Toward Heisenberg-Limited Rabi Spectroscopy

Ravid Shaniv, Tom Manovitz, Yotam Shapira, Nitzan Akerman, and Roee Ozeri

Phys. Rev. Lett. 120, 243603 – Published 14 June 2018

Abstract

The use of entangled states was shown to improve the fundamental limits of spectroscopy to beyond the standard-quantum limit. Here, rather than probing the free evolution of the phase of an entangled state with respect to a local oscillator, we probe the evolution of an initially separable two-atom register under an Ising spin Hamiltonian with a transverse field. The resulting correlated spin-rotation spectrum is twice as narrow as that of an uncorrelated rotation. We implement this ideally Heisenberg-limited Rabi spectroscopy scheme on the optical-clock electric-quadrupole transition of ${}^{88}{Sr}^{+}$ using a two-ion crystal. We further show that depending on the initial state, correlated rotation can occur in two orthogonal subspaces of the full Hilbert space, yielding entanglement-enhanced spectroscopy of either the average transition frequency of the two ions or their difference from the mean frequency. The use of correlated spin rotations can potentially lead to new paths for clock stability improvement.

Received 29 August 2017
Revised 25 December 2017

DOI:https://doi.org/10.1103/PhysRevLett.120.243603

Physics Subject Headings (PhySH)

Coherent control Quantum entanglement Quantum gates Quantum measurements Quantum metrology Quantum simulation

Trapped ions

Spectroscopy

Quantum Information, Science & TechnologyAtomic, Molecular & OpticalGeneral Physics

Authors & Affiliations

Ravid Shaniv, Tom Manovitz, Yotam Shapira, Nitzan Akerman, and Roee Ozeri

Department of Physics of Complex Systems, Weizmann Institute of Science, Rehovot 7610001, Israel

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Vol. 120, Iss. 24 — 15 June 2018

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Images

Figure 1
Coupling in the two metrological subspaces using Mølmer-Sørensen interaction. (a), (b) Experimental results of resonant Rabi nutation between $| ↓ ↓ ⟩ \leftrightarrow | ↑ ↑ ⟩$ and $| ↑ ↓ ⟩ \leftrightarrow | ↓ ↑ ⟩$ , respectively. Inset: Rotations illustrated on a Bloch sphere representing each subspace. (c) Laser frequencies and the configuration of spin and motion energy levels coupled by the Mølmer-Sørensen operation for both the $| ↓ ↓ ⟩ \leftrightarrow | ↑ ↑ ⟩$ (left) and $| ↑ ↓ ⟩ \leftrightarrow | ↓ ↑ ⟩$ (right) transitions. In the limit of large $ϵ$ , this operation approximates the Hamiltonian $H$ in Eq. (1). Insets: A diagrammatic representation of $δ_{1}$ and $δ_{2}$ scans in the two subspaces.
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Figure 2
Broad scan of $δ_{1}$ in both subspaces. (a) $δ_{1}$ scan results when initializing in $| ↓ ↑ ⟩$ and measuring the population in $| ↑ ↓ ⟩$ (yellow) and $| ↓ ↑ ⟩$ (purple). (b) $δ_{1}$ scan results when initializing in $| ↓ ↓ ⟩$ and measuring the population in $| ↑ ↑ ⟩$ (blue) and $| ↓ ↓ ⟩$ (orange). The data were shifted to be symmetric around $δ_{1} = 0$ , and the solid lines are simulation results with no fit parameters. The discrepancy between simulation and data in (a) is due to in population leaking to $| ↓ ↓ ⟩, | ↑ ↑ ⟩$ owing to experimental imperfections. The inset of (b) is a magnification of the shaded area in (b). The $δ_{1} / ϵ$ interval is shaded in (a) for comparison. In both measurements, $ϵ = 2 π \times 25.5 kHz \approx 10 η Ω$ .
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Figure 3
Sensitivity of correlated rotations to $δ_{1}$ and $δ_{2}$ . (a) Population of $| ↑ ↑ ⟩$ when initializing in $| ↓ ↓ ⟩$ . (b) Population of $| ↑ ↓ ⟩$ when initializing in $| ↓ ↑ ⟩$ . We shift the resonance frequency of one of the two ions by $Δ f_{ls}$ using a tightly focused, off-resonance laser beam. As a result, $δ_{1}$ is shifted by $| (Δ f_{ls} / 2) |$ and $δ_{2}$ is shifted by $\pm (Δ f_{ls} / 2)$ , where the sign depends on which ion is shifted. For each value of $Δ f_{ls}$ we scan the laser frequency $ω_{L}$ with respect to an arbitrary offset and measure populations. As shown, a resonance in the odd subspace appears every time $δ_{2} = 0$ regardless of $ω_{L}$ , which only shifts $δ_{1}$ . On the other hand, in the even subspace, $Δ f_{ls}$ shifts the $δ_{1}$ resonance symmetrically when either of the ions is light-shifted, indicating that the associated change in $δ_{2}$ does not affect this subspace. The full two-spin-state populations corresponding to this experiment can be found in the Supplemental Material [24].
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Figure 4
Heisenberg-limited spectroscopy. (a) Comparison between the $δ_{1}$ spectra for the cases of a single ion (red and orange full circles for the right and left ion, respectively) and of two ions correlated by spectroscopy in the even subspace (blue full circles). All spectra are shifted to center around $δ_{1} = 0$ due to small shifts (tens of hertz) and fitted to Eq. (3) (solid blue line). (b) Comparison between the $δ_{2}$ spectra of two ions in an uncorrelated (red full circles) and a correlated (blue full circles) measurement in the odd subspace. In the uncorrelated case, the average frequency of the two ions was measured on each ion separately as a function of $δ_{2}$ , and both populations were averaged to give the measurement result (See Supplemental Material [24]). The spectra were also shifted to be centered at $δ_{2} = 0$ . Insets: Comparison between the normalized uncorrelated case fit, the product of the two single-ion fits in the uncorrelated case, and the normalized correlated fit for both $δ_{1}$ and $δ_{2}$ scans (red, purple, and blue solid lines, respectively).
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