A Double-Well System Composed of Phonons in a 

Pair of Trapped Ions

by Patricia J. Lee

ARL-TR-5781 September 2011

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Army Research Laboratory

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ARL-TR-5781 September 2011

A Double-Well System Composed of Phonons in a 

Pair of Trapped Ions

Patricia J. Lee

Sensors and Electron Devices Directorate, ARL

Approved for public release; distribution unlimited. 
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A Double-Well System Composed of Phonons in a Pair of Trapped Ions

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Patricia J. Lee

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14. ABSTRACT

This report presents a theoretical framework using a double-well system to describe the transverse phonon coupling between a 

pair of trapped ions. The double-well system is a special case of the more generalized Bose-Hubbard Model, with exactly two 

sites available to store particles. Analogous double-well systems in solid-state junctions and in Bose-Einstein condensates 

have been used for interferometric sensing, as well as quantum information processing, and many exotic quantum phenomena 

such as coherent tunneling and the Josephson Effect also arise naturally in such systems. This perspective offers insight into 

possible methods to control phonons in trapped ions, which are commonly used as a resource for quantum computation and 

quantum simulation.

15. SUBJECT TERMS

Atmom interferometry, sensors, quantum information, double well, Bose-Hubbard model

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Patricia J. Lee

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iii

Contents

List of Figures iv

1. Introduction 1

2. Transverse Phonons in a Pair of Trapped Ions 1

3. Experimental Considerations 3

4. Conclusion 4

5. Bibliography 5

Distribution List 6
iv

List of Figures

Figure 1. Two ions in a linear RF Paul trap. Phonons are the ion’s vibrational excitation in a 

harmonic potential, and are coupled between the ions by the Coulomb force. .........................2
1

1. Introduction

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. An example of a physical system containing a double-well potential is a Josephson 

junction containing a thin insulating barrier between two superconductors (1), and 

superconducting quantum interference devices (SQUIDs) based on superconducting loops 

containing Josephson junctions have been used as sensitive magnetometers (2). SQUIDs can 

also be used to store qubits for quantum computation (3). In atomic physics, recent 

demonstrations of matter-wave interference with Bose-Einstein condensates in double-well traps 

have also shown great promise for applications in precise inertial and gravitational field sensing

(4).

Given the usefulness of double-well systems, this report will examine the transverse phonon 

coupling between a pair of trapped ions and show a direct mapping to a double-well system. 

Such a comparison has never been presented in the literature previously, but should be extremely 

valuable in gaining an understanding of what quantum behaviors can be expected and how to 

control and manipulate such a system. This system is of interest to many quantum applications 

since transverse phonons in trapped ions are now regularly used in the laboratory as a data bus 

for quantum computation and for quantum simulation (5). Previous work has already shown that 

the transverse phonons in a long string of trapped ions can be modeled by the Bose-Hubbard 

Hamiltonian (6). This report will follow the same methodology to analyze the special case with 

exactly two ions in the trap. 

2. Transverse Phonons in a Pair of Trapped Ions

Let us consider a pair of trapped ions confined by an oscillating RF field in a Paul trap exerting

mutually repulsive Coulomb force on each other, as shown in figure 1. The time-averaged 

potential around the equilibrium points is harmonic in the lowest order, with a corresponding 

Hamiltonian

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

where the first term is the kinetic energy of the ions, the second term is the trapping potential, 

and the last term is the Coulomb interaction. and are the momentum and the displacement 

of the i-th ion, respectively, and , , are the harmonic frequencies of the trap in the 

respective x, y, and z directions. The ions line up along the weakest trapping (axial) direction z, 
2

and the trapping potential is stronger in the orthogonal (transverse) directions x and y. For 

simplicity, we require that the trapping frequencies in all three directions are non-degenerate, so 

that motion along x, y, or z do not couple to one another. 

Figure 1. Two ions in a linear RF Paul trap. Phonons are the ion’s vibrational excitation in a 

harmonic potential, and are coupled between the ions by the Coulomb force. 

Here we will focus only on vibration in one particular transverse direction x. At low 

temperature, the ion displacement from equilibrium point is small compared to the distance 

between ions, and under realistic experimental conditions, the Coulomb energy is small 

compared to the potential energy. Phonons are defined in the usual way, as in the i-th ion has 

phonons in the x direction if the vibrational state of the i-th ion is in the -th Fock state 

 in 

the corresponding harmonic potential. The transverse phonon potential in the x direction can be 

approximated to lowest order by

 

 

 

 

 

 

 

 

 

where

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

The term accounts for the spatially dependent shift of the trapping frequency, is the 

tunneling energy, and 

 

is the equilibrium position in z of the i-th ion. The operators 

 

and 

are the creation and annihilation operators defined by the relation 

 

). 

The first term of the Hamiltonian is the energy associated with the number of phonons in 

each site, according the number operator 

 

 . The second term of the Hamiltonian is the 

tunneling energy, where a phonon is destroyed from one site and created in the other site by the 

operators 

 

 . In the basis 

 

 

 , the Hamiltonian is represented by the matrix

 

 

 

 

The phonons in the two sites (double-well) are coupled by the off-diagonal term , and an 

initial phonon population in one site will oscillate between the two sites at frequency (7). 
3

This is directly analogous to Rabi oscillation in a two-level system or to Josephson oscillation 

where there is a barrier between two superconductors.

