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arXiv:1005.4160 · Physical Review B

Quantum simulation and phase diagram of the transverse-field Ising model with three atomic spins

E.E. Edwards, S. É. Korenblit, Kihwan Kim, Rajibul Islam, M.-S. Chang, James K. Freericks, Guin-Dar Lin, Luming Duan, Christopher R. Monroe

Published Aug 27, 2010• 106 citations
arXivHTMLPDF

Abstract

We perform a quantum simulation of the Ising model with a transverse field using a collection of three trapped atomic ion spins. By adiabatically manipulating the Hamiltonian, we directly probe the ground state for a wide range of fields and form of the Ising couplings, leading to a phase diagram of magnetic order in this microscopic system. The technique is scalable to much larger numbers of trapped ion spins, where phase transitions approaching the thermodynamic limit can be studied in cases where theory becomes intractable.

DOI
10.1103/physrevb.82.060412

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We perform a quantum simulation of the Ising model with a transverse field using a collection of three trapped atomic ion spins. By adiabatically manipulating the Hamiltonian, we directly probe the ground state for a wide range of fields and form of the Ising couplings, leading to a phase diagram of magnetic order in this microscopic system. The technique is scalable to much larger numbers of trapped ion spins, where phase transitions approaching the thermodynamic limit can be studied in cases where theory becomes intractable.
In a proof-of-principle experiment, we simulate the dynamics of a quantum spin system in an ion trap. Following a theoretical proposal by D. Porras and I. Cirac, we use a system of ground state cooled trapped ions to simulate and study the dynamics of a quantum-mechanical system, or more precisely, the dynamics of a quantum spin Hamiltonian. We implement the smallest non-trivial quantum spin Hamiltonian, the quantum Ising model for two spins. Each spin is represented by two hyperfine ground levels of trapped 25Mg+ ions. The interaction with an external magnetic field is simulated by coherently coupling these hyperfine levels via laser and radiofrequency radiation. The spin-spin interaction is simulated via optical dipole forces, where the effective interaction is mediated by the phonons of the linear ion chain. We demonstrate the adiabatic evolution from a paramagnetically ordered system to ferromagnetic order. The final state of this adiabatic transition is a superposition state of the two degenerate spin configurations of ferromagnetic order Psi_final = 1/sqrt(2) (|up up> + |down down>) with a quantum magnetisation of 98%. We also show the transition from paramagnetic to antiferromagnetic order with the final state Psi_final = 1/sqrt(2) (|up down> + |down up>). Moreover, we prove that this transition which is to become a quantum phase transition in the thermodynamic limit of infinitely many spins, is driven by quantum fluctuations which dominate the dynamics of such systems at the absolute zero-point of temperature rather than thermal fluctuations which are absent at 0 K. This is verified by the fact that the final state of our adiabatic evolution is entangled, close to a Bell state at a fidelity exceeding 88%. The set of tools presented in this thesis might serve as a basis for larger scale quantum simulations which might help in gaining insight into many-particle effects that are intractable on classical computers such as spin frustration in triangular lattices or high-Tc superconductivity.
The quantum simulation of complex many body systems holds promise for understanding the origin of emergent properties of strongly correlated systems, such as high-Tc superconductors and spin liquids. Cold atomic systems provide an almost ideal platform for quantum simulation due to their excellent quantum coherence, initialization and readout properties, and their ability to support several forms of interactions. In this thesis, I present experiments on the quantum simulation of long range Ising models in the presence of transverse magnetic fields with a chain of up to sixteen ultracold 171-Yb+ ions trapped in a linear radio frequency Paul trap. Two hyperfine levels in each of the 171-Yb+ ions serve as the spin-1/2 systems. We detect the spin states of the individual ions by observing state-dependent fluorescence with single site resolution, and can directly measure any possible spin correlation function. The spin-spin interactions are engineered by applying dipole forces from precisely tuned lasers whose beatnotes induce stimulated Raman transitions that couple virtually to collective phonon modes of the ion motion. The Ising couplings are controlled, both in sign and strength with respect to the effective transverse field, and adiabatically manipulated to study various aspects of this spin model, such as the emergence of a quantum phase transition in the ground state and spin frustration due to competing antiferromagnetic interactions. Spin frustration often gives rise to a massive degeneracy in the ground state, which can lead to entanglement in the spin system. We detect and characterize this frustration induced entanglement in a system of three spins, demonstrating the first direct experimental connection between frustration and entanglement. With larger numbers of spins we also vary the range of the antiferromagnetic couplings through appropriate laser tunings and observe that longer range interactions reduce the excitation energy and thereby frustrate the ground state order. This system can potentially be scaled up to study a wide range of fully connected spin networks with a few dozens of spins, where the underlying theory becomes intractable on a classical computer.
We control quantum fluctuations to create the ground state magnetic phases of a classical Ising model with a tunable longitudinal magnetic field using a system of 6 to 10 atomic ion spins. Because of the long-range Ising interactions, the various ground state spin configurations are separated by multiple first-order phase transitions, which in our zero temperature system cannot be driven by thermal fluctuations. We instead use a transverse magnetic field as a quantum catalyst to observe the first steps of the complete fractal devil's staircase, which emerges in the thermodynamic limit and can be mapped to a large number of many-body and energy-optimization problems.
Motivated by recent experiments with a Penning ion trap quantum simulator, we perform numerically exact Stochastic Series Expansion quantum Monte Carlo simulations of long-range transverse-field Ising models on a triangular lattice for different decay powers $\ensuremath{\alpha}$ of the interactions. The phase boundary for the ferromagnet is obtained as a function of $\ensuremath{\alpha}$. For antiferromagnetic interactions, there is strong indication that the transverse field stabilizes a clock ordered phase with sublattice magnetization $(M,\ensuremath{-}\frac{M}{2},\ensuremath{-}\frac{M}{2})$ with unsaturated M<1M<1M<1 in a process known as ``order by disorder'' similar to the nearest-neighbor antiferromagnet on the triangular lattice. Connecting the known limiting cases of nearest-neighbor and infinite-range interactions, a semiquantitative phase diagram is obtained. Magnetization curves for the ferromagnet for experimentally relevant system sizes and with open boundary conditions are presented.
Despite the fact that a complete theoretical description of critical phenomena in connection with phase transitions has been well established through the renormalization-group theory, the microscopic nature of the phase transitions remains to be understood in a satisfactory way. For example, how does the interaction between particle/spins drive the system across distinct phases as parameters vary, and how do these particle/spins respond to parameter variations during this transition process? In this work, we investigate these questions through the paradigmatic example of quantum phase transitions (QPT) in the one-dimensional transverse Ising model (TIM). We first introduce 2L collective structures, referred to as patterns, for the TIM with L ferromagnetically interacting spins, and then analyze the contributions of these patterns to the system's states, e.g., the ground state, the first excited state, and so on, from which the analog of the QPT process between the disordered phase in the weakly coupling regime and the ferromagnetic phase in the strongly coupling regime is clearly identified at the interaction strength J_{c}=1. We systematically explore this process for small lattice sizes of L=6,8,10,12, whose ground-state energies are identical to those obtained by directly numerical exact diagonalization. Increasing the system size up to L=128, the actual QPT point located at J_{c}=1 in the thermodynamical limit is gradually approached. Our results show that the pattern picture is not only able to provide a microscopic process of phase transitions, but also of practical interest in analyzing analogues of QPT in diverse quantum simulation platforms.