Atom lithography with near-resonant standing waves

We study the optimal focusing of two-level atoms with a near-resonant standing wave light, using both classical and quantum treatments of the problem in the thin- and thick-lens regimes. It is found that the near-resonant standing wave focuses the atoms with a reduced background in comparison with far-detuned light fields. For some parameters, the quantum atomic distribution shows even better localization than the classical one. Spontaneous emission effects are included via the technique of quantum Monte Carlo wave function simulations. We investigate the extent to which nonadiabatic and spontaneous emission effects limit the achievable minimal size of the deposited structures.

Figure 1

Schematic representation of laser focusing of atoms by a SW light. (a) A collimated atomic beam impinges on the near-resonant SW light whose intensity varies sinusoidally along the $x$ direction. (b) The interaction with the light induces the adiabatic potential [either plus or minus in Eq. (4)] for the external motion of atoms depending upon the sign of the detuning. For a positive (blue) detuning, the atoms localize near the minima of the light intensity, while for a negative (red) detuning they get localized near the light intensity maxima. The dashed curves represent the quadratic approximation to the adiabatic potentials near their minima.

Figure 2

(a) Localization factor of the atomic distribution as a function of the dimensionless time $t$ for ${\sigma }_t=0.07$ and $\Delta /{\Omega }_0=-0.125$ (solid curve), $\Delta /{\Omega }_0=0.125$ (dashed curve), $\Delta /{\Omega }_0=5$ (dotted curve). The global minimum values of $L(t)$ are 0.17 (solid curve), 0.63 (dashed curve), and 0.42 (dotted curve). (b) Probability density (P) of the atomic distribution at the time $t=t_m$ of the best atomic localization. The parameters are same as those of (a) with $t_m=0.82$ (solid curve), $t_m=0.52$ (dashed curve), and $t_m=7.38$ (dotted curve). For the sake of comparison, the solid curve has been displaced by 0.25 units along the $X$ axis. The times $t_m$ correspond to the global minima of the localization factor in (a).

Figure 3

Classical treatment. Minimal localization factor of the atomic distribution as a function of the detuning $\Delta /{\Omega }_0$ for (a) thin lens, ${\sigma }_t=0.07$, and (b) thick lens, ${\sigma }_t=4$.

Figure 4

(a) Localization factor of the atomic distribution as a function of the dimensionless time $t$ for ${\sigma }_t=4$ and $\Delta /{\Omega }_0=-0.125$ (solid curve), $\Delta /{\Omega }_0=0.125$ (dashed curve), $\Delta /{\Omega }_0=1$ (dotted curve). The global minimum values of $L(t)$ are 0.1 (solid curve), 0.36 (dashed curve), and 0.31 (dotted curve). (b) Probability density ($P$) of the atomic distribution at the time $t=t_m$ of the best atomic localization. The parameters are same as those of (a) with $t_m=0.2$ (solid curve), $t_m=0.58$ (dashed curve), and $t_m=2$ (dotted curve). For the sake of comparison, the solid curve has been displaced by 0.25 units along the $x$ axis. The times $t_m$ correspond to the global minima of the localization factor in (a).

Figure 5

Localization factor of the atomic distribution as a function of the dimensionless time $t$ for the parameters ${\sigma }_t=0.0006$, ${\Omega }_0=1.92\times {10}^5$, and (a) $\Delta /{\Omega }_0=-0.125$, (b) $\Delta /{\Omega }_0=0.125$ (solid curve), $\Delta /{\Omega }_0=5$ (long-dashed curve). The global minimum values of $L(t)$ are (a) 0.15, and (b) 0.2 (solid curve), 0.42 (long-dashed curve).

Figure 6

Probability density $(P=|\Psi (x,t)|{}^2)$ of the atomic distribution at the time $t=t_m$ of the best atomic localization. The parameters are ${\sigma }_t=0.0006$, ${\Omega }_0=1.92\times {10}^5$, and $\Delta /{\Omega }_0=-0.125$, $t_m=7\times {10}^{-3}$ (solid curve), $\Delta /{\Omega }_0=0.125$, $t_m=0.778$ (dashed curve), $\Delta /{\Omega }_0=5$, $t_m=0.72$ (long-dashed curve). For the sake of comparison, the solid curve has been displaced by 0.25 units along the $x$ axis. The times $t_m$ correspond to the global minima of the localization factor in Fig. 5.

Figure 7

Quantum treatment. Minimal localization factor of the atomic distribution as a function of the detuning $\Delta /{\Omega }_0$ for (a) thin lens, ${\sigma }_t=0.0006$, and (b) thick lens, ${\sigma }_t=0.01$.

Figure 8

(a) Localization factor of the atomic distribution as a function of the dimensionless time $t$ for the parameters ${\sigma }_t=0.01$, ${\Omega }_0=4\times {10}^4$, and $\Delta /{\Omega }_0=-0.125$ (solid curve), $\Delta /{\Omega }_0=0.125$ (dashed curve), $\Delta /{\Omega }_0=1$ (long-dashed curve). The global minimum values of $L(t)$ are 0.1 (solid curve), 0.37 (dashed curve), and 0.31 (long-dashed curve). (b) Probability density $(P=|\Psi (x,t)|{}^2)$ of the atomic distribution at the time $t=t_m$ of the best atomic localization. The parameters are same as those of (a) with $t_m=0$ (solid curve), $t_m=1.5\times {10}^{-3}$ (dashed curve), and $t_m=5.3\times {10}^{-3}$ (long-dashed curve). For the sake of comparison, we have displaced the solid curve by 0.25 units along the $x$ axis and the dashed curve by 20 units along the $P$ axis. The times $t_m$ correspond to the global minima of the localization factor in (a).

Figure 9

Probability density $(P={|\Psi (x,t)|}^2)$ of the atomic distribution for the parameters ${\sigma }_t=0.01$, ${\Omega }_0=2\times {10}^4$, $t=0$, and $\Delta /{\Omega }_0={10}^{-2}$.

Figure 10

(a) Localization factor of the atomic distribution as a function of the dimensionless time $t$ around its global minimum for the parameters ${\sigma }_t=0.0006$, ${\Omega }_0=1.92\times {10}^5$, $\Delta /{\Omega }_0=-0.125$, and $\Gamma =238$ (solid curve), $\Gamma =0$ (dashed curve). The minimum values of $L(t)$ are 0.175 (solid curve) and 0.15 (dashed curve). (b) Probability density $(P={|\Psi (x,t)|}^2)$ of the atomic distribution at the time $t=t_m$ of the best atomic localization. The parameters are same as those of (a) with $t_m=7\times {10}^{-3}$. The solid curve is almost indistinguishable from the dashed curve on the scale shown. For clarity, the close-up to the right of origin $(x=0)$ is shown in the inset.

Figure 11

(a) Localization factor of the atomic distribution as a function of the dimensionless time $t$ around its global minimum for the parameters ${\sigma }_t=0.01$, ${\Omega }_0=4\times {10}^4$, $\Delta /{\Omega }_0=-0.125$, and $\Gamma =238$ (solid curve), $\Gamma =0$ (dashed curve). The minimum values of $L(t)$ are 0.18 (solid curve) and 0.1 (dashed curve). (b) Probability density $(P={|\Psi (x,t)|}^2)$ of the atomic distribution at the time $t=t_m$ of the best atomic localization. The parameters are same as those of (a) with $t_m=0$. For clarity, the close-up to the right of origin $(x=0)$ is shown in the inset.