Spectral response of magnetically trapped Bose gases to weak microwave fields

P. Federsel, C. Rogulj, T. Menold, J. Fortágh, and A. Günther

Phys. Rev. A 92, 033601 – Published 1 September 2015

Abstract

Microwave fields can be used to drive local spin transitions in quantum gases and for outcoupling of cold atomic beams from magnetic traps. In this paper, we derive an analytic theory for the outcoupling rate as a response to weak microwave fields of varying frequency and power. The theory holds for thermal clouds and Bose-Einstein condensates. It allows for calculating transition rates in arbitrary magnetic trap geometries and includes the effect of gravity. We verify our theory by measuring the flux of outcoupled atoms at the single-particle level. The derived spectral response is important for magnetic noise spectroscopy with quantum gases, and for probing quantum gas dynamics with single atom detectors in real time.

Received 29 May 2015

DOI:https://doi.org/10.1103/PhysRevA.92.033601

Authors & Affiliations

P. Federsel, C. Rogulj, T. Menold, J. Fortágh, and A. Günther^*

Physikalisches Institut der Universität Tübingen, Auf der Morgenstelle 14, D-72076 Tübingen, Germany

^*a.guenther@uni-tuebingen.de

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Vol. 92, Iss. 3 — September 2015

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Figure 1
Microwave outcoupling of ${}^{87}{Rb}$ atoms. (a) Outcoupling scheme: Atoms are initially prepared in the $5 S_{1 / 2}$ , $| F = 2, m_{F} = 2 〉$ state ( $g_{F} = 1 / 2$ ), where they are trapped in a magnetic potential $U_{mag} (\vec{r})$ . Energy selective outcoupling is achieved by microwave (mw) coupling to the nontrapped $| F = 1, m_{F} = 1 〉$ state ( $g_{F} = - 1 / 2$ ). The coupling rate depends strongly on the intensity and frequency of the applied microwave and allows for tomographic investigation of quantum gases. (b) Dressed state picture for mw coupling of the initial and final bare states, $| i 〉$ and $| f 〉$ . The microwave coupling causes degeneracy at the resonance positions and a spatial dependent detuning (black dashed line). Due to the coupling strength the dressed state energies change slightly (red line), resulting in avoided crossings at the resonance positions. (c),(d) Microwave resonance surfaces and density distributions for a thermal cloud of 190 nK (c) and a BEC of 8000 atoms (d) in a harmonic trap with $ω_{x / y / z} = 2 π \times 85, 70, 16 Hz$ . Resonance surfaces are shown for microwave frequency spacings of $2 π \times 20 kHz$ . They are elliptically shaped and centered around the magnetic field minimum. Both density distributions are displaced by the gravitational sag, amounting to $y_{0} \approx 50 μ m$ . For small cloud extensions (BEC), the resonance surfaces can be approximated as flat.
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Figure 2
Spectral response of thermal clouds and a BEC with $10 000$ atoms in a harmonic trap with trapping frequencies $ω_{x / y / z} = 2 π \times 85, 70, 16 Hz$ , as calculated from Eqs. (22) and (24). For $T > 1 μ K$ the spectral shape is dominated by the cloud temperature, giving access to the particles' energy distribution. For $T < 1 μ K$ the spectral response depends directly on the cloud's density profile and the gravitational acceleration, allowing for spatial tomography of quantum gases.
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Figure 3
Spectral response of a thermal cloud (red points) and a BEC (blue points), measured by sweeping the mw frequency through the cloud, while detecting the ion count rate. For a single sweep through the thermal cloud ( $N = 730 \times 10^{3}$ , $T = 160 nK$ ) and the BEC ( $N = 8200$ , $T \approx 30 nK$ ), the total number of detected ion amounts are 4000 and 50, respectively. To reduce noise, the BEC measurement is shown as average of four sweeps. Microwave frequencies are shown relative to the magnetic trap center (black dashed line), located at 876 and 873 mG for the thermal cloud and BEC. From the measurements we deduce the FWHM (full width half maximum) of the spectral response to be 69 kHz for the thermal cloud and 18 kHz for the BEC. The results from the theoretical ab initio calculations following Eqs. (22) and (24) are shown as solid lines.
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Figure 4
Response of a thermal cloud (red points) and a BEC (blue points) to different mw amplitudes and Rabi frequencies. At each amplitude the ion count rate is measured for 800 ms on individual clouds. Data points show the initial ionization rate as extracted from an exponential fit to the measurement time. The ab initio theory from Eqs. (22) and (24) is shown with zero (dashed lines) and finite (solid lines) offset count rates. The offset values for the thermal cloud ( $N = 5.8 \times 10^{5}$ , $T = 240 nK$ ) and the BEC ( $N = 10 \times 10^{3}$ ) have been set to 700 and $15 Hz$ , respectively. The offset is purely technical and describes atomic spin flips driven by an off-resonant excitation from the 778-nm laser. In general (without offset) the sensitivity limit is reached at the dark count level amounting to $14.7 p T / \sqrt{Hz}$ for the thermal cloud.
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