3. Experimental Considerations

The transverse phonon energy for ions in an RF Paul trap depends on the amplitude of the 

oscillating electric fields at the ions’ locations, and can be tuned by changing the RF power and 

electrode distances. The coupling strength depends on the strength of the axial confinement, 

which, for a linear trap, can be tuned with DC voltages at endcap electrodes. Let us consider two 

different regimes: (i) a decoupled system, where the tunneling energy is negligible compared to 

phonon energy, i.e., ; and (ii) a strongly-coupled system, where the tunneling energy is 

significant compared to phonon energy, i.e. . In the first case, the ions are far apart from 

each other such that the Coulomb interaction is small compared to the phonon energy. Phonons 

can be created or destroyed independently at each site using Raman sideband transitions (8), 

assuming the phonon excitations can be coherently driven on a time scale much faster than . 

The internal spin of the ions is typically used for adding exactly a fixed number of phonons to 

the system at a time (9), but coherent states can also be generated using a or force (10). 

Individual addressing of the ions should be trivial given that the separation between the ions is 

large in this regime. In the second case, the strong coupling between the phonons at two sites 

means that tunneling cannot be ignored. By diagonalizing the Hamiltonian, we find that the 

stationary states are comprised of the collective motion of both ions and have energies , 

which are respectively labeled as the center-of-mass mode for the symmetric motion and the 

stretch mode for the antisymmetric motion. In this regime, it is best to directly excite these 

common modes of motion using Raman sideband transition, instead of exciting the uncoupled 

phonon modes since the tunneling time is on the order of the sideband transition time and will 

make clean transitions to the uncoupled phonon modes difficult. 

The system can, in principle, switch between the two cases previously described by ramping the 

axial trap strength, and with negligible unintended transverse phonon excitation in the adiabatic 

regime, where during the transition the rate of change in the Hamiltonian is much less than the 

transverse phonon excitation energy, i.e. 

 

. The major difference between the 

trapped ion system and other double-well systems such as Josephson junctions and Bose-Einstein 

condensates is that phonons in a harmonic trap do not interact with each other and, therefore, has 

no additional energy cost to place multiple phonons at the same site. This is in sharp contrast to 

electrons or neutral atoms, which do interact with each other and will result in an energy shift 

that depends on the number of particles located at the same site. With neutral atoms, it is 

possible to turn off atom-atom interaction through a Feshbach resonance, but requires additional 

effort to reach high magnetic fields with fine tuning and high stability. Therefore, we expect a 

Rabi-like oscillation in the phonon population of each site in the trapped ion system, while 
4

previous experiments with Bose-Einstein condensates in a double-well have observed a 

macroscopic quantum self-trapping due to inter-particle interaction (11). 

For a more direct analogy between trapped ions and other double-well systems, on-site energy 

shift can be artificially generated by adding anharmonicity to the trapping potential or by 

inducing a light shift at the ions’ locations, as described in reference (6). In such a setup, where 

tunneling and on-site energy 

 

 

 

can be independently controlled, if the system 

transforms from the strongly coupled regime to the decoupled regime adiabatically with respect 

to both and , we expect to observe phonon number squeezing analogous to the superfluid to 

Mott-Insulator quantum phase transition predicted for a long string of ions (6). This 

phenomenon has already been observed with neutral bosonic atoms in a double-well optical 

lattice (12), where atoms initially in a single well are adiabatically transformed to an equal 

number of atoms in each site of the double-well. Using the same principle, we also expect to 

create a coherent state of phonons in the two sites if the system is transformed from the strongly

coupled regime to the decoupled regime diabatically. This phenomenon also arises from the 

Bose-Hubbard theory (13) and has been observed in neutral atoms (12).

4. Conclusion 

This report has shown a direct mapping of the transverse phonon coupling between two trapped 

ions to a double-well system, and described how to experimentally access some of the rich 

quantum phenomenology in trapped ions that arises from double-well physics. Besides being 

able to control the phonons through number squeezing, further studies of this system may yield

other potential quantum applications. 
5

5. Bibliography

1. Josephson, B. D. Possible New Effects in Superconductive Tunnelling. Physics Letters

1962, 1, 251.

2. Jaklevic, R. C., et al. Quantum Interference Effects in Josephson Tunneling. Physical 

Review Letters 1964, 12, 159.

3. Bouchiat, V., et al. Quantum Coherence with a Single Cooper Pair. Physica Scripta 1998, 

T76, 165.

4. Schumm, T, et al. Matter-wave Interferometry in a Double Well on an Atom Chip. Nature 

Physics 2005, 1, 57–62.

5. Kim, K., et al. Entanglement and Tunable Spin-Spin Couplings between Trapped Ions Using 

Multiple Transverse Modes. Physical Review Letters 2009, 103, 120502.

6. Porras, D.; Cirac, J. I. Bose-Einstein Condensation and Strong-correlation Behavior of 

Phonons in Ion Traps. Physical Review Letters 2004, 93, 263602.

7. Smerzi, A, et al. Quantum Coherent Atomic Tunneling between Two Trapped Bose-Einstein 

Condensates. 1997, 79, 4950.

8. Deslauriers, L., et al. Zero-point Cooling and Low Heating of Trapped 111Cd+ Ions.

Physical Review A 2004, 70, 043408.

9. Monroe, C., et al. Resolved-sideband Raman Cooling of a Bound Atom to the 3D Zero-point 

Energy. Physical Review Letters 1995, 75, 4011.

10. Lee, P. J., et al. Phase Control of Trapped Ion Quantum Gates. Jounal of Optics B 2005, 7, 

S371.

11. Albiez, Michael, et al. Direct Observation of Tunneling and Nonlinear Self-trapping in a 

Single Bosonic Josephson Junction. Physical Review Letters 2005, 95, 010402.

12. Sebby-Strabley, J., et al. Preparing and Probing Atomic Number States with an Atom 

Interferometer. Physical Review Letters 2007, 98, 200405.

13. Greiner, Markus, et al. Collapse and Revival of Matter Wave Field of a Bose-Einstein 

Condensate. Nature 2002, 419, 51.
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