Long-range Heisenberg models in quasiperiodically driven crystals of trapped ions

A. Bermudez, L. Tagliacozzo, G. Sierra, and P. Richerme

Phys. Rev. B 95, 024431 – Published 30 January 2017

Abstract

We introduce a theoretical scheme for the analog quantum simulation of long-range XYZ models using current trapped-ion technology. In order to achieve fully tunable Heisenberg-type interactions, our proposal requires a state-dependent dipole force along a single vibrational axis, together with a combination of standard resonant and detuned carrier drivings. We discuss how this quantum simulator could explore the effect of long-range interactions on the phase diagram by combining an adiabatic protocol with the quasiperiodic drivings, and test the validity of our scheme numerically. At the isotropic Heisenberg point, we show that the long-range Hamiltonian can be mapped onto a nonlinear sigma model with a topological term that is responsible for its low-energy properties, and we benchmark our predictions with matrix-product-state numerical simulations.

Received 29 September 2016
Revised 5 January 2017

DOI:https://doi.org/10.1103/PhysRevB.95.024431

Physics Subject Headings (PhySH)

Quantum simulation Trapped ions

Heisenberg model Matrix product states XYZ model

Quantum Information, Science & TechnologyAtomic, Molecular & Optical

Authors & Affiliations

A. Bermudez^1,2, L. Tagliacozzo³, G. Sierra⁴, and P. Richerme⁵

¹Department of Physics, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom
²Instituto de Física Fundamental, IFF-CSIC, Madrid E-28006, Spain
³Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, United Kingdom
⁴Instituto de Física Teórica (IFT), UAM-CSIC, Madrid, Spain
⁵Department of Physics, Indiana University, Bloomington, Indiana 47405, USA

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Vol. 95, Iss. 2 — 1 January 2017

