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Engineered 2D Ising interactions on a trapped-ion quantum simulator with hundreds of spins

Joseph W. Britton, Brian C. Sawyer, Adam C. Keith, C. -C. Joseph Wang, James K. Freericks, Hermann Uys, Michael J. Biercuk, John. J. Bollinger

arXiv:1204.5789v1quant-phcond-mat.str-elphysics.comp-ph

TL;DR

The paper develops controlled spin-spin interactions in a large 2D ion crystal and characterizes the conditions enabling quantum simulations. The setup uses a tunable spin-dependent optical dipole force, with demonstrated coupling strength and confinement benchmarks, while spin-state benchmarking remains global rather than ion-resolved.

  • Problem

    Mean-field validity and quantum simulations of interacting ion-crystal spins require understanding engineered interactions and their experimental operating limits.

  • Method

    A spin-dependent optical dipole force from off-resonant laser beams generates controlled forces on qubit states in a rotating 2D Coulomb crystal.

  • Results

    For detunings |µR −ω1| ≲10 kHz, the setup achieves ¯J ≫Γ; at 2 kHz detuning, ¯J ∼ 2π × 0.5 kHz and Γ/¯J ∼0.06.

  • Takeaways & Limitations

    The measured interaction-to-emission regime supports simulations beyond mean-field theory, including spin squeezing and spin depolarization from many-body interactions.

  • Takeaways & Limitations

    Spin-precession benchmarking currently uses global fluorescence measurements, while time-resolved top-view imaging is anticipated for individual-ion spin-state readout.

Abstract

from arXiv · show

The presence of long-range quantum spin correlations underlies a variety of physical phenomena in condensed matter systems, potentially including high-temperature superconductivity. However, many properties of exotic strongly correlated spin systems (e.g., spin liquids) have proved difficult to study, in part because calculations involving N-body entanglement become intractable for as few as N~30 particles. Feynman divined that a quantum simulator - a special-purpose "analog" processor built using quantum particles (qubits) - would be inherently adept at such problems. In the context of quantum magnetism, a number of experiments have demonstrated the feasibility of this approach. However, simulations of quantum magnetism allowing controlled, tunable interactions between spins localized on 2D and 3D lattices of more than a few 10's of qubits have yet to be demonstrated, owing in part to the technical challenge of realizing large-scale qubit arrays. Here we demonstrate a variable-range Ising-type spin-spin interaction J_ij on a naturally occurring 2D triangular crystal lattice of hundreds of spin-1/2 particles (9Be+ ions stored in a Penning trap), a computationally relevant scale more than an order of magnitude larger than existing experiments. We show that a spin-dependent optical dipole force can produce an antiferromagnetic interaction J_ij ~ 1/d_ij^a, where a is tunable over 0<a<3; d_ij is the distance between spin pairs. These power-laws correspond physically to infinite-range (a=0), Coulomb-like (a=1), monopole-dipole (a=2) and dipole-dipole (a=3) couplings. Experimentally, we demonstrate excellent agreement with theory for 0.05<a<1.4. This demonstration coupled with the high spin-count, excellent quantum control and low technical complexity of the Penning trap brings within reach simulation of interesting and otherwise computationally intractable problems in quantum magnetism.

METHODS

The Penning-trap parameters create a planar triangular Coulomb crystal, while a spin-dependent optical dipole force couples spins through transverse motional modes.

  • METHODS: β ≪1 allows 100 ≲N ≲300 ions to relax into a single 2D plane and form a triangular Coulomb crystal.The crystal has N transverse eigenmodes, with the center-of-mass mode at the highest frequency.
  • METHODS: The spin-dependent optical dipole force is generated by two off-resonant beams with angular separation θR ∼4.8° and beat frequency µR.Their traveling optical lattice propagates along z and produces a state-dependent force on the qubit states.
  • METHODS: The force amplitude is chosen with F↑≈−F↓, reducing spin-motion entanglement under the stated operating conditions.The methods account for a typical distribution of composite spin states and finite-temperature occupation corrections.

SUPPLEMENTARY INFORMATION

The supplementary information situates the engineered Ising interaction within prior work on Penning-trap Coulomb crystals and quantum-information control.

