Quasithermalization of collisionless particles in quadrupole potentials

We analyze several puzzling features of a recent experiment with a noninteracting gas of atoms in a quadrupole trap. After an initial momentum kick, the system reaches a stationary, quasithermal state even without collisions, due to the dephasing of individual particle trajectories. Surprisingly, the momentum distribution remains anisotropic at long times, characterized by different temperatures along the different directions. In particular, there is no transfer of the kick energy between the axial and radial trap directions. To understand these effects we discuss and solve two closely related models: a spherically symmetric trap $V\left(r\right)\simeq r^{\alpha }$ and a strongly confined gas along one direction (a pancake trap). We find that in the isotropic trap, the gas unexpectedly also preserves the anisotropy of the kick at long times, which we are able to explain using the conservation of angular momentum and the virial theorem. Depending on the value of $\alpha$ we find that the kick can cool or heat the orthogonal directions. The pancake trap case is quantitatively similar to the quadrupole one. We show that for the former, the temperature anisotropy and memory of the kick direction are due to the change in the two-dimensional effective potential resulting from the kick, thereby also explaining the quadrupole experimental results.

Figure 1

Numerical simulation of the relaxation dynamics in a quadrupole trap. (a) Kinetic energy per atom along different directions as a function of time after a kick $q=1$ along $x$ at $t=0$. As in all following graphs, times is expressed in units of $t_0=\sqrt{m{k}_B{T}_0}/{\mu }_{B}b$. The energies seem to reach a stationary state for $t\ge 80$. Along $z$ the average kinetic energy is almost unchanged from its initial value $\sim 1$, but along $x$ and $y$ the corresponding values increase by the same amount to a final value of $\sim 1.2$. (b) Momentum distribution $n{p}_z$ of the steady state. The solid line ${\mathcal{G}}_{\mathrm{eq}}$ is a Gaussian distribution with the same variance. Similar results are found along all three directions.

Figure 2

The long time Boltzmann distribution $f$ after a kick along $z$ plotted as a function of $p_z$ keeping all other variables $x,y,z,p_x,p_y$ fixed and for different values of $x$. To obtain a nonzero number of atoms in the six-dimensional volume we considered a narrow region in phase space given by the coordinates in the figure and divided it into bins. We plot the number of atoms in each bin averaged over time. It can be seen that $f$ does not resemble a Gaussian thermal function and that the three peaks become more prominent for $x$ closer to the center. Similar results are found by plotting along all other coordinates.

Figure 3

Poincaré map of the quadrupole potential. We study trajectories with the same energy but different initial phase-space coordinates. The values of $x$ and $p_x$ are recorded whenever $z=0$ and $p_z>0$. We see the appearance of small islands denoting invariant torii close to which quasi-integrable trajectories evolve, separated by contiguous regions of chaotic dynamics.

Figure 4

Orbit of atom in a plane with a central potential $V\left(r\right)=r$ after increasingly long times from left to right. Since the trajectory never closes, according to Bertrand's theorem, it fills the annular region between $r_{\max }$ and $r_{\min }$ in an isotropic, dense fashion as $t\to \infty$.

Figure 5

Comparison of the simulation results for $T_i={\langle p_i^2\rangle }_{t\to \infty }$ ($i=x,y,z$) with the analytical predictions (35) and (36) for an isotropic potential (21) with $\alpha =1$ for different kick strengths $q_z$ along the $z$ direction. Note that the predicted ${\langle p_x^2\rangle }_{\infty }$ and ${\langle p_y^2\rangle }_{\infty }$ are identical. The numerical errors are smaller than the size of the symbols.

Figure 6

Comparing the different terms of $\langle p_x^2\rangle$ in (44) with $\langle p_z^2\rangle$ (36) for different values of momentum kick $q_z$ with $m=k_{B}T=1$.

Figure 7

Using Eqs. (33) and (34) to plot ${\langle E\rangle }_{\mathrm{plane}}\left(\theta \right)$ for different values of momentum kick $q_z$.

Figure 8

Using Eqs. (33) and (34) to plot ${\langle E\rangle }_{\mathrm{plane}}\left(\theta \right)/|cos\theta |$ for different values of momentum kick $q_z$.

Figure 9

Using Eqs. (33) and (34) to plot the ratio between $\langle p_z^2\rangle$ and $\langle p_x^2\rangle$ for different values of momentum kick $q_z$.

Figure 10

Behavior of the final temperatures $\mathrm{\Delta }{T}_i/\mathrm{\Delta }E$ as a function of the anisotropy $\varepsilon$ near the spherical limit after a kick along $x$. Temperatures are estimated as the standard deviation of a ${10}^5$ particles distribution, leading to a relative error of $\sim 0.3\%$. At $\varepsilon =0$, $T_z=T_y$ after which there is a discontinuous change in the temperatures due to the breaking of spherical symmetry along $z$. Data are obtained by numerical simulation over $100000$ atoms. Horizontal dashed, dotted, and solid lines correspond to the theoretical expectations of the fully isotropic (39), (40), almost isotropic (38) and (51), and quadrupole geometries, respectively.

Figure 11

Equilibration time as a function of the trap anisotropy $\varepsilon$. The equilibration time is defined as the time required for $T_y$ to reach 99% of the steady $T_x$ value. The solid line is a fit $A\times {\varepsilon }^{-1}$ following the expression (47). The best fit leads to an $R$-squared value of 0.99.

Figure 12

Comparison of the 2D potential given by (57) and the pancake potential. Here $\varepsilon =100$, but larger $\varepsilon$ values produced consistent results. The plots show $T_x=\langle p_x^2\rangle$ as a function of $q_z^2$ (top), and as a function of $q_x^2$ (bottom). Solid and dashed lines represent guides to the eye. The numerical errors are smaller than the size of the symbols. Note that the two potentials give almost identical results for the heating along the kick direction and a small discrepancy for the orthogonal cooling.

Figure 13

Comparison of the three types of potentials: spherical ($\varepsilon =0$), quadrupole ($\varepsilon =3$) and pancake ($\varepsilon =100$) showing the much better agreement between the quadrupole and the pancake compared with the spherical case. The top panel shows $T_z=\langle p_z^2\rangle$ and the bottom panel $T_x=\langle p_x^2\rangle$ as a function of the momentum kick $q_z^2$ in $z$ direction. The blue solid lines are the analytical predictions (35) and (36) for the spherical potential. Dashed and dotted lines for the other potentials are guides to the eye. The numerical errors are smaller than the size of the symbols.