Intrinsic anharmonic effects on the phonon frequencies and effective spin-spin interactions in a quantum simulator made from trapped ions in a linear Paul trap

M. McAneny and J. K. Freericks

Phys. Rev. A 90, 053405 – Published 6 November 2014

Abstract

The Coulomb repulsion between ions in a linear Paul trap gives rise to anharmonic terms in the potential energy when expanded about the equilibrium positions. We examine the effect of these anharmonic terms on the accuracy of a quantum simulator made from trapped ions. To be concrete, we consider a linear chain of ${Yb}^{171 +}$ ions stabilized close to the zigzag transition. We find that for typical experimental temperatures, frequencies change by no more than a factor of $0.01 %$ due to the anharmonic couplings. Furthermore, shifts in the effective spin-spin interactions (driven by a spin-dependent optical dipole force) are also, in general, less than $0.01 %$ for detunings to the blue of the transverse center-of-mass frequency. However, detuning the spin interactions near other frequencies can lead to non-negligible anharmonic contributions to the effective spin-spin interactions. We also examine an odd behavior exhibited by the harmonic spin-spin interactions for a range of intermediate detunings, where nearest-neighbor spins with a larger spatial separation on the ion chain interact more strongly than nearest neighbors with a smaller spatial separation.

Received 20 June 2014

DOI:https://doi.org/10.1103/PhysRevA.90.053405

Authors & Affiliations

M. McAneny and J. K. Freericks^*

Department of Physics, Georgetown University, 37th and O Streets NW, Washington, DC 20057, USA

^*freericks@physics.georgetown.edu

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Issue

Vol. 90, Iss. 5 — November 2014

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Images

Figure 1
Anharmonic frequency shifts as a function of the temperature of the phonon modes measured relative to the transverse center-of-mass phonon frequency $ω_{c . m .}$ . The center-of-mass mode is shown in black in all panels, while the transverse zigzag mode is shown in dark gray (red) in (a) and (c). All other modes are shown in light gray (cyan). (a) Frequency shifts of transverse modes in the Doppler cooling limit, where both transverse and longitudinal phonons are at temperature $T$ . (b) Shifts of the longitudinal modes in the Doppler cooling limit. (c) Shifts of transverse modes with Doppler and sideband cooling (zero transverse phonon occupation, so transverse phonons are in the ground state, while longitudinal ones are at temperature $T$ ). (d) Shifts of longitudinal modes with Doppler and sideband cooling (zero transverse phonon occupation).
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Figure 2
(a) Proportional shifts in spin-spin interactions for detunings above the center-of-mass mode. (b) The same for detunings above the fifth lowest transverse frequency mode. Both plots are shown for $δ = 10^{- 1}$ , $δ = 10^{- 2}$ , $δ = 10^{- 3}$ , $δ = 10^{- 4}$ , $δ = 10^{- 5}$ , and $δ = 10^{- 6}$ . Symbols show the shifts for all ( $24 \times 23 / 2 = 276$ ) spin-spin interactions $J_{i j}$ . Solid horizontal lines show the corresponding average shift.
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Figure 3
Dependence of the spin-spin interactions detuned above the center-of-mass mode as a function of the system's temperature. Left: Doppler cooling only. Right: Doppler plus sideband cooling. Spin-spin interactions plotted are $J_{1 j}$ , that is, spin interactions between the leftmost ion in the chain and every other ion, and we use three detunings, corresponding to the three colors. The most significant trend in this plot is that the shifts are nearly linear as a function of temperature. Furthermore, in (a) and (b), and even in (c) and (d), there are clear outliers that shift much more than the other lines. As expected, these lines plot spin interactions between ions on either side of the chain; the shifts are proportionally large because the spin-spin interactions between distant sites are quite small to begin with, hence the effects of these shifts on the dynamics of the system, even though they appear to be large, are likely to be rather small.
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Figure 4
Harmonic spin-spin interactions as a function of the distance between interacting spins. Note that for intermediate detuning, $δ = 10^{- 2}$ [(red) square] and $δ = 10^{- 3}$ [(blue) triangle] curves, and for small separations between spins, there can be increasing spin interactions for increasing separation between the ions. This suggests that ions towards either end of the chain interact more strongly with their neighbors than those in the middle of the chain do, a surprising result since ions in the middle of the chain are closer together. For larger distances, spin-spin couplings become approximate power laws, as expected.
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