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arXiv:1408.0944 · Physical Review A

Magnetic tensor gradiometry using Ramsey interferometry of spinor condensates

A. A. Wood, L. M. Bennie, A. Duong, M. Jasperse, Lincoln D. Turner, R. P. Anderson

Published Nov 5, 2015• 22 citations
arXivHTMLPDF

Abstract

We have realized a magnetic tensor gradiometer by interferometrically measuring the relative phase between two spatially separated Bose-Einstein condensates (BECs). We perform simultaneous Ramsey interferometry of the proximate $^{87}\mathrm{Rb}$ spin-1 condensates in free fall and infer their relative Larmor phase, and thus the differential magnetic-field strength, with a common-mode phase noise suppression exceeding $50\phantom{\rule{0.28em}{0ex}}\mathrm{dB}$. By appropriately biasing the magnetic field and separating the BECs along orthogonal directions, we measure in vacuo the magnetic-field-gradient tensor of ambient and applied magnetic fields with a nominal precision of $0.30\phantom{\rule{0.28em}{0ex}}\mathrm{nT}\phantom{\rule{0.16em}{0ex}}{\mathrm{mm}}^{\ensuremath{-}1}$ and a sensor volume of $2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\phantom{\rule{0.28em}{0ex}}{\mathrm{mm}}^{3}$. We predict a spin-projection noise-limited magnetic energy resolution of order $\ensuremath{\sim}10\ensuremath{\hbar}$ for typical Zeeman coherence times of trapped condensates with this scheme, even with the low measurement duty cycle of current BEC experiments.

