Quantum Simulationis of Condensed Matter Systems Using Trapped Ions

D.J. Berkeland, G.H. Ogin, W.R. Scarlett (P-21), M.G. Blain, C.P. Tigges (SNL)
Excerpted from LA-14202-PR

Introduction

When Richard Feynman famously proposed a quantum computer, his intended application was actually to simulate quantum dynamical systems. This problem is difficult because as the number of elements of a quantum system increases linearly, the complexity of the equations modeling it grows exponentially. For example, to completely describe the dynamics of just 40 spin-1/2 particles requires 2⁴⁰ × 2⁴⁰ = 10²⁴ matrix elements, orders of magnitude greater than what can be stored on any classical supercomputer. This complexity is why we cannot even determine the correct theoretical behavior of some important systems. Our understanding of quantum phenomena such as superconductivity, antiferromagnetism, behavior of f-electrons in solids, and so on, is seriously limited.

Feynman’s proposed solution to this problem was to simulate one quantum-mechanical system with another. The states of the simulator would follow the same equations of motion as the real system, yet would be directly accessible so the state evolution could be monitored. A decade later, Seth Lloyd showed that Feynman’s conjectured solution was correct. He pointed out that “a mere 30 or 40 quantum bits would suffice to perform quantum simulations of multidimensional fermionic systems such as the Hubbard model,” which is believed to explain phenomena such as high-temperature superconductivity, “that have proved resistant to conventional computational techniques.”¹

A first-principles understanding of the behavior of materials would profoundly affect academia, defense, and industry. In spite of this, little experimental effort has been extended towards quantum simulations of condensed-matter systems. This is because theoretical proposals for quantum simulations have been cast in general terms. Only recently, quantum information theorists have begun to map these problems onto experimentally accessible atomic systems, urged on by correspondingly recent advances in coherent manipulations of those systems. For example, E. Jané proposed that several paradigms of condensed-matter physics can be modeled in trapped-ion systems and neutral atoms in optical lattices.² Subsequently, more detailed simulations were proposed for trapped ions in Porras’ and Barjaktarevic’s studies.^3,4

These proposals rely on several key concepts. A single atom can simulate a spin system (for example, an electron) which is the fundamental building block of real material. Laser light interacts with the atoms to simulate site-specific potentials (such as magnetic fields). State-dependent optical forces manipulate atomic positions to simulate interactions between the spin systems. Put together, these components map atomic systems onto equivalent arrays of quantum spins found in some condensed-matter systems. Yet, our atomic spin systems can be free of complicating impurities and defects, we can precisely control the interatomic interactions, and the system evolution can be characterized and measured more easily than in real materials.

We will describe experiments we are just beginning, using arrays strontium-88(+) ions confined in our linear radio-frequency trap.⁵ This work is part of a larger LANL project on quantum simulations in Theoretical Division and Chemistry Division.

Basic Interactions

We confine ions in the trap shown in Figure 1. Figure 2 shows the relevant optical transition and quantum-mechanical states of strontium-88(+). The Zeeman levels of the ion’s ground state simulate the spin-up and spin-down states of a spin-1/2 particle: |↓〉 = |S_1/2, m_J = −1/2〉 and |↑〉 = |S_1/2, m_J = +1/2〉. The ion’s wavefunction is described by the equation ψ = c↑ |↑〉 + c_↓ |↓〉, where c_↑ and c_↓ are complex numbers with a relative phase φ between them. We can determine ψ using a series of laser and magnetic-field pulses not described here. As Figure 3 illustrates, this spin system of |↓〉 and |↑〉 states can be visualized as a vector. The vector’s direction depends on the relative probability of the ion being in one state or the other and on the phase difference between c_↑ and c_↓.