  • SUPPLEMENTARY INFORMATION: The reported engineered Ising interaction builds on previous experiments demonstrating high-fidelity quantum control with planar ion arrays in Penning traps.The supplementary material also points to prior theoretical work on 2D Coulomb crystals for quantum information and computation.

I. SPIN INITIALIZATION, CONTROL, AND MEASUREMENT

The experiment initializes, controls, and measures 9Be+ spin qubits in a Penning-trap crystal, while current benchmarking uses global rather than individually resolved spin detection.

  • I. SPIN INITIALIZATION, CONTROL, AND MEASUREMENT: The valence-electron states |↑⟩ and |↓⟩ form the qubit, with nuclear spins optically pumped to mI = +3/2 and ions Doppler-cooled to ∼1 mK.The qubit splitting at 4.46 T is approximately Ω0 = 2π × 124 GHz.
  • I. SPIN INITIALIZATION, CONTROL, AND MEASUREMENT: Low-phase-noise 124 GHz microwaves provide global spin rotations, with π-pulse fidelity greater than 99.9%.The microwave field is predominantly perpendicular to the trap’s magnetic-field axis.
  • I. SPIN INITIALIZATION, CONTROL, AND MEASUREMENT: State-dependent resonance fluorescence distinguishes bright |↑⟩ ions from dark |↓⟩ ions at the end of each experimental sequence.For the reported precession measurements, fluorescence was collected for global spin-state detection.
  • I. SPIN INITIALIZATION, CONTROL, AND MEASUREMENT: The experiment repeated measurements approximately 100 times and averaged them, yielding a few-percent uncertainty from shot noise in P(↑).Short detection periods of approximately 500 µs produced about one detected photon per bright state.
  • I. SPIN INITIALIZATION, CONTROL, AND MEASUREMENT: Future time-resolved top-view imaging is intended to recover individual-ion spin states and enable spin-spin correlation measurements in crystals of about 300 ions.The anticipated detector system could support high-fidelity measurement in approximately 10 ms.
  • I. SPIN INITIALIZATION, CONTROL, AND MEASUREMENT: Crystal-orientation stick-slip motion can occur, but it can be tracked and corrected during imaging.Other issues discussed include ion loss and background-gas collisions; hydrogen collisions produce BeH+ that collects at the crystal perimeter.

II. OPTICAL-DIPOLE-FORCE LASER SETTINGS

The optical-dipole-force setup uses detuned, polarized beams to cancel single-beam Stark shifts while producing opposite forces on the two qubit states; stronger configurations yield sizable tunable couplings.

  • II. OPTICAL-DIPOLE-FORCE LASER SETTINGS: The qubit states experience opposite ODF forces, F↑=−F↓, reducing sensitivity to laser-intensity fluctuations.If the forces are unequal, the interaction includes terms linear in σz.
  • II. OPTICAL-DIPOLE-FORCE LASER SETTINGS: The ODF beams are arranged in the y-z plane at angles ±θR/2 and use different linear polarization angles relative to vertical polarization.This geometry is designed to create a polarization gradient and state-dependent force.
  • II. OPTICAL-DIPOLE-FORCE LASER SETTINGS: The beam waists provide less than 10% intensity variation across crystals with N < 250.The vertical and horizontal waists are approximately 110 µm and 1 mm, respectively.
  • II. OPTICAL-DIPOLE-FORCE LASER SETTINGS: At ΔR = −63.8 GHz, the single-beam AC Stark shift cancels near φp ≃±65°, and the two beam polarizations are chosen at +65° and −65°.The operating detunings are +15.6 GHz and −26.1 GHz relative to the relevant transitions.
  • II. OPTICAL-DIPOLE-FORCE LASER SETTINGS: For θR = 35° and a 20 mW/beam configuration, the predicted coupling is Ji,j/(2πN) ∼(560 Hz)(d0/di,j)^1.7.Here N = 217, d0 ∼20 µm, and the required detuning is µR−ω1 = 2π × 100 kHz.

