Multi-loop atomic Sagnac interferometry
Christian Schubert, Sven Abend, Matthias Gersemann, Martina Gebbe, Dennis Schlippert, Peter Berg, Ernst M. Rasel
MMulti-loop atomic Sagnac interferometry
Christian Schubert a ,1,2 , Sven Abend , Matthias Gersemann , Martina Gebbe ,Dennis Schlippert , Peter Berg , Ernst M. Rasel a [email protected] Deutsches Zentrum f¨ur Luft- und Raumfahrt e.V. (DLR), Institut f¨ur Satellitengeod¨asie undInertialsensorik, c/o Leibniz Universit¨at Hannover, DLR-SI, Callinstraße 36, 30167 Hannover, Germany Institut f¨ur Quantenoptik, Gottfried Wilhelm Leibniz Universit¨at Hannover, Welfengarten 1,D-30167 Hannover, Germany Zentrum f¨ur angewandte Raumfahrttechnologie und Mikrogravitation (ZARM),Universit¨at Bremen, Am Fallturm, D-28359 Bremen, Germany
Abstract
The sensitivity of light and matter-wave interferometers to rotations is based on the Sagnaceffect and increases with the area enclosed by the interferometer. In the case of light, thelatter can be enlarged by forming multiple fibre loops, whereas the equivalent for matter-waveinterferometers remains an experimental challenge. We present a concept for a multi-loop atominterferometer with a scalable area formed by light pulses. Our method will offer sensitivities ashigh as 2 · − rad/s at 1 s in combination with the respective long-term stability as requiredfor Earth rotation monitoring. Keywords:
Atom interferometer, Matter-wave interferometry, Rotation measurement
Rotation measurements are utilised for inertial navi-gation and earth observation exploiting the large en-closed area in fibre-optical and meter-scale ring lasergyroscopes [1–3]. Atom interferometry offers a dif-ferent approach for providing absolute measurementsof inertial forces with high long-term stability. More-over, achieving the necessary areas for competitiveperformance with matter waves is a long-standingchallenge [4–10].We propose an atom interferometer performingmultiple loops in free fall. Our setup opens the per-spective for sensitivities as high as 2 · − rad/s at1 s, comparable to the results of the ring laser gyro-scope at the geodetic observatory Wettzell [1, 2].The interferometric Sagnac phase shift [11] in-duced by a rotation (cid:126) Ω depends linearly on the areavector (cid:126) A as described in the following equation ∆ φ Sagnac = 4 π E (cid:126) c (cid:126) A (cid:126) Ω (1) where E is the energy associated with the atom E at = mc or photon E ph = (cid:126) ω , m is the massof the atom, ω the angular frequency of the lightfield, and c the speed of light. Since E at (cid:29) E ph , itscales favourably for atoms, motivating early experi-ments [12–14], while much larger areas were demon-strated for light [2].In Sagnac sensors based on laser cooled atoms infree fall, light fields driving Raman or Bragg transi-tions coherently split, deflect, and recombine atomicwave packets in interferometers based on three [6, 8]or four pulses [7, 9] with an area of up to 11 cm [5].Another avenue for rotation measurements arewave guides or traps moving atoms in loops, espe-cially for applications requiring compact setups [3].Multiple loops were created in ring traps employ-ing quantum degenerate gases for different pur-poses including the investigation of superconductiveflows [15, 16]. Exploiting them for guided atomicSagnac interferometers in optical or magnetic trapsremains an experimental challenge [17–20].1 a r X i v : . [ phy s i c s . a t o m - ph ] F e b z gAΩ v rl S D + ¯ hk m − ¯ hk m ac bd Figure 1:
Trajectories of the free falling atoms in the inter-ferometer. Red arrows denote the light fields for splitting,redirecting, and recombination. Orange arrows indicate thelight fields for relaunching the atoms against gravity g witha velocity v rl enabling operation with a single beam splittingaxis (red) and closing the interferometer at its starting point.The atoms start at (a) where a beam splitting pulse leadsto a coherent superposition of two momentum states (blue,green) that separate symmetrically with a recoil velocity of ± (cid:126) k / (2 m ). Here, k denotes the effective wave number of thebeam splitter (red) and m the atomic mass. One momentumstate follows the green arrows and the second one the dashedblue arrows according to the numbering. The state deflectedin positive x-direction (green) propagates from (a) to (b), (c),(d), and back to (a). Similarly, but with inverted momentum,the other state (blue) proceeds from (a) to (d), (c), (b), andback to (a), closing one loop. As a consequence, the inter-ferometer encloses the area A (grey shaded area), rendering itsensitive to rotations Ω . Both trajectories meet at (c) wherethe two momentum states are relaunched at the same time.Input (up) and output ports (down) of the interferometer areindicated by black arrows below (a). The maximum wavepacket separation is indicated by S and the drop distance by D . In our geometry (fig. 1), atoms are coherently ma-nipulated by two perpendicular light gratings (redand orange dashed lines) to form a multi-loop in-terferometer [21]. Pulsed light fields enable sym-metric beam splitting of the atomic wave packetsin the horizontal axis (red arrows) [22–25] and re-launching in vertical direction (orange arrows) [26].This approach offers a variety of advantages: (i) thefree-fall time can be tuned to scale the area, (ii)the area is well defined by velocities imprinted dur-ing the coherent atom-light interactions, and (iii)the geometry utilises a single axis for beam split-ting which avoids the requirement for relative align-ment [5, 21, 27]. (iv) It enables multiple loops,(v) and its symmetry suppresses biases due to lightshifts associated with the atom-light interaction. (vi) tx a c a c ab b b bd d d d0 T T T T T T T T π/ π relaunch π π/ orrelaunch π relaunch π π/ tI bs T T T T T T tI rl T T T Figure 2:
Space-time diagram and pulse timings of a multi-loop interferometer. The upper diagram shows the timing ofthe π/ π mirror pulsesat position (b) and (d) in fig. 1, and relaunch at position (c)in fig. 1 as well as the recombination ( π/
2) pulse at (a). Non-opaque lines indicate an implementation with the minimum oftwo loops ( n = 1) and opaque lines a four-loop interferometer( n = 2). The interferometer can be extended to n loops closedwith a π/ nT by introducing relaunches at r · T for r ∈ (1, 2, ..., n −
1) (position (a) in fig. 1). The lowerdiagram shows the time-dependent intensities of the beamsplitting pulses I bs and relaunches I rl . Diagrams are not toscale, neglect pulse shaping and the initial launch before theatoms enter the interferometer. Moreover, our concept in principle allows incorpora-tion of several additional measurements such as localgravity [28, 29] and tilt of the apparatus [23] with re-spect to gravity. Dual or multi-loop interferometershave also been proposed in the context of terres-trial and space-borne infrasound gravitational wavedetection designed for measuring strain rather thanrotations [21, 30–33].
