Onset of a quantum phase transition with a trapped ion quantum simulator

Abstract

A quantum simulator is a well-controlled quantum system that can follow the evolution of a prescribed model whose behaviour may be difficult to determine. A good example is the simulation of a set of interacting spins, where phase transitions between various spin orders can underlie poorly understood concepts such as spin liquids. Here we simulate the emergence of magnetism by implementing a fully connected non-uniform ferromagnetic quantum Ising model using up to 9 trapped 171 Yb + ions. By increasing the Ising coupling strengths compared with the transverse field, the crossover from paramagnetism to ferromagnetic order sharpens as the system is scaled up, prefacing the expected quantum phase transition in the thermodynamic limit. We measure scalable order parameters appropriate for large systems, such as various moments of the magnetization. As the results are theoretically tractable, this work provides a critical benchmark for the simulation of intractable arbitrary fully connected Ising models in larger systems.

Intrinsic phonon effects on analog quantum simulators with ultracold trapped ions

“…This procedure is theoretically identical to the ground state passage of the spin polarized state | ↓ y ... ↓ y ... ↓ y to the ground state of the negative of the Ising Hamiltonian −H Is (t) with the system Hamiltonian being modified as H(t) → −H(t) [11]. In a typical trapped ion quantum simulator, the frequency µ is sufficiently far from any phonon frequencies such that the condition η Xν Ω i |µ − ω Xν | holds to avoid the heating of the system away from the initial phonon vacuum state during the simulation.…”

“…In recent years, there has been significant success in trying to achieve this goal by quantum simulation of desired spin models through analogous cold atom systems [6][7][8]. We focus here on one platform for performing analog quantum computation, the simulation of interacting quantum spins via manipulation of hyperfine states of ions in a linear Paul trap [9][10][11][12][13] although many ideas presented here can be generalized * Electronic address: joseph@physics.georgetown.edu to adiabatic quantum state computation in the two dimensional Penning trap as well [8]. In the Paul trap systems, clock states of the ions (states with no net zcomponent of angular momentum) are the pseudospin states, which can be manipulated independently by a pseudospin-dependent force driven by laser beams.…”

“…19 for a review). It has been known experimentally that moderate phonon creation is commonplace (on the order of one phonon per mode) [11], even when the system is cooled to essentially the phonon ground state prior to the start of the simulation. In addition, the role phonons play are intrinsic and essential for the mediated spin-spin interaction in trapped ion systems especially in the presence of noncommuting magnetic field Hamiltonian in addition to the spin Hamiltonian of interest.…”

“…Sufficiently small systems can be treated numerically in an exact fashion by truncating the phonon basis and taking into account all possible quantum states in the solution of the time-dependent Schrödinger equation. Experimentally, ion traps have been used to simulate the transverse-field Ising model with a small number of ions [9,11,12] based on simulated quantum annealing [18] (see Ref. 19 for a review).…”

Adiabatic Mach-Zehnder interferometer via an array of trapped ions

“…Because only two ionic hyperfine states are used for implementing the MZ interferometry, each ion can be regarded as a spin-1/2 particle of two spin states |↓ and |↑ . By globally addressing all the ions with particular lasers, the system obeys a transverse-field quantum Ising Hamiltonian [14][15][16][17],where σ i x,z are Pauli operators for the i-th ion, B is the transverse field, and J ij = J/ |i − j| 3 is the effective Ising interaction between ions i and j with J ≥ 0 denoting the nearest-neighboring interaction. For convenience, we consider the dimensionless model in units of = 1 before the discussions of experimental possibility.…”