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Creation of two-dimensional Coulomb crystals of ions in oblate Paul traps for quantum simulations
© Yoshimura et al.; licensee Springer on behalf of EPJ 2014
Received: 2 June 2014
Accepted: 22 October 2014
Published: 4 January 2015
We develop the theory to describe the equilibrium ion positions and phonon modes for a trapped ion quantum simulator in an oblate Paul trap that creates two-dimensional Coulomb crystals in a triangular lattice. By coupling the internal states of the ions to laser beams propagating along the symmetry axis, we study the effective Ising spin-spin interactions that are mediated via the axial phonons and are less sensitive to ion micromotion. We find that the axial mode frequencies permit the programming of Ising interactions with inverse power law spin-spin couplings that can be tuned from uniform to with DC voltages. Such a trap could allow for interesting new geometrical configurations for quantum simulations on moderately sized systems including frustrated magnetism on triangular lattices or Aharonov-Bohm effects on ion tunneling. The trap also incorporates periodic boundary conditions around loops which could be employed to examine time crystals.
Using a digital computer to predict the ground state of complex many-body quantum systems, such as frustrated magnets, becomes an intractable problem when the number of spins becomes too large. The constraints on the system’s size become even more severe if one is interested in the (nonequilibrium) quantum dynamics of the system. Feynman proposed the use of a quantum-mechanical simulator to efficiently solve these types of quantum problems . One successful platform for simulating lattice spin systems is the trapped ion quantum simulator, which has already been used to simulate a variety of scenarios [2–13]. In one realization , ions are cooled in a trap to form a regular array known as a Coulomb crystal and the quantum state of each simulated spin can be encoded in the internal states of each trapped ion. Laser illumination of the entire crystal then can be used to program the simulation (spin-spin interactions, magnetic fields, etc.) via coupling to phonon modes, and readout of the internal ion states at the end of the simulation corresponds to a projective measurement of each simulated spin on the measurement basis.
To date, the largest number of spins simulated in this type of device is about 300 ions trapped in a rotating approximately-triangular lattice in a Penning trap . In that experiment, a spin-dependent optical dipole force was employed to realize an Ising-type spin-spin coupling with a tunable power law behavior. Although the Penning trap can implement a transverse magnetic field to rotate the ions, the Penning trap simulator was not able to apply a time-dependent transverse magnetic field to perform certain desirable tasks such as the adiabatic state preparation of the ground state of a frustrated magnet. The reason why the transverse field has not been tried yet is that the Penning trap cannot be cooled to the ground state for the phonons, and the presence of phonons causes problems with the quantum coherence of the system. Improvements of the experiment are currently in progress which may allow for a transverse field Ising model simulation in the near future. The complexity of the Penning trap apparatus also creates a barrier to adoption and therefore does not seem to be as widespread as radio-frequency (RF) Paul traps.
Experiments in linear Paul traps have already performed a wide range of different quantum simulations within a one-dimensional linear crystal [2–9]. The linear Paul trap is a mature architecture for quantum information processing, and quantum simulations in linear chains of ions have benefited from a vast toolbox of techniques that have been developed over the years. Initially, the basic protocol  was illustrated in a two-ion trap , which was followed by a study of the effects of frustration in a three-ion trap . These experiments were scaled up to larger systems first for the ferromagnetic case  and then for the antiferromagnetic case . Effective spin Hamiltonians were also investigated using a Trotter-like stroboscopic approach . As it became clear that the adiabatic state preparation protocol was difficult to achieve in these experiments, ion trap simulators turned to spectroscopic measurements of excited states  and global quench experiments to examine Lieb-Robinson bounds and how they change with long-range interactions (in both Ising and XY models) [8, 9].
It is therefore desirable to be able to use the demonstrated power of the Paul trap systems to extend access to the 2D physics that is native to the Penning trap systems. However, extension of this technology to higher dimensions is hampered by the fact that most ions in 2D and 3D Coulomb crystals no longer sit on the RF null. This leads to significant micromotion at the RF frequency and can couple to the control lasers through Doppler shifts if the micromotion is not perpendicular to the laser-illumination direction, leading to heating and the congestion of the spectrum by micromotion sidebands. However, the micromotion may not cause as many problems as previously thought. It has recently been shown that a robust quantum gate can be implemented even in the presence of large micromotion .