The trajectories in our interferometer are detailed infig. 1. Initially, an atomic wave packet is launchedvertically. On its upward way, the wave packet inter-acts with the horizontal beam splitter with an effec-tive wave number k forming two wave packets drift-ing apart with a momentum of ± (cid:126) k / (2 m ). Aftera time T , the horizontally oriented light field (red)inverts the movement of the atoms on its axis. Ontheir way down due to gravity, the vertically orientedlight field (orange) relaunches the atoms [26] at thelowest point of the interferometer at 2 T reversingtheir momentum to move upward. The atoms passthe horizontal atom-light interaction zone again at3 T where they are deflected towards each other and2 able 1: Comparison of our multi-loop scheme with a four-pulse interferometer and performance estimation. We assumedevices based on rubidium atoms [34], a number of N detected atoms, an effective wave number k , a pulse separation time T (see fig. 2), and a contrast C , performing n loops. We denote A as the effectively enclosed area, which scales with n . Both thecalculation of A and the sensitivity neglect finite pulse durations. The maximum trajectory separation is given by S . For theestimation of the drop distance D with regard to a compact scenario we allow for additional time of 6 ms for the momentumtransfer. Our interferometer cycle time is denoted by t c and we state the sensitivity in the shot-noise limit according to eq. 4.In the first four rows we compare multi-loop interferometers to four-pulse geometries without a relaunch [5, 7, 9], emphasisingdifferences in the parameters in blue, and neglecting atom losses and contrast reduction due to imperfect beam splitters. Forthe lower two rows, we assume a simple model in which the contrast for multiple loops C ( n ) depends on the contrast for asingle loop C (1) and scales as C ( n ) = C (1) n . Furthermore, our model reduces the number of detected atoms by a factor l n − with l = 0.9 for additional loops to take inefficiencies in the atom-light interactions into account.sensor features N k T n C A t c S D sensitivity (cid:2) π
780 nm (cid:3) [ms] (cid:2) m (cid:3) [s] [m] [m] (cid:104) rad / s √ Hz (cid:105)
1: multi loop 10
40 10 10 1 4.6 · − · − · − · −
1: four pulse 10
350 10 - 1 4 · − · − · − · −
2: multi loop 4 ·
20 250 10 1 3.6 · − · − · −
2: four pulse 4 ·
28 189 - 1 2.1 · − · − · − compact 5.9 ·
40 10 6 0.53 2.8 · − · − · − · − high sensitivity 2.9 ·
20 250 4 0.66 1.4 · − · − · − cross falling downwards at 4 T completing their firstloop. In order to start the next loop, they are re-launched (see fig. 2). Repetition of the proceduredetermines the number of loops n . After the lastloop, the interferometer is closed by flashing a beamsplitter pulse instead of an upward acceleration.The resulting area determining the sensitivity torotations depends on the total time of the interfer-ometer 4 T , the effective wave vector (cid:126) k , local gravity (cid:126) g , and leads to the Sagnac phase ∆ φ Sagnac = n · (cid:126) k × (cid:126) g ) (cid:126) ΩT (2)calculated with the methods outlined in ref. [35–37]and similar as in refs. [5, 7, 9, 38]. The relaunchvelocity v rl = | (cid:126) v rl | = 3 g T with g = | (cid:126) g | is alignedparallel to gravity and is chosen to close the atominterferometer at its starting point. In this configu-ration, the area is given by A = n · (cid:126) km g T . (3)It can be enlarged by a higher transverse momen-tum (cid:126) k = (cid:126) | (cid:126) k | , e.g. by transferring more photonrecoils, and by increasing the free fall time 4 T of theinterferometer.Enlarging the number of loops by a factor n effec-tively increases the enclosed area without changingthe dimensions of the geometry, defined by the max-imum wave packet separation in the horizontal axis S = (cid:126) kT / m and the drop distance in the verticalaxis D = (3 T / · g /
