Subwavelength optical lattices induced by position-dependent dark states

A method for the generation of subwavelength optical lattices based on multilevel dark states is proposed. The dark state is formed by a suitable combination of standing wave light fields, leading to position-dependent populations of the ground states. An additional field coupling dispersively to one of the ground states translates this position dependence into a subwavelength optical potential. We provide two semiclassical approaches to understand the involved physics, and demonstrate that they lead to identical results in a certain meaningful limit. Then we apply a Monte Carlo simulation technique to study the full quantum dynamics of the subwavelength trapping. Finally, we discuss the relevant time scales for the trapping, optimum conditions, and possible implementations.

Figure 1

Examples for the level $N\times \Lambda$ structures considered. (a) $1\times \Lambda$ setup, (b) $2\times \Lambda$ scheme, and (c) general $N\times \Lambda$ system. Note that in all cases, an off-resonant field $S_0$ is added to provide a light shift potential.

Figure 2

Quantum Monte Carlo results for the position distribution at different times. The parameters are $\Delta =5$, $S_0=2\cos (\mathit{kz})$, $S_1=\sin (\mathit{kz})$, and $R_1=\cos (\mathit{kz})$. The results are averaged over $2400$ trajectories. In (a), the initial state is $|\varphi (0)\rangle =|g_1,0\rangle$. At short times like $\Gamma t=10$, there is only one peak in a period of $\pi$ because only $S_0$ is involved in the dispersive coupling. In (b), the initial state is $|\varphi (0)\rangle =|g_2,0\rangle$. At $\Gamma t=10$ there are two peaks because both $S_0$ and $S_1$ are required for the dispersive coupling from the initial state. As expected, at later times, the influence of the initial state diminishes.

Figure 3

Quantum Monte Carlo results for the position distribution at different times. The parameters are $\Delta =50$, $S_0=20\cos (\mathit{kz})$, $S_1=10\sin (\mathit{kz})$, and $R_1=10\cos (\mathit{kz})$. All results are averaged over $240$ trajectories. Thus, the Rabi frequencies and the detuning are $10$ times of those in Fig. 2. (a) shows the position distribution. Since the potential depth is $10$ times larger than in Fig. 2, the population peaks are narrower and higher. (b) shows the corresponding kinetic energy. In comparison to Fig. 2, the heating rate is increased by about $6$ times.

Figure 4

Quantum Monte Carlo results for the position distribution at different times. Parameters are $\Delta =20$, $S_0=2\cos (\mathit{kz})$, $S_1=\sin (\mathit{kz})$, and $R_1=\cos (\mathit{kz})$. Results are averaged over $2400$ trajectories. The Rabi frequencies are as in Fig. 2, but the detuning is larger. Thus, the potential depth is shallower, and the spontaneous emissions is significantly decreased. As a result, we find smooth and equally distributed position peaks in (a), and a low heating rate in (b).

Figure 5

Quantum Monte Carlo results for the position distribution at different times. Parameters are $\Delta =20$, $S_0=2\cos (\mathit{kz})$, $S_1=\sin (\mathit{kz})$, and $R_1=3$. Curves are averaged over $2400$ trajectories. In (a), we find that the amplitudes of the population peaks oscillate with time. Overall, the trapping time is longer. (b) shows that the heating rate is very small, and the kinetic energy exhibits an oscillatory behavior.

Figure 6

Quantum Monte Carlo results for the position distribution at different times for a $2\times \Lambda$ 6-level setup. Parameters are $\Delta =30$, $S_0=10\sin (\mathit{kz})$, $S_1=5\sin (\mathit{kz}+\pi /3)$, $S_2=5\sin (\mathit{kz}+2\pi /3)$, $R_1=5$, and $R_2=15$. The results are averaged over $1200$ trajectories. (a) Theoretical expectation for the the light-induced potential from our analytical results for the 6-level system. Compared to the 4-level system, a higher subwavelength resolution is achieved. (b) The corresponding population distribution obtained from the Monte Carlo simulation. Again, the peak heights oscillate with time.

Figure 7

(a) Potential realization of the 4-level $1\times \Lambda$-system in Fig. 1a via atomic states of the $D_1$ and $D_2$ line in ${}^{87}\text{Rb}$. The states $|g_1\rangle ,|g_2\rangle$ and $|e_1\rangle$ correspond to magnetic quantum numbers $m_F=-1,1,0$, respectively. Note that the transition $(F=1,m_F=0)\leftrightarrow (F^{\prime }=1,m_F=0)$ is dipole forbidden. All standing light fields and the magnetic field are applied along the trapping axis $z$. (b) Calculated light shift potential for the parameters of Fig. 4. The red dashed line represents the light shift potential according to Eq. (5). The solid black line corresponds to a numerical calculation via the dipole force approach [see Sec. 2b] and includes the unwanted couplings of $S_0$ to additional transitions.