In an effort to utilize the desirable features of the Paul trap system to study the 2D physics, arrays of Paul traps are being pursued [17–22]. It has also been shown that effective higher-dimensional models may be programmed into a simulator with a linear crystal if sufficient control of the laser fields can be achieved . In this article, we study an alternative approach to applying Paul traps to the simulation of frustrated 2D spin lattices. We consider a Paul trap with axial symmetry that forms an oblate potential, squeezing the ions into a 2D crystal. The micromotion in this case is purely radial due to symmetry, and lasers that propagate perpendicular to the crystal plane will therefore not be sensitive to Doppler shifts from micromotion. We study the parameter space of a particular model trap geometry to find the crystal structures, normal modes, and programmable spin-spin interactions for 2D triangular crystals in this trap. We find a wide parameter space for making such crystals, and an Ising spin-spin interaction with widely-tunable range, suitable for studying spin frustration and dynamics on triangular lattices with tens of ions.
In the future, we expect the simulation of larger systems to be made possible within either the Penning trap, or in linear Paul traps that can stably trap large numbers of ions. It is likely that spectroscopy of energy levels will continue to be examined, including new theoretical protocols . Designing adiabatic fast-passage experiments along the lines of what needs to be done for the nearest-neighbor transverse field Ising model  might improve the ability to create complex quantum ground states. Motional effects of the ions are also interesting, such as tunneling studies for motion in the quantum regime . It is also possible that novel ideas like time crystals [27, 28] could be tested (although designing such experiments might be extremely difficult).
The organization of the remainder of the paper is as follows: In Section 2, we introduce and model the potential energy and effective trapping pseudopotential for the oblate Paul trap. In Section 3, we determine the equilibrium positions and the normal modes of the trapped ions, with a focus on small systems and how the geometry of the system changes as ions are added in. In Section 4, we show representative numerical results for the equilibrium positions and the normal modes. We then show numerical results for the effective spin-spin interactions that can be generated by a spin-dependent optical dipole force. In Section 5, we provide our conclusions and outlook.
2 Oblate Paul trap
The quantum simulator architecture we study here is based on a Paul trap with an azimuthally symmetric trapping potential that has significantly stronger axial confinement than radial confinement, which we call an ‘oblate Paul trap’ since the resulting effective potential resembles an oblate ellipsoid. As we show below, this can create a Coulomb crystal of ions that is a (finite) two-dimensional triangular lattice. 2D Coulomb crystals in oblate Paul traps have been studied since the 1980’s and were used, for instance, by the NIST Ion Storage Group  to study spectroscopy [30, 31], quantum jumps , laser absorption [33, 34] and cooling processes . 2D crystals in oblate Paul traps have also been studied by other groups both experimentally [36, 37] and theoretically [38–44].
where A is a constant and are the perpendicular unit vectors with . Using a static electric field with a saddle-point, both Penning and radio-frequency (RF) Paul traps have successfully trapped charged particles in free space by applying an additional field. In the Penning trap, one applies a strong uniform magnetic field, such that the charged particles are confined to a circular orbit via the Lorentz force, . The RF Paul trap applies a time-varying voltage to its electrodes, which produce a saddle potential that oscillates sinusoidally as a function of time. This rapid change of sign allows for certain ions to be trapped because for particular charge to mass ratios, the effective focusing force is stronger than the defocussing force.
Since the effective potential energy is just a function of , it is rotationally symmetric around the -axis and we would expect there to be a zero frequency rotational mode in the phonon eigenvectors. That mode can be lifted from zero by breaking the symmetry, which can occur by adding additional fields that do not respect the cylindrical symmetry, and probably occur naturally due to imperfections in the trap, the optical access ports, stray fields, etc.
3 Equilibrium structure and normal modes
There are two sets of normal modes: eigenvectors of the matrix yield the ‘axial’ modes (those corresponding to motion perpendicular to the crystal plane) and eigenvectors of the matrix , , yield the ‘planar’ modes (those corresponding to ion motion in the crystal plane).
