Fast momentum-selective transport of Bose-Einstein condensates via controlled non-adiabatic dynamics in optical lattices

Abstract

We present a detailed numerical study of a protocol for momentum-selective transport of a Bose-Einstein condensate (BEC) in a one-dimensional optical lattice, achieving narrow momentum distributions through controlled non-adiabatic dynamics. The protocol consists of non-adiabatic loading into the lattice, coherent acceleration using a symmetric trapezoidal acceleration profile, and non-adiabatic release into free space. Using the time-dependent Gross-Pitaevskii equation, we simulate the full sequence and analyze the role of non-adiabatic excitations on the final momentum distribution. We identify the intra-site breathing dynamics as the dominant mechanism governing spectral purity under fast loading conditions. By tracking the condensate's spatial width during the evolution, we demonstrate a direct correlation with the final momentum spread. A variational model based on a Gaussian ansatz quantitatively reproduces the observed dynamics and provides physical insight into the breathing mechanism. Our results reveal the existence of "magic" times, i.e., specific loading or acceleration durations synchronized with the breathing oscillation period, where quasi-monochromatic momentum distributions can be achieved even with loading times as short as 100 microseconds. In the tight-binding regime, this approach offers speedup factors of 3 to 6 compared to adiabatic protocols while maintaining high transfer fidelities, providing a practical route to coherent transport for quantum sensors operating under stringent timing constraints.

Figures (7)

• Figure 1: Transport protocol and evolution of the condensate density. (a) Time-dependent lattice acceleration $a_\mathrm{OL}(t)$ following a symmetric trapezoidal profile with total duration $t_{\mathrm{acc}} = 0.7$ ms and maximum value $a_{\mathrm{max}} = 1790$ m/s$^2$. Colored markers indicate the time points corresponding to subplots (b)–(e). (b) Initial ground-state density of the BEC in the harmonic trap ($t = 0$), obtained for $N = 10^4$ atoms of $^{87}$Rb. The wave function is numerically computed using the stationary Gross–Pitaevskii equation. (c) Density profile after fast loading into the optical lattice ($t = t_L = 0.1$ ms). The lattice potential is ramped up while the harmonic trap is switched off over the same timescale. The inset shows the emergence of periodic density modulation at the lattice scale. (d) Condensate density after the acceleration phase ($t = t_L + t_{\mathrm{acc}}$), illustrating the motion of the cloud induced by the lattice. The inset reveals persistent internal structure. (e) Final density profile after a fast release from the lattice (also over a duration $t_L = 0.1$ ms). The central region remains relatively smooth, indicating small excitation. All density profiles are plotted in position space. Insets in (c)–(e) zoom into the central part of the wave packet to highlight local density modulations induced by the lattice.
• Figure 2: Momentum distribution of the condensate after the transport protocol, as a function of the loading time $t_L$. Each curve represents the momentum distribution $P(k)$ (with the maximum of the central peak normalized to 1) computed for a fixed acceleration duration $t_{\mathrm{acc}} = 0.7$ ms and varying $t_L$ values from 0.1 ms to 0.5 ms. For short loading durations ($t_L \lesssim 0.3$ ms), non-adiabatic excitations are visible in the form of pronounced side peaks at $k = 188\,k_L$ and $k = 192\,k_L$ (i.e., $\pm 2 k_L$ from the main peak), reflecting population transfer into neighboring momentum states. As $t_L$ increases, these sidebands are progressively suppressed and the distribution becomes increasingly concentrated around $k = 190\,k_L$, the expected final momentum corresponding to the total number of momentum kicks set by the acceleration ramp. This transition illustrates the onset of adiabatic dynamics and the emergence of a narrow, spectrally pure momentum distribution.
• Figure 3: (a) Momentum-state populations as a function of the acceleration time $t_{\mathrm{acc}}$, for a fixed loading duration $t_L = 0.1$ ms. The central peak population $P_0$ at $k = 190\,k_L$ (black curve) undergoes pronounced oscillations, periodically reaching values close to unity. The sideband populations $P_{-2}$ (orange squares) and $P_{+2}$ (green dashed line), corresponding to the first-order sideband peaks at $k = 188\,k_L$ and $192\,k_L$, oscillate out of phase with $P_0$, revealing coherent redistribution between momentum states as $t_{\mathrm{acc}}$ varies. Insets (b) and (c) show representative momentum spectra for two values of $t_{\mathrm{acc}}$: when $P_0$ is close to its maximum ($t_{\mathrm{acc}} = 973.2\ \mu$s, left inset b), the distribution is narrow and nearly monochromatic. In contrast, when $P_0$ is smaller ($t_{\mathrm{acc}} = 976.6\ \mu$s, right inset c), the sideband peaks are growing, illustrating the role of coherent non-adiabatic dynamics in shaping the final state.
• Figure 4: Evolution of the intra-site spatial width $\Delta x(t)$ of the condensate wave function during the transport protocol, for fixed loading and release times $t_L = 0.1$ ms and varying acceleration durations $t_{\mathrm{acc}}$. The width $\Delta x(t)$ is defined as the full width at half maximum (FWHM) of the density distribution, computed at each time step on a single central optical lattice site. Three successive stages are visible: (i) in panel (a) during loading $(t \leqslant t_L = 0.1\,\mathrm{ms})$, the condensate is compressed into the lattice wells, leading to a reduction in spatial extent; (ii) in panel (b) during the acceleration phase $(t_L \leqslant t \leqslant t_L+t_\mathrm{acc})$, $\Delta x(t)$ exhibits coherent oscillations, reflecting breathing-like motion within the lattice; (iii) in panel (c) during the release $(t > t_L+t_\mathrm{acc})$, the spatial width expands progressively. The two curves correspond to acceleration times $t_{\mathrm{acc}} = 973.2~\mu\mathrm{s}$ (solid blue line) and $t_{\mathrm{acc}} = 976.6~\mu\mathrm{s}$ (red dashed line). For both cases, $\Delta x(t)$ follows nearly identical dynamics during the loading and acceleration phases, indicating an identical coherent evolution. The amplitude of oscillations during the acceleration phase serves as a diagnostic of non-adiabatic excitations. A clear difference appears during the release phase, where the curves exhibit a phase shift and different final widths, reflecting dephasing and distinct expansion dynamics due to accumulated dynamical differences. The final value of $\Delta x(t)$ correlates with the spectral purity of the momentum distribution (see text for details).
• Figure 5: Comparison between the full numerical simulation and the variational model for the intra-site spatial dynamics of the condensate. The red dashed line shows the time-dependent width of the central density peak, $\Delta x(t)$, expressed as the full width at half maximum (FWHM), extracted from the Gross–Pitaevskii equation for $t_{\mathrm{acc}} = 976.6~\mu\mathrm{s}$. The black solid line corresponds to the evolution predicted by the variational model based on a Gaussian ansatz and assuming harmonic confinement within each lattice site. The variational propagation is initialized in panel (a) at the time $t_i$ of the fourth local maximum of $\Delta x(t)$, as obtained from the GPE solution. The corresponding value of $\sigma(t_i)$ is set from the numerical result to ensure accurate matching at the starting point. Excellent agreement is observed between the two approaches during the second half of the loading ramp in panel (a) and throughout the acceleration phase shown in panel (b), where the model captures both the amplitude and phase of the intra-site breathing oscillations. Discrepancies emerge at the end of the release stage in panel (c), when the wave function freely expands and begins to overlap with neighboring sites, breaking the validity of the single-site variational approximation. These results confirm the applicability of the variational model in the tight-binding regime and support the use of $\Delta x(t)$ as a robust observable of internal dynamics.