2. The relaunch at 4 T (ormultiples of 4 T for more than four loops, position (a) in fig. 1) reuses the same light field (orange ar-rows) as for the first relaunch at 2 T (position (c) infig. 1) and does thus not add complexity.Typically, the cycle of an atom interferometer con-sists of the generation and preparation of the atomicensembles during the time t prep , the interferometertime which for our geometry reads n · T , and de-tecting the population of the output ports withinthe time t det . This leads to a total cycle time of t c = t prep + n · T + t det . Using the phase shiftgiven in eq. 2, the shot-noise limited sensitivity forrotations Ω y with N atoms and a contrast C is givenby σ Ω ( t ) = 1 C √ N · n · (4 kg T ) (cid:114) t prep + n · T + t det t (4)for an averaging time t corresponding to multiples ofthe cycle time t c . Consequently, an interferometerwith a small free fall time 4 T (cid:28) t prep + t det benefitsmore from multiple loops with a scaling of ∼ / n in eq. 4 than other scenarios with 4 T ≈ t prep + t det that scale as ∼ / √ n . Implementing an interfer-ometer time n · T > t prep can enable a continuousscheme by sharing π/ T with an appro-priate n . In multi-loop interferometers, the sensitivity to DCaccelerations and phase errors depending on the ini-3 able 2:
Requirements on the pointing of the relaunch. Theangles α δτ , α Γ and β are calculated to induce contributions(eqs. 5, 6, and 7) by a factor of 10 · n below the shot-noiselimit for the scenarios in the lower rows of tab. 1 at 1 s. We as-sume δτ = 10 ns for a typical experiment control system [40],Earth’s gravity gradient Γ x = 1.5 · − s − , and Earth’s rota-tion rate Ω z = 7.27 · − rad / s [35, 36]. Values are given inrad. α δτ α Γ β compact 1.3 · − < · − high-sensitivity 6 · − · − · − tial position and velocity is suppressed [33, 38]. How-ever, a non-ideal pointing of the relaunch velocity (cid:126) v rl may introduce spurious phase shifts in a real setup.We consider small deviations α = | (cid:126) v rl × (cid:126) e x | / ( | (cid:126) v rl || (cid:126) e x | )and β = | (cid:126) v rl × (cid:126) e y | / ( | (cid:126) v rl || (cid:126) e y | ) in a double-loop con-figuration. Here, (cid:126) e x = (cid:126) k / | (cid:126) k | denotes the unit vectorin x-direction and (cid:126) e y = ( (cid:126) k × (cid:126) g ) / ( | (cid:126) k || (cid:126) g | ) denotes theunit vector in y-direction (see fig. 1).If the timing of the relaunch is not ideally centredaround 2 T , but shifted by δτ , coupling to non-zero α leads to the phase shift [21] ∆ φ α , τ = − kv rl αδτ = − kg T αδτ . (5)Provided the pointing of the relaunch velocity is ad-justable (e.g. with a tip-tilt mirror controlling thealignment of the light field for relaunching) α and δτ can be adjusted by iteratively scanning both.In addition, tilting the relaunch vector inducesphase shifts resembling those of a three-pulse orMach-Zehnder-like interferometer [35, 36] by cou-pling to gravity gradients Γ and rotations (cid:126) Ω . Thesecontributions read ∆ φ α , Γ = (cid:126) kΓ (cid:126) v rl T = 3 k α Γ x g T , (6)with Γ x = (cid:126) e x Γ (cid:126) e x and ∆ φ β , Ω = 2 (cid:16) (cid:126) k × (cid:126) v rl (cid:17) · (cid:126) ΩT = 6 k β g Ω z T , (7)corresponding to a spurious sensitivity to a rotation Ω z = (cid:126) e z · (cid:126) Ω .Scanning the interferometer time 4 T enables aniterative procedure to minimise spurious phase shiftsby optimising the contrast [27]. Apart from the (re-)launch, our scheme with n = 1resembles four-pulse interferometers [5, 7, 9, 38] ingeometry and scale factor (eqs. 2 and 3). Hence, weshow the advantages of our method with respect toa four-pulse interferometer for two design choices. For both interferometers in tab. 1 (upper four rows),we assume ideal contrast, no losses of atoms, aswell as shot-noise limited sensitivities (see eq. 4).(1) Matching the free-fall time T and sensitivity forthe multi-loop and four-pulse sensor, the four-pulseinterferometer requires a larger photon momentumtransfer in horizontal direction. Consequently, thesize S · D of the multi-loop geometry is by a factorof 15 smaller (emphasised in blue in tab. 1). (2)Aiming for similar dimensions S and D , we obtaina nearly an order of magnitude higher sensitivity forour multi-loop geometry at the cost of an increasedcycle time.Showcasing more realistic scenarios, we considera simple model for losses of atoms and reductionof contrast in dependence on the number of loopsas summarised in tab. 1 (lower rows). The lat-ter can result from