4.1 Equilibrium configurations
4.2 Normal modes
After determining the equilibrium positions, we can find the spring constants and then solve the eigenvalue problem to find the normal modes. Note that due to rotational symmetry, there always is a zero frequency planar mode corresponding to the free rotation of the crystal. In an actual experiment, however, we expect that the rotational symmetry of the trap will be broken by stray fields, the radial optical access tunnels, imperfections in the electrodes, etc., so that mode will be lifted from zero.
4.3 Ising spin-spin interaction
We calculate the spin-spin coupling, using Eq. (25), for a small number of ions in order to neglect the micromotion and assume the ions are at their average position. The micromotion will be approximately equal to the displacement of the ion from the RF null times the Mathieu q-parameter . The ions with the largest micromotion will occur in the outermost ‘ring’ of the crystal. For N ions and an average inter-ion spacing s, the ions in the outermost ring are a distance from the center, which means that the amplitude of the micromotion is approximately , and the ratio of this micromotion amplitude to the inter-ion spacing is therefore .
The trap could also be use the recent work on a controlled phase flip (CPF) quantum gate that includes the corrections due to the micromotion and is optimized to attain high fidelity . The high fidelity is obtained by dividing the time of the gate into m equal segments, during each segment of equal time a particular Rabi frequency is applied, ( ), that has been optimized for the βth segment. The optimized are determined by solving the motional dynamics, that explicitly take into account all the micromotion contributions. The motion of the ions are described by a set of coupled time-dependent Mathieu equations.
4.4 Quantum motional effects
The trap could also be used to examine different types of quantum motional effects of ions, similar to the recent work on the Aharonov-Bohm effect . In order to examine such effects, one would need to cool the system to nearly the ground state. This can be accomplished by including Raman side-band cooling after Doppler cooling the system for all modes except the soft rotational mode, at least when the potential is large enough that the mode frequencies are sizeable. To cool the rotational mode, one would need to add a perturbation to the system that lifts the phonon mode frequency, side-band cool it, and then adiabatically reduce the frequency by removing the perturbation. This procedure will cool off that phonon mode, which can yield quite small quanta in it . Once the system has been prepared in this state, then quantum tunneling effects, or coherent motional effects could be studied in the trap for a range of different ion configurations. It might also be interesting to extend these types of studies to cases where the ions no longer lie completely in one plane, but have deformed into a full three-dimensional structure (as long as the larger micromotion does not cause problems). Finally, many of these ideas would need to be used if one tried to examine time crystals, especially the cooling of the rotational mode to be able to see quantum effects [27, 28].
In this work, we have studied 2D ion crystals in an oblate Paul trap for use in quantum simulations. With this system, one can trap a modest number of ions in 2D planar structures that are likely to be highly frustrated without needing a Penning trap, providing a controlled way to study the onset of frustration effects in quantum simulations. We calculated the equilibrium positions and the phonon frequencies for the proposed oblate Paul trap over its stable region. The equilibrium positions with form a single ring configuration and could potentially be used to study periodic boundary conditions and the Aharonov-Bohm effect when (and possibly time crystals). Once , the equilibrium configurations have multiple rings that are nearly formed from triangular lattices. One can generate an effective Ising Hamiltonian by driving axial modes with a spin-dependent optical dipole force. When detuning is to the blue of the axial center-of-mass mode, the spin-spin coupling, , has an approximate power law that is within the expected range of 0 to 3. In the future, as this trap is tested and performs simulations of spin models with ions, the work presented here will be critical to determining the parameters of the Hamiltonian and for selecting the appropriate configurations to use in the different simulations.
We thank Dr. Philippe Bado, Dr. Mark Dugan and Dr. Christopher Schenck of Translume (Ann Arbor, MI) for valuable discussions. B.Y. acknowledges the Achievement Rewards for College Scientists Foundation for supporting this work. M.S. acknowledges the National Science Foundation under grant number DMR-1004268 for support. J.K.F. and B.Y. acknowledge the National Science Foundation under grant number PHY-1314295 for support. D.D. and W.C.C. acknowledge support from the U.S. Air Force Office of Scientific Research Young Investigator Program under grant number FA9550-13-1-0167 and support from the AFOSR STTR Program. J.K.F. also acknowledges support from the McDevitt bequest at Georgetown University.
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