Dynamics of cold atoms crossing a one-way barrier

We implemented an optical one-way potential barrier that allows ultracold ${}^{87}\text{R}\text{b}$ atoms to transmit through when incident on one side of the barrier but reflect from the other. This asymmetric barrier is a realization of Maxwell’s demon, which can be employed to produce phase-space compression and has implications for cooling atoms and molecules not amenable to standard laser-cooling techniques. The barrier comprises two focused Gaussian laser beams that intersect the focus of a far-off-resonant single-beam optical dipole trap that holds the atoms. The main barrier beam presents a state-dependent potential to incident atoms, while the repumping barrier beam optically pumps atoms to a trapped state. We investigated the robustness of the barrier asymmetry to changes in the barrier-beam separation, the initial atomic potential energy, the intensity of the second beam, and the detuning of the first beam. We performed simulations of the atomic dynamics in the presence of the barrier, showing that the initial three-dimensional momentum distribution plays a significant role, and that light-assisted collisions are likely the dominant loss mechanism. We also carefully examined the relationship to Maxwell’s demon and explicitly accounted for the apparent decrease in entropy for our particular system.

Figure 1

(Color online) Schematic diagram of the optical setup, showing the dipole trap, barrier beams, and imaging system.

Figure 2

Relevant atomic levels of ${}^{87}\text{R}\text{b}$ on the $D_2$ cooling transition (not to scale), shown with the off-resonant coupling of the main barrier beam. Atoms with $F=1$ see this field as red detuned, while $F=2$ atoms see a blue detuning. The repumping barrier beam is resonant with the $F=1\to F^{\prime }=2$ MOT repumping transition.

Figure 3

(Color online) Optical potentials seen by atoms along the axis of the dipole trap in our experiment. The overall trapping potential is from the main dipole-trap beam, while the narrow state-dependent feature results from the main barrier beam. The main barrier beam produces a repulsive potential for atoms in the reflecting $\left(F=2\right)$ ground state, and an attractive potential for atoms in the transmitting $\left(F=1\right)$ ground state.

Figure 4

Density of atoms along the dipole trap responding to the one-way barrier. The dipole-trap focus and barrier beams are located at the horizontal origins. Each curve represents an average of 78 [38 for column (e)] repetitions of the experiment. Column (a): atoms are initially in the $F=1$ (transmitting) state and to the left of the trap center, with no barrier present. Column (b): atoms are initially in the $F=1$ (transmitting) state on the left (transmitting) side of the barrier. Column (c): atoms are initially in the $F=2$ (reflecting) state on the left (transmitting) side of the barrier. Column (d): atoms are initially in the $F=1$ (transmitting) state on the right (reflecting) side of the barrier. Column (e): atoms are initially loaded relatively uniformly throughout the dipole trap (a “filled trap”) in the $F=1$ (transmitting) state.

Figure 5

(Color online) Populations on the left- and right-hand sides of the one-way barrier for a ‘‘filled-trap’’ experiment where atoms were initially distributed throughout the dipole trap. Error bars indicate statistical error from 38 repetitions.

Figure 6

(Color online) Schematic of a four-level atom that would allow much larger barrier-beam detunings for reduced scattering. Here, the main barrier beam is detuned from a different transition from the one with which the repumping barrier beam is resonant. If the main-barrier-beam transition has a substantially different frequency from the repumping-barrier-beam transition, then the main barrier beam may be detuned by an amount greater than the ground-state splitting, and still create an effective barrier for atoms in the reflecting ground state while having negligible effect on atoms in the transmitting state.

Figure 7

(Color online) Populations on the left- and right-hand sides of the one-way barrier after 100 ms as functions of barrier-beam separation. From left to right, top to bottom, atoms were initially loaded on the left in the $F=1$ ground state, on the left in the $F=2$ ground state, on the right in the $F=1$ ground state, and on the right in the $F=2$ ground state. Error bars indicate statistical error from at least 20 repetitions.