III. WAVEFRONT ALIGNMENT

Misaligned ODF lattice wave fronts make the force depend on each ion’s planar position, complicating the engineered interactions. A fluorescence-based alignment method mitigates this effect and achieves sub-0.05° alignment while preserving lattice stability.

  • Alignment challenge: Position-dependent ODF time dependence complicates the effective spin-spin interactions when wave fronts are not normal to the magnetic-field axis.The complication is adequately mitigated by careful alignment.
  • Alignment procedure: 0.05° is the achieved upper bound for ODF wave-front misalignment relative to the planar ion array.The alignment technique uses real-time top-view fluorescence imaging to optimize the beams.
  • Alignment procedure: Dark fluorescence bands identify high-intensity antinodes because AC Stark shifts move ions out of resonance with the Doppler cooling laser.The alignment measurement uses a stationary lattice with µR = 0 and background-subtracted top-view imaging.
  • Alignment validation: 1 s integration images indicate that the one-dimensional lattice remains stable during the integration period, with phase stability better than 1 s.The observed fringe pattern also provides information about misalignment angle and direction.
  • Alignment limitation: Direct fluorescence imaging of the lattice is not viable because resonantly scattered photons exert torque that rapidly changes the crystal’s rotation and radius.The alternative AC-Stark-shift method avoids this disturbance.

IV. MODELING MEAN FIELD SPIN PRECESSION

The experiment models the engineered Ising interaction as an effective mean field that drives spin precession about the z-axis. A spin-echo sequence isolates this interaction-dependent precession while canceling constant unwanted precession.

  • Mean-field model: The effective mean field H_MF describes each spin’s response to the engineered Ising interaction through an excess magnetic field along z.The resulting spin precession is proportional to the mean field generated by the other spins.
  • Mean-field model: The measured observable is the excess spin precession averaged over all spins, rather than ordinary Larmor precession.The mean-field equations describe precession about the z-axis at frequency B̄_j.
  • Spin-echo measurement: The spin-echo sequence cancels constant precession independent of ⟨σ_z⟩ while coherently adding precession proportional to ⟨σ_z⟩.The π pulse reverses the spin projection during the second arm, enabling the interaction-dependent signal to accumulate.
  • Spin-echo measurement: 2J̄ cos(θ1)·τ_arm is the mean-field precession angle accumulated during each interaction arm.After the π pulse reverses ⟨σ_z⟩, the two arms combine into a total precession measured by the final π/2 pulse.
  • Readout: The final π/2 pulse converts accumulated precession into excursions above or below the Bloch-sphere equator, which are detected through the |↑⟩ probability.The sequence evolution operator is constructed from the operators for each segment.

V. OPTICAL DIPOLE FORCE LASER INTENSITY CALIBRATION

Because the measured spin-spin coupling scales with the square of ODF electric-field intensity, each beam’s intensity is calibrated independently through the qubit AC Stark shift.

  • Intensity calibration: J̄ depends on the square of the ODF electric-field intensity I_R, making intensity calibration essential for benchmarking the interaction strength.The calibration is performed separately for each ODF beam.
  • Intensity calibration: The qubit AC Stark shift is measured as a function of polarization angle φ_p to determine each beam’s intensity.The qubit transition frequency is extracted by fitting the center of a Rabi-resonance profile.

VI. SPONTANEOUS EMISSION

Spontaneous emission is treated as decoherence of the qubit coherences and included in the spin-precession model. Its impact constrains achievable interaction-to-decoherence ratios, although the present setup reaches J̄ ≫ Γ near selected detunings.

  • Decoherence model: Spontaneous light scattering does not optically pump 9Be+ ions outside the two qubit levels.Its effect is therefore modeled primarily as decay of off-diagonal coherences.
  • Decoherence model: The measured spin-precession probability acquires an exp(−Γ·2τ_arm) factor describing spontaneous-emission-induced coherence decay.Γ includes contributions from Raman and elastic Rayleigh scattering and is fixed from independent intensity calibration.
  • Decoherence calibration: Γ = 82 s^-1 is calculated for Δ_R = −63.8 GHz, φ_p = ±65.3°, and I_R = 1 W/cm^2.The decoherence rate is related to ODF laser intensity through an atomic-physics calculation.
  • Operating regime: Γ/J̄ ∼ 0.06 is calculated for 4 mW per beam and |µ_R − ω_1| = 2 kHz, with J̄ ∼ 2π × 0.5 kHz.Under these conditions, the potential spin squeezing is 5 dB and is limited by spontaneous emission.
  • Limitation: Very large ODF detunings are unlikely to reduce spontaneous-emission impact because interaction strength and scattering both scale as 1/Δ^2.Different detunings may help, but they introduce the complication F_↑ ≠ −F_↓.