Figure 4 illustrates how we induce ion-ion interactions that simulate the spin-spin interactions in condensed-matter problems. A force on each ion pushes it in the α direction only if it is in the |↑〉_α state (|↑〉 in the basis in which α is the quantization axis). We derive this force from beams of 422 nm laser light that are tightly focused on points a few microns from each ion in the z-direction. This light is detuned Δ = 1 to 10 GHz below resonance with the S_1/2 ↔ P_1/2 transitions as shown in Figure 1. This detuning Δ is enough that spontaneously scattered light does not ruin the coherence between the spin states, but not so much that the laser no longer affects the ion. The atom-laser interaction creates an optical dipole force so that the ion moves towards the most intense part of the laser beam. Furthermore, the light is circularly polarized so that the |↑〉 takes part in the atom-light interaction, but not the |↓〉 state. If the ion is in a superposition of the |↓〉 and |↑〉 states, only its |↑〉 component moves along the z-axis. This part of its wavefunction accumulates a phase φ because it moves through the combined potential of the trap and the other ions.⁶ Changing the relative phase between the |↓〉 and |↑〉 wave-function components is equivalent to rotating the simulated spin about the z-axis. Because this rotation depends on the relative positions of the other ions in the trap, and because the state-dependent force of 422 nm laser beams also affects those positions, the rotation angle of one spin-1/2 system depends on the quantum states of the other ions.

It is not surprising, then, that these interactions result in a spin-spin interaction term in the equations describing this system. More rigorous calculations show that the interactions due to the 422 nm pushing laser can map onto the Ising model (in which just the vertical components of the particles’ spins interact), and the XYZ model (in which all three components of spins interact) if three sets of 422 nm beams propagate along the three orthogonal trap axes.³ These two models describe some of the most fundamental interactions in quantum many-body systems, from which many more complicated systems can be derived.

Quantum Simulations

All of our simulations will follow a similar basic procedure. The system will be initialized so that all ions are in the |↓〉 state. The lasers will then be turned on for up to ~ 100 μs. After this process, we will measure the state of each ion. As an example, we could search for a quantum phase transition in a one-dimensional array of trapped ions as the simulated spin-spin interaction grows relative to an applied magnetic field. At a particular value, we expect that the final state of the ion array changes from a disordered one to either one in which the spins are aligned with each other, or one in which the direction of the spins alternate along the ion array.

We can compare our experimental results to exact calculations when we use small numbers of ions and two sophisticated computer simulations using somewhat larger systems. This comparison of moderately large physical systems with computational results will let us check the precision of our experimental system and that of our theoretical techniques as they progress together. However, even the most sophisticated computer approximations cannot simulate the dynamics of one-dimensional systems with more than roughly 30 spins, and we expect to be able to perform such experiments in the next few years.

Two-dimensional arrays of trapped ions will allow more sophisticated and complex simulations. A good first choice for a two-dimensional spin array is a two-legged ladder such as that shown in Figure 5. Inducing XYZ and Ising interactions on such a ladder will let us investigate systems that display unusual phase transitions and systems that have possible connections to high-T_c superconductivity.⁷ In fact, we can make the interactions between opposite spins of a spin ladder XY-like, while coupling nearest neighbor spins with an Ising-like interaction. The resulting quantum-spin model is equivalent to a one-dimensional fermionic Hubbard model.⁸ This is the most widely used model of a strongly interacting system in condensed-matter physics, and the ability to simulate it vastly increases our ability to use our simulator to understand materials-science problems.

Through collaboration between LANL and Sandia National Laboratories, we are designing the new ion traps that are required for simulations with tens of ions in one and two dimensions. Such traps will be microfabricated in a unique tungsten deposition process that has already made arrays of millions of micron-sized ion traps for mass-spectroscopy applications.⁹ We will push this process to build the first two-dimensional traps, to confine two parallel, interacting ion chains (a spin ladder), and to confine ions in configurations such as a hexagonal close-packed array. These microfabrication techniques can build traps that are compatible with the trap-mounted photonics, electronics, and data-acquisition systems that will be necessary for our quantum simulations with larger ion arrays.