Figure 8

(Color online) Populations on the left- and right-hand sides of the one-way barrier after 100 ms as functions of initial loading position. The initial loading position is the position of the zero of the MOT magnetic fields relative to the focus of the dipole trap, in millimeters. Our canonical starting locations were about 0.65(8) mm away from the focus of the dipole trap and are marked by the two vertical gray regions. Vertical error bars indicate statistical error from 38 repetitions, and horizontal error bars indicate uncertainty in measuring the MOT center.

Figure 9

(Color online) Some sample time series that formed the data in Fig. 8, showing populations on the right- and left-hand sides of the barrier as functions of time. Shown are curves for initial loading positions of (a) 0.89(8), (b) 0.43(8), (c) 0.14(8), (d) $-0.51\left(8\right)$, and (e) $-0.90\left(8\right)\text{mm}$ from the barrier center. Error bars indicate statistical error from 38 repetitions.

Figure 10

(Color online) Populations on the left- and right-hand sides of the one-way barrier after 100 ms as functions of repumping-barrier-beam power. The left panel shows results where the atoms were initially on the left-hand (transmitting) side of the barrier, and the right panel shows results where the atoms were initially on the right-hand (reflecting) side. The vertical lines in the plots show our usual repumping-barrier-beam power of $0.36\mu \text{W}$. The two horizontal bars show the ending populations from data taken with no barriers [the one shown in column (a) of Fig. 4 and a similar data set where atoms were initially on the other side of the barrier]. The no-barrier cases show unequal left- and right-hand populations due to the oscillations of the atoms. Error bars indicate statistical error from at least 38 repetitions.

Figure 11

(Color online) Populations on the left- and right-hand sides of the one-way barrier for a few different repumping-barrier-beam powers as functions of time. The top two panels show data collected without barrier beams. The bottom two panels are generated from the same data used in Fig. 10. The data with the dotted line and squares correspond to the minimum beam power (about 2 nW) shown in Fig. 10. The data with the dashed line and diamonds correspond to approximately the same power used to collect our canonical data. The data with the solid line and triangles correspond to the maximum beam power (about $9\mu \text{W}$) shown in Fig. 10. The top curve in each panel shows the total population, while the other solid symbols show the right-hand side population and the open symbols show the left-hand side population.

Figure 12

(Color online) A schematic of the rubidium emission spectrum around the $D_2$ line, showing the detunings we tested for the main barrier laser. The horizontal axis measures detuning from the ${}^{87}\text{R}\text{b}$ $F=2\to F^{\prime }=3$ MOT trapping transition. Both ${}^{87}\text{R}\text{b}$ and ${}^{85}\text{R}\text{b}$ resonances are shown. Our canonical detuning coincides with the ${}^{85}\text{R}\text{b}$ $F=3\to F^{\prime }=3,4$ crossover transition, and the far-detuned value coincides with the ${}^{85}\text{R}\text{b}$ $F=2\to F^{\prime }=2$ transition. For the near-detuned value, we monitored the detuning with a Fabry-Pérot cavity. Each detuning from the ${}^{87}\text{R}\text{b}$ MOT trapping transition is shown with a solid arrow. This approximates the detuning seen by atoms in the $F=2$ (reflecting) state. A dotted arrow shows the detuning from the ${}^{87}\text{R}\text{b}$ MOT repumping transition, which approximates the detuning seen by atoms in the $F=1$ (transmitting) state.

Figure 13

(Color online) Populations on the left- and right-hand sides of the one-way barrier, comparing barrier performances when the main barrier laser operates at different frequencies. For the canonical, near-detuned, and far-detuned data, the main-barrier detunings were 1.05(5), 0.75(5), and 3.97(7) GHz blue of the ${}^{87}\text{R}\text{b}$ MOT transition, respectively. See Fig. 12 for details. The near-detuned and canonical data are very similar, while the far-detuned data show much more atom loss. Error bars indicate statistical error from at least 28 repetitions.

Figure 14

(Color online) Populations on the left- and right-hand sides of the one-way barrier, comparing data (points) to simulations (solid and dashed lines). The solid simulation curve has an extra light-assisted collisional loss mechanism.