DOI
10.1103/physreva.92.053604

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We have realized a magnetic tensor gradiometer by interferometrically measuring the relative phase between two spatially separated Bose-Einstein condensates (BECs). We perform simultaneous Ramsey interferometry of the proximate 87Rb^{87}\mathrm{Rb}87Rb spin-1 condensates in free fall and infer their relative Larmor phase, and thus the differential magnetic-field strength, with a common-mode phase noise suppression exceeding 50dB50\phantom{\rule{0.28em}{0ex}}\mathrm{dB}50dB. By appropriately biasing the magnetic field and separating the BECs along orthogonal directions, we measure in vacuo the magnetic-field-gradient tensor of ambient and applied magnetic fields with a nominal precision of 0.30nTmm\ensuremath−10.30\phantom{\rule{0.28em}{0ex}}\mathrm{nT}\phantom{\rule{0.16em}{0ex}}{\mathrm{mm}}^{\ensuremath{-}1}0.30nTmm\ensuremath−1 and a sensor volume of 2\ifmmode×\else\texttimes\fi10\ensuremath−5mm32\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}\phantom{\rule{0.28em}{0ex}}{\mathrm{mm}}^{3}2\ifmmode×\else\texttimes\fi10\ensuremath−5mm3. We predict a spin-projection noise-limited magnetic energy resolution of order \ensuremath∼10\ensuremathℏ\ensuremath{\sim}10\ensuremath{\hbar}\ensuremath∼10\ensuremathℏ for typical Zeeman coherence times of trapped condensates with this scheme, even with the low measurement duty cycle of current BEC experiments.
We demonstrate a precise magnetic microscope based on direct imaging of the Larmor precession of a 87Rb^{87}\mathrm{Rb}87Rb spinor Bose-Einstein condensate. This magnetometer attains a field sensitivity of 8.3  pT/Hz1/28.3\text{ }\text{ }\mathrm{pT}/{\mathrm{Hz}}^{1/2}8.3  pT/Hz1/2 over a measurement area of $120\text{ }\text{ }\ensuremath{\mu}{\mathrm{m}}^{2}$, an improvement over the low-frequency field sensitivity of modern SQUID magnetometers. The achieved phase sensitivity is close to the atom shot-noise limit, estimated as 0.15  pT/Hz1/20.15\text{ }\text{ }\mathrm{pT}/{\mathrm{Hz}}^{1/2}0.15  pT/Hz1/2 for a unity duty cycle measurement, suggesting the possibilities of spatially resolved spin-squeezed magnetometry. This magnetometer marks a significant application of degenerate atomic gases to metrology.
Spinor Bose-Einstein condensates (BECs) are multi-component superfluids with magnetic interactions, formed from dilute gases of atoms in the Zeeman sublevels mF of a single hyperfine ground state F. Spinor BECs are inherently magnetic quantum objects, and exhibit a multitude of features such as spin domains, magnon excitations, and distinct magnetic phases. At the heart of the magnetic nature of a spinor BEC is the spin-dependent collisional interaction c, which determines the overall magnetic behaviour of the superfluid. In conjunction with the quadratic Zeeman shift q, the magnitude and sign of c give rise to an intricate magnetic phase diagram. The magnetic phase of a spinor BEC is tunable, and c and q are the knobs. In addition to realising quantum simulators of magnetism, spinor BECs are also precise magnetic sensors. Atomic magnetometry measures the Larmor precession frequency of atoms, which is proportional to the magnetic field strength. The sensitivity of an atomic magnetometer is principally dependent on the duration of the measurement (interrogation time) and number of atoms participating in the measurement. BECs are highly suitable as atomic magnetometers, allowing for long interrogation times and high atomic density, with microscopic sensor volumes. The latter is an important consideration, as measurement of small magnetic sources demand equally microscopic sensors. Established forms of magnetometry (such as warm vapour magnetometers) use large sensing volumes and are not trivially miniaturised. The exquisite sensitivity of a spinor BEC to magnetic fields becomes a liability when one studies low energy (~ 1 Hz) spinor physics over long evolution times (100 ms) in a typical laboratory environment. To first order, the spin-collisional interaction is independent of the magnetic field. However, spatially inhomogeneous magnetic fields are problematic, exerting mechanical forces on the magnetically sensitive spin components, suppressing spin collisions and possibly introducing relaxation mechanisms that confound the observation of intrinsic equilibration. Precise characterisation of magnetic field gradients and their cancellation is thus of paramount importance. In this thesis, the interaction of a spinor BEC with its environment is studied. The construction, operation and optimisation of a BEC machine is described in detail, as well as the quantum state preparation and control techniques used. Spin-mixing collisions were studied by observing population oscillations, the characteristic hallmark of such coherent phenomena. The observed population dynamics showed strong evidence of the deleterious effects of magnetic field gradients. A series of experiments were performed using spin-echo pulse sequences to decouple the evolving spinor condensate from the inhomogeneous magnetic environment with encouraging results, but which posed further questions. The fidelity of spin-echo pulses in the strong radiative coupling regime was also studied. Spinor condensates were then employed as sensitive magnetometers to characterise the magnetic landscape they inhabit. A new method of measuring magnetic field gradients was developed, using a pair of spinor BECs separated in space that are simultaneously addressed by a Ramsey interferometry sequence. Combined with a re-orientable magnetic bias field, the resulting gradiometer was used to measure the magnetic field gradient tensor in vacuo. This scheme can be extended to realise a high-precision vector co-magnetometer, with spatiotemporal sensitivity comparable to established forms of magnetometry. Vector light shifts (VLSs) are optically-induced energy shifts that result in effective magnetic fields, which in our case originate from the laser beams used to trap the atoms. The VLS vanishes for purely linearly polarised light, which is experimentally difficult to obtain in vacuo. Small elliptical polarisation imperfections originating from birefringent optics and the glass vacuum cell result in a non-zero shift that contaminates magnetometry measurements. The VLS is spatially varying due to the intensity gradient of the trapping light. Due to gravitational sag of the trapping potential, the spatial variation of the VLS is accentuated, resulting in an effective magnetic field gradient that contributes to dephasing and component separation. The simultaneous Ramsey interferometry experiments are then adapted to measure the VLS-induced Zeeman shift and diagnose its elimination. Up to 99.96% of the VLS is eliminated using this method, and extensions to the technique are described.