inhomogeneities of the lightfields [22, 43, 64–68]. According to our estima-tions, our method would still lead to a compactsensor with a sensitivity of 1.2 · − (rad / s) / √ Hzwithin a volume of 20 mm for the interferome-ter, and a highly sensitive, but larger device with1.7 · − (rad / s) / √ Hz within a meter-sized vac-uum vessel, comparable to the performance of largering laser gyroscopes [2].Multiple experiments investigated beam splittingas well as relaunch operations as required for ourscheme. They realised the transfer of large mo-menta with subsequent pulses or higher order transi-tions [41–46] and their combination with Bloch os-cillations [47–49]. The implementation of symmet-ric splitting [22, 23, 25, 50, 51] was demonstratedwith an effective wave number corresponding to 408photon recoils in a twin-lattice atom interferome-ter [22]. A similar procedure enabled the relaunchof atoms [26]. The requirement for high efficiencyimplies using atomic ensembles with very low residualexpansion rates [52] as enabled by delta-kick collima-tion of evaporated atoms [40, 53] and Bose-Einsteincondensates [22, 26, 54–56]. In addition, interfer-ometers exploiting such ensembles may benefit fromthe suppression systematic of uncertainties [57–59].Fountain geometries utilised launch techniques com-patible with these ensembles [26, 40, 53, 56]. Rapidgeneration of Bose-Einstein condensates with 10 atoms was demonstrated [60, 61] and realised withatom chips in 1 s [62, 63] which we adopted for ourestimation.Reaching the shot-noise limited sensitivity impliesa restriction on tilt instability as detailed in tab. 2due to couplings in eqs. 5, 6, and 7. It is met atthe modest level of 0.1 mrad / √ Hz for the compact4cenario and at 7 nrad / √ Hz for high-sensitivity . We presented our concept for an atomic gyroscopecapable of performing multiple loops by exploitinglight pulses for beam splitting and relaunching atomswith the perspective of reaching unprecedented sen-sitivities for rotations. It offers unique scalabil-ity in a limited size of a sensor head. Key ele-ments as the symmetric beam splitting [22, 23],relaunch [26], as well preparation of the ultracoldatoms [53, 55, 62, 63, 71] have already been demon-strated. The tools for coherent manipulation in ourscheme additionally allow for the implementationof geometries for a tiltmeter [23] and a gravime-ter [26, 28, 29]. We showed the perspective for com-pact setups, which can be scaled up to compete withlarge ring laser gyroscopes [1, 2]. This might enablethe detection of multiple rotational components ina single set-up by adding a second orthogonal beamsplitting axis, and sensitivities as required for mea-suring the Lense-Thirring effect [72–75].
Acknowledgements
The presented work is supported by the CRC 1227DQmat within the projects B07 and B09, the CRC1464 TerraQ within the projects A01, A02 and A03,the German Space Agency (DLR) with funds pro-vided by the Federal Ministry of Economic Affairsand Energy (BMWi) due to an enactment of the Ger-man Bundestag under Grant No. DLR 50WM1952(QUANTUS-V-Fallturm), 50WP1700 (BECCAL),50RK1957 (QGYRO), and the Verein Deutscher In-genieure (VDI) with funds provided by the FederalMinistry of Education and Research (BMBF) un-der Grant No. VDI 13N14838 (TAIOL). Funded bythe Deutsche Forschungsgemeinschaft (DFG, Ger-man Research Foundation) under Germany’s Ex-cellence Strategy – EXC-2123 QuantumFrontiers –390837967. D.S. acknowledges support by the Fed-eral Ministry of Education and Research (BMBF)through the funding program Photonics ResearchGermany under contract number 13N14875. We ac-knowledge financial support from “Nieders¨achsischesVorab” through “F¨orderung von Wissenschaft undTechnik in Forschung und Lehre“ for the initialfunding of research in the new DLR-SI Institute Dedicated vibration isolation systems demonstrated anoise floor of 1 nrad / √ Hz in a frequency range of 1 Hz to100 Hz [69]. Alternatively, and similar as in a large ring lasergyroscope [1, 2], tiltmeters with a resolution of sub nrad [70]may enable post correction methods. and through the “Quantum- and Nano-Metrology(QUANOMET)” initiative within the project QT3.
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