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Physical Review A

Generation of entanglement between qubits in a one-dimensional harmonic oscillator

E. T. Owen, M. C. Dean, C. H. W. Barnes

Published Feb 14, 2012• 14 citations
Publisher

Abstract

We present a scheme for generating entanglement between two qubits associated with a pair of particles, which move and interact, in a one-dimensional harmonic-oscillator potential. The spatial wave functions of the particles are initially localized on either side of the oscillator. As they ``fall'' toward each other, we find that the interaction between the two particles produces an effective exchange interaction between the qubits. This allows us to produce a ``power-of-swap'' operation on the initial two-qubit quantum state without requiring any time-dependent control. We find that under certain conditions, we can generate maximally entangled states from a product state with a fidelity that is only weakly dependent on initialization errors.

DOI
10.1103/physreva.85.022319

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We present a scheme for generating entanglement between two qubits associated with a pair of particles, which move and interact, in a one-dimensional harmonic-oscillator potential. The spatial wave functions of the particles are initially localized on either side of the oscillator. As they fall'' toward each other, we find that the interaction between the two particles produces an effective exchange interaction between the qubits. This allows us to produce a power-of-swap'' operation on the initial two-qubit quantum state without requiring any time-dependent control. We find that under certain conditions, we can generate maximally entangled states from a product state with a fidelity that is only weakly dependent on initialization errors.
Quantum states can contain correlations which are stronger than is possible in classical systems. Quantum information technologies use these correlations, which are known as entanglement, as a resource for implementing novel protocols in a diverse range of fields such as cryptography, teleportation and computing. However, current methods for generating the required entangled states are not necessarily robust against perturbations in the proposed systems. In this thesis, techniques will be developed for robustly generating the entangled states needed for these exciting new technologies. The thesis starts by presenting some basic concepts in quantum information proccessing. In Ch. 2, the numerical methods which will be used to generate solutions for the dynamic systems in this thesis are presented. It is argued that using a GPU-accelerated staggered leapfrog technique provides a very efficient method for propagating the wave function. In Ch. 3, a new method for generating maximally entangled two-qubit states using a pair of interacting particles in a one-dimensional harmonic oscillator is proposed. The robustness of this technique is demonstrated both analytically and numerically for a variety of interaction potentials. When the two qubits are initially in the same state, no entanglement is generated as there is no direct qubit-qubit interaction. Therefore, for an arbitrary initial state, this process implements a root-of-swap entangling quantum gate. Some possible physical implementations of this proposal for low-dimensional semiconductor systems are suggested. One of the most commonly used qubits is the spin of an electron. However, in semiconductors, the spin-orbit interaction can couple this qubit to the electron's momentum. In order to incorporate this e ffect into our numerical simulations, a new discretisation of this interaction is presented in Ch. 4 which is signi ficantly more accurate than traditional methods. This technique is shown to be similar to the standard discretisation for magnetic fields. In Ch. 5, a simple spin-precession model is presented to predict the eff ect of the spin-orbit interaction on the entangling scheme of Ch. 3. It is shown that the root-of-swap quantum gate can be restored by introducing an additional constraint on the system. The robustness of the gate to perturbations in this constraint is demonstrated by presenting numerical solutions using the methods of Ch. 4.
We propose a scheme to generate entanglement between two distant qubits (two-level atom) which are separately trapped in their own (in general) non-Markovian dissipative cavities by utilizing entangling swapping, considering the case in which the qubits can move along their cavity axes rather than a static state of motion. We first examine the role of movement of the qubit by studying the entropy evolution for each subsystem. The average entropy over the initial states of the qubit is calculated. Then by performing a Bell state measurement on the fields leaving the cavities, we swap the entanglement between qubit-field in each cavity into qubit-qubit and field-field subsystems. The entangling power is used to measure the average amount of swapped entanglement over all possible pure initial states. Our results are presented in two weak and strong coupling regimes, illustrating the positive role of movement of the qubits on the swapped entanglement. It is revealed that by considering certain conditions for the initial state of qubits, it is possible to achieve a maximally long-leaving stationary entanglement (Bell state) which is entirely independent of the environmental variables as well as the velocity of qubits. This happens when the two qubits have the same velocities.
We show that Bell's-inequality experiments can be carried out using four harmonic oscillators, pairs of which interact with each other at suitable instants of time to allow for the exchange of energy. Between interactions, the oscillators are shifted in phase relative to each other in a controllable manner. Energy becomes distributed among the four harmonic oscillators in a manner that is incompatible with a local realism picture in which information is exchanged only when the oscillators interact with each other. Entanglement is dynamically generated from an initial product state.
We study dynamics of two interacting ultracold Bose atoms in a harmonic oscillator potential in one spatial dimension. Making use of the exact solution of the eigenvalue problem of a particle in the $\ensuremath{\delta}$-like potential, we study the time evolution of an initially separable state of two particles. The corresponding time-dependent single-particle density matrix is obtained and diagonalized, and single-particle orbitals are found. This allows us to study decoherence as well as creation of entanglement during the dynamics. The evolution of the orbital corresponding to the largest eigenvalue is then compared to the evolution according to the Gross-Pitaevskii equation. We show that if initially the center of mass and relative degrees of freedom are entangled, then the Gross-Pitaevskii equation fails to reproduce the exact dynamics and entanglement is produced dynamically. We stress that predictions of our study can be verified experimentally in an optical lattice in the low-tunneling limit.
We investigate finite-number effects in collisions between two states of an initially well-known number of identical bosons with contact interactions, oscillating in the presence of harmonic confinement in one dimension. We investigate two N/2N/2N/2 (interacting) ground states, which are initially displaced from the trap center, and the effects of varying interaction strength. The numerics focus on the simplest case of N=4N=4N=4. In the noninteracting case, such a system would display periodic oscillation with a half harmonic oscillator period (due to the left-right symmetry). With the addition of contact interactions between the bosons, collisions generate entanglement between each of the states and distribute energy into other modes of the oscillator. We study the system numerically via an exact diagonalization of the Hamiltonian with a finite basis, investigating left-right number uncertainty as our primary measure of entanglement. Additionally, we study the time evolution and equilibration of the single-body von Neumann entropy for both the attractive and repulsive cases. We identify parameter regimes for which attractive interactions create behavior qualitatively different from that of repulsive interactions, due to the presence of bound states (quantum solitons), and explain the processes behind this.
The Schrödinger--Newton (SN) equation provides a semiclassical framework for the evolution of self-gravitating of massive quantum systems. We propose a two-body Schrödinger--Newton model that separates local nonlinear self-localization from the nonseparable Newtonian pair potential. Analytically, we show that the nonlinear self-field preserves the Schmidt spectrum, whereas direct entanglement generation arises from the nonseparable pair potential. Using numerical simulations in a regularized one-dimensional geometry, we find that entanglement generation depends sensitively on the initial spatial configuration and on the mass ratio. Highly localized, self-bound wavepackets experience minimal entanglement growth during scattering. Spatial delocalization and kinetic dispersion broaden the interaction region, amplifying the entangling power of the pair potential and exciting higher-order spatial modes. For dispersive Gaussian initial states, mass asymmetry shatters the lighter particle, producing Wigner negativity and rapid entanglement growth, whereas stationary SN profiles strongly suppress this effect. Stationary SN profiles isolate the bare pair-potential contribution; dispersive Gaussian initial states inflate it.