Figure 15

(Color online) Populations on the left- and right-hand sides of the one-way barrier, comparing data (points) to simulations (solid and dashed lines). These simulations and data are the same as in Fig. 14, except that the simulation initial conditions had no transverse spread in either position or momentum. These simulations do not match the data as well as the simulations in Fig. 14, illustrating how basic barrier results, such as the time atoms take to reach the barrier and whether atoms oscillate across the barrier, are strongly affected by the initial conditions. The solid simulation curve has an extra light-assisted collisional loss mechanism.

Figure 16

(Color online) Populations on the left- and right-hand sides of the one-way barrier, comparing data (points) to simulations (solid and dashed lines) for a long experiment showing the barrier lifetime. The trap was initially loaded nearly symmetrically. The dashed curve shows a simulation without including light-assisted collisions, which models the data very poorly. The solid curve shows the same simulations including light-assisted collisions, which matches the data very closely. For each curve (data and both simulations), the curves from top to bottom show the total population, the population on the right side of the barrier, and the population on the left side of the barrier. Error bars indicate statistical error from 19 repetitions.

Figure 17

(Color online) Plot of Eq. (B7), showing the expected number of scattering events for a two-level atom passing through a Gaussian barrier. The barrier height $k$ is defined in Eq. (B4), and the number of scattering events is given as $N^{\prime }≔N{w}_{0}m{v}_0\Gamma /\hslash \left|\Delta \right|$, where $N$ is the actual number of scattering events.

Figure 18

(Color online) Plot of Eq. (B7), showing the expected number of scattering events for a quasi-two-level (ignoring state changes) ${}^{87}\text{R}\text{b}$ atom passing through a Gaussian barrier as a function of beam detuning from the $F=2\to F^{\prime }$ transitions. The curves that increase for larger detunings are for atoms in the $F=1$ transmitting state, and the curves that decrease for larger detunings are for atoms in the $F=2$ reflecting state. The effective detuning from the $F=1$ ground state for a beam tuned to the $F=2\to F^{\prime }$ transitions is about 6.53 GHz. Each $F=1$ and $F=2$ pair of curves represents a different barrier height. The dotted curves show the expected numbers of scattering events for a barrier height of $k=1.25$. This means that the beam intensity for each detuning is adjusted so that the potential barrier seen by atoms in the $F=2$ ground state is 1.25 times the kinetic energy of the atoms. The short-dashed curves are for $k=1.5$, the medium-dashed curves are for $k=2$, and the long-dashed curves are for $k=4$. Our canonical data had $k=2.3$ with an effective two-level detuning of 1.12 GHz.

Figure 19

(Color online) Plot of cylindrical coordinates for two orbits near the focus of a dipole trap using the same dipole-trap parameters as in Sec. 5, except without gravity. Longitudinal distances $\left(z\right)$ are in millimeters, and radial distances $\left(r\right)$ are in $10\mu \text{m}$ units. The solid curves show a simulation with 90% of the critical angular momentum required for the dipole-trap focus to become repelling. This trajectory shows an oscillation about the focus that is slowed due to the angular momentum. The dashed curves show the same initial conditions, but with a slightly larger $r$ value and azimuthal velocity, pushing the angular momentum to 110% of the critical value, and show how such a particle can appear to bounce off an attractive potential.

Figure 20

(Color online) Plots of cylindrical coordinates for three orbits in a dipole trap, using the same dipole-trap parameters as in Sec. 5, except without gravity. Longitudinal distances $\left(z\right)$ are in millimeters, and radial distances $\left(r\right)$ are in $10\mu \text{m}$ units. The solid curve shows a slight variation in the predicted saddle-point orbit, with the correct angular momentum but with $q$ slightly below unity. This curve shows the particle oscillating about the saddle-point orbit, but drifting along the $q=1$ manifold. The radial restoring force for this orbit in our trap is much stiffer than the longitudinal restoring force, so $r$ oscillates rapidly (resulting in a noncircular orbit). The dashed curve shows the predicted saddle-point orbit, but with 101% of the saddle-point angular momentum; this orbit diverges. The dotted curve shows the predicted saddle-point orbit, but with 99% of the saddle-point angular momentum. This trajectory falls away from the orbit and instead oscillates about the focus.