Observation and Uses of Position-Space Bloch Oscillations in an Ultracold Gas

We report the observation and characterization of position-space Bloch oscillations using cold atoms in a tilted optical lattice. While momentum-space Bloch oscillations are a common feature of optical lattice experiments, the real-space center-of-mass dynamics are typically unresolvable. In a regime of rapid tunneling and low force, we observe real-space Bloch oscillation amplitudes of hundreds of lattice sites, in both ground and excited bands. We demonstrate two unique capabilities enabled by tracking of Bloch dynamics in position space: measurement of the full position-momentum phase-space evolution during a Bloch cycle, and direct imaging of the lattice band structure. These techniques, along with the ability to exert long-distance coherent control of quantum gases without modulation, may open up new possibilities for quantum control and metrology.

Figure 1

Position-space Bloch oscillations. (a) A time sequence of in situ absorption images of atoms in a $5.4E_R$-deep optical lattice, subjected to a force corresponding to a Bloch frequency of 21.1 Hz. Each image corresponds to an individual experimental run with the indicated hold time. (b) Measured evolution of integrated density distribution as a function of axial position during a Bloch oscillation. (c) Numerical GPE prediction for (b), using the split-step Fourier method [21]. Asymmetrical width variation is due to force inhomogeneity.

Figure 2

Characterization of position-space Bloch oscillations. (a) Ensemble position versus time during Bloch oscillations at different forces with a constant tunneling rate $J=1.55\mathrm{kHz}$. (b) Position versus time during Bloch oscillations with different tunneling rates at a constant force of $F=23\mathrm{Hz}$. Solid black lines in (a) and (b) are fits to damped sinusoids; thicker solid red lines in (a) are the result of numerical GPE calculations. (c) Bloch oscillation amplitude as a function of force at $J=1.55\mathrm{kHz}$. (d) Bloch oscillation amplitude as a function of tunneling rate at $F=23\mathrm{Hz}$. Solid lines in (c) and (d) represent the predicted $l_{WS}=2J/F$ with no fit parameters.

Figure 3

Phase-space evolution during a Bloch oscillation. Here $J=1.55\mathrm{kHz}$ and $F=23\mathrm{Hz}$. (a) Absorption images of the atomic distribution after a 3 ms time of flight. The distribution at each hold time is a convolution of initial quasimomentum and position. (b) Absorption images after varying times of flight, for a hold time of 21 ms. The initial quasimomentum can be extracted via a linear fit. (c),(d) Ensemble position and momentum versus time during a Bloch oscillation. Where multiple momentum peaks can be distinguished, both are plotted. (e) Combining (c) and (d) yields the phase space evolution of the ensemble during the first Bloch period $T_B=1/f_B$. The color map corresponds to the hold time in units of $T_B$. Where multiple momentum peaks can be distinguished, both are plotted with symbol size indicating relative atom number. At $0.5T_B$, Bragg scattering is observed as a discontinuity of $2\hslash k_L$ in the measured momentum distribution.

Figure 4

Ensemble width evolution during Bloch oscillations. Measured time evolution of the second moment of the atomic distribution $w_z$ for different forces with $J=1.55\mathrm{kHz}$. Black lines correspond to semiclassical evolution, using the calculated band dispersion, of an ensemble with the measured initial width, and red lines to numerical GPE results. At lower values of $F$ (higher values of $l_{WS}$), the effects of force inhomogeneity are more pronounced.

Figure 5

Band imaging with position-space Bloch oscillations. (a) Time-series images of atomic ensembles undergoing Bloch oscillations in lattice depths of $6E_R$, $7.4E_R$, and $10.3E_R$. (b) Measured band structure $E(k)$ for the three lattice depths (circles), determined by scaling the measured center positions from (a) according to Eq. (5). Lines show independently-calculated band structures with no adjustable parameters.

Figure 6

Position-space Bloch oscillations in an excited band. The lattice depth is set to $9.5E_R$ and the Bloch frequency to 50.7 Hz. (a) Time-series absorption image of ground-band Bloch oscillations. (b) Time-series image of Bloch oscillations in the first excited band, which are characterized by the same frequency as ground-band oscillations, but exhibit a significantly larger amplitude due to the larger bandwidth. Also visible is a remnant fraction of atoms in the ground band, due to imperfect excitation. These Bloch oscillate independently.