Impact on Other Fields

What will we gain from these experiments? We have already described the sorts of first-principle understandings into materials science resulting from these physical simulations. These insights should lead to practical advances in real materials science. Historically, the ability to design and manufacture better materials has been crucial to military and industrial superiority. Developing such materials in fabrication labs with macroscopic samples is extremely expensive and time consuming, and investigating them in scientific labs is difficult because of imperfections and complications in real materials. Because of this, massive computational resources are applied to these problems, but these techniques falter when quantum-mechanical effects become important. A quantum simulator could ultimately be used to design advanced materials with new types of quantum-mechanical order. Even without using thousands of simulated spins, a quantum simulator could test the models that are used in the design of advanced materials before designers fully invest their funding and time.

Another significant benefit from these experiments is that they will represent the largest quantum computer test-bed to date. This special class of quantum computers is experimentally feasible because the stringent requirements of universal quantum computation are drastically reduced, largely because the interactions are simple and fast. The quantum simulator requires less laser stability than a universal quantum computer and less immunity from the external field noise that destroys quantum mechanical entanglement. The ions do not have to be prepared in their ground state of motion.^3,6 Indeed, the extreme demands of error correction on quantum computing are vastly reduced and probably unnecessary for simulations involving 30–50 ions.¹⁰ Consequently, we can leap from studying one and two ions for universal quantum computation to working with an order of magnitude more ions in more than one dimension. We therefore expect to gain crucial insight into the engineering and algorithmic demands of large-scale quantum computation.

Finally, even though a quantum simulator requires much less control over a physical system than a quantum computer does, it still uses the same technical resources. Both applications require traps that can hold large, two-dimensional arrays of ions. Both will ultimately require complex optical systems and compact data-acquisition systems that can function with thousands of ions, or more, simultaneously. Because the two systems share common needs, the technology developed while building a quantum simulator would lead to practical quantum computation.

In short, the work described here has the fortunate attributes of both basic, fundamental science and practical, applied technologies. We expect to contribute both to national-security interests and to the intellectual pursuits of condensed-matter and quantum-information scientists.

References

1. S. Lloyd, “Universal quantum simulators,” Science 273, 1073–1078 (1996).
2. E. Jané et al., “Simulation of quantum dynamics with quantum optical systems,” Quantum Information and Computation 3, 15–37 (2003).
3. D. Porras and J.I. Cirac, “Effective quantum spin systems with trapped ions,” Physical Review Letters 92, 207901 (2004).
4. J.P. Barjaktarevic, G.J. Milburn, and R.H. McKenzie, “Fast simulation of a quantum phase transition in an ion-trap realisable unitary map,” e-Print Archive preprint quant-ph/0401137 (22-Jan-04).
5. D.J. Berkeland, “Linear Paul trap for strontium ions,” Review of Scientific Instruments 73, 2856–2860 (2002).
6. T. Calarco, J.I. Cirac, and P. Zoller, “Entangling ions in arrays of microscopic traps,” Physical Review A 63, 062304-01–062304-20 (2001).
7. E. Dagotto and T.M. Rice, “Surprises on the way from one- to two-dimensional quantum magnets: The ladder materials,” Science 271, 618–623 (1996).
8. C.D. Batista and G. Ortiz, “Algebraic approach to interacting quantum systems,” Advances in Physics 53, 1–82 (2004).
9. M.G. Blain et al., “Towards the hand-held mass spectrometer: design considerations, simulations, and fabrication of micrometer-scaled cylindrical ion traps,” International Journal of Mass Spectrometry 236, 91–104 (2004).
10. J.I. Cirac and P. Zoller, “New Frontiers in Quantum Information with Atoms and Ions,” Physics Today 57, 38–44 (March 2004).

Acknowledgments

This work is funded through the Laboratory-Directed Research and Development program as part of 20050076DR, “Cold Atom Quantum Simulators.”

For further information, contact Dana Berkeland at 505-665-9148, djb@lanl.gov.