VII. LAMB-DICKE CONFINEMENT

The analysis quantifies how well the state-dependent force is spatially uniform across each ion’s wave function and accounts for thermal transverse modes and wavefront misalignment.

  • Confinement parameter: ηind,i quantifies the force’s spatial variation across an individual ion’s wave function, with zrms,i its axial rms extent and δk=2π/λR.The paper distinguishes this individual-ion confinement parameter from the usual ground-state Lamb-Dicke parameter.
  • Transverse-mode estimate: Summing all transverse modes yields zrms,i≃520 nm at the array center and ≃250 nm at the edge, corresponding to ηind,i≃0.89 and 0.42, respectively.The calculation uses N=217, ωr=2π×45.6 kHz, and a common mode temperature.
  • Wavefront alignment: For a typical 200-ion array with Rp≃200 µm and θerr≃0.05°, the wavefront-alignment parameter δk·2Rp sinθerr is approximately 0.6.This condition is intended to keep the time-dependent axial shift from invalidating the uniform-force approximation.

VIII. ION LATTICE CONFIGURATION AT EQUILIBRIUM AND TRANSVERSE NORMAL MODES

The Penning-trap ions form a calculated, rotating-frame triangular crystal whose equilibrium geometry determines the transverse phonon modes used in the interaction analysis.

  • Equilibrium configuration: The equilibrium positions form a triangular lattice with a radially increasing lattice constant, a smooth unfaceted edge, and degraded orientational order near the perimeter.The positions are obtained by minimizing the Euler-Lagrange action subject to planar confinement.
  • Transverse normal modes: Expanding the potential about equilibrium yields a stiffness matrix Kij, whose eigenmodes provide N transverse-mode eigenvalues and frequencies.The normal-mode calculation diagonalizes Kij/M after deriving the rotating-frame equations of motion.
  • Trap model: The trap includes a uniform axial magnetic field, harmonic axial and radial electric potentials, and a rotating-wall quadrupole potential controlling the rotation frequency.Ion-ion Coulomb separation enters the Penning-trap potential description.
  • Numerical solution: Equilibrium positions are found in a rotating frame seeded with a regular triangular lattice; closed-shell ion numbers aid consistent numerical convergence.For N=127, six closed shells provide an example, while the rotating wall produces overall ellipticity and peripheral deviations from perfect triangular order.

IX. LIMITS TO THE VALIDITY OF MEAN FIELD THEORY

The paper tests when the collective ˆJ2 interaction produces the same short-time spin precession as mean-field theory, finding agreement over a duration range that depends on interaction strength and spin number.

  • Mean-field criterion: Mean-field treatment of the ˆJ2 interaction is assessed by comparing its predicted transformation of collective-spin expectation values with an ideal z-axis rotation.The analysis uses states in the symmetric composite-spin subspace and examines the action of e−iφˆJz.
  • Uniform-coupling comparison: For uniform Ising coupling, the ˆJ2-predicted precession frequency 2χcosθ agrees with the mean-field prediction.The comparison applies in the stated near-resonant uniform-coupling regime.
  • Validity limit: The approximation requires short durations satisfying 2χt·2(ΔMJ)^2≪1, with the initial coherent spin state setting ΔMJ≲√N.This establishes a duration limit for replacing the ˆJ2 evolution with a z-axis rotation.
  • Exact benchmark: For N=5, 50, and 100 spins, exact calculations agree reasonably at χt=0.2, while deviations emerge at χt=1.6 and decrease as N increases.For N=50 and 100, agreement is excellent at χt=0.2 and 0.8; for N=5, χt=0.8 and 1.6 show disagreement.
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