Tunneling-induced angular momentum for single cold atoms

We study the generation of angular momentum carrying states for a single cold particle by breaking the symmetry of a spatial adiabatic passage process in a two-dimensional system consisting of three harmonic potential wells. By following a superposition of two eigenstates of the system, a single cold particle is completely transferred to the degenerate first excited states of the final trap, which are resonantly coupled via tunneling to the ground states of the initial and middle traps. Depending on the total time of the process, angular momentum is generated in the final trap, with values that oscillate between $\pm \hslash$. This process is discussed in terms of the asymptotic eigenstates of the individual wells and the results are checked by simulations of the full two-dimensional Schrödinger equation.

Figure 1

Schematic representation of the system of three harmonic traps with broken spatial symmetry. Traps $A$ and $B$ have a trapping frequency $\omega$ and trap $C$ has a trapping frequency $\omega /2$.

Figure 2

(a) Temporal evolution of the distances between traps $d_{AB}$, $d_{BC}$, and $d_{AC}$ (left panel) and the couplings $J^{AB}$, $J^{BC}$, $J_{1,0}^{AC}$, and $J_{0,1}^{AC}$ (right panel), during the spatial adiabatic passage process. (b) Energy eigenvalues as a function of time. The parameter values are $\beta =0.55\pi$, $\delta =0.2T$, and $d_{BC}$ and $d_{AB}$ with values between $10\alpha$ and $3.5\alpha$ and between $9\alpha$ and $2.5\alpha$, respectively, where $\alpha =\sqrt{\hslash /m\omega }$, $m$ is the mass of the particle, and $T$ is the total time of the process.

Figure 3

Temporal evolution of the population of each asymptotic level of the traps (${\psi }_{0,0}^A$, ${\psi }_{0,0}^B$, ${\psi }_{1,0}^C$, and ${\psi }_{0,1}^C$) for the four eigenstates of the system (${\Psi }_1$, ${\Psi }_2$, ${\Psi }_3$, and ${\Psi }_4$). The parameter values are the same as in Fig. 2.

Figure 4

Temporal evolution of the population distribution of the single particle in the system of three traps with broken spatial symmetry for $T=5183{\omega }^{-1}$ and the parameter values as in Fig. 2. The adiabatic following of the superposition of ${\Psi }_2\left(t\right)$ and ${\Psi }_3\left(t\right)$ can be observed by comparing with Fig. 3: From $t=0.42T$ to $0.48T$ oscillations corresponding to the phase difference between ${\Psi }_2(t<0.5T)=({\psi }_{0,0}^A+{\psi }_{0,1}^C)/\sqrt{2}$ and ${\Psi }_3(t<0.5T)=({\psi }_{0,0}^A-{\psi }_{0,1}^C)/\sqrt{2}$ are shown; for $t=0.58T$ and $0.6T$ oscillations due to the phase difference between ${\Psi }_2(t>0.5T)={\psi }_{0,1}^{\prime C}$ and ${\Psi }_3(t>0.5T)={\psi }_{1,0}^{\prime C}$ are also observed.

Figure 5

Generated angular momentum as a function of the total time of the process $T$ (upper plot) and final states in the $C$ trap and their phases for four different final total times $a$, $b$, $c$, and $d$ (lower plots). The parameter values are the same as in Fig. 2.