published Oct 18, 2018• 12 citations
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Abstract

We propose an interferometric measurement of weak forces using a single ion subjected to designed time-dependent spin-dependent forces. Explicit expressions of the relation between the unknown force and the final populations are found considering different scenarios, such as the weak force acting on two branches, acting on one branch, or having errors in the driving force. The flexibility to design the trap trajectories is used to minimize errors due to anharmonicities in the trap. The advantages of the approach are the use of geometrical phases, which provides stability, the possibility to design faster-than-adiabatic processes with sensitivity control, and the independence of the results on the motional states for the small-oscillations regime in which the effective potentials are purely harmonic.

DOI
10.1103/physreva.98.043622

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We propose an interferometric measurement of weak forces using a single ion subjected to designed time-dependent spin-dependent forces. Explicit expressions of the relation between the unknown force and the final populations are found considering different scenarios, such as the weak force acting on two branches, acting on one branch, or having errors in the driving force. The flexibility to design the trap trajectories is used to minimize errors due to anharmonicities in the trap. The advantages of the approach are the use of geometrical phases, which provides stability, the possibility to design faster-than-adiabatic processes with sensitivity control, and the independence of the results on the motional states for the small-oscillations regime in which the effective potentials are purely harmonic.
We reviewand update schemes for different measurements using STA-mediated guided interferometry with a single trapped particle. STA stands for "shortcuts to adiabaticity", a set of techniques to achieve the results of adiabatic dynamics in shorter times. In the first scheme we presented a protocol aimed at detecting weak unknown forces. It consisted of a single ion trapped in a harmonic potential and driven by time-and-spin-dependent forces generated via off-resonant lasers. Our approach provided stability and the independence of the results on the motional states for the small-oscillations regime. We could, also, design faster-than-adiabatic processes with sensitivity control. However, it required a rotation of the trapping potential at the moment the experiment starts. A much more practical and broadly applicable design was then developed, where no rotation is involved. Here, a single atom is driven by two moving spin-dependent trapping potentials where we guide the arms of the interferometer via shortcuts to adiabatic paths. In this paper, in addition to a brief review of these two previous proposals, we revisit the first scheme and present a new protocol where the spin-dependent driving force is generated via a "shaken" optical lattice. This opens the possibility for additional interferometric measurements beyond an unknown force, for example, the mass of the trapped ion, while still preserving the advantages of the previously proposed method.
We propose a compact atom interferometer to measure homogeneous constant forces guiding the arms via shortcuts to adiabatic paths.For a given sensitivity, which only depends on the space-time area of the guiding paths, the cycle time can be made fast without losing visibility.The atom is driven by spin-dependent trapping potentials moving in opposite directions, complemented by linear and time-dependent potentials that compensate the trap acceleration.Thus, the arm states are adiabatic in the moving frames and nonadiabatic in the laboratory frame.The trapping potentials may be anharmonic, e.g., optical lattices, and the interferometric phase does not depend on the initial motional state or on the pivot point for swaying the linear potentials.
