Spatial correlation of two-dimensional bosonic multimode condensates

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Spatial correlation of two-dimensional bosonic multimode condensates

Wolfgang H. Nitsche, Na Young Kim, Georgios Roumpos, Christian Schneider, Sven Höfling, Alfred Forchel, and Yoshihisa Yamamoto

Phys. Rev. A 93, 053622 – Published 23 May 2016

Abstract

The Berezinskii-Kosterlitz-Thouless (BKT) theorem predicts that two-dimensional bosonic condensates exhibit quasi-long-range order which is characterized by a slow decay of the spatial coherence. However previous measurements on exciton-polariton condensates revealed that their spatial coherence can decay faster than allowed under the BKT theory, and different theoretical explanations have already been proposed. Through theoretical and experimental study of exciton-polariton condensates, we show that the fast decay of the coherence can be explained through the simultaneous presence of multiple modes in the condensate.

Received 10 October 2015

DOI:https://doi.org/10.1103/PhysRevA.93.053622

This article is available under the terms of the Creative Commons Attribution 3.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Exciton polariton Quantum coherence & coherence measures

2-dimensional systems Bose-Einstein condensates

Atomic, Molecular & Optical

Authors & Affiliations

Wolfgang H. Nitsche^1,*, Na Young Kim^1,†, Georgios Roumpos^1,‡, Christian Schneider², Sven Höfling^2,3,4, Alfred Forchel², and Yoshihisa Yamamoto^1,4

¹E. L. Ginzton Laboratory, Stanford University, Stanford, California 94305, USA
²Technische Physik, Universität Würzburg, Am Hubland, 97074 Würzburg, Germany
³School of Physics & Astronomy, University of St Andrews, St Andrews KY16 9SS, United Kingdom
⁴National Institute of Informatics, Hitosubashi, Chiyoda-ku, Tokyo 101-8430, Japan

^*Corresponding author. Present address: Halliburton, Houston, Texas 77032. ; nitsche@stanford.edu
^†Present address: University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada.
^‡Present address: Google, Mountain View, California 94043.

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Vol. 93, Iss. 5 — May 2016

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Images

Figure 1
Simulated data. (a) A total density $n_{total} (r)$ compared to the $n_{wanted} (r) = H (10 μ m - r) \times 1 a . u$ . The contributions $| A_{s} Ψ_{s} (r, ϕ) |^{2}$ of the individual modes are also shown. (b) A total density $n_{total} (x, y)$ . (c) The interference signal as it would be measured by the camera of a Michelson interference setup. (d) The phase as it would be determined through evaluating interference data. (e) A correlation function $g_{total}^{(1)} (x, y, - x, y)$ as it would be calculated from the interference data. (f) All the values of (e) but plotted as a function of $x$ . Each gray point corresponds to one pixel of (e). The continuous black line shows the averages of all the gray data points with the same $x$ value. The oscillatory behavior can be understood if we assume that one of the higher modes dominates at longer distances since the lowest mode is only significant in the region close to the center.
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Figure 2
Measured data. (a)–(c) Measured phase (a) and visibility (b) and (c) of an exciton-polariton condensate with nearly constant density which has been created by pumping the sample with a top-hat profile at low temperature ( $\approx 7 K$ ). The decay of the visibility is significantly faster than measured (i) for a single-mode BKT condensate [4] with quasi-long-range order. Pseudovortices which appear as forks in (a) are clearly visible. Each gray point in (c) corresponds to one pixel of (b), and the continuous black line in (c) has been calculated by averaging all the gray data points with the same $x$ value. (d)–(f) The same kind of measurement (still with a top-hat pump profile) at $\approx 200 K$ where the coherence can only be explained by VCSEL lasing. Nevertheless, the behavior is the same as for (a)–(c), which confirms that the visibility is determined by the presence of several quantized modes rather than by the intrinsic coherence properties of a condensate. (g)–(i) The same kind of measurement at low temperatures ( $\approx 5 K$ ) with a Gaussian pump profile which excites a single-mode exciton-polariton condensate. No pseudovortices (forks) appear in the phase map (g).
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Figure 3
Measured dispersion. (a) The same experimental conditions as in Figs. 2–2 (exciton-polariton condensation with top-hat pump profile at $\approx 7 K$ ). Many quantized modes at different energy levels are clearly visible. (b) The same experimental conditions as in Figs. 2–2 (VCSEL lasing with top-hat pump profile at $\approx 200 K$ ). At least two quantized modes at different energies can be seen. (c) The same experimental conditions as in Figs. 2–2 (exciton-polariton condensate with Gaussian pump profile at $\approx 5 K$ ). Condensation occurs in a single mode at one energy level. (d) The same experimental conditions as in Fig. 4 (exciton-polariton condensation with top-hat pump profile at $\approx 8 K$ ). Many quantized modes at different energy levels are clearly visible, and the modes are sufficiently energetically separated to allow the selection of individual modes through spectroscopy as shown in Fig. 4.
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Figure 4
Spectrometric-resolved interference measurement (exciton-polariton condensate with nearly constant density, created at $\approx 8 K$ with top-hat pumping). The estimated detuning was −8 meV, which is far red-detuned. (a) A spectroscopic interference setup. (b) A position $x$ - and $y$ -resolved interference image (for one specific path-length difference $Δ L$ ). (c) A position $x$ - and energy $E$ -resolved interference image, again for one specific path-length difference. Here, we can identify at least ten individual modes at different energy levels. (d) Evaluation for the sixth mode. $I_{measured}^{(mode 6)}$ has been measured at one specific path-length difference, whereas $I_{A}^{(mode 6)}$ and $I_{B}^{(mode 6)}$ have been calculated by using the measured $I_{measured}^{(mode 6)}$ from many different path-length differences. (e) Visibilities $g_{(mode s)}^{(1)} = I_{A}^{(mode s)} / I_{B}^{(mode s)}$ for the ten individual modes. (f) The overall correlation (black line) which has been calculated from the position $x$ - and $y$ -resolved measurement where only data within the area between the black lines in (b) have been considered. For comparison we also show different approaches to calculate the overall correlation from the position $x$ - and energy $E$ -resolved interference image (c).
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