Single impurity atom embedded in a dipolar two-soliton molecule as a qubit

TL;DR

This work proposes a qubit realization based on a single impurity atom embedded in a dipolar two-soliton molecule that forms a self-induced double-well potential. A variational approach is used to obtain the soliton molecule profile, and a numerical solution of the impurity spectrum reveals a near degenerate ground and first excited state whose splitting provides the qubit frequency; a two-mode model then yields Josephson-like oscillations between the left and right wells that are confirmed by full simulations. The two lowest impurity states span a robust qubit subspace, with the observed dynamics matching the two-mode predictions to high accuracy. The setup offers tunable qubit frequency via the interaction strengths and shows exceptionally high state purity, suggesting feasibility in current dipolar BEC platforms such as Cr, Dy, or Er.

Abstract

We consider a single impurity atom trapped in a double well (DW) potential created by a dipolar two-soliton molecule in a quasi-one-dimensional geometry. By solving the eigenvalue problem for the impurity atom in the DW potential, we find that its ground and first excited states are well separated from higher excited states. This allows it to be approximated by a desirable two-level quantum system. Numerical simulations of the Schrödinger equation, governing impurity atom, demonstrate periodic oscillations in the probability of finding the impurity confined either to the ``left" or to the ``right" side of the DW potential. An analytic expression for the coherent oscillations of the population imbalance between the two wells of the DW potential has been derived using the two-mode approximation. Theoretical predictions of the mathematical model are in good agreement with the results of numerical simulations. Potential usage of the developed setup as a physical realization of ``qubit" has been discussed.

Figures (5)

• Figure 1: (a) The effective potential given by Eq. (\ref{['pot']}). (b) The wave profile of the two-soliton molecule obtained by numerical solution of the GPE (red line) and according to VA (blue dashed line). Parameter values $A=0.177$, $a=3.299$ follow from VA Eq. (\ref{['att']}) for $q=1$, $g=6$, $w=5$, $N=1$.
• Figure 2: (a) The wave profiles of the symmetric ground state with $\mu_0 = -0.326$ (blue), and the antisymmetric first excited state with $\mu_1=-0.312$ (yellow) of the impurity atom in the double well potential. (b) Localized combinations constructed from the linear superposition of the ground and first excited states of the system according to Eq. (\ref{['lr']}). For visual convenience the double well potential $V(x)=-\gamma |\psi(x)|^2$ with $\gamma=4$ is also shown by a dashed black line. The parameter values are similar to Fig. \ref{['fig1']}.
• Figure 3: (a) The DW potential $\sim |\psi(x,t)|^2$ created by the dipolar two-soliton molecule remains stationary as time advances. (b) The density plot for $|\phi(x,t)|^2$ shows regular oscillations of the probability of finding the impurity atom on the left/right well. (c) The corresponding population imbalance $z(t)=\int_{x<0}|\phi(x,t)|^{2}dx - \int_{x>0}|\phi(x,t)|^{2}dx$ demonstrates coherent Josephson-type oscillations. The numerically measured oscillation period $T_{\mathrm{meas}} = 223.506$ agrees closely with the theoretical prediction $T_{\mathrm{pred}} = 223.125$ obtained from the two-mode model [Eq. (20)]. The parameters used are $g = 10$, $q = 1$, $w=5$, $\gamma = 3$, and $N = 10$.
• Figure 4: The frequency of coherent oscillations of the impurity atom between the two wells of the double-well (DW) potential as a function of the interaction strengths in the dipolar condensate. (a) Dependence of the oscillation frequency on the contact interaction parameter $q$ for fixed $g = 10$, $w=5$, $\gamma = 3$, and $N = 10$. (b) Dependence on the dipolar (nonlocal) interaction strength $g$ for fixed $q = 1$, $w=5$, $\gamma = 3$, and $N = 5$. Blue circles represent the numerically obtained oscillation frequencies, while the yellow dashed curves correspond to the analytical predictions based on the variationally obtained width parameter $a_{0}$ from Eq. (8).
• Figure 5: Left panel: The Bloch sphere represents a qubit state. The north pole corresponds to the ground state $\phi_0$ and the south pole corresponds to the first excited state $\phi_1$. An arbitrary superposition of these basis states can be obtained by parametrization $\phi = \cos(\theta/2)|\phi_0\rangle + e^{i \varphi} \sin(\theta/2)|\phi_1\rangle$. Right panel: The Bloch components $r_x \simeq \cos(\Delta t)$, $r_y \simeq \sin(\Delta t)$, $r_z \simeq 0$ show circular equatorial motion (upper). The purity of the two-level system calculated as ${\rm Tr}(\rho^2)$, with $\rho$ being the density matrix, remains very close to unity indicating negligible de-coherence (lower). The parameters used are $g = 10$, $q = 1$, $w=5$, $\gamma = 3$, and $N = 10$.