Trapped ions are sensitive detectors of weak forces and electric fields that excite ion motion. Here measurements of the center-of-mass motion of a trapped-ion crystal that are phase coherent with an applied weak external force are reported. These experiments are conducted far from the trap motional frequency on a two-dimensional trapped-ion crystal of approximately 100 ions, and determine the fundamental measurement imprecision of our protocol free from noise associated with the center-of-mass mode. The driven sinusoidal displacement of the crystal is detected by coupling the ion crystal motion to the internal spin degree of freedom of the ions using an oscillating spin-dependent optical dipole force. The resulting induced spin precession is proportional to the displacement amplitude of the crystal, and is measured with near-projection-noise-limited resolution. A 49pm49\phantom{\rule{4pt}{0ex}}\mathrm{pm}49pm displacement is detected with a signal-to-noise ratio of 1 in a single experimental determination, which is an order-of-magnitude improvement over prior phase-incoherent experiments. This displacement amplitude is 40 times smaller than the zero-point fluctuations. With our repetition rate, an 8.4pm/Hz8.4\phantom{\rule{4pt}{0ex}}\mathrm{pm}/\sqrt{\mathrm{Hz}}8.4pm/Hz​ displacement sensitivity is achieved, which implies 12(yN/ion)/Hz12\phantom{\rule{0.16em}{0ex}}(\mathrm{yN}/\mathrm{ion})/\sqrt{\mathrm{Hz}}12(yN/ion)/Hz​ and $77\phantom{\rule{0.16em}{0ex}}(\ensuremath{\mu}\mathrm{V}/\mathrm{m})/\sqrt{\mathrm{Hz}}$ sensitivities to forces and electric fields, respectively. This displacement sensitivity, when applied on-resonance with the center-of-mass mode, indicates the possibility of weak force and electric field detection below ${10}^{\ensuremath{-}3}\phantom{\rule{4pt}{0ex}}\mathrm{yN}$/ion and 1nV1\phantom{\rule{4pt}{0ex}}\mathrm{nV}1nV/m, respectively.
We report a calibrated measurement of 174 yoctonewton (1× 10−24 N = 1 yN), using a cloud of 60 Be ion confined in a Penning ion trap. These measurements suggest that ion traps may form the basis of a new class of ultra-sensitive deployable force sensors. 1. Summary Measurement of extremely small forces is of interest in a variety of fields ranging from atomic-force-microscopy, to electron spin resonance, to tests of fundamental principles. State-of-the-art small force detection relies on using thin cantilevers of which the resonant vibrational response is probed in reaction to an external force. Such systems have achieved force sensitivities on the order of attonewtons [1]. It has been suggested that single trapped ions could achieve force sensitivities on the order of ∼ 1 yn and consequently be used as a new class of force sensor [2]. Here we demonstrate a calibrated, yoctonewton scale force measurement using 60 Be ions confined in a Penning ion trap. The ions are subjected to a small oscillating electric field and their resonant response is probed using phase sensitive detection of modulations in the fluorescence intensity of a probe beam near-resonant with the S1/2 −→ P3/2 transition. We detect forces as small as 174 yn corresponding to a force sensitivity of 390±150 yn/ √ Hz. Extrapolated sensitivities are consistent with ∼ 1.7 yn/√Hz in the single ion limit [3]. Figure 1: Resonant response of Be ions under the influence of weak, oscillating electric fields. Left-most column Experiment, modulations in fluorescence intensity as a function of driving force simulation for different drive times (indicated in right-most column). Central column Theoretical simulation corresponding to left column. Right-most column cross sections through data in left-most column with theoretical fits.
We experimentally implement circuits of one- and two-mode operations on two motional modes of a single trapped ion. This is achieved by implementing the required displacement, squeezing, two-mode squeezing, and beam-splitter operations using oscillating electric potentials applied to the trap electrodes. The resulting electric fields drive the modes resonantly or parametrically without the need for optical forces. As a demonstration, we implement SU(2) and SU(1,1) interferometers with phase sensitivities near the Cram\'er-Rao bound. We report a maximum sensitivity of a SU(2) interferometer within 0.67(5) dB of the standard quantum limit (SQL) as well as a single- and two-mode SU(1,1) sensitivity of 5.9(2) dB and 4.5(2) dB below the SQL, respectively.
Abstract We propose a scheme for detecting time-varying weak forces using a quantum probe consisting of a single spin and quantum oscillator under the effect of collective dissipation. We study the force estimation in the steady-state regime where the information of the force is extracted by measuring observable of the oscillator such as quadrature and mean phonon excitation. We quantify the force sensitivity in terms of quantum Fisher information and show that it diverges approaching the critical spin-boson coupling making the system sensitive to very small force perturbation. We show that close to the critical coupling the measurement of the oscillator quadrature is optimal in a sense that saturates the fundamental Cramer–Rao bound. Furthermore, we study the force estimation in the presence of phonon squeezing and show that it can significantly improve the sensitivity reaching minimal detectable force of order of xN (1xN = 10 −27 N).