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Figure 1
Numerical validation of the XYZ Heisenberg model: (a) Magnetization dynamics ${〈 σ_{i}^{y} (t) 〉}_{β}$ for a two-ion chain evolving under $β \in {exact, XYZ (2), QIM (6)}$ . The trapped-ion parameters for the MS (3) are $δ / 2 π = 500 kHz, Ω_{L} / 2 π = 0.9 MHz$ , and $ϕ_{s} = π / 3$ , whereas for the modulated carrier (7) in Table 1, $h_{0} / 2 π = 2.5 kHz, ξ = 0.09$ , and $Δ / 2 π = 10 kHz$ . We consider initial mean phonon numbers ${\bar{n}}_{z z} = 0.05$ and ${\bar{n}}_{com} = 0.047$ , and truncate the vibrational Hilbert space to 7 phonons per mode. To ease the visualization, we perform a spin-echo refocusing pulse at the middle $U_{s} = e^{i \frac{π}{2} \sum σ_{i}^{y}}$ for different evolution times given by multiples of $π / h_{0} = 4 π / Δ$ . (b), (c) Quantum channel error as a function of (b) the MS phase $ϕ_{s}$ , and (c) the mean number of phonons in the lowest mode, considering for the same parameters, but setting $Ω / 2 π = 0.5 MHz$ , and varying $h_{0} / 2 π \in {1.2, 1.8, 2.4, 3, 3.6, 4.2, 4.8, 5.3, 5.9} kHz$ , and the associated detunings $Δ = 4 h_{0}$ . The different lines correspond to the above values of $h_{0}$ increasing in the direction of the arrow. (d) Adiabatic evolution of $| ψ_{0} 〉 = | -_{y} +_{y} -_{y} +_{y} -_{y} +_{y} 〉$ for $N = 6$ ions, subjected to (i) an Ising linear ramp $t \in [0 t_{f} / 2]$ of the staggered field $h_{0} (t) = h_{0} (0) - δ h t$ , with $h_{0} (0) = 12 J_{12}$ , and $δ h = - 2 h_{0} (0) / t_{f}$ , followed by (ii) a Heisenberg linear ramp $t \in [t_{f} / 2, t_{f}]$ of the phase $ϕ_{s} (t) = δ ϕ t$ , where $δ ϕ = 2 ϕ_{f} / t_{f}$ , as described in the text. The adiabatic fidelity $F_{ad}$ is represented as a function of the total ramp time $t_{f}$ , which is set to be an integer multiple of $2 π / h_{0}$ , and the final phase $ϕ_{f}$ .
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Figure 2
Central charge in the long-range Heisenberg model: Entanglement entropies $S_{ℓ}$ of blocks of length $ℓ$ in a chain of $N = 128$ spins computed with a matrix-product-state ansatz for the ground state of the long-range Heisenberg model with couplings (13) using bond dimensions up to 200 to guarantee the convergence of the ground-state energy up to 12 digits. The ground state is obtained using the time-dependant variational algorithm originally presented in [51] and later reformulated for finite chains with long-range interactions in [52, 53]. The entanglement entropies, for large enough $ℓ$ , scale linearly with the variable $y_{ℓ} = ln [\frac{2 N}{π} sin (\frac{π ℓ}{N})]$ , as expected for a conformally invariant spin chain, for all the values considered $λ \in {0.1, 0.075, 0.05}$ . Quantitatively, the prefactor of the scaling exceeds the central charge $c = 1$ of the nearest-neighbor model (red line). The coefficients $c_{eff}$ are obtained by a linear fit of the larger $ℓ$ entropies to $S_{ℓ} = \frac{c_{eff}}{6} y_{ℓ} + a$ , and shown in the inset.
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Figure 3
Scheme of the quasiperiodically modulated Mølmer-Sørensen force: (a) Ions (green circles) forming a chain within the electrodes of a linear Paul trap. Two internal electronic levels of each ion form a pseudospin (thin arrows) that is coupled to the incident laser beams (wide arrows), the beat note of which is controlled by an acousto-optical modulator (AOM) through an arbitrary wave generator (AWG). By using the different modulation frequencies in the AWG listed in Table 1, we obtain a quasiperiodically modulated Mølmer-Sørensen force that couples the spins to the motion along the axis $e_{x} | | Δ k$ . (b) Scheme for the blue and red sidebands ( $ω_{0} \pm ω_{n, x}$ ) of the whole vibrational branch for the Mølmer-Sørensen beams, and their symmetric detuning $δ_{n}$ . We also represent with small arrows the comb of frequencies $m Δ$ , with $m \in Z$ , due to the additional drivings introduced in the scheme (A6), which could hit an undesired resonance (e.g., carrier transition). (c) Ratio of the coupling strength $R_{m}$ between the possible spurious resonance with the carrier due to the additional comb of frequencies $m Δ$ , and the desired MS sideband, as a function of $m \in {1, ..., 6}$ . We see that the high resonances (i.e., high $m$ ) required by the constraint (A7) lead to a vanishingly small ratio, such that these terms can be safely neglected for the time scales of interest of the experiment.
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Figure 4
Distance dependence of the spin-spin couplings: (a) Distance between neighboring ions $| z_{j}^{0} - z_{j + 1}^{0} |$ as a function of the lattice index $j \in {1, ..., 49}$ for a chain of $N = 50$ Yb ions. The blue circles are obtained from the exact equilibrium positions $z_{j}^{0} = ℓ_{z} u_{j}$ by solving $u_{j} - \sum_{k \neq j} (u_{j} - u_{k}) / {| u_{j} - u_{k} |}^{3} = 0$ , where $ℓ_{z} = {(e^{2} / 4 π ɛ_{0} m ω_{z}^{2})}^{1 / 3}$ . The red line stands for the theoretical model of a homogeneous ion chain $z_{j}^{0} = a_{0} j$ with constant lattice spacing given by $a_{0} = \min {| z_{j}^{0} - z_{j + 1}^{0} |}$ . (b) Normalized vibrational frequencies $ω_{n, x} / ω_{x}$ as a function of the normal-mode index $n \in {1, ..., 50}$ . The blue circles represent the exact normal modes obtained by solving $\sum_{i j} M_{i, n} K_{i, j} M_{j, m} = ω_{n, x}^{2} δ_{n, m}$ , where $K_{i, j} / ω_{x}^{2} = (1 - δ_{i, j}) β_{x} / | z_{i}^{0} - z_{j}^{0} |^{3} + δ_{i, j} (1 - \sum_{l \neq i} β_{x} / | z_{i}^{0} - z_{l}^{0} |^{3})$ is obtained from the numerical solution of the equilibrium positions in the inhomogeneous crystal. The red line stands for the theoretical model for a homogeneous periodic chain with the normal modes given in Eq. (C2). (c) Normalized spin-spin couplings $J_{i, j} / J_{i, i + 1}$ between the central ion $i = N / 2 = 25$ and its bulk neighbors $j = N / 2 + r$ , with $r \in {1, \dots, 16}$ , as a function of their respective distance $| z_{i}^{0} - z_{j}^{0} |$ , and for different values of the MS detuning $δ_{x} = ω_{x} - μ$ with respect to the center-of-mass mode, $δ_{x} / 2 π \in {62.5, 125, 250, 500, 1000} kHz$ , increasing in the direction of the arrow. The blue circles are obtained by solving Eq. (C1) using the previous normal modes and frequencies. The red lines stand for our analytical result (C6) without any fitting parameter, but rather using the microscopic values for Eqs. (C4) and (C8). The green dashed lines correspond to a power-law decay $J_{r} \propto 1 / r^{s}$ for two different exponents $s = 3$ and $s = 1$ .
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