Hamiltonian Ratchet for Matter-Wave Transport

We report on the design of a Hamiltonian ratchet exploiting periodically at rest integrable trajectories in the phase space of a modulated periodic potential, leading to the linear nondiffusive transport of particles. Using Bose-Einstein condensates in a modulated one-dimensional optical lattice, we make the first observations of this spatial ratchet, which provides way to coherently transport matter waves with possible applications in quantum technologies. In the semiclassical regime, the quantum transport strongly depends on the effective Planck constant due to Floquet state mixing. We also demonstrate the interest of quantum optimal control for efficient initial state preparation into the transporting Floquet states to enhance the transport periodicity.

Figure 1

Stroboscopic phase portraits and experimental images for $(\gamma ,ϵ,{\phi }_0)=(1.2,0.3,1.7)$ at subperiod observation times $t=(n+r)\times 2\pi$ [with $n\in \mathbb{N}$ and $r=0$, 0.25, 0.5, and 0.75 for (a) to (d) respectively]. Left: the ratcheting island and the trajectory starting in $(x_0,p_0,t_0)=(0,0,0)$ are in blue, the other regular structures are in gray, and the chaotic sea is in red. The area of the ratcheting island is $\mathcal{A}=0.21$. Right: corresponding time-of-flight absorption images, starting from the ground state of the lattice during the first period $n=0$ for $1/{\hslash }_{\mathrm{eff}}\approx 1.27$.

Figure 2

Eigenstate and transport dependences on the effective Planck constant. (a) Overlap between the ground state $|{\varphi }_0⟩$ of the lattice and the ratcheting Floquet state $|F_{\mathrm{rat}}⟩$ and (b) transport [Eq. (2)] of $|F_{\mathrm{rat}}⟩$ over one modulation period as a function of $1/{\hslash }_{\mathrm{eff}}$. (c)–(e) Stroboscopic phase portraits in the unit cell of system [Eq. (1)] and Husimi representations of $|{\varphi }_0⟩$ (top, purple) and $|F_{\mathrm{rat}}⟩$ (bottom, green) for the values of $1/{\hslash }_{\mathrm{eff}}=0.70$, 1.27, 1.56 respectively, identified by vertical lines on the panels (a)–(b). The color range for each Husimi function extends from zero to its maximum value, with a truncation to a quarter of this value in the outlined rectangular regions of panels (d) and (e) in order to reveal details (note that the Floquet states share the $x\to -x$ symmetry of the phase portrait).

Figure 3

Transport of the ground state. (a) Top: numerical simulation of the momentum distribution during the modulation as a function of time for $1/{\hslash }_{\mathrm{eff}}\approx 1.27$ [corresponding to Fig. 2]. Bottom: corresponding experimental integrated absorption images. (b) Same as (a) for $1/{\hslash }_{\mathrm{eff}}\approx 1.56$ [corresponding to Fig. 2]. (c) Expected numerical (solid red line) and experimental (blue markers) transport (see text) for data (a) and (b) as a function of time, and transport reversability (a’) for ${\phi }_0\to -{\phi }_0$ and $1/{\hslash }_{\mathrm{eff}}\approx 1.30$ (see text).

Figure 4

Transport of the ratcheting Floquet state prepared by QOC. (a)–(c) Example of QOC for ratchet transport at $1/{\hslash }_{\mathrm{eff}}\approx 1.56$. (a) Husimi representation of $|{\varphi }_0⟩$ (purple) in the phase space of the static lattice (solid black lines). The color range for the Husimi function extends from zero to its maximum value. (b) Phase of the lattice along time to drive the system from $|{\varphi }_0⟩$ to $|F_{\mathrm{rat}}⟩$. (c) Same as (a) for the prepared state $|{\psi }_{\mathrm{QOC}}⟩$. (d) Top: numerical simulation of the momentum distribution during the modulation as a function of time for $1/{\hslash }_{\mathrm{eff}}\approx 1.27$ [corresponding to Figs. 2 and 3]. Bottom: corresponding experimental integrated absorption images. (e) Same as (d) for $1/{\hslash }_{\mathrm{eff}}\approx 1.56$ [corresponding to panels (a)–(c) as well as Figs. 2 and 3].