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arXiv:cond-mat/0412639 · Physical Review A

Loading Bose-Einstein-condensed atoms into the ground state of an optical lattice

Paul S. Julienne, Carl J. Williams, Y. B. Band, Marek Trippenbach

Published Nov 16, 2005• 7 citations
arXivHTMLPDF

Abstract

We optimize the turning on of a one-dimensional optical potential, ${V}_{L}(x,t)=S(t){V}_{0}\phantom{\rule{0.2em}{0ex}}{\mathrm{cos}}^{2}(kx)$ to obtain the optimal turn-on function $S(t)$ so as to load a Bose-Einstein condensate into the ground state of the optical lattice of depth ${V}_{0}$. Specifically, we minimize interband excitations at the end of the turn-on of the optical potential at the final ramp time ${t}_{r}$, where $S({t}_{r})=1$, given that $S(0)=0$. Detailed numerical calculations confirm that a simple unit cell model is an excellent approximation when the turn-on time ${t}_{r}$ is long compared with the inverse of the band excitation frequency and short in comparison with nonlinear time $\ensuremath{\hbar}∕\ensuremath{\mu}$ where $\ensuremath{\mu}$ is the chemical potential of the condensate. We demonstrate using the Gross-Pitaevskii equation with an optimal turn-on function $S(t)$ that the ground state of the optical lattice can be loaded with no significant excitation even for times ${t}_{r}$ on the order of the inverse band excitation frequency.

