Observing the Onset of Effective Mass

Rockson Chang, Shreyas Potnis, Ramon Ramos, Chao Zhuang, Matin Hallaji, Alex Hayat, Federico Duque-Gomez, J. E. Sipe, and Aephraim M. Steinberg

Phys. Rev. Lett. 112, 170404 – Published 2 May 2014

Abstract

The response of a particle in a periodic potential to an applied force is commonly described by an effective mass, which accounts for the detailed interaction between the particle and the surrounding potential. Using a Bose-Einstein condensate of ${}^{87}{Rb}$ atoms initially in the ground band of an optical lattice, we experimentally show that the initial response of a particle to an applied force is in fact characterized by the bare mass. Subsequently, the particle response undergoes rapid oscillations and only over time scales that are long compared to those of the interband dynamics is the effective mass observed to be an appropriate description. Our results elucidate the role of the effective mass on short time scales, which is relevant for example in the interaction of few-cycle laser pulses with dielectric and semiconductor materials.

Received 5 March 2013

DOI:https://doi.org/10.1103/PhysRevLett.112.170404

Authors & Affiliations

Rockson Chang^1,*, Shreyas Potnis¹, Ramon Ramos¹, Chao Zhuang¹, Matin Hallaji¹, Alex Hayat^1,2, Federico Duque-Gomez¹, J. E. Sipe¹, and Aephraim M. Steinberg¹

¹Department of Physics and Institute of Optics, University of Toronto, 60 St. George Street, Toronto, Ontario M5S 1A7, Canada
²Department of Electrical Engineering, Technion, Haifa 32000, Israel

^*Present address: Laboratoire Charles Fabry, Institut d’Optique, CNRS, Univ Paris-Sud, 2 Avenue Augustin Fresnel, 91127, Palaiseau Cedex, France. rockson.chang@u-psud.fr

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Vol. 112, Iss. 17 — 2 May 2014

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Images

Figure 1
Illustration of effective mass dynamics. (a) Lattice dispersion relation showing the time evolution of an initially single-band wave packet under an abruptly applied force $F$ . (b) The corresponding acceleration $⟨ a ⟩$ . Over time scales that are long compared to one over the band-gap frequency, the acceleration is characterized by the lattice effective mass $m^{*}$ and leads to Bloch oscillations (dashed black line). However the initial response of the single-band state is characterized by the bare mass $m_{0}$ (bare mass response shown by green dash-dotted line), and then rapidly oscillates around the usual effective mass behaviour.
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Figure 2
Observation of effective mass dynamics. $s = 9.4$ and $F / m_{0} = 11.7 μ m / {ms}^{2}$ . (a) Composite of absorption images. Each row is a slice of an image taken after variable evolution time in the lattice ( $4 μ s$ resolution) and 20 ms TOF. Color scale: from blue to red, indicates increasing optical density. (b) Example absorption image, and one-dimensional profile and fit. (c)–(e) The velocity and amplitude of each peak, and the reconstructed average velocity of the distribution. The gap in the data at $650 μ s$ due to poor condensate preparation during these runs. The solid black curve is the result of low-pass filtering this data, serving as a guide to the eye. (f)–(h) Corresponding GPE simulation (see text).
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Figure 3
(a) Variation of dynamical mass frequency (upper) and Bloch frequency (lower) with applied force. The dynamical mass frequency, normalized by $Δ$ , is expected to be independent of the force. The Bloch frequency scales linearly with the applied force. (b) Variation in Bloch frequency (upper) and dynamical mass frequency (lower) with lattice depth. The Bloch frequency, normalized by $F / m_{0}$ , is expected to be a constant $m_{0} d / h$ (blue solid line). The dynamical mass frequency is compared to the band gap at $k = 0$ and $k = k_{r}$ (upper and lower solid blue lines, respectively), and to simulated data fitted to Eq. (5) (dashed black curve).
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Figure 4
Effective mass and dynamical mass results. The effective mass at the center of the ground band associated with the Bloch oscillation is plotted as green squares. As the lattice depth increases, the effective mass is expected to increase (dashed black line). Accounting for the finite imaging window skews the expected $m^{*}$ down (solid black line; see text). The initial ( $t = 0$ ) dynamical mass is extracted from the full dynamics (red circles), and is expected to be consistent with the bare mass response $m^{*} / m_{0} = 1$ , independent of lattice depth (solid blue line).
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