Quantum time-of-flight distribution for cold trapped atoms

The time-of-flight distribution for a cloud of cold atoms falling freely under gravity is considered. We generalize the probability current density approach to calculate the quantum arrival time distribution for the mixed state describing the Maxwell-Boltzmann distribution of velocities for the falling atoms. We find an empirically testable difference between the time-of-flight distribution calculated using the quantum probability current and that obtained from a purely classical treatment which is usually employed in analyzing time-of-flight measurements. The classical time-of-flight distribution matches with the quantum distribution in the large mass and high temperature limits.

Figure 1

(Color online) The classical $\left[{\Pi }_C\left(t\right)\right]$ and quantum $\left[{\Pi }_Q\left(t\right)\right]$ TOF distributions of the atomic cloud falling freely under gravity are plotted for varying mass of the atoms at a fixed temperature $T=2.5\times {10}^{-6}\mathrm{K}$ with ${\sigma }_0={10}^{-5}\mathrm{cm}$ and $Z=-30\mathrm{cm}$.

Figure 2

(Color online) The classical $\left[{\Pi }_C\left(t\right)\right]$ and quantum $\left[{\Pi }_Q\left(t\right)\right]$ TOF distributions of the atomic cloud falling freely under gravity are plotted for varying temperatures at a fixed mass of Rb atom $(m=85.4678\mathrm{amu})$ with ${\sigma }_0={10}^{-5}\mathrm{cm}$ and $Z=-30\mathrm{cm}$.

Figure 3

(Color online) The mass variation of the mean arrival times $\overline{{\tau }_Q}$ and $\overline{{\tau }_C}$ are shown in the figure for a fixed value of temperature $(T=1.41\times {10}^{-6}\mathrm{K})$ with $Z=-30\mathrm{cm}$. The quantum mean arrival time $\overline{{\tau }_Q}$ is plotted for two different values of ${\sigma }_0$. 2(a) $\overline{{\tau }_Q}$ for ${\sigma }_0={10}^{-5}\mathrm{cm}$, 2(b) $\overline{{\tau }_Q}$ for ${\sigma }_0=2\times {10}^{-5}\mathrm{cm}$. 2(c) The mean arrival time $\overline{{\tau }_C}$ calculated through the classical TOF distribution ${\Pi }_C\left(t\right)$ with $T=1.41\times {10}^{-6}\mathrm{K}$ and $Z=-30\mathrm{cm}$.

Figure 4

(Color online) The classical $\left[{\Pi }_C\left(t\right)\right]$ and quantum $\left[{\Pi }_Q\left(t\right)\right]$ TOF distributions of the atomic cloud falling freely under gravity are plotted for four different values of ${\sigma }_0$ at a fixed mass of Be atom $(m=9.01\mathrm{amu})$ and at a fixed temperature $T=3.0\times {10}^{-6}\mathrm{K}$ with $Z=-30\mathrm{cm}$.