Phase-space manipulations of many-body wave functions

We explore the manipulation in phase space of many-body wave functions that exhibit self-similar dynamics under the application of sudden force and/or in the presence of a constant acceleration field. For this purpose, we work out a common theoretical framework based on the Wigner function. We discuss squeezing in position space, phase-space rotation, and its implications in cooling for both noninteracting and interacting gases and time-reversal operation. We discuss various optical analogies and calculate the role of a spherical-like aberration in cooling protocols. We also present the equivalent of a spin-echo technique to improve the robustness of velocity dispersion reduction protocols.

Figure 1

Focalization of a wave packet by suddenly applying an attractive harmonic force in a very short amount of time.

Figure 2

To squeeze the wave packet along the $P$ axis, two successive steps are considered: (a) a free expansion, (b) followed by a $\delta$-kick force with the right strength to ensure the rotation by the appropriate angle ${\beta }^{*}$. (c) A rotation by twice this angle, simply obtained by an applied force two times larger than that used for the squeezing, reverses the position-velocity correlation and therefore acts as a time-reversal operator. The complete time-reversal process implies three steps: (a) free propagation during a finite time $\tau$, (c) rotation by an angle $2{\beta }^{*}$, and (d) free propagation over a duration equal to $\tau$.

Figure 3

(a) and (b) Using a linear kick force with an offset, the cooling and translation on the momentum axis can be obtained simultaneously. (c) and (d) The final position of the squeezed wave packet can be chosen in phase space by an appropriate sequence of evolution under a constant force followed by the application of a sudden force with or without offset.

Figure 4

Velocity dispersion just after the $\delta$-kick cooling protocol as a function of the time $t_{-}$ at which the rotation is performed and for various values of the ratio between the initial Thomas-Fermi radius $R_{\mathrm{TF}}$ and the harmonic length ${\sigma }_0$. The velocity dispersion is normalized to that of the Gaussian ground-state wave function in the same initial harmonic confinement, and the time is normalized to ${\omega }_0^{-1}$.

Figure 5

Asymptotic velocity dispersion as a function of the time $t_{-}$ at which the rotation is performed and for various values of the ratio between the initial Thomas-Fermi radius $R_{\mathrm{TF}}$ and the harmonic length ${\sigma }_0$. Same normalization as in Fig. 4.

Figure 6

(a) Strategy 1: simple phase-space rotation (dashed line). (b) Strategy 2: spin-echo-like phase-space manipulation (solid line). For the sake of simplicity, we only represent the long axis of the ellipse shape of the phase-space volume. (c) The relative error on the phase $|S_{\varepsilon }/S_0-1|$ as a function of the frequency mismatch $\varepsilon$.