DOI
10.1103/physreva.72.053615

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We optimize the turning on of a one-dimensional optical potential, VL(x,t)=S(t)V0cos2(kx){V}_{L}(x,t)=S(t){V}_{0}\phantom{\rule{0.2em}{0ex}}{\mathrm{cos}}^{2}(kx)VL​(x,t)=S(t)V0​cos2(kx) to obtain the optimal turn-on function S(t)S(t)S(t) so as to load a Bose-Einstein condensate into the ground state of the optical lattice of depth V0{V}_{0}V0​. Specifically, we minimize interband excitations at the end of the turn-on of the optical potential at the final ramp time tr{t}_{r}tr​, where S(tr)=1S({t}_{r})=1S(tr​)=1, given that S(0)=0S(0)=0S(0)=0. Detailed numerical calculations confirm that a simple unit cell model is an excellent approximation when the turn-on time tr{t}_{r}tr​ is long compared with the inverse of the band excitation frequency and short in comparison with nonlinear time \ensuremathℏ∕\ensuremathμ\ensuremath{\hbar}∕\ensuremath{\mu}\ensuremathℏ∕\ensuremathμ where \ensuremathμ\ensuremath{\mu}\ensuremathμ is the chemical potential of the condensate. We demonstrate using the Gross-Pitaevskii equation with an optimal turn-on function S(t)S(t)S(t) that the ground state of the optical lattice can be loaded with no significant excitation even for times tr{t}_{r}tr​ on the order of the inverse band excitation frequency.
We derive closed analytical expressions for the order parameter $\ensuremath{\Phi}(x)$ and for the chemical potential $\ensuremath{\mu}$ of a Bose-Einstein condensate loaded into a harmonically confined, one-dimensional optical lattice for sufficiently weak, repulsive, or attractive interaction and not too strong laser intensities. Our results are compared to exact numerical calculations in order to map out the range of validity of the perturbative analytical approach. We identify parameter values where the optical lattice compensates the interaction-induced nonlinearity, such that the condensate ground state coincides with a simple, single-particle harmonic-oscillator wave function.
Within the framework of a mean-field description, we investigate atomic Bose-Einstein condensates, with attraction between atoms, under the action of a strong transverse confinement and periodic [optical-lattice (OL)] axial potential. Using a combination of the variational approximation, one-dimensional (1D) nonpolynomial Schr\"odinger equation, and direct numerical solutions of the underlying 3D Gross-Pitaevskii equation, we show that the ground state of the condensate is a soliton belonging to the semi-infinite band gap of the periodic potential. The soliton may be confined to a single cell of the lattice or extended to several cells, depending on the effective self-attraction strength ggg (which is proportional to the number of atoms bound in the soliton) and depth of the potential, V0{V}_{0}V0​, the increase of V0{V}_{0}V0​ leading to strong compression of the soliton. We demonstrate that the OL is an effective tool to control the soliton's shape. It is found that, due to the 3D character of the underlying setting, the ground-state soliton collapses at a critical value of the strength, g=gcg={g}_{c}g=gc​, which gradually decreases with the increase of V0{V}_{0}V0​; under typical experimental conditions, the corresponding maximum number of 7Li^{7}\mathrm{Li}7Li atoms in the soliton, Nmax{N}_{\mathrm{max}}Nmax​, ranges between 8000 and 4000. Examples of stable multipeaked solitons are also found in the first finite band gap of the lattice spectrum. The respective critical value gc{g}_{c}gc​ again slowly decreases with the increase of V0{V}_{0}V0​, corresponding to ${N}_{\mathrm{max}}\ensuremath{\simeq}5000$.
We present an effective and fast (few microseconds) procedure for transferring a Bose–Einstein condensate from the ground state in a harmonic trap into the desired bands of an optical lattice. Our shortcut method is a designed pulse sequence where the time duration and the interval in each step are fully optimized in order to maximize robustness and fidelity of the final state with respect to the target state. The atoms can be prepared in a single band with even or odd parity, and superposition states of different bands can be prepared and manipulated. Furthermore, we extend this idea to the case of two-dimensional or three-dimensional optical lattices where the energies of excited states are degenerate. We experimentally demonstrate various examples and show very good agreement with the theoretical model. Efficient shortcut methods will find applications in the preparation of quantum systems, in quantum information processing, in precise measurement and as a starting point to investigate dynamics in excited bands.
A protocol is proposed for the rapid coherent loading of a Bose-Einstein condensate into the ground state of an optical lattice, without residual excitation associated with the breakdown of adiabaticity. The driving potential required to assist the rapid loading is derived using the fast-forward technique, and generates the ground state in any desired short time. We propose an experimentally feasible loading scheme using a bichromatic lattice potential, which approximates the fast-forward driving potential with high fidelity.
We present a shortcuts-to-adiabaticity protocol to load a spin-polarized ultracold Fermi gas from the ground ( s ) band into the first excited ( p ) band of a one-dimensional optical lattice. In contrast to a Bose-Einstein condensate in a shallow lattice—whose narrow quasimomentum distribution admits uniform lattice modulation—a noninteracting Fermi gas occupies the entire first Brillouin zone, rendering high-efficiency transfer across all momentum classes exceptionally challenging. To address this, we cast the loading process as a global, multiparameter optimization problem: We jointly optimize the timing of lattice-depth pulses and the corresponding phase shifts to maximize the ensemble-averaged transfer fidelity over all quasimomentum states. We demonstrate that fine phase tuning dramatically enhances loading efficiency, though the mean fidelity systematically decreases as more quasimomentum modes are occupied. Finally, we benchmark our shortcuts-to-adiabaticity protocol against established high-band-loading methods, underscoring its minimal hardware requirements and submillisecond timescales. Our work provides a practical path for rapid, high-fidelity preparation of Fermi gases in excited lattice orbitals, opening avenues for exploring orbital dynamics, anisotropic interactions, and many-body phenomena.
We discuss the Bloch-state solutions of the stationary Gross-Pitaevskii equation and of the Bogoliubov equations for a Bose-Einstein condensate in the presence of a one-dimensional optical lattice. The results for the compressibility, effective mass and velocity of sound are analysed as a function of the lattice depth and of the strength of the two-body interaction. The band structure of the spectrum of elementary excitations is compared with the one exhibited by the stationary solutions (Bloch bands). Moreover, the numerical calculations are compared with the analytic predictions of the tight binding approximation. We also discuss the role of quantum fluctuations and show that the condensate exhibits 3D, 2D or 1D features depending on the lattice depth and on the number of particles occupying each potential well. We finally show how, using a local density approximation, our results can be applied to study the behaviour of the gas in the presence of harmonic trapping.