Production and manipulation of wave packets from ultracold atoms in an optical lattice

Within the combined potential of an optical lattice and a harmonic magnetic trap, it is possible to form matter wave packets by intensity modulation of the lattice. An analysis of the production and motion of these wave packets provides a detailed understanding of the dynamical evolution of the system. The modulation technique also allows for a controllable transfer (deexcitation) of atoms from such wave packets to a state bound by the lattice. Thus, it acts as a beam splitter for matter waves that can selectively address different bands, enabling the preparation of atoms in localized states. The combination of wave packet creation and deexcitation closely resembles the well-known method of pump-probe spectroscopy. Here, we use the deexcitation for spectroscopy of the anharmonicity of the combined potential. Finally, we demonstrate that lattice modulation can be used to excite matter wave packets to even higher momenta, producing fast wave packets with potential applications in precision measurements.

Figure 1

(a) Calculated spectrum of the 1D single-particle Schrödinger equation for the combined magnetic trap and optical lattice potential. Each horizontal line represents the summed population density of the eigenstates within an energy bin of width $h\times 664$ Hz. The shading becomes darker with increasing density of states and white indicates forbidden regions. The dashed red line indicates the local maximum of the combined potential. Three experimental processes are shown: (1) A wave packet can be generated via a two-step excitation process. (2) A wave packet can be deexcited into lower-lying localized states. (3) A wave packet can be excited to higher states. (b) The same processes are shown in the conventional band picture represented in the first Brillouin zone versus quasimomentum $\hslash q$.

Figure 2

Number of atoms in the lower wave packet as a function of the modulation frequency for a pulse of $T=500\mu$s duration. The data were fitted with a Gaussian as an empirical model, yielding an optimal transfer frequency of ${\nu }_{\mathrm{mod}}=30.8$ kHz, and a $1/e^2$ width of 5.7 kHz. Inset: Number of atoms in the lower wave packet as a function of modulation duration $T$ for ${\nu }_{\mathrm{mod}}=33$ kHz (the same vertical axis units as in the main figure); the data were fitted with an exponential saturation function with time constant $\tau =130\mu$s. For both data sets, the relative modulation amplitude was $\epsilon =0.165$.

Figure 3

Wave packet motion in the quasicontinuum following modulation at 30 kHz, fitted with Eq. (2). Lower-wave-packet data and fit, filled circles, solid line; upper-wave-packet data and fit, open circles, dashed line. The wave packets oscillate harmonically but with turning regions caused by reflections at the energy gap between the second and third excited bands.

Figure 4

Wave packet motion for different modulation frequencies. (a) Upper-wave-packet displacement as a function of hold time for ${\nu }_{\mathrm{mod}}=28$ kHz (filled circles, solid line), 32 kHz (open squares, dotted line), and 36 kHz (open circles, dashed line); the data for each frequency have been fitted with Eq. (2). (b) Maximum wave packet displacement as a function of modulation frequency; data and errors were obtained from fits as in (a). Solid line, fit to data using Eq. (3).

Figure 5

Fraction of the initial wave packet deexcited to localized states as a function of deexcitation frequency for a pulse amplitude of $\epsilon =0.15$ and a duration of 500 $\mu$s. The data show the mean fraction in the upper localized state in a 2 kHz bin. The data have been fitted with a triple Gaussian function, yielding resonances at 19.1, 31.7, and 47.7 kHz; these correspond to deexcitation into the second excited, first excited, and lowest bands, respectively. The error bars show the standard error of the mean for a given bin.

Figure 6

Fraction of atoms deexcited from the lower wave packet as a function of pulse amplitude. The experiment was performed for three deexcitation frequencies: 17 kHz (blue, open squares, dotted line), 34 kHz (black, filled circles, solid line), and 49 kHz (red, open circles, dashed line). The data were corrected for off-resonant transfer to lower bands (see text). Linear fits were performed to guide the eye. Inset: Fraction of atoms in a localized state as a function of deexcitation pulse duration at a frequency of 32 kHz and $\epsilon =0.17$. The data were fitted with an exponential saturation function with time constant $\tau =132\mu$s.

Figure 7

Resonant deexcitation frequency as a function of position. The extracted background trapping potential (see text) is shown on the other vertical axis. The data were fitted with quadratic (dashed line) and cubic (solid line) functions to show the anharmonicity of the potential. The fitted offset ($c$) was subtracted from the data points and the fits.

Figure 8

Motion of multiply excited wave packets. (a) Lower-wave-packet data and fit, filled circles, solid line; upper-wave-packet data and fit, open circles, dashed line. (b) Absorption image showing the multiply excited and the initial wave packets at a hold time of 4.5 ms.

Figure 9

Atom number in the highly excited wave packet as a function of the excitation frequency for four values of the delay between the modulation pulses: 0.6 ms (red filled circles), 2.0 ms (black circles), 2.9 ms (blue diamonds), and 3.3 ms (green squares). Solid lines indicate Lorentzian fits to each data set. The shift of the resonance frequency and the broadening of the excitation profile can be understood by considering the movement of the wave packet within the band structure.

Figure 10

Resonant transition frequency for the second excitation as a function of the delay between modulation pulses. The red dashed line is a fit to the data using Eq. (4). The blue solid line is a fit to a model based on a numerical calculation of the band structure for the combined trap. It shows a small kink at the gap between the third and fourth excited bands, which is not resolved in the data; a closeup of this region is shown in the inset. The error bars correspond to the uncertainty of the resonant frequency extracted from the Lorentzian fits in Fig. 9.