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

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.

Figure 1

Energy shift of the even states of an harmonic oscillator due to a perturbing interaction, $V\left(r\right)=U\delta \left(r\right)$.

Figure 2

Fidelity to the maximally entangled state, $|{\Psi }^{-i}\rangle$, for a range of contact interaction potentials, $V\left(r\right)=U\delta \left(r\right)$. Note that the “dark spots” are actually entangled states with high fidelity, but to the orthogonal state $|{\Psi }^{+i}\rangle$. For large potentials, after a few oscillations, the wave function scatters out of the basis $\left\{\right|L_{{\varsigma }_1}{R}_{{\varsigma }_2}\rangle ,|L_{{\varsigma }_2}{R}_{{\varsigma }_1}\rangle \}$, so the periodic behavior is suppressed.

Figure 3

Fidelity to the maximally entangled state, $|{\Psi }^{-i}\rangle$, after a single oscillation for (a) Gaussian and (b) softened Coulomb interactions with varying characteristic length scales. The contact interaction is also shown as the limit where the interaction length tends to zero. A maximally entangled state is achieved for a broad range of potentials with high fidelity. The classical limit is given by the horizontal line, which is equivalent to the initial product state. Note that to let the interaction energy vary for a Coulomb potential, the frequency of the harmonic oscillator is varied. This changes the width and displacement of the initial state.