A Bose-Einstein condensate is used as an atomic source for a high precision sensor. A $5\ifmmode\times\else\texttimes\fi{}1{0}^{6}\text{ }\text{ }\mathrm{atom}$ F=1F=1F=1 spinor condensate of 87Rb^{87}\mathrm{Rb}87Rb is released into free fall for up to 750 ms and probed with a T=130  msT=130\text{ }\text{ }\mathrm{ms}T=130  ms Mach-Zehnder atom interferometer based on Bragg transitions. The Bragg interferometer simultaneously addresses the three magnetic states $|{m}_{f}=1,0,\ensuremath{-}1⟩$, facilitating a simultaneous measurement of the acceleration due to gravity with a 1000 run precision of $\mathrm{\ensuremath{\Delta}}g/g=1.45\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}9}$ and the magnetic field gradient to a precision of 120 pT/m120\text{ }\mathrm{pT}/\mathrm{m}120 pT/m.
A precision inertial sensor based on the interference of matter waves has been designed and implemented. The apparatus is capable of producing both cold thermal and ultracold Bose-Einstein condensate (BEC) atomic ensembles as the inertial test mass. BECs of up to 5x10^6 Rubidium 87 atoms with an effective temperature of 50nK are produced every 13 seconds using a combination of two and three dimensional magneto-optical traps (2D and 3D MOT) and evaporative cooling in a hybrid magnetic-quadrupole and crossed optical dipole trap. The atomic cloud is then propagated through a vertically oriented Mach-Zehnder interferometer utilizing Bragg diffraction beam splitters. An 2.6m drop tube allows for up to 750ms of free fall and interferometer times of up to 250ms. The drop has four regions of imaging corresponding to 0, 25, 220, 530, and 750ms of expansion. Standard absorption imaging is used for the upper two imaging regions and frequency modulation imaging (FMI) is used for the lower regions. FMI exhibits near atom-shot noise limited signal-to-noise of 948 on a 1x10^6 atom cloud. The inertial reference is provided by the Bragg beam retro-reflector which is suspended via a geometric anti-spring passive vibration isolation system providing 65 dB of isolation at 70 Hz. Prior to the first interferometer pulse the ensemble is placed into a spin superposition of the |F=1> ground state manifold using horizontal co-propagating Raman beams. The three internal spin states remain orthogonal throughout the interferometer sequence resulting in three simultaneous interferometer with separable magnetic gradient and gravitational acceleration signals. The effects of meanfield energy on the interferometric phase uncertainty is explored under current trap conditions. It is found that 30 ms of free expansion prior to the first beam splitter decreases the internal energy such that the expected meanfield phase noise is below 1mrad. No significant decrease in the interferometer contrast is observed for all available T, whereas a thermal test mass interferometer of equivalent longitudinal temperature produces no interference at interferometer times > 100ms. A direct comparison of test mass spatial coherence length in an optically transversely confined interferometer is investigated. The visibility and contrast of a BEC and three thermal sources with varying spatial coherence are compared as a function of interferometer time. At short times, the fringe visibility of a BEC source approaches 100%, nearly independent of p pulse efficiency, while thermal sources have fringe visibilities limited to the p pulse efficiency. More importantly for precision measurement systems, the BEC source maintains interference at interferometer times significantly beyond that of the thermal source. A BEC test mass is used for the simultaneous precision measurement of gravity and magnetic field gradients. An 8 hour data run of a 130ms interferometer shows good agreement between the interferometer output and a theoretical model of the solid earth tides. The residual of the theoretical model and the experimental data show a 1000 run precision of dg/g = 1.45x10^9 corresponding to a phase noise of 3.8mrad. The integrated phase noise of the apparatus shows a sensitivity to magnetic field gradients of 120pT/m. By varying the central spatial position of a 40ms interferometer the magnetic field gradient along a portion of the drop tube is measured.
Magnetic field sensing utilizing Bose-condensed atomic gases can achieve both high sensitivity and spatial resolution. Employing two-photon transitions from F=1, mF=1→F=2, mF=-1 (F is the quantum number of total atomic angular momentum, mF is the magnetic quantum number of F level) of Bose-condensed 87Rb atomic gases in one-dimensional optical lattice, the precision magnetic field measurement with phase Ramsey interference has been demonstrated. Utilizing single-photon transitions from F=1, mF=1→F=2, mF=0 of Bose-condensed 87Rb atomic gases in a crossed optical dipole trap, we experimentally study frequency domain Ramsey interference, time domain Ramsey interference, and phase domain Ramsey interference. Based on phase Ramsey interference, the sensitivity and spatial resolution of static magnetic field measurement are evaluated. The comparison of these three typical Ramsey interferences reveals that phase Ramsey interference can effectively avoid the fringe distortion in frequency-domain interference and the decoherence effects in time-domain interference, thereby improving the sensitivity and accuracy of the static magnetic field measurement. Utilizing Bose-condensed 87Rb atomic gases with the volume of 1.31±0.03×10-9 cm3 in an unshielded environment, a sensitivity of 130 nT/Hz for static magnetic fields can be achieved. By reducing the preparation time of the atomic gases and increasing the signal interrogation time, the sensitivity has the potential to be further improved.
We study the time-evolution of an optically trapped spinor Bose-Einstein condensate under the influence of a dominating magnetic bias field in the z-direction, and a perpendicular smaller field that couples the spinor states. We show that if the bias field depends quadratically on time, the relative phases of the spinor components affect the populations of the final state. This allows one to measure the differences in the time-evolution of the relative phases, thereby realizing a multi-arm interferometer in a spinor Bose-Einstein condensate.