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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Phonons in a double-well: transverse vibrations in a pair of trapped ions

Patricia J. Lee

Published May 1, 2012• 0 citations
Publisher

Abstract

Double-well potentials have extensive applications in many branches of physics and have resulted in devices that are useful for precision sensing as well as quantum information processing. Here, we present a theoretical analysis of the transverse phonon coupling between a pair of trapped ions as described by a double-well system. Typically the phonons in trapped ions are used as an intermediary in quantum information processing, but here we focus on the dynamics of the phonons themselves. The double-well system is a special case of the more generalized Bose-Hubbard Model, with exactly two sites available to store particles. We draw an analogy to another double-well system in solid state physics: a Josephson junction containing a thin insulating barrier between two superconductors. Applications of the Josephson junction include superconducting quantum interference devices (SQUIDs), which have emerged as sensitive magnetometers and also as qubits for quantum computation. Another analogy is the recent demonstrations of matter-wave interference with Bose-Einstein condensates in double-well traps, which have also shown great promise for applications in precise inertial and gravitational field sensing. We will also discuss experimental techniques that can be applied to control and manipulate this double-well system.

DOI
10.1117/12.918940

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Double-well potentials have extensive applications in many branches of physics and have resulted in devices that are useful for precision sensing as well as quantum information processing. Here, we present a theoretical analysis of the transverse phonon coupling between a pair of trapped ions as described by a double-well system. Typically the phonons in trapped ions are used as an intermediary in quantum information processing, but here we focus on the dynamics of the phonons themselves. The double-well system is a special case of the more generalized Bose-Hubbard Model, with exactly two sites available to store particles. We draw an analogy to another double-well system in solid state physics: a Josephson junction containing a thin insulating barrier between two superconductors. Applications of the Josephson junction include superconducting quantum interference devices (SQUIDs), which have emerged as sensitive magnetometers and also as qubits for quantum computation. Another analogy is the recent demonstrations of matter-wave interference with Bose-Einstein condensates in double-well traps, which have also shown great promise for applications in precise inertial and gravitational field sensing. We will also discuss experimental techniques that can be applied to control and manipulate this double-well system.
We theoretically investigate the properties of a double-well bosonic Josephson junction coupled to a single trapped ion. We find that the coupling between the wells can be controlled by the internal state of the ion, which can be used for studying mesoscopic entanglement between the two systems and to measure their interaction with high precision. As a particular example we consider a single ^{87}Rb atom and a small Bose-Einstein condensate controlled by a single 171Yb+ ion. We calculate interwell coupling rates reaching hundreds of Hz, while the state dependence amounts to tens of Hz for plausible values of the currently unknown s-wave scattering length between the atom and the ion. The analysis shows that it is possible to induce either the self-trapping or the tunneling regime, depending on the internal state of the ion. This enables the generation of large scale ion-atomic wave packet entanglement within current technology.
One of the most striking manifestations of the quantum collective behavior of matter is the Josephson effect: it allows for a classically forbidden current to flow without dissipation [125]. Traditional Josephson junctions are built by means of superconductors separated by thin non-conducting oxide barriers and they have drawn an intense interest for their striking phenomenology. A bosonic analog is the so-called Bosonic Josephson junction (BJJ) where two macroscopic populations of bosons are trapped in a double well geometry. Bose-Einstein condensation of dilute atomic gases has allowed for the demonstration of BJJs by trapping two condensates in double wells [126],[127].
We apply modern techniques from quantum optics and quantum information science to Bose-Einstein condensates (BECs) in order to study, for the first time, the quantum decoherence of phonons of isolated BECs.In the last few years, major advances in the manipulation and control of phonons have highlighted their potential as carriers of quantum information in quantum technologies, particularly in quantum processing and quantum communication.Although most of these studies have focused on trapped ion and crystalline systems, another promising system that has remained relatively unexplored is BECs.The potential benefits in using this system have been emphasized recently with proposals of relativistic quantum devices that exploit quantum states of phonons in BECs to achieve, in principle, superior performance over standard nonrelativistic devices.Quantum decoherence is often the limiting factor in the practical realization of quantum technologies, but here we show that quantum decoherence of phonons is not expected to heavily constrain the performance of these proposed relativistic quantum devices.
In this thesis, I present the first experimental implementation of a Josephson junction for Bose-Einstein condensates. The weak link between two BECs constitutes the nonlinear generalization of the well known Josephson junction of weakly-coupled superconductors that are separated by a thin insulating barrier. In our experiment, the required overlap of two macroscopic wavefunctions is provided by loading a BEC into an optical double well potential. It is realized by a superposition of a one-dimensional optical lattice with a focused laser beam optical dipole trap. The tunneling dynamics between the two potential wells exhibits two distinct dynamical regimes. For small initial population imbalances of the two wells we observe nearly sinusoidal Josephson tunneling oscillations, which are characterized by an oscillating population and relative phase. The situation changes drastically, if the initial population imbalance is chosen above a critical value. In this case, resonant tunneling between the two wells is prohibited because the difference between the on-site particle interaction energies in the two wells exceeds the tunneling energy splitting. As a consequence, the atomic distribution becomes self-locked and the relative phase evolves unbound in time. This regime of prohibited tunneling, which has no analogon in superconducting Josephson junctions, is called “macroscopic quantum self-trapping”.
We study the dynamics of matter waves in an effectively one-dimensional Bose-Einstein condensate in a double well potential. We consider in particular the case when one of the double wells confines excited states. Similarly to the known ground state oscillations, the states can tunnel between the wells experiencing the physics known for electrons in a Josephson junction, or be self-trapped. As the existence of dark solitons in a harmonic trap are continuations of such non-ground state excitations, one can view the Josephson-like oscillations as tunnelings of dark solitons. Numerical existence and stability analysis based on the full equation is performed, where it is shown that such tunneling can be stable. Through a numerical path following method, unstable tunneling is also obtained in different parameter regions. A coupled-mode system is derived and compared to the numerical observations. Regions of (in)stability of Josephson tunneling are discussed and highlighted. Finally, we outline an experimental scheme designed to explore such dark soliton dynamics in the laboratory.
Bose Einstein condensates (BECs) in double well potentials have drawn a lot of attention for the possibilities they offer to study fundamental quantum mechanical effects at the macroscopic level as well as for potential applications like interferometry, high precision measurements or thermometry. In contrast to Josephson junctions realized in superconductors and superfluids, in BEC the interatomic interactions play a crucial role leading to anharmonic Josephson oscillations, if the initial population imbalance of the two wells is below a critical value, and to macroscopic quantum self-trapping i.e, inhibition of large amplitude Josephson oscillations above the threshold for the population imbalance. In this paper, a novel technique to adiabatically control of BEC in a double-well potential is proposed. The adiabatic dynamics of a BEC in a double well potential can be described in terms of a dark variable resulting from a combination of the population imbalance and the spatial atomic coherence.