Classical aspects of ultracold atom wave packet motion through microstructured waveguide bends

The properties of low-density, low-energy matter wave packets propagating through waveguide bends are investigated. Time-dependent quantum-mechanical calculations using simple harmonic oscillator confining potentials are performed for a range of parameters close to those accessible by recent “atom chip”-based experiments. We compare classical calculations based on Ehrenfest’s theorem to these results to determine whether classical mechanics can predict the amount of transverse excitation as measured by the transverse heating. The present results thus elucidate some of the limits for which matter wave propagation through microstructures can be reliably considered using classical particle motion.

Figure 1

Potential-energy surface for a ${\rho }_0=20$, ${\varphi }_0=90°$ SHO-based circular bend. Both the energy and coordinates are given in oscillator units. The coordinates are those relative to the $z⩽0$ region of Eq. (2).

Figure 2

Example non-Cartesian, nonuniform grid used for the Crank-Nicolson with finite-differences calculations for a ${\varphi }_0=90°$ circular bend with ${\rho }_0=20$. (a) shows the grid points translated to a Cartesian coordinate system. (b) shows the grid points used in the calculations: where the coordinate transverse to the direction of propagation in both the leads and bend is given by $x$, while $z^{\prime }$ is an auxillary coordinate that measures the distance from the middle of the bend ($z^{\prime }=0$ at $\varphi ={\varphi }_0∕2$) along an equipotential line. The grid points located within the bend are connected with the dashed lines for clarity.

Figure 3

Propagation of a wave packet with $v_z=3$ through three $90°$ circular bends $\left({\rho }_0=15,25,35\right)$ with the SHO waveguide center marked at $15\pm 1$, $25\pm 1$, and $35\pm 1$. Three snapshots at roughly the same time are shown, with each contour line corresponding to a logarithmic decrease of ${\mid \Psi \left(x,z,t\right)\mid }^2$ from ${10}^{-2}$ to ${10}^{-5}$ (for the initial wave packet, the peak probability on the grid was ${\mid \Psi \left(x,z,t_0\right)\mid }^2=3.18\times {10}^{-2}$). No reflections were observed at this level of detail. Inset: ${\rho }_0=15$ wave packet superimposed on its Ehrenfest trajectory (dashed line).

Figure 4

Ehrenfest trajectories (E.T., dashed lines) through $90°$ bends for various ${\rho }_0$, with fixed incoming velocity $v_z=3$ (incoming from the bottom of the figure). The solid lines correspond to the path of $⟨x⟩\left(t\right)$,$⟨z⟩\left(t\right)$ for the time-dependent wave-packet calculations (Q.M.). The dotted lines are the positions of the SHO minima ${\rho }_0$ and ${\rho }_0\pm 1$.

Figure 5

Transverse heating (in oscillator units) for a $90°$ bend with fixed incoming $v_z=3$ as a function of ${\rho }_0$. The diamonds are from the time-dependent wave-packet calculations, while the Ehrenfest trajectory results form the solid line. The dashed line corresponds to an analytic result using a harmonic approximation $E_h^{H.A.}$. The dotted and dot-dashed lines correspond to the analytic results of Ref. 33, $⟨E_h^{BZ1}⟩$ and $⟨E_h^{BZ2}⟩$, respectively.

Figure 6

Transverse heating (in oscillator units) for a $90°$ bend with fixed ${\rho }_0=10$ as a function of the incoming propagation velocity. The diamonds are from the time-dependent wave-packet calculations, while the Ehrenfest trajectory results form the solid line. The dashed line corresponds to an analytic result using a harmonic approximation $E_h^{H.A.}$. The dotted and dot-dashed lines correspond to the BZ analytic results, $⟨E_h^{BZ1}⟩$ and $⟨E_h^{BZ2}⟩$, respectively.

Figure 7

$⟨V⟩$ (in oscillator units) as a function of time for a $90°$ bend with fixed ${\rho }_0=10$ for incident wave packets with varying initial velocity, $v_z$, ranging from $2.5$ to $8.0$. Each wave packet is initially in the ground state of the SHO $[⟨V⟩\left(t=0\right)\approx 0.25]$, and transverse heating by the bend is seen as an increase in $⟨V⟩$. The time axis has been rescaled by $v_z$ so that the effects due to the bend [seen as the peak in $⟨V⟩\left(t\right)$] occurs at roughly the same point in the figure.